A Hypocoercive LBM–PINN Framework for High-Mach Rotating Thermal-Hydraulics in Nuclear Reactor Components

1. Introduction

The modelling of high-speed rotating turbulent flows remains one of the most formidable challenges in computational fluid dynamics, with applications spanning from the intricate combustion dynamics inside gas turbine engines to the colossal accretion disks that power active galactic nuclei. The fundamental difficulty lies in the intricate, multi-scale interplay between compressibility, rotation, turbulence, and heat transfer — a combination that systematically frustrates classical modelling approaches. This introduction provides a concise survey of the evolution of turbulence modelling, tracing the intellectual lineage from early phenomenological closures through the rigorous mathematical foundations of kinetic theory and hypocoercivity, to the recent emergence of data-driven paradigms. We identify the critical gaps that motivate the present work and establish the scientific context for our contributions.

1.1. Historical Development of Turbulence Modelling

The modern era of turbulence modelling began with the systematic development of second-moment closures, epitomised by the Reynolds-stress transport equations introduced by Launder, Reece, and Rodi [7]. These formulations represented a significant advance over the simpler eddy-viscosity hypothesis, capturing the anisotropic nature of turbulent stresses in non-equilibrium flows. However, the closure still required empirical modelling of the pressure-strain and dissipation terms, introducing coefficients calibrated against canonical flows — a limitation that persisted across subsequent generations of models. These formulations were swiftly extended to capture the complex dynamics of astrophysical accretion disks by Begelman, Blandford, and Rees [11], while parallel experimental campaigns by Andereck, Liu, and Swinney [12] mapped the rich phenomenology of rotating instabilities in Taylor–Couette systems, establishing benchmark configurations for testing closure models.

Beyond these canonical astrophysical and aerospace applications, high-Mach rotating turbulent flows are of direct relevance to nuclear engineering, particularly in the thermal-hydraulic analysis of advanced reactor systems. In gas-cooled reactors (e.g., high-temperature gas-cooled reactors, HTGRs) and high-speed helium circulators, the clearance gaps between rotor and stator in canned motor pumps give rise to Taylor–Couette-like flows under extreme rotational speeds and significant temperature gradients [35,59]. The predictive accuracy of turbulence closures in such narrow, rotating annular passages directly impacts the assessment of coolant temperature distributions, flow resistance, and ultimately the safety margins of the reactor core. Classical eddy-viscosity models, calibrated for incompressible flows, are known to struggle in these regimes, motivating the development of structure-preserving, data-augmented closures with rigorous stability guarantees — a gap that the present work directly addresses.

The high computational cost of Reynolds-stress transport models gave rise to more accessible two-equation frameworks, most notably the k-$\epsilon$ model systematised by Wilcox [20]. While enormously successful in industrial practice, its reliance on the isotropic eddy-viscosity hypothesis fundamentally limits its ability to capture anisotropic turbulence in rotating flows, curved geometries, or regions of strong mean strain. Concurrently, the theoretical underpinnings of fluid behaviour were being rigorously re-examined through kinetic theory. Boltzmann’s equation [5] elegantly captures non-equilibrium phenomena beyond Newtonian constitutive relations, while Grad [2] provided a systematic method for deriving macroscopic equations through moment expansions. The Bhatnagar–Gross–Krook (BGK) approximation [1] dramatically simplified the kinetic equation while rigorously preserving conservation laws and the H-theorem. Lundgren [3] extended kinetic ideas to turbulence by deriving statistical distribution function equations, laying the groundwork for PDF-based modelling. The comprehensive treatises by Chapman and Cowling [13] and Glassey [18] established the rigorous mathematical foundations of the Boltzmann equation, inspiring later developments in hypocoercivity. Sarkar et al. [14] and Lele [17] systematically characterised compressibility effects on turbulence, identifying the key mechanisms that any credible compressible turbulence model must capture. Bertin’s comprehensive treatment of hypersonic aerothermodynamics [16] further underscored the importance of capturing high-Mach effects for aerospace and reactor applications.

The late 1990s witnessed two transformative developments. First, the Lattice Boltzmann Method (LBM), pioneered by Chen and Doolen [19], emerged as a numerical framework that explicitly solves a discretised kinetic equation, eliminating the need for an explicit pressure Poisson equation and offering exceptional parallelism. Second, Spalart and Allmaras [15] introduced a one-equation turbulence model tailored for aerodynamic applications. Succi’s comprehensive monograph [25] cemented LBM as a versatile computational technique. However, standard LBM formulations could violate the discrete H-theorem, leading to numerical instabilities at high Reynolds numbers. Boghosian et al. [22] addressed this by introducing entropic LBM formulations that rigorously enforce the H-theorem at the discrete level. Pope’s landmark textbook [24] synthesised the state of the art in turbulence modelling, while Heinz [26] bridged kinetic descriptions and engineering turbulence modelling by deriving a thermodynamic foundation for two-equation closures. By this period, RANS closures were deployed in increasingly challenging engineering applications, including combustion phenomena inside gas turbine engines [34]. Large Eddy Simulation (LES) emerged as a higher-fidelity alternative, with Bazilevs and Akkerman [32] developing residual-based variational multiscale implementations. Guo and Shu’s monograph [37] further extended LBM applications, while Fei and Luo [41] introduced thermal cascaded LBM formulations for compressible and non-isothermal regimes.

1.2. The Emergence of Hypocoercivity and Experimental Challenges

Recognising the need for rigorous convergence guarantees, a parallel mathematical thread emerged through the theory of hypocoercivity. Hérau [27] established exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Mouhot and Neumann [28] provided quantitative perturbative studies of convergence to equilibrium for collisional kinetic models on the torus, while Mouhot [29] derived rate-of-convergence results for the spatially homogeneous Boltzmann equation. Dolbeault, Mouhot, and Schmeiser [30] developed a unified framework based on an augmented energy functional that mixes the $L^2$ norm with its spatial gradient. Villani’s influential memoir [31] definitively unified these developments into a coherent mathematical framework, providing a systematic methodology for proving exponential convergence for a large class of kinetic equations. Concurrently, Jin [33] provided a systematic review of asymptotic-preserving (AP) schemes for multiscale kinetic equations, demonstrating how numerical methods can transition seamlessly between kinetic and hydrodynamic regimes. Leonov and Kuznetsov [42] offered complementary tools for analysing the stability of reduced-order models through Lyapunov dimension analysis.

However, systematic experimental investigations delivered a decisive blow to the isotropic eddy-viscosity hypothesis. Guillerm et al. [39] conclusively demonstrated that even refined LES approaches catastrophically fail to capture heat transfer physics in rotating, stratified Taylor–Couette flows, producing qualitatively incorrect temperature profiles and significantly mispredicting Nusselt numbers. Broader assessments under high-Mach-number rotating conditions by Acquaye [40] and Morgan et al. [46] confirmed this was a qualitative breakdown of predictive capability across both RANS and LES paradigms. These experimental benchmarks serve as stringent testbeds for any proposed improvement and directly motivate our numerical validation.

1.3. The Rise of Data-Driven Turbulence Modelling

In response to these persistent structural inadequacies, the turbulence community has increasingly turned to data-driven paradigms. Ling, Kurzawski, and Templeton [43] pioneered the use of deep neural networks with embedded Galilean invariance to learn Reynolds-stress anisotropies, achieving superior performance on test flows but lacking rigorous stability guarantees. Wu et al. [47] developed a comprehensive physics-informed machine learning framework for augmenting turbulence models. Maulik et al. [48] applied neural networks to subgrid-scale modelling for two-dimensional turbulence, again without mathematical stability certificates. The emergence of Physics-Informed Neural Networks (PINNs), introduced by Raissi, Perdikaris, and Karniadakis [49], represented a paradigm shift by embedding governing PDEs directly into the loss function. This philosophy was generalised by Karniadakis et al. [51], while Kochkov and collaborators [52] demonstrated machine learning-accelerated CFD at scale.

Targeted efforts to correct rotational turbulence closures have yielded promising neural-network-based models. Lee, Kim, and Park [55] developed neural closure models for rotating flows, demonstrating improved Reynolds stress prediction. Zhang, Wang, and Chen [56] introduced structure-preserving neural networks for kinetic equations, explicitly enforcing conservation laws and entropy dissipation. McConkey, Kalia, Yee, and Lien [57] introduced realisability-informed machine learning for turbulence anisotropy mappings. Most recently, Li, Liu, and Xu [58] demonstrated spectral decomposition PINN-LBM hybrids for high-Reynolds-number turbulence simulation. Briant [54] provided a perturbative analysis of hypocoercivity for confined Boltzmann-type collisional equations, extending the mathematical toolkit for certifying the stability of kinetic-based models. The Scale-Adaptive Simulation (SAS) method, developed by Menter and Egorov [36], emerged as a hybrid RANS-LES approach that adjusts turbulent viscosity based on local length scales, offering an intermediate-cost alternative for unsteady flow predictions. However, as our results demonstrate, even such sophisticated hybrid approaches fail at extreme Mach numbers, underscoring the need for the structure-preserving neural augmentation proposed herein.

1.4. The Gap: Missing Mathematical Guarantees

Yet, despite these sweeping empirical advances, a critical vulnerability persists across the vast majority of hybrid and data-driven models: they lack rigorous mathematical guarantees regarding stability, convergence, and error propagation over long-time integration. There is often no proof that the neural correction will not destabilise the simulation, no quantification of how the approximation error propagates through temporal integration, and no systematic criterion for designing networks that respect the underlying entropic and dissipative structure of the kinetic equations. Specifically, Ling et al. [43] provide no analysis of whether their corrections preserve the positive definiteness of the Reynolds stress tensor. Wu et al. [47] enforce physics-informed constraints but do not provide stability guarantees. Maulik et al. [48] demonstrate empirical stability but offer no mathematical proof of long-time stability. Lee et al. [55] and Li et al. [58] report improved numerical results but do not provide hypocoercive or Lyapunov stability analyses. Even state-of-the-art hybrid RANS-LES approaches like SAS [36], while offering improved resolution of unsteady structures, lack rigorous stability certificates for the extreme compressible regimes considered here.

Crucially, the theory of hypocoercivity — as developed by Gross [6], Villani [31], Hérau [27], Mouhot [28,29], Dolbeault, Mouhot, and Schmeiser [30], and extended by Briant [54] — provides precisely the missing mathematical language to analyse and guarantee the stability of neural-network-augmented kinetic closures. Yet it remains conspicuously absent from current machine-learning turbulence models. This gap — between the empirical success of neural turbulence models and the rigorous mathematical certification required for reliable predictive simulation — is the central problem addressed by the present work.

1.5. The Present Contribution

Motivated by these converging lines of inquiry — the entropic and hypocoercive guarantees of kinetic theory, the efficiency of AP-LBM discretisations, and the representational capacity of PINNs — the present research develops a novel structure-preserving closure framework for high-speed rotating flows that rigorously integrates all three pillars. Unlike existing neural closures for rotating turbulence [55] and LBM-PINN hybrids [58], our approach explicitly regularises the neural components with hypocoercive penalty terms derived from Villani’s framework [31], ensuring that the overall scheme remains provably dissipative, realisable, and convergent to the correct hydrodynamic limit.

Specifically, we make the following contributions:

First, we derive a kinetic model for compressible turbulent fluctuations from the Lundgren-Monin-Novikov hierarchy [3,8], closed by a BGK operator [1] that rigorously preserves the collision invariants and the H-theorem. A learnable neural correction ${\mathcal{N}}_{\theta }$ is introduced and constrained to be orthogonal to the five collision invariants, ensuring exact conservation of mass, momentum, and energy — a property absent from previous data-driven closures [43,47,55,58].

Second, we provide a rigorous hypocoercive stability analysis for the nonlinear kinetic equation with neural correction. Following the augmented energy methodology of Villani [31], we prove global well-posedness, uniqueness, and exponential convergence to the turbulent Maxwellian for sufficiently small initial perturbations. The explicit decay rate, $\lambda ={\lambda }_0/\left(4\tau \right)-C_2{L}_{\theta }$, directly links the neural Lipschitz constant $L_{\theta }$ to the spectral gap ${\lambda }_0=1/\tau$ of the linearised collision operator. In terms of Reynolds and Mach numbers ($\tau \sim \mathrm{Ma}/\mathrm{Re}$), this becomes $\lambda \sim ({\lambda }_0/4)(\mathrm{Re}/\mathrm{Ma})-C_2{L}_{\theta }$, showing that high Reynolds numbers stabilise the system while large Lipschitz constants or high Mach numbers may destabilise it.

Third, we perform a systematic Chapman-Enskog expansion (following Chapman and Cowling [13] and Glassey [18]) that recovers the compressible RANS equations at first order, with the explicit eddy-viscosity closure ${\nu }_t=\tau \overline{k}$ and turbulent thermal diffusivity ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$. The neural correction contributes only at Burnett order ($\mathcal{O}\left({\tau }^2\right)$), thus not affecting the leading-order macroscopic closure.

Fourth, we develop a semi-discrete LBM–PINN scheme and provide rigorous convergence estimates. Using Gauss-Hermite quadrature of degree m [19,25,37] and a second-order finite difference spatial discretisation, we prove that the reconstruction error is bounded by

$${\parallel e\left(t\right)\parallel }_{L^2}\le C_T\left(\Delta x^2+{(\Delta v)}^{m+1}+\tau +{\delta }_{\mathrm{NN}}\right).$$

This is the first error estimate for a structure-preserving LBM–PINN hybrid scheme, establishing that the neural approximation error ${\delta }_{\mathrm{NN}}$ appears additively, not multiplicatively.

Fifth, for the linearised kinetic equation in a rotating reference frame, we derive a new hypocoercive inequality — the Santos-Andrade inequality — that explicitly incorporates the angular velocity $\mathit{\omega }$, the Mach number, and the neural Lipschitz constant:

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta }.$$

This inequality quantifies the competition between turbulent relaxation (stabilising), the destabilising effect of rotation, and the neural Lipschitz constant, providing the first result that combines learnable corrections, rotation, and compressibility in a rigorous hypocoercive framework. The stability condition ${\lambda }_{\mathrm{SA}}>0$ provides a practical, mathematically certified threshold for designing neural closures that are both accurate and provably stable for rotating compressible flows.

Sixth and finally, we validate our framework through extensive numerical experiments on compressible Taylor-Couette flow at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$. A comprehensive comparison across Smagorinsky, RANS, SAS, and their PINN-augmented variants reveals that classical closures fail catastrophically at $\mathrm{Ma}=10.0$, producing unphysical constant temperature and pressure fields that violate the hydrostatic balance. In stark contrast, the Smagorinsky+PINN scheme uniquely restores a realistic radial temperature gradient and yields a physically plausible Nusselt number $\mathrm{Nu}=17.14$. The observed Lipschitz constant $L_{\theta }\approx 2.96$ remains well below the theoretical threshold, confirming the practical utility of the Santos-Andrade inequality. The failure of RANS+PINN and SAS+PINN models underscores that the PINN correction must be specifically calibrated for the underlying closure; generic application without model-specific training fails to restore physical gradients.

By anchoring our data-driven corrections in first-principle kinetic theory, we provide a mathematically certified pathway to predictive simulation of rotating compressible turbulence — moving decisively beyond the qualitative breakdowns that plague both classical and modern empirical closures.

1.6. Paper Outline

The remainder of this paper is organised as follows.

Section 2 lays the mathematical foundation, introducing the kinetic framework based on the Boltzmann equation and its BGK approximation [1,13], establishing the functional setting, the local Maxwellian equilibrium, the structural properties of the collision operator, and the spectral gap ${\lambda }_0=1/\tau$. This section also formalises the assumptions on the structure-preserving neural correction ${\mathcal{N}}_{\theta }$, including conservation constraints and Lipschitz continuity with constant $L_{\theta }$.

Section 3 extends the kinetic description to a rotating frame with angular velocity $\mathit{\omega }$, deriving the rotating Maxwellian $M_{\omega }$, the modified transport operator ${\mathcal{T}}_{\omega }$, and the linearised collision operator ${\mathcal{L}}_{\omega }$. A Chapman-Enskog expansion recovers the compressible Navier-Stokes equations with explicit eddy-viscosity closure ${\nu }_t=\tau \overline{k}$ and turbulent thermal diffusivity ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$. The commutator estimate $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ quantifies the rotation-dependent terms in the hypocoercive analysis.

Section 4 presents the Santos–Andrade inequality in full rigour, the central theoretical contribution of this work. We introduce the augmented energy functional, prove the exponential decay estimate

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -2{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right),{\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

and establish the stability condition ${\lambda }_{\mathrm{SA}}>0$ as a mathematically certified threshold for designing stable neural closures. A nonlinear generalisation via entropy methods is also presented, along with a comparison with Villani’s classical hypocoercive framework.

Section 5 details the numerical discretisation using a hybrid Lattice Boltzmann Method with PINN correction. The velocity space is discretised via Gauss–Hermite quadrature, and the spatial domain via a second-order skew-symmetric finite difference scheme. We present the semi-discrete LBM–PINN scheme, prove the discrete hypocoercive inequality, and establish the main convergence theorem:

$${\parallel e\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\le C_T\left(\Delta x^2+{(\Delta v)}^{m+1}+\tau +{\delta }_{\mathrm{NN}}\right),$$

where ${\delta }_{\mathrm{NN}}$ is the neural approximation error. Practical implementation guidelines, including spectral normalisation and the stability condition $L_{\theta }<1/\left(4C_1\tau \right)$, are also provided.

Section 6 presents the numerical validation on the canonical Taylor-Couette flow configuration at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$. The section describes the rotor–stator geometry, flow parameters, thermal boundary conditions, and the $300\times 300$ grid. The experimental methodology covers offline PINN training, online rotor simulations for two turbulence models (Smagorinsky and Smagorinsky+PINN), and post-processing extraction of diagnostics. The results validate the Santos–Andrade inequality through systematic parameter sweeps, present radial profiles of velocity and temperature (yielding $\mathrm{Nu}=26.39$ at $\mathrm{Ma}=5.0$ and $\mathrm{Nu}=17.14$ at $\mathrm{Ma}=10.0$), and demonstrate the catastrophic failure of the Smagorinsky model at $\mathrm{Ma}=10.0$ versus the restoration of physical fidelity by the PINN-augmented scheme.

Section 7 discusses the applicability of the framework to nuclear thermal-hydraulics and outlines future research directions. These include extension to three-dimensional geometries with axial flow, non-uniform gap widths, and multi-component coolant mixtures; integration with reactor system codes (RELAP, TRACE, CATHARE) for high-fidelity component-scale modelling and digital twin applications; advancement of the neural correction framework through adaptive Lipschitz control, transfer learning, and uncertainty quantification; and application to other reactor thermal-hydraulic phenomena such as natural circulation, two-phase flow, thermal stratification, and turbulent mixing in fuel assemblies. The pathway to industrial deployment through verification, validation, HPC implementation, and standardised workflows is also discussed.

Section 8 discusses the principal constraints of the current study with a constructive perspective. These include the phenomenological nature of the BGK closure and the theoretical scope of the hypocoercive analysis; the two-dimensional geometric idealisation and the modest but sufficient $300\times 300$ grid resolution; the small-perturbation assumption underlying the linearised analysis (mitigated by the nonlinear entropy generalisation and validated numerically across the full Mach range); the global Lipschitz continuity assumption on the neural correction (practically enforced via spectral normalisation with $L_{\theta }\approx 2.96$, comfortably below the threshold); the reliance on synthetic BGK training data; and the validation scope limited to the Taylor–Couette benchmark.

Section 9 presents the quantitative findings from the numerical experiments, organised into two main parts: validation of the Santos-Andrade inequality through systematic parameter sweeps, and comparative assessment of the Smagorinsky and Smagorinsky+PINN models across the compressible regime. The results include energy spectra exhibiting Kolmogorov $-5/3$ scaling, Nusselt numbers ($\mathrm{Nu}=26.39$ at $\mathrm{Ma}=5.0$ and $\mathrm{Nu}=17.14$ at $\mathrm{Ma}=10.0$), velocity profiles with $2.5\%$ maximum deviation from the analytical Couette solution, Strouhal numbers $\mathrm{St}\approx 0.28$ consistent with experimental benchmarks [12], and the catastrophic failure of the Smagorinsky model at $\mathrm{Ma}=10.0$ (producing $\mathrm{Nu}\approx 2.34$) versus the restoration of physical fidelity by the PINN-augmented scheme.

Finally, Section 10 offers concluding remarks, summarising the theoretical and numerical contributions, discussing broader implications for turbulence modelling and nuclear thermal-hydraulics, and reinforcing the message that the Santos-Andrade inequality provides a mathematically certified pathway to predictive simulation of rotating compressible turbulence.

2. Mathematical Background

Before introducing the new inequality or the numerical scheme, it is essential to establish a common ground. This section provides a self-contained account of the kinetic theory and hypocoercive framework upon which our work is built. We revisit the Boltzmann equation, the BGK collision operator, and Villani’s augmented energy, but with a specific focus: we are ultimately interested in how these structures behave in a rotating reference frame and how they can be discretised without losing their fundamental properties.

This presentation follows a systematic progression: we first establish the rigorous functional-analytic setting for the Boltzmann equation, then introduce the BGK simplification, and finally develop the hypocoercive framework that underpins our stability analysis. Throughout, we emphasise the structural properties — conservation laws, entropy dissipation, and spectral gaps — that are essential for the subsequent development of the Santos–Andrade criterion.

2.1. The Boltzmann–BGK Framework

We consider a monatomic gas described by the distribution function $f(\mathbf{x},\mathbf{v},t)$, defined on the phase space $(\mathbf{x},\mathbf{v})\in \Omega \times {\mathbb{R}}^3$, where $\Omega ={\mathbb{T}}^3\cong {[0,1)}^3$ is the three-dimensional torus with periodic boundary conditions, and $\mathbf{v}\in {\mathbb{R}}^3$ denotes the microscopic velocity. The quantity

$$f(\mathbf{x},\mathbf{v},t)d\mathbf{x}d\mathbf{v}$$

represents the expected number of particles in the infinitesimal phase-space volume $d\mathbf{x}d\mathbf{v}$ centered at $(\mathbf{x},\mathbf{v})$ at time $t\ge 0$. The distribution function is non-negative, $f\ge 0$, and satisfies the normalisation condition

$${\int }_{\Omega \times {\mathbb{R}}^3}f(\mathbf{x},\mathbf{v},t)d\mathbf{x}d\mathbf{v}=N,$$

where N is the total number of particles in the system.

Remark 1.

The periodic torus ${\mathbb{T}}^3$ eliminates boundary effects, allowing us to focus on the interior dynamics and the spectral properties of the collision operator. This choice is standard in the hypocoercive literature [31] and is justified for studying the intrinsic stability properties of the kinetic equations without the complication of boundary layers.

2.1.1. Functional Setting and Regularity Assumptions

To provide a rigorous foundation for the analysis, we specify the precise functional spaces in which the distribution function and its perturbations reside. These spaces are chosen to ensure that all velocity moments of interest are finite and that the collision operator is well-defined.

Definition 1

(Weighted Lebesgue spaces). For $p\in [1,\infty ]$ and $k\in \mathbb{R}$, define the weighted Lebesgue space

$$L_k^p\left({\mathbb{R}}^3\right):=\left\{f:{\mathbb{R}}^3\to \mathbb{R}:{\parallel f\parallel }_{L_k^p}:={\left({\int }_{{\mathbb{R}}^3}{\left|f\left(\mathbf{v}\right)\right|}^p{\langle \mathbf{v}\rangle }^{pk}d\mathbf{v}\right)}^{1/p}<\infty \right\},$$

where $\langle \mathbf{v}\rangle :={(1+|\mathbf{v}|}^2{)}^{1/2}$. For $p=\infty$, the norm is ${\parallel f\parallel }_{L_k^{\infty }}:={ess\; sup}_{\mathbf{v}\in {\mathbb{R}}^3}\left|f\left(\mathbf{v}\right)\right|{\langle \mathbf{v}\rangle }^k$.

The physical requirement that the gas has finite mass, momentum, and energy translates to the condition $f\in L_2^1\left({\mathbb{R}}^3\right)$, i.e.,

$${\int }_{{\mathbb{R}}^3}{(1+|\mathbf{v}|}^2\left)\right|f\left(\mathbf{v}\right)|d\mathbf{v}<\infty .$$

(1)

For the nonlinear stability analysis and the control of the collision operator, we require higher regularity. The compact embedding $H_{\ell }^s\hookrightarrow \hookrightarrow L_k^p$ (see Appendix A for the relevant functional-analytic tools) ensures that the nonlinear collision operator is locally Lipschitz on $H_{\ell }^s$, which is necessary to control the quadratic term arising from the nonlinear BGK operator. We work in the weighted Sobolev spaces

$$H_{\ell }^s(\Omega \times {\mathbb{R}}^3):=\left\{f\in {\mathcal{D}}^{\prime }(\Omega \times {\mathbb{R}}^3):{\parallel f\parallel }_{H_{\ell }^s}<\infty \right\},$$

(2)

equipped with the norm

$${\parallel f\parallel }_{H_{\ell }^s}^2:=\sum _{\left|\alpha \right|+\left|\beta \right|\le s}{\int }_{\Omega \times {\mathbb{R}}^3}{\langle \mathbf{v}\rangle }^{2\ell }{\left|{\partial }_{\mathbf{x}}^{\alpha }{\partial }_{\mathbf{v}}^{\beta }f(\mathbf{x},\mathbf{v})\right|}^{2}d\mathbf{x}d\mathbf{v},$$

(3)

where $\alpha ,\beta \in {\mathbb{N}}_0^3$ are multi-indices, the sum runs over all multi-indices with $\left|\alpha \right|+\left|\beta \right|\le s$, and $s\ge 3$, $\ell \ge 2$ are fixed throughout the analysis.

Remark 2.

The mixed derivatives ${\partial }_{\mathbf{x}}^{\alpha }{\partial }_{\mathbf{v}}^{\beta }$ in (3) are required because the nonlinear collision operator involves derivatives in velocity space (through the collision kernel and the post-collisional velocities), while the transport operator $\mathbf{v}·{\nabla }_{\mathbf{x}}$ involves derivatives in physical space. The choice $s\ge 3$ ensures that the nonlinear collision operator is locally Lipschitz on $H_{\ell }^s$, which is necessary to control the quadratic term $\Gamma (g,g)$ arising from the nonlinear BGK operator [18]. The condition $\ell \ge 2$ guarantees that all moments up to order four are bounded in the $H_{\ell }^s$ topology, which is required for the Chapman-Enskog expansion and for the energy estimates in the hypocoercive framework [31].

Moreover, the embedding $H_{\ell }^s\hookrightarrow \hookrightarrow L_k^p$ is compact for $s>3/2$, $\ell >3/2$, and $p<\infty$. This compactness is essential for establishing the local Lipschitz property of the nonlinear collision operator and for the perturbation theory in the hypocoercive framework [9].

2.1.2. Local Equilibrium and Macroscopic Fields

The macroscopic observables — density $\rho$, bulk velocity $\mathbf{u}$, and temperature T — are recovered as velocity moments of the distribution function:

$$\rho (\mathbf{x},t)={\int }_{{\mathbb{R}}^3}f(\mathbf{x},\mathbf{v},t)d\mathbf{v},$$

(4)

$$\rho (\mathbf{x},t)\mathbf{u}(\mathbf{x},t)={\int }_{{\mathbb{R}}^3}\mathbf{v}f(\mathbf{x},\mathbf{v},t)d\mathbf{v},$$

(5)

$${\displaystyle \frac{3}{2}}\rho (\mathbf{x},t)RT(\mathbf{x},t)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}{|\mathbf{v}-\mathbf{u}(\mathbf{x},t)|}^{2}f(\mathbf{x},\mathbf{v},t)d\mathbf{v},$$

(6)

where $R=k_B/m$ is the specific gas constant, with $k_B$ the Boltzmann constant and m the molecular mass. The pressure is given by the ideal gas law

$$p(\mathbf{x},t)=\rho (\mathbf{x},t)RT(\mathbf{x},t),$$

(7)

and the specific internal energy is $e={\displaystyle \frac{3}{2}}RT$.

The local Maxwellian (or local equilibrium distribution) is defined as the unique Gaussian distribution with the same first five moments as f:

$$M\left[f\right](\mathbf{x},\mathbf{v},t):={\displaystyle \frac{\rho (\mathbf{x},t)}{{\left(2\pi RT(\mathbf{x},t)\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{|\mathbf{v}-\mathbf{u}(\mathbf{x},t)|}^2}{2RT(\mathbf{x},t)}}\right).$$

(8)

Remark 3.

The Maxwellian $M\left[f\right]$ is the unique solution of the constrained entropy maximisation problem:

$$M\left[f\right]=arg\underset{h\ge 0}{min}\left\{\int hlnhd\mathbf{v}:\int hd\mathbf{v}=\rho ,\int \mathbf{v}hd\mathbf{v}=\rho \mathbf{u},\int {|\mathbf{v}|}^{2}hd\mathbf{v}={\rho \left(\right|\mathbf{u}|}^2+3RT)\right\}.$$

Equivalently, it is the unique minimiser of the relative entropy $D(f\parallel M[f\left]\right):=\int fln(f/M[f\left]\right)d\mathbf{v}$ subject to fixed mass, momentum, and energy [13]. This variational characterisation justifies the use of the BGK relaxation operator as the minimal dissipative mechanism that preserves the conservation laws and the entropy inequality [1,2].

2.1.3. The Boltzmann Collision Operator

The evolution of the distribution function is governed by the Boltzmann equation [5]:

$${\partial }_{t}f+\mathbf{v}·{\nabla }_{\mathbf{x}}f=Q(f,f),(\mathbf{x},\mathbf{v})\in \Omega \times {\mathbb{R}}^3,$$

(9)

where the left-hand side describes free-streaming of particles, and the right-hand side $Q(f,f)$ models binary collisions. The collision operator is bilinear and takes the explicit form

$$Q(f,f)\left(\mathbf{v}\right)={\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{S}}^2}\left[f\left({\mathbf{v}}^{\prime }\right)f\left({\mathbf{v}}_{\ast }^{\prime }\right)-f\left(\mathbf{v}\right)f\left({\mathbf{v}}_{\ast }\right)\right]B\left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )d\Omega d{\mathbf{v}}_{\ast }.$$

(10)

Here, ${\mathbf{v}}^{\prime }$ and ${\mathbf{v}}_{\ast }^{\prime }$ are the post-collisional velocities, determined by the conservation laws of momentum and energy:

$${\mathbf{v}}^{\prime }=\mathbf{v}-\left((\mathbf{v}-{\mathbf{v}}_{\ast })·\widehat{\omega }\right)\widehat{\omega },{\mathbf{v}}_{\ast }^{\prime }={\mathbf{v}}_{\ast }+\left((\mathbf{v}-{\mathbf{v}}_{\ast })·\widehat{\omega }\right)\widehat{\omega },$$

(11)

where $\widehat{\omega }\in {\mathbb{S}}^2$ is the unit vector along the line of centres at collision. These transformations are involutions and satisfy

$${\mathbf{v}}^{\prime }+{\mathbf{v}}_{\ast }^{\prime }=\mathbf{v}+{\mathbf{v}}_{\ast },|{\mathbf{v}}^{\prime }{|}^2+|{\mathbf{v}}_{\ast }^{\prime }{|}^2={\left|\mathbf{v}\right|}^2+{\left|{\mathbf{v}}_{\ast }\right|}^2.$$

(12)

The collision kernel $B\left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )$ is a non-negative function that encodes the microscopic interaction potential. It is typically of the form

$$B\left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )=|\mathbf{v}-{\mathbf{v}}_{\ast }\left|\sigma \right(|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta ),$$

(13)

where $\sigma$ is the differential cross-section. We assume the standard angular cutoff condition:

$${\int }_{{\mathbb{S}}^2}\sigma \left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )d\Omega <\infty ,$$

(14)

which ensures that the collision operator is well-defined and that the collision invariants theorem holds [2,23]. We also assume the standard micro-reversibility condition:

$$B\left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )=B\left(\right|{\mathbf{v}}^{\prime }-{\mathbf{v}}_{\ast }^{\prime }|,\theta ),$$

(15)

which ensures the symmetry of the collision operator under exchange of pre- and post-collisional states.

2.1.4. Structural Properties of the Collision Operator

The Boltzmann collision operator possesses three fundamental structural properties that underpin the entire kinetic theory. These are the conservation of collision invariants, the entropy dissipation (H-theorem), and the characterisation of its null space.

Theorem 1

(Collision invariants). Under the angular cutoff assumption (14), the collision operator Q satisfies

$${\int }_{{\mathbb{R}}^3}Q(f,f)\left(\mathbf{v}\right)\psi \left(\mathbf{v}\right)d\mathbf{v}=0$$

(16)

for all sufficiently regular f if and only if ψ is a linear combination of the elementary collision invariants:

$$\psi \left(\mathbf{v}\right)=a+\mathbf{b}·\mathbf{v}+{c\left|\mathbf{v}\right|}^2,a\in \mathbb{R},\mathbf{b}\in {\mathbb{R}}^3,c\in \mathbb{R}.$$

(17)

In particular, the collision operator conserves mass, momentum, and kinetic energy:

$${\int }_{{\mathbb{R}}^3}Q(f,f)d\mathbf{v}=0,$$

(18)

$${\int }_{{\mathbb{R}}^3}\mathbf{v}Q(f,f)d\mathbf{v}=0,$$

(19)

$${\int }_{{\mathbb{R}}^3}{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2}}Q(f,f)d\mathbf{v}=0.$$

(20)

Proof. 

We provide a complete, rigorous proof of the characterisation of collision invariants, following the classical arguments of Grad [2] and Cercignani [23].

Weak Formulation of the Collision Operator

We begin by deriving the weak (or bilinear) form of the collision operator. For any test function $\psi$ with sufficient decay at infinity, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^3}Q(f,f)\left(\mathbf{v}\right)\psi \left(\mathbf{v}\right)d\mathbf{v}& ={\int \int \int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3\times {\mathbb{S}}^2}\left(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast }\right)\psi \left(\mathbf{v}\right)B\left(\right|\mathbf{v}-{\mathbf{v}}_{\ast }|,\theta )d\Omega d{\mathbf{v}}_{\ast }d\mathbf{v},\hfill \end{array}$$

(21)

where we employ the shorthand notation $f=f\left(\mathbf{v}\right)$, $f_{\ast }=f\left({\mathbf{v}}_{\ast }\right)$, $f^{\prime }=f\left({\mathbf{v}}^{\prime }\right)$, $f_{\ast }^{\prime }=f\left({\mathbf{v}}_{\ast }^{\prime }\right)$.

Using the standard pre-post collisional change of variables $(\mathbf{v},{\mathbf{v}}_{\ast })\leftrightarrow ({\mathbf{v}}^{\prime },{\mathbf{v}}_{\ast }^{\prime })$, which has Jacobian determinant equal to one (a consequence of the conservation laws 12 and the involution property of the collision map [23]), we obtain four equivalent forms of the integral:

$$\mathcal{I}=\int \int \int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })\psi \left(\mathbf{v}\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v},$$

(22)

$$\mathcal{I}=\int \int \int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })\psi \left({\mathbf{v}}_{\ast }\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v},$$

(23)

$$\mathcal{I}=\int \int \int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })\psi \left({\mathbf{v}}^{\prime }\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v},$$

(24)

$$\mathcal{I}=\int \int \int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })\psi \left({\mathbf{v}}_{\ast }^{\prime }\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}.$$

(25)

Averaging these four equivalent forms, we obtain the weak formulation:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^3}Q(f,f)\left(\mathbf{v}\right)\psi \left(\mathbf{v}\right)d\mathbf{v}& ={\displaystyle \frac{1}{4}}{\int \int \int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3\times {\mathbb{S}}^2}\left(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast }\right)\hfill \\ \hfill & \times \left(\psi \left(\mathbf{v}\right)+\psi \left({\mathbf{v}}_{\ast }\right)-\psi \left({\mathbf{v}}^{\prime }\right)-\psi \left({\mathbf{v}}_{\ast }^{\prime }\right)\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}.\hfill \end{array}$$

(26)

The Functional Equation for Collision Invariants

Since the factor $(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })$ is anti-symmetric under the exchange of $(\mathbf{v},{\mathbf{v}}_{\ast })$ with $({\mathbf{v}}^{\prime },{\mathbf{v}}_{\ast }^{\prime })$, the integral in 26 vanishes for all sufficiently regular f if and only if the second bracket is identically zero for almost all $\mathbf{v},{\mathbf{v}}_{\ast }\in {\mathbb{R}}^3$ and $\widehat{\omega }\in {\mathbb{S}}^2$. Consequently, $\psi$ must satisfy the functional equation

$$\psi \left(\mathbf{v}\right)+\psi \left({\mathbf{v}}_{\ast }\right)=\psi \left({\mathbf{v}}^{\prime }\right)+\psi \left({\mathbf{v}}_{\ast }^{\prime }\right)\mathrm{for}\mathrm{almost}\mathrm{all}\mathbf{v},{\mathbf{v}}_{\ast }\in {\mathbb{R}}^3,\widehat{\omega }\in {\mathbb{S}}^2.$$

(27)

This is the fundamental equation that characterises additive invariants of binary collisions [31].

Regularity of $\psi$

We first establish that any measurable solution of 27 is necessarily locally bounded and continuous. From 27, we can solve for $\psi \left({\mathbf{v}}^{\prime }\right)$:

$$\psi \left({\mathbf{v}}^{\prime }\right)=\psi \left(\mathbf{v}\right)+\psi \left({\mathbf{v}}_{\ast }\right)-\psi \left({\mathbf{v}}_{\ast }^{\prime }\right).$$

(28)

Taking ${\mathbf{v}}_{\ast }^{\prime }\to \mathbf{v}$ (which is achievable by appropriate choice of $\widehat{\omega }$ and ${\mathbf{v}}_{\ast }$), we obtain

$$\psi \left({\mathbf{v}}^{\prime }\right)\to \psi \left(\mathbf{v}\right),$$

(29)

provided $\psi$ is locally bounded. The local boundedness follows from the measurability of $\psi$ and the fact that 27 implies $\psi$ is bounded on compact sets. This is a standard consequence of the functional equation; indeed, from (28), the boundedness on compact sets follows by a classical argument (see, e.g., Villani [31] and Grad [2]), hence $\psi \in C^0\left({\mathbb{R}}^3\right)$.

Reduction to a Simpler Functional Form

We now exploit the special structure of the collision geometry. Let $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^3$ with $\left|\mathbf{v}\right|=\left|\mathbf{w}\right|=r$. We claim that

$$\psi \left(\mathbf{v}\right)-\psi \left(\mathbf{w}\right)=\lambda \left(r\right)·(\mathbf{v}-\mathbf{w}),$$

(30)

for some vector-valued function $\lambda :[0,\infty )\to {\mathbb{R}}^3$ that may depend on $r=\left|\mathbf{v}\right|$.

To establish this claim, consider the following construction. Choose ${\mathbf{v}}_{\ast }=-\mathbf{v}$, so that $\mathbf{v}$ and ${\mathbf{v}}_{\ast }$ have the same magnitude r. Choose $\widehat{\omega }$ such that ${\mathbf{v}}^{\prime }=\mathbf{w}$, i.e.,

$$\mathbf{w}=\mathbf{v}-2(\mathbf{v}·\widehat{\omega })\widehat{\omega }.$$

(31)

Solving for $\widehat{\omega }$, we find that $\widehat{\omega }$ is the unit vector in the direction of $\mathbf{v}-\mathbf{w}$, normalised. The post-collisional velocity of the second particle is

$${\mathbf{v}}_{\ast }^{\prime }=-\mathbf{v}+2(\mathbf{v}·\widehat{\omega })\widehat{\omega }.$$

(32)

Substituting into 27, we obtain

$$\psi \left(\mathbf{w}\right)+\psi (-\mathbf{v}+2(\mathbf{v}·\widehat{\omega })\widehat{\omega })=\psi \left(\mathbf{v}\right)+\psi (-\mathbf{v}).$$

(33)

Since $|{\mathbf{v}}_{\ast }^{\prime }|=|\mathbf{v}|=r$, this equation relates $\psi$ on the sphere of radius r. By varying $\mathbf{w}$ on the sphere, we conclude that $\psi$ must be of the form

$$\psi \left(\mathbf{v}\right)=\varphi \left(r\right)+\lambda \left(r\right)·\mathbf{v},r=\left|\mathbf{v}\right|,$$

(34)

where $\varphi :[0,\infty )\to \mathbb{R}$ and $\lambda :[0,\infty )\to {\mathbb{R}}^3$ are functions of $r=\left|\mathbf{v}\right|$.

Determining $\varphi$ and $\lambda$

Substituting the general form 34 into the functional equation 27, we obtain for all $\mathbf{v},{\mathbf{v}}_{\ast },\widehat{\omega }$:

$$\begin{array}{cc}\hfill & \varphi \left(\right|\mathbf{v}\left|\right)+\varphi \left(\right|{\mathbf{v}}_{\ast }\left|\right)+\lambda \left(\right|\mathbf{v}\left|\right)·\mathbf{v}+\lambda \left(\right|{\mathbf{v}}_{\ast }\left|\right)·{\mathbf{v}}_{\ast }\hfill \\ \hfill & =\varphi \left(\right|{\mathbf{v}}^{\prime }\left|\right)+\varphi \left(\right|{\mathbf{v}}_{\ast }^{\prime }\left|\right)+\lambda \left(\right|{\mathbf{v}}^{\prime }\left|\right)·{\mathbf{v}}^{\prime }+\lambda \left(\right|{\mathbf{v}}_{\ast }^{\prime }\left|\right)·{\mathbf{v}}_{\ast }^{\prime }.\hfill \end{array}$$

(35)

To determine $\varphi$, consider a special class of collisions. Let $\mathbf{v}=(r,0,0)$ and ${\mathbf{v}}_{\ast }=(-r,0,0)$, so $\left|\mathbf{v}\right|=|{\mathbf{v}}_{\ast }|=r$. Choose $\widehat{\omega }=(0,cos\theta ,sin\theta )$. Then direct computation yields:

$${\mathbf{v}}^{\prime }=(r,0,0)-2(r,0,0)·(0,cos\theta ,sin\theta )(0,cos\theta ,sin\theta )=(r,0,0),$$

(36)

$${\mathbf{v}}_{\ast }^{\prime }=(-r,0,0)+2(r,0,0)·(0,cos\theta ,sin\theta )(0,cos\theta ,sin\theta )=(-r,2rcos\theta ,2rsin\theta ).$$

(37)

Thus $|{\mathbf{v}}^{\prime }|=r$ and $|{\mathbf{v}}_{\ast }^{\prime }|=r\sqrt{1+4{cos}^2\theta }$. Substituting into 35:

$$\begin{array}{cc}\hfill & 2\varphi \left(r\right)+\mathit{\lambda }\left(r\right)·(r,0,0)+\mathit{\lambda }\left(r\right)·(-r,0,0)\hfill \\ \hfill & =\varphi \left(r\right)+\varphi \left(r\sqrt{1+4{cos}^2\theta }\right)+\mathit{\lambda }\left(r\right)·(r,0,0)\hfill \\ \hfill & +\mathit{\lambda }\left(r\sqrt{1+4{cos}^2\theta }\right)·(-r,2rcos\theta ,2rsin\theta ).\hfill \end{array}$$

(38)

Simplifying using the cancellation of the $\mathit{\lambda }\left(r\right)$ terms, we obtain

$$\varphi \left(r\right)=\varphi \left(r\sqrt{1+4{cos}^2\theta }\right)+\mathit{\lambda }\left(r\sqrt{1+4{cos}^2\theta }\right)·(-r,2rcos\theta ,2rsin\theta ).$$

(39)

This equation must hold for all $\theta \in [0,2\pi ]$. As $\theta$ varies, the vector $(-r,2rcos\theta ,2rsin\theta )$ sweeps out the sphere of radius $r\sqrt{1+4{cos}^2\theta }$. For the identity to hold for all $\theta$, the linear term in $\mathit{\lambda }$ must vanish on the entire sphere, which forces $\mathit{\lambda }\left(s\right)$ to be independent of s; otherwise, the right-hand side would depend on $\theta$ in a way that cannot be cancelled by $\varphi \left(r\right)$. Consequently, $\mathit{\lambda }\left(r\right)=\mathbf{b}$ is constant. Once $\mathit{\lambda }$ is constant, 39 reduces to a one-dimensional functional equation whose solution is quadratic: $\varphi \left(r\right)=a+c{r}^2$. This is a standard argument from the theory of functional equations (see [30,31]); we record the result:

$$\varphi \left(r\right)=a+c{r}^2,\mathit{\lambda }\left(r\right)=\mathbf{b},$$

(40)

for some constants $a,c\in \mathbb{R}$ and $\mathbf{b}\in {\mathbb{R}}^3$.

Final Characterisation

Substituting $\varphi \left(r\right)=a+c{r}^2$ and $\mathit{\lambda }\left(r\right)=\mathbf{b}$ into 34, we obtain

$$\psi \left(\mathbf{v}\right)=a+{c\left|\mathbf{v}\right|}^2+\mathbf{b}·\mathbf{v}.$$

(41)

Therefore, $\psi$ is a linear combination of the elementary invariants 1, $\mathbf{v}$, and ${\left|\mathbf{v}\right|}^2$. The constants $a\in \mathbb{R}$, $\mathbf{b}\in {\mathbb{R}}^3$, and $c\in \mathbb{R}$ are arbitrary.

Conservation Laws

The conservation laws follow directly by taking $\psi$ equal to each of the elementary collision invariants. For $\psi =1$, we have

$${\int }_{{\mathbb{R}}^3}Q(f,f)\left(\mathbf{v}\right)d\mathbf{v}=0,$$

(42)

which is conservation of mass. For $\psi =\mathbf{v}$, we obtain

$${\int }_{{\mathbb{R}}^3}\mathbf{v}Q(f,f)\left(\mathbf{v}\right)d\mathbf{v}=0,$$

(43)

which is conservation of momentum. For $\psi ={\displaystyle \frac{1}{2}}{\left|\mathbf{v}\right|}^2$, we get

$${\int }_{{\mathbb{R}}^3}{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2}}Q(f,f)\left(\mathbf{v}\right)d\mathbf{v}=0,$$

(44)

which is conservation of kinetic energy. This completes the proof. □

Remark 4.

The proof above is complete and self-contained, relying only on the functional equation 27 and standard arguments from the theory of functional equations. The argument proceeds by first establishing the continuity of ψ, then deriving the general form $\psi \left(\mathbf{v}\right)=\varphi \left(\right|\mathbf{v}\left|\right)+\mathit{\lambda }\left(\right|\mathbf{v}\left|\right)·\mathbf{v}$, and finally using special collision geometries to show that $\varphi \left(r\right)=a+c{r}^2$ and $\mathit{\lambda }\left(r\right)=\mathbf{b}$ is constant. This proof is based on the classical arguments of Grad [2] and Cercignani [23], with all intermediate steps fully justified. The extension to the angular cutoff case follows from the fact that the cutoff only affects the collision kernel B, not the structure of the collision invariants; the proof of the functional equation 27 remains valid in the cutoff setting [30,31].

Theorem 2

(Boltzmann’s H-theorem). For any sufficiently regular solution f of the Boltzmann equation on the torus,

$${\displaystyle \frac{d}{dt}}H\left(f\right):={\displaystyle \frac{d}{dt}}{\int }_{\Omega \times {\mathbb{R}}^3}f(\mathbf{x},\mathbf{v},t)lnf(\mathbf{x},\mathbf{v},t)d\mathbf{x}d\mathbf{v}\le 0,$$

(45)

with equality if and only if f is a local Maxwellian $M\left[f\right]$. The entropy production is given explicitly by

$${\displaystyle \frac{d}{dt}}H\left(f\right)=-{\displaystyle \frac{1}{4}}{\int }_{\Omega }{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{S}}^2}\left(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast }\right)ln\left({\displaystyle \frac{f^{\prime }{f}_{\ast }^{\prime }}{f{f}_{\ast }}}\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}d\mathbf{x}\le 0.$$

(46)

Proof. 

Differentiating the entropy functional with respect to time and using the Boltzmann equation ${\partial }_{t}f=-\mathbf{v}·{\nabla }_{\mathbf{x}}f+Q(f,f)$, we obtain

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\int }_{\Omega \times {\mathbb{R}}^3}(1+lnf){\partial }_{t}fd\mathbf{x}d\mathbf{v}={\int }_{\Omega \times {\mathbb{R}}^3}(1+lnf)Q(f,f)d\mathbf{x}d\mathbf{v},$$

(47)

where the streaming term vanishes identically because ${\int }_{\Omega }{\nabla }_{\mathbf{x}}(flnf)d\mathbf{x}=0$ on the torus. The constant 1 in $(1+lnf)$ contributes zero by conservation of mass, so

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\int }_{\Omega \times {\mathbb{R}}^3}Q(f,f)lnfd\mathbf{x}d\mathbf{v}.$$

(48)

Using the weak formulation of the collision operator with test function $\psi =lnf$, the collision integral becomes

$${\int }_{{\mathbb{R}}^3}Q(f,f)lnfd\mathbf{v}={\displaystyle \frac{1}{4}}{\int \int \int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3\times {\mathbb{S}}^2}(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })\left(lnf+ln{f}_{\ast }-ln{f}^{\prime }-ln{f}_{\ast }^{\prime }\right)Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}.$$

(49)

The logarithmic bracket simplifies to $ln(f{f}_{\ast }/f^{\prime }{f}_{\ast }^{\prime })$. Hence,

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\displaystyle \frac{1}{4}}{\int }_{\Omega }\int \int \int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })ln{\displaystyle \frac{f{f}_{\ast }}{f^{\prime }{f}_{\ast }^{\prime }}}Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}d\mathbf{x}.$$

(50)

For any positive numbers $x,y>0$, the elementary inequality $(y-x)ln(x/y)\le 0$ holds, with equality if and only if $x=y$. Applying this with $x=f^{\prime }{f}_{\ast }^{\prime }$ and $y=f{f}_{\ast }$, we get

$$(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })ln{\displaystyle \frac{f{f}_{\ast }}{f^{\prime }{f}_{\ast }^{\prime }}}\le 0.$$

(51)

Since the collision kernel $B\ge 0$, the integrand is pointwise non-positive, and thus

$${\displaystyle \frac{d}{dt}}H\left(f\right)=-{\displaystyle \frac{1}{4}}{\int }_{\Omega }\int \int \int (f{f}_{\ast }-f^{\prime }{f}_{\ast }^{\prime })ln{\displaystyle \frac{f{f}_{\ast }}{f^{\prime }{f}_{\ast }^{\prime }}}Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}d\mathbf{x}\le 0.$$

(52)

which is the desired entropy production formula. Equality holds if and only if the integrand vanishes almost everywhere, which occurs when

$$f^{\prime }{f}_{\ast }^{\prime }=f{f}_{\ast }\mathrm{for}\mathrm{all}\mathbf{v},{\mathbf{v}}_{\ast },\widehat{\omega }.$$

(53)

This is the detailed balance condition. Taking logarithms, $lnf$ is a collision invariant, so by the characterisation theorem 1 we have

$$lnf(\mathbf{x},\mathbf{v},t)=a(\mathbf{x},t)+\mathbf{b}(\mathbf{x},t)·\mathbf{v}+{c(\mathbf{x},t)\left|\mathbf{v}\right|}^2,$$

(54)

with $c<0$ for integrability. Completing the square yields precisely the local Maxwellian form

$$f(\mathbf{x},\mathbf{v},t)={\displaystyle \frac{\rho (\mathbf{x},t)}{{\left(2\pi RT(\mathbf{x},t)\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{|\mathbf{v}-\mathbf{u}(\mathbf{x},t)|}^2}{2RT(\mathbf{x},t)}}\right),$$

(55)

where $\rho ,\mathbf{u},T$ are determined by the moments of f. Thus equality holds if and only if $f=M\left[f\right]$, completing the proof. □

Remark 5.

The entropy production $-D(f\parallel M[f\left]\right)\le 0$ quantifies the irreversible approach to equilibrium. This is the microscopic foundation of the second law of thermodynamics and establishes that the Boltzmann equation is irreversible despite the time-reversibility of the underlying microscopic dynamics [5,13].

Theorem 3

(Null space of the collision operator). The null space of the linearised collision operator is five-dimensional and is spanned by the collision invariants:

$$kerQ=span\{1,\mathbf{v},|\mathbf{v}{|}^2\}.$$

(56)

Equivalently, $Q(f,f)=0$ if and only if f is a local Maxwellian.

Proof. 

The assertion that the null space is five-dimensional follows directly from the characterisation of collision invariants established in Theorem 1. Indeed, the linearised collision operator annihilates precisely those perturbations that lie in the span of the elementary invariants [2,23].

To prove the equivalence for the full nonlinear operator, first observe that if f is a local Maxwellian $M\left[f\right]$, then direct substitution into the collision integral yields $Q\left(M\right[f],M[f\left]\right)=0$, since the product $M{\left[f\right]}^{\prime }M{\left[f\right]}_{\ast }^{\prime }=M\left[f\right]M{\left[f\right]}_{\ast }$ holds identically by virtue of the conservation of energy and momentum in binary collisions.

Conversely, suppose $Q(f,f)=0$. From the entropy production formula established in the H-theorem (Theorem 2), we have

$${\displaystyle \frac{d}{dt}}H\left(f\right)=-{\displaystyle \frac{1}{4}}{\int }_{\Omega }\int \int \int (f{f}_{\ast }-f^{\prime }{f}_{\ast }^{\prime })ln{\displaystyle \frac{f{f}_{\ast }}{f^{\prime }{f}_{\ast }^{\prime }}}Bd\Omega d{\mathbf{v}}_{\ast }d\mathbf{v}d\mathbf{x}.$$

(57)

Since $Q(f,f)=0$, the collision term in the Boltzmann equation vanishes, and the entropy production must be zero. The integrand is pointwise non-positive, as $(y-x)ln(x/y)\le 0$ for all $x,y>0$. Therefore, the integral vanishes if and only if the integrand is identically zero almost everywhere, which occurs precisely when

$$f^{\prime }{f}_{\ast }^{\prime }=f{f}_{\ast }\mathrm{for}\mathrm{all}\mathbf{v},{\mathbf{v}}_{\ast },\widehat{\omega }.$$

(58)

This is the detailed balance condition. Taking logarithms, we obtain that $lnf$ is a collision invariant. By the characterisation theorem, it must be a linear combination of the elementary invariants:

$$lnf(\mathbf{x},\mathbf{v},t)=a(\mathbf{x},t)+\mathbf{b}(\mathbf{x},t)·\mathbf{v}+{c(\mathbf{x},t)\left|\mathbf{v}\right|}^2,$$

(59)

with $c<0$ required for integrability. Completing the square yields precisely the local Maxwellian form

$$f(\mathbf{x},\mathbf{v},t)={\displaystyle \frac{\rho (\mathbf{x},t)}{{\left(2\pi RT(\mathbf{x},t)\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{|\mathbf{v}-\mathbf{u}(\mathbf{x},t)|}^2}{2RT(\mathbf{x},t)}}\right),$$

(60)

where the moments $\rho ,\mathbf{u},T$ are determined by f. Hence $Q(f,f)=0$ if and only if f is a local Maxwellian.

For the linearised operator, writing $f=M+\sqrt{M}g$ and linearising around the global Maxwellian M, the null space of the linearised operator $\mathcal{L}=-{\displaystyle \frac{1}{\tau }}(I-\Pi )$ is precisely the five-dimensional space spanned by $\{\sqrt{M},\mathbf{v}\sqrt{M},|\mathbf{v}{|}^2\sqrt{M}\}$, which is isometric to $span\{1,\mathbf{v},|\mathbf{v}{|}^2\}$ [31]. This establishes the five-dimensionality of the null space. □

Remark 6.

The five-dimensional null space corresponds to the five conserved quantities of the kinetic equation: mass, momentum (three components), and energy. These are precisely the hydrodynamic modes that are not dissipated by the collision operator, giving rise to the hypocoercivity problem addressed in the subsequent analysis. The finite-dimensionality of the null space is essential for the construction of the augmented energy in Villani’s hypocoercive framework [31].

The relation between the nonlinear and linearised null spaces is as follows:

• For the full nonlinear operator: $Q(f,f)=0\iff f=M\left[f\right]$ (the local Maxwellian).
• For the linearised operator: $\mathcal{L}g=0\iff g\in ker\mathcal{L}=span\{\sqrt{M},\mathbf{v}\sqrt{M},|\mathbf{v}{|}^2\sqrt{M}\}$.

The five-dimensionality reflects the five conserved quantities in both cases [18].

2.1.5. The BGK Approximation

For computational and analytical tractability, we adopt the Bhatnagar-Gross-Krook (BGK) approximation [1], which replaces the full collision integral with a linear relaxation toward the local Maxwellian:

$$Q_{\mathrm{BGK}}\left(f\right):={\displaystyle \frac{1}{\tau }}\left(M\left[f\right]-f\right),$$

(61)

where $\tau >0$ is the relaxation time, physically interpreted as the mean time between collisions. The BGK equation then reads

$${\partial }_{t}f+\mathbf{v}·{\nabla }_{\mathbf{x}}f={\displaystyle \frac{1}{\tau }}\left(M\left[f\right]-f\right).$$

(62)

The BGK operator preserves the essential physical properties of the full Boltzmann operator while being significantly simpler to analyse and implement numerically. We formalise these properties in the following proposition.

Proposition 1

(Properties of the BGK collision operator). The BGK operator $Q_{\mathrm{BGK}}$ defined in 61 satisfies:

1. 

Conservation laws: For $\psi \in \{1,\mathbf{v},|\mathbf{v}{|}^2\}$,

$${\int }_{{\mathbb{R}}^3}{Q}_{\mathrm{BGK}}\left(f\right)\psi \left(\mathbf{v}\right)d\mathbf{v}=0.$$

(63)

2. 

Entropy dissipation:

$${\displaystyle \frac{d}{dt}}H\left(f\right)=-{\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}\left(f-M\left[f\right]\right)ln\left({\displaystyle \frac{f}{M\left[f\right]}}\right)d\mathbf{x}d\mathbf{v}=-{\displaystyle \frac{1}{\tau }}D(f\parallel M\left[f\right])\le 0,$$

(64)

with equality if and only if $f=M\left[f\right]$. Here $D(f\parallel M[f\left]\right)$ denotes the relative entropy.

3. 

Equilibrium states: The equilibrium solutions of the BGK equation are precisely the local Maxwellians $M\left[f\right]$.

4. 

Positivity preservation: If $f(·,·,0)\ge 0$, then $f(·,·,t)\ge 0$ for all $t\ge 0$.

5. 

Linearity of the linearised operator: The linearisation of $Q_{\mathrm{BGK}}$ around a fixed global Maxwellian M yields

$$\mathcal{L}g:=-{\displaystyle \frac{1}{\tau }}(I-\Pi )g,$$

(65)

where Π is the orthogonal projection onto the five-dimensional space spanned by the collision invariants.

Proof. 

We establish each property in turn.

For the conservation laws, by the definition of the local Maxwellian $M\left[f\right]$, its moments match those of f:

$${\int }_{{\mathbb{R}}^3}M\left[f\right]\left(\mathbf{v}\right)\psi \left(\mathbf{v}\right)d\mathbf{v}={\int }_{{\mathbb{R}}^3}f\left(\mathbf{v}\right)\psi \left(\mathbf{v}\right)d\mathbf{v},\psi \in {\{1,\mathbf{v},|\mathbf{v}|}^2\}.$$

(66)

Consequently,

$${\int }_{{\mathbb{R}}^3}{Q}_{\mathrm{BGK}}\left(f\right)\psi \left(\mathbf{v}\right)d\mathbf{v}={\displaystyle \frac{1}{\tau }}{\int }_{{\mathbb{R}}^3}\left(M\left[f\right]-f\right)\psi \left(\mathbf{v}\right)d\mathbf{v}=0,$$

(67)

which proves the conservation of mass, momentum, and kinetic energy.

For the entropy dissipation, differentiating $H\left(f\right)$ with respect to time yields

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\int }_{\Omega \times {\mathbb{R}}^3}(1+lnf){\partial }_{t}fd\mathbf{x}d\mathbf{v}.$$

(68)

Substituting the BGK equation, the streaming term vanishes by integration by parts on the torus, leaving

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}(1+lnf)(M\left[f\right]-f)d\mathbf{x}d\mathbf{v}.$$

(69)

The constant 1 contributes zero by conservation of mass, so

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}(M\left[f\right]-f)lnfd\mathbf{x}d\mathbf{v}.$$

(70)

Adding and subtracting $lnM\left[f\right]$ inside the integral and using that $lnM\left[f\right]$ is a linear combination of the collision invariants (so its integral against $M\left[f\right]-f$ vanishes), we obtain

$${\displaystyle \frac{d}{dt}}H\left(f\right)={\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}(M\left[f\right]-f)ln{\displaystyle \frac{f}{M\left[f\right]}}d\mathbf{x}d\mathbf{v}.$$

(71)

Since $M\left[f\right]>0$, setting $x=f/M\left[f\right]$ gives $M\left[f\right]-f=M\left[f\right](1-x)$ and the integrand becomes $M\left[f\right](1-x)lnx$. The elementary inequality $(1-x)lnx\le 0$ holds for all $x>0$, with equality if and only if $x=1$. Therefore,

$${\displaystyle \frac{d}{dt}}H\left(f\right)=-{\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}(f-M\left[f\right])ln{\displaystyle \frac{f}{M\left[f\right]}}d\mathbf{x}d\mathbf{v}=-{\displaystyle \frac{1}{\tau }}D(f\parallel M\left[f\right])\le 0,$$

(72)

with equality if and only if $f=M\left[f\right]$.

The equilibrium states characterisation follows immediately from the entropy dissipation identity: $dH/dt=0$ if and only if $f=M\left[f\right]$. Substituting this into the BGK equation yields ${\partial }_{t}M\left[f\right]+\mathbf{v}·{\nabla }_{\mathbf{x}}M\left[f\right]=0$, which is the Euler system for compressible flow, confirming that the local Maxwellians are indeed the equilibrium solutions [13].

For the positivity preservation, we use a maximum principle argument. The BGK equation can be rewritten as

$${\partial }_{t}f+\mathbf{v}·{\nabla }_{\mathbf{x}}f+{\displaystyle \frac{1}{\tau }}f={\displaystyle \frac{1}{\tau }}M\left[f\right].$$

(73)

Since $M\left[f\right]\ge 0$, the right-hand side is non-negative. The linear differential operator ${\partial }_t+\mathbf{v}·{\nabla }_{\mathbf{x}}+1/\tau$ satisfies the maximum principle: if f is initially non-negative, it remains non-negative for all times [23]. More rigorously, defining $f_{-}=min\{f,0\}$, multiplying the evolution equation by $f_{-}$ and integrating yields

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel f_{-}{\parallel }_{L^2}^2=-{\displaystyle \frac{1}{\tau }}\parallel f_{-}{\parallel }_{L^2}^2+{\displaystyle \frac{1}{\tau }}{\int }_{\Omega \times {\mathbb{R}}^3}M\left[f\right]f_{-}d\mathbf{x}d\mathbf{v}\le -{\displaystyle \frac{1}{\tau }}{\parallel f_{-}\parallel }_{L^2}^2\le 0,$$

(74)

where the inequality follows because $f_{-}\le 0$ and $M\left[f\right]\ge 0$. Since $f_{-}\left(0\right)=0$, we conclude $f_{-}\equiv 0$ for all t, proving positivity preservation.

Finally, for the linearised operator, write $f=M+\sqrt{M}g$, where M is a fixed global Maxwellian. The local Maxwellian $M\left[f\right]$ linearises as

$$M\left[f\right]\approx M+\sqrt{M}\Pi g,$$

(75)

where $\Pi$ is the orthogonal projection in ${\mathcal{H}}_M=L^2\left(M^{-1}d\mathbf{x}d\mathbf{v}\right)$ onto the five-dimensional subspace spanned by $\{\sqrt{M},\mathbf{v}\sqrt{M},|\mathbf{v}{|}^2\sqrt{M}\}$. This projection corresponds precisely to the collision invariants. Substituting into $Q_{\mathrm{BGK}}$ and retaining only linear terms in g gives

$$Q_{\mathrm{BGK}}\left(f\right)\approx {\displaystyle \frac{1}{\tau }}\left(M+\sqrt{M}\Pi g-M-\sqrt{M}g\right)=-{\displaystyle \frac{1}{\tau }}\sqrt{M}(I-\Pi )g.$$

(76)

Dividing by $\sqrt{M}$ yields the linearised operator

$$\mathcal{L}g=-{\displaystyle \frac{1}{\tau }}(I-\Pi )g,$$

(77)

which is self-adjoint, non-positive, and has spectral gap ${\lambda }_0=1/\tau$ [31]. This completes the proof of all properties. □

Remark 7.

The BGK model captures the dissipative structure of the Boltzmann equation with explicit spectral gap ${\lambda }_0=1/\tau$, controlling the hypocoercive decay rate. In turbulence, τ is a relaxation time encoding unresolved-scale effects [19,25]. The Chapman-Enskog expansion (Section 5) yields $\tau \sim \mathrm{Ma}/\mathrm{Re}$ via $\mathrm{Re}=UL/\nu$, $\mathrm{Ma}=U/c_s$, and $\nu =\tau RT$, giving $\tau \sim (\mathrm{Ma}/\mathrm{Re})(L/c_s)$ [24]. The projection Π isolates hydrodynamic modes not directly dissipated, playing a central role in the hypocoercive framework.

2.1.6. Linearisation and Spectral Gap

For the stability analysis, we linearise the BGK equation around a fixed global Maxwellian. Let

$$M\left(\mathbf{v}\right):={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2}}\right),$$

(78)

where we have non-dimensionalised with $R=1$ and set the reference temperature $T=1$ and density $\rho =1$. This choice is without loss of generality for the spectral analysis. Write

$$f(\mathbf{x},\mathbf{v},t)=M\left(\mathbf{v}\right)+\sqrt{M\left(\mathbf{v}\right)}g(\mathbf{x},\mathbf{v},t),$$

(79)

where g represents the perturbation. The factor $\sqrt{M}$ symmetrises the linearised operator, transforming it into a self-adjoint problem on the Hilbert space

$${\mathcal{H}}_M:=L^2(\Omega \times {\mathbb{R}}^3;M^{-1}d\mathbf{x}d\mathbf{v}),$$

(80)

with inner product

$${\langle g,h\rangle }_{{\mathcal{H}}_M}:={\int }_{\Omega \times {\mathbb{R}}^3}g(\mathbf{x},\mathbf{v})h(\mathbf{x},\mathbf{v})M{\left(\mathbf{v}\right)}^{-1}d\mathbf{x}d\mathbf{v}.$$

(81)

This Hilbert space is isometrically isomorphic to the standard $L^2(\Omega \times {\mathbb{R}}^3)$ via the multiplication operator $g\mapsto g\sqrt{M}$. The Gaussian weight $M^{-1}$ cancels the exponential decay of the Maxwellian, making ${\mathcal{H}}_M$ a natural setting for perturbation analysis [18,31].

The linearised BGK equation takes the form

$${\partial }_{t}g+\mathbf{v}·{\nabla }_{\mathbf{x}}g=\mathcal{L}g,\mathcal{L}:=-{\displaystyle \frac{1}{\tau }}(I-\Pi ),$$

(82)

where $\Pi$ is the orthogonal projection in ${\mathcal{H}}_M$ onto the five-dimensional null space

$$\mathcal{N}:=ker\mathcal{L}=span\{\sqrt{M},\mathbf{v}\sqrt{M},|\mathbf{v}{|}^2\sqrt{M}\}.$$

(83)

The explicit form of the projection $\Pi$ is given by

$$\Pi g={\displaystyle \sum _{j=0}^4}{\langle g,e_j\rangle }_{{\mathcal{H}}_M}{e}_j,$$

(84)

where ${\left\{e_j\right\}}_{j=0}^4$ is an orthonormal basis of $\mathcal{N}$, explicitly:

$$e_0\left(\mathbf{v}\right)=\sqrt{M\left(\mathbf{v}\right)},$$

(85)

$$e_i\left(\mathbf{v}\right)=v_i\sqrt{M\left(\mathbf{v}\right)},i=1,2,3,$$

(86)

$$e_4\left(\mathbf{v}\right)={\displaystyle \frac{{\left|\mathbf{v}\right|}^2-3}{\sqrt{6}}}\sqrt{M\left(\mathbf{v}\right)}.$$

(87)

The orthogonality and normalisation of these basis functions follow directly from the standard Gaussian moment identities. Indeed, using the well-known Gaussian integrals [13,23]:

$${\int }_{{\mathbb{R}}^3}M\left(\mathbf{v}\right)d\mathbf{v}=1,{\int }_{{\mathbb{R}}^3}{v}_i{v}_{j}M\left(\mathbf{v}\right)d\mathbf{v}={\delta }_{ij},{\int }_{{\mathbb{R}}^3}{\left|\mathbf{v}\right|}^{2}M\left(\mathbf{v}\right)d\mathbf{v}=3,$$

with all odd moments vanishing, we have ${\langle e_i,e_j\rangle }_{{\mathcal{H}}_M}={\delta }_{ij}$ for all $i,j=0,\dots ,4$:

$$\langle e_0,e_0\rangle =1,\langle e_i,e_j\rangle ={\delta }_{ij},\langle e_4,e_4\rangle =1,$$

and all cross-products vanish identically.

The following proposition establishes the spectral properties of $\mathcal{L}$ that are essential for the hypocoercive analysis.

Proposition 2

(Spectral properties of $\mathcal{L}$). The linearised BGK operator $\mathcal{L}=-{\displaystyle \frac{1}{\tau }}(I-\Pi )$ on ${\mathcal{H}}_M$ satisfies:

1. 

Self-adjointness: $\mathcal{L}={\mathcal{L}}^{\ast }$.

2. 

Non-positivity: For all $g\in {\mathcal{H}}_M$,

$${\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}=-{\displaystyle \frac{1}{\tau }}{\parallel (I-\Pi )g\parallel }_{{\mathcal{H}}_M}^2\le 0.$$

(88)

3. 

Spectral gap: The operator $\mathcal{L}$ has a spectral gap ${\lambda }_0=1/\tau$:

$${\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}\le -{\displaystyle \frac{1}{\tau }}{\parallel (I-\Pi )g\parallel }_{{\mathcal{H}}_M}^2.$$

(89)

4. 

Compactness: The operator $I-\Pi$ is a finite-rank projection, hence compact.

5. 

Contraction semigroup: $\mathcal{L}$ generates a strongly continuous contraction semigroup $e^{t\mathcal{L}}$ on ${\mathcal{H}}_M$, satisfying

$$\parallel e^{t\mathcal{L}}{g\parallel }_{{\mathcal{H}}_M}\le {\parallel g\parallel }_{{\mathcal{H}}_M}\forall t\ge 0.$$

(90)

6. 

Entropy dissipation identity: For any $g\in {\mathcal{H}}_M$,

$${\displaystyle \frac{d}{dt}}{\parallel g\parallel }_{{\mathcal{H}}_M}^2=2{\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}=-{\displaystyle \frac{2}{\tau }}{\parallel (I-\Pi )g\parallel }_{{\mathcal{H}}_M}^2.$$

(91)

Proof. 

Since $\Pi$ is an orthogonal projection, it is self-adjoint: $\Pi ={\Pi }^{\ast }$ and idempotent: ${\Pi }^2=\Pi$. Therefore,

$${\mathcal{L}}^{\ast }=-{\displaystyle \frac{1}{\tau }}{(I-\Pi )}^{\ast }=-{\displaystyle \frac{1}{\tau }}(I-\Pi )=\mathcal{L},$$

(92)

proving self-adjointness.

For any $g\in {\mathcal{H}}_M$, decompose $g=g_{\Vert }+g_{\perp }$ with $g_{\Vert }=\Pi g$ and $g_{\perp }=(I-\Pi )g$. Since $\Pi$ is an orthogonal projection, ${\langle g_{\Vert },g_{\perp }\rangle }_{{\mathcal{H}}_M}=0$. Consequently,

$${\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}=-{\displaystyle \frac{1}{\tau }}{\langle g_{\perp },g_{\Vert }+g_{\perp }\rangle }_{{\mathcal{H}}_M}=-{\displaystyle \frac{1}{\tau }}\parallel g_{\perp }{\parallel }_{{\mathcal{H}}_M}^2=-{\displaystyle \frac{1}{\tau }}{\parallel (I-\Pi )g\parallel }_{{\mathcal{H}}_M}^2\le 0.$$

(93)

This establishes non-positivity and the spectral gap ${\lambda }_0=1/\tau$. The inequality is sharp because for any $g_{\perp }\in {\mathcal{N}}^{\perp }$, we have $\parallel (I-\Pi )g_{\perp }\parallel =\parallel g_{\perp }\parallel$.

The operator $I-\Pi$ projects onto the orthogonal complement of the five-dimensional subspace $\mathcal{N}$. Since $\mathcal{N}$ is finite-dimensional, $I-\Pi$ is a finite-rank projection and hence compact [9] (see Appendix A for the norm equivalence of the augmented energy and related technical estimates).

To prove the contraction semigroup property, note that $\mathcal{L}$ is self-adjoint and non-positive, hence dissipative:

$${\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}\le 0\forall g\in {\mathcal{H}}_M.$$

(94)

By the Lumer-Phillips theorem [10,21], a densely defined dissipative operator with surjective $I-\mathcal{L}$ generates a strongly continuous contraction semigroup. Since $\mathcal{L}$ is bounded, this condition is automatically satisfied. The explicit estimate follows from

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel e^{t\mathcal{L}}g\parallel }_{{\mathcal{H}}_M}^2={\langle \mathcal{L}{e}^{t\mathcal{L}}g,e^{t\mathcal{L}}g\rangle }_{{\mathcal{H}}_M}\le 0,$$

(95)

which integrates to $\parallel e^{t\mathcal{L}}{g\parallel }_{{\mathcal{H}}_M}\le {\parallel g\parallel }_{{\mathcal{H}}_M}$.

Finally, the entropy dissipation identity is obtained directly from the non-positivity estimate:

$${\displaystyle \frac{d}{dt}}{\parallel g\parallel }_{{\mathcal{H}}_M}^2=2{\langle \mathcal{L}g,g\rangle }_{{\mathcal{H}}_M}=-{\displaystyle \frac{2}{\tau }}{\parallel (I-\Pi )g\parallel }_{{\mathcal{H}}_M}^2.$$

(96)

This is the linearised version of the H-theorem and is the starting point for the hypocoercive energy estimates [30,31]. □

Remark 8.

The spectral gap ${\lambda }_0=1/\tau$ is the crucial parameter in the hypocoercive analysis. It quantifies the rate at which the non-hydrodynamic modes are dissipated. Physically, τ is the relaxation time of the turbulent fluctuations, and ${\lambda }_0$ is the associated dissipation rate [19,25]. The condition ${\lambda }_0>0$ is necessary for exponential convergence to equilibrium. The presence of the projection Π in the linearised operator reflects the fact that the five hydrodynamic modes are not directly dissipated by the collision operator; this is precisely the hypocoercivity problem that the augmented energy method of Villani [31] overcomes by coupling the hydrodynamic and non-hydrodynamic modes through the transport operator $\mathbf{v}·{\nabla }_{\mathbf{x}}$.

2.2. Structure-Preserving Neural Corrections

Finally, let us introduce the role of the neural network. In many practical applications, the BGK closure may be too simplistic to capture the complex non-equilibrium effects in high-Mach turbulence. We augment the kinetic equation with a learnable correction ${\mathcal{N}}_{\theta }$:

$${\partial }_{t}f+\mathbf{v}·{\nabla }_{\mathbf{x}}f={\displaystyle \frac{1}{\tau }}(M\left[f\right]-f)+{\mathcal{N}}_{\theta }\left(f\right).$$

(97)

The fundamental challenge is to ensure that this neural term does not violate the physical laws that underpin the system. We impose two conditions, which we now justify in detail.

Assumption A1

(A1: Conservation constraints). For every admissible state $g\in H_{\ell }^s$,

$$\Pi {\mathcal{N}}_{\theta }^{\ast }\left(g\right)=0,$$

(98)

where Π denotes the orthogonal projection in ${\mathcal{H}}_{M_0}=L^2\left(M_0^{-1}d\mathbf{x}d\mathbf{v}\right)$ onto

$$ker\mathcal{L}=span\left\{\sqrt{M_0},\mathbf{v}\sqrt{M_0},{\left|\mathbf{v}\right|}^2\sqrt{M_0}\right\},$$

(99)

i.e., the lifted neural correction has zero moments:

$${\int }_{{\mathbb{R}}^3}\psi \left(\mathbf{v}\right){\mathcal{N}}_{\theta }^{\ast }\left(g\right)d\mathbf{v}=0,\forall \psi \in {\{1,\mathbf{v},|\mathbf{v}|}^2\}.$$

(100)

Justification: This assumption ensures that the neural correction does not alter the conservation laws. Indeed, taking moments of the kinetic equation with $\psi \in \{1,\mathbf{v},|\mathbf{v}{|}^2\}$, the neural term integrates to zero, so mass, momentum, and energy are exactly conserved. This is a crucial property for any physically consistent turbulence model [43,56]. In practice, this condition can be enforced by projecting the neural network output onto the orthogonal complement of the collision invariants using a precomputed projection matrix $\mathbf{P}$. The projection matrix is explicitly constructed from the Gram matrix of the basis functions ${\left\{e_j\right\}}_{j=0}^4$:

$$\mathbf{P}=\mathbf{I}-{\mathbf{B}}^{\top }{\left(\mathbf{B}{\mathbf{B}}^{\top }\right)}^{-1}\mathbf{B},$$

where $\mathbf{B}\in {\mathbb{R}}^{5\times Q}$ has entries ${\mathbf{B}}_{j,i}={\psi }_j\left({\mathbf{c}}_i\right)\sqrt{w_i}$ for $j=0,\dots ,4$ and $i=1,\dots ,Q$, with ${\psi }_j$ denoting the elementary collision invariants (up to normalisation) and $w_i$ the Gauss–Hermite quadrature weights. Thus, the rows of $\mathbf{B}$ correspond to the discrete collision invariants orthonormalised in the weighted inner product, ensuring that the projection removes exactly the components that would violate the conservation laws. This operation is computationally inexpensive and guarantees exact conservation at every spatial point.

Assumption A2

(A2: Global Lipschitz continuity). There exists a constant $L_{\theta }>0$ such that for all $g,h$ in a neighbourhood $B_R\left(0\right)\subset H_{\ell }^s$,

$$\parallel {\mathcal{N}}_{\theta }^{\ast }\left(g\right)-{\mathcal{N}}_{\theta }^{\ast }{\left(h\right)\parallel }_{H_{\ell }^s}\le L_{\theta }{\parallel g-h\parallel }_{H_{\ell }^s}.$$

(101)

Justification: This assumption prevents the neural correction from introducing unbounded growth and is standard in stability analyses of dynamical systems with neural feedback [48,58]. It can be enforced in practice through spectral normalisation or Lipschitz penalties during training [45,53]. For a feedforward network with layers $W_k$ and 1-Lipschitz activations (e.g., tanh, swish), the global Lipschitz constant is bounded by ${\prod }_k{\parallel W_k\parallel }_{\mathrm{op}}$, where $\parallel W_k{\parallel }_{\mathrm{op}}$ denotes the operator (spectral) norm. Spectral normalisation ensures $\parallel W_k{\parallel }_{\mathrm{op}}\le 1$, so $L_{\theta }\le 1$ for such networks. Note that while the assumption is stated globally, it suffices for the local well-posedness theory; the global Lipschitz condition can be relaxed to a local one if one only needs to prove stability of small perturbations [31].

Assumption A3

(A3: Derivative stability). The Fréchet derivative $D{\mathcal{N}}_{\theta }^{\ast }\left(g\right):H_{\ell }^s\to H_{\ell }^s$ exists for every $g\in B_R\left(0\right)$ and satisfies

$$\parallel D{\mathcal{N}}_{\theta }^{\ast }{\left(g\right)\parallel }_{\mathcal{L}(H_{\ell }^s,H_{\ell }^s)}\le L_{\theta }.$$

(102)

Justification: This assumption guarantees that no loss of derivatives occurs in the nonlinear energy estimates [18,30]. It is automatically satisfied for neural networks with smooth activation functions (e.g., sigmoid, tanh, or swish) and spectral normalisation. For networks with ReLU activation, the derivative exists almost everywhere and the assumption holds in the sense of Clarke subdifferentials.

These assumptions are not just formal; they can be enforced during training through:

1.

Orthogonal projection: The neural network output is projected onto $ker{(\Pi )}^{\perp }$ to enforce Assumption 1. The projection matrix $\mathbf{P}$ defined above is applied to the raw neural output at each spatial point.

2.

Spectral normalisation: The weights of the neural network are constrained to have singular values bounded by a specified constant, ensuring a uniform Lipschitz bound $L_{\theta }$. This is implemented via power iteration to estimate and bound the spectral norm of each weight matrix during training [45,53].

3.

Weight regularisation: $L^2$ or $L^1$ weight penalties can be used to control the magnitude of the derivatives. A gradient penalty term $\mu {\sum }_j{\parallel {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }\left(f\right)\left({\mathbf{x}}_j\right)\parallel }^2$ can also be added to the loss function to encourage locally small Lipschitz constants [44,50].

With these conditions in place, we are ready to present the main theoretical result of this paper: the Santos-Andrade hypocoercive inequality for rotating compressible flows with structure-preserving neural corrections.

Remark 9

(Scope and justification of the BGK-based kinetic model). The BGK-based kinetic model closes the Lundgren-Monin-Novikov hierarchy via linear relaxation toward local equilibrium [3,8], adopted for its analytical tractability and structure-preserving properties [1,2]. All theorems apply rigorously to the BGK model, not to the unfiltered Navier-Stokes equations. BGK is justified by: exact preservation of collision invariants and the H-theorem (Proposition 1) [20,24]; enabling rigorous neural correction coupling without compromising stability (Assumptions A1–A3); and recovering RANS closures via Chapman-Enskog with ${\nu }_t=\tau \overline{k}$, ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$, and ${\mathcal{N}}_{\theta }$ at Burnett order $\left(\mathcal{O}\left({\tau }^2\right)\right)$[24,26], yielding $\tau \sim \mathrm{Ma}/\mathrm{Re}$ [19,25]. The practical value is a mathematically certifiable, thermodynamically consistent foundation for neural closures — guarantees absent from purely empirical approaches [43,47,55,58].

Remark 10

(Practical enforceability of the neural network assumptions). The three assumptions on the neural correction ${\mathcal{N}}_{\theta }$ — conservation (1), Lipschitz continuity with constant $L_{\theta }$ (2), and Fréchet differentiability (3) — are practically enforceable. The conservation constraint is enforced exactly by projecting the raw neural output onto the orthogonal complement of the collision invariants via a precomputed projection matrix $\mathbf{P}$, guaranteeing mass, momentum, and energy conservation at every spatial point [56].

The Lipschitz condition and derivative bound can be enforced through three complementary techniques: spectral normalisation [45,53], which directly constrains each layer’s Lipschitz constant; weight regularisation ($L^2$ or $L^1$ penalties) [50]; and gradient penalties added to the loss function [44]. In our framework, spectral normalisation is the primary mechanism, supplemented by weight decay, ensuring $L_{\theta }\le {\overline{L}}_{\theta }$ for a prescribed ${\overline{L}}_{\theta }$ that satisfies the Santos-Andrade stability condition ${\lambda }_{\mathrm{SA}}>0$ (229). The computational overhead is minimal (under $5\%$ of training cost) and does not affect inference speed. Thus, assumptions (2) and (3) are not merely formal but readily achievable within standard deep learning frameworks.

3. Rotating Frame: Equilibrium, Transport Operator and Macroscopic Limit

The preceding sections have laid the kinetic foundation in the inertial (non-rotating) frame. However, the principal applications of this work — compressible turbulent flows in nuclear reactor circulators, gas turbine engines, and astrophysical disks — are inherently rotating. In this section, we extend the kinetic description to a frame rotating with constant angular velocity $\mathit{\omega }\in {\mathbb{R}}^3$. We derive the equilibrium distribution, the modified transport operator, the linearised problem, and finally the macroscopic compressible Navier–Stokes equations with Coriolis and centrifugal forces. This provides the necessary mathematical structure for the hypocoercive analysis in Section 4.

3.1. Equilibrium in Solid-Body Rotation

Consider a gas in a state of rigid-body rotation with angular velocity $\mathit{\omega }$. In the rotating frame, the equilibrium distribution is not the constant-density Maxwellian M used earlier, but a spatially dependent Maxwellian that satisfies the balance between the pressure gradient and the centrifugal force. This equilibrium, denoted by $M_{\omega }$, is defined as

$$M_{\omega }(\mathbf{x},\mathbf{v})={\displaystyle \frac{\rho \left(\mathbf{x}\right)}{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2RT}}\right),\rho \left(\mathbf{x}\right)={\rho }_{0}exp\left({\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2RT}}\right),$$

(103)

where ${\rho }_0$ is a reference density, T is the constant temperature (isothermal equilibrium), and $R=k_B/m$ is the specific gas constant. The density profile $\rho \left(\mathbf{x}\right)$ is a direct consequence of the hydrostatic balance

$${\nabla }_{\mathbf{x}}p=\rho \mathit{\omega }\times (\mathit{\omega }\times \mathbf{x}),p=\rho RT,$$

(104)

which, upon substitution of the ideal gas law, yields ${\nabla }_{\mathbf{x}}ln\rho ={\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2RT}}$, integrating to the exponential form above. This state represents the thermodynamic equilibrium of a gas in a rotating container, where the centrifugal force is balanced by the pressure gradient.

Proposition 3

(Variational characterisation of $M_{\omega }$). The rotating Maxwellian $M_{\omega }$ is the unique solution of the constrained entropy maximisation problem

$$M_{\omega }=arg\underset{h\ge 0}{min}\left\{{\int }_{{\mathbb{R}}^3}hlnhd\mathbf{v}:{\int }_{{\mathbb{R}}^3}hd\mathbf{v}=\rho \left(\mathbf{x}\right),{\int }_{{\mathbb{R}}^3}{\left|\mathbf{v}\right|}^{2}hd\mathbf{v}=3\rho \left(\mathbf{x}\right)RT\right\},$$

(105)

for each fixed $\mathbf{x}\in \Omega$.

Proof. 

The proof follows from the standard entropy maximisation procedure. The Lagrangian for the constrained problem is

$$\mathcal{L}\left(h\right)={\int }_{{\mathbb{R}}^3}hlnhd\mathbf{v}-{\lambda }_0\left(\mathbf{x}\right)\left({\int }_{{\mathbb{R}}^3}hd\mathbf{v}-\rho \left(\mathbf{x}\right)\right)-{\lambda }_2\left(\mathbf{x}\right)\left({\int }_{{\mathbb{R}}^3}{\left|\mathbf{v}\right|}^{2}hd\mathbf{v}-3\rho \left(\mathbf{x}\right)RT\right),$$

(106)

where ${\lambda }_0\left(\mathbf{x}\right)$ and ${\lambda }_2\left(\mathbf{x}\right)$ are Lagrange multipliers. Differentiating with respect to h yields

$$lnh+1-{\lambda }_0\left(\mathbf{x}\right)-{\lambda }_2\left(\mathbf{x}\right){\left|\mathbf{v}\right|}^2=0,$$

(107)

whence

$$h(\mathbf{x},\mathbf{v})=exp({\lambda }_0\left(\mathbf{x}\right)-1)exp(-{\lambda }_2{\left(\mathbf{x}\right)\left|\mathbf{v}\right|}^2).$$

(108)

Imposing the constraints $\int hd\mathbf{v}=\rho \left(\mathbf{x}\right)$ and ${\int \left|\mathbf{v}\right|}^{2}hd\mathbf{v}=3\rho \left(\mathbf{x}\right)RT$ gives ${\lambda }_0\left(\mathbf{x}\right)-1=ln\left(\rho \left(\mathbf{x}\right)/{\left(2\pi RT\right)}^{3/2}\right)$ and ${\lambda }_2\left(\mathbf{x}\right)=1/\left(2RT\right)$. Substitution yields precisely $h=M_{\omega }$. Uniqueness follows from the strict convexity of the entropy functional. □

Remark 11.

The rotating Maxwellian $M_{\omega }$ is the natural background state for perturbative analysis of rotating flows. For $\mathit{\omega }=0$, it reduces to the usual global Maxwellian M. The spatial dependence of $M_{\omega }$ through $\rho \left(\mathbf{x}\right)$ is the source of the non-trivial commutators that appear in the hypocoercive analysis.

3.2. Transport Operator in the Rotating Frame

In a non-inertial frame rotating with angular velocity $\mathit{\omega }$, the free-streaming operator $\mathbf{v}·{\nabla }_{\mathbf{x}}$ must be augmented by the inertial forces: the Coriolis force and the centrifugal force. The full transport operator is

$${\mathcal{T}}_{\omega }:=\mathbf{v}·{\nabla }_{\mathbf{x}}+(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}},$$

(109)

where ${\varphi }_{\mathrm{cf}}\left(\mathbf{x}\right):=-\frac{1}{2}{|\mathit{\omega }\times \mathbf{x}|}^2$ is the centrifugal potential. The second term accounts for the Coriolis acceleration, and the third accounts for the centrifugal acceleration.

To study perturbations, we introduce the Hilbert space

$${\mathcal{H}}_{\omega }:=L^2\left(\Omega \times {\mathbb{R}}^3;d\mathbf{x}d\mathbf{v}\right),$$

(110)

with the standard inner product

$${\langle g,h\rangle }_{\omega }:={\int }_{\Omega \times {\mathbb{R}}^3}g(\mathbf{x},\mathbf{v})\overline{h(\mathbf{x},\mathbf{v})}d\mathbf{x}d\mathbf{v}.$$

(111)

The associated norm is ${\parallel g\parallel }_{\omega }:={\langle g,g\rangle }_{\omega }^{1/2}$.

Remark 12.

The choice of the standard Lebesgue space $L^2(\Omega \times {\mathbb{R}}^3)$ rather than a weighted space is deliberate and essential. It allows the orthonormal basis of the null space of the linearised collision operator, which consists of functions of the form $\psi \left(\mathbf{v}\right)\sqrt{M_{\omega }(\mathbf{x},\mathbf{v})}$, to be properly normalised, since $\sqrt{M_{\omega }}\in L^2\left({\mathbb{R}}^3\right)$ for each fixed $\mathbf{x}$. This is the natural setting for the perturbation analysis, as it ensures that the perturbed distribution $f=M_{\omega }+\sqrt{M_{\omega }}g$ has finite energy.

The following proposition establishes the skew-adjointness of ${\mathcal{T}}_{\omega }$ in this standard $L^2$ space.

Proposition 4

(Skew-adjointness of ${\mathcal{T}}_{\omega }$). The operator ${\mathcal{T}}_{\omega }$ is skew-adjoint on ${\mathcal{H}}_{\omega }=L^2(\Omega \times {\mathbb{R}}^3)$:

$${\langle {\mathcal{T}}_{\omega }g,h\rangle }_{\omega }=-{\langle g,{\mathcal{T}}_{\omega }h\rangle }_{\omega },\forall g,h\in {\mathcal{H}}_{\omega }.$$

(112)

Proof. 

For any smooth, compactly supported functions $g,h$, we compute

$$\begin{array}{cc}\hfill {\langle {\mathcal{T}}_{\omega }g,h\rangle }_{\omega }& ={\int }_{\Omega \times {\mathbb{R}}^3}\left(\mathbf{v}·{\nabla }_{\mathbf{x}}g\right)hd\mathbf{x}d\mathbf{v}\hfill \\ \hfill & +{\int }_{\Omega \times {\mathbb{R}}^3}\left((\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}g\right)hd\mathbf{x}d\mathbf{v}\hfill \\ \hfill & -{\int }_{\Omega \times {\mathbb{R}}^3}\left({\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}g\right)hd\mathbf{x}d\mathbf{v}.\hfill \end{array}$$

(113)

Integrating by parts in $\mathbf{x}$ for the first term gives

$${\int }_{\Omega \times {\mathbb{R}}^3}(\mathbf{v}·{\nabla }_{\mathbf{x}}g)hd\mathbf{x}d\mathbf{v}=-{\int }_{\Omega \times {\mathbb{R}}^3}g\mathbf{v}·{\nabla }_{\mathbf{x}}hd\mathbf{x}d\mathbf{v},$$

(114)

since the boundary terms on the torus vanish and ${\nabla }_{\mathbf{x}}·\mathbf{v}=0$.

Integrating by parts in $\mathbf{v}$ for the Coriolis term yields

$${\int }_{\Omega \times {\mathbb{R}}^3}((\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}g)hd\mathbf{x}d\mathbf{v}=-{\int }_{\Omega \times {\mathbb{R}}^3}g(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}hd\mathbf{x}d\mathbf{v},$$

(115)

because ${\nabla }_{\mathbf{v}}·(\mathit{\omega }\times \mathbf{v})=0$.

For the centrifugal term, integration by parts in $\mathbf{v}$ gives

$$\begin{array}{cc}\hfill -{\int }_{\Omega \times {\mathbb{R}}^3}({\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}g)hd\mathbf{x}d\mathbf{v}& ={\int }_{\Omega \times {\mathbb{R}}^3}g{\nabla }_{\mathbf{v}}·\left(h{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}\right)d\mathbf{x}d\mathbf{v}\hfill \\ \hfill & ={\int }_{\Omega \times {\mathbb{R}}^3}g({\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}h)d\mathbf{x}d\mathbf{v},\hfill \end{array}$$

(116)

since ${\nabla }_{\mathbf{v}}·{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}=0$.

Combining (114), (115), and (116), we obtain

$${\langle {\mathcal{T}}_{\omega }g,h\rangle }_{\omega }=-{\int }_{\Omega \times {\mathbb{R}}^3}g{\mathcal{T}}_{\omega }hd\mathbf{x}d\mathbf{v}=-{\langle g,{\mathcal{T}}_{\omega }h\rangle }_{\omega },$$

(117)

which proves the proposition for smooth functions. The result extends to all $g,h\in {\mathcal{H}}_{\omega }$ by density. □

Remark 13.

The skew-adjointness of ${\mathcal{T}}_{\omega }$ in the standard $L^2$ space is a direct consequence of the fact that the vector field

$$\mathbf{V}(\mathbf{x},\mathbf{v})=\left(\mathbf{v},\mathit{\omega }\times \mathbf{v}-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}\right)$$

(118)

is divergence-free in the phase space:

$${\nabla }_{\mathbf{x}}·\mathbf{v}+{\nabla }_{\mathbf{v}}·(\mathit{\omega }\times \mathbf{v}-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}})=0.$$

(119)

This property is independent of the density profile $\rho \left(\mathbf{x}\right)$, unlike the weighted-space formulation. This is a significant simplification and is the correct mathematical setting for the perturbation analysis.

3.3. Linearisation Around the Rotating Maxwellian

We now linearise the BGK equation around the rotating Maxwellian $M_{\omega }$. Write

$$f(\mathbf{x},\mathbf{v},t)=M_{\omega }(\mathbf{x},\mathbf{v})+\sqrt{M_{\omega }(\mathbf{x},\mathbf{v})}g(\mathbf{x},\mathbf{v},t),$$

(120)

with $g\in {\mathcal{H}}_{\omega }=L^2(\Omega \times {\mathbb{R}}^3)$. Substituting into the BGK equation 62 and retaining only linear terms in g, we obtain

$${\partial }_{t}g+{\mathcal{T}}_{\omega }g={\mathcal{L}}_{\omega }g,$$

(121)

where the linearised collision operator is

$${\mathcal{L}}_{\omega }:=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega }),$$

(122)

and ${\Pi }_{\omega }$ is the orthogonal projection in ${\mathcal{H}}_{\omega }$ onto the five-dimensional null space

$$ker{\mathcal{L}}_{\omega }=span\{\sqrt{M_{\omega }},\mathbf{v}\sqrt{M_{\omega }}{,\left|\mathbf{v}\right|}^2\sqrt{M_{\omega }}\}.$$

(123)

The projection ${\Pi }_{\omega }$ has the explicit representation

$${\Pi }_{\omega }g={\displaystyle \sum _{j=0}^4}{\langle g,e_j^{\omega }\rangle }_{\omega }{e}_j^{\omega },$$

(124)

with the orthonormal basis

$$e_0^{\omega }={\displaystyle \frac{\sqrt{M_{\omega }}}{\parallel \sqrt{M_{\omega }}{\parallel }_{\omega }}},e_i^{\omega }={\displaystyle \frac{v_i\sqrt{M_{\omega }}}{\parallel v_i\sqrt{M_{\omega }}{\parallel }_{\omega }}}(i=1,2,3),e_4^{\omega }={\displaystyle \frac{{\left(\right|\mathbf{v}|}^2-3)\sqrt{M_{\omega }}}{\sqrt{6}{\parallel \left(\right|\mathbf{v}|}^2-3{)\sqrt{M_{\omega }}\parallel }_{\omega }}},$$

(125)

where the normalisation constants are given explicitly by

$$\parallel \sqrt{M_{\omega }}{\parallel }_{\omega }^2=\rho \left(\mathbf{x}\right),\parallel v_i\sqrt{M_{\omega }}{\parallel }_{\omega }^2={\rho \left(\mathbf{x}\right)RT,\parallel \left(\right|\mathbf{v}|}^2-3{)\sqrt{M_{\omega }}\parallel }_{\omega }^2=6\rho \left(\mathbf{x}\right){\left(RT\right)}^2.$$

(126)

These constants are obtained from the standard Gaussian moment identities:

$${\int }_{{\mathbb{R}}^3}{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right),{\int }_{{\mathbb{R}}^3}{v}_i{v}_j{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right)RT{\delta }_{ij},$$

(127)

and

$${\int }_{{\mathbb{R}}^3}{\left(\right|\mathbf{v}|}^2{-3)}^2{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=6\rho \left(\mathbf{x}\right){\left(RT\right)}^2.$$

(128)

Remark 14.

Unlike the non-rotating case, the basis functions $e_j^{\omega }$ depend on $\mathbf{x}$ through $\rho \left(\mathbf{x}\right)$. This spatial dependence introduces non-trivial commutators $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ that are absent in the inertial frame. These commutators will play a central role in the Santos-Andrade inequality, as they generate the rotation-dependent terms in the decay rate ${\lambda }_{\mathrm{SA}}$.

Proposition 5

(Spectral properties of ${\mathcal{L}}_{\omega }$). The operator ${\mathcal{L}}_{\omega }$ satisfies the following properties on ${\mathcal{H}}_{\omega }$:

1. 

Self-adjointness: ${\mathcal{L}}_{\omega }={\mathcal{L}}_{\omega }^{\ast }$.

2. 

Non-positivity: For all $g\in {\mathcal{H}}_{\omega }$,

$${\langle {\mathcal{L}}_{\omega }g,g\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}{\parallel (I-{\Pi }_{\omega })g\parallel }_{\omega }^2\le 0.$$

(129)

3. 

Spectral gap: ${\mathcal{L}}_{\omega }$ has spectral gap ${\lambda }_0=1/\tau$:

$${\langle {\mathcal{L}}_{\omega }g,g\rangle }_{\omega }\le -{\displaystyle \frac{1}{\tau }}{\parallel (I-{\Pi }_{\omega })g\parallel }_{\omega }^2.$$

(130)

Proof. 

The projection ${\Pi }_{\omega }$ is orthogonal, hence ${\Pi }_{\omega }={\Pi }_{\omega }^{\ast }$ and ${\Pi }_{\omega }^2={\Pi }_{\omega }$. Therefore, ${\mathcal{L}}_{\omega }^{\ast }=-{\displaystyle \frac{1}{\tau }}{(I-{\Pi }_{\omega })}^{\ast }=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega })={\mathcal{L}}_{\omega }$, proving self-adjointness.

For any $g\in {\mathcal{H}}_{\omega }$, decompose $g=g_{\Vert }+g_{\perp }$ with $g_{\Vert }={\Pi }_{\omega }g$ and $g_{\perp }=(I-{\Pi }_{\omega })g$. Since ${\langle g_{\Vert },g_{\perp }\rangle }_{\omega }=0$, we have

$${\langle {\mathcal{L}}_{\omega }g,g\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}{\langle g_{\perp },g_{\Vert }+g_{\perp }\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}\parallel g_{\perp }{\parallel }_{\omega }^2=-{\displaystyle \frac{1}{\tau }}{\parallel (I-{\Pi }_{\omega })g\parallel }_{\omega }^2\le 0,$$

(131)

establishing non-positivity and the spectral gap ${\lambda }_0=1/\tau$. □

3.4. Chapman–Enskog Expansion in the Rotating Frame

To connect the kinetic description with macroscopic fluid dynamics, we perform a Chapman–Enskog expansion [2,13] in the rotating frame. We introduce the small parameter $\epsilon =\tau /t_0$, where $t_0$ is a characteristic macroscopic time scale (equivalently, $\epsilon$ is the turbulent Knudsen number). We expand the distribution function as

$$f=f^{\left(0\right)}+\epsilon f^{\left(1\right)}+{\epsilon }^2{f}^{\left(2\right)}+\dots ,$$

(132)

with $f^{\left(0\right)}=M\left[f\right]$, the local Maxwellian constructed from the macroscopic fields $\rho ,\mathbf{u},T$. In the rotating frame, the transport operator ${\mathcal{T}}_{\omega }$ acts on both $\mathbf{x}$ and $\mathbf{v}$, and the force terms are already included.

Theorem 4

(Chapman-Enskog limit in the rotating frame). In the hydrodynamic limit $\epsilon \to 0$, the BGK equation (62) in the rotating frame yields the compressible Navier–Stokes equations

$${\partial }_t\rho +{\nabla }_{\mathbf{x}}·\left(\rho \mathbf{u}\right)=0,$$

(133)

$${\partial }_t\left(\rho \mathbf{u}\right)+{\nabla }_{\mathbf{x}}·(\rho \mathbf{u}\otimes \mathbf{u}+p\mathbf{I})=-2\rho \mathit{\omega }\times \mathbf{u}-\rho \mathit{\omega }\times (\mathit{\omega }\times \mathbf{x})+{\nabla }_{\mathbf{x}}·\mathit{\tau },$$

(134)

$${\partial }_{t}E+{\nabla }_{\mathbf{x}}·\left[(E+p)\mathbf{u}\right]=-{\nabla }_{\mathbf{x}}·\mathbf{q}+{\nabla }_{\mathbf{x}}·(\mathit{\tau }·\mathbf{u}),$$

(135)

where $E=\rho e+\frac{1}{2}\rho {\left|\mathbf{u}\right|}^2$ is the total energy density, $e={\displaystyle \frac{3}{2}}RT$ is the internal energy, and the stress tensor and heat flux are given by

$$\tau =-\mu \left({\nabla }_{\mathbf{x}}\mathbf{u}+{\left({\nabla }_{\mathbf{x}}\mathbf{u}\right)}^{\top }-{\displaystyle \frac{2}{3}}({\nabla }_{\mathbf{x}}·\mathbf{u})\mathbf{I}\right),\mathbf{q}=-\kappa {\nabla }_{\mathbf{x}}T,$$

(136)

with

$$\mu =\tau p,\kappa ={\displaystyle \frac{5}{2}}\tau pR.$$

(137)

Proof. 

We follow the standard Chapman-Enskog procedure [2,13]. The BGK equation is written as

$${\partial }_{t}f+{\mathcal{T}}_{\omega }f={\displaystyle \frac{1}{\tau }}(M\left[f\right]-f),$$

(138)

where $M\left[f\right]$ denotes the local Maxwellian constructed from the macroscopic fields $\rho ,\mathbf{u},T$.

We introduce the small parameter $\epsilon =\tau /t_0$, where $t_0$ is a characteristic macroscopic time scale, and expand the distribution function as

$$f=f^{\left(0\right)}+\epsilon f^{\left(1\right)}+{\epsilon }^2{f}^{\left(2\right)}+\dots ,$$

(139)

with $f^{\left(0\right)}=M\left[f\right]$. Substituting (139) into (138) and equating powers of $\epsilon$, we obtain the following hierarchy.

Order ${\epsilon }^{-1}$

At order ${\epsilon }^{-1}$, we have:

$$M\left[f^{\left(0\right)}\right]-f^{\left(0\right)}=0,$$

(140)

hence $f^{\left(0\right)}=M\left[f^{\left(0\right)}\right]$, the local Maxwellian. The solvability condition for the next order requires that $f^{\left(0\right)}$ has the same first five moments as f, which is automatically satisfied by construction of the local Maxwellian.

Order ${\epsilon }^0$

At order ${\epsilon }^0$, the equation is

$${\partial }_t{f}^{\left(0\right)}+{\mathcal{T}}_{\omega }{f}^{\left(0\right)}=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{M\left[f\right]})f^{\left(1\right)},$$

(141)

where ${\Pi }_{M\left[f\right]}$ denotes the orthogonal projection in $L^2\left(M{\left[f\right]}^{-1}d\mathbf{v}\right)$ onto the five-dimensional space spanned by the collision invariants $\{1,\mathbf{v},|\mathbf{v}{|}^2\}$, i.e.,

$${\Pi }_{M\left[f\right]}h={\displaystyle \sum _{j=0}^4}{\langle h,e_j^{M\left[f\right]}\rangle }_{M\left[f\right]}{e}_j^{M\left[f\right]},$$

(142)

with ${\left\{e_j^{M\left[f\right]}\right\}}_{j=0}^4$ an orthonormal basis of the collision invariants with respect to the local Maxwellian $M\left[f\right]$. This projection is the standard one in the Chapman–Enskog theory and should be distinguished from ${\Pi }_{\omega }$, the projection onto the collision invariants of the rotating Maxwellian $M_{\omega }$ introduced in Section 3.

Taking moments of (141) with the collision invariants $\{1,\mathbf{v},|\mathbf{v}{|}^2\}$, the right-hand side vanishes because $(I-{\Pi }_{M\left[f\right]})f^{\left(1\right)}$ is orthogonal to the collision invariants. This yields the Euler equations:

$${\partial }_t\rho +{\nabla }_{\mathbf{x}}·\left(\rho \mathbf{u}\right)=0,$$

(143)

$${\partial }_t\left(\rho \mathbf{u}\right)+{\nabla }_{\mathbf{x}}·(\rho \mathbf{u}\otimes \mathbf{u}+p\mathbf{I})=-2\rho \omega \times \mathbf{u}-\rho \omega \times (\omega \times \mathbf{x}),$$

(144)

$${\partial }_{t}E+{\nabla }_{\mathbf{x}}·\left[(E+p)\mathbf{u}\right]=0.$$

(145)

These are the compressible Euler equations with Coriolis and centrifugal forces.

Order ${\epsilon }^1$

At order ${\epsilon }^1$, the equation is

$${\partial }_t{f}^{\left(1\right)}+{\mathcal{T}}_{\omega }{f}^{\left(1\right)}+{\mathcal{T}}_{\omega }{f}^{\left(0\right)}=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{M\left[f\right]})f^{\left(2\right)}.$$

(146)

The solvability condition for $f^{\left(2\right)}$ requires that the left-hand side of (146) be orthogonal to the collision invariants. This condition yields the viscous corrections to the Euler equations. To compute these corrections, we solve for $f^{\left(1\right)}$ from (141):

$$(I-{\Pi }_{M\left[f\right]})f^{\left(1\right)}=-\tau \left({\partial }_t{f}^{\left(0\right)}+{\mathcal{T}}_{\omega }{f}^{\left(0\right)}\right).$$

(147)

Using the Euler equations (143) – (145) to eliminate time derivatives, the right-hand side of (147) can be expressed in terms of spatial gradients of $\mathbf{u}$ and T. The first-order distribution function is then

$$f^{\left(1\right)}=-{\displaystyle \frac{\tau }{p}}M\left[f^{\left(0\right)}\right]\left({\displaystyle \frac{1}{2}}\left({\mathbf{v}}^{\prime }\otimes {\mathbf{v}}^{\prime }-{\displaystyle \frac{|{\mathbf{v}}^{\prime }{|}^2}{3}}\mathbf{I}\right):{\nabla }_{\mathbf{x}}\mathbf{u}+\left({\displaystyle \frac{|{\mathbf{v}}^{\prime }{|}^2}{2RT}}-{\displaystyle \frac{5}{2}}\right){\mathbf{v}}^{\prime }·{\nabla }_{\mathbf{x}}lnT\right),$$

(148)

where ${\mathbf{v}}^{\prime }=\mathbf{v}-\mathbf{u}$.

Substituting (148) into the solvability condition for (146) (i.e., taking moments with the collision invariants) gives the viscous corrections. The moment of the collision term gives

$${\int }_{{\mathbb{R}}^3}(I-{\Pi }_{M\left[f\right]})f^{\left(2\right)}\psi \left(\mathbf{v}\right)d\mathbf{v}=0,\psi \in {\{1,\mathbf{v},|\mathbf{v}|}^2\},$$

(149)

which yields the Navier–Stokes equations (133)–(135). The stress tensor and heat flux are obtained from the moments of $f^{\left(1\right)}$:

$$\mathit{\tau }={\int }_{{\mathbb{R}}^3}{\mathbf{v}}^{\prime }\otimes {\mathbf{v}}^{\prime }{f}^{\left(1\right)}d{\mathbf{v}}^{\prime },\mathbf{q}={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}{\left|{\mathbf{v}}^{\prime }\right|}^2{\mathbf{v}}^{\prime }{f}^{\left(1\right)}d{\mathbf{v}}^{\prime }.$$

(150)

Evaluating these integrals using the Gaussian moment identities from Appendix B yields

$$\mathit{\tau }=-\mu \left({\nabla }_{\mathbf{x}}\mathbf{u}+{\left({\nabla }_{\mathbf{x}}\mathbf{u}\right)}^{\top }-{\displaystyle \frac{2}{3}}({\nabla }_{\mathbf{x}}·\mathbf{u})\mathbf{I}\right),\mathbf{q}=-\kappa {\nabla }_{\mathbf{x}}T,$$

(151)

with

$$\mu =\tau p,\kappa ={\displaystyle \frac{5}{2}}\tau pR.$$

(152)

This completes the proof. □

Remark 15.

The Chapman-Enskog expansion confirms BGK reproduces the compressible Navier-Stokes equations with rotation in the hydrodynamic limit. The neural correction ${\mathcal{N}}_{\theta }$ appears only at Burnett order, leaving leading-order conservation laws intact; Coriolis and centrifugal terms provide the physical interpretation of the rotation-dependent terms in the Santos-Andrade inequality. The Prandtl number is $\mathrm{Pr}=1$ for BGK ($2/3$ for full Boltzmann). We distinguish two projections: ${\Pi }_{M\left[f\right]}$ (Chapman-Enskog, hydrodynamic limit) and ${\Pi }_{\omega }$ (hypocoercive analysis, stability around rotating equilibrium, Section 3).

3.5. The Commutator $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ and Its Role

A crucial ingredient for the hypocoercive analysis in Section 4 is the commutator between the spatial gradient and the projection ${\Pi }_{\omega }$. Because ${\Pi }_{\omega }$ depends on $\mathbf{x}$ through $\rho \left(\mathbf{x}\right)$ and the normalisation constants, this commutator does not vanish.

Lemma 1

(Commutator estimate). There exists a constant $C_{\Pi }>0$, depending only on the domain Ω, such that for all $g\in H^1\left({\mathcal{H}}_{\omega }\right)$,

$$\parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }\le C_{\Pi }\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel g\parallel }_{\omega },$$

(153)

where $c_s=\sqrt{\gamma RT}$ is the speed of sound and $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]g:={\nabla }_{\mathbf{x}}\left({\Pi }_{\omega }g\right)-{\Pi }_{\omega }\left({\nabla }_{\mathbf{x}}g\right)$.

Proof. 

Since ${\Pi }_{\omega }$ is a finite-rank projection, we have:

$$[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]g={\displaystyle \sum _{j=0}^4}\left({\langle g,{\nabla }_{\mathbf{x}}{e}_j^{\omega }\rangle }_{\omega }{e}_j^{\omega }+{\langle g,e_j^{\omega }\rangle }_{\omega }{\nabla }_{\mathbf{x}}{e}_j^{\omega }\right),$$

(154)

where ${\left\{e_j^{\omega }\right\}}_{j=0}^4$ is the orthonormal basis defined in (125). From the definition of $M_{\omega }$, we have:

$${\nabla }_{\mathbf{x}}{e}_j^{\omega }=e_j^{\omega }\left({\displaystyle \frac{1}{2}}{\nabla }_{\mathbf{x}}ln\rho \left(\mathbf{x}\right)-{\displaystyle \frac{\partial {\psi }_j}{\partial \mathbf{v}}}·{\nabla }_{\mathbf{x}}\mathbf{u}\left(\mathbf{x}\right)\right)+\mathcal{O}\left(\right|{\nabla }_{\mathbf{x}}\mathbf{u}{|}^2),$$

(155)

where ${\psi }_j$ are the polynomials in $\mathbf{v}$ such that $e_j^{\omega }={\psi }_j\left(\mathbf{v}\right)\sqrt{M_{\omega }}$ after normalisation.

Using the hydrostatic balance (367), we obtain:

$${\displaystyle \frac{1}{2}}|{\nabla }_{\mathbf{x}}ln\rho \left(\mathbf{x}\right)|\le C_{\Omega }{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}},$$

(156)

with $C_{\Omega }=diam(\Omega )/\left(2\gamma \right)$. Furthermore, in the rotating frame, the mean velocity satisfies $\parallel {\nabla }_{\mathbf{x}}{\mathbf{u}\parallel }_{L^{\infty }(\Omega )}\le C\left|\mathit{\omega }\right|$ for some $C>0$ depending only on the geometry. Therefore,

$$\parallel {\nabla }_{\mathbf{x}}{e}_j^{\omega }{\parallel }_{\omega }\le C\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel e_j^{\omega }\parallel }_{\omega }.$$

(157)

Applying the Cauchy–Schwarz inequality to (154) and using $\parallel e_j^{\omega }{\parallel }_{\omega }=1$ gives

$$\begin{array}{cc}\hfill \parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }& \le {\displaystyle \sum _{j=0}^4}\left(\parallel {\nabla }_{\mathbf{x}}{e}_j^{\omega }{\parallel }_{\omega }{\parallel g\parallel }_{\omega }+\parallel {\nabla }_{\mathbf{x}}{e}_j^{\omega }{\parallel }_{\omega }{\parallel g\parallel }_{\omega }\right)\hfill \\ \hfill & \le {2\parallel g\parallel }_{\omega }{\displaystyle \sum _{j=0}^4}{\parallel {\nabla }_{\mathbf{x}}{e}_j^{\omega }\parallel }_{\omega }.\hfill \end{array}$$

(158)

Substituting (157) and defining $C_{\Pi }=2C{\sum }_{j=0}^4{\parallel e_j^{\omega }\parallel }_{\omega }$ yields (153). □

Remark 16.

The linear term $\left|\mathit{\omega }\right|/c_s$ arises from the dependence of the basis functions on the mean velocity $\mathbf{u}$ (through ${\nabla }_{\mathbf{x}}\mathbf{u}\sim \left|\mathit{\omega }\right|$), while the quadratic term ${\left|\mathit{\omega }\right|}^2/c_s^2$ arises from the density gradient ${\nabla }_{\mathbf{x}}\rho$. This commutator is the mathematical origin of the rotation-dependent term in the Santos–Andrade decay rate (Theorem 5). In the non-rotating limit $\mathit{\omega }\to 0$, the commutator vanishes and Villani’s classical hypocoercive result is recovered.

3.6. Summary of Rotating-Frame Structures

For convenience, we collect here the key objects introduced in this section, which will be used in the hypocoercive analysis of Section 4.

• Rotating Maxwellian:

  $$M_{\omega }(\mathbf{x},\mathbf{v})={\displaystyle \frac{\rho \left(\mathbf{x}\right)}{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2RT}}\right),\rho \left(\mathbf{x}\right)={\rho }_{0}exp\left({\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2RT}}\right).$$

  (159)
• Transport operator:

  $${\mathcal{T}}_{\omega }=\mathbf{v}·{\nabla }_{\mathbf{x}}+(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}+{\nabla }_{\mathbf{x}}\left({\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2}}\right)·{\nabla }_{\mathbf{v}}.$$

  (160)
• Hilbert space:

  $${\mathcal{H}}_{\omega }=L^2(\Omega \times {\mathbb{R}}^3),{\langle g,h\rangle }_{\omega }={\int }_{\Omega \times {\mathbb{R}}^3}ghd\mathbf{x}d\mathbf{v}.$$

  (161)
• Linearised collision operator:

  $${\mathcal{L}}_{\omega }=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega }),ker{\mathcal{L}}_{\omega }=span\{\sqrt{M_{\omega }},\mathbf{v}\sqrt{M_{\omega }}{,\left|\mathbf{v}\right|}^2\sqrt{M_{\omega }}\}.$$

  (162)
• Commutator estimate:

  $$\parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }\le C_{\Pi }\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel g\parallel }_{\omega }.$$

  (163)

Remark 17.

The choice of the standard $L^2$ space is the key mathematical correction in this section. It ensures that:

1. 

The orthonormal basis of $ker{\mathcal{L}}_{\omega }$ is well-defined and normalisable.

2. 

The transport operator ${\mathcal{T}}_{\omega }$ is skew-adjoint without requiring any special density profile.

3. 

The hypocoercive framework of Villani [31] applies directly, with the commutator estimate providing the rotation-dependent terms.

This formulation is mathematically consistent and aligns with the standard theory of hypocoercivity for kinetic equations with non-constant equilibrium states.

This section has established the kinetic and macroscopic foundations for rotating compressible flows. The next section will use these tools to derive the Santos-Andrade inequality, which provides a rigorous stability criterion for the LBM–PINN framework in the presence of rotation, compressibility, and neural corrections.

4. The Santos–Andrade Hypocoercive Inequality

We now present the central theoretical result of this work: a hypocoercive inequality for the linearised kinetic equation in a rotating reference frame, augmented with a structure-preserving neural correction. This inequality provides an explicit decay rate that quantifies the competing effects of turbulent relaxation, rotation, and the neural network’s Lipschitz constant.

Remark 18

(Overview of the proof strategy). The proof of the Santos-Andrade inequality follows the augmented energy methodology pioneered by Villani [31], but with four crucial adaptations to the rotating frame:

1. 

The equilibrium Maxwellian $M_{\omega }$ acquires spatial dependence through the density profile $\rho \left(\mathbf{x}\right)={\rho }_0{exp\left(\right|\mathit{\omega }\times \mathbf{x}|}^2/\left(2RT\right))$, introducing non-trivial commutators $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ that yield the rotation-dependent terms.

2. 

The neural correction ${\mathcal{N}}_{\theta }^{\ast }$ is controlled via its Lipschitz constant $L_{\theta }$, leading to the stability threshold $L_{\theta }<(1/\left(4\tau \right)-C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|\left)\right)/C_2$.

3. 

The cross term ${\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle }_{\omega }$ couples the hydrodynamic and non-hydrodynamic modes, enabling the dissipation of $g_{\perp }$ to control $g_{\Vert }$ through a refined Poincaré-type inequality.

4. 

The rotation operator $\mathcal{R}=\mathit{\omega }\times \mathbf{v}·{\nabla }_{\mathbf{v}}$ is recognised as generating a Lie algebra, allowing a sharper spectral estimate that interpolates between the linear and quadratic regimes.

The resulting decay rate ${\lambda }_{\mathrm{SA}}$ explicitly quantifies the competition between turbulent relaxation (stabilising), rotation (destabilising), and neural corrections (potentially destabilising).

4.1. Functional Setting for Rotating Frames

Let $\Omega ={\mathbb{T}}^3$ be the three-dimensional torus. We consider a gas in a state of solid-body rotation with constant angular velocity $\mathit{\omega }\in {\mathbb{R}}^3$. Following the corrected formulation established in Section 3, the equilibrium distribution in the rotating frame is the rotating Maxwellian

$$M_{\omega }(\mathbf{x},\mathbf{v})={\displaystyle \frac{\rho \left(\mathbf{x}\right)}{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2RT}}\right),\rho \left(\mathbf{x}\right)={\rho }_{0}exp\left({\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2RT}}\right),$$

(164)

where $R=k_B/m$ is the specific gas constant, T is the temperature, and the density profile $\rho \left(\mathbf{x}\right)$ reflects the balance between the centrifugal force and the pressure gradient. The speed of sound is defined by $c_s:=\sqrt{\gamma RT}$, where $\gamma =5/3$ is the adiabatic index for a monatomic gas.

The density profile $\rho \left(\mathbf{x}\right)$ satisfies the equilibrium condition

$${\nabla }_{\mathbf{x}}p=\rho \mathit{\omega }\times (\mathit{\omega }\times \mathbf{x}),$$

(165)

which is the hydrostatic balance between the pressure gradient and the centrifugal force. Substituting the ideal gas law $p=\rho RT$ into 165 yields ${\nabla }_{\mathbf{x}}ln\rho ={|\mathit{\omega }\times \mathbf{x}|}^2/\left(2RT\right)$, which integrates to the expression in 164.

Remark 19

(Physical interpretation of the rotating Maxwellian). The rotating Maxwellian $M_{\omega }$ represents the thermodynamic equilibrium of a gas in solid-body rotation. The density profile $\rho \left(\mathbf{x}\right)={\rho }_0{exp\left(\right|\mathit{\omega }\times \mathbf{x}|}^2/\left(2RT\right))$ reflects the fact that the centrifugal force pushes denser gas toward the outer regions of the rotating system. This spatially inhomogeneous equilibrium is the key distinction from the non-rotating case and is the source of the rotation-dependent terms in the hypocoercive inequality.

We write the distribution function as a perturbation around the rotating Maxwellian:

$$f(\mathbf{x},\mathbf{v},t)=M_{\omega }(\mathbf{x},\mathbf{v})+\sqrt{M_{\omega }(\mathbf{x},\mathbf{v})}g(\mathbf{x},\mathbf{v},t),$$

(166)

where g denotes the perturbation. The natural Hilbert space for the perturbation analysis is the standard Lebesgue space

$${\mathcal{H}}_{\omega }:=L^2\left(\Omega \times {\mathbb{R}}^3\right),$$

(167)

equipped with the standard inner product

$${\langle g,h\rangle }_{\omega }:={\int }_{\Omega \times {\mathbb{R}}^3}g(\mathbf{x},\mathbf{v})\overline{h(\mathbf{x},\mathbf{v})}d\mathbf{x}d\mathbf{v}.$$

(168)

The associated norm is ${\parallel g\parallel }_{\omega }:={\langle g,g\rangle }_{\omega }^{1/2}$.

Remark 20.

The choice of the standard Lebesgue space $L^2(\Omega \times {\mathbb{R}}^3)$ rather than a weighted space is deliberate and essential. It allows the orthonormal basis of the null space of the linearised collision operator, which consists of functions of the form $\psi \left(\mathbf{v}\right)\sqrt{M_{\omega }(\mathbf{x},\mathbf{v})}$, to be properly normalised, since $\sqrt{M_{\omega }}\in L^2\left({\mathbb{R}}^3\right)$ for each fixed $\mathbf{x}$. This is the natural setting for the perturbation analysis, as it ensures that the perturbed distribution $f=M_{\omega }+\sqrt{M_{\omega }}g$ has finite energy. This formulation is consistent with the corrected framework of Section 3.

In the rotating frame, the transport operator takes the form

$${\mathcal{T}}_{\omega }:=\mathbf{v}·{\nabla }_{\mathbf{x}}+(\omega \times \mathbf{v})·{\nabla }_{\mathbf{v}}-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}},$$

(169)

where,

$${\varphi }_{\mathrm{cf}}\left(\mathbf{x}\right):=-{\displaystyle \frac{1}{2}}{|\mathit{\omega }\times \mathbf{x}|}^2$$

(170)

is the centrifugal potential. The operator ${\mathcal{T}}_{\omega }$ is skew-adjoint on ${\mathcal{H}}_{\omega }$ with respect to the standard inner product 168:

$${\langle {\mathcal{T}}_{\omega }g,h\rangle }_{\omega }=-{\langle g,{\mathcal{T}}_{\omega }h\rangle }_{\omega }\forall g,h\in {\mathcal{H}}_{\omega },$$

(171)

which follows from integration by parts on the torus and the fact that the velocity-space divergence satisfies ${\nabla }_{\mathbf{v}}·(\mathit{\omega }\times \mathbf{v})=0$ and ${\nabla }_{\mathbf{v}}·\left({\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}\right)=0$. A direct verification of 171 is provided in Appendix D.

Remark 21

(Interpretation of the transport operator). The transport operator ${\mathcal{T}}_{\omega }$ consists of three terms:

• $\mathbf{v}·{\nabla }_{\mathbf{x}}$: the standard streaming term.
• $(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}$: the Coriolis force, which deflects particles in the rotating frame.
• $-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}$: the centrifugal force, which pushes particles outward.

The skew-adjointness of ${\mathcal{T}}_{\omega }$ ensures that the transport operator does not contribute to the energy growth of the system, a property that is essential for the hypocoercive analysis.

The linearised collision operator in the rotating frame is the BGK operator

$${\mathcal{L}}_{\omega }:=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega }),$$

(172)

where $\tau >0$ is the turbulent relaxation time and ${\Pi }_{\omega }$ is the orthogonal projection in ${\mathcal{H}}_{\omega }$ onto the five-dimensional null space

$$ker{\mathcal{L}}_{\omega }=span\{\sqrt{M_{\omega }},\mathbf{v}\sqrt{M_{\omega }}{,\left|\mathbf{v}\right|}^2\sqrt{M_{\omega }}\}.$$

(173)

The projection ${\Pi }_{\omega }$ has the explicit representation

$${\Pi }_{\omega }g={\displaystyle \sum _{j=0}^4}{\langle g,e_j^{\omega }\rangle }_{\omega }{e}_j^{\omega },$$

(174)

where ${\left\{e_j^{\omega }\right\}}_{j=0}^4$ is an orthonormal basis of $ker{\mathcal{L}}_{\omega }$:

$$e_0^{\omega }={\displaystyle \frac{\sqrt{M_{\omega }}}{\parallel \sqrt{M_{\omega }}{\parallel }_{\omega }}},e_i^{\omega }={\displaystyle \frac{v_i\sqrt{M_{\omega }}}{\parallel v_i\sqrt{M_{\omega }}{\parallel }_{\omega }}}(i=1,2,3),e_4^{\omega }={\displaystyle \frac{{\left(\right|\mathbf{v}|}^2-3)\sqrt{M_{\omega }}}{\sqrt{6}{\parallel \left(\right|\mathbf{v}|}^2-3{)\sqrt{M_{\omega }}\parallel }_{\omega }}},$$

(175)

with the normalisation constants given explicitly by

$$\parallel \sqrt{M_{\omega }}{\parallel }_{\omega }^2=\rho \left(\mathbf{x}\right),\parallel v_i\sqrt{M_{\omega }}{\parallel }_{\omega }^2={\rho \left(\mathbf{x}\right)RT,\parallel \left(\right|\mathbf{v}|}^2-3{)\sqrt{M_{\omega }}\parallel }_{\omega }^2=6\rho \left(\mathbf{x}\right){\left(RT\right)}^2.$$

(176)

These constants are obtained from the standard Gaussian moment identities:

$${\int }_{{\mathbb{R}}^3}{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right),{\int }_{{\mathbb{R}}^3}{v}_i{v}_j{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right)RT{\delta }_{ij},$$

(177)

and

$${\int }_{{\mathbb{R}}^3}{\left(\right|\mathbf{v}|}^2{-3)}^2{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=6\rho \left(\mathbf{x}\right){\left(RT\right)}^2.$$

(178)

Remark 22.

The key difference between the rotating and non-rotating settings is the spatial dependence of the Maxwellian $M_{\omega }$ through the density $\rho \left(\mathbf{x}\right)$. This spatial dependence introduces non-trivial commutators between the spatial gradient ${\nabla }_{\mathbf{x}}$ and the projection operator ${\Pi }_{\omega }$, which are the source of the rotation-dependent terms in the Santos-Andrade inequality. In the non-rotating case, ρ is constant, Π is independent of $\mathbf{x}$, and the commutator $[{\nabla }_{\mathbf{x}},\Pi ]=0$ vanishes identically.

The following lemma establishes a crucial bound on the commutator between the spatial gradient and the projection operator. This bound is the technical heart of the rotating-frame analysis and quantifies the additional dissipation required to counteract the destabilising effects of rotation. The proof is provided in Appendix C.

Lemma 2

(Commutator estimate for rotating frame). There exists a constant $C_{\Pi }>0$, depending only on the spatial domain Ω, such that for all $g\in H^1\left({\mathcal{H}}_{\omega }\right)$,

$$\parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }\le C_{\Pi }\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel g\parallel }_{\omega },$$

(179)

where $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]g:={\nabla }_{\mathbf{x}}\left({\Pi }_{\omega }g\right)-{\Pi }_{\omega }\left({\nabla }_{\mathbf{x}}g\right)$.

Remark 23.

The commutator estimate 179 is the key technical result that distinguishes the rotating frame from the non-rotating case. The linear term $\left|\mathit{\omega }\right|/c_s$ arises from the dependence of the basis functions on the mean velocity $\mathbf{u}$ (through ${\nabla }_{\mathbf{x}}\mathbf{u}\sim \left|\mathit{\omega }\right|$), while the quadratic term ${\left|\mathit{\omega }\right|}^2/c_s^2$ arises from the density gradient ${\nabla }_{\mathbf{x}}\rho$ and the centrifugal potential. Both terms are controlled by the speed of sound $c_s$, which provides the natural velocity scale for compressible flows. In the non-rotating limit $\mathit{\omega }\to 0$, the commutator vanishes and we recover the standard hypocoercive framework of Villani [31]. A complete derivation is given in Appendix C.

4.2. Statement of the Santos-Andrade Inequality

We now state the main theorem of this section.

Theorem 5

(Santos-Andrade hypocoercive inequality). Let $g\in H^1\left({\mathcal{H}}_{\omega }\right)$ be a solution of the linearised kinetic equation in a rotating reference frame with structure-preserving neural correction:

$${\partial }_{t}g+{\mathcal{T}}_{\omega }g={\mathcal{L}}_{\omega }g+{\mathcal{N}}_{\theta }^{\ast }\left(g\right),$$

(180)

where ${\mathcal{T}}_{\omega }$ is defined by 169, ${\mathcal{L}}_{\omega }$ by 172, and the neural correction ${\mathcal{N}}_{\theta }^{\ast }$ satisfies Assumptions A1, A2, and A3 with Lipschitz constant $L_{\theta }>0$. Define the augmented energy functional

$${\mathcal{E}}_{\omega }\left(g\right):={\parallel g\parallel }_{\omega }^2+\alpha {\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle }_{\omega }+\beta {\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,$$

(181)

with parameters $0<\alpha ,\beta \ll 1$ chosen as

$$\alpha ={\displaystyle \frac{\tau }{4}},\beta ={\displaystyle \frac{{\tau }^2}{4}}.$$

(182)

Then there exist constants $C_1,C_2>0$, with explicit expressions

$$C_1={\displaystyle \frac{\tau }{16}}{\displaystyle \sum _{\left|\alpha \right|=1}^2}{\displaystyle \frac{\parallel P_{\alpha }{\Pi }_{\omega }\mathbf{v}·{\nabla }_{\mathbf{x}}{H}_{\alpha }{\parallel }_{\omega }}{\parallel H_{\alpha }{\parallel }_{\omega }}},C_2={\displaystyle \frac{1}{2}}\underset{{\parallel g\parallel }_{\omega }=1}{sup}\underset{{\parallel h\parallel }_{\omega }=1}{sup}{\displaystyle \frac{\parallel {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }{\left(g\right)·\mathbf{v}h\parallel }_{\omega }}{\parallel {\nabla }_{\mathbf{x}}{g\parallel }_{\omega }{\parallel h\parallel }_{\omega }}},$$

(183)

where ${\left\{H_{\alpha }\right\}}_{\alpha \in {\mathbb{N}}^3}$ is the Hermite polynomial basis, and $P_{\alpha }$ is the projection onto the α-th Hermite mode. These constants depend only on the projection ${\Pi }_{\omega }$, the domain Ω, and the spectral properties of the collision operator.

Then

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -2{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right),$$

(184)

where the explicit decay rate is given by

$${\lambda }_{\mathrm{SA}}:={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta }.$$

(185)

Consequently, if ${\lambda }_{\mathrm{SA}}>0$, the perturbation decays exponentially:

$${\parallel g\left(t\right)\parallel }_{\omega }\le C_0{e}^{-{\lambda }_{\mathrm{SA}}t}{\parallel g\left(0\right)\parallel }_{H_{\omega }^1},$$

(186)

for some constant $C_0>0$ independent of t.

Proof. 

The proof proceeds by estimating the time derivative of each term in the augmented energy ${\mathcal{E}}_{\omega }\left(g\right)$. We begin by recalling the decomposition $g=g_{\Vert }+g_{\perp }$, where $g_{\Vert }:={\Pi }_{\omega }g$ and $g_{\perp }:=(I-{\Pi }_{\omega })g$.

Time Derivative of the $L^2$ Norm

Using 180 and the skew-adjointness of ${\mathcal{T}}_{\omega }$ 171, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel g\parallel }_{\omega }^2& ={\langle {\mathcal{L}}_{\omega }g,g\rangle }_{\omega }+{\langle {\mathcal{N}}_{\theta }^{\ast }\left(g\right),g\rangle }_{\omega }\hfill \\ \hfill & =-{\displaystyle \frac{1}{\tau }}{\parallel g_{\perp }\parallel }_{\omega }^2+{\langle {\mathcal{N}}_{\theta }^{\ast }\left(g\right),g_{\perp }\rangle }_{\omega },\hfill \end{array}$$

(187)

where we used ${\mathcal{L}}_{\omega }=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega })$ and the fact that the neural correction has zero projection onto $ker{\mathcal{L}}_{\omega }$ (Assumption A1), so ${\langle {\mathcal{N}}_{\theta }^{\ast }\left(g\right),g_{\Vert }\rangle }_{\omega }=0$.

By the Lipschitz estimate (Assumption A2) and the Cauchy-Schwarz inequality,

$${\langle {\mathcal{N}}_{\theta }^{\ast }\left(g\right),g_{\perp }\rangle }_{\omega }\le \parallel {\mathcal{N}}_{\theta }^{\ast }{\left(g\right)\parallel }_{\omega }\parallel g_{\perp }{\parallel }_{\omega }\le L_{\theta }{\parallel g\parallel }_{\omega }{\parallel g_{\perp }\parallel }_{\omega }.$$

(188)

Applying Young’s inequality $ab\le {\displaystyle \frac{\epsilon }{2}}{a}^2+{\displaystyle \frac{1}{2\epsilon }}{b}^2$ with $\epsilon =1/\left(4L_{\theta }\right)$ when $L_{\theta }>0$, and using a uniform estimate that remains valid for $L_{\theta }=0$, we obtain:

$${\langle {\mathcal{N}}_{\theta }^{\ast }\left(g\right),g_{\perp }\rangle }_{\omega }\le {\displaystyle \frac{1}{8\tau }}\parallel g_{\perp }{\parallel }_{\omega }^2+2\tau L_{\theta }^2{\parallel g\parallel }_{\omega }^2,$$

(189)

which is valid for all $L_{\theta }\ge 0$.

Substituting into 187, we get

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel g\parallel }_{\omega }^2\le -{\displaystyle \frac{7}{8\tau }}\parallel g_{\perp }{\parallel }_{\omega }^2+2\tau L_{\theta }^2{\parallel g\parallel }_{\omega }^2.$$

(190)

Time Derivative of the Cross Term

Define $C_{\omega }\left(g\right):={\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle }_{\omega }$. Differentiating with respect to time and using 180, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{C}_{\omega }\left(g\right)& ={\langle {\nabla }_{\mathbf{x}}{\partial }_{t}g,\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\partial }_{t}g\rangle }_{\omega }\hfill \\ \hfill & =-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\omega }g,\mathbf{v}g\rangle }_{\omega }-{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{T}}_{\omega }g\rangle }_{\omega }\hfill \\ \hfill & +{\langle {\nabla }_{\mathbf{x}}{\mathcal{L}}_{\omega }g,\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{L}}_{\omega }g\rangle }_{\omega }\hfill \\ \hfill & +{\langle {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }\left(g\right),\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{N}}_{\theta }^{\ast }\left(g\right)\rangle }_{\omega }.\hfill \end{array}$$

(191)

We estimate each group of terms separately.

Transport Term

The transport operator decomposes as ${\mathcal{T}}_{\omega }={\mathcal{T}}_0+{\mathcal{T}}_{\mathrm{Cor}}+{\mathcal{T}}_{\mathrm{cf}}$, where

$${\mathcal{T}}_0=\mathbf{v}·{\nabla }_{\mathbf{x}},{\mathcal{T}}_{\mathrm{Cor}}=(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}},{\mathcal{T}}_{\mathrm{cf}}=-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}.$$

(192)

For ${\mathcal{T}}_0$, integration by parts gives

$$-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{0}g,\mathbf{v}g\rangle }_{\omega }-{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{T}}_{0}g\rangle }_{\omega }\le C_{\mathrm{str}}{\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2,$$

(193)

with $C_{\mathrm{str}}>0$ independent of $\mathit{\omega }$. This estimate follows from the skew-adjointness of ${\mathcal{T}}_0$ and the fact that the commutator $[{\nabla }_{\mathbf{x}},{\mathcal{T}}_0]=0$ vanishes identically.

For the Coriolis term ${\mathcal{T}}_{\mathrm{Cor}}=(\mathit{\omega }\times \mathbf{v})·{\nabla }_{\mathbf{v}}$, we exploit the Lie algebra structure of the rotation operator. The operator $\mathcal{R}=\mathit{\omega }\times \mathbf{v}·{\nabla }_{\mathbf{v}}$ acts diagonally on the Hermite polynomial basis ${\left\{H_{\alpha }\right\}}_{\alpha \in {\mathbb{N}}^3}$:

$$\mathcal{R}{H}_{\alpha }=i\sum _{\beta :\left|\beta \right|\le \left|\alpha \right|}{c}_{\alpha \beta }{H}_{\beta },$$

(194)

with $c_{\alpha \beta }=\mathcal{O}\left(\right|\mathit{\omega }\left|\right)$ and the spectral norm ${\parallel \mathcal{R}\parallel }_{\mathcal{L}\left({\mathcal{H}}_{\omega }\right)}=\left|\mathit{\omega }\right|$. The commutator with the projection gives

$$[{\nabla }_{\mathbf{x}},{\mathcal{T}}_{\mathrm{Cor}}]=\mathit{\omega }\times {\nabla }_{\mathbf{v}},$$

(195)

which is a first-order differential operator in $\mathbf{v}$. After integration by parts in both $\mathbf{x}$ and $\mathbf{v}$, and using the Hermite spectral decomposition, one obtains the sharp estimate

$$\left|-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\mathrm{Cor}}g,\mathbf{v}g\rangle }_{\omega }-{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{T}}_{\mathrm{Cor}}g\rangle }_{\omega }\right|\le C_{\mathrm{Cor}}{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,$$

(196)

where $C_{\mathrm{Cor}}>0$ depends only on the projection ${\Pi }_{\omega }$ and the domain. The interpolating factor ${\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}$ arises from the combination of the spectral norm of $\mathcal{R}$ and the commutator contribution.

For the centrifugal term ${\mathcal{T}}_{\mathrm{cf}}=-{\nabla }_{\mathbf{x}}{\varphi }_{\mathrm{cf}}·{\nabla }_{\mathbf{v}}$, the commutator is

$$[{\nabla }_{\mathbf{x}},{\mathcal{T}}_{\mathrm{cf}}]=-\left({\nabla }_{\mathbf{x}}^2{\varphi }_{\mathrm{cf}}\right)·{\nabla }_{\mathbf{v}},$$

(197)

which involves second derivatives of ${\varphi }_{\mathrm{cf}}$ of order ${\left|\mathit{\omega }\right|}^2$. The estimate is

$$\left|-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\mathrm{cf}}g,\mathbf{v}g\rangle }_{\omega }-{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{T}}_{\mathrm{cf}}g\rangle }_{\omega }\right|\le C_{\mathrm{cf}}{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,$$

(198)

with $C_{\mathrm{cf}}>0$ depending only on the domain.

Combining 193, 196, and 198, and using ${\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\le {\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}·{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}$ for $\left|\mathit{\omega }\right|\le c_s$, we obtain the unified estimate

$$-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\omega }g,\mathbf{v}g\rangle }_{\omega }-{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{T}}_{\omega }g\rangle }_{\omega }\le C_{\mathrm{str}}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,$$

(199)

where ${\tilde{C}}_1:=C_{\mathrm{Cor}}+C_{\mathrm{cf}}$ absorbs the constants.

Collision Term

For the BGK operator ${\mathcal{L}}_{\omega }=-{\displaystyle \frac{1}{\tau }}(I-{\Pi }_{\omega })$, we compute

$$\begin{array}{cc}\hfill & {\langle {\nabla }_{\mathbf{x}}{\mathcal{L}}_{\omega }g,\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{L}}_{\omega }g\rangle }_{\omega }\hfill \\ \hfill & =-{\displaystyle \frac{1}{\tau }}\left({\langle {\nabla }_{\mathbf{x}}{g}_{\perp },\mathbf{v}{g}_{\perp }\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}{g}_{\perp },\mathbf{v}{g}_{\Vert }\rangle }_{\omega }\right.\hfill \\ \hfill & \left.+{\langle {\nabla }_{\mathbf{x}}{g}_{\Vert },\mathbf{v}{g}_{\perp }\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{L}}_{\omega }g\rangle }_{\omega }\right),\hfill \end{array}$$

(200)

where $g_{\Vert }:={\Pi }_{\omega }g$ and $g_{\perp }:=(I-{\Pi }_{\omega })g$. We now estimate each term.

The first term satisfies

$${\langle {\nabla }_{\mathbf{x}}{g}_{\perp },\mathbf{v}{g}_{\perp }\rangle }_{\omega }\le {\displaystyle \frac{1}{2}}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\displaystyle \frac{1}{2}}{\parallel \mathbf{v}{g}_{\perp }\parallel }_{\omega }^2,$$

(201)

where $\parallel \mathbf{v}{g}_{\perp }{\parallel }_{\omega }\le C_v{\parallel g_{\perp }\parallel }_{\omega }$ with $C_v:={max}_{\mathbf{x}\in \Omega }{\parallel \mathbf{v}\parallel }_{L^{\infty }\left({\mathbb{R}}^3\right)}$.

The cross terms are bounded using the commutator estimate Lemma 2:

$$\begin{array}{cc}\hfill {\langle {\nabla }_{\mathbf{x}}{g}_{\perp },\mathbf{v}{g}_{\Vert }\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}{g}_{\Vert },\mathbf{v}{g}_{\perp }\rangle }_{\omega }& \le \parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }\parallel \mathbf{v}{g}_{\Vert }{\parallel }_{\omega }+\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }{\parallel \mathbf{v}{g}_{\perp }\parallel }_{\omega }\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2+C\left(\parallel g_{\perp }{\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }\parallel }_{\omega }^2\right),\hfill \end{array}$$

(202)

where the constant C absorbs $C_v$ and the commutator constants.

We now use the refined Poincaré-type inequality for the hydrodynamic part. The correct inequality is:

$$\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2\le C_P\left(\parallel g_{\perp }{\parallel }_{\omega }^2+\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\parallel g_{\Vert }\parallel }_{\omega }^2\right),$$

(203)

where $C_P>0$ is the Poincaré constant. This follows from the fact that the only functions in $ker{\Pi }_{\omega }$ that are spatially constant are the global invariants. The term $\parallel g_{\Vert }{\parallel }_{\omega }^2$ on the right-hand side accounts for the possibility that ${\nabla }_{\mathbf{x}}{g}_{\Vert }$ is small while $g_{\Vert }$ remains large (e.g., spatially constant functions). The full elliptic regularity estimate for the mixed problem ${\Pi }_{\omega }g=g_{\Vert }$ provides the stated control. □

Remark 24

(Origin of the refined Poincaré-type inequality). The refined Poincaré-type inequality 203 is a key ingredient in the hypocoercive framework. It states that the gradient of the hydrodynamic part $g_{\Vert }$ is controlled by the non-hydrodynamic part $g_{\perp }$, its gradient, and the $L^2$ norm of $g_{\Vert }$ itself. The term $\parallel g_{\Vert }{\parallel }_{\omega }^2$ is necessary because the kernel of ${\Pi }_{\omega }$ contains spatially constant functions (the global invariants), for which ${\nabla }_{\mathbf{x}}{g}_{\Vert }=0$ but $\parallel g_{\Vert }{\parallel }_{\omega }^2>0$. In the proof of the hypocoercive estimate, this term will be controlled by the $L^2$ norm of g through the norm equivalence of the augmented energy. In the rotating frame, the spatial dependence of ${\Pi }_{\omega }$ introduces additional commutator terms that are controlled by Lemma 2.

Using 203, we obtain

$${\langle {\nabla }_{\mathbf{x}}{\mathcal{L}}_{\omega }g,\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{L}}_{\omega }g\rangle }_{\omega }\le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+C_3\parallel g_{\perp }{\parallel }_{\omega }^2+C_4\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+C_4^{\prime }{\parallel g_{\Vert }\parallel }_{\omega }^2,$$

(204)

where $C_3,C_4,C_4^{\prime }>0$ depend on the projection ${\Pi }_{\omega }$ and the constant $C_P$.

Neural Term

For the neural correction, using Assumptions A2 and A3, we have the refined estimate

$$\begin{array}{cc}\hfill & {\langle {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }\left(g\right),\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{N}}_{\theta }^{\ast }\left(g\right)\rangle }_{\omega }\hfill \\ \hfill & \le C_v{L}_{\theta }\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right)+C_{\mathrm{comm}}{L}_{\theta }{\parallel g\parallel }_{\omega }{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega },\hfill \end{array}$$

(205)

where $C_v:={max}_{\mathbf{x}\in \Omega }{\parallel \mathbf{v}\parallel }_{L^{\infty }\left({\mathbb{R}}^3\right)}$ and $C_{\mathrm{comm}}$ accounts for the commutator $[{\nabla }_{\mathbf{x}},{\mathcal{N}}_{\theta }^{\ast }]$, which is non-zero in general. Defining $C_5:=C_v+C_{\mathrm{comm}}/2$ and using Young’s inequality, we have

$${\langle {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }\left(g\right),\mathbf{v}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}{\mathcal{N}}_{\theta }^{\ast }\left(g\right)\rangle }_{\omega }\le C_5{L}_{\theta }\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right).$$

(206)

Combining 199, 204, and 206, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{C}_{\omega }\left(g\right)& \le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+C_3\parallel g_{\perp }{\parallel }_{\omega }^2+C_4\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+C_4^{\prime }{\parallel g_{\Vert }\parallel }_{\omega }^2\hfill \\ \hfill & +{\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2+C_5{L}_{\theta }\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right),\hfill \end{array}$$

(207)

Time Derivative of the Gradient Norm

Differentiating the gradient norm with respect to time and using 180, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2& ={\langle {\nabla }_{\mathbf{x}}{\partial }_{t}g,{\nabla }_{\mathbf{x}}g\rangle }_{\omega }\hfill \\ \hfill & =-{\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\omega }g,{\nabla }_{\mathbf{x}}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}{\mathcal{L}}_{\omega }g,{\nabla }_{\mathbf{x}}g\rangle }_{\omega }+{\langle {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }\left(g\right),{\nabla }_{\mathbf{x}}g\rangle }_{\omega },\hfill \end{array}$$

(208)

The transport term vanishes by skew-adjointness of ${\mathcal{T}}_{\omega }$: ${\langle {\nabla }_{\mathbf{x}}{\mathcal{T}}_{\omega }g,{\nabla }_{\mathbf{x}}g\rangle }_{\omega }=0$.

For the collision term,

$${\langle {\nabla }_{\mathbf{x}}{\mathcal{L}}_{\omega }g,{\nabla }_{\mathbf{x}}g\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\displaystyle \frac{1}{\tau }}{\langle {\nabla }_{\mathbf{x}}{g}_{\Vert },{\nabla }_{\mathbf{x}}{g}_{\perp }\rangle }_{\omega }\le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\displaystyle \frac{1}{2\tau }}{\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }\parallel }_{\omega }^2.$$

(209)

For the neural term, using Assumption A3,

$${\langle {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }\left(g\right),{\nabla }_{\mathbf{x}}g\rangle }_{\omega }\le L_{\theta }{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2.$$

(210)

Therefore,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel {\nabla }_{\mathbf{x}}{g\parallel }_{\omega }^2\le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+L_{\theta }{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2.$$

(211)

Combining the Estimates

With $\alpha =\tau /4$ and $\beta ={\tau }^2/4$, we form the augmented energy ${\mathcal{E}}_{\omega }\left(g\right)={\parallel g\parallel }_{\omega }^2+\alpha C_{\omega }\left(g\right)+\beta {\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2$. Using 190, 207, and 211, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)& \le -{\displaystyle \frac{7}{8\tau }}\parallel g_{\perp }{\parallel }_{\omega }^2+2\tau L_{\theta }^2{\parallel g\parallel }_{\omega }^2\hfill \\ \hfill & +\alpha \left[-{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+C_3\parallel g_{\perp }{\parallel }_{\omega }^2+C_4\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+C_4^{\prime }{\parallel g_{\Vert }\parallel }_{\omega }^2\right.\hfill \\ \hfill & \left.+{\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2+C_5{L}_{\theta }\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right)\right]\hfill \\ \hfill & +\beta \left[-{\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+2L_{\theta }{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right],\hfill \end{array}$$

(212)

where we have used $2·{\displaystyle \frac{1}{2}}=1$ for the gradient norm term.

Grouping terms by $\parallel g_{\perp }{\parallel }_{\omega }^2$, $\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2$, $\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2$, and $\parallel g_{\Vert }{\parallel }_{\omega }^2$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)& \le -\left({\displaystyle \frac{7}{8\tau }}-\alpha C_3\right)\parallel g_{\perp }{\parallel }_{\omega }^2-\left({\displaystyle \frac{\alpha }{2\tau }}+{\displaystyle \frac{\beta }{\tau }}\right){\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2\hfill \\ \hfill & +\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+\alpha C_4^{\prime }{\parallel g_{\Vert }\parallel }_{\omega }^2\hfill \\ \hfill & +\left(2\tau L_{\theta }^2+\alpha C_5{L}_{\theta }\right){\parallel g\parallel }_{\omega }^2\hfill \\ \hfill & +\left(\alpha {\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}+\alpha C_5{L}_{\theta }+2\beta L_{\theta }\right){\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,\hfill \end{array}$$

(213)

Using the refined Poincaré inequality 203 to eliminate $\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2$, we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)& \le -\left({\displaystyle \frac{7}{8\tau }}-\alpha C_3-C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)\right){\parallel g_{\perp }\parallel }_{\omega }^2\hfill \\ \hfill & -\left({\displaystyle \frac{\alpha }{2\tau }}+{\displaystyle \frac{\beta }{\tau }}-C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)\right){\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2\hfill \\ \hfill & +\left(\alpha C_4^{\prime }+C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)\right){\parallel g_{\Vert }\parallel }_{\omega }^2\hfill \\ \hfill & +\left(2\tau L_{\theta }^2+\alpha C_5{L}_{\theta }\right){\parallel g\parallel }_{\omega }^2\hfill \\ \hfill & +\left(\alpha {\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}+\alpha C_5{L}_{\theta }+2\beta L_{\theta }\right){\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2,\hfill \end{array}$$

(214)

Choose $\tau$ sufficiently small and $\alpha =\tau /4$, $\beta ={\tau }^2/4$ so that the coefficients of $\parallel g_{\perp }{\parallel }_{\omega }^2$ and $\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2$ are positive. This requires

$$\alpha C_3+C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)<{\displaystyle \frac{7}{8\tau }},$$

(215)

and

$$C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)<{\displaystyle \frac{\alpha }{2\tau }}+{\displaystyle \frac{\beta }{\tau }}.$$

(216)

For $\tau$ small enough, these conditions are satisfied. Then there exists $\gamma >0$ such that

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -\gamma \left(\parallel g_{\perp }{\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2\right)+{\gamma }_1{\parallel g_{\Vert }\parallel }_{\omega }^2+{\gamma }_2\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right),$$

(217)

where,

$$\gamma :=min\left\{{\displaystyle \frac{7}{8\tau }}-\alpha C_3-C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right),{\displaystyle \frac{\alpha }{2\tau }}+{\displaystyle \frac{\beta }{\tau }}-C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right)\right\}>0,$$

(218)

$${\gamma }_1:=\alpha C_4^{\prime }+C_P\left(\alpha C_4+{\displaystyle \frac{\beta }{\tau }}\right),$$

(219)

and

$${\gamma }_2:=max\left\{2\tau L_{\theta }^2+\alpha C_5{L}_{\theta },\alpha {\tilde{C}}_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}+\alpha C_5{L}_{\theta }+2\beta L_{\theta }\right\}.$$

(220)

By the norm equivalence of the augmented energy (which follows from the Cauchy-Schwarz inequality and the choice of $\alpha =\tau /4$, $\beta ={\tau }^2/4$), there exist constants $c_0,C_0>0$ such that

$$c_0\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right)\le {\mathcal{E}}_{\omega }\left(g\right)\le C_0\left({\parallel g\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}g\parallel }_{\omega }^2\right).$$

(221)

Using the decomposition ${\parallel g\parallel }_{\omega }^2=\parallel g_{\Vert }{\parallel }_{\omega }^2+{\parallel g_{\perp }\parallel }_{\omega }^2$ and the refined Poincaré inequality 203, we have:

$$\parallel g_{\Vert }{\parallel }_{\omega }^2\le {\displaystyle \frac{1}{C_P}}\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2+\parallel g_{\perp }{\parallel }_{\omega }^2+{\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }\parallel }_{\omega }^2.$$

(222)

Substituting into 217 and using the norm equivalence 221, we obtain:

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -2{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right),$$

(223)

with

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{\gamma }{2C_0}}-{\displaystyle \frac{{\gamma }_1}{2c_0{C}_P}}-{\displaystyle \frac{{\gamma }_2}{2c_0}}.$$

(224)

Now, using the explicit expressions for $\gamma$, ${\gamma }_1$, ${\gamma }_2$, $\alpha =\tau /4$, and $\beta ={\tau }^2/4$, we simplify 224 to

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

(225)

where the constants $C_1,C_2>0$ are defined as

$$C_1:={\displaystyle \frac{\alpha {\tilde{C}}_1}{2C_0}},C_2:={\displaystyle \frac{\alpha C_5+2\beta }{2c_0}}+{\displaystyle \frac{2\tau }{c_0}}+{\displaystyle \frac{\alpha C_4^{\prime }+C_P(\alpha C_4+\beta /\tau )}{2c_0{C}_P}}.$$

(226)

The condition ${\lambda }_{\mathrm{SA}}>0$ ensures exponential decay of the augmented energy. By the norm equivalence 221, this implies exponential decay of the $H^1$ norm. The exponential decay estimate 186 follows by integrating 223:

$${\mathcal{E}}_{\omega }\left(g\left(t\right)\right)\le e^{-2{\lambda }_{\mathrm{SA}}t}{\mathcal{E}}_{\omega }\left(g\left(0\right)\right).$$

(227)

Using the norm equivalence 221, we obtain

$${\parallel g\left(t\right)\parallel }_{\omega }^2\le {\displaystyle \frac{C_0}{c_0}}{e}^{-2{\lambda }_{\mathrm{SA}}t}{\parallel g\left(0\right)\parallel }_{H_{\omega }^1}^2,$$

(228)

which gives 186 with $C_0=\sqrt{C_0/c_0}$. This completes the proof. □

4.3. Discussion of the Santos-Andrade Inequality

The decay rate 352 encapsulates the competition between three key effects:

1.

Turbulent relaxation: The term $1/\left(4\tau \right)$ represents the stabilising effect of the BGK collision operator. A smaller relaxation time $\tau$ (i.e., faster relaxation) leads to a larger decay rate. This term originates from the spectral gap ${\lambda }_0=1/\tau$ of the linearised collision operator, which quantifies the rate at which non-hydrodynamic modes are dissipated.

2.

Rotation: The term $-C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|)$ quantifies the destabilising influence of rotation. This expression provides a sharper interpolation between the linear regime ($\left|\mathit{\omega }\right|\ll c_s$) and the saturated regime ($\left|\mathit{\omega }\right|\gg c_s$). The linear behaviour arises from the Coriolis force, which introduces a precession of the velocity field that can inhibit the relaxation to equilibrium. The saturation reflects the fact that the rotation operator has a finite spectral norm, and the commutator with the projection contributes the quadratic term through the density gradient and centrifugal potential.

3.

Neural Lipschitz constant: The term $-C_2{L}_{\theta }$ represents the potential destabilising effect of the neural correction. A larger Lipschitz constant $L_{\theta }$ reduces the decay rate. This provides a quantitative design criterion: for stable simulations, the neural network must be sufficiently contractive, i.e.,

$$L_{\theta }<{\displaystyle \frac{1}{C_2}}\left({\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}\right).$$

(229)

Remark 25

(Physical interpretation of the stability threshold). The stability threshold 229 has a clear physical interpretation: the neural correction must be sufficiently contractive (small $L_{\theta }$) relative to the stabilising effect of turbulent relaxation (measured by $1/\left(4\tau \right)$) and the destabilising effect of rotation (measured by $C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|)$). In the non-rotating case, this reduces to $L_{\theta }<1/\left(4C_2\tau \right)$, which is a constraint on the neural network’s Lipschitz constant relative to the turbulent relaxation time. In strongly rotating flows, the stability threshold is more stringent, requiring a smaller $L_{\theta }$ to compensate for the destabilising effect of rotation.

Remark 26.

The Santos-Andrade inequality is the first result that rigorously quantifies the combined influence of rotation, compressibility, and neural-network corrections on the stability of kinetic-based turbulence models. It provides a practical, mathematically certified criterion for designing neural closures that are both accurate and provably stable for rotating compressible flows.

Remark 27

(Explicit constants and their estimation). The constants $C_1$ and $C_2$ in 226 can be estimated a priori from the spectral properties of the collision operator and the geometry of the domain. For the BGK model, these constants are of order unity when non-dimensionalised appropriately. The explicit expressions in 183 allow for numerical evaluation:

$$C_1={\displaystyle \frac{\tau }{16}}{\displaystyle \sum _{\left|\alpha \right|=1}^2}{\displaystyle \frac{\parallel P_{\alpha }{\Pi }_{\omega }\mathbf{v}·{\nabla }_{\mathbf{x}}{H}_{\alpha }{\parallel }_{\omega }}{\parallel H_{\alpha }{\parallel }_{\omega }}},C_2={\displaystyle \frac{1}{2}}\underset{{\parallel g\parallel }_{\omega }=1}{sup}\underset{{\parallel h\parallel }_{\omega }=1}{sup}{\displaystyle \frac{\parallel {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }^{\ast }{\left(g\right)·\mathbf{v}h\parallel }_{\omega }}{\parallel {\nabla }_{\mathbf{x}}{g\parallel }_{\omega }{\parallel h\parallel }_{\omega }}}.$$

(230)

The Hermite polynomial basis ${\left\{H_{\alpha }\right\}}_{\alpha \in {\mathbb{N}}^3}$ diagonalises the rotation operator $\mathcal{R}$, with eigenvalues proportional to $\left|\mathit{\omega }\right|$ times angular momentum quantum numbers. This spectral decomposition provides a systematic way to compute $C_1$ for any given velocity discretisation. The constant $C_2$ depends on the neural network architecture and can be bounded by the product of the spectral norms of the weight matrices via spectral normalisation.

Using the estimates from Proposition 6, we have $C_0\sim 1+\alpha C_v/2+\beta$, $c_0\sim 1/2$ for small τ, and $C_P\sim 1$ on the torus. Thus, $C_1\sim (\tau /4){\tilde{C}}_1/\left(2\right)=\tau {\tilde{C}}_1/8$ and $C_2\sim (\tau C_5/4+{\tau }^2/2)/1+2\tau +\mathcal{O}\left(\tau \right)$, which are both $\mathcal{O}\left(\tau \right)$ for small τ. This confirms that the stability threshold is primarily controlled by the ratio $\tau /\left(\right|\mathit{\omega }|/(c_s+\left|\mathit{\omega }\right|\left)\right)$, with the neural term contributing at the same order.

The Santos-Andrade inequality establishes a rigorous foundation for the numerical scheme developed in the next section. The stability condition 229 will be used to guide the design of the PINN correction, ensuring that the neural network does not destabilise the LBM simulation.

4.4. Nonlinear Generalisation via Entropy Methods

A significant advancement of the Santos-Andrade framework is its extension to the nonlinear kinetic equation. This subsection establishes that the hypocoercive decay rate derived for the linearised problem also controls the entropy dissipation in the fully nonlinear setting.

Consider the nonlinear kinetic equation with structure-preserving neural correction:

$${\partial }_{t}f+{\mathcal{T}}_{\omega }f=Q_{\mathrm{BGK}}\left(f\right)+{\mathcal{N}}_{\theta }\left(f\right),$$

(231)

where $Q_{\mathrm{BGK}}\left(f\right)={\displaystyle \frac{1}{\tau }}(M\left[f\right]-f)$ is the BGK collision operator, and ${\mathcal{N}}_{\theta }$ satisfies Assumptions A1–A3.

Define the relative entropy (Kullback–Leibler divergence) between the distribution function f and the rotating Maxwellian $M_{\omega }$:

$$\mathcal{D}(f\parallel M_{\omega }):={\int }_{\Omega \times {\mathbb{R}}^3}fln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}-{\int }_{\Omega \times {\mathbb{R}}^3}(f-M_{\omega })d\mathbf{x}d\mathbf{v},$$

(232)

where the second term ensures that $\mathcal{D}\ge 0$ and $\mathcal{D}=0$ if and only if $f=M_{\omega }$.

The following theorem establishes the nonlinear hypocoercive estimate.

Theorem 6

(Nonlinear Santos–Andrade entropy inequality). Let f be a sufficiently regular solution of the nonlinear kinetic equation 231 with neural correction satisfying Assumptions A1–A3. Suppose the initial perturbation is sufficiently small so that $\parallel f\left(0\right)-M_{\omega }{\parallel }_{H_{\ell }^s}\le {\epsilon }_0$ for some ${\epsilon }_0>0$ sufficiently small. Then there exist constants $C>0$ and ${\lambda }_{\mathrm{SA}}>0$ such that the relative entropy satisfies

$${\displaystyle \frac{d}{dt}}\mathcal{D}(f\parallel M_{\omega })\le -2{\lambda }_{\mathrm{SA}}\mathcal{D}(f\parallel M_{\omega })+C{\parallel {\mathcal{N}}_{\theta }\left(f\right)\parallel }_{\omega }^2,$$

(233)

where ${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|)-C_2{L}_{\theta }$ is the same decay rate as in the linear case.

Proof. 

We begin by establishing two fundamental estimates that relate the relative entropy to the $L^2$ norm of the perturbation $g=(f-M_{\omega })/\sqrt{M_{\omega }}$.

Entropy-to-Norm Equivalence

For sufficiently small perturbations, the relative entropy admits the expansion

$$\mathcal{D}(f\parallel M_{\omega })={\displaystyle \frac{1}{2}}{\parallel g\parallel }_{\omega }^2+{\mathcal{O}(\parallel g\parallel }_{\omega }^3),$$

(234)

which follows from the standard Taylor expansion of the logarithm. Therefore, there exist constants $c_1,c_2>0$ such that

$$c_1{\parallel g\parallel }_{\omega }^2\le \mathcal{D}(f\parallel M_{\omega })\le c_2{\parallel g\parallel }_{\omega }^2,$$

(235)

provided ${\parallel g\parallel }_{\omega }$ is sufficiently small. These constants depend only on the background Maxwellian and the domain.

Logarithmic Sobolev Inequality for the BGK Operator

The BGK collision operator satisfies the following logarithmic Sobolev inequality:

$${\int }_{\Omega \times {\mathbb{R}}^3}(f-M\left[f\right])ln\left({\displaystyle \frac{f}{M\left[f\right]}}\right)d\mathbf{x}d\mathbf{v}\ge {\displaystyle \frac{1}{2}}{\parallel (I-{\Pi }_{\omega })g\parallel }_{\omega }^2,$$

(236)

which is a classical result for the BGK model [31]. This inequality follows from the fact that the BGK operator has a spectral gap of size $1/\tau$ and that the relative entropy $D(f\parallel M[f\left]\right)$ controls the $L^2$ norm of the non-hydrodynamic part of the perturbation.

Time Derivative of the Relative Entropy

Differentiating $\mathcal{D}(f\parallel M_{\omega })$ with respect to time and using the nonlinear kinetic equation 231, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\mathcal{D}(f\parallel M_{\omega })& ={\int }_{\Omega \times {\mathbb{R}}^3}{\partial }_{t}fln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}\hfill \\ \hfill & =\int \left(-{\mathcal{T}}_{\omega }f+Q_{\mathrm{BGK}}\left(f\right)+{\mathcal{N}}_{\theta }\left(f\right)\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}.\hfill \end{array}$$

(237)

The streaming term vanishes because ${\mathcal{T}}_{\omega }$ is skew-adjoint and ${\mathcal{T}}_{\omega }{M}_{\omega }=0$:

$$\int {\mathcal{T}}_{\omega }fln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}=\int f{\mathcal{T}}_{\omega }ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}=\int f{\displaystyle \frac{{\mathcal{T}}_{\omega }f}{f}}d\mathbf{x}d\mathbf{v}=\int {\mathcal{T}}_{\omega }fd\mathbf{x}d\mathbf{v}=0.$$

(238)

For the BGK term, we use the identity

$$ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)=ln\left({\displaystyle \frac{f}{M\left[f\right]}}\right)+ln\left({\displaystyle \frac{M\left[f\right]}{M_{\omega }}}\right).$$

(239)

Since $ln(M\left[f\right]/M_{\omega })$ is a linear combination of the collision invariants (mass, momentum, and energy), and $M\left[f\right]$ has the same first five moments as f, we have

$$\int (f-M\left[f\right])ln\left({\displaystyle \frac{M\left[f\right]}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}=0.$$

(240)

Therefore, the BGK contribution to the entropy production is

$$\int Q_{\mathrm{BGK}}\left(f\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}=-{\displaystyle \frac{1}{\tau }}\int (f-M\left[f\right])ln\left({\displaystyle \frac{f}{M\left[f\right]}}\right)d\mathbf{x}d\mathbf{v}.$$

(241)

Applying the logarithmic Sobolev inequality 236, we obtain

$$\int Q_{\mathrm{BGK}}\left(f\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}\le -{\displaystyle \frac{1}{2\tau }}{\parallel (I-{\Pi }_{\omega })g\parallel }_{\omega }^2.$$

(242)

For the neural term, we use Assumption A2 (Lipschitz continuity) and the Cauchy-Schwarz inequality:

$$\int {\mathcal{N}}_{\theta }\left(f\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}\le {\parallel {\mathcal{N}}_{\theta }\left(f\right)\parallel }_{\omega }{∥ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)∥}_{\omega }.$$

(243)

For small perturbations, the logarithmic term satisfies $\parallel ln(f/M_{\omega }){\parallel }_{\omega }\le C_L{\parallel g\parallel }_{\omega }$ for some constant $C_L>0$. This follows from the expansion $ln(1+x)=x+\mathcal{O}\left(x^2\right)$ and the fact that $f/M_{\omega }=1+g/\sqrt{M_{\omega }}$. Therefore,

$$\int {\mathcal{N}}_{\theta }\left(f\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}\le C_L{L}_{\theta }{\parallel g\parallel }_{\omega }^2+C_L\parallel {\mathcal{N}}_{\theta }{\left(0\right)\parallel }_{\omega }{\parallel g\parallel }_{\omega }.$$

(244)

Since ${\mathcal{N}}_{\theta }\left(0\right)=0$ by Assumption A1 (the neural correction vanishes for the equilibrium state), the second term vanishes. Applying Young’s inequality, we obtain

$$\int {\mathcal{N}}_{\theta }\left(f\right)ln\left({\displaystyle \frac{f}{M_{\omega }}}\right)d\mathbf{x}d\mathbf{v}\le C_L{L}_{\theta }{\parallel g\parallel }_{\omega }^2+{\delta }_{\mathrm{NN}}{\parallel g\parallel }_{\omega },$$

(245)

where ${\delta }_{\mathrm{NN}}$ accounts for any residual neural approximation error.

Combining 242 and 245, we get

$${\displaystyle \frac{d}{dt}}\mathcal{D}(f\parallel M_{\omega })\le -{\displaystyle \frac{1}{2\tau }}\parallel (I-{\Pi }_{\omega }){g\parallel }_{\omega }^2+C_L{L}_{\theta }{\parallel g\parallel }_{\omega }^2.$$

(246)

Relating the Entropy Dissipation to the Hypocoercive Energy

We now recall the linear hypocoercive estimate from Theorem 5. For the linearised equation ${\partial }_{t}g+{\mathcal{T}}_{\omega }g={\mathcal{L}}_{\omega }g+{\mathcal{N}}_{\theta }^{\ast }\left(g\right)$, the augmented energy ${\mathcal{E}}_{\omega }\left(g\right)$ satisfies

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -2{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right)+C_{\mathcal{N}}{\delta }_{\mathrm{NN}}^2.$$

(247)

For the nonlinear equation, the additional quadratic terms are small when ${\parallel g\parallel }_{\omega }$ is small. More precisely, the nonlinearity $Q_{\mathrm{BGK}}\left(f\right)-{\mathcal{L}}_{\omega }g$ is of order $\mathcal{O}(\parallel g{\parallel }_{\omega }^2)$, and the neural term is already linearised. By a standard perturbation argument, the same hypocoercive estimate holds for the nonlinear equation up to a small error:

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -2{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right)+C_{\mathcal{N}}{\delta }_{\mathrm{NN}}^2+C_{\mathrm{NL}}{\parallel g\parallel }_{\omega }^3,$$

(248)

where $C_{\mathrm{NL}}>0$ is a constant depending on the nonlinearity.

Since the initial perturbation is sufficiently small, we have ${\parallel g\left(t\right)\parallel }_{\omega }\le \epsilon$ for all $t\ge 0$ by a standard continuation argument. The cubic term $C_{\mathrm{NL}}{\parallel g\parallel }_{\omega }^3$ is then bounded by ${\lambda }_{\mathrm{SA}}{\parallel g\parallel }_{\omega }^2$ for $\epsilon$ small enough. Therefore,

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega }\left(g\right)\le -{\lambda }_{\mathrm{SA}}{\mathcal{E}}_{\omega }\left(g\right)+C_{\mathcal{N}}{\delta }_{\mathrm{NN}}^2.$$

(249)

By the norm equivalence 221, there exist constants ${\kappa }_1,{\kappa }_2>0$ such that ${\kappa }_1{\parallel g\parallel }_{\omega }^2\le {\mathcal{E}}_{\omega }\left(g\right)\le {\kappa }_2{\parallel g\parallel }_{\omega }^2$. Combining this with 246 and the entropy-norm equivalence 235, we obtain

$${\displaystyle \frac{d}{dt}}\mathcal{D}(f\parallel M_{\omega })\le -2{\lambda }_{\mathrm{SA}}\mathcal{D}(f\parallel M_{\omega })+C{\parallel {\mathcal{N}}_{\theta }\left(f\right)\parallel }_{\omega }^2,$$

(250)

for some constant $C>0$ that absorbs the neural approximation error and the entropy-to-norm conversion factors. This completes the proof. □

Remark 28

(Significance of the nonlinear generalisation). The nonlinear entropy estimate 233 extends the Santos-Andrade inequality to the fully nonlinear setting, showing that the hypocoercive decay rate controls entropy dissipation for sufficiently small perturbations. This is crucial for practical turbulence simulations far from equilibrium. The constant C depends on ${\delta }_{\mathrm{NN}}$, reflecting that inaccurate neural corrections can generate entropy, rigorously justifying the structure-preserving constraints (Assumptions 1 – 3) that prevent uncontrolled entropy production.

4.5. Comparison with Villani’s Hypocoercive Framework

To clarify the novelty and positioning of the Santos-Andrade inequality within the existing mathematical literature, we provide a systematic comparison with the classical hypocoercive framework developed by Villani [31], which itself synthesised and extended the contributions of Hérau [27], Mouhot [28,29], and Dolbeault, Mouhot, and Schmeiser [30]. This comparison serves three purposes: (i) to acknowledge the foundational work upon which our analysis builds, (ii) to articulate precisely what is new in our contribution, and (iii) to demonstrate that our inequality recovers Villani’s classical result as a special case.

4.5.1. The Villani Framework: Augmented Energy and Hypocoercivity

Villani’s hypocoercive framework addresses the fundamental problem that the linearised collision operator $\mathcal{L}=-{\displaystyle \frac{1}{\tau }}(I-\Pi )$ dissipates only the non-hydrodynamic part of the perturbation, while the transport operator $\mathcal{T}=\mathbf{v}·{\nabla }_{\mathbf{x}}$ is skew-adjoint and dissipates no $L^2$ norm. To overcome this, Villani constructs an augmented energy of the form

$${\mathcal{E}}_{\mathrm{V}}\left(g\right):={\parallel g\parallel }^2+\alpha \langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle +\beta {\parallel {\nabla }_{\mathbf{x}}g\parallel }^2,$$

(251)

with $\alpha =\tau /4$ and $\beta ={\tau }^2/4$, and proves that for the linearised BGK equation without forcing,

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\mathrm{V}}\left(g\right)\le -2{\lambda }_{\mathrm{V}}{\mathcal{E}}_{\mathrm{V}}\left(g\right),{\lambda }_{\mathrm{V}}={\displaystyle \frac{1}{4\tau }}.$$

(252)

This yields exponential decay of the perturbation at rate ${\lambda }_{\mathrm{V}}=1/\left(4\tau \right)$. The key insight is that the cross term $\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle$ couples the hydrodynamic and non-hydrodynamic modes, allowing the dissipation of $g_{\perp }$ to indirectly control $g_{\Vert }$ through the Poincaré-type inequality for the hydrodynamic part.

4.5.2. What Is New in the Santos-Andrade Inequality

The Santos-Andrade inequality extends Villani’s framework in four distinct directions, each representing a genuine theoretical advance:

1.

Explicit rotation dependence via commutator estimates and Lie algebra structure.

When the kinetic equation is formulated in a rotating reference frame, the equilibrium Maxwellian acquires spatial dependence through the density profile $\rho \left(\mathbf{x}\right)={\rho }_0{exp\left(\right|\mathit{\omega }\times \mathbf{x}|}^2/\left(2RT\right))$. This introduces non-trivial commutators $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ between the spatial gradient and the projection operator. The derivation of the refined bound

$$\parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }\le C_{\Pi }\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel g\parallel }_{\omega }$$

(253)

(Lemma 2) is non-trivial and yields the explicit rotation-dependent term in the decay rate:

$$-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}.$$

(254)

The linear term arises from the dependence of the basis functions on the mean velocity $\mathbf{u}$ (which satisfies ${\nabla }_{\mathbf{x}}\mathbf{u}\sim \left|\mathit{\omega }\right|$), while the quadratic term arises from the density gradient and the centrifugal potential. The improved interpolation ${\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}$ is obtained by recognising the Lie algebra structure of the rotation operator, providing a sharper estimate than the previous separated form. These terms are absent from Villani’s framework, which assumes a spatially homogeneous Maxwellian.

2.

Neural correction with explicit Lipschitz constant.

Villani’s framework does not address data-driven or learnable corrections. The Santos-Andrade inequality introduces a structure-preserving neural correction ${\mathcal{N}}_{\theta }$ constrained by Assumptions 1–3 and proves that its effect on stability is quantified by the term

$$-C_2{L}_{\theta },$$

(255)

where $L_{\theta }$ is the Lipschitz constant of the neural operator. This provides the first rigorous stability certificate for neural-network-augmented kinetic models, directly addressing the critical gap identified in Section 1: the absence of mathematical guarantees in existing data-driven closures [43,47,48,55,58].

3.

Discrete hypocoercivity for LBM–PINN schemes.

Villani’s framework is formulated at the continuous level. The Santos-Andrade inequality is extended to a semi-discrete setting in Lemma 4 and Theorem 7, accounting for spatial discretisation ($\Delta x$), velocity quadrature ($\Delta v$), and neural approximation errors (${\delta }_{\mathrm{NN}}$). The discrete decay rate

$${\lambda }_Q={\displaystyle \frac{{\lambda }_0}{4\tau }}-C_1{L}_{\theta }$$

(256)

and the convergence estimate

$${\parallel e\left(t\right)\parallel }_{L^2}\le C_T\left(\Delta x^2+{(\Delta v)}^{m+1}+\tau +{\delta }_{\mathrm{NN}}\right)$$

(257)

provide the first rigorous error analysis for a structure-preserving LBM–PINN hybrid scheme.

4.

Nonlinear generalisation via entropy methods.

Villani’s framework is primarily linear. The Santos–Andrade inequality is extended to the nonlinear setting in Theorem 6, establishing that the same decay rate controls the relative entropy dissipation for sufficiently small initial perturbations. This provides a rigorous justification for the stability of the nonlinear kinetic equation with neural correction.

4.5.3. Recovery of Villani’s Result as a Special Case

To further demonstrate the consistency and generality of our framework, we note that the Santos–Andrade inequality reduces exactly to Villani’s classical result when the new contributions are removed:

• Non-rotating frame: If $\mathit{\omega }=0$, the rotation-dependent term vanishes, and the commutator $[{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]$ reduces to zero (since $\Pi$ is independent of $\mathbf{x}$ in the non-rotating case). The decay rate 352 becomes

  $${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_2{L}_{\theta }.$$

  (258)
• Without neural correction: If $L_{\theta }=0$ (i.e., no neural network is used), the neural term vanishes and we recover

  $${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}={\lambda }_{\mathrm{V}},$$

  (259)

  which is precisely Villani’s decay rate from 252. Moreover, when $\mathit{\omega }=0$ and $L_{\theta }=0$, the augmented energy inequality 184 reduces to ${\displaystyle \frac{d}{dt}}\mathcal{E}\left(g\right)\le -{\displaystyle \frac{1}{2\tau }}\mathcal{E}\left(g\right)$, which is exactly Villani’s classical result.
• Continuous limit: In the limit $\Delta x,\Delta v,{\delta }_{\mathrm{NN}}\to 0$, the discrete hypocoercive inequality converges to the continuous one, and the convergence estimate 257 recovers the expected exponential decay with rate ${\lambda }_{\mathrm{V}}$.

Thus, Villani’s classical result is obtained as the special case of our framework when there is no rotation and no neural correction. The Santos–Andrade inequality generalises Villani’s work by incorporating these practically relevant extensions.

4.5.4. Why This Matters for Turbulence Modelling

The practical significance of this comparison is twofold. First, it establishes that our framework is built on solid mathematical foundations: we do not discard or replace Villani’s rigorous theory, but rather extend it in directions that are essential for practical turbulence simulations. Second, it provides a clear benchmark: if a neural closure satisfies the Santos-Andrade stability condition, it is guaranteed to be at least as stable as the baseline BGK model in the non-rotating case. This certification is absent from previous data-driven approaches, which rely on empirical validation without mathematical guarantees.

Remark 29

(Summary of theoretical advances). In summary, while the augmented energy methodology is inherited from Villani [31] (see 251 and 252) and the spectral gap analysis follows Mouhot [28,29] and Hérau [27], the Santos-Andrade inequality represents a genuine theoretical advance by:

1.

Deriving explicit rotation-dependent terms (254) through careful commutator estimates (253) that quantify the additional dissipation required to counteract Coriolis and centrifugal forces, with the improved interpolation ${\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}$ obtained via the Lie algebra structure of the rotation operator.

2.

Incorporating the neural Lipschitz constant as a quantitative stability parameter (255), bridging the gap between abstract hypocoercive theory and the practical design of data-driven closures.

3.

Extending the framework to the semi-discrete LBM–PINN setting with rigorous convergence guarantees (257) that account for spatial discretisation, velocity quadrature, and neural approximation errors.

4.

Providing the first nonlinear generalisation via entropy methods (Theorem 6), establishing that the hypocoercive decay rate controls the relative entropy dissipation in the fully nonlinear setting.

The inequality is therefore not merely an "engineering estimate” but a mathematically certified criterion that connects first-principle kinetic theory with practical neural-network design.

5. Numerical Discretisation: A Convergent LBM–PINN Scheme

We now discretise the kinetic model 97 using a hybrid Lattice Boltzmann Method (LBM) with a Physics-Informed Neural Network (PINN) correction. The scheme preserves the structure of the continuous problem: mass, momentum, and energy are exactly conserved, the H-theorem is enforced at the discrete level through the BGK relaxation, and the Santos-Andrade inequality 352 provides a stability condition that guides the choice of the neural network’s Lipschitz constant.

The discretisation strategy proceeds in two stages. First, we discretise the velocity space using a Gauss-Hermite quadrature, which preserves the Gaussian structure of the equilibrium distribution. Second, we discretise the spatial domain using a second-order finite difference scheme, which is skew-symmetric and thus preserves the hypocoercive structure at the discrete level. The neural correction is evaluated pointwise in physical space and then projected onto the orthogonal complement of the collision invariants, ensuring exact conservation laws.

This section is organised as follows. Section 5.2 presents the velocity discretisation via Gauss-Hermite quadrature and establishes the quadrature error estimate. Section 5.3 introduces the semi-discrete LBM–PINN scheme and the structure-preserving projection of the neural correction. Section 5.4 defines the discrete augmented energy and proves the discrete hypocoercive inequality. Section 5.5 states and proves the main convergence theorem. Finally, Section 5.6 discusses practical implementation aspects, including the stability condition and neural network design.

5.1. Structure-Preserving Discretisation Philosophy

Before presenting the technical details, we articulate the guiding principles that underpin our discretisation strategy.

Remark 30

(Structure-preserving discretisation philosophy). The discretisation is designed to preserve three fundamental properties of the continuous kinetic model:

1. 

Conservation laws: Mass, momentum, and energy are exactly conserved at the discrete level through the projection operator ${\Pi }_Q$ and the orthogonal projection of the neural correction. This is achieved by ensuring that the discrete equilibrium and the neural correction share the same first five velocity moments as their continuous counterparts.

2. 

Entropy dissipation: The BGK relaxation preserves the H-theorem at the discrete level, ensuring thermodynamic consistency. The discrete collision operator satisfies a discrete entropy inequality analogous to 64.

3. 

Hypocoercive stability: The skew-symmetric spatial discretisation and the spectral gap of the discrete collision operator guarantee that the discrete augmented energy decays exponentially, provided the neural Lipschitz constant satisfies the Santos-Andrade stability condition.

This structure-preserving approach ensures that the numerical scheme inherits the stability and convergence properties of the continuous model, avoiding the spurious oscillations and instabilities that plague purely empirical closures.

5.2. Velocity Discretisation: Gauss–Hermite Quadrature

The continuous velocity space ${\mathbb{R}}^3$ is replaced by a discrete set of velocities ${\left\{{\mathbf{c}}_i\right\}}_{i=1}^Q$, where Q is the number of quadrature points. These points and associated weights ${\left\{w_i\right\}}_{i=1}^Q$ are chosen so that the Gauss–Hermite quadrature rule is exact for all polynomials of degree at most m:

$${\displaystyle \sum _{i=1}^Q}{w}_{i}p\left({\mathbf{c}}_i\right)={\int }_{{\mathbb{R}}^3}p\left(\mathbf{v}\right)M_0\left(\mathbf{v}\right)d\mathbf{v},\forall p\in {\mathbb{P}}_m,$$

(260)

where ${\mathbb{P}}_m$ denotes the space of polynomials of degree at most m, and $M_0$ is a fixed reference Maxwellian:

$$M_0\left(\mathbf{v}\right):={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2}}\right).$$

(261)

Remark 31

(Choice of quadrature degree and velocity set). For three-dimensional Hermite quadrature, the number of points Q scales as ${(m+1)}^3$, and the effective velocity spacing is $\Delta v\sim Q^{-1/3}$. The choice $m\ge 4$ is necessary and sufficient to ensure that:

• All moments up to order four are exactly captured, which is required to preserve the conservation laws and the energy balance.
• The discrete equilibrium has the same first five moments as the continuous Maxwellian.
• The spectral gap of the discrete collision operator is preserved up to quadrature error of order ${(\Delta v)}^{m+1}$.

In practice, $m=4$ (corresponding to $Q=5^3=125$ velocities in three dimensions) is the minimal choice for compressible flows, while $m=6$ or $m=8$ provides higher accuracy at increased computational cost. For compressible flows at high Mach numbers ($Ma>3$), the standard D3Q19 or D3Q27 lattice sets are insufficient because they cannot capture the higher-order moments required for the energy equation. The Gauss–Hermite quadrature with $m\ge 4$ yields a discrete velocity set specifically designed to preserve moments up to order four. The D3Q125 lattice ($m=4$, $Q=125$) is the minimal set that correctly captures the compressible Navier–Stokes equations in the Chapman–Enskog limit.

Define the projection operator ${\mathcal{P}}_Q:L^2\left({\mathcal{H}}_{M_0}\right)\to {\mathbb{R}}^Q$ by

$${\left({\mathcal{P}}_{Q}f\right)}_i:=w_i{M}_0{\left({\mathbf{c}}_i\right)}^{-1/2}f\left({\mathbf{c}}_i\right),i=1,\dots ,Q,$$

(262)

where the factor $M_0{\left({\mathbf{c}}_i\right)}^{-1/2}$ ensures that the discrete inner product approximates the continuous inner product in the symmetrised Hilbert space ${\mathcal{H}}_{M_0}=L^2\left(M_0^{-1}d\mathbf{v}\right)$.

The adjoint operator ${\mathcal{P}}_Q^{\ast }:{\mathbb{R}}^Q\to L^2\left({\mathcal{H}}_{M_0}\right)$ is defined by the relation

$${\langle {\mathcal{P}}_{Q}f,\mathbf{g}\rangle }_{{\mathbb{R}}^Q}={\langle f,{\mathcal{P}}_Q^{\ast }\mathbf{g}\rangle }_{{\mathcal{H}}_{M_0}},$$

(263)

for all $f\in L^2\left({\mathcal{H}}_{M_0}\right)$ and $\mathbf{g}\in {\mathbb{R}}^Q$, where ${\langle ·,·\rangle }_{{\mathbb{R}}^Q}$ is the Euclidean inner product with weights $w_i$. A direct computation yields

$$\left({\mathcal{P}}_Q^{\ast }\mathbf{g}\right)\left(\mathbf{v}\right)=M_0{\left(\mathbf{v}\right)}^{1/2}{\displaystyle \sum _{i=1}^Q}{g}_i\delta (\mathbf{v}-{\mathbf{c}}_i).$$

(264)

Remark 32

(Consistency of the quadrature approximation). The operators ${\mathcal{P}}_Q$ and ${\mathcal{P}}_Q^{\ast }$ satisfy ${\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q=I$ on the space of polynomials of degree at most m by the exactness of the quadrature rule. This property is essential for the convergence analysis, as it ensures that the projection error is entirely controlled by the quadrature error.

The following lemma establishes the spectral accuracy of the velocity discretisation.

Lemma 3

(Quadrature error). For any $f\in H_v^{m+1}\left({\mathbb{R}}^3\right)$ with ${\parallel f\parallel }_{H_v^{m+1}}<\infty$, there exists a constant $C_{\mathrm{quad}}>0$, independent of Q, such that

$$\parallel f-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q{f\parallel }_{L^2\left(M_0^{-1}\right)}\le C_{\mathrm{quad}}{(\Delta v)}^{m+1}{\parallel f\parallel }_{H_v^{m+1}},$$

(265)

where $\Delta v\sim Q^{-1/3}$ is the effective velocity spacing.

Proof. 

The proof proceeds by constructing a polynomial interpolant of degree m that matches f at the quadrature points. Let ${\mathcal{I}}_{m}f$ denote the interpolation polynomial of degree at most m in each variable. By the Bramble–Hilbert lemma applied to the interpolation error on the grid of quadrature points, we have

$$\parallel f-{\mathcal{I}}_m{f\parallel }_{L^2\left(M_0^{-1}\right)}\le C_1{(\Delta v)}^{m+1}{\parallel f\parallel }_{H_v^{m+1}}.$$

(266)

Since the quadrature is exact for polynomials of degree at most m, we have ${\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q{\mathcal{I}}_{m}f={\mathcal{I}}_{m}f$. Therefore,

$$\begin{array}{cc}\hfill \parallel f-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q{f\parallel }_{L^2\left(M_0^{-1}\right)}& \le \parallel f-{\mathcal{I}}_m{f\parallel }_{L^2\left(M_0^{-1}\right)}+{\parallel {\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q(f-{\mathcal{I}}_{m}f)\parallel }_{L^2\left(M_0^{-1}\right)}\hfill \\ \hfill & \le (1+\parallel {\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q{\parallel }_{\mathcal{L}(L^2,L^2)})\parallel f-{\mathcal{I}}_m{f\parallel }_{L^2\left(M_0^{-1}\right)}\hfill \\ \hfill & \le C_{\mathrm{quad}}{(\Delta v)}^{m+1}{\parallel f\parallel }_{H_v^{m+1}},\hfill \end{array}$$

(267)

where the operator norm $\parallel {\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q{\parallel }_{\mathcal{L}(L^2,L^2)}$ is bounded uniformly in Q for Gauss-Hermite quadrature [37]. This completes the proof. □

Remark 33

(Spectral convergence of the velocity discretisation). The quadrature error estimate 265 is spectral in nature: for smooth functions f, the error decays faster than any polynomial order as $Q\to \infty$. This property is essential for high-accuracy simulations, as it ensures that the velocity discretisation does not introduce significant errors even when the number of quadrature points is moderate. In practice, the quadrature error is typically negligible compared to the spatial discretisation error for well-resolved simulations, allowing the use of relatively small velocity sets ($Q\sim 125$ to $Q\sim 729$) for most applications.

5.3. Semi-Discrete LBM–PINN Scheme

Let $N_x$ be the number of spatial grid points, and let $\Delta x>0$ be the spatial grid spacing. We discretise the spatial domain $\Omega ={\mathbb{T}}^3$ with a uniform Cartesian grid ${\left\{{\mathbf{x}}_j\right\}}_{j=1}^{N_x}$, and use a second-order centered finite difference scheme for the streaming operator $\mathbf{v}·{\nabla }_{\mathbf{x}}$:

$${\mathbf{c}}_i·{\nabla }_{\mathbf{x}}{f}_i\left({\mathbf{x}}_j\right)={\displaystyle \sum _{k=1}^3}{c}_{i,k}{\displaystyle \frac{f_i({\mathbf{x}}_j+\Delta x{\mathbf{e}}_k)-f_i({\mathbf{x}}_j-\Delta x{\mathbf{e}}_k)}{2\Delta x}}+\mathcal{O}(\Delta x^2),$$

(268)

where ${\mathbf{e}}_k$ denotes the unit vector in the k-th coordinate direction.

Remark 34

(Skew-symmetry of the spatial discretisation). The finite difference scheme 268 is skew-symmetric: for any two grid functions $u,v$,

$${\displaystyle \sum _{j=1}^{N_x}}v\left({\mathbf{x}}_j\right)({\mathbf{c}}_i·{\nabla }_{\mathbf{x}}u\left({\mathbf{x}}_j\right))\Delta x^3=-{\displaystyle \sum _{j=1}^{N_x}}u\left({\mathbf{x}}_j\right)({\mathbf{c}}_i·{\nabla }_{\mathbf{x}}v\left({\mathbf{x}}_j\right))\Delta x^3,$$

(269)

which is essential for preserving the hypocoercive structure at the discrete level. The consistency error of the finite difference scheme is $\mathcal{O}(\Delta x^2)$ for sufficiently smooth functions.

Remark 35

(Courant–Friedrichs–Lewy condition). The finite difference scheme 268 is conditionally stable. For the LBM, the CFL condition requires $\Delta t\le C\Delta x/{max}_i\left|{\mathbf{c}}_i\right|$ for some constant $C>0$. For the standard D3Q125 lattice with $m=4$, the maximum velocity magnitude is ${max}_i\left|{\mathbf{c}}_i\right|=\sqrt{3}{c}_s$, where $c_s=1/\sqrt{3}$ in lattice units. Thus, the CFL condition becomes $\Delta t\le C\Delta x$. In the semi-discrete setting considered here, we assume the temporal discretisation is sufficiently small to satisfy this condition, and we focus on the spatial and velocity discretisation errors. The fully discrete scheme with a suitable time integration method (e.g., explicit Euler or Runge–Kutta) preserves the stability properties established here, provided the CFL condition is satisfied.

The semi-discrete LBM–PINN scheme reads:

$${\partial }_t{f}_i({\mathbf{x}}_j,t)+{\mathbf{c}}_i·{\nabla }_{\mathbf{x}}{f}_i({\mathbf{x}}_j,t)={\displaystyle \frac{1}{\tau }}\left(f_i^{\mathrm{eq}}({\mathbf{x}}_j,t)-f_i({\mathbf{x}}_j,t)\right)+{\mathcal{N}}_{\theta ,i}({\mathbf{x}}_j,t),$$

(270)

where $i=1,\dots ,Q$ indexes the discrete velocities ${\mathbf{c}}_i$, and $f_i({\mathbf{x}}_j,t):={\mathcal{P}}_{Q}f({\mathbf{x}}_j,{\mathbf{c}}_i,t)$ are the discrete distribution functions.

The discrete equilibrium distributions are defined by

$$f_i^{\mathrm{eq}}({\mathbf{x}}_j,t):={\mathcal{P}}_{Q}M\left[f\right]({\mathbf{x}}_j,{\mathbf{c}}_i,t),$$

(271)

where $M\left[f\right]$ is the local Maxwellian 8. By construction, the discrete equilibrium has the same first five velocity moments as the continuous equilibrium:

$${\displaystyle \sum _{i=1}^Q}{w}_i{f}_i^{\mathrm{eq}}=\rho ,{\displaystyle \sum _{i=1}^Q}{w}_i{\mathbf{c}}_i{f}_i^{\mathrm{eq}}=\rho \mathbf{u},{\displaystyle \sum _{i=1}^Q}{w}_i|{\mathbf{c}}_i{|}^2{f}_i^{\mathrm{eq}}={\rho \left(\right|\mathbf{u}|}^2+3RT).$$

(272)

The discrete neural correction ${\mathcal{N}}_{\theta ,i}$ is defined by

$${\mathcal{N}}_{\theta ,i}({\mathbf{x}}_j,t):={\mathcal{P}}_Q{\mathcal{N}}_{\theta }\left(f\right)({\mathbf{x}}_j,{\mathbf{c}}_i,t).$$

(273)

By Assumption A1, the neural correction is orthogonal to the collision invariants at the discrete level:

$${\displaystyle \sum _{i=1}^Q}{w}_i{\mathcal{N}}_{\theta ,i}({\mathbf{x}}_j,t)=0,{\displaystyle \sum _{i=1}^Q}{w}_i{\mathbf{c}}_i{\mathcal{N}}_{\theta ,i}({\mathbf{x}}_j,t)=\mathbf{0},{\displaystyle \sum _{i=1}^Q}{w}_i{\left|{\mathbf{c}}_i\right|}^2{\mathcal{N}}_{\theta ,i}({\mathbf{x}}_j,t)=0.$$

(274)

These orthogonality conditions ensure that mass, momentum, and energy are exactly conserved at the discrete level.

Remark 36

(Exact conservation at the discrete level). The discrete orthogonality conditions 274 are the discrete analogues of the continuous conservation laws. They are enforced exactly by the projection matrix $\mathbf{P}$, which is precomputed from the quadrature points and weights. This is a crucial advantage over empirical closures (such as Smagorinsky or k-ε), which do not guarantee conservation of energy at the discrete level. The exact conservation of mass, momentum, and energy ensures that the numerical scheme is free from unphysical drifts and that the long-time behaviour of the simulation is physically meaningful.

The projection onto the orthogonal complement of the collision invariants is implemented using a precomputed projection matrix $\mathbf{P}\in {\mathbb{R}}^{Q\times Q}$:

$${\mathbf{N}}_{\theta }^{\perp }=\mathbf{P}{\mathbf{N}}_{\theta },$$

(275)

where ${\mathbf{N}}_{\theta }\in {\mathbb{R}}^Q$ is the raw neural network output at each spatial point. The matrix $\mathbf{P}$ satisfies the following properties:

1.

Idempotence: ${\mathbf{P}}^2=\mathbf{P}$, ensuring that the projection is well-defined.

2.

Self-adjointness: $\mathbf{P}={\mathbf{P}}^{\ast }$ with respect to the weight matrix $\mathbf{W}=diag(w_1,\dots ,w_Q)$.

3.

Orthogonality to invariants: $\mathbf{P}\mathbf{1}=\mathbf{0}$, $\mathbf{P}{\mathbf{c}}_i=\mathbf{0}$, $\mathbf{P}|{\mathbf{c}}_i{|}^2=\mathbf{0}$.

These properties guarantee that the projected neural correction does not alter the conserved quantities.

Remark 37

(Construction of the projection matrix). The projection matrix $\mathbf{P}$ can be constructed explicitly from the orthonormal basis ${\left\{{\mathbf{e}}_j\right\}}_{j=0}^4$ of the collision invariants in the discrete space. Let $\mathbf{E}\in {\mathbb{R}}^{Q\times 5}$ be the matrix whose columns are the basis vectors, normalised with respect to the weight matrix $\mathbf{W}$. Then the projection onto the collision invariants is ${\Pi }_Q=\mathbf{E}{\mathbf{E}}^{\top }\mathbf{W}$, and the projection onto the orthogonal complement is $\mathbf{P}=\mathbf{I}-{\Pi }_Q$. This construction ensures that $\mathbf{P}$ is idempotent, self-adjoint, and orthogonal to the invariants. In practice, $\mathbf{P}$ can be precomputed once for a given velocity set and stored for use in all subsequent simulations.

5.4. Discrete Augmented Energy and Hypocoercive Inequality

To analyse the stability of the discrete scheme, we define the discrete $L^2$ norm and the discrete gradient norm:

$${\parallel \mathbf{f}\parallel }_{L_w^2}^2:={\displaystyle \sum _{j=1}^{N_x}}{\displaystyle \sum _{i=1}^Q}{w}_i{\left|f_i\left({\mathbf{x}}_j\right)\right|}^2\Delta x^3,$$

(276)

$$\parallel {\nabla }_{\mathbf{x}}{\mathbf{f}\parallel }_{L_w^2}^2:={\displaystyle \sum _{j=1}^{N_x}}{\displaystyle \sum _{i=1}^Q}{w}_i{\left|{\nabla }_{\mathbf{x}}{f}_i\left({\mathbf{x}}_j\right)\right|}^2\Delta x^3,$$

(277)

where the spatial gradient ${\nabla }_{\mathbf{x}}{f}_i\left({\mathbf{x}}_j\right)$ is computed using the centered finite difference scheme 268.

The discrete augmented energy is defined analogously to the continuous case:

$${\mathcal{E}}_Q\left(\mathbf{f}\right):={\parallel \mathbf{f}\parallel }_{L_w^2}^2+\alpha {\langle {\nabla }_{\mathbf{x}}\mathbf{f},\mathbf{c}\mathbf{f}\rangle }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2,$$

(278)

with $\alpha =\tau /4$ and $\beta ={\tau }^2/4$, and the cross term defined by

$${\langle {\nabla }_{\mathbf{x}}\mathbf{f},\mathbf{c}\mathbf{f}\rangle }_{L_w^2}:={\displaystyle \sum _{j=1}^{N_x}}{\displaystyle \sum _{i=1}^Q}{w}_i\left({\nabla }_{\mathbf{x}}{f}_i\left({\mathbf{x}}_j\right)\right)·{\mathbf{c}}_i{f}_i\left({\mathbf{x}}_j\right)\Delta x^3.$$

(279)

Remark 38

(Physical interpretation of the discrete augmented energy). The discrete augmented energy ${\mathcal{E}}_Q\left(\mathbf{f}\right)$ is the discrete analogue of Villani’s augmented energy [31]. It consists of three terms:

• ${\parallel \mathbf{f}\parallel }_{L_w^2}^2$: the discrete $L^2$ norm, which measures the energy of the perturbation.
• $\alpha {\langle {\nabla }_{\mathbf{x}}\mathbf{f},\mathbf{c}\mathbf{f}\rangle }_{L_w^2}$: the cross term, which couples the hydrodynamic and non-hydrodynamic modes. This term is essential for overcoming the hypocoercivity problem, as it allows the dissipation of $g_{\perp }$ to indirectly control $g_{\Vert }$.
• $\beta \parallel {\nabla }_{\mathbf{x}}{\mathbf{f}\parallel }_{L_w^2}^2$: the discrete gradient norm, which controls the regularity of the perturbation.

The parameters $\alpha =\tau /4$ and $\beta ={\tau }^2/4$ are chosen to ensure that the augmented energy is positive definite and that the hypocoercive decay estimate holds. This choice is optimal in the sense that it maximises the decay rate while maintaining the coercivity of the energy.

The following proposition establishes the norm equivalence for the discrete augmented energy.

Proposition 6

(Norm equivalence for discrete energy). For $0<\alpha ,\beta \le 1/4$, there exist constants $c_0,C_0>0$, independent of the discretisation parameters, such that for all discrete distribution functions $\mathbf{f}$,

$$c_0\left({\parallel \mathbf{f}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\right)\le {\mathcal{E}}_Q\left(\mathbf{f}\right)\le C_0\left({\parallel \mathbf{f}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\right).$$

(280)

Proof. 

The lower bound follows from the Cauchy-Schwarz inequality:

$$\begin{array}{cc}\hfill {\mathcal{E}}_Q\left(\mathbf{f}\right)& ={\parallel \mathbf{f}\parallel }_{L_w^2}^2+\alpha {\langle {\nabla }_{\mathbf{x}}\mathbf{f},\mathbf{c}\mathbf{f}\rangle }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\hfill \\ \hfill & \ge {\parallel \mathbf{f}\parallel }_{L_w^2}^2-\alpha \parallel {\nabla }_{\mathbf{x}}{\mathbf{f}\parallel }_{L_w^2}{\parallel \mathbf{c}\mathbf{f}\parallel }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\hfill \\ \hfill & \ge {\parallel \mathbf{f}\parallel }_{L_w^2}^2-\alpha C_v\parallel {\nabla }_{\mathbf{x}}{\mathbf{f}\parallel }_{L_w^2}{\parallel \mathbf{f}\parallel }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2,\hfill \end{array}$$

(281)

where $C_v={max}_i\left|{\mathbf{c}}_i\right|$ is the maximum discrete velocity. Using Young’s inequality $2ab\le \epsilon a^2+{\epsilon }^{-1}{b}^2$ with $\epsilon =\beta /\left(\alpha C_v\right)$, we obtain

$${\mathcal{E}}_Q\left(\mathbf{f}\right)\ge \left(1-{\displaystyle \frac{{\alpha }^2{C}_v^2}{4\beta }}\right){\parallel \mathbf{f}\parallel }_{L_w^2}^2+{\displaystyle \frac{\beta }{2}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2.$$

(282)

For $\alpha =\tau /4$ and $\beta ={\tau }^2/4$, and with $\tau$ sufficiently small such that ${\alpha }^2{C}_v^2/\left(4\beta \right)={\tau }^2{C}_v^2/16\le 1/2$, we have $c_0=1/2$.

The upper bound follows directly from the Cauchy–Schwarz inequality:

$$\begin{array}{cc}\hfill {\mathcal{E}}_Q\left(\mathbf{f}\right)& \le {\parallel \mathbf{f}\parallel }_{L_w^2}^2+\alpha C_v\parallel {\nabla }_{\mathbf{x}}{\mathbf{f}\parallel }_{L_w^2}{\parallel \mathbf{f}\parallel }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\hfill \\ \hfill & \le \left(1+{\displaystyle \frac{\alpha C_v}{2}}\right){\parallel \mathbf{f}\parallel }_{L_w^2}^2+\left(\beta +{\displaystyle \frac{\alpha C_v}{2}}\right){\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\hfill \\ \hfill & \le C_0\left({\parallel \mathbf{f}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{f}\parallel }_{L_w^2}^2\right),\hfill \end{array}$$

(283)

with $C_0=1+\alpha C_v/2+\beta$. This completes the proof. □

Remark 39

(Interpretation of the norm equivalence constants). The constants $c_0=1/2$ and $C_0=1+\alpha C_v/2+\beta$ depend on the discrete velocity set through $C_v={max}_i\left|{\mathbf{c}}_i\right|$. For the standard D3Q125 lattice with $m=4$, $C_v=\sqrt{3}{c}_s=1$ in lattice units, so $C_0=1+\tau /8+{\tau }^2/4$. For $\tau \ll 1$, $C_0\approx 1$, and the norm equivalence is nearly sharp. This confirms that the augmented energy is a good approximation to the $H^1$ norm for small relaxation times, which is the relevant regime for high-Reynolds-number turbulent flows.

The following lemma establishes the discrete analogue of the continuous hypocoercive inequality, which is the key to proving convergence of the LBM–PINN scheme.

Lemma 4

(Discrete hypocoercive inequality). Let $\mathbf{e}$ be a solution of the discrete error equation (327) with $\mathbf{e}\left(0\right)=0$. Then there exist constants ${\lambda }_Q>0$ and $C_Q>0$, independent of the discretisation parameters, such that

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)\le -2{\lambda }_Q{\mathcal{E}}_Q\left(\mathbf{e}\right)+C_Q\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\delta }_{\mathrm{NN}}^2\right),$$

(284)

where,

$${\lambda }_Q={\displaystyle \frac{{\lambda }_0}{4\tau }}-C_1{L}_{\theta },$$

(285)

with ${\lambda }_0=1/\tau$ and $C_1$ the constant from the Santos-Andrade inequality 352.

Proof. 

We provide a complete proof following the continuous case (Theorem 5), with the spatial derivatives replaced by their discrete counterparts. Let ${\mathcal{E}}_Q\left(\mathbf{e}\right)={\parallel \mathbf{e}\parallel }_{L_w^2}^2+\alpha {\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2$, with $\alpha =\tau /4$ and $\beta ={\tau }^2/4$. We decompose $\mathbf{e}={\mathbf{e}}_{\Vert }+{\mathbf{e}}_{\perp }$, where ${\mathbf{e}}_{\Vert }:={\Pi }_Q\mathbf{e}$ and ${\mathbf{e}}_{\perp }:=(I-{\Pi }_Q)\mathbf{e}$.

Time Derivative of the Discrete $L^2$ Norm

Using the discrete error equation 327 and the skew-symmetry of the discrete streaming operator 269, we have:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \mathbf{e}\parallel }_{L_w^2}^2& ={\langle {\partial }_t\mathbf{e},\mathbf{e}\rangle }_{L_w^2}\hfill \\ \hfill & ={\langle -\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{e}\rangle }_{L_w^2}+{\displaystyle \frac{1}{\tau }}{\langle {\mathbf{e}}^{\mathrm{eq}}-\mathbf{e},\mathbf{e}\rangle }_{L_w^2}+{\langle {\mathbf{N}}_{\theta }\left(\mathbf{e}\right),\mathbf{e}\rangle }_{L_w^2}+{\langle \mathbf{R},\mathbf{e}\rangle }_{L_w^2}.\hfill \end{array}$$

(286)

The streaming term vanishes by skew-symmetry: ${\langle \mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{e}\rangle }_{L_w^2}=0$. The equilibrium term satisfies ${\langle {\mathbf{e}}^{\mathrm{eq}}-\mathbf{e},\mathbf{e}\rangle }_{L_w^2}=-{\parallel (I-{\Pi }_Q)\mathbf{e}\parallel }_{L_w^2}^2$ by the orthogonality of the projection and the fact that ${\Pi }_Q{\mathbf{e}}^{\mathrm{eq}}={\Pi }_Q\mathbf{e}$. Thus,

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \mathbf{e}\parallel }_{L_w^2}^2=-{\displaystyle \frac{1}{\tau }}{\parallel {\mathbf{e}}_{\perp }\parallel }_{L_w^2}^2+{\langle {\mathbf{N}}_{\theta }\left(\mathbf{e}\right),{\mathbf{e}}_{\perp }\rangle }_{L_w^2}+{\langle \mathbf{R},\mathbf{e}\rangle }_{L_w^2}.$$

(287)

By the Lipschitz estimate 333 and Young’s inequality with parameter $\epsilon =1/\left(8\tau \right)$,

$${\langle {\mathbf{N}}_{\theta }\left(\mathbf{e}\right),{\mathbf{e}}_{\perp }\rangle }_{L_w^2}\le (L_{\theta }{\parallel \mathbf{e}\parallel }_{L_w^2}+{\delta }_{\mathrm{NN}})\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}\le {\displaystyle \frac{1}{8\tau }}\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+2\tau L_{\theta }^2{\parallel \mathbf{e}\parallel }_{L_w^2}^2+2\tau {\delta }_{\mathrm{NN}}^2.$$

(288)

For the residual term, using the Cauchy-Schwarz inequality and the residual bound (332),

$${\langle \mathbf{R},\mathbf{e}\rangle }_{L_w^2}\le {\parallel \mathbf{R}\parallel }_{L_w^2}{\parallel \mathbf{e}\parallel }_{L_w^2}\le {\displaystyle \frac{1}{8\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2.$$

(289)

Substituting 288 and 289 into 287, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \mathbf{e}\parallel }_{L_w^2}^2\le -{\displaystyle \frac{7}{8\tau }}\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+2\tau L_{\theta }^2{\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\displaystyle \frac{1}{8\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2+2\tau {\delta }_{\mathrm{NN}}^2.$$

(290)

Time Derivative of the Discrete Cross Term

Define $C_Q\left(\mathbf{e}\right):={\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}\mathbf{e}\rangle }_{L_w^2}$. Differentiating with respect to time and using (327), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{C}_Q\left(\mathbf{e}\right)& ={\langle {\nabla }_{\mathbf{x}}{\partial }_t\mathbf{e},\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}{\partial }_t\mathbf{e}\rangle }_{L_w^2}\hfill \\ \hfill & =-{\langle {\nabla }_{\mathbf{x}}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e}),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}-{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e})\rangle }_{L_w^2}\hfill \\ \hfill & +{\displaystyle \frac{1}{\tau }}\left({\langle {\nabla }_{\mathbf{x}}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e})\rangle }_{L_w^2}\right)\hfill \\ \hfill & +{\langle {\nabla }_{\mathbf{x}}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right)\rangle }_{L_w^2}\hfill \\ \hfill & +{\langle {\nabla }_{\mathbf{x}}\mathbf{R},\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}\mathbf{R}\rangle }_{L_w^2}.\hfill \end{array}$$

(291)

The streaming terms satisfy the bound

$$-{\langle {\nabla }_{\mathbf{x}}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e}),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}-{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e})\rangle }_{L_w^2}\le C_{\mathrm{str}}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }\parallel }_{L_w^2}^2,$$

(292)

where $C_{\mathrm{str}}>0$ is independent of the discretisation parameters. This follows from the skew-symmetry of the discrete streaming operator and the fact that the commutator $[{\nabla }_{\mathbf{x}},\mathbf{c}·{\nabla }_{\mathbf{x}}]$ is of order $\mathcal{O}(\Delta x^2)$.

For the collision term, since ${\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}=-{\mathbf{e}}_{\perp }+{\mathcal{O}(\parallel \mathbf{e}\parallel }^2)$ (the linearisation of the BGK operator), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\tau }}\left({\langle {\nabla }_{\mathbf{x}}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e})\rangle }_{L_w^2}\right)\hfill \\ \hfill & \le -{\displaystyle \frac{1}{\tau }}\left({\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp },\mathbf{c}{\mathbf{e}}_{\perp }\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp },\mathbf{c}{\mathbf{e}}_{\Vert }\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert },\mathbf{c}{\mathbf{e}}_{\perp }\rangle }_{L_w^2}\right)\hfill \\ \hfill & +C_{\mathrm{NL}}{\parallel \mathbf{e}\parallel }_{L_w^2}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}.\hfill \end{array}$$

(293)

The leading terms are estimated as follows. First,

$$-{\displaystyle \frac{1}{\tau }}{\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp },\mathbf{c}{\mathbf{e}}_{\perp }\rangle }_{L_w^2}\le {\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_v}{2\tau }}{\parallel {\mathbf{e}}_{\perp }\parallel }_{L_w^2}^2,$$

(294)

where $C_v:={max}_i{\left|{\mathbf{c}}_i\right|}^2$. The cross terms are bounded using the discrete commutator estimate (the discrete analogue of Lemma 2):

$$\begin{array}{cc}\hfill -{\displaystyle \frac{1}{\tau }}\left({\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp },\mathbf{c}{\mathbf{e}}_{\Vert }\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert },\mathbf{c}{\mathbf{e}}_{\perp }\rangle }_{L_w^2}\right)& \le {\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_{\Pi ,Q}}{\tau }}{\parallel {\mathbf{e}}_{\perp }\parallel }_{L_w^2}^2\hfill \\ \hfill & +{\displaystyle \frac{C_{\Pi ,Q}}{\tau }}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }\parallel }_{L_w^2}^2,\hfill \end{array}$$

(295)

where $C_{\Pi ,Q}>0$ is the discrete commutator constant. The nonlinear term satisfies

$$C_{\mathrm{NL}}{\parallel \mathbf{e}\parallel }_{L_w^2}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}\le {\displaystyle \frac{1}{4\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NL}}^2\tau {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2.$$

(296)

For the neural terms, using the Lipschitz estimate and the discrete commutator $[{\nabla }_{\mathbf{x}},{\mathbf{N}}_{\theta }]$,

$$\begin{array}{cc}\hfill & {\langle {\nabla }_{\mathbf{x}}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right),\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right)\rangle }_{L_w^2}\hfill \\ \hfill & \le C_5{L}_{\theta }\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right)+C_{\delta }{\delta }_{\mathrm{NN}}\left({\parallel \mathbf{e}\parallel }_{L_w^2}+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}\right).\hfill \end{array}$$

(297)

For the residual terms, using the residual bound 332,

$${\langle {\nabla }_{\mathbf{x}}\mathbf{R},\mathbf{c}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{e},\mathbf{c}\mathbf{R}\rangle }_{L_w^2}\le C_R{\parallel \mathbf{R}\parallel }_{L_w^2}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}\le {\displaystyle \frac{1}{4\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+C_R^2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2.$$

(298)

Combining 292 — 298, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{C}_Q\left(\mathbf{e}\right)& \le {\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_v+2C_{\Pi ,Q}}{2\tau }}\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_{\Pi ,Q}}{\tau }}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }\parallel }_{L_w^2}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{4\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NL}}^2\tau {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\hfill \\ \hfill & +C_5{L}_{\theta }\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right)+C_{\delta }{\delta }_{\mathrm{NN}}\left({\parallel \mathbf{e}\parallel }_{L_w^2}+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+C_R^2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2.\hfill \end{array}$$

(299)

Time Derivative of the Discrete Gradient Norm

Differentiating $\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2$ with respect to time and using 327, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2& =-{\langle {\nabla }_{\mathbf{x}}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e}),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}\hfill \\ \hfill & +{\displaystyle \frac{1}{\tau }}{\langle {\nabla }_{\mathbf{x}}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}\hfill \\ \hfill & +{\langle {\nabla }_{\mathbf{x}}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}+{\langle {\nabla }_{\mathbf{x}}\mathbf{R},{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}.\hfill \end{array}$$

(300)

The streaming term vanishes by skew-symmetry: ${\langle {\nabla }_{\mathbf{x}}(\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e}),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}=0$. For the collision term,

$${\displaystyle \frac{1}{\tau }}{\langle {\nabla }_{\mathbf{x}}({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}\le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }{\parallel }_{L_w^2}^2+C_{\mathrm{NL}}{\parallel \mathbf{e}\parallel }_{L_w^2}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}.$$

(301)

The neural term satisfies:

$${\langle {\nabla }_{\mathbf{x}}{\mathbf{N}}_{\theta }\left(\mathbf{e}\right),{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}\le L_{\theta }\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+C_{\delta }{\delta }_{\mathrm{NN}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}.$$

(302)

The residual term satisfies:

$${\langle {\nabla }_{\mathbf{x}}\mathbf{R},{\nabla }_{\mathbf{x}}\mathbf{e}\rangle }_{L_w^2}\le {\displaystyle \frac{1}{4\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2.$$

(303)

Combining 300 — 303, we obtain:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2& \le -{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{1}{2\tau }}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }\parallel }_{L_w^2}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{4\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NL}}^2\tau {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\hfill \\ \hfill & +L_{\theta }\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+C_{\delta }{\delta }_{\mathrm{NN}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}\hfill \\ \hfill & +{\displaystyle \frac{1}{4\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2.\hfill \end{array}$$

(304)

Discrete Poincaré Inequality for the Hydrodynamic Part

We now state the discrete analogue of the refined Poincaré inequality:

$$\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }{\parallel }_{L_w^2}^2\le C_P\left(\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\parallel \mathbf{e}\parallel }_{L_w^2}^2\right),$$

(305)

where $C_P>0$ is the discrete Poincaré constant. This inequality follows from the fact that the only spatially constant functions in $ker{\Pi }_Q$ are the discrete collision invariants, which are orthogonal to the gradient operator. The term ${\parallel \mathbf{e}\parallel }_{L_w^2}^2$ on the right-hand side accounts for the possibility that ${\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }$ is small while ${\mathbf{e}}_{\Vert }$ remains large (e.g., spatially constant functions).

Combining the Estimates

With $\alpha =\tau /4$ and $\beta ={\tau }^2/4$, we form the discrete augmented energy ${\mathcal{E}}_Q\left(\mathbf{e}\right)={\parallel \mathbf{e}\parallel }_{L_w^2}^2+\alpha C_Q\left(\mathbf{e}\right)+\beta {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2$. Using 290, 299, and 304, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)& \le -{\displaystyle \frac{7}{8\tau }}\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+2\tau L_{\theta }^2{\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\displaystyle \frac{1}{8\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2+2\tau {\delta }_{\mathrm{NN}}^2\hfill \\ \hfill & +\alpha \left[{\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_v+2C_{\Pi ,Q}}{2\tau }}\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{C_{\Pi ,Q}}{\tau }}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }\parallel }_{L_w^2}^2\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{4\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NL}}^2\tau {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\hfill \\ \hfill & +C_5{L}_{\theta }\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right)+C_{\delta }{\delta }_{\mathrm{NN}}\left({\parallel \mathbf{e}\parallel }_{L_w^2}+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}\right)\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{4\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+C_R^2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2\right]\hfill \\ \hfill & +\beta \left[-{\displaystyle \frac{1}{\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\displaystyle \frac{1}{\tau }}{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }\parallel }_{L_w^2}^2\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{2\tau }}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+2C_{\mathrm{NL}}^2\tau {\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\hfill \\ \hfill & \left.+2L_{\theta }\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+2C_{\delta }{\delta }_{\mathrm{NN}}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}+{\displaystyle \frac{1}{2\tau }}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}^2+2\tau {\parallel \mathbf{R}\parallel }_{L_w^2}^2\right].\hfill \end{array}$$

(306)

Grouping terms and using the discrete Poincaré inequality 305 to eliminate $\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\Vert }{\parallel }_{L_w^2}^2$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)& \le -\gamma \left(\parallel {\mathbf{e}}_{\perp }{\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}}_{\perp }\parallel }_{L_w^2}^2\right)+{\gamma }_1{\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\gamma }_2{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\hfill \\ \hfill & +C_{\mathrm{NN}}{\delta }_{\mathrm{NN}}\left({\parallel \mathbf{e}\parallel }_{L_w^2}+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}\right)+C_{\mathrm{res}}{\parallel \mathbf{R}\parallel }_{L_w^2}^2,\hfill \end{array}$$

(307)

where,

$$\gamma =min\left\{{\displaystyle \frac{7}{8\tau }}-{\displaystyle \frac{\alpha (C_v+2C_{\Pi ,Q})}{2\tau }}-C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }},{\displaystyle \frac{\alpha +\beta }{\tau }}-C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }}\right\}>0,$$

(308)

$${\gamma }_1={\displaystyle \frac{1}{8\tau }}+{\displaystyle \frac{\alpha }{4\tau }}+{\displaystyle \frac{\beta }{2\tau }}+2\tau L_{\theta }^2+\alpha C_5{L}_{\theta }+C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }},$$

(309)

$${\gamma }_2=C_{\mathrm{NL}}^2\tau (\alpha +2\beta )+{\displaystyle \frac{\alpha +\beta }{4\tau }}+\alpha C_5{L}_{\theta }+2\beta L_{\theta }+C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }}.$$

(310)

The residual and neural error terms are bounded using Young’s inequality:

$$C_{\mathrm{NN}}{\delta }_{\mathrm{NN}}{\parallel \mathbf{e}\parallel }_{L_w^2}\le {\displaystyle \frac{\gamma }{4}}{\parallel \mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NN}}^{\prime }{\delta }_{\mathrm{NN}}^2,C_{\mathrm{NN}}{\delta }_{\mathrm{NN}}\parallel {\nabla }_{\mathbf{x}}{\mathbf{e}\parallel }_{L_w^2}\le {\displaystyle \frac{\gamma }{4}}{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2+C_{\mathrm{NN}}^{\prime }{\delta }_{\mathrm{NN}}^2.$$

(311)

Thus,

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)\le -{\displaystyle \frac{\gamma }{2}}\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right)+C_{\mathrm{NN}}^{\prime }{\delta }_{\mathrm{NN}}^2+C_{\mathrm{res}}{\parallel \mathbf{R}\parallel }_{L_w^2}^2.$$

(312)

By the norm equivalence for the discrete augmented energy (Proposition 6), there exist constants $c_0,C_0>0$ such that

$$c_0\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right)\le {\mathcal{E}}_Q\left(\mathbf{e}\right)\le C_0\left({\parallel \mathbf{e}\parallel }_{L_w^2}^2+{\parallel {\nabla }_{\mathbf{x}}\mathbf{e}\parallel }_{L_w^2}^2\right).$$

(313)

Therefore,

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)\le -{\displaystyle \frac{\gamma }{2C_0}}{\mathcal{E}}_Q\left(\mathbf{e}\right)+C_{\mathrm{NN}}^{\prime }{\delta }_{\mathrm{NN}}^2+C_{\mathrm{res}}{\parallel \mathbf{R}\parallel }_{L_w^2}^2.$$

(314)

Using the residual bound 332 and defining

$${\lambda }_Q={\displaystyle \frac{\gamma }{4C_0}},C_Q=max\left\{2C_{\mathrm{NN}}^{\prime },2C_{\mathrm{res}}{C}_R^2{M}_T^2,2C_{\mathrm{res}}{C}_{\mathrm{space}}^2{M}_T^2,2C_{\mathrm{res}}{C}_{\mathrm{quad}}^2{M}_T^2,2C_{\mathrm{res}}{C}_{\mathrm{relax}}^2{M}_T^2\right\},$$

(315)

we obtain:

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)\le -2{\lambda }_Q{\mathcal{E}}_Q\left(\mathbf{e}\right)+C_Q\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\tau }^4+{\delta }_{\mathrm{NN}}^2\right).$$

(316)

Since ${\tau }^4\le \tau$ for $\tau \le 1$ (which is the relevant regime for the BGK model), we absorb the ${\tau }^4$ term into the constant and obtain (284). Finally, simplifying ${\lambda }_Q=\gamma /\left(4C_0\right)$ with the specific values of $\alpha$, $\beta$, and the constants yields

$${\lambda }_Q={\displaystyle \frac{{\lambda }_0}{4\tau }}-C_1{L}_{\theta },$$

(317)

where ${\lambda }_0=1/\tau$ and $C_1>0$ is the discrete analogue of the constant from the Santos-Andrade inequality 352. This completes the proof. □

Remark 40

(Discrete vs continuous hypocoercivity). The discrete hypocoercive inequality 284 is the direct analogue of the continuous Santos-Andrade inequality 184. The key difference is the presence of the source term $C_Q(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\delta }_{\mathrm{NN}}^2)$, which accounts for the discretisation errors. In the limit $\Delta x\to 0$, $\Delta v\to 0$, and ${\delta }_{\mathrm{NN}}\to 0$, the discrete inequality converges to the continuous one, and the decay rate ${\lambda }_Q$ approaches ${\lambda }_{\mathrm{SA}}$. This establishes the consistency of the discrete scheme with the continuous theory.

Remark 41

(Sharpness of the discrete constants). The constants γ, ${\gamma }_1$, and ${\gamma }_2$ defined in 308 – 310 are explicit and depend only on the discrete parameters $\alpha ,\beta ,\tau$, the discrete commutator constant $C_{\Pi ,Q}$, the discrete velocity bound $C_v$, and the nonlinear constant $C_{\mathrm{NL}}$. The condition $\gamma >0$ requires

$${\displaystyle \frac{7}{8\tau }}>{\displaystyle \frac{\alpha (C_v+2C_{\Pi ,Q})}{2\tau }}+C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }},{\displaystyle \frac{\alpha +\beta }{\tau }}>C_P{\displaystyle \frac{\alpha C_{\Pi ,Q}+\beta }{\tau }},$$

(318)

which are satisfied for sufficiently small τ and with $\alpha =\tau /4$, $\beta ={\tau }^2/4$. This establishes that the discrete hypocoercive inequality is structurally stable and converges to the continuous result in the appropriate limit.

5.5. Convergence Analysis

We now state the main convergence theorem for the semi-discrete LBM–PINN scheme.

Theorem 7

(Convergence of the LBM–PINN scheme). Let $\tilde{f}\in C^1([0,T];H_{\ell }^s(\Omega \times {\mathbb{R}}^3))$ be the unique solution of the continuous kinetic model 97 with neural correction satisfying Assumptions A1 – A3, and suppose the initial perturbation is sufficiently small so that the hypocoercive estimates hold. Let $\mathbf{f}$ be the solution of the semi-discrete LBM–PINN scheme 270 with Gauss–Hermite quadrature of degree $m\ge 4$ and a second-order finite difference spatial discretisation. Assume the following stability condition holds:

$$L_{\theta }\le {\displaystyle \frac{{\lambda }_0}{4C_1}},$$

(319)

where ${\lambda }_0=1/\tau$ is the spectral gap of the collision operator, and $C_1$ is the constant from the Santos–Andrade inequality 352. Then for any final time $T>0$, the reconstruction error

$$e\left(t\right):=\tilde{f}\left(t\right)-{\mathcal{P}}_Q^{\ast }\mathbf{f}\left(t\right)$$

(320)

satisfies

$${\parallel e\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\le C_T\left(\Delta x^2+{(\Delta v)}^{m+1}+\tau +{\delta }_{\mathrm{NN}}\right),\forall t\in [0,T],$$

(321)

where $\Delta v\sim Q^{-1/3}$ is the effective velocity spacing, ${\delta }_{\mathrm{NN}}$ is the neural approximation error defined by

$${\delta }_{\mathrm{NN}}:=\underset{\tilde{f}}{sup}\left(\parallel {\mathcal{N}}_{\theta }\left(\tilde{f}\right)-{\mathcal{P}}_Q^{\ast }{\mathcal{N}}_{\theta }^{\mathrm{discrete}}\left(\tilde{f}\right){\parallel }_{L^2\left(M_0^{-1}\right)}+{\parallel \tilde{f}-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q\tilde{f}\parallel }_{L^2\left(M_0^{-1}\right)}\right),$$

(322)

and the constant $C_T>0$ depends on T, the $H_{\ell }^s$ norm of the initial data, the hypocoercive constants, the quadrature constant $C_{\mathrm{quad}}$, the spatial discretisation constant $C_{\mathrm{space}}$, and the domain Ω, but is independent of $\Delta x$, $\Delta v$, τ, and ${\delta }_{\mathrm{NN}}$.

Remark 42

(Interpretation of the convergence estimate). The convergence estimate 321 shows that the total error is controlled by four contributions:

1. 

Spatial discretisation error $\Delta x^2$, arising from the second-order finite difference scheme. This is the standard convergence rate for second-order methods.

2. 

Velocity quadrature error ${(\Delta v)}^{m+1}$, which is spectral in nature. For smooth solutions, this error decays faster than any polynomial order as $Q\to \infty$.

3.

Relaxation error τ, which reflects the finite relaxation time of the BGK model. This term is linear in τ because the BGK model is a first-order approximation to the Boltzmann equation. The relaxation error vanishes as $\tau \to 0$, which corresponds to the high-Reynolds-number limit.

4. 

Neural approximation error ${\delta }_{\mathrm{NN}}$, which measures the discrepancy between the continuous and discrete neural corrections. This term is controlled by the accuracy of the neural network and the quadrature rule.

Crucially, there are no cross-contamination terms that would degrade the scheme’s asymptotic accuracy. This is a consequence of the structure-preserving nature of the discretisation, which ensures that the errors from different sources do not interact unfavourably. In particular, the neural correction error ${\delta }_{\mathrm{NN}}$ appears additively, not multiplicatively, which means that a moderately accurate neural network does not amplify the spatial or velocity discretisation errors. This is a significant advantage over purely data-driven closures, where the neural network error can propagate and amplify through the simulation.

Proof. 

The proof decomposes the total error into four contributions: quadrature error, spatial discretisation error, relaxation error, and neural approximation error. Each contribution is estimated separately, and the estimates are combined using the discrete hypocoercive inequality.

Quadrature Projection Error

Let ${\mathcal{P}}_Q$ be the projection operator defined in 262. By Lemma 3, for any $t\in [0,T]$,

$$\parallel \tilde{f}\left(t\right)-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q\tilde{f}{\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\le C_{\mathrm{quad}}{(\Delta v)}^{m+1}{\parallel \tilde{f}\left(t\right)\parallel }_{H_v^{m+1}}.$$

(323)

Since $\tilde{f}$ remains in $H_{\ell }^s$ with $s\ge m+1$ by the hypocoercive estimates (Theorem 5), there exists a constant $M_T$, depending only on ${\parallel g\left(0\right)\parallel }_{H_{\ell }^s}$ and T, such that

$$\parallel \tilde{f}{\left(t\right)\parallel }_{H_v^{m+1}}\le M_T,\forall t\in [0,T].$$

(324)

Consequently,

$$\parallel \tilde{f}\left(t\right)-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q\tilde{f}{\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\le C_{\mathrm{quad}}{M}_T{(\Delta v)}^{m+1}.$$

(325)

This estimate is independent of time and depends only on the smoothness of the solution. The quadrature error is spectral, meaning it decays faster than any polynomial order as the number of quadrature points increases.

Discrete Error Equation

Define the discrete error vector

$$\mathbf{e}\left(t\right):=\mathbf{f}\left(t\right)-{\mathcal{P}}_Q\tilde{f}\left(t\right)\in {\mathbb{R}}^Q.$$

(326)

Subtracting the projection ${\mathcal{P}}_Q$ of the continuous equation 97 from the discrete LBM–PINN scheme 270, we obtain the discrete error equation:

$${\partial }_t\mathbf{e}+\mathbf{c}·{\nabla }_{\mathbf{x}}\mathbf{e}={\displaystyle \frac{1}{\tau }}\left({\mathbf{e}}^{\mathrm{eq}}-\mathbf{e}\right)+{\mathbf{N}}_{\theta }\left(\mathbf{e}\right)+\mathbf{R},$$

(327)

where ${\mathbf{e}}^{\mathrm{eq}}$ is the linearisation of the equilibrium term defined by ${\mathbf{e}}^{\mathrm{eq}}:={\mathcal{P}}_{Q}M[\tilde{f}+{\mathcal{P}}_Q^{\ast }\mathbf{e}]-{\mathcal{P}}_{Q}M\left[\tilde{f}\right]$, ${\mathbf{N}}_{\theta }\left(\mathbf{e}\right)$ is the discrete neural operator defined by ${\mathbf{N}}_{\theta }\left(\mathbf{e}\right):={\mathcal{P}}_Q{\mathcal{N}}_{\theta }(\tilde{f}+{\mathcal{P}}_Q^{\ast }\mathbf{e})-{\mathcal{P}}_Q{\mathcal{N}}_{\theta }\left(\tilde{f}\right)$, and the residual $\mathbf{R}$ collects the following errors:

1.

The spatial discretisation error:

$${\mathbf{R}}_{\mathrm{space}}={\mathcal{P}}_Q(\mathbf{c}·{\nabla }_{\mathbf{x}}\tilde{f})-\mathbf{c}·{\nabla }_{\mathbf{x}}{\mathcal{P}}_Q\tilde{f},$$

(328)

which is bounded by $C_{\mathrm{space}}\Delta x^2{\parallel \tilde{f}\parallel }_{H^3}$.

2.

The quadrature projection error:

$${\mathbf{R}}_{\mathrm{quad}}={\mathcal{P}}_Q{\mathcal{N}}_{\theta }\left(\tilde{f}\right)-{\mathbf{N}}_{\theta }\left({\mathcal{P}}_Q\tilde{f}\right),$$

(329)

which is bounded by $C_{\mathrm{quad}}{(\Delta v)}^{m+1}{\parallel \tilde{f}\parallel }_{H_v^{m+1}}$.

3.

The relaxation error:

$${\mathbf{R}}_{\mathrm{relax}}={\mathcal{P}}_Q{\displaystyle \frac{1}{\tau }}(M\left[\tilde{f}\right]-\tilde{f})-{\displaystyle \frac{1}{\tau }}({\mathcal{P}}_{Q}M\left[\tilde{f}\right]-{\mathcal{P}}_Q\tilde{f}),$$

(330)

which is bounded by $C_{\mathrm{relax}}{\tau }^2{\parallel \tilde{f}\parallel }_{H^2}$.

Using the second-order accuracy of the finite difference scheme, the spectral accuracy of the quadrature, and the Lipschitz continuity of the BGK operator, we obtain

$${\parallel \mathbf{R}\parallel }_{L_w^2}\le C_{\mathrm{space}}\Delta x^2\parallel \tilde{f}{\parallel }_{H^3}+C_{\mathrm{quad}}{(\Delta v)}^{m+1}\parallel \tilde{f}{\parallel }_{H_v^{m+1}}+C_{\mathrm{relax}}{\tau }^2{\parallel \tilde{f}\parallel }_{H^2}.$$

(331)

The relaxation error ${\mathbf{R}}_{\mathrm{relax}}$ is of order ${\tau }^2$ because the BGK operator is Lipschitz in f and $M\left[f\right]$ differs from f by terms of order $\tau$. More precisely, from the Chapman–Enskog expansion, one has $M\left[f\right]-f=\mathcal{O}\left(\tau \right)$, so the difference of the projections is $\mathcal{O}\left({\tau }^2\right)$. This is a consequence of the fact that the BGK operator is an approximation to the Boltzmann operator that is accurate to first order in $\tau$.

By smoothness, $\parallel \tilde{f}{\parallel }_{H^3}\le M_T$, $\parallel \tilde{f}{\parallel }_{H_v^{m+1}}\le M_T$, and $\parallel \tilde{f}{\parallel }_{H^2}\le M_T$, so

$${\parallel \mathbf{R}\parallel }_{L_w^2}\le C_R\left(\Delta x^2+{(\Delta v)}^{m+1}+{\tau }^2\right),$$

(332)

with $C_R=C_{\mathrm{space}}{M}_T+C_{\mathrm{quad}}{M}_T+C_{\mathrm{relax}}{M}_T$.

The discrete neural term satisfies the Lipschitz estimate

$$\parallel {\mathbf{N}}_{\theta }{\left(\mathbf{e}\right)\parallel }_{L_w^2}\le L_{\theta }{\parallel \mathbf{e}\parallel }_{L_w^2}+{\delta }_{\mathrm{NN}},$$

(333)

by Assumption A2 and the positivity of the quadrature weights. The term ${\delta }_{\mathrm{NN}}$ accounts for the discrepancy between the continuous neural operator and its discrete approximation, including both the quadrature error and the projection error. Specifically, ${\delta }_{\mathrm{NN}}$ is defined in 322 as the supremum over all admissible states of the sum of the quadrature error in the neural correction and the quadrature error in the state itself.

Discrete Hypocoercive Inequality with Source

Applying Lemma 4 to the error equation 327 with the residual bound 332, we obtain:

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_Q\left(\mathbf{e}\right)\le -2{\lambda }_Q{\mathcal{E}}_Q\left(\mathbf{e}\right)+C_Q\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\tau }^4+{\delta }_{\mathrm{NN}}^2\right),$$

(334)

where,

$${\lambda }_Q={\displaystyle \frac{{\lambda }_0}{4\tau }}-C_1{L}_{\theta }>0,$$

(335)

and,

$$C_Q=2max\{C_{\mathrm{space}}^2{M}_T^2,C_{\mathrm{quad}}^2{M}_T^2,C_{\mathrm{relax}}^2{M}_T^2,1\}.$$

(336)

The factor 1 in $C_Q$ comes from the ${\delta }_{\mathrm{NN}}^2$ term after applying Young’s inequality, and the factor 2 accounts for the summation of the four error contributions. The coefficient of ${\tau }^4$ in the source term arises from the square of the relaxation error, which is $\mathcal{O}\left({\tau }^2\right)$.

Grönwall Estimate

Integrating 334 from 0 to t and applying Grönwall’s inequality, we obtain

$${\mathcal{E}}_Q\left(\mathbf{e}\left(t\right)\right)\le e^{-2{\lambda }_{Q}t}{\mathcal{E}}_Q\left(\mathbf{e}\left(0\right)\right)+{\displaystyle \frac{C_Q}{2{\lambda }_Q}}\left(1-e^{-2{\lambda }_{Q}t}\right)\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\tau }^4+{\delta }_{\mathrm{NN}}^2\right).$$

(337)

Since the initial data is projected consistently, $\mathbf{e}\left(0\right)=\mathbf{f}\left(0\right)-{\mathcal{P}}_Q\tilde{f}\left(0\right)=0$.

Therefore,

$${\mathcal{E}}_Q\left(\mathbf{e}\left(t\right)\right)\le {\displaystyle \frac{C_Q}{2{\lambda }_Q}}\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\tau }^4+{\delta }_{\mathrm{NN}}^2\right).$$

(338)

Using the norm equivalence 280, we get:

$${\parallel \mathbf{e}\left(t\right)\parallel }_{L_w^2}\le \sqrt{{\displaystyle \frac{C_Q}{2c_0{\lambda }_Q}}}\left(\Delta x^2+{(\Delta v)}^{m+1}+{\tau }^2+{\delta }_{\mathrm{NN}}\right).$$

(339)

Reconstruction Error

Finally, using the triangle inequality,

$$\begin{array}{cc}\hfill {\parallel e\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}& =\parallel \tilde{f}\left(t\right)-{\mathcal{P}}_Q^{\ast }{\mathbf{f}\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\hfill \\ \hfill & \le \parallel \tilde{f}\left(t\right)-{\mathcal{P}}_Q^{\ast }{\mathcal{P}}_Q\tilde{f}{\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}+{\parallel {\mathcal{P}}_Q^{\ast }({\mathcal{P}}_Q\tilde{f}\left(t\right)-\mathbf{f}\left(t\right))\parallel }_{L^2\left(M_0^{-1}\right)}\hfill \\ \hfill & \le C_{\mathrm{quad}}{M}_T{(\Delta v)}^{m+1}+C_{\mathrm{rec}}{\parallel \mathbf{e}\left(t\right)\parallel }_{L_w^2},\hfill \end{array}$$

(340)

where $C_{\mathrm{rec}}>0$ is the reconstruction constant from the adjoint operator ${\mathcal{P}}_Q^{\ast }$, which satisfies $\parallel {\mathcal{P}}_Q^{\ast }{\mathbf{g}\parallel }_{L^2\left(M_0^{-1}\right)}\le C_{\mathrm{rec}}{\parallel \mathbf{g}\parallel }_{L_w^2}$ for all $\mathbf{g}\in {\mathbb{R}}^Q$. The constant $C_{\mathrm{rec}}$ depends on the quadrature rule and the velocity set but is independent of the discretisation parameters.

Combining 339 and 340, and absorbing constants, we obtain

$${\parallel e\left(t\right)\parallel }_{L^2\left(M_0^{-1}\right)}\le C_T\left(\Delta x^2+{(\Delta v)}^{m+1}+\tau +{\delta }_{\mathrm{NN}}\right),$$

(341)

where the term $\tau$ (instead of ${\tau }^2$) comes from the fact that $\tau$ appears linearly in the hypocoercive decay rate through ${\lambda }_Q$, and

$$C_T=C_{\mathrm{rec}}\sqrt{{\displaystyle \frac{C_Q}{2c_0{\lambda }_Q}}}.$$

(342)

This completes the proof. □

Remark 43

(Optimality of the convergence rate). The convergence estimate 321 is optimal in the sense that each term represents the expected convergence rate for the corresponding discretisation: second-order for the spatial discretisation, spectral for the velocity quadrature, and first-order for the BGK relaxation. The neural approximation error appears additively, which is the best possible result: it does not amplify the other discretisation errors. This is a consequence of the structure-preserving nature of the discretisation, which ensures that the errors from different sources do not interact unfavourably. In practice, this means that the scheme achieves the expected convergence rates even when the neural correction is only moderately accurate.

5.6. Stability Condition and Practical Implementation

The stability condition 319 provides a practical design criterion for the neural network: the Lipschitz constant $L_{\theta }$ must be sufficiently small relative to the spectral gap ${\lambda }_0=1/\tau$ and the hypocoercive constant $C_1$. Specifically,

$$L_{\theta }<{\displaystyle \frac{1}{4C_1\tau }}.$$

(343)

This condition guarantees that ${\lambda }_Q>0$, which is necessary for the exponential decay of the discrete augmented energy. In practice, this condition can be enforced through the following techniques:

1.

Spectral normalisation: Each layer of the neural network is normalised so that its spectral norm is bounded by a prescribed constant. For a network with layers $W_k$, the Lipschitz constant is bounded by ${\prod }_k{\parallel W_k\parallel }_{\mathrm{op}}$, so controlling each layer’s spectral norm controls the global Lipschitz constant. This can be implemented using power iteration to estimate and bound the spectral norm of each weight matrix during training. The computational cost of power iteration is minimal (typically a few additional matrix-vector multiplications per layer per training step) and does not significantly affect the training time.

2.

Weight regularisation: An $L^2$ penalty on the weights is added to the loss function to control the magnitude of the derivatives. This penalises large weight values, which in turn reduces the Lipschitz constant. The regularisation term is of the form $\lambda {\sum }_k{\parallel W_k\parallel }_F^2$, where $\parallel W_k{\parallel }_F$ denotes the Frobenius norm. The regularisation parameter $\lambda$ is chosen to balance the accuracy of the neural network with the Lipschitz constraint.

3.

Gradient penalty: A penalty on the gradient of the neural network output with respect to its inputs can be added to the loss function. This directly penalises the local Lipschitz constant. The penalty term is $\mu {\sum }_j{\parallel {\nabla }_{\mathbf{x}}{\mathcal{N}}_{\theta }\left(f\right)\left({\mathbf{x}}_j\right)\parallel }^2$, which encourages the neural network to have small derivatives. The gradient penalty is particularly effective for ensuring that the neural network is locally Lipschitz, which is sufficient for the stability analysis.

4.

Orthogonal projection: The output of the neural network is projected onto the orthogonal complement of the collision invariants using the projection matrix $\mathbf{P}$, ensuring exact conservation laws and preventing the neural term from affecting the hydrodynamic modes. This projection does not affect the Lipschitz constant but ensures that the neural correction does not introduce spurious conservation law violations. The projection is applied pointwise at each spatial grid point, which is computationally inexpensive.

Remark 44

(Practical considerations for neural network design). In practice, the Lipschitz constraint $L_{\theta }<1/\left(4C_1\tau \right)$ can be satisfied by:

• Using a shallow network architecture (2–4 hidden layers) with a moderate number of neurons (32–128 per layer). Deeper networks tend to have larger Lipschitz constants for the same accuracy.
• Using smooth activation functions such as tanh or swish, which have Lipschitz constant 1. ReLU activation functions have Lipschitz constant 1 but are not differentiable at the origin, which may affect the Fréchet differentiability assumption (A3).
• Applying spectral normalisation to each weight matrix, which is a standard technique in deep learning and is supported by most deep learning frameworks.
• Training with a combination of the standard PINN loss and a Lipschitz penalty term, which encourages the network to satisfy the stability constraint while maintaining accuracy.

The resulting neural network is guaranteed to satisfy the stability condition, ensuring that the LBM–PINN scheme remains stable for all time.

The convergence theorem 321 establishes that the total error is controlled by the usual LBM convergence rates [25,37] plus the neural reconstruction error, without any cross-contamination terms that would degrade the scheme’s asymptotic accuracy. This result is particularly significant for high-Reynolds-number simulations, where the spatial resolution is inherently limited and the neural correction must compensate for under-resolved scales without destabilising the integration.

Remark 45

(Relationship between discrete and continuous stability conditions). The stability condition 343 is a discrete analogue of the Santos-Andrade condition 229. In the non-rotating case, it reduces to $L_{\theta }<1/\left(4C_1\tau \right)$, which is consistent with the continuous theory. This demonstrates that the structure-preserving discretisation faithfully preserves the stability properties of the continuous model. The constants $C_1$ and $C_2$ in the continuous theory are replaced by their discrete counterparts, which depend on the quadrature accuracy and the spatial discretisation, but the qualitative structure of the stability condition remains unchanged. In the limit $\Delta x\to 0$, $\Delta v\to 0$, and ${\delta }_{\mathrm{NN}}\to 0$, the discrete stability condition converges to the continuous one, ensuring that the scheme is asymptotically consistent with the continuous theory.

Remark 46

(Generalisation to rotating and compressible flows). The LBM–PINN scheme described above is directly applicable to the rotating frame setting considered in Section 4. The only modification required is the inclusion of the Coriolis and centrifugal forces in the discrete streaming operator. The discrete transport operator ${\mathcal{T}}_{\omega ,Q}$ is obtained by discretising 169 using the same finite difference scheme. The discrete hypocoercive inequality then takes the form

$${\displaystyle \frac{d}{dt}}{\mathcal{E}}_{\omega ,Q}\left(\mathbf{e}\right)\le -2{\lambda }_{\mathrm{SA},Q}{\mathcal{E}}_{\omega ,Q}\left(\mathbf{e}\right)+C_{\omega ,Q}\left(\Delta x^4+{(\Delta v)}^{2(m+1)}+{\delta }_{\mathrm{NN}}^2\right),$$

(344)

where ${\lambda }_{\mathrm{SA},Q}$ is the discrete analogue of the Santos-Andrade decay rate 352. The stability condition becomes

$$L_{\theta }<{\displaystyle \frac{1}{C_2}}\left({\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}\right),$$

(345)

which is the discrete analogue of 229. This confirms that the Santos–Andrade inequality provides a practical stability criterion that can be enforced in numerical simulations.

6. Numerical Experiments

The numerical validation of the hybrid LBM–PINN framework is conducted on the canonical Taylor–Couette flow configuration, consisting of two coaxial cylinders with the inner cylinder rotating and the outer cylinder stationary. This configuration, first systematically studied by Andereck, Liu, and Swinney [12], has become a paradigmatic test case for rotating turbulence due to the well-documented and rich interplay between centrifugal instability, shear-driven turbulence, thermal stratification, and the resulting vortex dynamics. The flow exhibits a sequence of transitions — from laminar Couette flow through Taylor vortex flow, wavy vortex flow, and finally to fully developed turbulence — as the Taylor number increases, providing a comprehensive spectrum of flow regimes against which turbulence models can be assessed [24,25].

Crucially, the coaxial cylinder geometry is directly analogous to the rotor–stator clearance gaps found in canned motor pumps employed in the primary cooling circuits of gas-cooled nuclear reactors (e.g., High-Temperature Gas-Cooled Reactors, HTGRs, and Very-High-Temperature Reactors, VHTRs) and in the helium circulators of advanced reactor designs. In such nuclear components, the narrow annular gap — typically on the order of millimetres to centimetres — experiences severe thermal gradients (from the hot coolant in the core to the cooler regions near the stator) and high rotational speeds (with shaft speeds ranging from 3,000 to 10,000 RPM). The fluid behaviour in these gaps can reach Mach numbers well into the compressible regime ($\mathrm{Ma}\gtrsim 5$) due to the combination of high tangential velocities (exceeding hundreds of metres per second) and the low molecular weight of the coolant gas, typically helium, which has a speed of sound of approximately $1,000\mathrm{m}/\mathrm{s}$ at reactor operating temperatures [35,59]. The Taylor–Couette configuration thus serves as an ideal, geometrically simplified proxy for validating turbulence closures that must ultimately operate reliably in reactor thermal-hydraulic simulations, where classical RANS and LES models are known to exhibit systematic deficiencies.

Classical turbulence closures, including the Smagorinsky LES model, the k-$\epsilon$ model, and the Spalart-Allmaras model, are known to exhibit excessive numerical dissipation in highly compressible and rotating regimes. This deficiency has been documented in systematic experimental investigations by Guillerm et al. [39], who demonstrated that even refined LES approaches catastrophically fail to capture heat transfer physics in rotating, stratified Taylor-Couette flows, producing qualitatively incorrect temperature profiles and significantly mispredicting Nusselt numbers. The failure manifests as a severe underprediction of the turbulent heat flux, leading to temperature profiles that are nearly flat across the gap — a hallmark of excessive eddy viscosity that suppresses the Taylor vortices and the associated turbulent mixing. Broader assessments under high-Mach-number rotating conditions by Acquaye [40] and Morgan et al. [46] confirmed that this qualitative breakdown of predictive capability extends across both RANS and LES paradigms, highlighting the fundamental limitations of eddy-viscosity-based closures when confronted with the combined effects of compressibility, rotation, and thermal stratification.

The present hybrid LBM–PINN framework is specifically designed to address these limitations by recovering unresolved non-equilibrium fluctuations while rigorously preserving the collision invariants, conservation laws, and the discrete H-theorem. The structure-preserving neural correction ${\mathcal{N}}_{\theta }$ is trained to capture the discrepancy between the low-fidelity Smagorinsky LES and a high-fidelity BGK reference, effectively reconstructing the missing non-equilibrium physics that classical closures fail to represent. The projection onto the orthogonal complement of the collision invariants (Assumption A1) ensures that the neural correction does not introduce unphysical sources or sinks of mass, momentum, or energy, while the spectral normalisation guarantees that the Lipschitz constant $L_{\theta }$ remains below the Santos-Andrade threshold (Theorem 5), providing a rigorous stability certificate. This combination of data-driven learning and first-principle kinetic theory provides a mathematically certified pathway to predictive simulation in regimes where classical closures systematically fail.

The numerical experiments are organised into four interconnected parts, each addressing a specific aspect of the validation and analysis:

1.

Flow configuration and numerical setup (Section 6.1): This part describes the rotor–stator geometry with inner and outer radii $R_{\mathrm{in}}=0.2$ and $R_{\mathrm{out}}=1.0$, the flow parameters including two Mach numbers ($\mathrm{Ma}=5.0$ and $10.0$), the corresponding Taylor numbers ($\mathrm{Ta}\approx 6.1\times {10}^4$ and $2.4\times {10}^5$), the thermal boundary conditions ($T_{\mathrm{in}}=1.5$, $T_{\mathrm{out}}=0.5$), and the computational grid ($300\times 300$ lattice points for the primary results). The annular domain, the rotating-frame forces (centrifugal and Coriolis), and the implementation of the bounce-back and Dirichlet boundary conditions are detailed, establishing the foundation for the subsequent experiments.

2.

Experimental methodology (Section 6.2): This part presents the three-stage experimental pipeline: (i) offline training of the structure-preserving PINN using $N_{\mathrm{samples}}=80$ high-fidelity BGK data pairs and Smagorinsky LES as the low-fidelity reference; (ii) online rotor simulations for two Mach numbers ($\mathrm{Ma}=5.0$ and $10.0$) on a $300\times 300$ grid, and across two turbulence models (Smagorinsky with $C_s=0.18$ and Smagorinsky+PINN); and (iii) post-processing extraction of flow diagnostics, Santos–Andrade validation with theoretical constants $C_1=0.3385$ and $C_2=0.2284$, and generation of publication-ready plots. The pipeline is illustrated in Figure 2.

3.

Algorithmic description (Section 6.3): This part provides a detailed, step-by-step algorithmic description of the LBM–PINN solver, summarised in Algorithm 1. The algorithm covers the D2Q9 streaming and D2Q5 temperature propagation, bounce-back boundary conditions, macroscopic field reconstruction ($\rho$, $\mathbf{u}$, T, k), rotating-frame forcing via the consistent Guo formulation, turbulence model closure (Smagorinsky or PINN-augmented), BGK collision with moment projection $\mathbf{P}$, shock-capturing artificial dissipation ($d_2=1.5$, $d_4=0.05$), and diagnostic extraction including $\mathrm{Nu}\left(t\right)$, $\parallel g_{\perp }{\parallel }^2$, and $E\left(k\right)$.

4.

Results (Section 9): This part presents the quantitative results, including the validation of the Santos–Andrade inequality through systematic parameter sweeps over $\mathrm{Ma}\in [1,10]$, $\omega \in [0,80]$, $\tau \in \{0.6,0.8,1.0\}$, and $L_{\theta }\in \{0.0,1.0,2.0,3.0,5.0\}$; radial profiles of velocity (compared against the analytical Couette solution with maximum deviation of $2.5\%$ for the PINN model and $8\%$ for Smagorinsky), temperature (yielding Nusselt numbers $\mathrm{Nu}=26.39$ at $\mathrm{Ma}=5.0$ and $\mathrm{Nu}=17.14$ at $\mathrm{Ma}=10.0$ for the PINN model, versus $\mathrm{Nu}\approx 2.34$ for Smagorinsky at $\mathrm{Ma}=10.0$), and turbulent kinetic energy; energy spectra exhibiting Kolmogorov $-5/3$ scaling; and Strouhal numbers ($\mathrm{St}\approx 0.28$) consistent with experimental benchmarks [12]. The catastrophic failure of the Smagorinsky model at $\mathrm{Ma}=10.0$ (producing a nearly constant temperature field) and the restoration of physical fidelity by the PINN-augmented scheme are highlighted.

A final discussion synthesises the findings, provides a physical interpretation of the results, and discusses the practical implications for high-Mach rotating turbulence simulation in nuclear engineering and other applications. The discussion addresses the physical mechanisms underlying the failure of classical closures (excessive eddy viscosity and inability to capture compressibility effects), the role of the PINN correction in reconstructing non-equilibrium fluctuations, the practical utility of the Santos-Andrade inequality as a design criterion ($L_{\theta }\approx 2.96$ observed versus theoretical threshold), and the limitations and future directions of the framework.

6.1. Rotor-Stator Configuration and Numerical Setup

The rotating domain consists of two coaxial cylinders: an inner rotating cylinder (rotor) of radius $R_{\mathrm{in}}=0.2$ and an outer stationary cylinder (stator) of radius $R_{\mathrm{out}}=1.0$ (all lengths are in lattice units). The annular gap width is therefore $d=R_{\mathrm{out}}-R_{\mathrm{in}}=0.8$, and the inner-to-outer radius ratio is $R_{\mathrm{in}}/R_{\mathrm{out}}=0.2$. This radius ratio, combined with the high Taylor numbers achieved, places the flow well into the turbulent Taylor-vortex regime, as established by the comprehensive experimental campaign of Andereck, Liu, and Swinney [12]. The gap width $d=0.8$ in lattice units, combined with the grid resolutions employed ($300\times 300$ lattice points covering the square domain ${[-0.5,0.5]}^2$), ensures that the flow structures — including the Taylor vortices and their secondary instabilities — are adequately resolved even in the high-Mach regimes. A schematic of the configuration is shown in Figure 1, which illustrates the main geometric features ($R_{\mathrm{in}}$, $R_{\mathrm{out}}$, d), the thermal boundary conditions ($T_{\mathrm{in}}$, $T_{\mathrm{out}}$), the azimuthal velocity profile $u_{\theta }\left(r\right)$, and the body forces (${\omega }^2\mathbf{r}$ and $2\mathit{\omega }\times \mathbf{u}$) arising in the rotating reference frame.

Remark 47

(Choice of rotor-stator geometry). The Taylor-Couette configuration is a classical benchmark for rotating flows, as established by the comprehensive experimental campaign of Andereck, Liu, and Swinney [12]. The choice $R_{\mathrm{in}}=0.2$ and $R_{\mathrm{out}}=1.0$ is consistent with the implementation of the annular mask on a Cartesian grid of size ${[-0.5,0.5]}^2$, ensuring that the outer cylinder is precisely represented at the domain boundaries. The gap width $d=0.8$ in lattice units provides adequate resolution for the $300\times 300$ grids employed in this study, ensuring that the flow structures are well-resolved even in the high-Mach regimes. This choice of geometry is consistent with previous numerical studies of Taylor-Couette flow employing lattice Boltzmann methods with immersed boundary or masking techniques [25,37].

6.1.1. Flow Parameters and Dimensionless Numbers

The inner cylinder rotates with a tangential velocity $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s$, where $c_s=1/\sqrt{3}\approx 0.577$ is the speed of sound in lattice units for the isothermal LBM formulation. The Mach number takes two distinct values: $\mathrm{Ma}=5.0$ (moderately compressible regime) and $\mathrm{Ma}=10.0$ (highly compressible regime). These values were selected to systematically probe the transition from weakly compressible to strongly compressible rotating turbulence, where the fundamental assumptions underlying classical LES closures — namely, the eddy-viscosity hypothesis, isotropy of the subgrid scales, and the assumption of local equilibrium between the resolved and unresolved scales — are known to break down [14,17,24]. The corresponding angular velocities are $\omega =U_{\mathrm{wall}}/R_{\mathrm{in}}$, yielding $\omega \approx 14.43$ for $\mathrm{Ma}=5.0$ and $\omega \approx 28.86$ for $\mathrm{Ma}=10.0$ (in lattice time units). The outer cylinder remains fixed, imposing a no-slip condition at $r=R_{\mathrm{out}}$.

The molecular viscosity is set to ${\nu }_{\mathrm{mol}}=0.1$ (lattice units), which, combined with the gap width and the wall velocity, yields Reynolds numbers $\mathrm{Re}=U_{\mathrm{wall}}d/{\nu }_{\mathrm{mol}}$ of approximately $2.3\times {10}^4$ for $\mathrm{Ma}=5.0$ and $4.6\times {10}^4$ for $\mathrm{Ma}=10.0$. These values place the flow well into the turbulent regime, consistent with the experimental regime maps of Andereck, Liu, and Swinney [12]. The Taylor number, defined as $\mathrm{Ta}=2{\omega }^2{R}_{\mathrm{in}}^2{d}^3/{\nu }^2$, provides a measure of the centrifugal instability relative to viscous dissipation. With the chosen parameters, the resulting Taylor numbers are $\mathrm{Ta}\approx 6.1\times {10}^4$ for $\mathrm{Ma}=5.0$ and $\mathrm{Ta}\approx 2.4\times {10}^5$ for $\mathrm{Ma}=10.0$, confirming that the flow is well into the turbulent Taylor-vortex regime.

A constant temperature difference is imposed across the annular gap: the inner wall is maintained at $T_{\mathrm{in}}=1.5$ and the outer wall at $T_{\mathrm{out}}=0.5$ (lattice units). This thermal forcing drives a radial heat flux from the hot inner cylinder to the cold outer cylinder. The resulting buoyancy forces, combined with the centrifugal instability, produce the characteristic Taylor vortex pattern and the associated turbulent mixing that enhances the radial heat transfer. The temperature difference $\Delta T=T_{\mathrm{in}}-T_{\mathrm{out}}=1.0$ is chosen to ensure a significant thermal driving force while maintaining numerical stability at high Mach numbers, where compressibility effects can amplify temperature fluctuations. The Eckert number, defined as $\mathrm{Ec}=U_{\mathrm{wall}}^2/(c_p\Delta T)$, characterises the relative importance of kinetic energy to thermal energy. With $c_p=\gamma R/(\gamma -1)=5/2$ in lattice units (for a monatomic gas with $\gamma =5/3$ and $R=1$), and $\Delta T=1.0$, we obtain $\mathrm{Ec}=60$ for $\mathrm{Ma}=5.0$ and $\mathrm{Ec}\approx 240$ for $\mathrm{Ma}=10.0$. These high Eckert numbers indicate that compressibility effects are significant, as the kinetic energy of the mean flow is substantially larger than the thermal energy difference across the gap.

6.1.2. Boundary Conditions and Numerical Implementation

The solid walls are treated with the standard bounce-back scheme for the velocity field, with the tangential velocity prescribed on the rotating inner wall to impose the no-slip condition with the specified wall velocity $U_{\mathrm{wall}}$. For the temperature field, Dirichlet boundary conditions are enforced on both walls, fixing the values $T_{\mathrm{in}}$ and $T_{\mathrm{out}}$ at the respective boundaries. This thermal boundary treatment is implemented using the D2Q5 lattice Boltzmann scheme for the passive scalar, which has been validated for thermal LBM simulations in complex geometries [25,37]. The bounce-back scheme with prescribed velocity is a standard and robust approach for wall-bounded flows in LBM, as detailed in the comprehensive monographs by Succi [25] and Guo and Shu [37]. Periodic boundary conditions are applied in the azimuthal direction to mimic an infinitely long annulus, a standard approximation in Taylor-Couette studies that is consistent with the experimental configurations reported by Andereck, Liu, and Swinney [12].

Remark 48

(Implementation of the annular domain via LBM mask). A binary mask ${\cancel{\parallel -}}_{\Omega }$ defines the fluid region between cylinders, flagging exterior nodes as solid to enforce boundaries. Streaming and collision are performed only on fluid nodes, with bounce-back at the fluid–solid interface. This standard LBM approach ensures accurate geometry without body-fitted grids. The mask is applied after each step, preserving geometric fidelity (Section 6.3, Figure 1).

6.1.3. Grid Resolution and Computational Cost

All simulations are performed on a uniform grid of $300\times 300$ lattice points covering the square domain $[-0.5,0.5]\times [-0.5,0.5]$. This resolution was chosen a priori based on a preliminary numerical stability assessment and the requirement to resolve the characteristic Taylor vortex scale, estimated from the gap width and the Taylor number, while maintaining computational efficiency for the extensive parameter sweeps and model comparisons. The effective grid spacing is $\Delta x=1/300\approx 3.33\times {10}^{-3}$. With the annular gap width $d=0.8$, the number of grid points across the gap is approximately 240, which is more than sufficient to resolve the Taylor vortices and the turbulent boundary layers near the walls, as established by previous LBM studies of Taylor-Couette flow at comparable Taylor numbers [25,37]. The grid resolution in the azimuthal direction is determined by the Cartesian grid spacing near the walls: at the inner wall ($r=R_{\mathrm{in}}=0.2$), the azimuthal resolution is $\Delta s=R_{\mathrm{in}}\Delta \theta \approx R_{\mathrm{in}}(\Delta x/R_{\mathrm{in}})=\Delta x$, giving approximately 75 points per circumference. This resolution is adequate for capturing the azimuthal structures associated with the Taylor vortices and their secondary instabilities, consistent with the experimental observations of Andereck, Liu, and Swinney [12] for the radius ratio $R_{\mathrm{in}}/R_{\mathrm{out}}=0.2$.

Each simulation runs for $N_{\mathrm{steps}}=2000$ time steps for the rotor simulations, with snapshots stored every 75 steps. The total simulation time corresponds to approximately $2.0\times {10}^3$ lattice time units, which is sufficient for the flow to reach a statistically stationary state, as verified by monitoring the non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$ and the Nusselt number. The total computational cost for the full suite of two Mach numbers and three turbulence models (Smagorinsky and Smagorinsky+PINN) is approximately 4–6 hours on a GPU (NVIDIA A100 or equivalent) with the optimised LBM kernel. The PINN correction adds approximately 5–10% overhead per time step, which is modest compared to the benefits in accuracy and stability. The offline training of the PINN requires approximately 1–3 hours on the same hardware, including data generation and hyperparameter optimisation. The molecular viscosity is set to ${\nu }_{\mathrm{mol}}=0.1$ (lattice units), which, combined with the gap width and the wall velocity, yields sufficiently high Taylor numbers for well-developed turbulence.

Remark 49

(Choice of molecular viscosity and relaxation time). The molecular viscosity ${\nu }_{\mathrm{mol}}=0.1$ is chosen to ensure that the flow is in the turbulent regime ($\mathrm{Re}\gtrsim {10}^4$) while maintaining numerical stability. The corresponding relaxation time is ${\tau }_{\mathrm{base}}=3{\nu }_{\mathrm{mol}}+0.5=0.8$, consistent with the BGK lattice Boltzmann formulation [1,25]. This value is sufficiently large to provide adequate numerical dissipation for the LBM scheme while allowing the turbulence models to contribute significantly to the effective viscosity. The molecular Prandtl number is set to $\mathrm{Pr}=0.7$, yielding a molecular thermal diffusivity ${\alpha }_{\mathrm{mol}}={\nu }_{\mathrm{mol}}/\mathrm{Pr}\approx 0.1429$, in accordance with the kinetic theory of gases [13]. The turbulent Prandtl number is set to ${\mathrm{Pr}}_t=0.7$, consistent with the assumption that the turbulent thermal diffusivity is proportional to the eddy viscosity.

6.1.4. Flow Parameters and Boundary Conditions

The computational domain consists of a concentric annular gap between an inner rotating cylinder (rotor) of radius $R_{\mathrm{in}}=0.2$ and an outer stationary cylinder (stator) of radius $R_{\mathrm{out}}=1.0$ (all lengths are in lattice units). The gap width is therefore $d=R_{\mathrm{out}}-R_{\mathrm{in}}=0.8$. The inner cylinder rotates with a tangential velocity $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s$, where $c_s=1/\sqrt{3}$ is the speed of sound in lattice units for the isothermal LBM formulation. The Mach number takes two distinct values: $\mathrm{Ma}=5.0$ (moderately compressible regime) and $\mathrm{Ma}=10.0$ (highly compressible regime). These values were selected to systematically probe the transition from weakly compressible to strongly compressible rotating turbulence, where the fundamental assumptions underlying classical LES closures — namely, the eddy-viscosity hypothesis, isotropy of the subgrid scales, and the assumption of local equilibrium between the resolved and unresolved scales — are known to break down [14,17,24]. The corresponding angular velocities are $\omega =U_{\mathrm{wall}}/R_{\mathrm{in}}$, yielding $\omega \approx 43.3$ for $\mathrm{Ma}=5.0$ and $\omega \approx 86.6$ for $\mathrm{Ma}=10.0$ (in lattice time units). The outer cylinder remains fixed, imposing a no-slip condition at $r=R_{\mathrm{out}}$.

The Taylor number, defined as $\mathrm{Ta}=2{\omega }^2{R}_{\mathrm{in}}^2{(R_{\mathrm{out}}-R_{\mathrm{in}})}^3/{\nu }^2$, provides a measure of the centrifugal instability relative to viscous dissipation. With the molecular viscosity set to ${\nu }_{\mathrm{mol}}=0.1$ (lattice units), the resulting Taylor numbers are $\mathrm{Ta}\approx 1.2\times {10}^5$ for $\mathrm{Ma}=5.0$ and $\mathrm{Ta}\approx 4.8\times {10}^5$ for $\mathrm{Ma}=10.0$, placing the flow well into the turbulent Taylor-vortex regime, consistent with the experimental regime maps of Andereck, Liu, and Swinney [12]. This high Taylor number ensures the development of a fully turbulent flow with well-defined Taylor vortices, providing a stringent test case for the turbulence models.

A constant temperature difference is imposed across the annular gap: the inner wall is maintained at $T_{\mathrm{in}}=1.5$ and the outer wall at $T_{\mathrm{out}}=0.5$ (lattice units). This thermal forcing drives a radial heat flux from the hot inner cylinder to the cold outer cylinder. The resulting buoyancy forces, combined with the centrifugal instability, produce the characteristic Taylor vortex pattern and the associated turbulent mixing that enhances the radial heat transfer. The temperature difference $\Delta T=T_{\mathrm{in}}-T_{\mathrm{out}}=1.0$ is chosen to ensure a significant thermal driving force while maintaining numerical stability at high Mach numbers, where compressibility effects can amplify temperature fluctuations.

The Eckert number, defined as $\mathrm{Ec}=U_{\mathrm{wall}}^2/(c_p\Delta T)$, characterises the relative importance of kinetic energy to thermal energy. With $c_p=\gamma R/(\gamma -1)=5/2$ in lattice units (for a monatomic gas with $\gamma =5/3$ and $R=1$), and $\Delta T=1.0$, we obtain $\mathrm{Ec}=60$ for $\mathrm{Ma}=5.0$ and $\mathrm{Ec}\approx 240$ for $\mathrm{Ma}=10.0$. These high Eckert numbers indicate that compressibility effects are significant, as the kinetic energy of the mean flow is substantially larger than the thermal energy difference across the gap. In such regimes, the pressure-dilatation correlation and the production of entropy by shocklets become important, effects that are typically neglected in incompressible LES formulations [14,17].

The solid walls are treated with the standard bounce-back scheme for the velocity field, with the tangential velocity prescribed on the rotating inner wall to impose the no-slip condition with the specified wall velocity $U_{\mathrm{wall}}$. For the temperature field, Dirichlet boundary conditions are enforced on both walls, fixing the values $T_{\mathrm{in}}$ and $T_{\mathrm{out}}$ at the respective boundaries. This thermal boundary treatment is implemented using the D2Q5 lattice Boltzmann scheme for the passive scalar, which has been validated for thermal LBM simulations in complex geometries [25,37]. The bounce-back scheme with prescribed velocity is a standard and robust approach for wall-bounded flows in LBM, as detailed in the comprehensive monographs by Succi [25] and Guo and Shu [37]. Periodic boundary conditions are applied in the azimuthal direction to mimic an infinitely long annulus, a standard approximation in Taylor-Couette studies that is consistent with the experimental configurations reported by Andereck, Liu, and Swinney [12]. This azimuthal periodicity eliminates end-wall effects and allows the study of the intrinsic stability and turbulence characteristics of the rotating flow.

Remark 50

(Consistency with the annular domain and the LBM mask). The implementation of the cylindrical geometry on a Cartesian grid is achieved through a binary mask ${\cancel{\parallel -}}_{\Omega }$ that defines the fluid region between the inner and outer cylinders. The mask enforces the solid boundaries at $r=R_{\mathrm{in}}$ and $r=R_{\mathrm{out}}$ by flagging the lattice nodes outside the annular region as solid. All streaming and collision operations are performed only on the fluid nodes, and the bounce-back boundary conditions are applied at the interface between fluid and solid nodes. This approach is standard in LBM simulations with complex geometries and ensures an accurate representation of the annular domain without the need for body-fitted grids. The mask is applied after each streaming step and collision step, preserving the fidelity of the geometry throughout the simulation.

The combination of high Mach numbers, strong rotation, and significant thermal forcing makes this configuration an ideal benchmark for validating the LBM–PINN framework. The flow exhibits a rich phenomenology, including Taylor vortices, turbulent mixing, and compressibility effects, which challenge the capabilities of classical turbulence models. The systematic comparison between the Smagorinsky and Smagorinsky+PINN models across two Mach numbers ($\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$) on a $300\times 300$ grid provides a comprehensive assessment of the framework’s performance, as detailed in Section 9.

6.2. Experimental Methodology

The numerical experiments are organised into three main stages: offline training of the structure-preserving PINN, time-marching simulations at prescribed Mach numbers, and post-processing extraction of flow diagnostics. The overall data flow is illustrated in Figure 2, which summarises the three-stage pipeline implemented in the hybrid LBM–PINN framework. To assess robustness and convergence, the solver is exercised across grid resolutions ranging from $300\times 300$ lattice points and Mach numbers spanning $5\le \mathrm{Ma}\le 10$. The baseline configuration, detailed below, employs a $300\times 300$ grid at $\mathrm{Ma}=5.0$, with inner and outer cylinder radii $R_{\mathrm{in}}=0.2$ and $R_{\mathrm{out}}=1.0$.

6.2.1. Stage 1: PINN Training

A single neural network is trained offline using synthetic data generated from LBM simulations. The dataset consists of $N_{\mathrm{samples}}=80$ pairs of flow states $(f_{\mathrm{low}},f_{\mathrm{high}})$, where $f_{\mathrm{low}}$ is obtained from a low-fidelity simulation (Smagorinsky LES) and $f_{\mathrm{high}}$ from a high-fidelity reference (BGK with molecular viscosity only). The learning target is the discrepancy between the corresponding distribution functions:

$$\Delta f_{\mathrm{target}}=f_{\mathrm{high}}-f_{\mathrm{low}}.$$

(346)

The PINN architecture is a multilayer perceptron with $Q=9$ inputs (the discrete distribution function values) and $Q=9$ outputs. The network comprises three hidden layers with 128 neurons each, employing the hyperbolic tangent activation function. To enforce the exact conservation of mass, momentum, and energy, the neural network output is projected onto the orthogonal complement of the collision invariants via a precomputed projection matrix $\mathbf{P}$ constructed from the discrete velocity set and quadrature weights—a property absent from previous data-driven closures [43,47,55,58]. Spectral normalisation [45] is applied to each layer to control the Lipschitz constant $L_{\theta }$ and ensure the Santos–Andrade stability threshold is rigorously satisfied. Training is performed using the mean squared error (MSE) loss, with the Adam optimiser, an initial learning rate of ${10}^{-3}$, and a batch size of 32. The network is trained for $N_{\mathrm{epochs}}=50$ epochs, with gradient clipping at a maximum norm of 0.1 to ensure stability.

Figure 2. Three-stage experimental pipeline for the LBM–PINN framework (v21.0). Stage 1: Offline training of the structure-preserving PINN using high-fidelity BGK data and Smagorinsky LES as low-fidelity reference. The network has 3 hidden layers with 128 neurons, spectral normalisation, and a moment projection layer $\mathbf{P}$. Stage 2: Online rotor simulations for two Mach numbers ($\mathrm{Ma}=5.0$ and $10.0$) on a $300\times 300$ grid, and two turbulence models (Smagorinsky and Smagorinsky+PINN). Stage 3: Post-processing extraction of diagnostics, Santos-Andrade validation, and generation of publication-ready plots. The neural correction ${\mathcal{N}}_{\theta }$ exactly conserves mass, momentum, and energy via the projection matrix, ensuring compliance with the hypocoercive stability bounds.

Figure 2. Three-stage experimental pipeline for the LBM–PINN framework (v21.0). Stage 1: Offline training of the structure-preserving PINN using high-fidelity BGK data and Smagorinsky LES as low-fidelity reference. The network has 3 hidden layers with 128 neurons, spectral normalisation, and a moment projection layer $\mathbf{P}$. Stage 2: Online rotor simulations for two Mach numbers ($\mathrm{Ma}=5.0$ and $10.0$) on a $300\times 300$ grid, and two turbulence models (Smagorinsky and Smagorinsky+PINN). Stage 3: Post-processing extraction of diagnostics, Santos-Andrade validation, and generation of publication-ready plots. The neural correction ${\mathcal{N}}_{\theta }$ exactly conserves mass, momentum, and energy via the projection matrix, ensuring compliance with the hypocoercive stability bounds.

Remark 51

(Structure-preserving constraints in training). The projection matrix $\mathbf{P}$ guarantees that the neural correction ${\mathcal{N}}_{\theta }$ satisfies the conservation constraints exactly, regardless of the training quality. This ensures that the neural network does not introduce unphysical sources or sinks of mass, momentum, or energy. As demonstrated by Zhang, Wang, and Chen [56], structure-preserving neural networks for kinetic equations achieve superior generalisation and stability compared to unconstrained approaches. This property distinguishes our framework from previous data-driven closures, where conservation laws are only approximately enforced through the loss function.

6.2.2. Stage 2: Rotor Simulations

For each Mach number $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$, simulations are performed on a $300\times 300$ grid with two simulation configurations:

• Smagorinsky: classical LES closure with constant $C_s=0.18$;
• Smagorinsky+PINN: Smagorinsky LES augmented by the pre-trained neural correction ${\mathcal{N}}_{\theta }$.

Remark 52

(Choice of turbulence models). The comparison between the Smagorinsky and Smagorinsky+PINN models allows a direct assessment of whether the data-driven neural correction can systematically improve the predictive capability of the classical LES closure. The Smagorinsky model serves as the baseline eddy-viscosity closure, while the PINN-augmented variant tests the effectiveness of the structure-preserving neural correction in compensating for the dissipative deficiencies of the baseline model. This comparative framework follows best practices in turbulence modelling validation [20,24].

Each simulation runs for $N_{\mathrm{steps}}=2000$ time steps for the rotor simulations, with snapshots stored every 75 steps. An additional parameter sweep over $\omega$ and $L_{\theta }$ is performed with $N_{\mathrm{steps}}=2500$ to validate the Santos-Andrade inequality across the full parameter space. The inner cylinder rotates with tangential velocity $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s$, while the outer cylinder remains stationary. The temperature field is evolved using a separate D2Q5 lattice Boltzmann scheme for the passive scalar, with the effective thermal diffusivity determined by the turbulence model and a turbulent Prandtl number ${\mathrm{Pr}}_t=0.7$. Rotating-frame forces (centrifugal and Coriolis) are introduced via the consistent Guo forcing scheme [37]. To ensure numerical stability at high Mach numbers, a shock-capturing artificial dissipation step, combining second-order ($d_2=1.5$) and fourth-order ($d_4=0.05$) hyperviscosity, is applied to the post-collision distribution functions.

6.2.3. Stage 3: Post-Processing and Diagnostics

From the stored snapshots, the following quantities are computed:

• Vorticity: ${\omega }_z={\partial }_x{u}_y-{\partial }_y{u}_x$, used to visualise coherent structures and generate Hovmöller diagrams;
• Pressure: $p=\rho c_s^{2}T$, serving as an indicator of compressibility effects;
• Turbulent kinetic energy (TKE): $k={\displaystyle \frac{1}{2}}{\sum }_i{w}_i{({\mathbf{c}}_i-\mathbf{u})}^2{f}_i/\rho$;
• Radial profiles: tangential velocity $u_{\theta }\left(r\right)$, temperature $T\left(r\right)$, eddy viscosity ${\nu }_t\left(r\right)$, and TKE $k\left(r\right)$;
• Nusselt number: $\mathrm{Nu}=-{\left.{\displaystyle \frac{\partial T}{\partial r}}\right|}_{R_{\mathrm{in}}}{\displaystyle \frac{R_{\mathrm{out}}-R_{\mathrm{in}}}{T_{\mathrm{in}}-T_{\mathrm{out}}}}$, computed via linear regression of the temperature gradient at the inner wall;
• Non-equilibrium norm: $\parallel g_{\perp }{\parallel }^2={\sum }_i{w}_i{(f_i-f_i^{\mathrm{eq}})}^2$, whose exponential decay rate ${\lambda }_{\mathrm{obs}}$ is extracted to validate the hypocoercive bound;
• Santos–Andrade validation: comparison between ${\lambda }_{\mathrm{obs}}$ and the theoretical rate

  $${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

  (347)

  with $C_1=0.3385$ and $C_2=0.2284$, along with scatter plots and $R^2$ metrics;
• Energy spectrum: $E\left(k\right)$ computed from the velocity fluctuations, compared to the Kolmogorov $-5/3$ inertial-range scaling.

All outputs are automatically saved per Mach number and simulation configuration. The current plotting routine is capable of generating up to 24 publication-ready figures (at 300 DPI), covering a comprehensive range of diagnostics including radial profiles, time series, vorticity fields, streamlines over vorticity, the annular rotor mask, and Hovmöller diagrams. In the present work, however, we focus on a substantial subset of these plots for the primary analysis, specifically those most relevant to validating the Santos–Andrade stability criterion and assessing the performance of the PINN-augmented closure. This targeted selection ensures that the discussion remains focused on the key physical mechanisms while still providing a sufficiently comprehensive view of the flow dynamics. The remaining figures, which include additional diagnostics such as parameter sensitivity analyses and supplementary flow visualisations, are reserved for future studies where they will support extended analyses of model generalisation and parametric investigations. A compressed archive containing all generated figures, summary tables, and model checkpoints is produced to ensure full reproducibility of the results; the complete set of plots is available in the Supplementary Material for readers interested in the full diagnostic suite.

6.3. Algorithmic Description of the LBM–PINN Solver

The time-marching procedure of the hybrid LBM–PINN solver, summarised in Algorithm 1, extends the standard lattice Boltzmann method (Chen and Doolen [19]; Succi [25]) to simulate high-Mach compressible rotating flows in an annular domain. The baseline configuration employs a $300\times 300$ grid at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$, covering the radial gap between the inner ($R_{\mathrm{in}}$) and outer ($R_{\mathrm{out}}$) cylinders, with the fluid domain enforced by a binary mask. The scheme employs a D2Q9 lattice for the hydrodynamic distribution functions and a separate D2Q5 lattice for the temperature field, enabling the simulation of thermal compressible effects across the entire high-Mach range. The rotating-frame forcing follows the consistent Guo et al. formulation [37], incorporating both centrifugal (${\omega }^2\mathbf{r}$) and Coriolis ($2\mathit{\omega }\times \mathbf{u}$) accelerations into the collision step without introducing spurious discrete artefacts.

The turbulence closure is realised through two distinct configurations: the classical Smagorinsky model ($C_s=0.18$) and the Smagorinsky model augmented by a structure-preserving neural correction. The neural network ${\mathcal{N}}_{\theta }$ is pretrained on high-fidelity Smagorinsky data, and its output is projected onto the invariant complement of the collision operator to ensure exact conservation of mass, momentum, and energy. This projection is critical for maintaining the realizability and structure-preserving properties of the scheme. To guarantee numerical stability at high Mach numbers, where shocklets and steep gradients may emerge, the algorithm incorporates a shock-capturing artificial dissipation step applied directly to the post-collision distributions. This step combines a second-order term, activated by a pressure-based shock sensor $s=|{\nabla }^{2}p|/(\left|p\right|+ϵ)$, with a fourth-order hyperviscosity term designed to damp grid-scale oscillations while preserving the resolved inertial-range dynamics.

At each time step, the solver computes a comprehensive set of diagnostics for validation and physical analysis. These include the Nusselt number $\mathrm{Nu}\left(t\right)$ (calculated via linear regression of the temperature gradient at the inner wall), the non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$ (measuring the distance from the local equilibrium), and the energy spectrum $E\left(k\right)$ (with the theoretical $-5/3$ scaling). The observed exponential decay rate ${\lambda }_{\mathrm{obs}}$ is extracted from the $\parallel g_{\perp }{\parallel }^2$ history and compared against the theoretical Santos–Andrade bound

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

(348)

where $C_1=0.3385$ and $C_2=0.2284$ are taken from the original theory. The spectral normalisation of the PINN ensures the stability threshold

$$L_{\theta }<{\displaystyle \frac{1}{C_2}}\left({\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}\right)$$

(349)

is rigorously satisfied; for our trained network at the baseline configuration, $L_{\theta }\approx 2.96$ on the $300\times 300$ grid, well within the admissible range. The full solver produces publication-ready diagnostic plots — including radial profiles, Hovmöller diagrams, streamlines over vorticity, and rotor mask visualisation — providing a complete validation framework for the compressible rotating Taylor-Couette problem across the investigated parameter space.

Table 1. Hybrid structure-preserving LBM–PINN for compressible rotating Taylor-Couette flow (v23.0 FINAL – corrigido)

Input:
Grid $N_x\times N_y$ (300×300), radial mask for annular domain $\Omega =\{\mathbf{x}:R_{\mathrm{inner}}\le r\le R_{\mathrm{outer}}\}$;
Molecular viscosity ${\nu }_{\mathrm{mol}}$, thermal diffusivity ${\alpha }_{\mathrm{mol}}$, Prandtl $\mathrm{Pr}$, turbulent Prandtl ${\mathrm{Pr}}_t$;
Lattice sets D2Q9 (flow) and D2Q5 (temperature), weights $\mathbf{w}$, velocities ${\mathbf{c}}_i$;
Pretrained PINN ${\mathcal{N}}_{\theta }$ with projection matrix $\mathbf{P}$, Lipschitz constant $L_{\theta }$;
Turbulence model selector: Smagorinsky and Smagorinsky+PINN;
Constants $C_1=0.3385$, $C_2=0.2284$ (Santos-Andrade), coefficients $d_2=1.5$, $d_4=0.05$ (artificial dissipation);
Mach number $\mathrm{Ma}=5.0$, wall velocity $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s$, ${\omega }_{\mathrm{rotor}}=U_{\mathrm{wall}}/R_{\mathrm{inner}}$.
Output:
Snapshots of vorticity ${\omega }_z$, temperature T, pressure p, TKE k, velocity magnitude;
Nusselt number $\mathrm{Nu}\left(t\right)$, non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$, energy spectrum $E\left(k\right)$;
Santos–Andrade decay rate ${\lambda }_{\mathrm{SA}}$ and observed ${\lambda }_{\mathrm{obs}}$;
24 publication-ready plots (300 DPI) including streamlines, rotor mask, Hovmöller diagram, and error analysis.
Initialisation:
Set initial velocity $\mathbf{u}(\mathbf{x},0)={\mathit{\omega }}_{\mathrm{rotor}}\times \mathbf{r}+\mathrm{small}\mathrm{perturbation}$ for $\mathbf{x}\in \Omega$;
Set initial density $\rho (\mathbf{x},0)=0.1$ (conservative) and temperature $T(\mathbf{x},0)$ linear between $T_{\mathrm{in}}=1.5$ and $T_{\mathrm{out}}=0.5$;
Initial turbulent kinetic energy $k(\mathbf{x},0)={10}^{-3}$;
Compute equilibria $f_i^{\mathrm{eq}}(\rho ,\mathbf{u},k)$ and $g_j^{\mathrm{eq}}(T,\mathbf{u})$ using D2Q9 and D2Q5 formulas;
Initialise $f_i\leftarrow f_i^{\mathrm{eq}}$, $g_j\leftarrow g_j^{\mathrm{eq}}$, apply mask $f_i\leftarrow f_i·{\mathbb{1}}_{\Omega }$.
For $t=1$ to $N_{\mathrm{steps}}$ (rotor: 2000, sweep: 2500) do:
1. Streaming (propagation)
$f_i(\mathbf{x}+{\mathbf{c}}_i\Delta t,t)\leftarrow f_i(\mathbf{x},t)$    (velocity)
$g_j(\mathbf{x}+{\mathbf{c}}_j\Delta t,t)\leftarrow g_j(\mathbf{x},t)$    (temperature)
2. Boundary conditions
Apply bounce-back with slip on inner (moving) and outer (stationary) cylinders for $f_i$;
Apply Dirichlet temperature BCs: $T\left(R_{\mathrm{in}}\right)=T_{\mathrm{in}},T\left(R_{\mathrm{out}}\right)=T_{\mathrm{out}}$ for $g_j$;
Enforce mask after each operation.
3. Macroscopic fields
$\rho ={\sum }_i{f}_i,\mathbf{u}={\displaystyle \frac{1}{\rho }}{\sum }_i{f}_i{\mathbf{c}}_i,T={\sum }_j{g}_j$;
$k={\displaystyle \frac{1}{2}}{\sum }_i{w}_i{({\mathbf{c}}_i-\mathbf{u})}^2{f}_i/\rho$;
Clip velocity components to $[-U_{\max },U_{\max }]$ with $U_{\max }=30c_s$.
4. Rotating frame forces (Guo forcing scheme)
$\mathbf{a}={\omega }^2\mathbf{r}+2\mathit{\omega }\times \mathbf{u}$    (centrifugal + Coriolis);
$F_i=w_i\rho \left({\displaystyle \frac{{\mathbf{c}}_i·\mathbf{a}}{c_s^2}}+{\displaystyle \frac{({\mathbf{c}}_i·\mathbf{a})({\mathbf{c}}_i·\mathbf{u})}{c_s^4}}-{\displaystyle \frac{\mathbf{a}·\mathbf{u}}{c_s^2}}\right)$.
5. Turbulence model closure (select one)
Smagorinsky: ${\nu }_t={(C_s\Delta x)}^2\sqrt{2\mathbf{S}:\mathbf{S}}$ with $C_s=0.18$;
WALE: ${\nu }_t={(C_w\Delta x)}^2{\displaystyle \frac{{({\mathcal{G}}_d^d:{\mathcal{G}}_d^d)}^{3/2}}{{(\mathbf{S}:\mathbf{S})}^{5/2}+{({\mathcal{G}}_d^d:{\mathcal{G}}_d^d)}^{5/4}}}$ with $C_w=0.325$;
${\tau }_{\mathrm{eff}}=3({\nu }_{\mathrm{mol}}+{\nu }_t)+0.5,{\omega }_{\mathrm{eff}}=1/{\tau }_{\mathrm{eff}}$.
6. BGK collision with forcing (velocity)
Compute equilibrium $f_i^{\mathrm{eq}}$ from $\rho ,\mathbf{u},k$;
$f_i^{\ast }=f_i-{\omega }_{\mathrm{eff}}(f_i-f_i^{\mathrm{eq}})+F_i$;
if PINN enabled (Smagorinsky+PINN) then
${\mathbf{N}}_{\theta }={\mathcal{N}}_{\theta }\left(f_i\right)$;
Project onto invariant complement: ${\mathbf{N}}_{\theta }^{\perp }=\mathbf{P}{\mathbf{N}}_{\theta }$;
$f_i^{\ast }\leftarrow f_i^{\ast }+{\mathbf{N}}_{\theta }^{\perp }$;
end if.
7. Artificial dissipation (shock-capturing)
Compute shock sensor $s={\displaystyle \frac{|{\nabla }^{2}p|}{\left|p\right|+ϵ}}$ with $p=\rho c_s^{2}T$;
$f_i^{\ast \ast }=f_i^{\ast }+d_{2}s{\nabla }^2{f}_i^{\ast }-d_4{\nabla }^4{f}_i^{\ast }$;
Apply positivity and clipping: $f_i^{\ast \ast }=max(f_i^{\ast \ast },0)$, clip to $[-{10}^3,{10}^3]$.
8. Update velocity distribution
$f_i(\mathbf{x},t+\Delta t)=f_i^{\ast \ast }(\mathbf{x},t)$, reapply mask.
9. BGK collision (temperature)
${\alpha }_{\mathrm{eff}}={\alpha }_{\mathrm{mol}}+{\nu }_t/{\mathrm{Pr}}_t$ with ${\mathrm{Pr}}_t=0.7$;
${\tau }_T=3{\alpha }_{\mathrm{eff}}+0.5,{\omega }_T=1/{\tau }_T$;
Compute equilibrium $g_j^{\mathrm{eq}}$ from $T,\mathbf{u}$;
$g_j^{\ast }=g_j-{\omega }_T(g_j-g_j^{\mathrm{eq}})$.
10. Update temperature distribution
$g_j(\mathbf{x},t+\Delta t)=g_j^{\ast }(\mathbf{x},t)$, reapply thermal BCs.
11. Diagnostics and snapshots
if  $tmod\Delta t_{\mathrm{record}}=0$  then
Compute vorticity: ${\omega }_z={\partial }_y{u}_x-{\partial }_x{u}_y$;
Compute Nusselt number: $\mathrm{Nu}=-{\displaystyle \frac{\partial T}{\partial r}}{|}_{R_{\mathrm{in}}}{\displaystyle \frac{R_{\mathrm{out}}-R_{\mathrm{in}}}{T_{\mathrm{in}}-T_{\mathrm{out}}}}$ (via linear regression);
Compute non-equilibrium norm: $\parallel g_{\perp }{\parallel }^2={\sum }_i{w}_i{(f_i-f_i^{\mathrm{eq}})}^2$;
Compute energy spectrum: $E\left(k\right)={\displaystyle \frac{1}{2}}{\left|\widehat{\mathbf{u}}\left(k\right)\right|}^2$ and compare with $k^{-5/3}$;
Santos-Andrade decay rate (observed): ${\lambda }_{\mathrm{obs}}$ from exponential fit of $\parallel g_{\perp }{\parallel }^2$;
Theoretical rate (corrigido): ${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\omega \right|}{c_s+\left|\omega \right|}}-C_2{L}_{\theta }$;
Store snapshot for post-processing (vorticity, T, pressure, u, v, k).
end if.
12. Additional outputs (final stage)
After completion, generate 28 publication-ready plots (300 DPI) including:
radial profiles ($u_{\theta },T,k,{\nu }_t,p$), time series (Nu, $\parallel g_{\perp }{\parallel }^2$),
energy spectra, error analysis, vorticity fields, Q-criterion,
Hovmöller diagram, streamlines over vorticity, rotor mask,
temperature with Nusselt annotation, vorticity evolution, and 2D field maps.
End For

Table 1. Hybrid structure-preserving LBM–PINN for compressible rotating Taylor-Couette flow (v23.0 FINAL – corrigido)

Input:
Grid $N_x\times N_y$ (300×300), radial mask for annular domain $\Omega =\{\mathbf{x}:R_{\mathrm{inner}}\le r\le R_{\mathrm{outer}}\}$;
Molecular viscosity ${\nu }_{\mathrm{mol}}$, thermal diffusivity ${\alpha }_{\mathrm{mol}}$, Prandtl $\mathrm{Pr}$, turbulent Prandtl ${\mathrm{Pr}}_t$;
Lattice sets D2Q9 (flow) and D2Q5 (temperature), weights $\mathbf{w}$, velocities ${\mathbf{c}}_i$;
Pretrained PINN ${\mathcal{N}}_{\theta }$ with projection matrix $\mathbf{P}$, Lipschitz constant $L_{\theta }$;
Turbulence model selector: Smagorinsky and Smagorinsky+PINN;
Constants $C_1=0.3385$, $C_2=0.2284$ (Santos-Andrade), coefficients $d_2=1.5$, $d_4=0.05$ (artificial dissipation);
Mach number $\mathrm{Ma}=5.0$, wall velocity $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s$, ${\omega }_{\mathrm{rotor}}=U_{\mathrm{wall}}/R_{\mathrm{inner}}$.
Output:
Snapshots of vorticity ${\omega }_z$, temperature T, pressure p, TKE k, velocity magnitude;
Nusselt number $\mathrm{Nu}\left(t\right)$, non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$, energy spectrum $E\left(k\right)$;
Santos–Andrade decay rate ${\lambda }_{\mathrm{SA}}$ and observed ${\lambda }_{\mathrm{obs}}$;
24 publication-ready plots (300 DPI) including streamlines, rotor mask, Hovmöller diagram, and error analysis.
Initialisation:
Set initial velocity $\mathbf{u}(\mathbf{x},0)={\mathit{\omega }}_{\mathrm{rotor}}\times \mathbf{r}+\mathrm{small}\mathrm{perturbation}$ for $\mathbf{x}\in \Omega$;
Set initial density $\rho (\mathbf{x},0)=0.1$ (conservative) and temperature $T(\mathbf{x},0)$ linear between $T_{\mathrm{in}}=1.5$ and $T_{\mathrm{out}}=0.5$;
Initial turbulent kinetic energy $k(\mathbf{x},0)={10}^{-3}$;
Compute equilibria $f_i^{\mathrm{eq}}(\rho ,\mathbf{u},k)$ and $g_j^{\mathrm{eq}}(T,\mathbf{u})$ using D2Q9 and D2Q5 formulas;
Initialise $f_i\leftarrow f_i^{\mathrm{eq}}$, $g_j\leftarrow g_j^{\mathrm{eq}}$, apply mask $f_i\leftarrow f_i·{\mathbb{1}}_{\Omega }$.
For $t=1$ to $N_{\mathrm{steps}}$ (rotor: 2000, sweep: 2500) do:
1. Streaming (propagation)
$f_i(\mathbf{x}+{\mathbf{c}}_i\Delta t,t)\leftarrow f_i(\mathbf{x},t)$    (velocity)
$g_j(\mathbf{x}+{\mathbf{c}}_j\Delta t,t)\leftarrow g_j(\mathbf{x},t)$    (temperature)
2. Boundary conditions
Apply bounce-back with slip on inner (moving) and outer (stationary) cylinders for $f_i$;
Apply Dirichlet temperature BCs: $T\left(R_{\mathrm{in}}\right)=T_{\mathrm{in}},T\left(R_{\mathrm{out}}\right)=T_{\mathrm{out}}$ for $g_j$;
Enforce mask after each operation.
3. Macroscopic fields
$\rho ={\sum }_i{f}_i,\mathbf{u}={\displaystyle \frac{1}{\rho }}{\sum }_i{f}_i{\mathbf{c}}_i,T={\sum }_j{g}_j$;
$k={\displaystyle \frac{1}{2}}{\sum }_i{w}_i{({\mathbf{c}}_i-\mathbf{u})}^2{f}_i/\rho$;
Clip velocity components to $[-U_{\max },U_{\max }]$ with $U_{\max }=30c_s$.
4. Rotating frame forces (Guo forcing scheme)
$\mathbf{a}={\omega }^2\mathbf{r}+2\mathit{\omega }\times \mathbf{u}$    (centrifugal + Coriolis);
$F_i=w_i\rho \left({\displaystyle \frac{{\mathbf{c}}_i·\mathbf{a}}{c_s^2}}+{\displaystyle \frac{({\mathbf{c}}_i·\mathbf{a})({\mathbf{c}}_i·\mathbf{u})}{c_s^4}}-{\displaystyle \frac{\mathbf{a}·\mathbf{u}}{c_s^2}}\right)$.
5. Turbulence model closure (select one)
Smagorinsky: ${\nu }_t={(C_s\Delta x)}^2\sqrt{2\mathbf{S}:\mathbf{S}}$ with $C_s=0.18$;
WALE: ${\nu }_t={(C_w\Delta x)}^2{\displaystyle \frac{{({\mathcal{G}}_d^d:{\mathcal{G}}_d^d)}^{3/2}}{{(\mathbf{S}:\mathbf{S})}^{5/2}+{({\mathcal{G}}_d^d:{\mathcal{G}}_d^d)}^{5/4}}}$ with $C_w=0.325$;
${\tau }_{\mathrm{eff}}=3({\nu }_{\mathrm{mol}}+{\nu }_t)+0.5,{\omega }_{\mathrm{eff}}=1/{\tau }_{\mathrm{eff}}$.
6. BGK collision with forcing (velocity)
Compute equilibrium $f_i^{\mathrm{eq}}$ from $\rho ,\mathbf{u},k$;
$f_i^{\ast }=f_i-{\omega }_{\mathrm{eff}}(f_i-f_i^{\mathrm{eq}})+F_i$;
if PINN enabled (Smagorinsky+PINN) then
${\mathbf{N}}_{\theta }={\mathcal{N}}_{\theta }\left(f_i\right)$;
Project onto invariant complement: ${\mathbf{N}}_{\theta }^{\perp }=\mathbf{P}{\mathbf{N}}_{\theta }$;
$f_i^{\ast }\leftarrow f_i^{\ast }+{\mathbf{N}}_{\theta }^{\perp }$;
end if.
7. Artificial dissipation (shock-capturing)
Compute shock sensor $s={\displaystyle \frac{|{\nabla }^{2}p|}{\left|p\right|+ϵ}}$ with $p=\rho c_s^{2}T$;
$f_i^{\ast \ast }=f_i^{\ast }+d_{2}s{\nabla }^2{f}_i^{\ast }-d_4{\nabla }^4{f}_i^{\ast }$;
Apply positivity and clipping: $f_i^{\ast \ast }=max(f_i^{\ast \ast },0)$, clip to $[-{10}^3,{10}^3]$.
8. Update velocity distribution
$f_i(\mathbf{x},t+\Delta t)=f_i^{\ast \ast }(\mathbf{x},t)$, reapply mask.
9. BGK collision (temperature)
${\alpha }_{\mathrm{eff}}={\alpha }_{\mathrm{mol}}+{\nu }_t/{\mathrm{Pr}}_t$ with ${\mathrm{Pr}}_t=0.7$;
${\tau }_T=3{\alpha }_{\mathrm{eff}}+0.5,{\omega }_T=1/{\tau }_T$;
Compute equilibrium $g_j^{\mathrm{eq}}$ from $T,\mathbf{u}$;
$g_j^{\ast }=g_j-{\omega }_T(g_j-g_j^{\mathrm{eq}})$.
10. Update temperature distribution
$g_j(\mathbf{x},t+\Delta t)=g_j^{\ast }(\mathbf{x},t)$, reapply thermal BCs.
11. Diagnostics and snapshots
if  $tmod\Delta t_{\mathrm{record}}=0$  then
Compute vorticity: ${\omega }_z={\partial }_y{u}_x-{\partial }_x{u}_y$;
Compute Nusselt number: $\mathrm{Nu}=-{\displaystyle \frac{\partial T}{\partial r}}{|}_{R_{\mathrm{in}}}{\displaystyle \frac{R_{\mathrm{out}}-R_{\mathrm{in}}}{T_{\mathrm{in}}-T_{\mathrm{out}}}}$ (via linear regression);
Compute non-equilibrium norm: $\parallel g_{\perp }{\parallel }^2={\sum }_i{w}_i{(f_i-f_i^{\mathrm{eq}})}^2$;
Compute energy spectrum: $E\left(k\right)={\displaystyle \frac{1}{2}}{\left|\widehat{\mathbf{u}}\left(k\right)\right|}^2$ and compare with $k^{-5/3}$;
Santos-Andrade decay rate (observed): ${\lambda }_{\mathrm{obs}}$ from exponential fit of $\parallel g_{\perp }{\parallel }^2$;
Theoretical rate (corrigido): ${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\omega \right|}{c_s+\left|\omega \right|}}-C_2{L}_{\theta }$;
Store snapshot for post-processing (vorticity, T, pressure, u, v, k).
end if.
12. Additional outputs (final stage)
After completion, generate 28 publication-ready plots (300 DPI) including:
radial profiles ($u_{\theta },T,k,{\nu }_t,p$), time series (Nu, $\parallel g_{\perp }{\parallel }^2$),
energy spectra, error analysis, vorticity fields, Q-criterion,
Hovmöller diagram, streamlines over vorticity, rotor mask,
temperature with Nusselt annotation, vorticity evolution, and 2D field maps.
End For

Remark 53

(Artificial dissipation and shock capturing). At high Mach numbers ($\mathrm{Ma}\ge 5$), compressibility effects generate shocklets that require stabilisation. To maintain numerical stability of the LBM–PINN framework without compromising the turbulence closures, we apply a shock-capturing artificial dissipation scheme directly to the post-collision distribution functions:

$$f_i^{\ast \ast }=f_i^{\ast }+d_{2}s{\nabla }^2{f}_i^{\ast }-d_4{\nabla }^4{f}_i^{\ast },s={\displaystyle \frac{|{\nabla }^{2}p|}{\left|p\right|+ϵ}},ϵ={10}^{-12},$$

(350)

where $p=\rho c_s^{2}T$ is the pressure. The sensor s activates only in regions of steep pressure gradients, remaining near zero in fully turbulent regions. The coefficients, calibrated for the $300\times 300$ grid, are $d_2=1.5$ (second-order smoothing) and $d_4=0.05$ (fourth-order hyperviscosity for dealiasing). Following dissipation, positivity and boundedness are enforced via clipping ($f_i^{\ast \ast }=max(f_i^{\ast \ast },0)$, $min(f_i^{\ast \ast },{10}^3)$), and the annular mask is reapplied. This approach preserves the inertial energy cascade and the fidelity of the $E\left(k\right)\sim k^{-5/3}$ power-law spectra.

6.4. Validation

This subsection presents the quantitative results from the numerical experiments, organised into the following parts: (i) validation of the Santos-Andrade inequality through parameter sweeps, (ii) radial profiles of velocity and temperature, (iii) energy spectra and turbulence scales, (iv) pressure fields and compressibility effects, (v) Nusselt numbers and heat transfer enhancement, (vi) velocity profiles and Strouhal numbers, (vii) azimuthal correlation and spectral analysis, (viii) entropy production at high Mach numbers, and (ix) radial pressure profiles and Reynolds stresses at $\mathrm{Ma}=10.0$. The results are presented for $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$ on the $300\times 300$ grid, with comparisons drawn across the Smagorinsky and Smagorinsky+PINN turbulence models.

6.4.1. Validation of the Santos–Andrade Inequality

The central theoretical result of this work — the Santos–Andrade hypocoercive inequality — is validated through a systematic parameter sweep designed to probe the combined effects of compressibility, rotation, and neural-network corrections on the stability of the kinetic model. A comprehensive sweep was performed over the Mach number $\mathrm{Ma}\in [1,10]$, the angular velocity $\omega \in [0,80]$, the relaxation time $\tau \in [0.6,0.8,1.0]$, and the neural Lipschitz constant $L_{\theta }\in [0.0,1.0,2.0,3.0,5.0]$, with each combination simulated for $N_{\mathrm{steps}}=2000$ steps. The observed decay rate ${\lambda }_{\mathrm{obs}}$ was extracted from the exponential decay of the non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$, defined as:

$$\parallel g_{\perp }{\parallel }^2={\displaystyle \sum _{i=1}^Q}{w}_i{\left(f_i-f_i^{\mathrm{eq}}\right)}^2,$$

(351)

which measures the distance of the distribution function from the local equilibrium state. The exponential decay of this norm is a direct consequence of the hypocoercive structure of the linearised kinetic equation, as established in Theorem 5.

Figure 3 presents the most comprehensive validation of the Santos-Andrade inequality, showing the observed decay rate ${\lambda }_{\mathrm{obs}}$ as a function of Mach number $\mathrm{Ma}\in [1,10]$ for fixed $\omega =40$, $\tau =0.8$, and $L_{\theta }=1.0$. The observed decay rate follows the theoretical prediction:

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

(352)

with the constants $C_1=0.3385$ and $C_2=0.2284$ determined from the linear regression analysis presented below. The excellent agreement across the entire compressible regime confirms that the inequality accurately captures the combined effects of rotation, compressibility, and neural corrections on the stability of the kinetic model.

The data, summarised in Table 2, reveals several important features. At low Mach numbers ($\mathrm{Ma}\le 3$), the observed decay rate is approximately constant at ${\lambda }_{\mathrm{obs}}\approx 9.0$, reflecting the dominance of the BGK relaxation term $1/\left(4\tau \right)\approx 0.3125$ over the rotation and neural contributions. As the Mach number increases beyond $\mathrm{Ma}=3$, the decay rate exhibits a monotonic decrease, consistent with the destabilising effect of compressibility captured by the term $C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|)$. The speed of sound $c_s$ decreases with Mach number in the normalised units ($c_s=1/\mathrm{Ma}$ for this parameter sweep), so the ratio $\left|\mathit{\omega }\right|/(c_s+|\mathit{\omega }\left|\right)$ increases, reducing the decay rate. This behaviour is physically interpreted as the weakening of the collision operator’s dissipative effect relative to the transport operator as the flow becomes more compressible.

The relative error between the observed and theoretical decay rates remains below $5\%$ for $\mathrm{Ma}\le 8$, with the closest agreement at $\mathrm{Ma}=2$ ($0.0\%$ error) and $\mathrm{Ma}=5$ ($1.3\%$ error). At $\mathrm{Ma}=9$ and 10, the errors increase to $10\%$ and $14\%$, respectively. This increased discrepancy at very high Mach numbers is attributed to the onset of strong compressibility effects that are not fully captured by the linear hypocoercive theory, particularly the nonlinear coupling between the acoustic and vortical modes. These effects become significant at $\mathrm{Ma}\gtrsim 8$, consistent with the findings of Sarkar et al. [14] and Lele [17], who demonstrated that compressibility effects introduce non-negligible modifications to the turbulence dynamics at high Mach numbers.

Notably, the observed decay rate at $\mathrm{Ma}=10.0$ (${\lambda }_{\mathrm{obs}}=5.7$) remains positive and well above the stability threshold ${\lambda }_{\mathrm{SA}}>0$, confirming that the LBM–PINN scheme remains stable even under extreme compressibility. This is a critical result, as it demonstrates that the neural correction, with Lipschitz constant $L_{\theta }=1.0$, does not destabilise the simulation at high Mach numbers. The Santos–Andrade stability condition:

$$L_{\theta }<{\displaystyle \frac{1}{C_2}}\left({\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}\right)$$

(353)

is satisfied for all Mach numbers considered, providing a rigorous mathematical certificate for the stability of the numerical scheme. This condition is particularly stringent at high Mach numbers, where the term $\left|\mathit{\omega }\right|/(c_s+|\mathit{\omega }\left|\right)$ approaches unity, reducing the available stability margin. The successful validation of the inequality across the entire Mach range confirms that the Santos–Andrade criterion provides a practical, mathematically certified guideline for designing neural closures that are both accurate and provably stable for rotating compressible flows.

The fitted constants $C_1=0.3385$ and $C_2=0.2284$ are of order unity, consistent with the theoretical analysis in Section 4. The coefficient of $1/\tau$ is found to be $0.2498$, in excellent agreement with the theoretical value $0.25$ from Villani’s hypocoercive framework [31]. This agreement confirms that the augmented energy methodology (181) correctly captures the dissipative structure of the BGK operator, and that the neural correction contributes a linear term $-C_2{L}_{\theta }$ to the decay rate, as predicted by Assumption A2. These results provide strong empirical support for the Santos–Andrade inequality and establish it as a robust, mathematically certified stability criterion for structure-preserving neural closures in compressible rotating turbulence simulations.

This systematic validation across the full parameter space provides a rigorous foundation for the LBM–PINN framework. The excellent agreement between the observed and theoretical decay rates confirms that the hypocoercive theory developed in Section 4 accurately captures the complex interplay between rotation, compressibility, and neural corrections. The practical utility of the stability condition (353) is demonstrated by its ability to guide the design of neural closures that maintain stability even under extreme flow conditions, a property that is essential for reliable predictive simulation in nuclear thermal-hydraulics and other demanding applications.

6.4.2. Energy Spectra and Turbulence Scales

The energy spectra $E\left(k\right)$ of the velocity fluctuations provide critical insight into the turbulent kinetic energy distribution across scales and the ability of the turbulence models to capture the inertial-range dynamics. Figure 4 presents the energy spectra for $\mathrm{Ma}=5.0$ on the $300\times 300$ grid, comparing the laminar (BGK) reference, the Smagorinsky model, and the Smagorinsky+PINN model against the Kolmogorov $-5/3$ inertial-range scaling.

At $\mathrm{Ma}=5.0$, all models exhibit an inertial range with a slope close to the Kolmogorov $-5/3$ scaling, indicating the presence of a forward energy cascade characteristic of fully developed turbulence. The laminar (BGK) simulation shows the lowest energy content across all wavenumbers, as expected for a non-turbulent reference. The absence of a well-defined $-5/3$ range in the laminar spectrum confirms that the turbulent cascade is driven by the unresolved fluctuations captured by the turbulence models.

The Smagorinsky model captures the general $-5/3$ scaling but exhibits a slightly shallower slope ($-1.58$ compared to the theoretical $-1.67$) and reduced energy at intermediate wavenumbers ($k\in [0.1,0.4]$). This reduction is attributed to the dissipative nature of the eddy-viscosity closure, which dampens high-wavenumber fluctuations and suppresses the inertial-range energy transfer. The excessive dissipation is particularly evident in the near-dissipation range, where the Smagorinsky spectrum falls below the Kolmogorov reference by approximately $10\%$ at $k=0.4$.

The Smagorinsky+PINN model recovers the $-5/3$ slope more accurately ($-1.67$) and maintains higher energy content across the entire inertial range, with the spectrum closely following the Kolmogorov reference line. This improvement is a direct consequence of the neural correction, which compensates for the deficiencies of the Smagorinsky closure by providing the missing non-equilibrium fluctuations. The PINN correction effectively counteracts the excessive dissipation of the LES closure, allowing the turbulent cascade to extend further into the dissipative range. This is consistent with the higher turbulent kinetic energy levels observed in the Smagorinsky+PINN model and confirms that the neural correction promotes a more realistic turbulence structure.

The improved spectral fidelity of the Smagorinsky+PINN model is essential for accurate prediction of the Reynolds stresses and the turbulent heat flux, which are critical for the thermal-hydraulic analysis of high-Mach rotating flows in nuclear applications. The results at $\mathrm{Ma}=5.0$ provide strong evidence that the structure-preserving PINN correction effectively compensates for the dissipative shortcomings of the classical Smagorinsky closure, yielding spectral energy levels that closely approach the Kolmogorov reference.

6.4.3. Pressure Fields and Compressibility Effects

The pressure field $p(\mathbf{x},t)=\rho c_s^{2}T$ provides a direct measure of compressibility effects and the hydrodynamic response of the flow to the centrifugal and Coriolis forces. Figure 5 presents the two-dimensional pressure fields for $\mathrm{Ma}=5.0$ on the $300\times 300$ grid, comparing the laminar (BGK) reference, the Smagorinsky model, and the Smagorinsky+PINN model.

All three models exhibit a pronounced radial pressure gradient, with higher pressure near the inner rotating cylinder ($r=R_{\mathrm{in}}$) and lower pressure near the outer stationary wall ($r=R_{\mathrm{out}}$). This radial pressure distribution is a direct consequence of the centrifugal acceleration ${\omega }^2\mathbf{r}$, which drives fluid outward and creates a pressure gradient that balances the centrifugal force. In the rotating frame, the hydrostatic balance is given by:

$${\nabla }_{\mathbf{x}}p=\rho \mathit{\omega }\times (\mathit{\omega }\times \mathbf{x}),$$

(354)

which integrates to the pressure distribution:

$$p\left(r\right)=p\left(R_{\mathrm{in}}\right)exp\left({\displaystyle \frac{{\omega }^2}{2RT}}\left(r^2-R_{\mathrm{in}}^2\right)\right),$$

(355)

assuming isothermal conditions. The approximately exponential radial increase in pressure observed in all three models is consistent with this theoretical prediction, confirming the physical consistency of the LBM implementation.

The pressure fields for the three models are qualitatively similar, reflecting the dominant role of the centrifugal force in establishing the pressure distribution. The laminar (BGK) reference exhibits the sharpest radial pressure gradient, with a clear distinction between the high-pressure region near the inner wall and the low-pressure region near the outer wall. The pressure contours are approximately axisymmetric, confirming the statistical stationarity of the flow and the adequacy of the azimuthal averaging.

The Smagorinsky model produces a slightly smoother pressure field compared to the laminar reference, particularly in the near-wall region. This smoothing is attributed to the enhanced turbulent viscosity introduced by the LES closure, which diffuses momentum and pressure fluctuations more effectively than the molecular viscosity alone. The smoother pressure gradients are consistent with the reduced velocity gradients observed in the Smagorinsky model and the enhanced thermal diffusivity indicated by the temperature profiles.

The Smagorinsky+PINN model produces pressure fields that closely resemble the laminar reference, with sharper radial gradients and more defined pressure contours. This is consistent with the behaviour observed in the velocity and temperature profiles, where the neural correction effectively counteracts the excessive dissipation of the Smagorinsky closure. The preservation of the pressure gradients by the PINN-corrected model indicates that the neural correction does not introduce spurious pressure fluctuations, a critical property for maintaining the accuracy of the hydrodynamic balance.

Remark 54

(Physical interpretation of the pressure field). The radial pressure gradient is a direct manifestation of the centrifugal force in the rotating frame, with pressure increasing outward to maintain hydrostatic balance according to $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$, which for an ideal gas yields an exponential profile. The excellent agreement between the simulated pressure fields and this theoretical prediction confirms the accuracy of the LBM implementation and the correct treatment of the rotating-frame forces via the Guo forcing scheme. The preservation of the pressure gradients by the Smagorinsky+PINN model demonstrates that the neural correction does not introduce unphysical momentum sources that would perturb the hydrodynamic balance.

The minimal differences between the three models in the pressure field, compared to the more pronounced differences observed in the temperature and velocity profiles, reflect the fact that pressure is primarily determined by the centrifugal balance rather than by turbulent fluctuations. At $\mathrm{Ma}=5.0$, the turbulent pressure fluctuations are secondary to the mean pressure gradient established by the centrifugal force, explaining the qualitative similarity of the pressure fields across the models. This is consistent with the findings of Sarkar et al. [14], who demonstrated that the pressure-dilatation correlation becomes significant only at very high Mach numbers ($\mathrm{Ma}\gtrsim 8$), where the acoustic modes couple with the turbulent fluctuations.

The pressure field analysis at $\mathrm{Ma}=5.0$ provides further validation of the LBM–PINN framework. The excellent agreement between the simulated pressure fields and the theoretical hydrostatic balance confirms that the rotating-frame forces are correctly implemented and that the structure-preserving neural correction does not perturb the fundamental hydrodynamic balance. This is essential for the predictive capability of the framework, as accurate pressure prediction is critical for the calculation of the pressure-dilatation correlation and the turbulent kinetic energy budget, particularly in high-Mach compressible flows.

6.4.4. Nusselt Number and Heat Transfer Enhancement: Comparative Analysis at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$

The Nusselt number $\mathrm{Nu}$ is a fundamental dimensionless parameter that quantifies the efficiency of convective heat transfer relative to pure conduction. For the Taylor–Couette configuration, $\mathrm{Nu}$ is defined as:

$$\mathrm{Nu}=-{\left.{\displaystyle \frac{\partial T}{\partial r}}\right|}_{R_{\mathrm{in}}}{\displaystyle \frac{R_{\mathrm{out}}-R_{\mathrm{in}}}{T_{\mathrm{in}}-T_{\mathrm{out}}}},$$

(356)

where the temperature gradient at the inner wall is evaluated via linear regression of the radial temperature profile. This definition follows the standard convention in Taylor–Couette heat transfer studies [32,39]. The Nusselt number directly quantifies the turbulent heat flux enhancement: $\mathrm{Nu}=1$ corresponds to pure conduction, while $\mathrm{Nu}>1$ indicates the presence of turbulent convection.

Figure 6 and Figure 7 present the radial temperature profiles and the corresponding Nusselt number calculations for the Smagorinsky and Smagorinsky+PINN models at $\mathrm{Ma}=5.0$, while Figure 8 presents the results for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. All cases employ the same thermal boundary conditions ($T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$, $\Delta T=2.8$) to enable direct comparison.

Smagorinsky Model at $\mathrm{Ma}=5.0$

Figure 6 shows the temperature profile for the Smagorinsky model at $\mathrm{Ma}=5.0$. The profile exhibits a monotonic decrease with a steep gradient in the near-wall region ($r\in [0.2,0.4]$), characteristic of the thermal boundary layer in turbulent Taylor–Couette flow. The steep gradient near the inner wall reflects the intense turbulent mixing driven by the shear and centrifugal instability, which efficiently transports heat from the hot inner cylinder to the cold outer cylinder. The Nusselt number, computed from the inner-wall gradient via linear regression, yields $\mathrm{Nu}\approx 30.0$. This value represents a significant enhancement of convective heat transfer relative to pure conduction, with the turbulent heat flux exceeding the conductive flux by a factor of approximately 30. The high Nusselt number is consistent with the high Taylor number regime ($\mathrm{Ta}\approx {10}^5$) explored in this study, where the centrifugal instability produces intense Taylor vortices that efficiently mix the fluid across the annular gap.

The temperature profile exhibits a characteristic boundary layer structure, with the steepest gradient confined to the region $r\in [0.2,0.4]$. Beyond $r\approx 0.4$, the temperature gradient becomes more gradual, indicating that the turbulent mixing is most intense near the rotating inner wall where the shear is strongest. This behaviour is consistent with the classical theory of turbulent boundary layers in rotating flows, where the thermal boundary layer is thinner than the momentum boundary layer due to the high Prandtl number ($\mathrm{Pr}=0.7$ in this study).

Smagorinsky+PINN Model at $\mathrm{Ma}=5.0$

Figure 7 shows the temperature profile for the Smagorinsky+PINN model at $\mathrm{Ma}=5.0$. The profile is qualitatively similar to the Smagorinsky model, with a slightly smoother gradient near the inner wall. The Nusselt number is $\mathrm{Nu}\approx 26.39$, which is approximately $12\%$ lower than the Smagorinsky model ($\mathrm{Nu}\approx 30.0$). This apparent reduction in $\mathrm{Nu}$ is noteworthy and requires careful physical interpretation.

The reduction in $\mathrm{Nu}$ from $30.0$ (Smagorinsky) to $26.39$ (Smagorinsky+PINN) at $\mathrm{Ma}=5.0$ may initially appear counterintuitive, as the PINN correction was expected to improve the predictive capability of the LES closure. However, this result is physically consistent with the known behaviour of the Smagorinsky model: the Smagorinsky closure systematically overestimates the eddy viscosity in regions of strong shear, leading to excessive turbulent diffusion and artificially steep temperature gradients. The PINN correction, by learning the discrepancy between the Smagorinsky model and the high-fidelity BGK reference, effectively reduces this excessive diffusion, resulting in a more physical (i.e., less steep) temperature gradient and a correspondingly lower Nusselt number. The value $\mathrm{Nu}\approx 26.39$ is therefore more physically accurate than the Smagorinsky value $\mathrm{Nu}\approx 30.0$, as it removes the artificial enhancement of heat transfer caused by the excessive eddy viscosity.

This interpretation is supported by the fundamental physics of the Smagorinsky closure. The eddy viscosity in the Smagorinsky model is proportional to the resolved strain rate:

$${\nu }_t={(C_s\Delta x)}^2\left|\overline{\mathbf{S}}\right|,$$

(357)

where $|\overline{\mathbf{S}}|=\sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}}$. In rotating flows, the strain rate tensor does not distinguish between shear-driven and rotation-driven turbulence, leading to an overestimation of ${\nu }_t$ in regions of strong rotation. The PINN correction, by providing a data-driven closure that is not constrained by this assumption, effectively reduces the excessive eddy viscosity and produces a more physical temperature gradient. The lower Nusselt number in the PINN-corrected model is therefore a consequence of the reduced artificial dissipation, not a degradation of predictive capability.

Smagorinsky model at $\mathrm{Ma}=10.0$

At $\mathrm{Ma}=10.0$, the Smagorinsky model produces a nearly constant temperature field across the entire gap, completely failing to capture the radial temperature gradient. For the standard configuration ($\Delta T=1.0$), the Nusselt number is $\mathrm{Nu}\approx 2.34$, representing a catastrophic failure of the closure. This qualitative breakdown indicates that the standard Smagorinsky closure, calibrated for incompressible flows, is unable to represent the complex physics of high-Mach rotating flows.

The failure mechanism can be understood in terms of two fundamental deficiencies. First, the eddy-viscosity hypothesis assumes that the subgrid-scale stress tensor is proportional to the resolved strain rate tensor. In rotating flows, the strain rate tensor does not distinguish between shear-driven and rotation-driven turbulence. The Coriolis force modifies the turbulence structure by suppressing vertical velocity fluctuations and enhancing the anisotropy of the Reynolds stresses. This anisotropy is not captured by the scalar eddy viscosity, leading to an overestimation of the dissipation rate and a suppression of the turbulent heat flux.

Second, at high Mach numbers, compressibility effects introduce significant deviations from the incompressible assumptions underlying the Smagorinsky model. The pressure-dilatation correlation $\overline{p^{\prime }{\partial }_k{u}_k^{\prime }}$ and the production of entropy by shocklets become important, modifying the energy cascade and the turbulence structure. The Smagorinsky model, calibrated for incompressible flows, does not account for these compressibility effects, leading to a severe underprediction of the turbulent kinetic energy and the turbulent heat flux.

Smagorinsky+PINN Model at $\mathrm{Ma}=10.0$

Figure 8 presents the temperature profile for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ with $\Delta T=2.8$. Remarkably, despite the extreme compressibility, the temperature profile remains well-resolved with a similar boundary layer structure, yielding $\mathrm{Nu}\approx 29.15$. The close agreement between the Nusselt numbers at $\mathrm{Ma}=5.0$ ($\mathrm{Nu}\approx 30.0$ for Smagorinsky, $\mathrm{Nu}\approx 26.39$ for Smagorinsky+PINN) and $\mathrm{Ma}=10.0$ ($\mathrm{Nu}\approx 29.15$ for Smagorinsky+PINN) demonstrates that the PINN-corrected model maintains high heat transfer efficiency even under extreme compressibility.

For the standard configuration ($\Delta T=1.0$), the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ yields $\mathrm{Nu}=17.14\pm 0.25$. This value represents a dramatic improvement over the Smagorinsky model ($\mathrm{Nu}\approx 2.34$), restoring the correct heat transfer physics and yielding a physically plausible Nusselt number for a high-Reynolds-number, high-Mach turbulent Taylor–Couette flow. The Nusselt number $\mathrm{Nu}=17.14$ indicates that the turbulent heat flux is approximately 17 times greater than the conductive flux, a physically plausible enhancement for this regime.

The PINN correction addresses the deficiencies of the Smagorinsky model by providing a data-driven closure that is not constrained by the eddy-viscosity assumption. By learning the discrepancy between the Smagorinsky model and the high-fidelity BGK reference, the neural network effectively reconstructs the missing non-equilibrium physics. The projection onto the collision invariants (Assumption A1) ensures that the correction does not violate the conservation laws, while spectral normalisation guarantees that the Lipschitz constant remains below the Santos–Andrade threshold. The result is a model that accurately captures the anisotropy of the subgrid-scale stresses, the compressibility effects, and the non-equilibrium fluctuations that are essential for accurate heat transfer prediction at high Mach numbers.

Comparative Physical Interpretation

The Nusselt number results reveal a clear hierarchy of model performance and provide insight into the physical mechanisms underlying the heat transfer enhancement. Table 6 summarises the Nusselt numbers for the four configurations.

Table 3. Summary of Nusselt numbers for the four configurations with $T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$ ($\Delta T=2.8$).

Model	$\mathbf{Ma}=5.0$	$\mathbf{Ma}=10.0$
Smagorinsky	$30.00$	—
Smagorinsky+PINN	$26.39$	$29.15$

Table 3. Summary of Nusselt numbers for the four configurations with $T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$ ($\Delta T=2.8$).

Model	$\mathbf{Ma}=5.0$	$\mathbf{Ma}=10.0$
Smagorinsky	$30.00$	—
Smagorinsky+PINN	$26.39$	$29.15$

At $\mathrm{Ma}=5.0$, the Smagorinsky model overestimates the Nusselt number ($\mathrm{Nu}\approx 30.0$) due to excessive eddy viscosity, while the PINN-corrected model produces a more physical value ($\mathrm{Nu}\approx 26.39$). At $\mathrm{Ma}=10.0$, the Smagorinsky model fails catastrophically ($\mathrm{Nu}\approx 2.34$ for $\Delta T=1.0$), while the PINN-corrected model maintains high heat transfer efficiency ($\mathrm{Nu}\approx 29.15$ for $\Delta T=2.8$, $\mathrm{Nu}=17.14$ for $\Delta T=1.0$).

The consistency of the PINN-corrected model across different thermal boundary conditions ($\Delta T=2.8$ and $\Delta T=1.0$) and Mach numbers ($\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$) demonstrates its robustness and generalisability. The Nusselt number scales approximately linearly with the temperature difference: $\mathrm{Nu}\propto \Delta T$, which is consistent with the expected behaviour for turbulent convection where the heat flux is proportional to the temperature difference. This linear scaling confirms that the PINN-corrected model captures the correct physics of turbulent heat transfer, in contrast to the Smagorinsky model, which fails to maintain any meaningful temperature gradient at $\mathrm{Ma}=10.0$.

It is important to clarify the role of the two thermal boundary conditions employed in this study. The results presented in Figure 6, Figure 7, and Figure 8 use $\Delta T=2.8$ ($T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$) primarily to demonstrate the robustness of the PINN-corrected model across a wider range of thermal driving forces and to facilitate a direct visual comparison of the temperature profiles. The corresponding Nusselt numbers ($\mathrm{Nu}\approx 30.0$ for Smagorinsky at $\mathrm{Ma}=5.0$, $\mathrm{Nu}\approx 26.39$ for Smagorinsky+PINN at $\mathrm{Ma}=5.0$, and $\mathrm{Nu}\approx 29.15$ for Smagorinsky+PINN at $\mathrm{Ma}=10.0$) serve as qualitative indicators of the model’s ability to maintain a thermal gradient under varying conditions. However, the primary quantitative benchmark used for comparison with the literature and for assessing the physical plausibility of the heat transfer enhancement is the standard configuration with $\Delta T=1.0$ ($T_{\mathrm{in}}=1.5$, $T_{\mathrm{out}}=0.5$), which yields $\mathrm{Nu}=17.14\pm 0.25$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The scaling of $\mathrm{Nu}$ with $\Delta T$ is approximately linear, as expected for turbulent convection, which further supports the physical consistency of the PINN-corrected model.

The Nusselt number analysis provides compelling evidence for the practical utility of the Santos-Andrade inequality and the structure-preserving PINN correction. The ability of the Smagorinsky+PINN model to maintain high heat transfer efficiency at $\mathrm{Ma}=10.0$ demonstrates that the neural correction, constrained by the hypocoercive stability condition, can extend the predictive capability of classical LES closures to extreme compressible regimes where they would otherwise fail catastrophically. This has significant implications for nuclear thermal-hydraulics, where accurate prediction of heat transfer in high-Mach rotating flows is essential for safety assessments and design optimisation.

Remark 55

(Physical interpretation of the Nusselt number results). At $\mathrm{Ma}=5.0$, the reduction in $\mathrm{Nu}$ from $30.0$ (Smagorinsky) to $26.39$ (Smagorinsky+PINN) reflects the neural correction’s ability to reduce excessive eddy viscosity. The Smagorinsky overestimates diffusivity because ${\nu }_t={(C_s\Delta x)}^2\left|\overline{\mathbf{S}}\right|$ does not distinguish shear- from rotation-driven turbulence. At $\mathrm{Ma}=10.0$, the PINN model preserves $\mathrm{Nu}$ by reconstructing non-equilibrium fluctuations essential for heat transfer, while the Smagorinsky model suppresses Taylor vortices and collapses the thermal boundary layer. These findings are consistent with Guillermet al. [39] and Sarkaret al. [14].

Remark 56

(Interpretation of Nusselt numbers in two dimensions). The Nusselt numbers reported in this section are obtained from two-dimensional simulations of the Taylor–Couette configuration. While the two-dimensional setup captures the dominant azimuthal dynamics and provides a meaningful comparative assessment of turbulence closures, it inherently lacks the three-dimensional vortex stretching and the full energy cascade that characterise real turbulent flows in industrial components. Consequently, the absolute values of $\mathrm{Nu}$ should not be directly extrapolated to three-dimensional reactor geometries; they are best interpreted as relative indicators of model performance under controlled conditions. The primary value of these results lies in the clear distinction between the catastrophic failure of classical closures at $\mathrm{Ma}=10.0$ and the restoration of a physically consistent thermal gradient by the Smagorinsky+PINN model. Future work will extend the validation to three-dimensional annular geometries with axial flow, where quantitative Nusselt number predictions can be compared against experimental data for reactor-relevant configurations.

6.4.5. Velocity Profiles and Vortex Dynamics: Validation Against Analytical Solutions

The radial velocity profiles and the vortex shedding dynamics provide critical validation of the LBM implementation and the turbulence models. The tangential velocity profile $u_{\theta }\left(r\right)$ is compared against the analytical Couette solution for incompressible Taylor–Couette flow, while the Strouhal number quantifies the vortex shedding frequency and provides insight into the coherent structures in the turbulent flow.

Figure 9. Radial profiles of the tangential velocity $u_{\theta }\left(r\right)$ at Ma = 5.0 on the 300 × 300 grid. The laminar (BGK) simulation closely follows the analytical Couette solution:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}},$$

(358)

confirming the accuracy of the LBM implementation and the no-slip boundary conditions. The Smagorinsky model exhibits slight deviations near the inner wall (r ∈ [0.2, 0.35]), where the velocity gradient is overestimated due to excessive eddy viscosity. The Smagorinsky+PINN model recovers the analytical profile with greater accuracy, demonstrating the corrective effect of the neural closure. The profiles are averaged over the last 500 time steps, with shaded regions representing one standard deviation of the temporal fluctuations.

Figure 9. Radial profiles of the tangential velocity $u_{\theta }\left(r\right)$ at Ma = 5.0 on the 300 × 300 grid. The laminar (BGK) simulation closely follows the analytical Couette solution:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}},$$

(358)

confirming the accuracy of the LBM implementation and the no-slip boundary conditions. The Smagorinsky model exhibits slight deviations near the inner wall (r ∈ [0.2, 0.35]), where the velocity gradient is overestimated due to excessive eddy viscosity. The Smagorinsky+PINN model recovers the analytical profile with greater accuracy, demonstrating the corrective effect of the neural closure. The profiles are averaged over the last 500 time steps, with shaded regions representing one standard deviation of the temporal fluctuations.

Figure 9 presents the radial profiles of the tangential velocity $u_{\theta }\left(r\right)$ at $\mathrm{Ma}=5.0$ for the laminar (BGK) reference, the Smagorinsky model, and the Smagorinsky+PINN model. The laminar simulation closely follows the analytical Couette solution for incompressible flow between concentric cylinders:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}},$$

(359)

where $\omega =U_{\mathrm{wall}}/R_{\mathrm{in}}$ is the angular velocity of the inner cylinder. The excellent agreement between the laminar simulation and the analytical solution confirms the accuracy of the LBM implementation, the correct treatment of the no-slip boundary conditions via the bounce-back scheme, and the proper representation of the annular geometry through the binary mask.

The Smagorinsky model exhibits slight deviations from the analytical solution near the inner wall ($r\in [0.2,0.35]$), where the velocity gradient is overestimated by approximately $8\%$. This deviation is attributed to the excessive eddy viscosity of the Smagorinsky closure in regions of strong shear. The eddy viscosity is given by:

$${\nu }_t={(C_s\Delta x)}^2\left|\overline{\mathbf{S}}\right|,$$

(360)

where $|\overline{\mathbf{S}}|=\sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}}$ is the magnitude of the resolved strain rate tensor. Near the inner wall, the strain rate is largest, leading to high eddy viscosity and enhanced momentum diffusion. This artificial diffusion smooths the velocity profile and reduces the velocity gradient, resulting in the observed deviation from the analytical solution.

The Smagorinsky+PINN model recovers the analytical profile with greater accuracy, with a maximum deviation of approximately $2.5\%$ from the Couette solution. This improvement is a direct consequence of the neural correction, which compensates for the deficiencies of the Smagorinsky closure by providing the missing non-equilibrium fluctuations. The PINN correction effectively reduces the excessive eddy viscosity near the inner wall, restoring the correct velocity gradient and improving the agreement with the analytical solution. This result is consistent with the findings of Lee, Kim, and Park [55], who demonstrated that neural closure models can improve Reynolds stress prediction in rotating flows.

The velocity profiles at $\mathrm{Ma}=5.0$ confirm that the Smagorinsky+PINN model provides a more accurate representation of the mean flow than the classical Smagorinsky closure. The improved agreement with the analytical solution validates the structure-preserving nature of the neural correction and demonstrates its ability to enhance the predictive capability of LES closures without introducing spurious effects. This is particularly important for the accurate prediction of the Reynolds stresses and the turbulent kinetic energy budget, which are essential for the thermal-hydraulic analysis of high-Mach rotating flows.

Figure 10 presents the time history of the vorticity and the corresponding frequency spectrum for the Smagorinsky+PINN model at $\mathrm{Ma}=5.0$. The vorticity time series, extracted at the mid-gap location ($r=0.6$), exhibits periodic oscillations characteristic of coherent vortex shedding in turbulent Taylor-Couette flow. The oscillations reflect the passage of Taylor vortices and their secondary instabilities past the measurement point. The amplitude of the oscillations is approximately $0.005$–$0.007$ in the normalised vorticity units, consistent with the turbulent fluctuations observed in the TKE profiles.

The frequency spectrum, obtained via Fast Fourier Transform (FFT) of the vorticity time series, reveals a dominant frequency $f_{\mathrm{dom}}\approx 0.0045$ (in lattice time units), corresponding to the vortex shedding frequency. The Strouhal number, defined as:

$$\mathrm{St}={\displaystyle \frac{f_{\mathrm{dom}}d}{U_{\mathrm{wall}}}},$$

(361)

where $d=R_{\mathrm{out}}-R_{\mathrm{in}}=0.8$ is the gap width and $U_{\mathrm{wall}}=\mathrm{Ma}{c}_s=5.0\times 0.577\approx 2.887$ is the wall velocity, yields $\mathrm{St}\approx 0.28$. This value falls within the experimental range reported by Andereck, Liu, and Swinney [12] for $R_i/R_o=0.2$ ($\mathrm{St}\approx 0.25$–$0.35$) and is in excellent agreement with the LES results of Bazilevs and Akkerman [32] ($\mathrm{St}\approx 0.27$–$0.30$).

Remark 57

(Physical interpretation of the Strouhal number). The Strouhal number $\mathrm{St}$ characterises the vortex shedding frequency relative to the mean flow velocity in Taylor-Couette flow. The value $\mathrm{St}\approx 0.28$ obtained in this study is consistent with the wavy vortex and turbulent vortex regimes identified by Andereck, Liu, and Swinney [12], confirming that the Smagorinsky+PINN model accurately captures the coherent vortex dynamics and shedding frequency essential for correct turbulent heat flux and Nusselt number prediction.

The excellent agreement between the Smagorinsky+PINN model and the analytical Couette solution for the velocity profile, combined with the accurate prediction of the Strouhal number, provides strong validation of the LBM–PINN framework. The velocity profile confirms that the neural correction effectively reduces the excessive dissipation of the Smagorinsky closure, while the Strouhal number confirms that the model captures the correct vortex shedding dynamics. These results establish the credibility of the framework for predicting the mean flow, the turbulent fluctuations, and the coherent structures in high-Mach rotating flows.

The validation against the analytical Couette solution is particularly significant, as it demonstrates that the LBM implementation correctly reproduces the fundamental physics of laminar Taylor–Couette flow. The excellent agreement between the Smagorinsky+PINN model and the analytical solution at $\mathrm{Ma}=5.0$ provides confidence that the neural correction does not introduce spurious effects that would perturb the mean flow. This is essential for the predictive capability of the framework, as it ensures that the neural correction only acts on the unresolved scales and does not affect the resolved hydrodynamic balance.

The Strouhal number analysis provides additional validation of the turbulence models’ ability to capture the coherent structures in the flow. The agreement with the experimental and LES benchmarks confirms that the Smagorinsky+PINN model accurately represents the vortex shedding dynamics, which are critical for the correct prediction of the turbulent heat flux. This is consistent with the findings of Guillerm et al. [39], who demonstrated that accurate prediction of the coherent structures is essential for capturing the heat transfer physics in rotating, stratified Taylor-Couette flows.

6.4.6. Azimuthal Correlation and Spectral Analysis of Coherent Structures at $\mathrm{Ma}=10.0$

The azimuthal correlation function and the azimuthal energy spectrum provide complementary insights into the spatial organisation and the characteristic scales of the coherent structures in the turbulent Taylor-Couette flow at extreme Mach numbers. The azimuthal correlation function quantifies the spatial coherence of the velocity field in the angular direction, while the azimuthal energy spectrum reveals the distribution of turbulent kinetic energy among the azimuthal modes. Together, these diagnostics offer a comprehensive characterisation of the vortex dynamics and the turbulent cascade in the rotating flow under highly compressible conditions.

The azimuthal correlation function $R_{uu}(\Delta \theta )$ is defined as:

$$R_{uu}(\Delta \theta )={\displaystyle \frac{\langle u_{\theta }^{\prime }(r,\theta ,t)u_{\theta }^{\prime }(r,\theta +\Delta \theta ,t)\rangle }{\langle u_{\theta }^{\prime 2}\rangle }},$$

(362)

where $u_{\theta }^{\prime }=u_{\theta }-\langle u_{\theta }\rangle$ is the fluctuating component of the azimuthal velocity, $\langle ·\rangle$ denotes averaging over time and over the radial coordinate r, and $\Delta \theta$ is the angular separation. The correlation function measures the degree of spatial coherence of the velocity fluctuations: $R_{uu}\left(0\right)=1$ indicates perfect correlation at zero separation, while $R_{uu}(\Delta \theta )\to 0$ as $\Delta \theta \to \infty$ indicates loss of correlation at large angular separations.

Figure 11 presents the azimuthal correlation function $R_{uu}(\Delta \theta )$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The correlation function exhibits a rapid decay from $R_{uu}\left(0\right)=1$ at zero angular separation, followed by a characteristic oscillatory pattern with distinct positive and negative lobes. The first zero-crossing occurs at $\Delta \theta \approx \pm {40}^{\circ }$ (approximately $0.70$ radians), which defines the angular correlation length $L_{\theta }\approx {40}^{\circ }$. This correlation length corresponds to the characteristic angular extent of the coherent structures, providing an estimate of the typical azimuthal size of the Taylor vortices under extreme compressibility.

The oscillatory behaviour of the correlation function is particularly revealing. After the initial decay, $R_{uu}(\Delta \theta )$ becomes negative, reaching a minimum of approximately $-0.75$ at $\Delta \theta \approx \pm {75}^{\circ }$ and $\pm {100}^{\circ }$. This negative correlation indicates that the velocity fluctuations are anti-correlated at these angular separations, meaning that a positive fluctuation at a given angular position is associated with a negative fluctuation approximately ${75}^{\circ }$–${100}^{\circ }$ away. This anti-correlation is a direct signature of coherent vortex structures with alternating vorticity: the Taylor vortices, which consist of pairs of counter-rotating rolls, produce velocity fluctuations of opposite sign at diametrically opposed positions. The correlation returns to positive values at $\Delta \theta \approx \pm {150}^{\circ }$, indicating the presence of a second vortex pair with the same sign of vorticity.

The oscillatory pattern with multiple positive and negative lobes suggests the presence of a well-defined dominant azimuthal wavelength. The distance between successive zero-crossings (from positive to negative to positive) is approximately ${80}^{\circ }$–${100}^{\circ }$, corresponding to a dominant azimuthal wavenumber $m\approx {360}^{\circ }/{80}^{\circ }\approx 4.5$, or more precisely, the fundamental mode $m=4$ or $m=5$. This is consistent with the presence of 4–5 pairs of Taylor vortices distributed around the annulus, which is characteristic of the turbulent Taylor-vortex regime at the Taylor numbers explored in this study, even at $\mathrm{Ma}=10.0$. The correlation function does not fully decay to zero even at $\Delta \theta =\pm {150}^{\circ }$, indicating that the flow retains some degree of spatial coherence over large angular separations, consistent with the presence of large-scale structures spanning the entire annulus.

Figure 12 presents the azimuthal energy spectrum $E\left(m\right)$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The spectrum, obtained by Fourier transforming the azimuthal velocity field at a fixed radius ($r=0.6$, mid-gap), reveals the distribution of turbulent kinetic energy among the azimuthal wavenumbers m. The spectrum exhibits a distinct oscillatory pattern: modes with odd m ($m=1,3,5,\dots$) consistently have higher energy ($E\approx 0.25$) than adjacent even modes ($E\approx 0.15$). This alternating pattern is a clear signature of a flow organisation with a fundamental azimuthal periodicity of ${180}^{\circ }$, corresponding to a dominant pattern with two-fold symmetry (or its harmonics). In Taylor–Couette flow, this pattern is associated with the presence of an even number of Taylor vortices, typically 4 or 6, which produce a periodic modulation of the velocity field with a ${180}^{\circ }$ period. The highest energy in the fundamental mode $m=1$ (energy $E\approx 0.25$) indicates that the largest-scale structure corresponds to a global azimuthal mode spanning the entire annulus, which persists even at $\mathrm{Ma}=10.0$.

The energy is concentrated in the low-wavenumber modes ($m\le 10$), with a gradual decay towards higher wavenumbers. The decay follows an approximate power-law scaling $E\left(m\right)\sim m^{-p}$. From the envelope of the spectrum, the exponent is estimated to be $p\approx 2.5$, which is consistent with the expected scaling for two-dimensional turbulence ($p=3$) [4] or for the inverse cascade regime in rotating flows. The slightly shallower exponent ($p\approx 2.5$) reflects the three-dimensional nature of the flow, where the energy cascade is modified by the strong rotation and the compressibility effects at $\mathrm{Ma}=10.0$. The energy in the odd modes remains significantly higher than in the even modes up to $m\approx 50$, indicating that the reflection symmetry of the flow is preserved across a wide range of scales, despite the extreme compressibility.

The dominant modes $m=1$ and $m=3$ (the odd modes) have the highest energy content, indicating that the large-scale flow is dominated by a global azimuthal mode with $m=1$ (a large-scale vortex) and its harmonics. The presence of significant energy in $m=1$ is characteristic of flows with a strong mean shear and centrifugal instability, where the largest structures span the entire annulus. The energy in $m=3$ and higher odd modes reflects the organisation of the Taylor vortices into a pattern of alternating rolls. The energy in $m=5,7,9,\dots$ continues the alternating pattern, indicating that the vortex organisation persists across multiple harmonics, consistent with the complex dynamics of turbulent Taylor–Couette flow at extreme Mach numbers.

Remark 58

(Physical interpretation of the azimuthal spectrum and correlation at $\mathrm{Ma}=10.0$). The alternating odd/even mode pattern in the azimuthal spectrum reflects Taylor vortex organisation, with odd-mode dominance indicating reflection symmetry under ${180}^{\circ }$ rotation and an even number of vortex pairs. The $m=1$ mode combined with odd-mode structure ($m=3,5,\dots$) reveals multi-scale organisation persisting at $\mathrm{Ma}=10.0$, confirming the Smagorinsky+PINN model preserves coherent dynamics under extreme compressibility. The correlation function confirms this: zero-crossing at $\Delta \theta \approx {40}^{\circ }$ gives $L_{\theta }\approx 0.70$ rad ($L\approx 0.42$), consistent with the Taylor vortex scale ($d=0.8$); negative lobes confirm alternating vorticity; positive correlation at $\pm {150}^{\circ }$ indicates global $m=1$ coherence. This persistence at $\mathrm{Ma}=10.0$ shows the neural correction preserves coherent structures where Smagorinsky fails catastrophically.

The consistency between the azimuthal correlation function and the azimuthal spectrum reinforces the physical interpretation. The correlation function provides a real-space measure of the structure size, while the spectrum provides the corresponding Fourier-space representation. The correlation function indicates a dominant angular separation of approximately ${80}^{\circ }$–${100}^{\circ }$ between correlated structures (from zero-crossing to zero-crossing), while the spectrum shows energy concentration in odd modes with a fundamental period of ${180}^{\circ }$. This relationship is quantitatively consistent: a pattern with a fundamental period of ${180}^{\circ }$ corresponds to a characteristic angular separation of ${90}^{\circ }$ between adjacent vortex pairs, which matches the correlation function’s indication of ${80}^{\circ }$–${100}^{\circ }$. The agreement between the two diagnostics confirms that the flow is dominated by a pattern of alternating vortices with a characteristic angular scale of approximately ${40}^{\circ }$–${50}^{\circ }$ per vortex, corresponding to 4–5 vortex pairs distributed around the annulus.

These results are consistent with the experimental observations of Andereck, Liu, and Swinney [12] and the numerical studies of Bazilevs and Akkerman [32], who reported similar azimuthal structures in turbulent Taylor-Couette flow. The ability of the Smagorinsky+PINN model to reproduce these fine-scale structures at $\mathrm{Ma}=10.0$, despite the dissipative nature of the LES closure and the extreme compressibility, demonstrates the effectiveness of the neural correction in preserving the coherent dynamics of the flow. This further validates the LBM–PINN framework as a robust tool for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulics applications, where accurate prediction of coherent structures is essential for the correct estimation of turbulent heat transfer and safety margins.

6.4.7. Energy Spectra and Entropy Production at $\mathrm{Ma}=10.0$

The energy spectrum and the entropy production rate provide complementary diagnostics for the turbulent cascade and the dissipative mechanisms at extreme Mach numbers. The energy spectrum $E\left(k\right)$ reveals the distribution of turbulent kinetic energy across spatial scales, while the entropy production rate quantifies the irreversible dissipation of kinetic energy into thermal energy. At $\mathrm{Ma}=10.0$, the classical Smagorinsky model fails catastrophically (as discussed in Section 9), but the Smagorinsky+PINN model maintains physical fidelity. The analysis presented here focuses on the Smagorinsky+PINN model, which successfully captures the turbulent dynamics at this extreme Mach number.

Figure 13 presents the energy spectrum $E\left(k\right)$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ on the $300\times 300$ grid. The spectrum exhibits a well-defined inertial range spanning approximately $k\in [1,10]$, with a slope that closely follows the Kolmogorov $-5/3$ scaling:

$$E\left(k\right)\sim C{\epsilon }^{2/3}{k}^{-5/3},$$

(363)

where $\epsilon$ is the turbulent kinetic energy dissipation rate and $C\approx 1.5$ is the Kolmogorov constant. The dashed line in Figure 13 represents the theoretical $-5/3$ reference, and the excellent agreement confirms that the Smagorinsky+PINN model sustains the turbulent cascade even under extreme compressibility.

The inertial range extends from $k\approx 1$ to $k\approx 10$, corresponding to length scales ranging from the integral scale ($L\sim 1$ in lattice units) down to approximately one-tenth of the integral scale. This range is consistent with the expected inertial range for turbulent Taylor-Couette flow at the Taylor numbers explored in this study. The presence of a well-defined $-5/3$ range confirms that the energy is transferred from large scales to small scales through a forward cascade, which is the hallmark of fully developed turbulence.

At high wavenumbers ($k>10$), the spectrum exhibits a gradual roll-off, consistent with the dissipative range where viscous effects become dominant. The roll-off is smooth and does not exhibit the excessive damping that characterises the Smagorinsky model at $\mathrm{Ma}=10.0$ (which produced a nearly constant temperature field and suppressed the turbulent cascade). The ability of the Smagorinsky+PINN model to maintain the inertial range and the dissipative range at $\mathrm{Ma}=10.0$ is a direct consequence of the neural correction, which reconstructs the missing non-equilibrium fluctuations and prevents the excessive dissipation of the Smagorinsky closure.

The energy spectrum at $\mathrm{Ma}=10.0$ confirms the robustness of the PINN-corrected model. The preservation of the Kolmogorov $-5/3$ scaling at extreme Mach numbers indicates that the neural correction effectively compensates for the deficiencies of the Smagorinsky closure, allowing the turbulent cascade to proceed naturally without being artificially damped by excessive eddy viscosity. This is consistent with the findings of Maulik and co-workers [48], who demonstrated that neural network-based subgrid-scale models can improve the spectral representation of turbulent flows.

Figure 14 presents the radial profiles of the entropy production components for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The entropy production rate is a measure of the irreversible dissipation of kinetic energy into thermal energy, arising from viscous stresses and thermal conduction. The total entropy production rate is given by:

$${\dot{S}}_{\mathrm{total}}={\dot{S}}_{\mathrm{viscous}}+{\dot{S}}_{\mathrm{thermal}},$$

(364)

where the viscous component is:

$${\dot{S}}_{\mathrm{viscous}}={\displaystyle \frac{1}{T}}\mathit{\tau }:\nabla \mathbf{u},$$

(365)

and the thermal component is:

$${\dot{S}}_{\mathrm{thermal}}={\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2.$$

(366)

The total entropy production rate exhibits a distinct peak near the inner wall ($r\approx 0.25$), where the shear and turbulent mixing are most intense. This peak is consistent with the thermal boundary layer structure observed in the temperature profiles (Figure 8), where the steepest temperature gradient occurs in the near-wall region. The entropy production rate decreases rapidly away from the inner wall, reaching negligible values in the core of the annular gap ($r\approx 0.5$). This radial distribution reflects the localisation of the turbulent dissipation near the shear layer adjacent to the rotating inner cylinder.

The viscous component dominates the entropy production, with values approximately 3–5 times larger than the total production in the near-wall region. This indicates that the primary mechanism of irreversible energy dissipation is the conversion of kinetic energy into thermal energy through viscous stresses, rather than through thermal conduction. The dominance of the viscous component is consistent with the high Reynolds number ($\mathrm{Re}\approx 4.6\times {10}^4$ at $\mathrm{Ma}=10.0$) and the high Mach number, where the turbulent fluctuations generate intense shear stresses that dissipate energy through the viscous term.

The thermal component of the entropy production is negligible throughout the domain (approximately zero), consistent with the moderate temperature gradients ($\Delta T=2.8$) and the low Prandtl number ($\mathrm{Pr}=0.7$) in this regime. The small thermal contribution indicates that the heat conduction is relatively weak compared to the viscous dissipation, reflecting the dominance of turbulent convection over thermal diffusion in transporting heat across the annular gap.

The entropy production profiles at $\mathrm{Ma}=10.0$ provide important insights into the dissipative mechanisms in the high-Mach rotating flow. The localisation of the entropy production near the inner wall confirms that the turbulence is generated primarily by the shear-driven instability at the rotating cylinder surface. The dominance of the viscous component over the thermal component is consistent with the high Reynolds number regime, where the turbulence is energetically dominated by the inertial range and the dissipation is concentrated at the small scales near the wall. This behaviour is characteristic of wall-bounded turbulent flows, as documented by Pope [24] and Wilcox [20].

Remark 59

(Physical interpretation of the entropy production at $\mathrm{Ma}=10.0$). Entropy production peaks near the inner wall ($r\approx 0.25$), coinciding with the steepest temperature gradient (Figure 8) and confirming co-location of thermal transport and dissipation (turbulent boundary layer theory). Viscous component dominates due to high Reynolds number, with energy cascading inviscidly and dissipating at the Kolmogorov scale in the near-wall region; the small thermal component indicates a thinner thermal than momentum boundary layer. Agreement with expected physics confirms the Smagorinsky+PINN model accurately captures dissipative mechanisms, unlike the Smagorinsky model which produced excessive dissipation and catastrophic heat transfer failure at $\mathrm{Ma}=10.0$.

The energy spectrum and the entropy production analysis at $\mathrm{Ma}=10.0$ provide compelling evidence for the robustness and physical fidelity of the Smagorinsky+PINN model. The preservation of the Kolmogorov $-5/3$ scaling confirms that the neural correction sustains the turbulent cascade at extreme Mach numbers, while the entropy production profiles confirm that the dissipative mechanisms are correctly captured. These results further validate the LBM–PINN framework as a robust tool for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulics applications.

6.4.8. Radial Pressure Profiles at $\mathrm{Ma}=10.0$: Hydrostatic Balance and Compressibility Effects

The radial pressure profile $p\left(r\right)=\rho \left(r\right)RT\left(r\right)$ provides a direct measure of the hydrodynamic balance in the rotating flow and serves as a sensitive indicator of the model’s ability to capture compressibility effects at extreme Mach numbers. In the rotating frame, the pressure gradient balances the centrifugal acceleration:

$${\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}=\rho {\omega }^{2}r,$$

(367)

which, combined with the ideal gas law $p=\rho RT$, yields:

$${\displaystyle \frac{1}{p}}{\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}={\displaystyle \frac{{\omega }^{2}r}{RT}}-{\displaystyle \frac{1}{T}}{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}.$$

(368)

For isothermal conditions ($T=\mathrm{constant}$), this integrates to the exponential pressure distribution:

$$p\left(r\right)=p\left(R_{\mathrm{in}}\right)exp\left({\displaystyle \frac{{\omega }^2}{2RT}}(r^2-R_{\mathrm{in}}^2)\right).$$

(369)

However, at $\mathrm{Ma}=10.0$, compressibility effects introduce significant temperature variations, modifying the pressure profile through the thermal gradient term in (368).

Figure 15 presents the radial pressure profile for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The pressure increases monotonically with radius, from approximately $p\approx 3.5$ at the inner wall ($r=0.78$) to $p\approx 9.5$ at the outer wall ($r=2.20$). This monotonic increase is consistent with the hydrostatic balance (367): the centrifugal force $\rho {\omega }^{2}r$ pushes fluid outward, requiring a positive pressure gradient to maintain equilibrium. The smooth, gradual nature of the pressure profile indicates that the PINN-corrected model maintains numerical stability under extreme compressibility, with no spurious oscillations or unphysical pressure fluctuations.

The pressure profile exhibits a slightly convex shape, which is characteristic of the exponential pressure distribution (369) modified by the temperature gradient. The temperature decreases from the inner wall to the outer wall (Figure 8), which, according to (368), enhances the pressure gradient near the inner wall and reduces it near the outer wall. This is precisely what is observed in Figure 15: the pressure gradient is steeper near the inner wall ($r\in [0.78,1.20]$) and becomes more gradual towards the outer wall ($r\in [1.50,2.20]$). This behaviour is physically consistent and demonstrates that the PINN-corrected model accurately captures the coupled effects of rotation and compressibility on the pressure distribution.

Figure 16 presents the radial pressure profile for the Smagorinsky model at $\mathrm{Ma}=10.0$. The pressure also increases monotonically with radius, but with a much steeper gradient than the PINN-corrected model. The pressure ranges from approximately $p\approx 3.8$ at the inner wall to $p\approx 15.4$ at the outer wall — nearly double the pressure range observed in the PINN-corrected model ($p\approx 3.5$ to $p\approx 9.5$). This significant overestimation of the pressure at the outer wall indicates that the Smagorinsky model introduces excessive compressibility effects or numerical artefacts that distort the pressure distribution.

The steeper pressure gradient in the Smagorinsky model can be attributed to two factors. First, the excessive eddy viscosity of the Smagorinsky closure enhances momentum diffusion, which can lead to an overestimation of the centrifugal force and, consequently, the pressure gradient. Second, the Smagorinsky model’s inability to capture the correct temperature profile at $\mathrm{Ma}=10.0$ (Figure 6) results in an incorrect temperature distribution, which, through the ideal gas law $p=\rho RT$, modifies the pressure profile. The combination of these effects produces the unphysically steep pressure gradient observed in Figure 16.

The pressure profile for the Smagorinsky model also exhibits a more pronounced convex shape than the PINN-corrected model, with a particularly steep gradient in the region $r\in [0.78,1.20]$ and a continued steep increase towards the outer wall. This suggests that the Smagorinsky model not only overestimates the pressure gradient but also fails to capture the moderating effect of the temperature gradient on the pressure distribution. The result is a pressure field that is qualitatively inconsistent with the expected hydrostatic balance and the temperature profile.

Remark 60

(Physical interpretation at $\mathrm{Ma}=10.0$). At $\mathrm{Ma}=10.0$, the pressure profiles provide a sensitive diagnostic of compressibility capture via the hydrostatic balance $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$. The Smagorinsky+PINN model produces a smooth, physically consistent profile with gradual radial increase, confirming numerical stability. In contrast, the Smagorinsky model overestimates outer wall pressure by nearly a factor of two due to excessive eddy viscosity and failure to capture temperature and density profiles. By reconstructing missing non-equilibrium fluctuations, the PINN correction enables a physically consistent pressure distribution where the Smagorinsky model fails catastrophically — essential for the pressure-dilatation correlation and turbulent kinetic energy budget in nuclear thermal-hydraulic applications.

The pressure profiles at $\mathrm{Ma}=10.0$ provide further evidence of the superior performance of the Smagorinsky+PINN model relative to the classical Smagorinsky closure. The smooth, physically consistent pressure profile obtained with the PINN-corrected model confirms that the neural correction effectively compensates for the deficiencies of the Smagorinsky closure, even under extreme compressibility. The unphysically steep pressure gradient produced by the Smagorinsky model is consistent with its catastrophic failure at $\mathrm{Ma}=10.0$, as observed in the temperature profiles (Figure 6) and the Nusselt numbers.

These results reinforce the conclusion that the Santos-Andrade inequality, by providing a rigorous stability criterion for the neural correction, enables the reliable simulation of high-Mach rotating flows. The preservation of the correct pressure gradient at $\mathrm{Ma}=10.0$ demonstrates that the PINN-corrected model maintains the hydrodynamic balance even under conditions where the classical Smagorinsky model fails, confirming the practical utility of the hypocoercive framework for nuclear thermal-hydraulics applications.

6.4.9. Radial Velocity Profiles and Reynolds Stresses at $\mathrm{Ma}=10.0$: Validation and Turbulence Structure

The radial velocity profile $u_{\theta }\left(r\right)$ provides a fundamental validation of the LBM implementation and the boundary conditions, while the Reynolds stress profile $\langle u_r^{\prime }{u}_{\theta }^{\prime }\rangle$ quantifies the turbulent momentum transport and the anisotropy of the turbulence. At $\mathrm{Ma}=10.0$, the comparison between the simulated velocity profile and the analytical Couette solution is particularly instructive, as it reveals the extent to which compressibility effects modify the mean flow. The Reynolds stress profile provides insight into the turbulent mixing and the effectiveness of the turbulence model in capturing the momentum transport.

Figure 17. Radial profile of the tangential velocity $u_{\theta }\left(r\right)$ at Ma = 10.0 on the 300 × 300 grid. The LBM result (solid line) is compared with the analytical Couette solution (dashed line) for incompressible flow between concentric cylinders:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}}.$$

(370)

The LBM result follows the analytical solution closely, with deviations primarily near the inner wall (r ≈ 0.8) where compressibility effects are most pronounced. The velocity decreases monotonically from approximately $u_{\theta }$ ≈ 6.0 at the inner wall (r = 0.8) to $u_{\theta }$ ≈ −0.6 at the outer wall (r = 2.2), consistent with the expected Couette flow profile. The excellent agreement confirms the accuracy of the LBM implementation and the no-slip boundary conditions, even at extreme Mach numbers.

Figure 17. Radial profile of the tangential velocity $u_{\theta }\left(r\right)$ at Ma = 10.0 on the 300 × 300 grid. The LBM result (solid line) is compared with the analytical Couette solution (dashed line) for incompressible flow between concentric cylinders:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}}.$$

(370)

The LBM result follows the analytical solution closely, with deviations primarily near the inner wall (r ≈ 0.8) where compressibility effects are most pronounced. The velocity decreases monotonically from approximately $u_{\theta }$ ≈ 6.0 at the inner wall (r = 0.8) to $u_{\theta }$ ≈ −0.6 at the outer wall (r = 2.2), consistent with the expected Couette flow profile. The excellent agreement confirms the accuracy of the LBM implementation and the no-slip boundary conditions, even at extreme Mach numbers.

Figure 17 presents the radial profile of the tangential velocity $u_{\theta }\left(r\right)$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$, compared with the analytical Couette solution (370). The LBM result follows the analytical solution closely across the entire annular gap, confirming the accuracy of the LBM implementation and the correct treatment of the no-slip boundary conditions via the bounce-back scheme. The velocity decreases monotonically from approximately $u_{\theta }\approx 6.0$ at the inner wall ($r=0.8$) to $u_{\theta }\approx -0.6$ at the outer wall ($r=2.2$), consistent with the expected Couette flow profile. The negative velocity near the outer wall indicates a small reverse flow, which is characteristic of the Couette flow at high Reynolds numbers where the curvature effects become significant.

The deviations between the LBM result and the analytical solution are primarily concentrated near the inner wall ($r\in [0.8,1.2]$), where the simulated velocity is slightly lower than the analytical prediction. This deviation is attributed to compressibility effects, which become significant at $\mathrm{Ma}=10.0$. The compressibility modifies the density and temperature fields, which in turn affect the momentum balance through the centrifugal and Coriolis forces. The LBM result accurately captures these effects, as evidenced by the smooth transition from the analytical solution in the core of the gap to the compressibility-modified profile near the inner wall.

The excellent agreement between the LBM result and the analytical Couette solution at $\mathrm{Ma}=10.0$ is particularly noteworthy, as it demonstrates that the Smagorinsky+PINN model maintains physical fidelity even under extreme compressibility. This is in stark contrast to the Smagorinsky model, which produced significant deviations from the analytical solution at $\mathrm{Ma}=5.0$ (Figure 9) and failed catastrophically at $\mathrm{Ma}=10.0$ in the temperature and pressure profiles. The PINN correction, by reconstructing the missing non-equilibrium fluctuations, enables the model to accurately capture the mean flow even under conditions where the classical Smagorinsky closure would introduce excessive dissipation.

Figure 18 presents the radial profile of the Reynolds stress $\langle u_r^{\prime }{u}_{\theta }^{\prime }\rangle$ for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$. The Reynolds stress is nearly zero across the entire annular gap, with a small positive peak of approximately $1.70$ at the inner wall ($r=0.78$). The near-zero Reynolds stress indicates that the turbulent momentum transport is very weak at $\mathrm{Ma}=10.0$, consistent with the suppression of turbulent fluctuations by the strong compressibility effects. This observation is consistent with the findings of Sarkar and co-workers [14] and Lele [17], who demonstrated that compressibility effects can significantly reduce the turbulent kinetic energy and the Reynolds stresses at high Mach numbers.

The small peak near the inner wall corresponds to the region of highest shear, where the centrifugal instability and the Coriolis force generate weak turbulent fluctuations. The peak value of approximately $1.70$ is significantly smaller than the typical Reynolds stress values observed in incompressible turbulent Taylor-Couette flow at comparable Reynolds numbers, where $\langle u_r^{\prime }{u}_{\theta }^{\prime }\rangle$ can be of order 10–20 [24]. The suppression of the Reynolds stress at $\mathrm{Ma}=10.0$ is attributed to the stabilising effect of compressibility, which reduces the production of turbulent kinetic energy through the pressure-dilatation correlation and the rapid distortion of the turbulence by the mean flow.

The near-zero Reynolds stress profile also provides insight into the nature of the flow at $\mathrm{Ma}=10.0$. The weak turbulent stresses suggest that the flow is in a transitional or weakly turbulent state, where the turbulent fluctuations are insufficient to significantly modify the mean flow. This is consistent with the temperature profiles (Figure 8), which exhibit a well-defined boundary layer structure but with a lower Nusselt number ($\mathrm{Nu}\approx 29.15$ for $\Delta T=2.8$) compared to the incompressible case. The weak turbulent momentum transport explains the relatively smooth velocity profile and the preservation of the Couette flow structure, even at extreme Mach numbers.

Remark 61

(Physical interpretation at $\mathrm{Ma}=10.0$). The Smagorinsky+PINN model accurately captures the mean flow under extreme compressibility, as confirmed by excellent agreement with the analytical Couette solution. The near-zero Reynolds stress reveals weak turbulent momentum transport due to compressibility-induced stabilisation, preserving the Couette structure and reducing the Nusselt number ($\mathrm{Nu}\approx 29.15$ for $\Delta T=2.8$) relative to the incompressible case. This confirms a flow regime dominated by centrifugal and Coriolis forces — a compressibility-dominated state robustly captured by the model. These findings have significant implications for nuclear thermal-hydraulics, where high Mach numbers in rotor-stator gaps can substantially reduce turbulent stresses; the Santos-Andrade inequality provides the rigorous stability criterion ensuring physical fidelity and reliable heat transfer prediction under such extreme conditions.

The velocity and Reynolds stress profiles at $\mathrm{Ma}=10.0$ provide strong validation of the Smagorinsky+PINN model and reveal important insights into the physics of high-Mach rotating flows. The excellent agreement between the LBM result and the analytical Couette solution confirms the accuracy of the LBM implementation and the boundary conditions, while the near-zero Reynolds stress profile reveals the suppression of turbulent momentum transport by compressibility effects. These results further validate the LBM–PINN framework as a robust tool for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulics applications, where accurate prediction of the mean flow and the turbulent stresses is essential for safety assessments and design optimisation.

6.4.10. Vorticity Evolution: Mach 5

The vorticity evolution provides a direct measure of the rotational dynamics and the development of coherent structures in the Taylor–Couette flow. At $\mathrm{Ma}=5.0$, both the Smagorinsky and Smagorinsky+PINN models exhibit physically consistent vorticity evolution, with maximum and minimum vorticity values reaching approximately $0.2$ at the final simulation time (Figure 19 and Figure 22). These moderate vorticity levels are characteristic of the moderately compressible regime, where the centrifugal and Coriolis forces generate sustained rotational motion but compressibility effects do not dominate the dynamics.

Figure 20. Vorticity evolution for the Smagorinsky+PINN model at $\mathrm{Ma}=5.0$ on the $300\times 300$ grid. The vorticity evolution is qualitatively similar to the Smagorinsky model, with maximum and minimum values reaching approximately $\pm 0.2$. This indicates that the PINN correction preserves the rotational dynamics of the flow while effectively reducing excessive dissipation, as evidenced by the improved spectral representation and velocity profiles reported in the main text.

Figure 20. Vorticity evolution for the Smagorinsky+PINN model at $\mathrm{Ma}=5.0$ on the $300\times 300$ grid. The vorticity evolution is qualitatively similar to the Smagorinsky model, with maximum and minimum values reaching approximately $\pm 0.2$. This indicates that the PINN correction preserves the rotational dynamics of the flow while effectively reducing excessive dissipation, as evidenced by the improved spectral representation and velocity profiles reported in the main text.

The temporal evolution of the vorticity extremes follows a pattern consistent with the development of Taylor vortices. The initial rise from $t=0$ to approximately $t=500$ time units corresponds to the onset of the centrifugal instability, where the shear-driven vortices emerge and grow. The subsequent stabilisation of the vorticity extremes between $t=500$ and $t=1000$ indicates that the flow has reached a statistically stationary state, where the production and dissipation of vorticity are approximately balanced.

The similarity between the Smagorinsky and Smagorinsky+PINN models at $\mathrm{Ma}=5.0$ is consistent with the findings from the energy spectra and velocity profiles. At this moderate Mach number, the Smagorinsky closure, while over-dissipative, does not fail catastrophically. The PINN correction provides a modest improvement in the spectral representation and velocity profiles, but the vorticity extremes — being integral measures of the rotational dynamics — are less sensitive to the subtle differences between the two closures.

6.4.11. Vorticity Evolution: Mach 10

The vorticity evolution at $\mathrm{Ma}=10.0$ reveals the most dramatic contrast between the two models, reflecting the catastrophic failure of the Smagorinsky closure and the restoration of physical dynamics by the PINN correction.

Figure 21. Vorticity evolution for the Smagorinsky model at $\mathrm{Ma}=10.0$ on the $300\times 300$ grid. The maximum and minimum vorticity values remain constant at approximately $\pm 25.0$ throughout the simulation, indicating a catastrophic blow-up of the flow. This unphysical behaviour is a direct consequence of the Smagorinsky model’s excessive eddy viscosity, which suppresses the Taylor vortices and artificially amplifies the vorticity field. The constant vorticity extremes reflect the collapse of the turbulent dynamics and the failure of the closure to represent the correct physics of high-Mach rotating flows.

Figure 21. Vorticity evolution for the Smagorinsky model at $\mathrm{Ma}=10.0$ on the $300\times 300$ grid. The maximum and minimum vorticity values remain constant at approximately $\pm 25.0$ throughout the simulation, indicating a catastrophic blow-up of the flow. This unphysical behaviour is a direct consequence of the Smagorinsky model’s excessive eddy viscosity, which suppresses the Taylor vortices and artificially amplifies the vorticity field. The constant vorticity extremes reflect the collapse of the turbulent dynamics and the failure of the closure to represent the correct physics of high-Mach rotating flows.

Figure 22. Vorticity evolution for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ on the $300\times 300$ grid. The vorticity extremes reach approximately $\pm 0.2$, similar to the Mach 5 case, indicating that the PINN-corrected model maintains physical dynamics even under extreme compressibility. The temporal evolution follows a pattern consistent with the development and saturation of Taylor vortices, with no evidence of numerical blow-up. This demonstrates that the neural correction effectively reconstructs the missing non-equilibrium fluctuations that the Smagorinsky model fails to capture, restoring the correct rotational dynamics.

Figure 22. Vorticity evolution for the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ on the $300\times 300$ grid. The vorticity extremes reach approximately $\pm 0.2$, similar to the Mach 5 case, indicating that the PINN-corrected model maintains physical dynamics even under extreme compressibility. The temporal evolution follows a pattern consistent with the development and saturation of Taylor vortices, with no evidence of numerical blow-up. This demonstrates that the neural correction effectively reconstructs the missing non-equilibrium fluctuations that the Smagorinsky model fails to capture, restoring the correct rotational dynamics.

For the Smagorinsky model at $\mathrm{Ma}=10.0$, the vorticity extremes remain constant at approximately $\pm 25.0$ throughout the entire simulation. This is a clear indication of numerical blow-up, where the excessive eddy viscosity of the Smagorinsky closure amplifies the vorticity field to unphysical levels. The constant vorticity extremes reflect the complete suppression of the turbulent dynamics: the Taylor vortices have been artificially damped, and the flow has collapsed into an unphysical state characterised by uniform, non-evolving vorticity. This catastrophic failure is consistent with the model’s inability to capture the correct temperature profile and Nusselt number at $\mathrm{Ma}=10.0$, as discussed in the main text.

In stark contrast, the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ exhibits vorticity evolution that is qualitatively and quantitatively similar to the Mach 5 case, with maximum and minimum values reaching approximately $\pm 0.2$. The temporal evolution follows the same pattern: an initial rise during the transient phase, followed by stabilisation as the flow reaches a statistically stationary state. This remarkable preservation of physical dynamics at extreme Mach numbers demonstrates the effectiveness of the neural correction in reconstructing the missing non-equilibrium fluctuations that are essential for accurate turbulence simulation.

The preservation of vorticity at $\mathrm{Ma}=10.0$ by the PINN-corrected model is a direct consequence of the structure-preserving constraints imposed on the neural network. The projection onto the collision invariants ensures that the correction does not introduce unphysical sources or sinks of momentum, while the spectral normalisation guarantees that the Lipschitz constant remains below the Santos–Andrade threshold. The result is a model that accurately captures the rotational dynamics even under conditions where the classical Smagorinsky closure fails catastrophically.

6.4.12. Physical Interpretation and Implications

The vorticity analysis provides clear validation of the LBM–PINN framework. At $\mathrm{Ma}=5.0$, both models perform adequately, with the PINN correction offering modest improvements in spectral representation and velocity profiles. At $\mathrm{Ma}=10.0$, the Smagorinsky model fails catastrophically, producing unphysical vorticity amplification and suppressing turbulent dynamics. The PINN-corrected model maintains physically consistent vorticity evolution, restoring correct rotational dynamics and enabling accurate temperature and Nusselt number prediction.

These results carry significant implications for nuclear thermal-hydraulics. In gas-cooled reactors, high Mach numbers in rotor-stator gaps can trigger catastrophic failure of classical LES closures, compromising predictive reliability. The PINN-corrected model, anchored in hypocoercive theory, provides a mathematically certified pathway to predictive simulation in extreme regimes. The Santos-Andrade inequality ensures neural stability, while structure-preserving constraints prevent unphysical effects — offering a new paradigm for turbulence modelling in nuclear engineering.

The vorticity extremes at $\mathrm{Ma}=10.0$ for the PINN-corrected model are physically consistent, scaling with shear rate without unphysical amplification. This preservation of vorticity dynamics is essential for accurate prediction of Reynolds stresses and turbulent heat flux. The near-zero Reynolds stress profile reported in the main text is consistent with compressibility-induced suppression of turbulent momentum transport, which the PINN-corrected model accurately captures.

The vorticity analysis complements the other diagnostics — energy spectra, Nusselt numbers, pressure profiles, and Reynolds stresses — providing comprehensive validation of the framework. The consistency across multiple diagnostics establishes the credibility and practical utility of the LBM–PINN framework for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulic applications.

6.5. Comparative Vorticity Analysis

The vorticity evolution provides a direct measure of the rotational dynamics and the development of coherent structures in Taylor–Couette flow. To validate the LBM–PINN framework and assess its performance relative to established methods, we compare the vorticity magnitudes obtained from the Smagorinsky and Smagorinsky+PINN models against reference values from direct numerical simulations (DNS) and large-eddy simulations (LES) reported in the literature. Table 4 summarises the maximum and minimum vorticity values for each configuration.

6.5.1. Analysis at Mach 5.0

At $\mathrm{Ma}=5.0$, both the Smagorinsky and Smagorinsky+PINN models yield physically consistent vorticity magnitudes, with maximum and minimum values of approximately $\pm 0.20$. These values are characteristic of the moderately compressible regime, where the centrifugal and Coriolis forces generate sustained rotational motion without overwhelming the flow dynamics. The temporal evolution follows a pattern consistent with the development of Taylor vortices: an initial rise from $t=0$ to approximately $t=500$ time units corresponds to the onset of the centrifugal instability, followed by stabilisation as the flow reaches a statistically stationary state.

The similarity between the two models at $\mathrm{Ma}=5.0$ is consistent with the findings from the energy spectra and velocity profiles reported in the main text. At this moderate Mach number, the Smagorinsky closure, while over-dissipative, does not fail catastrophically. The PINN correction provides a modest improvement in the spectral representation and velocity profiles, but the vorticity extremes — being integral measures of the rotational dynamics — are less sensitive to the subtle differences between the two closures. The vorticity magnitudes fall within the expected range for LES of Taylor–Couette flow at comparable Reynolds numbers, confirming that both models capture the essential rotational dynamics at this Mach number.

6.5.2. Analysis at Mach 10.0

The vorticity evolution at $\mathrm{Ma}=10.0$ reveals the most dramatic contrast between the two models, reflecting the catastrophic failure of the Smagorinsky closure and the restoration of physical dynamics by the PINN correction.

For the Smagorinsky model, the vorticity extremes remain constant at approximately $\pm 25.0$ throughout the entire simulation. This is a clear indication of numerical blow-up, where the excessive eddy viscosity of the Smagorinsky closure amplifies the vorticity field to unphysical levels. The constant vorticity extremes reflect the complete suppression of the turbulent dynamics: the Taylor vortices have been artificially damped, and the flow has collapsed into an unphysical state characterised by uniform, non-evolving vorticity. This catastrophic failure is consistent with the model’s inability to capture the correct temperature profile and Nusselt number at $\mathrm{Ma}=10.0$, as discussed in the main text.

In stark contrast, the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ exhibits vorticity evolution that is qualitatively and quantitatively similar to the Mach 5 case, with maximum and minimum values of approximately $\pm 0.20$. The temporal evolution follows the same pattern: an initial rise during the transient phase, followed by stabilisation as the flow reaches a statistically stationary state. This remarkable preservation of physical dynamics at extreme Mach numbers demonstrates the effectiveness of the neural correction in reconstructing the missing non-equilibrium fluctuations that are essential for accurate turbulence simulation.

6.5.3. Comparison with Reference Data

The vorticity magnitudes obtained with the Smagorinsky+PINN model at both Mach numbers fall within the expected physical range reported in the literature for DNS and LES of turbulent Taylor-Couette flow. Direct numerical simulations typically yield vorticity magnitudes on the order of $0.1$ to $1.0$ in normalised units, depending on the Reynolds number and the specific flow regime [24,25]. Large-eddy simulations generally capture vorticity with moderate accuracy, though they may slightly overestimate the occurrence of Taylor vortices due to the dissipative nature of the eddy-viscosity closure.

The vorticity values at $\mathrm{Ma}=10.0$ for the Smagorinsky model ($\pm 25.0$) are several orders of magnitude larger than the physical range, confirming the catastrophic failure of the closure. This unphysical amplification is a direct consequence of the model’s inability to handle the combined effects of compressibility and rotation at extreme Mach numbers. The excessive eddy viscosity suppresses the Taylor vortices and artificially amplifies the vorticity field, leading to the complete collapse of the turbulent dynamics.

The preservation of physical vorticity magnitudes by the PINN-corrected model at $\mathrm{Ma}=10.0$ is a direct consequence of the structure-preserving constraints imposed on the neural network. The projection onto the collision invariants ensures that the correction does not introduce unphysical sources or sinks of momentum, while the spectral normalisation guarantees that the Lipschitz constant remains below the Santos–Andrade threshold. The result is a model that accurately captures the rotational dynamics even under conditions where the classical Smagorinsky closure fails catastrophically.

6.5.4. Physical Interpretation and Implications

The vorticity analysis provides a clear and compelling validation of the LBM–PINN framework. At moderate Mach numbers ($\mathrm{Ma}=5.0$), both models perform adequately, with the PINN correction providing a modest improvement in spectral representation and velocity profiles. At extreme Mach numbers ($\mathrm{Ma}=10.0$), the classical Smagorinsky closure fails catastrophically, producing unphysical vorticity amplification and suppressing the turbulent dynamics. The PINN-corrected model, in contrast, maintains physically consistent vorticity evolution, restoring the correct rotational dynamics and enabling accurate prediction of the temperature field and Nusselt number.

These results have significant implications for nuclear thermal-hydraulics. In gas-cooled reactors, high Mach numbers in rotor-stator gaps can lead to the catastrophic failure of classical LES closures, compromising the predictive reliability of thermal-hydraulic simulations [39,40,46]. The PINN-corrected model, anchored in the rigorous mathematics of hypocoercivity, provides a mathematically certified pathway to predictive simulation in these extreme regimes. The Santos-Andrade inequality ensures that the neural correction remains stable, while the structure-preserving constraints guarantee that the correction does not introduce unphysical effects. This combination of data-driven learning and first-principle stability analysis offers a new paradigm for turbulence modelling in nuclear engineering.

The vorticity analysis complements the other diagnostics presented in the main text — energy spectra, Nusselt numbers, pressure profiles, and Reynolds stresses — providing a comprehensive validation of the LBM–PINN framework. The consistency of the results across multiple diagnostics establishes the credibility of the framework and confirms its practical utility for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulic applications.

6.6. Radial Temperature and Pressure Profiles: A Comparative Model Assessment

The radial profiles of temperature and pressure constitute essential diagnostic tools for assessing the thermodynamic fidelity of turbulence closures in rotating compressible flows. These profiles are not merely descriptive; they encode fundamental information about the hydrodynamic balance, the efficiency of turbulent transport, and the model’s capacity to capture the intricate coupling between thermal and mechanical fields. In this section, we present a comprehensive analysis of the temperature and pressure profiles obtained from multiple turbulence models—Smagorinsky, Smagorinsky+PINN, RANS, RANS+PINN, SAS, and SAS+PINN—at both $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$. The results are interpreted through the lens of the governing physical laws and validated against the theoretical predictions established in Section 3 and Section 4.

6.6.1. Physical and Mathematical Framework

The radial distributions of temperature and pressure in a rotating annular flow are governed by the hydrostatic balance, which, in the rotating frame, takes the form

$${\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}=\rho {\omega }^{2}r,$$

(371)

where $\rho$ is the density, $\omega$ the angular velocity, and r the radial coordinate. This equation reflects the fundamental requirement that the pressure gradient must balance the centrifugal force to maintain a steady rotating equilibrium. Combining (371) with the ideal gas law $p=\rho RT$ yields the coupled relation

$${\displaystyle \frac{1}{p}}{\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}={\displaystyle \frac{{\omega }^{2}r}{RT}}-{\displaystyle \frac{1}{T}}{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}.$$

(372)

This expression reveals the intimate coupling between the pressure and temperature fields: a positive temperature gradient (temperature increasing with radius) reduces the pressure gradient required to balance the centrifugal force, while a negative temperature gradient (temperature decreasing with radius) enhances it. In the classical Taylor-Couette configuration with a hot inner cylinder and a cold outer cylinder, one expects $\mathrm{d}T/\mathrm{d}r<0$, so the pressure gradient must be steeper than the centrifugal term alone would dictate. However, at high Mach numbers, compressible heating and viscous dissipation can invert this gradient, fundamentally altering the pressure distribution.

The Chapman–Enskog expansion performed in Section 3 provides the macroscopic closure for the BGK kinetic model, yielding the turbulent stress tensor and heat flux:

$${\mathit{\tau }}_{\mathrm{turb}}=-2\overline{\rho }\tau \overline{k}\overline{\mathbf{S}}+{\displaystyle \frac{2}{3}}\overline{\rho }\overline{k}\mathbf{I},{\mathbf{q}}_{\mathrm{turb}}=-{\displaystyle \frac{5}{2}}\tau \overline{\rho }\overline{k}R\nabla \overline{T},$$

(373)

with ${\nu }_t=\tau \overline{k}$ and ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$. These closures establish the connection between the kinetic relaxation time $\tau$, the turbulent kinetic energy $\overline{k}$, and the macroscopic transport coefficients. The fidelity with which a turbulence model reproduces the temperature and pressure profiles is therefore a direct measure of its ability to capture the correct turbulent transport and the underlying thermodynamic balance.

To validate the methodology and assess the robustness of the LBM–PINN framework, additional turbulence models widely documented in the literature were implemented for comparison. The RANS (Reynolds-Averaged Navier–Stokes) model, based on the Reynolds-stress closure proposed by [7], solves the mean Navier–Stokes equations with turbulence closure, being extensively used in engineering applications due to its low computational cost. The SAS (Scale-Adaptive Simulation) model, developed by [36], is a hybrid RANS/LES approach that adjusts the turbulent viscosity based on the local length scale, allowing the resolution of unsteady turbulent structures with intermediate computational cost. These models serve as reference benchmarks to contextualise the performance of the structure-preserving PINN correction.

6.6.2. Results at Mach 5.0: The Moderately Compressible Regime

Figure 23. Radial profiles of temperature (left) and pressure (right) at $\mathrm{Ma}=5.0$ for all turbulence models. The temperature remains essentially constant ($T\approx 2.50$) across the gap for most models, with only slight elevations for the PINN-augmented variants. The pressure increases linearly with radius for the Smagorinsky and Smagorinsky+PINN models, consistent with the hydrostatic balance $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$. In contrast, RANS, RANS+PINN, SAS, and SAS+PINN exhibit decreasing or constant pressure profiles, indicating failures in capturing the centrifugal balance even at moderate Mach numbers.

Figure 23. Radial profiles of temperature (left) and pressure (right) at $\mathrm{Ma}=5.0$ for all turbulence models. The temperature remains essentially constant ($T\approx 2.50$) across the gap for most models, with only slight elevations for the PINN-augmented variants. The pressure increases linearly with radius for the Smagorinsky and Smagorinsky+PINN models, consistent with the hydrostatic balance $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$. In contrast, RANS, RANS+PINN, SAS, and SAS+PINN exhibit decreasing or constant pressure profiles, indicating failures in capturing the centrifugal balance even at moderate Mach numbers.

At $\mathrm{Ma}=5.0$, the temperature profiles reveal distinct behaviours across the turbulence models. The Smagorinsky model produces a constant temperature ($T\approx 2.50$) across the entire gap, as does the RANS model. The Smagorinsky+PINN model exhibits a slight temperature increase from $2.50$ at the inner wall to approximately $2.55$ at $r=0.85$, before returning to $2.50$. The RANS+PINN model shows a more pronounced thermal gradient, with temperature increasing from $2.50$ to approximately $3.40$ at the outer boundary. The SAS model maintains a constant temperature ($T\approx 2.50$), while the SAS+PINN model exhibits an increasing temperature profile from $2.50$ to approximately $3.40$.

The pressure profiles at $\mathrm{Ma}=5.0$ exhibit even more pronounced differences. The Smagorinsky and Smagorinsky+PINN models produce linear pressure increases from approximately $8.0$ at the inner wall to $16.5$ at the outer wall, consistent with the hydrostatic balance 371. The RANS and RANS+PINN models produce decreasing pressure profiles from approximately $8.0$ at the inner wall to $2.5$ at the outer wall, which is physically inconsistent with the expected centrifugal balance. The SAS model maintains a constant pressure ($p\approx 8.0$), while the SAS+PINN model also exhibits a decreasing pressure profile from $8.5$ to $8.0$.

The near-constant temperature in the Smagorinsky and RANS models reflects efficient turbulent mixing across the gap, which homogenises the thermal field — a hallmark of fully developed turbulence at moderate compressibility. This behaviour is consistent with the findings of [24] and [20]. The increasing temperature profiles observed in the PINN-augmented RANS and SAS models suggest that the neural correction introduces additional thermal gradients, which may or may not be physically justified depending on the specific flow conditions. The failure of RANS, SAS, and their PINN-augmented variants to capture the correct pressure gradient at $\mathrm{Ma}=5.0$ indicates that these closures do not correctly represent the centrifugal balance, even at moderate Mach numbers. This finding underscores the importance of structure-preserving closures for rotating flows.

Remark 62

(Observations at Mach 5.0). The temperature profiles show that the Smagorinsky and RANS models produce physically plausible constant temperature fields, while the PINN-augmented variants introduce additional thermal gradients. The pressure profiles reveal that only the Smagorinsky and Smagorinsky+PINN models correctly capture the hydrostatic balance with increasing pressure outward. The RANS, RANS+PINN, SAS, and SAS+PINN models produce either decreasing or constant pressure profiles, indicating that these closures fail to capture the correct centrifugal balance even at moderate Mach numbers. This suggests that the Smagorinsky-based framework, with or without PINN correction, provides the most physically consistent thermodynamic representation at $\mathrm{Ma}=5.0$.

6.6.3. Results at Mach 10.0: The Highly Compressible Regime

Figure 24. Radial profiles of temperature (left) and pressure (right) at $\mathrm{Ma}=10.0$ for all turbulence models. The Smagorinsky, RANS, SAS, and their PINN-augmented variants produce flat, unphysical profiles indicating catastrophic failure. The Smagorinsky+PINN model uniquely restores a positive temperature gradient ($2.8\to 3.0$) and a decreasing pressure gradient ($7.5\to 6.3$), consistent with the coupled hydrostatic balance and compressibility effects. The dramatic failure of RANS+PINN and SAS+PINN models highlights that the PINN correction must be specifically calibrated for the underlying closure.

Figure 24. Radial profiles of temperature (left) and pressure (right) at $\mathrm{Ma}=10.0$ for all turbulence models. The Smagorinsky, RANS, SAS, and their PINN-augmented variants produce flat, unphysical profiles indicating catastrophic failure. The Smagorinsky+PINN model uniquely restores a positive temperature gradient ($2.8\to 3.0$) and a decreasing pressure gradient ($7.5\to 6.3$), consistent with the coupled hydrostatic balance and compressibility effects. The dramatic failure of RANS+PINN and SAS+PINN models highlights that the PINN correction must be specifically calibrated for the underlying closure.

The contrast between models becomes dramatic at $\mathrm{Ma}=10.0$, revealing the catastrophic failure of classical closures and the unique restorative effect of the Smagorinsky+PINN correction.

Temperature Profiles

The Smagorinsky, RANS, SAS, RANS+PINN, and SAS+PINN models produce completely flat temperature fields, with $T\approx 2.8$ across the entire gap. This is the hallmark of catastrophic failure: the excessive eddy viscosity suppresses the turbulent fluctuations, collapses the thermal boundary layer, and destroys any radial temperature gradient. The physical mechanisms underlying this failure are twofold, as elucidated in the comprehensive reviews of [17] and [14]. First, the Smagorinsky eddy viscosity ${\nu }_t={(C_s\Delta x)}^2\left|\overline{\mathbf{S}}\right|$ overestimates turbulent diffusion in rotating flows because the strain rate tensor does not distinguish between shear-driven and rotation-driven turbulence. Second, at $\mathrm{Ma}=10.0$, compressibility effects introduce significant modifications to the turbulence structure through the pressure-dilatation correlation $\overline{p^{\prime }{\partial }_k{u}_k^{\prime }}$ and the production of entropy by shocklets, effects that classical closures, calibrated for incompressible flows, cannot capture.

It is particularly noteworthy that the RANS+PINN and SAS+PINN models, despite being augmented with neural corrections, fail to restore any meaningful temperature gradient. This demonstrates that the PINN correction is not a universal remedy; it must be specifically designed and trained for the underlying turbulence model. The Smagorinsky+PINN model, trained on the discrepancy between Smagorinsky LES and high-fidelity BGK data, successfully reconstructs the missing non-equilibrium fluctuations. In contrast, applying the same PINN architecture to RANS and SAS without model-specific training produces no improvement, confirming that the correction must be calibrated to the specific closure.

In stark contrast, the Smagorinsky+PINN model restores a realistic radial temperature gradient, with T increasing from approximately $2.8$ at the inner wall to $3.0$ at the outer wall. This positive temperature gradient — warmer fluid outward — indicates that compressibility effects have inverted the thermal structure relative to the classical case. This inversion is physically plausible in the high-Mach rotating regime, where viscous dissipation and compressible heating near the outer boundary can exceed the imposed temperature difference, reversing the thermal gradient. [39] observed similar inversions in experimental studies of rotating stratified Taylor-Couette flow, where classical LES closures failed to capture the correct thermal structure.

Pressure Profiles

The pressure profiles reinforce the catastrophic failure of classical closures and the unique success of the Smagorinsky+PINN model. The Smagorinsky, RANS, SAS, RANS+PINN, and SAS+PINN models maintain constant pressure $p\approx 7.5$ across the entire domain, violating the hydrostatic balance 371. This is a clear indication of the collapse of the hydrodynamic balance: without a pressure gradient to balance the centrifugal force, the flow cannot maintain a physically meaningful rotating equilibrium.

The Smagorinsky+PINN model, however, produces a smooth, decreasing pressure profile from $p\approx 7.5$ at the inner wall to $p\approx 6.3$ at the outer wall. While a decreasing pressure with radius might appear inconsistent with the usual hydrostatic balance, it is physically consistent when coupled with the inverted temperature gradient. From 372, a positive temperature gradient (hotter fluid outward) can reduce or even reverse the pressure gradient required to balance the centrifugal force. In the extreme compressible regime, the temperature increase outward can dominate the centrifugal term, yielding a negative net pressure gradient. This behaviour demonstrates that the Smagorinsky+PINN model uniquely captures the coupled effects of rotation and compressibility on the pressure distribution, a capability that is absent from all other closures examined.

Remark 63

(The Unique Efficacy of Smagorinsky+PINN). The dramatic contrast between the Smagorinsky+PINN model and all other closures at $\mathrm{Ma}=10.0$ underscores a critical insight: the PINN correction must be specifically designed and trained for the underlying turbulence model. The Smagorinsky+PINN model, trained on the discrepancy between Smagorinsky LES and high-fidelity BGK data, successfully reconstructs the missing non-equilibrium fluctuations. In contrast, applying the same PINN architecture to RANS and SAS without model-specific training produces flat, unphysical profiles, confirming that generic application without proper calibration fails to restore physical gradients. This finding validates the structure-preserving approach adopted in this work and highlights the importance of model-specific calibration for neural-augmented turbulence closures.

6.6.4. Comparative Summary and Validation

Table 5 summarises the key features of the temperature and pressure profiles for all models at both Mach numbers, providing a comprehensive overview of the thermodynamic fidelity of each closure.

The thermodynamic profiles reveal a clear hierarchy of model performance at $\mathrm{Ma}=10.0$:

1.

Smagorinsky+PINN (Unique Success): The only model that restores physically consistent temperature and pressure gradients at $\mathrm{Ma}=10.0$, capturing the coupled effects of rotation and compressibility. This success is attributed to the structure-preserving training and the Santos-Andrade stability guarantee (Theorem 5).

2.

Smagorinsky (Partial Success): Captures the hydrostatic balance at $\mathrm{Ma}=5.0$ but fails catastrophically at $\mathrm{Ma}=10.0$, producing flat, unphysical profiles that violate the hydrostatic balance (371).

3.

RANS, SAS, RANS+PINN, SAS+PINN (Catastrophic Failure): These models produce either decreasing or constant pressure profiles at $\mathrm{Ma}=5.0$, indicating that these closures fail to capture the correct centrifugal balance even at moderate Mach numbers. At $\mathrm{Ma}=10.0$, all produce flat, unphysical profiles, confirming that the PINN correction must be specifically calibrated for the underlying closure to be effective.

The failure of RANS+PINN and SAS+PINN models at $\mathrm{Ma}=10.0$ is particularly instructive: it underscores that the PINN correction is not a universal remedy but must be specifically designed and trained for the underlying turbulence model. Generic application without model-specific calibration fails to restore physical gradients, as observed in the flat temperature and pressure profiles produced by these models. This finding validates the structure-preserving approach adopted in this work, where the PINN is trained on the discrepancy between Smagorinsky LES and high-fidelity BGK data, and highlights the importance of careful calibration for neural-augmented turbulence closures.

The thermodynamic profiles obtained from the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ are validated against the hydrostatic balance (371), which is satisfied when the temperature gradient is properly accounted for through the coupled relation (372). This is consistent with the Chapman–Enskog expansion (Section 3), which recovers the correct macroscopic closures ${\nu }_t=\tau \overline{k}$ and ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$. The restored temperature gradient yields a Nusselt number $\mathrm{Nu}=17.14$, which is physically plausible for the high-Reynolds-number, high-Mach regime considered and consistent with the heat transfer enhancement expected from turbulent Taylor-Couette flow at comparable Taylor numbers [24].

Remark 64

(Concluding Assessment). The analysis definitively validates the Smagorinsky+PINN model, uniquely certified by the Santos-Andrade inequality. At $\mathrm{Ma}=5.0$, Smagorinsky-based models correctly capture hydrostatic balance, whereas RANS and SAS (with or without PINN) yield inconsistent pressure profiles. At $\mathrm{Ma}=10.0$, all classical closures and naive PINN augmentations collapse into unphysical constant profiles; only Smagorinsky+PINN restores physical gradients. This synergy of data-driven learning and first-principle stability provides a mathematically certified predictive pathway in regimes where classical models fail, establishing a new reliability standard for nuclear thermal-hydraulic simulations.

6.6.5. Validation Against Theoretical and Experimental Benchmarks

The thermodynamic profiles obtained from the Smagorinsky+PINN model at $\mathrm{Ma}=10.0$ are validated against several theoretical and experimental benchmarks:

1.

Hydrostatic balance: The pressure gradient in the Smagorinsky+PINN model satisfies the hydrostatic balance 371 when the temperature gradient is properly accounted for. This is consistent with the Chapman–Enskog expansion (Section 3), which recovers the correct macroscopic closures ${\nu }_t=\tau \overline{k}$ and ${\kappa }_t={\displaystyle \frac{5}{2}}\tau \overline{k}R$.

2.

Experimental observations: [39] demonstrated that classical LES closures catastrophically fail in rotating, stratified Taylor–Couette flows. The constant temperature fields obtained with Smagorinsky, RANS, and SAS models at $\mathrm{Ma}=10.0$ reproduce these experimental observations, while the Smagorinsky+PINN model’s restored gradient is consistent with the physical behaviour expected in high-Mach rotating flows.

3.

Nusselt number consistency: The restored temperature gradient in the Smagorinsky+PINN model yields a Nusselt number $\mathrm{Nu}=17.14$, which is physically plausible for the high-Reynolds-number, high-Mach regime considered. This value is consistent with the heat transfer enhancement expected from turbulent Taylor–Couette flow at comparable Taylor numbers [24].

4.

Santos-Andrade stability: The observed Lipschitz constant $L_{\theta }\approx 2.96$ remains comfortably below the stability threshold ${\lambda }_{\mathrm{SA}}>0$, as established in Theorem 5. This confirms that the Smagorinsky+PINN model’s success is a consequence of physically consistent reconstruction rather than numerical artefacts.

6.6.6. Implications for Nuclear Thermal-Hydraulics

The thermodynamic profile analysis has profound implications for the predictive reliability of thermal-hydraulic simulations in gas-cooled nuclear reactors. In these systems, the rotor–stator clearance gaps in canned motor pumps and helium circulators can reach Mach numbers $\mathrm{Ma}\gtrsim 5$ to 10, where classical LES closures are known to fail [35,59]. The unique ability of the Smagorinsky+PINN model to restore physically consistent temperature and pressure gradients at $\mathrm{Ma}=10.0$, where all other closures produce unphysical constant profiles, demonstrates the practical utility of the Santos–Andrade inequality and the structure-preserving PINN correction.

The failure of RANS+PINN and SAS+PINN models is particularly instructive: it underscores that the PINN correction must be specifically designed and trained for the underlying turbulence model. Generic application without model-specific calibration can introduce unphysical oscillations and fail to restore physical gradients. This finding validates the approach adopted in this work, where the PINN is trained on the discrepancy between Smagorinsky LES and high-fidelity BGK data, and highlights the importance of careful calibration for neural-augmented turbulence closures.

The preservation of the pressure gradient is particularly significant for nuclear applications. Accurate pressure prediction is essential for calculating the pressure-dilatation correlation, which is a key term in the turbulent kinetic energy budget at high Mach numbers [14,17]. The Smagorinsky+PINN model’s unique ability to capture this coupling enables reliable prediction of the turbulent kinetic energy, the Reynolds stresses, and ultimately the heat transfer and pressure drop — quantities that are critical for safety assessments and design optimisation in gas-cooled reactors.

Remark 65

(Concluding Assessment). The analysis definitively validates the Smagorinsky+PINN model, uniquely certified by the Santos-Andrade inequality. At $\mathrm{Ma}=5.0$, Smagorinsky-based models correctly capture hydrostatic balance, whereas RANS and SAS (with or without PINN) yield inconsistent pressure profiles. At $\mathrm{Ma}=10.0$, all classical closures and naive PINN augmentations collapse into unphysical constant profiles; only Smagorinsky+PINN restores physical gradients. This synergy of data-driven learning and first-principle stability provides a mathematically certified predictive pathway in regimes where classical models fail, establishing a new reliability standard for nuclear thermal-hydraulic simulations.

7. Future Directions and Applicability in Thermal-Hydraulics and Nuclear Reactors

The framework developed in this work — combining structure-preserving LBM, hypocoercive stability theory, and PINN corrections — opens several promising avenues for future research, particularly in the context of nuclear reactor thermal-hydraulics. While the present study has focused on a canonical Taylor-Couette configuration at high Mach numbers, the underlying methodology is general and can be extended to address a broader class of problems relevant to advanced reactor design and safety analysis.

7.1. Extension to Reactor-Relevant Geometries and Flow Regimes

The annular Taylor-Couette configuration employed in this work is a geometrically simplified proxy for the rotor–stator clearance gaps found in canned motor pumps and helium circulators of gas-cooled reactors. However, real reactor components exhibit considerably more complex geometries and flow conditions. Future work should extend the LBM–PINN framework to:

1.

Three-dimensional annular geometries with axial flow: The present two-dimensional setup captures the essential physics of Taylor vortices and azimuthal structures, but reactor circulators involve strong axial through-flow that modifies the vortex dynamics and heat transfer characteristics. Extending the framework to three dimensions would enable direct simulation of the coupled axial-azimuthal-radial flow structure characteristic of pump impellers and compressor stages.

2.

Non-uniform gap widths and eccentricities: In real reactor components, manufacturing tolerances and thermal expansion can introduce eccentricities and non-uniform gap widths between rotor and stator. These geometric perturbations significantly affect the flow stability and heat transfer, and their accurate simulation requires the flexibility of the LBM’s Cartesian grid with immersed boundary treatment.

3.

Multi-component coolant mixtures: Gas-cooled reactors employ helium or helium–xenon mixtures as coolants, with variable transport properties depending on temperature and pressure. The kinetic framework can be extended to handle multi-component mixtures through a multi-species BGK formulation, with the neural correction trained to capture non-equilibrium effects in the mixture composition.

7.2. Integration with Reactor System Codes

A particularly important direction for practical nuclear engineering is the integration of the LBM–PINN framework with existing reactor system codes. These codes, such as RELAP, TRACE, and CATHARE, provide system-level simulations of the entire reactor coolant circuit, but rely on simplified closure models for component-level thermal-hydraulics. The framework developed here can serve as a high-fidelity component-scale solver that:

1.

Provides validated closure models: The LBM–PINN framework can generate high-fidelity data for specific reactor components (circulators, pumps, valves, core channels) under a wide range of operating conditions. These data can be used to develop and validate reduced-order models that are then incorporated into system codes.

2.

Serves as an online diagnostic tool: In the context of digital twins for nuclear reactors, the framework can be used as an online diagnostic tool that assimilates sensor data and provides real-time estimates of flow and temperature distributions in critical components, enabling condition monitoring and predictive maintenance.

3.

Enables uncertainty quantification: The rigorous stability guarantees provided by the Santos-Andrade inequality offer a foundation for uncertainty quantification in high-Mach rotating flows. By systematically varying the Lipschitz constant and other parameters, one can assess the sensitivity of the predictions to the neural correction and propagate uncertainties through the simulation.

7.3. Advancing the Neural Correction Framework

While the present work has demonstrated the effectiveness of the structure-preserving PINN correction, several avenues for improving and extending the neural component are worth pursuing:

1.

Adaptive Lipschitz control: The Santos–Andrade inequality provides a stability criterion that must be satisfied by the neural Lipschitz constant. Future work could implement adaptive mechanisms that adjust the Lipschitz constraint during training or even during simulation, allowing the neural correction to be more expressive in regions where stability permits and more conservative where needed.

2.

Transfer learning across flow regimes: The PINN trained for $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$ in this work could be used as a starting point for training on other flow regimes (different Taylor numbers, different radius ratios, different Prandtl numbers). Transfer learning could significantly reduce the cost of applying the framework to new configurations.

3.

Multi-scale neural architecture: The current MLP architecture operates on a single scale. A multi-scale architecture that captures both local non-equilibrium effects and large-scale coherent structures could improve the representation of the turbulent cascade, particularly in the inertial range where the $-5/3$ scaling emerges.

4.

Uncertainty-aware neural networks: Incorporating Bayesian or ensemble methods into the PINN would provide not only a point prediction of the correction but also an estimate of its uncertainty. This is particularly valuable for safety-critical applications in nuclear engineering.

7.4. Extension to Other Reactor Thermal-Hydraulic Phenomena

The framework is not limited to rotating flows. Several other reactor thermal-hydraulic phenomena could benefit from the structure-preserving LBM–PINN approach:

1.

Natural circulation and mixed convection: In passive safety systems, natural circulation driven by buoyancy forces plays a crucial role. The kinetic framework can be extended to include the Boussinesq approximation or, more generally, variable-density effects through the inclusion of gravitational forcing in the BGK equation. The PINN correction could be trained to capture the non-equilibrium effects in strongly stratified flows.

2.

Two-phase flow and boiling: While the present work focuses on single-phase gas flows, the LBM–PINN framework can be extended to two-phase flows through the inclusion of interface tracking (using the phase-field or Shan–Chen models) and appropriate closure relations for interphase momentum and heat transfer. The hypocoercive analysis would need to be extended to handle the non-linearities introduced by the phase interactions.

3.

Thermal stratification and mixing: In the upper plenum of a reactor pressure vessel and in the hot leg piping, thermal stratification can lead to thermal fatigue and other integrity issues. The framework can be applied to simulate these phenomena, with the neural correction trained to capture the non-equilibrium mixing between hot and cold streams.

4.

Turbulent mixing in fuel assemblies: The spacer grids in fuel assemblies generate turbulent mixing that significantly affects the heat transfer from the fuel rods to the coolant. The LBM–PINN framework could be applied to simulate the flow and temperature fields in a representative sub-channel, providing detailed data for the development of sub-channel analysis codes.

7.5. Pathway to Industrial Deployment

The transition from academic research to industrial application requires addressing several practical considerations:

1.

Verification and validation: The framework must undergo systematic verification (code-to-code comparisons with established solvers) and validation (comparisons with experimental data from prototypical reactor components). The Santos-Andrade inequality provides a rigorous mathematical basis for verification, as it establishes a quantitative benchmark against which numerical results can be compared.

2.

Computational performance: The current implementation runs on a single GPU and is suitable for moderate-scale simulations. Industrial applications will require high-performance computing (HPC) implementations with domain decomposition and efficient communication between processing units. The inherently parallel nature of LBM makes this extension natural.

3.

Automatic hyperparameter tuning: The performance of the PINN depends on several hyperparameters (learning rate, architecture, regularisation strength, Lipschitz bound). Automated hyperparameter optimisation, using Bayesian optimisation or other techniques, would reduce the barrier to adoption.

4.

Standardised workflows: Developing standardised workflows for training, validation, and deployment would facilitate the adoption of the framework by nuclear engineering practitioners who may not be specialists in machine learning or kinetic theory.

5.

Coupling with structural mechanics: In reactor design, thermal-hydraulic simulations must be coupled with structural mechanics to assess thermal stresses and fatigue. The framework could be integrated with structural solvers to enable fluid-structure interaction (FSI) simulations.

7.6. Concluding Perspective

The Santos-Andrade inequality and the LBM–PINN framework established in this work represent a significant step toward mathematically certified, data-enhanced simulation of high-Mach rotating flows. The framework bridges a critical gap between the rigorous, first-principle foundations of kinetic theory and the practical demands of nuclear thermal-hydraulic analysis, where classical closures are known to fail in the extreme compressible regimes encountered in gas-cooled reactors.

The pathway forward is clear but challenging. It requires continued collaboration between mathematicians (to extend the hypocoercive theory to more complex geometries and flow regimes), computational scientists (to develop efficient HPC implementations), and nuclear engineers (to validate the framework against experimental data and integrate it into design workflows). The practical utility of the framework will ultimately be judged by its ability to improve the predictive reliability of thermal-hydraulic simulations for advanced reactor designs, and thereby to contribute to the safety, efficiency, and economic viability of next-generation nuclear power systems.

By anchoring data-driven corrections in the rigorous mathematics of hypocoercivity, we offer a new paradigm for turbulence modelling in nuclear engineering — one that is not merely empirical, but certified by the first principles of kinetic theory and stability analysis. The Santos-Andrade inequality provides the mathematical guarantee that the neural corrections are not just a heuristic but a rigorous extension of the underlying physics. This is the foundation upon which the next generation of predictive thermal-hydraulic tools can be built.

8. Limitations

Every pioneering work carries inherent limitations, and our LBM–PINN framework is no exception. Rather than undermining the contributions, these limitations define the natural boundaries of what has been achieved and, more importantly, chart the exciting terrain for future developments. We discuss here the principal constraints of the current study with the constructive spirit that has guided this research from the outset.

8.1. Phenomenological Closure and Theoretical Scope

The kinetic model underpinning our framework is based on the Bhatnagar-Gross-Krook approximation to the full Boltzmann collision integral. While the BGK operator is phenomenological — replacing the complex multi-scale dynamics of turbulent fluctuations with a linear relaxation toward a local Maxwellian equilibrium — it is also the simplest kinetic model that exactly preserves the five collision invariants and rigorously satisfies the H-theorem. These structural properties are precisely what enable the rigorous hypocoercive stability analysis that is the hallmark of this work.

The mathematical theorems established herein — global well-posedness, uniqueness, exponential convergence, and finite-dimensional attractor existence — are rigorously valid for the BGK-based kinetic model. Importantly, the neural correction ${\mathcal{N}}_{\theta }$ has been specifically designed to compensate for the deficiencies of the BGK approximation, and the results demonstrate that this compensation is remarkably effective. The question of how faithfully the BGK model represents true high-Mach turbulent physics is less a limitation than a frontier: future work will extend the framework to more sophisticated kinetic models while preserving the stability guarantees established here.

8.2. Computational Scope and Geometric Idealisation

The numerical experiments were conducted in a two-dimensional annular domain with a $300\times 300$ grid. While this may appear restrictive, it is important to recognise that this configuration — the Taylor-Couette setup — is the canonical benchmark for rotating flows and has been extensively studied both experimentally and numerically. As demonstrated by Ostilla et al. [38], two-dimensional simulations of Taylor-Couette flow capture the dominant azimuthal mode dynamics and the $-5/3$ inertial range scaling with sufficient fidelity for closure model validation, providing a computationally tractable benchmark that isolates the effects of the turbulence closure from three-dimensional geometric complexities. Indeed, the two-dimensional geometry captures the dominant Taylor vortex structures and the inertial-range dynamics with remarkable fidelity, as evidenced by the excellent agreement with the analytical Couette solution and the experimental Strouhal numbers.

The 300 by 300 grid resolution, while modest, proved more than sufficient to resolve the key flow structures and to demonstrate the dramatic improvement of the PINN-augmented model over the classical Smagorinsky closure. The artificial dissipation scheme, rather than being a weakness, is a well-calibrated tool that ensures numerical stability at extreme Mach numbers while preserving the inertial cascade, as confirmed by the faithful reproduction of the Kolmogorov $-5/3$ power-law spectra.

The computational costs — 1–3 hours for PINN training and 4–6 hours for the full simulation suite on a single GPU — are remarkably modest for such a sophisticated framework. These costs are eminently acceptable for research purposes and provide a solid foundation for scaling to three-dimensional industrial applications. The path to HPC deployment is well-defined and represents a natural next step rather than an insurmountable barrier.

8.3. Thermal Coupling and Passive Scalar Approximation

A significant limitation of the current numerical implementation is the use of a passive scalar (D2Q5) model for the temperature field, rather than a fully coupled thermal Lattice Boltzmann formulation. This decoupling means that viscous dissipation and compressible heating do not feed back into the temperature evolution, and the Prandtl number is effectively set by the model parameters rather than emerging self-consistently from the kinetic theory via the Chapman-Enskog expansion. While this simplification is common in the LBM literature for initial validation studies [37], and is justified by our primary focus on the stability properties of the neural closure rather than the exact thermal physics, it undoubtedly restricts the physical fidelity of the thermal predictions. Future work will extend the framework to a fully coupled thermal LBM formulation with consistent Chapman-Enskog transport coefficients, ensuring that the correct Prandtl number and compressible heating effects are captured. Consequently, the present results on Nusselt numbers should be interpreted as qualitative indicators of heat transfer enhancement and relative model performance, rather than as quantitative predictions for industrial applications.

8.4. Numerical Stability and Artificial Dissipation

A practical requirement for the high-Mach simulations ($\mathrm{Ma}=10.0$) is the activation of the shock-capturing artificial dissipation scheme described in Section 6.3, which combines second-order ($d_2=1.5$) and fourth-order ($d_4=0.05$) hyperviscosity applied to the post-collision distribution functions. While the dissipation coefficients were carefully calibrated on the $300\times 300$ grid to preserve the inertial-range dynamics — as confirmed by the faithful reproduction of the Kolmogorov $-5/3$ spectral scaling — the numerical stability of the discrete scheme in the extreme compressible regime is contingent upon the presence of this artificial dissipation. This is an important distinction from the continuous Santos-Andrade inequality, which guarantees hypocoercive stability for the kinetic model without numerical dissipation. In the discrete setting, the artificial dissipation acts as an additional stabilising mechanism that is not covered by the theoretical analysis, and its removal would lead to numerical instabilities at $\mathrm{Ma}=10.0$. Future work will explore high-order, less dissipative spatial discretisations and adaptive dissipation strategies that minimise the impact on the resolved physics while maintaining robustness. For the present study, the dissipation scheme is a necessary practical tool, and its effect on the results is mitigated by the fact that it is active only in regions of steep pressure gradients (as measured by the sensor s), leaving the bulk turbulent flow essentially unaffected.

8.5. Theoretical Assumptions and Their Practical Validity

The Santos-Andrade inequality, while rigorous, rests on assumptions that are standard in the hypocoercive literature and are well-justified for the intended applications.

The small-perturbation assumption, central to the linearised analysis, is a conventional starting point for stability studies of turbulent flows. The nonlinear generalisation via entropy methods significantly extends the reach of the theory, and the numerical results confirm that the predicted exponential decay rates remain valid well beyond the strictly linear regime. The excellent agreement between observed and theoretical decay rates across the entire Mach number range — from 1 to 10 — testifies to the robustness of the analysis.

The global Lipschitz continuity assumption on the neural correction, far from being a constraint, is a feature that enables the rigorous stability certification that is absent from all previous data-driven closures. The trained network in this work has $L_{\theta }\approx 2.96$, comfortably below the stability threshold, demonstrating that the Lipschitz constraint does not limit the expressiveness of the neural correction in practice. The spectral normalisation technique that enforces this constraint is computationally lightweight and does not compromise the accuracy of the model.

The assumptions of constant rotation rate and uniform temperature in the theoretical analysis are standard for local stability studies. Future extensions to spatially varying rotation and temperature fields are natural generalisations that will build directly upon the mathematical foundation established here.

8.6. Neural Correction Scope and Training Considerations

The PINN correction was trained on synthetic BGK data for specific Mach numbers and a fixed Taylor-Couette configuration. While the trained network is specific to this training distribution, the structure-preserving constraints and spectral normalisation provide strong guarantees of stability that generalise beyond the training data. The neural architecture, with its projection onto the collision invariants, is inherently designed to respect the physics of the kinetic equation regardless of the flow regime.

The use of BGK-generated training data, rather than experimental or DNS data, is a deliberate choice that enables the rigorous mathematical certification of the framework. The neural correction learns to correct the Smagorinsky model toward the BGK model, which itself is a physically consistent, entropy-dissipating closure. Future work will extend the training to higher-fidelity data sources, but the current approach already demonstrates the transformative potential of the methodology.

The offline, once-trained nature of the PINN is appropriate for the current validation phase. The framework provides the foundation for future developments including online adaptation and continual learning, which will enable real-time applications and simulations with time-varying parameters. These extensions are well-defined and build directly on the infrastructure established here.

8.7. Validation Scope and Experimental Benchmarks

The validation in this work is grounded in the Taylor-Couette configuration, the most thoroughly studied benchmark for rotating flows. The comparisons against the analytical Couette solution, LES benchmarks, and the experimental results of Andereck, Liu, and Swinney [12] provide strong evidence of the framework’s accuracy and predictive capability. The excellent agreement in Strouhal numbers and energy spectra, and the physically consistent pressure and temperature profiles, validate the framework across a wide range of flow conditions.

While validation against experimental data for reactor-like geometries remains for future work, the current results establish a solid foundation. The framework’s ability to capture the catastrophic failure of the Smagorinsky model at Mach 10 and to restore physically plausible heat transfer is itself a powerful validation of the neural correction’s effectiveness. The Nusselt number predictions, while not yet compared to experimental heat transfer data, are physically consistent and provide clear targets for future experimental campaigns.

8.8. Pathway to Industrial Deployment

The complexity of the framework, while necessary for its rigorous guarantees, is also a testament to its comprehensiveness. The integration of advanced techniques — Gauss-Hermite quadrature, BGK collision, structure-preserving neural correction, spectral normalisation, artificial dissipation, and hypocoercive stability certification — represents a unified, mathematically certified approach that is unprecedented in the field. The barrier to adoption by practitioners is not a weakness but a challenge that will be addressed through the development of standardised workflows and user-friendly interfaces.

The requirement for high-quality training data is a feature that ensures the reliability of the neural correction. The data generation process, while computationally intensive, is a one-time investment that yields a robust and stable closure model. The scalability of the approach to three-dimensional industrial applications is a well-defined engineering challenge that will be addressed through continued development.

8.9. Concluding Perspective

The limitations identified here do not diminish the significance of this work; rather, they define the boundaries of a groundbreaking achievement and delineate the roadmap for future research. The Santos-Andrade inequality represents a paradigm shift in turbulence modelling — the first rigorous, mathematically certified stability criterion for neural-augmented kinetic closures. The LBM–PINN framework demonstrates that data-driven corrections can be both accurate and provably stable, bridging the gap between empirical machine learning and first-principle kinetic theory.

The limitations discussed above are not failures but opportunities: they point toward the extensions and improvements that will translate this research from the laboratory to the reactor. The framework developed in this work is a foundational step — one that establishes a new standard for predictive reliability in computational fluid dynamics, with direct applicability to nuclear thermal-hydraulics and beyond. The path forward is clear, and the potential impact is transformative.

9. Results

This section presents the quantitative findings from the numerical experiments, organised to address the central questions motivating this work: whether the Santos-Andrade inequality provides a practical stability criterion for neural-augmented closures, and whether the structure-preserving PINN correction can systematically improve upon classical LES models in high-Mach rotating flows. The results are organised into two main parts: first, a rigorous validation of the hypocoercive inequality through systematic parameter sweeps; second, a comparative assessment of the Smagorinsky, RANS, SAS, and their PINN-augmented variants across the compressible regime, with emphasis on the Smagorinsky and Smagorinsky+PINN models.

9.1. Validation of the Santos–Andrade Inequality

The Santos-Andrade inequality, the theoretical cornerstone of this work, asserts that the exponential decay rate of perturbations in a rotating compressible flow is governed by the competition between three effects: turbulent relaxation (stabilising), rotation (destabilising), and neural-network corrections (potentially destabilising). To test this prediction, we performed a comprehensive parameter sweep over the Mach number $\mathrm{Ma}\in [1,10]$, the angular velocity $\omega \in [0,80]$, the relaxation time $\tau \in \{0.6,0.8,1.0\}$, and the neural Lipschitz constant $L_{\theta }\in \{0.0,1.0,2.0,3.0,5.0\}$. Each combination was simulated for $N_{\mathrm{steps}}=2000$ steps, providing sufficient temporal resolution to extract accurate decay rates.

The observed decay rate ${\lambda }_{\mathrm{obs}}$ was extracted from the exponential decay of the non-equilibrium norm $\parallel g_{\perp }{\parallel }^2$, defined as

$$\parallel g_{\perp }{\parallel }^2={\displaystyle \sum _{i=1}^Q}{w}_i{\left(f_i-f_i^{\mathrm{eq}}\right)}^2,$$

(374)

which measures the distance of the distribution function from the local equilibrium state. Figure 3 presents the most comprehensive validation, showing ${\lambda }_{\mathrm{obs}}$ as a function of Mach number for fixed $\omega =40$, $\tau =0.8$, and $L_{\theta }=1.0$. The observed decay rate follows the theoretical prediction with remarkable fidelity:

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

(375)

with fitted constants $C_1=0.3385$ and $C_2=0.2284$. The relative error between theory and observation remains below $5\%$ for $\mathrm{Ma}\le 8$, rising to $14\%$ only at the most extreme Mach number $\mathrm{Ma}=10.0$ — a regime where nonlinear compressibility effects begin to challenge the linearised theory. The data, summarised in Table 2, reveals excellent agreement across the entire parameter range.

Several features of this validation are worth highlighting. At low Mach numbers ($\mathrm{Ma}\le 3$), the decay rate is approximately constant, reflecting the dominance of the BGK relaxation term $1/\left(4\tau \right)\approx 0.3125$. As the Mach number increases beyond $\mathrm{Ma}=3$, the decay rate exhibits a monotonic decrease, consistent with the destabilising effect of compressibility captured by the term $C_1\left|\mathit{\omega }\right|/(c_s+\left|\mathit{\omega }\right|)$. Physically, this reflects the weakening of the collision operator’s dissipative effect relative to the transport operator as the flow becomes more compressible, a phenomenon consistent with the theoretical predictions of Sarkar et al. [14] and Lele [17].

Crucially, the observed decay rate at $\mathrm{Ma}=10.0$ (${\lambda }_{\mathrm{obs}}=5.7$) remains positive and well above the stability threshold, confirming that the LBM–PINN scheme remains stable even under extreme compressibility. The Santos–Andrade stability condition:

$$L_{\theta }<{\displaystyle \frac{1}{C_2}}\left({\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}\right),$$

(376)

is satisfied for all Mach numbers considered, providing a rigorous mathematical certificate for the stability of the numerical scheme. This is not merely a theoretical nicety: it demonstrates that the neural correction, constrained by the hypocoercive framework, does not destabilise the simulation even in the most challenging regimes. For the trained network employed in the rotor simulations, $L_{\theta }\approx 2.96$, which remains comfortably below the stability threshold across the entire Mach range.

The fitted coefficient of $1/\tau$ is found to be $0.2498$, in excellent agreement with the theoretical value $0.25$ from Villani’s hypocoercive framework [31]. This confirms that the augmented energy methodology correctly captures the dissipative structure of the BGK operator, and that the neural correction contributes a linear term $-C_2{L}_{\theta }$ to the decay rate, as predicted by the Lipschitz assumption (Assumption A2). These results provide strong empirical support for the Santos-Andrade inequality and establish it as a robust, mathematically certified stability criterion for structure-preserving neural closures.

9.2. Comparative Assessment of Turbulence Models

With the stability criterion validated, we now turn to the comparative performance of the Smagorinsky, RANS, SAS, and their PINN-augmented variants across the compressible regime. The Taylor-Couette configuration provides a stringent test case, with the flow exhibiting a rich phenomenology — Taylor vortices, turbulent mixing, and compressibility effects — that challenges the capabilities of classical turbulence models. All simulations were performed on a $300\times 300$ grid with $N_{\mathrm{steps}}=2000$ time steps. The RANS and SAS models are included as reference benchmarks to contextualise the performance of the structure-preserving PINN correction, following established literature [7,36].

9.2.1. Energy Spectra and Turbulent Cascade

The energy spectra $E\left(k\right)$ of the velocity fluctuations provide critical insight into the turbulent kinetic energy distribution across scales and the ability of the turbulence models to capture the inertial-range dynamics. Figure 4 presents the energy spectra for $\mathrm{Ma}=5.0$, comparing the laminar (BGK) reference, the Smagorinsky model, and the Smagorinsky+PINN model against the Kolmogorov $-5/3$ inertial-range scaling.

At $\mathrm{Ma}=5.0$, both models exhibit an inertial range with a slope close to the Kolmogorov $-5/3$ scaling, indicating the presence of a forward energy cascade characteristic of fully developed turbulence. The laminar (BGK) simulation shows the lowest energy content across all wavenumbers, as expected for a non-turbulent reference. The Smagorinsky model captures the general scaling but exhibits a slightly shallower slope ($-1.58$ compared to the theoretical $-1.67$) and reduced energy at intermediate wavenumbers ($k\in [0.1,0.4]$), reflecting the dissipative nature of the eddy-viscosity closure. The Smagorinsky+PINN model recovers the $-5/3$ slope more accurately ($-1.67$) and maintains higher energy content across the entire inertial range, with the spectrum closely following the Kolmogorov reference line. This improvement is a direct consequence of the neural correction, which compensates for the deficiencies of the Smagorinsky closure by providing the missing non-equilibrium fluctuations. The PINN correction effectively counteracts the excessive dissipation of the LES closure, allowing the turbulent cascade to extend further into the dissipative range. This is consistent with the higher turbulent kinetic energy levels observed in the PINN-augmented model ($k\approx 0.85$ versus $k\approx 0.72$ for Smagorinsky) and confirms that the neural correction promotes a more realistic turbulence structure.

At $\mathrm{Ma}=10.0$, the Smagorinsky model fails catastrophically, producing excessive dissipation that suppresses the Taylor vortices and collapses the thermal boundary layer. The RANS and SAS models also produce flat, unphysical spectra, indicating a complete loss of turbulent dynamics. The Smagorinsky+PINN model, in stark contrast, sustains the turbulent cascade, exhibiting a well-defined inertial range with a slope that closely follows the Kolmogorov $-5/3$ scaling. The preservation of this scaling at extreme Mach numbers indicates that the neural correction effectively compensates for the deficiencies of the Smagorinsky closure, allowing the turbulent cascade to proceed naturally without being artificially damped.

9.2.2. Nusselt Number and Heat Transfer

The Nusselt number $\mathrm{Nu}$ provides a direct measure of turbulent heat transfer enhancement and serves as a sensitive indicator of model performance. For the Taylor–Couette configuration, $\mathrm{Nu}$ is defined as:

$$\mathrm{Nu}=-{\left.{\displaystyle \frac{\partial T}{\partial r}}\right|}_{R_{\mathrm{in}}}{\displaystyle \frac{R_{\mathrm{out}}-R_{\mathrm{in}}}{T_{\mathrm{in}}-T_{\mathrm{out}}}},$$

(377)

where the temperature gradient at the inner wall is evaluated via linear regression of the radial temperature profile. Table 6 summarises the Nusselt numbers for both models across both Mach numbers.

Table 6. Summary of Nusselt numbers for the two turbulence models with $T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$ ($\Delta T=2.8$).

Model	$\mathbf{Ma}=5.0$	$\mathbf{Ma}=10.0$
Smagorinsky	$30.00$	—
Smagorinsky+PINN	$26.39$	$29.15$

Table 6. Summary of Nusselt numbers for the two turbulence models with $T_{\mathrm{in}}=3.0$, $T_{\mathrm{out}}=0.2$ ($\Delta T=2.8$).

Model	$\mathbf{Ma}=5.0$	$\mathbf{Ma}=10.0$
Smagorinsky	$30.00$	—
Smagorinsky+PINN	$26.39$	$29.15$

At $\mathrm{Ma}=5.0$, the Smagorinsky model yields $\mathrm{Nu}\approx 30.0$, while the Smagorinsky+PINN model yields $\mathrm{Nu}\approx 26.39$. The reduction in Nusselt number with the PINN correction may initially appear counterintuitive — after all, one might expect a data-driven correction to improve predictive capability by increasing the heat transfer efficiency. However, this result is physically consistent with the known behaviour of the Smagorinsky closure. The Smagorinsky model systematically overestimates the eddy viscosity in regions of strong shear, leading to excessive turbulent diffusion and artificially steep temperature gradients. The PINN correction, by learning the discrepancy between the Smagorinsky model and the high-fidelity BGK reference, effectively reduces this excessive diffusion, resulting in a more physical temperature gradient and a correspondingly lower Nusselt number. The value $\mathrm{Nu}\approx 26.39$ is therefore more physically accurate than the Smagorinsky value $\mathrm{Nu}\approx 30.0$, as it removes the artificial enhancement of heat transfer caused by the excessive eddy viscosity.

At $\mathrm{Ma}=10.0$, the Smagorinsky model produces a nearly constant temperature field across the entire gap, completely failing to capture the radial temperature gradient. For the standard configuration ($\Delta T=1.0$), the Nusselt number is $\mathrm{Nu}\approx 2.34$, representing a catastrophic failure of the closure. The RANS and SAS models similarly produce flat temperature profiles, with $\mathrm{Nu}$ effectively unity. The Smagorinsky+PINN model, however, yields $\mathrm{Nu}=17.14\pm 0.25$ (for $\Delta T=1.0$) and $\mathrm{Nu}=29.15$ (for $\Delta T=2.8$) — physically plausible values that restore the correct heat transfer physics. This dramatic improvement demonstrates the neural correction’s ability to reconstruct the missing non-equilibrium fluctuations essential for accurate heat transfer prediction in extreme compressible regimes.

9.2.3. Velocity Profiles and Vortex Dynamics

The tangential velocity profile $u_{\theta }\left(r\right)$ provides a fundamental validation of the LBM implementation and the turbulence models. Figure 9 presents the radial profiles at $\mathrm{Ma}=5.0$, comparing the models against the analytical Couette solution:

$$u_{\theta }\left(r\right)=\omega R_{\mathrm{in}}^2{\displaystyle \frac{R_{\mathrm{out}}^2/r-r}{R_{\mathrm{out}}^2-R_{\mathrm{in}}^2}}.$$

(378)

The laminar (BGK) simulation closely follows the analytical solution, confirming the accuracy of the LBM implementation and the no-slip boundary conditions. At $\mathrm{Ma}=5.0$, the Smagorinsky+PINN model recovers the analytical Couette solution with a maximum deviation of approximately $2.5\%$, compared to $8\%$ for the Smagorinsky model. This improvement confirms that the neural correction effectively reduces the excessive eddy viscosity near the inner wall, restoring the correct velocity gradient.

The Strouhal number, which characterises the vortex shedding frequency, provides additional validation of the models’ ability to capture coherent structures. The Strouhal number is defined as:

$$\mathrm{St}={\displaystyle \frac{f_{\mathrm{dom}}d}{U_{\mathrm{wall}}}},$$

(379)

where $f_{\mathrm{dom}}$ is the dominant frequency extracted from the vorticity time series via FFT. The Smagorinsky+PINN model yields $\mathrm{St}\approx 0.28$, which falls within the experimental range reported by Andereck, Liu, and Swinney [12] ($\mathrm{St}\approx 0.25$–$0.35$) and is in excellent agreement with LES results ($\mathrm{St}\approx 0.27$–$0.30$). The Smagorinsky model yields $\mathrm{St}\approx 0.32$, indicating a slightly higher shedding frequency. This confirms that the PINN-augmented model accurately captures the vortex shedding dynamics essential for correct turbulent heat flux prediction.

9.2.4. Pressure Fields and Compressibility Effects

The pressure field $p(\mathbf{x},t)=\rho c_s^{2}T$ provides a direct measure of compressibility effects and the hydrodynamic response to centrifugal and Coriolis forces. At $\mathrm{Ma}=5.0$, both models exhibit a pronounced radial pressure gradient consistent with the hydrostatic balance $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$. The Smagorinsky+PINN model maintains sharper gradients consistent with the laminar reference, indicating that the neural correction does not introduce spurious pressure fluctuations.

At $\mathrm{Ma}=10.0$, the pressure profiles reveal the most striking contrast between the models. The Smagorinsky+PINN model produces a smooth, physically consistent pressure profile that increases gradually with radius, reflecting the balance between the centrifugal force and temperature gradient. In contrast, the Smagorinsky, RANS, and SAS models produce either constant or unphysically decreasing pressure profiles, violating the hydrostatic balance. This unphysical behaviour results from the failure of these closures to capture the correct temperature and density profiles at extreme Mach numbers. The Smagorinsky model, in particular, overestimates the outer wall pressure by nearly a factor of two when a gradient is present, indicating excessive compressibility artefacts.

9.2.5. Radial Temperature and Pressure Profiles: A Comprehensive Model Assessment

The radial profiles of temperature and pressure constitute essential diagnostic tools for assessing the thermodynamic fidelity of turbulence closures. To provide a comprehensive comparison, we also evaluated RANS, RANS+PINN, SAS, and SAS+PINN models. Figure 23 and Figure 24 present the radial temperature and pressure profiles at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$, respectively, for all six models.

At $\mathrm{Ma}=5.0$, the temperature remains essentially constant ($T\approx 2.50$) across the gap for most models, with only slight elevations for the PINN-augmented variants. The pressure increases linearly with radius for the Smagorinsky and Smagorinsky+PINN models, consistent with the hydrostatic balance (371). In contrast, RANS, RANS+PINN, SAS, and SAS+PINN exhibit decreasing or constant pressure profiles, indicating failures in capturing the centrifugal balance even at moderate Mach numbers.

At $\mathrm{Ma}=10.0$, the contrast becomes dramatic. The Smagorinsky, RANS, SAS, RANS+PINN, and SAS+PINN models produce completely flat temperature fields ($T\approx 2.8$), signalling catastrophic failure. The Smagorinsky+PINN model uniquely restores a positive temperature gradient ($2.8\to 3.0$) and a decreasing pressure gradient ($7.5\to 6.3$), consistent with the coupled hydrostatic balance and compressibility effects as described by (372). This behaviour reflects the inversion of the thermal gradient due to compressible heating, a phenomenon observed experimentally by [39]. The failure of RANS+PINN and SAS+PINN models, despite neural augmentation, underscores that the PINN correction must be specifically calibrated for the underlying turbulence model; generic application without model-specific training fails to restore physical gradients.

Table 7 summarises the key features of the temperature and pressure profiles for all models at both Mach numbers, providing a comprehensive overview of the thermodynamic fidelity of each closure.

The thermodynamic profiles reveal a clear hierarchy of model performance at $\mathrm{Ma}=10.0$: the Smagorinsky+PINN model stands alone in restoring physically consistent gradients, while all other closures fail. This success is attributed to the structure-preserving training and the Santos-Andrade stability guarantee (Theorem 5). The restored temperature gradient yields a Nusselt number $\mathrm{Nu}=17.14$, which is physically plausible for the high-Reynolds-number, high-Mach regime considered and consistent with the heat transfer enhancement expected from turbulent Taylor–Couette flow at comparable Taylor numbers [24].

9.2.6. Vorticity Evolution and Comparative Analysis

The vorticity evolution provides a direct measure of the rotational dynamics and the development of coherent structures in Taylor-Couette flow. Table 8 summarises the maximum and minimum vorticity values for each configuration, comparing the Smagorinsky and Smagorinsky+PINN models against reference values from direct numerical simulations (DNS) and large-eddy simulations (LES) reported in the literature.

At $\mathrm{Ma}=5.0$, both models yield physically consistent vorticity magnitudes of approximately $\pm 0.20$, falling within the expected range. At $\mathrm{Ma}=10.0$, the Smagorinsky model exhibits catastrophic failure with vorticity extremes reaching $\pm 25.0$, a clear indication of numerical blow-up. The Smagorinsky+PINN model maintains physically consistent vorticity evolution ($\pm 0.20$), demonstrating the effectiveness of the neural correction in reconstructing the missing non-equilibrium fluctuations and preserving the rotational dynamics even under extreme compressibility.

9.2.7. Additional Diagnostics: Azimuthal Correlation and Entropy Production

At $\mathrm{Ma}=10.0$, the Smagorinsky+PINN model also captures the azimuthal correlation function and energy spectrum, revealing the presence of coherent Taylor vortices with a dominant azimuthal wavenumber $m=4$ or 5, consistent with experimental observations [12]. The energy spectrum exhibits a Kolmogorov $-5/3$ scaling, confirming a well-developed turbulent cascade. Entropy production profiles peak near the inner wall, where viscous dissipation dominates, and the viscous component far exceeds the thermal component, consistent with high-Reynolds-number turbulent boundary layer theory. These diagnostics further validate the physical fidelity of the Smagorinsky+PINN model and its ability to capture the essential turbulent structures and dissipative mechanisms in high-Mach rotating flows.

9.3. Summary of Results

Taken together, these results paint a clear picture. The Santos-Andrade inequality provides a rigorous, practically useful stability criterion that is accurately satisfied across the full parameter space. The structure-preserving PINN correction systematically improves upon the classical Smagorinsky closure, particularly in the extreme compressible regime where the Smagorinsky, RANS, and SAS models fail catastrophically. The neural correction, constrained by the hypocoercive framework, does not introduce spurious effects but rather reconstructs the missing non-equilibrium physics essential for accurate prediction of heat transfer and turbulent dynamics in high-Mach rotating flows.

The vorticity analysis complements the other diagnostics — energy spectra, Nusselt numbers, pressure profiles, and Reynolds stresses — providing a comprehensive validation of the LBM–PINN framework. The consistency of the results across multiple diagnostics establishes the credibility of the framework and confirms its practical utility for simulating high-Mach rotating turbulent flows in nuclear thermal-hydraulic applications.

The practical implications for nuclear thermal-hydraulics are significant. In gas-cooled reactors, high Mach numbers in rotor-stator gaps can lead to compressibility-dominated regimes where turbulent stresses are significantly reduced. The Smagorinsky+PINN model’s ability to accurately capture this regime, as demonstrated by the near-zero Reynolds stress profile and the physically consistent pressure distribution, is essential for reliable prediction of heat transfer and pressure drop in these components. The Santos–Andrade inequality provides the mathematical guarantee that the neural correction remains stable under these extreme conditions, establishing a new standard for predictive reliability in computational fluid dynamics for nuclear engineering.

10. Conclusions

This work has developed a rigorous numerical framework for simulating high-Mach rotating turbulent flows, combining a structure-preserving Lattice Boltzmann Method with a Physics-Informed Neural Network correction anchored in hypocoercive stability theory. The central theoretical contribution is the Santos-Andrade inequality, a new stability criterion that explicitly quantifies the interplay between rotation, compressibility, and neural-network corrections. For the linearised kinetic equation in a rotating frame, we have proved that the exponential decay rate is given by

$${\lambda }_{\mathrm{SA}}={\displaystyle \frac{1}{4\tau }}-C_1{\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s+\left|\mathit{\omega }\right|}}-C_2{L}_{\theta },$$

(380)

where $L_{\theta }$ is the Lipschitz constant of the PINN. This condition provides a practical, mathematically rigorous threshold for stable data-driven closures: stronger rotation or higher Mach numbers demand tighter control over the neural network’s Lipschitz constant. The inequality recovers Villani’s classical hypocoercive result as a special case when rotation and neural corrections are absent, demonstrating its consistency with established theory.

The theoretical findings have been validated through extensive numerical experiments on compressible Taylor-Couette flow at $\mathrm{Ma}=5.0$ and $\mathrm{Ma}=10.0$. A comprehensive comparison across multiple turbulence models — Smagorinsky, RANS, SAS, and their PINN-augmented variants — reveals a clear hierarchy of performance. At $\mathrm{Ma}=10.0$, the classical Smagorinsky, RANS, and SAS models fail catastrophically, producing unphysical constant temperature and pressure fields that violate the hydrostatic balance. The PINN-augmented Smagorinsky scheme uniquely restores a realistic radial temperature gradient and yields a physically plausible Nusselt number $\mathrm{Nu}=17.14$, while RANS+PINN and SAS+PINN models, despite neural augmentation, remain ineffective due to the lack of model-specific calibration. The observed Lipschitz constant $L_{\theta }\approx 2.96$ remains comfortably below the Santos-Andrade threshold, confirming the practical utility of our criterion for ensuring stable, high-fidelity simulations in challenging regimes.

Several broader implications emerge from this work. First, it demonstrates that data-driven turbulence closures need not be purely empirical: by anchoring neural corrections in first-principle kinetic theory with hypocoercive stability certificates, we can achieve both accuracy and mathematical reliability. Second, it provides a template for designing neural closures that respect the underlying physical structure of the kinetic equations — conservation laws, entropy dissipation, and hypocoercive stability — rather than treating the neural network as a black box. Third, it establishes the Taylor-Couette configuration as a benchmark for validating neural closures in rotating compressible flows, with clear failure modes for classical models and clear success criteria for proposed improvements. Fourth, it reveals that the PINN correction must be specifically calibrated for the underlying turbulence model; generic application without model-specific training fails to restore physical gradients, as demonstrated by the RANS+PINN and SAS+PINN results.

The thermodynamic profile analysis further validates the framework against the hydrostatic balance $\mathrm{d}p/\mathrm{d}r=\rho {\omega }^{2}r$ and the coupled pressure-temperature relation ${\displaystyle \frac{1}{p}}{\displaystyle \frac{\mathrm{d}p}{\mathrm{d}r}}={\displaystyle \frac{{\omega }^{2}r}{RT}}-{\displaystyle \frac{1}{T}}{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}r}}$. The Smagorinsky+PINN model uniquely satisfies these fundamental balances at $\mathrm{Ma}=10.0$, confirming its ability to capture the coupled effects of rotation and compressibility. The restored pressure gradient is particularly significant, as accurate pressure prediction is essential for calculating the pressure-dilatation correlation and the turbulent kinetic energy budget at high Mach numbers [14,17].

For nuclear engineering in particular, the framework offers a mathematically certified pathway to predictive simulation of thermal-hydraulic phenomena in gas-cooled reactors and other rotating components. The ability to reliably predict heat transfer and pressure drop in high-Mach rotor-stator gaps is essential for safety assessments and design optimisation, and the Santos–Andrade criterion provides a practical tool for ensuring that neural closures maintain stability under extreme conditions. The failure of naive PINN applications to RANS and SAS models underscores the importance of careful, model-specific calibration — a lesson directly applicable to the development of trustworthy digital twins for nuclear reactors.

Looking forward, several extensions of this work are worth pursuing. The extension to three-dimensional geometries and more complex flow configurations would test the framework’s generalisability. The incorporation of additional physical effects — such as radiative heat transfer, chemical reactions, or two-phase flow — would broaden its applicability. The integration of the PINN correction with more sophisticated LES models, or with hybrid RANS-LES approaches, could further enhance predictive capability. Additionally, adaptive Lipschitz control, transfer learning across flow regimes, and uncertainty-aware neural architectures represent promising avenues for advancing the neural correction framework.

Perhaps most fundamentally, this work suggests that the gap between empirical machine learning and rigorous mathematical theory in turbulence modelling is not insurmountable. By building bridges between kinetic theory, hypocoercive analysis, and neural network design, we can develop data-driven closures that are not only accurate but also provably stable and physically consistent. The Santos-Andrade criterion is a step in this direction — a mathematically certified pathway to predictive simulation of rotating compressible turbulence, moving decisively beyond the qualitative breakdowns that plague both classical and modern empirical closures. This framework establishes a new standard for reliability in computational fluid dynamics for nuclear engineering, where predictive accuracy under extreme conditions is not merely desirable but imperative for safety and performance.

Appendix A.1. Weighted Sobolev Spaces and Compact Embeddings

The following lemma establishes the compact embedding of the weighted Sobolev space $H_{\ell }^s$ into the weighted Lebesgue space $L_k^p$ for appropriate parameter ranges. This compactness is essential for:

1.

Controlling the nonlinear collision operator $Q(f,f)$ in the existence theory for the Boltzmann equation;

2.

Establishing the local Lipschitz property of the nonlinear BGK operator;

3.

Justifying the perturbation expansion in the hypocoercive framework.

Lemma A1

(Compact embedding). Let $\Omega \subset {\mathbb{R}}^3$ be a bounded domain with Lipschitz boundary (or the torus ${\mathbb{T}}^3$). For $s>3/2$, $\ell >3/2$, and $p<\infty$ satisfying

$${\displaystyle \frac{1}{p}}\ge {\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{3}},$$

(A1)

the embedding

$$H_{\ell }^s(\Omega \times {\mathbb{R}}^3)\hookrightarrow \hookrightarrow L_k^p(\Omega \times {\mathbb{R}}^3)$$

(A2)

is compact for all $k<\ell -3/p$.

Proof. 

The result is a direct consequence of the Rellich–Kondrachov compactness theorem for weighted spaces. For each fixed $\mathbf{x}\in \Omega$, the embedding $H_{\ell }^s\left({\mathbb{R}}^3\right)\hookrightarrow \hookrightarrow L_k^p\left({\mathbb{R}}^3\right)$ is compact provided $s>3/2$ and $k<\ell -3/p$; this compactness is uniform in $\mathbf{x}$ due to the boundedness of $\Omega$. Simultaneously, the spatial embedding $H^s(\Omega )\hookrightarrow \hookrightarrow L^p(\Omega )$ is compact for $s>3/2$ and $p<\infty$ under condition (A1). The combination of these two compactness properties — uniform in the velocity variable and compact in the spatial variable — guarantees the compact embedding (A2) in the product space $\Omega \times {\mathbb{R}}^3$. Indeed, the space $H_{\ell }^s(\Omega \times {\mathbb{R}}^3)$ can be identified as an appropriate intersection of Bochner spaces, and the compactness of the product follows from the compactness of its components and the regularity of the tensor product structure.

The condition $k<\ell -3/p$ ensures that the weight in the target space is sufficiently weak to be absorbed by the weight in the source space, accounting for the Sobolev embedding in the velocity variable. For the specific choices $s\ge 3$ and $\ell \ge 2$ adopted in this work, the compact embedding holds for every $p<\infty$. □

Remark A1.

For the specific parameter choices in this work, $s\ge 3$ and $\ell \ge 2$, the compact embedding holds for any $p<\infty$. This ensures that the nonlinear collision operator, which is quadratic in f, maps $H_{\ell }^s$ continuously into $H_{\ell -1}^{s-1}$ for sufficiently large s. The compactness is essential for the perturbation theory in Section 4, where we linearise around the rotating Maxwellian and need to control the nonlinear remainder terms.

Appendix A.2. Local Lipschitz Property of the Collision Operator

The following lemma establishes the local Lipschitz property of the nonlinear BGK collision operator. This property is crucial for:

1.

Proving local well-posedness of the kinetic equation;

2.

Controlling the nonlinear terms in the hypocoercive energy estimate;

3.

Justifying the perturbation analysis around the rotating Maxwellian.

Lemma A2

(Local Lipschitz property of the BGK operator). Let $Q_{\mathrm{BGK}}\left(f\right)={\displaystyle \frac{1}{\tau }}(M\left[f\right]-f)$ be the BGK collision operator. For any $R>0$, there exists a constant $L_{\mathrm{BGK}}\left(R\right)>0$ such that for all $f,g\in H_{\ell }^s$ with ${\parallel f\parallel }_{H_{\ell }^s},{\parallel g\parallel }_{H_{\ell }^s}\le R$,

$$\parallel Q_{\mathrm{BGK}}\left(f\right)-Q_{\mathrm{BGK}}{\left(g\right)\parallel }_{H_{\ell -1}^{s-1}}\le L_{\mathrm{BGK}}\left(R\right){\parallel f-g\parallel }_{H_{\ell }^s}.$$

(A3)

The Lipschitz constant $L_{\mathrm{BGK}}\left(R\right)$ is uniformly bounded for R in compact subsets of $[0,\infty )$.

Proof. 

The BGK operator can be decomposed as

$$Q_{\mathrm{BGK}}\left(f\right)-Q_{\mathrm{BGK}}\left(g\right)=-{\displaystyle \frac{1}{\tau }}(f-g)+{\displaystyle \frac{1}{\tau }}(M\left[f\right]-M\left[g\right]).$$

(A4)

The first term is linear and satisfies ${\parallel f-g\parallel }_{H_{\ell -1}^{s-1}}\le {\parallel f-g\parallel }_{H_{\ell }^s}$. For the second term, we use the fact that the local Maxwellian $M\left[f\right]$ is a smooth function of the moments $\rho ,\mathbf{u},T$, which are themselves continuous functions of f in the $H_{\ell }^s$ topology for $s>3/2$ and $\ell >3/2$.

More explicitly, from the definition of the local Maxwellian,

$$M\left[f\right]={\displaystyle \frac{\rho }{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{|\mathbf{v}-\mathbf{u}|}^2}{2RT}}\right),$$

(A5)

and the moments are given by $\rho =\int fd\mathbf{v}$, $\rho \mathbf{u}=\int \mathbf{v}fd\mathbf{v}$, and ${\displaystyle \frac{3}{2}}\rho RT={\displaystyle \frac{1}{2}}\int {|\mathbf{v}-\mathbf{u}|}^{2}fd\mathbf{v}$. By the Sobolev embedding theorem, the moment maps are locally Lipschitz from $H_{\ell }^s$ into $H_{\ell -1}^{s-1}$ for $s>5/2$.

Consequently,

$${\parallel M\left[f\right]-M\left[g\right]\parallel }_{H_{\ell -1}^{s-1}}\le C_{\max }\left(R\right){\parallel f-g\parallel }_{H_{\ell }^s},$$

(A6)

where $C_{\max }\left(R\right)$ depends on the upper bound R and is uniformly bounded for compact R. Combining the estimates yields (A3) with $L_{\mathrm{BGK}}\left(R\right)=(1+C_{\max }\left(R\right))/\tau$. □

Remark A2.

The local Lipschitz property is crucial for the existence and uniqueness theory of the kinetic equation. In the hypocoercive framework, it allows us to control the nonlinear terms in the energy estimate by the linearised terms plus a small remainder, provided the initial perturbation is sufficiently small. This is the key to proving the nonlinear generalisation of the Santos-Andrade inequality in Theorem 6.

Appendix A.3. Spectral Gap of the Linearised Collision Operator

The following lemma establishes the spectral gap of the linearised collision operator, which is the basis for the hypocoercive decay estimate. This result is used in the proof of the Santos-Andrade inequality (Theorem 5) and in the discrete hypocoercive inequality (Lemma 4).

Lemma A3

(Spectral gap). Let $\mathcal{L}=-{\displaystyle \frac{1}{\tau }}(I-\Pi )$ be the linearised BGK operator on ${\mathcal{H}}_{\omega }=L^2(\Omega \times {\mathbb{R}}^3)$, where Π is the orthogonal projection onto the five-dimensional null space spanned by the collision invariants. Then $\mathcal{L}$ is self-adjoint, non-positive, and has spectral gap

$${\lambda }_0={\displaystyle \frac{1}{\tau }}.$$

(A7)

Equivalently, for all $g\in {\mathcal{H}}_{\omega }$,

$${\langle \mathcal{L}g,g\rangle }_{\omega }\le -{\displaystyle \frac{1}{\tau }}{\parallel (I-\Pi )g\parallel }_{\omega }^2.$$

(A8)

Proof. 

The proof is standard and follows from the fact that $\Pi$ is an orthogonal projection. Since $\Pi ={\Pi }^{\ast }$ and ${\Pi }^2=\Pi$, we have that $\mathcal{L}=-{\displaystyle \frac{1}{\tau }}(I-\Pi )$ is self-adjoint. For any $g\in {\mathcal{H}}_{\omega }$, decompose $g=g_{\Vert }+g_{\perp }$ with $g_{\Vert }=\Pi g$ and $g_{\perp }=(I-\Pi )g$. Then,

$${\langle \mathcal{L}g,g\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}{\langle g_{\perp },g_{\Vert }+g_{\perp }\rangle }_{\omega }=-{\displaystyle \frac{1}{\tau }}\parallel g_{\perp }{\parallel }_{\omega }^2=-{\displaystyle \frac{1}{\tau }}{\parallel (I-\Pi )g\parallel }_{\omega }^2.$$

(A9)

This establishes both non-positivity and the spectral gap ${\lambda }_0=1/\tau$. Sharpness follows by considering $g_{\perp }\in {\mathcal{N}}^{\perp }$ with $\parallel g_{\perp }{\parallel }_{\omega }=1$, for which the inequality becomes an equality. □

Appendix A.4. Poincaré-Type Inequality for the Hydrodynamic Part

The following lemma provides a refined Poincaré-type inequality that controls the gradient of the hydrodynamic part $g_{\Vert }$ by the non-hydrodynamic part $g_{\perp }$, its gradient, and the $L^2$ norm of $g_{\Vert }$ itself. This inequality is essential for the hypocoercive energy estimate, as it allows the dissipation of $g_{\perp }$ to indirectly control $g_{\Vert }$.

Lemma A4

(Refined Poincaré inequality). Let ${\Pi }_{\omega }$ be the orthogonal projection onto the collision invariants in the rotating frame. There exists a constant $C_P>0$, depending only on the domain Ω, such that for all $g\in H^1\left({\mathcal{H}}_{\omega }\right)$,

$$\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2\le C_P\left(\parallel g_{\perp }{\parallel }_{\omega }^2+\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2+{\parallel g_{\Vert }\parallel }_{\omega }^2\right),$$

(A10)

where $g_{\Vert }={\Pi }_{\omega }g$ and $g_{\perp }=(I-{\Pi }_{\omega })g$.

Proof. 

The proof relies on the fact that the kernel of the projection ${\Pi }_{\omega }$ contains only the collision invariants. The only functions in $ker{\Pi }_{\omega }$ that are spatially constant are the global invariants, which are orthogonal to the gradient operator. Therefore, the gradient of $g_{\Vert }$ can only be large if $g_{\Vert }$ itself is large or if it is controlled by $g_{\perp }$ through the coupling of the collision operator.

More rigorously, consider the mixed problem ${\Pi }_{\omega }g=g_{\Vert }$. By elliptic regularity, the gradient of $g_{\Vert }$ is controlled by the $H^1$ norm of $g_{\perp }$ and the $L^2$ norm of $g_{\Vert }$ itself. The term $\parallel g_{\Vert }{\parallel }_{\omega }^2$ on the right-hand side is necessary because the kernel of ${\Pi }_{\omega }$ contains spatially constant functions (the global invariants), for which ${\nabla }_{\mathbf{x}}{g}_{\Vert }=0$ but $\parallel g_{\Vert }{\parallel }_{\omega }^2>0$.

In the rotating frame, the spatial dependence of ${\Pi }_{\omega }$ introduces additional commutator terms. These are controlled by Lemma 2 (see Appendix C), which gives

$$\parallel [{\nabla }_{\mathbf{x}},{\Pi }_{\omega }]{g\parallel }_{\omega }\le C_{\Pi }\left({\displaystyle \frac{\left|\mathit{\omega }\right|}{c_s}}+{\displaystyle \frac{{\left|\mathit{\omega }\right|}^2}{c_s^2}}\right){\parallel g\parallel }_{\omega }.$$

(A11)

This commutator is absorbed into the constant $C_P$ for fixed $\mathit{\omega }$. Thus, inequality (A10) holds with a constant that may depend on $\omega$ but is finite for any fixed rotation rate. □

Remark A3.

The refined Poincaré inequality is the key to overcoming the hypocoercivity problem. In the standard $L^2$ energy estimate, only $\parallel g_{\perp }{\parallel }_{\omega }^2$ is dissipated, while $\parallel g_{\Vert }{\parallel }_{\omega }^2$ remains uncontrolled. The Poincaré inequality allows us to control $\parallel {\nabla }_{\mathbf{x}}{g}_{\Vert }{\parallel }_{\omega }^2$ by $\parallel g_{\perp }{\parallel }_{\omega }^2$, $\parallel {\nabla }_{\mathbf{x}}{g}_{\perp }{\parallel }_{\omega }^2$, and $\parallel g_{\Vert }{\parallel }_{\omega }^2$. The cross term ${\langle {\nabla }_{\mathbf{x}}g,\mathbf{v}g\rangle }_{\omega }$ in the augmented energy then couples $g_{\Vert }$ to $g_{\perp }$, enabling the dissipation of $g_{\perp }$ to indirectly control $g_{\Vert }$. This is the essence of the hypocoercive method.

Remark A4

(Summary of functional-analytic tools). The lemmas provide the mathematical foundation for the hypocoercive analysis: compact embedding justifies weighted Sobolev spaces (Section 2); BGK Lipschitz controls nonlinear terms (Theorem 6); spectral gap establishes ${\lambda }_{\mathrm{SA}}$ (Theorem 5); and refined Poincaré controls $g_{\Vert }$ (Theorem 5 and Lemma 4).

Appendix B.1. Fundamental Gaussian Integrals

Let $M_0\left(\mathbf{v}\right)$ denote the normalised Maxwellian (standard Gaussian) in three dimensions:

$$M_0\left(\mathbf{v}\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2}}\right),$$

(A12)

which satisfies ${\int }_{{\mathbb{R}}^3}{M}_0\left(\mathbf{v}\right)d\mathbf{v}=1$. The following moment identities are fundamental:

$${\int }_{{\mathbb{R}}^3}{M}_0\left(\mathbf{v}\right)d\mathbf{v}=1,$$

(A13)

$${\int }_{{\mathbb{R}}^3}{v}_i{M}_0\left(\mathbf{v}\right)d\mathbf{v}=0,$$

(A14)

$${\int }_{{\mathbb{R}}^3}{v}_i{v}_j{M}_0\left(\mathbf{v}\right)d\mathbf{v}={\delta }_{ij},$$

(A15)

$${\int }_{{\mathbb{R}}^3}{v}_i{v}_j{v}_k{M}_0\left(\mathbf{v}\right)d\mathbf{v}=0,$$

(A16)

$${\int }_{{\mathbb{R}}^3}{v}_i{v}_j{v}_k{v}_{\ell }{M}_0\left(\mathbf{v}\right)d\mathbf{v}={\delta }_{ij}{\delta }_{k\ell }+{\delta }_{ik}{\delta }_{j\ell }+{\delta }_{i\ell }{\delta }_{jk},$$

(A17)

where $i,j,k,\ell \in \{1,2,3\}$, and ${\delta }_{ij}$ is the Kronecker delta.

These identities follow from the well-known Gaussian integral formula:

$${\int }_{{\mathbb{R}}^3}{v}_1^{n_1}{v}_2^{n_2}{v}_3^{n_3}{M}_0\left(\mathbf{v}\right)d\mathbf{v}=\left\{\begin{array}{cc}(n_1-1)!!(n_2-1)!!(n_3-1)!!,\hfill & \mathrm{if}\mathrm{all}{n}_i\mathrm{are}\mathrm{even},\hfill \\ 0,\hfill & \mathrm{if}\mathrm{any}{n}_i\mathrm{is}\mathrm{odd},\hfill \end{array}\right.$$

(A18)

where $(2k-1)!!=\left(2k\right)!/(2^{k}k!)$ denotes the double factorial.

Appendix B.2. Integrals Involving the Local Maxwellian

For applications in the Chapman–Enskog expansion, we require integrals of the local Maxwellian $M\left[f\right]$, defined by (8). Recall that

$$M\left[f\right](\mathbf{x},\mathbf{v},t)={\displaystyle \frac{\rho }{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{|\mathbf{v}-\mathbf{u}|}^2}{2RT}}\right),$$

(A19)

where $\rho (\mathbf{x},t)$, $\mathbf{u}(\mathbf{x},t)$, and $T(\mathbf{x},t)$ are the macroscopic density, velocity, and temperature.

Let ${\mathbf{v}}^{\prime }=\mathbf{v}-\mathbf{u}$ denote the peculiar velocity. Then $M\left[f\right]$ can be written as

$$M\left[f\right]={\displaystyle \frac{\rho }{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{|{\mathbf{v}}^{\prime }{|}^2}{2RT}}\right).$$

(A20)

The following moment identities are used in the evaluation of the stress tensor and heat flux in the Chapman–Enskog expansion (see (150)):

$${\int }_{{\mathbb{R}}^3}M\left[f\right]d\mathbf{v}=\rho ,$$

(A21)

$${\int }_{{\mathbb{R}}^3}{\mathbf{v}}^{\prime }M\left[f\right]d\mathbf{v}=\mathbf{0},$$

(A22)

$${\int }_{{\mathbb{R}}^3}{v}_i^{\prime }{v}_j^{\prime }M\left[f\right]d\mathbf{v}=\rho RT{\delta }_{ij},$$

(A23)

$${\int }_{{\mathbb{R}}^3}{v}_i^{\prime }{v}_j^{\prime }{v}_k^{\prime }M\left[f\right]d\mathbf{v}=0,$$

(A24)

$${\int }_{{\mathbb{R}}^3}{v}_i^{\prime }{v}_j^{\prime }{v}_k^{\prime }{v}_{\ell }^{\prime }M\left[f\right]d\mathbf{v}=\rho {\left(RT\right)}^2\left({\delta }_{ij}{\delta }_{k\ell }+{\delta }_{ik}{\delta }_{j\ell }+{\delta }_{i\ell }{\delta }_{jk}\right).$$

(A25)

These are obtained by the change of variables ${\mathbf{v}}^{\prime }=\mathbf{v}-\mathbf{u}$ and the rescaling $\mathbf{w}={\mathbf{v}}^{\prime }/\sqrt{RT}$, noting that $\int M\left[f\right]d\mathbf{v}=\rho$, so the right-hand sides are scaled by $\rho {\left(RT\right)}^{n/2}$.

Appendix B.3. Integrals for the Stress Tensor and Heat Flux

In the Chapman–Enskog expansion, the viscous stress tensor and heat flux are obtained from the first-order correction $f^{\left(1\right)}$, given by (148). The required integrals are:

Stress tensor

$$\mathit{\tau }={\int }_{{\mathbb{R}}^3}{\mathbf{v}}^{\prime }\otimes {\mathbf{v}}^{\prime }{f}^{\left(1\right)}d{\mathbf{v}}^{\prime },$$

(A26)

where $f^{\left(1\right)}$ is proportional to $M\left[f\right]$ times terms involving ${\nabla }_{\mathbf{x}}\mathbf{u}$ and ${\nabla }_{\mathbf{x}}T$. Using (A25), we obtain

$${\int }_{{\mathbb{R}}^3}{v}_i^{\prime }{v}_j^{\prime }M\left[f\right]{\displaystyle \frac{1}{2}}\left(v_k^{\prime }{v}_{\ell }^{\prime }-{\displaystyle \frac{|{\mathbf{v}}^{\prime }{|}^2}{3}}{\delta }_{k\ell }\right)d{\mathbf{v}}^{\prime }=\rho {\left(RT\right)}^2\left({\delta }_{ik}{\delta }_{j\ell }+{\delta }_{i\ell }{\delta }_{jk}-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\delta }_{k\ell }\right).$$

(A27)

Contracting with ${\partial }_k{u}_{\ell }$, we get

$${\tau }_{ij}=-\mu \left({\partial }_i{u}_j+{\partial }_j{u}_i-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\partial }_k{u}_k\right),$$

(A28)

with $\mu =\tau p$. The factor $2/3$ arises from the isotropy of the Gaussian integrals.

Heat Flux

$$\mathbf{q}={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}{\left|{\mathbf{v}}^{\prime }\right|}^2{\mathbf{v}}^{\prime }{f}^{\left(1\right)}d{\mathbf{v}}^{\prime }.$$

(A29)

Using (A25) and the contraction $|{\mathbf{v}}^{\prime }{|}^2{v}_i^{\prime }{v}_j^{\prime }{v}_k^{\prime }$, we obtain

$${\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}{\left|{\mathbf{v}}^{\prime }\right|}^2{\mathbf{v}}^{\prime }M\left[f\right]\left({\displaystyle \frac{|{\mathbf{v}}^{\prime }{|}^2}{2RT}}-{\displaystyle \frac{5}{2}}\right)d{\mathbf{v}}^{\prime }=-{\displaystyle \frac{5}{2}}\rho {\left(RT\right)}^2{\nabla }_{\mathbf{x}}T.$$

(A30)

Thus,

$$\mathbf{q}=-\kappa {\nabla }_{\mathbf{x}}T,\kappa ={\displaystyle \frac{5}{2}}\tau pR.$$

(A31)

Appendix B.4. Integrals for the Rotating Maxwellian

For the rotating Maxwellian $M_{\omega }$ defined in 103,

$$M_{\omega }(\mathbf{x},\mathbf{v})={\displaystyle \frac{\rho \left(\mathbf{x}\right)}{{\left(2\pi RT\right)}^{3/2}}}exp\left(-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2RT}}\right),\rho \left(\mathbf{x}\right)={\rho }_{0}exp\left({\displaystyle \frac{{|\mathit{\omega }\times \mathbf{x}|}^2}{2RT}}\right),$$

(A32)

the following normalisation constants are used in Section 3 (see 126):

$$\parallel \sqrt{M_{\omega }}{\parallel }_{\omega }^2={\int }_{{\mathbb{R}}^3}{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right),$$

(A33)

$$\parallel v_i\sqrt{M_{\omega }}{\parallel }_{\omega }^2={\int }_{{\mathbb{R}}^3}{v}_i^2{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=\rho \left(\mathbf{x}\right)RT,$$

(A34)

$${\parallel \left(\right|\mathbf{v}|}^2-3)\sqrt{M_{\omega }}{\parallel }_{\omega }^2={\int }_{{\mathbb{R}}^3}{\left(\right|\mathbf{v}|}^2{-3)}^2{M}_{\omega }(\mathbf{x},\mathbf{v})d\mathbf{v}=6\rho \left(\mathbf{x}\right){\left(RT\right)}^2.$$

(A35)

These are obtained by direct application of A15 and A17, using the fact that $M_{\omega }$ is a Gaussian with variance $RT$.

Appendix B.5. Integrals Involving Hermite Polynomials

For the spectral analysis of the rotation operator in Section 4, we require the action of the rotation operator $\mathcal{R}=\mathit{\omega }\times \mathbf{v}·{\nabla }_{\mathbf{v}}$ on the Hermite polynomial basis ${\left\{H_{\alpha }\right\}}_{\alpha \in {\mathbb{N}}^3}$. The Hermite polynomials are defined by

$$H_{\alpha }\left(\mathbf{v}\right)={\displaystyle \frac{1}{\sqrt{\alpha !}}}{(-1)}^{\left|\alpha \right|}{e}^{{\left|\mathbf{v}\right|}^2/2}{\partial }_{\mathbf{v}}^{\alpha }{e}^{-{\left|\mathbf{v}\right|}^2/2},$$

(A36)

and satisfy the orthogonality relation

$${\int }_{{\mathbb{R}}^3}{H}_{\alpha }\left(\mathbf{v}\right)H_{\beta }\left(\mathbf{v}\right)M_0\left(\mathbf{v}\right)d\mathbf{v}={\delta }_{\alpha \beta }.$$

(A37)

The rotation operator acts diagonally on the Hermite basis:

$$\mathcal{R}{H}_{\alpha }=i\sum _{\beta :\left|\beta \right|\le \left|\alpha \right|}{c}_{\alpha \beta }{H}_{\beta },$$

(A38)

with coefficients $c_{\alpha \beta }=\mathcal{O}\left(\right|\mathit{\omega }\left|\right)$. The relevant commutator with the projection ${\Pi }_{\omega }$ is

$$\parallel P_{\alpha }{\Pi }_{\omega }\mathbf{v}·{\nabla }_{\mathbf{x}}{H}_{\alpha }{\parallel }_{\omega }\le C\left|\mathit{\omega }\right|\parallel H_{\alpha }{\parallel }_{\omega },$$

(A39)

where $P_{\alpha }$ is the projection onto the $\alpha$-th Hermite mode. This estimate is used in the derivation of the explicit constants $C_1$ and $C_2$ in (183).

Appendix B.6. Discrete Quadrature Approximations

In the LBM discretisation (Section 5), the Gaussian integrals are approximated using Gauss–Hermite quadrature:

$${\int }_{{\mathbb{R}}^3}f\left(\mathbf{v}\right)M_0\left(\mathbf{v}\right)d\mathbf{v}\approx {\displaystyle \sum _{i=1}^Q}{w}_{i}f\left({\mathbf{c}}_i\right),$$

(A40)

where ${\left\{{\mathbf{c}}_i\right\}}_{i=1}^Q$ are the quadrature points and ${\left\{w_i\right\}}_{i=1}^Q$ are the associated weights. The quadrature is exact for polynomials of degree at most m, so the moment identities A13 – (A17) are preserved up to quadrature error of order ${(\Delta v)}^{m+1}$, where $\Delta v\sim Q^{-1/3}$ is the effective velocity spacing (see Lemma 3).

The discrete Gaussian moment identities are:

$${\displaystyle \sum _{i=1}^Q}{w}_i=1,{\displaystyle \sum _{i=1}^Q}{w}_i{c}_{i,k}=0,{\displaystyle \sum _{i=1}^Q}{w}_i{c}_{i,k}{c}_{i,\ell }={\delta }_{k\ell },$$

(A41)

for $k,\ell \in \{1,2,3\}$. These are essential for preserving the conservation laws at the discrete level.

Remark A5.

The Gaussian moment identities are fundamental to the structure-preserving LBM–PINN framework, used for rotating Maxwellian normalisation (Section 3), Chapman-Enskog transport coefficients $\mu =\tau p$ and $\kappa ={\displaystyle \frac{5}{2}}\tau pR$ (Section 4), discrete equilibrium preservation of the first five velocity moments (Section 5), and rotating frame basis normalisation (Appendix C).