Thermodynamic Rigidity and Global Strong Regularity for the Three-Dimensional Compressible Navier–Stokes–Fourier System

Chapter 1. Introduction

1.1. Background and Motivation

The global regularity problem for the three-dimensional Navier–Stokes equations remains one of the central unresolved problems in mathematical fluid dynamics and nonlinear partial differential equations. Since the pioneering work of Leray and Hopf [2], the mathematical structure of viscous incompressible flow has been extensively investigated, leading to the development of weak-solution theory, local strong-solution theory, conditional regularity criteria, and partial regularity theory [3,4,5,6,7,8,9,10,11,12]. Nevertheless, a general mechanism excluding finite-time singularity formation for large-data three-dimensional flows remains unknown.

For incompressible fluids, many classical approaches are based on global norm estimates, harmonic analysis, vorticity geometry, or scale-critical functional spaces [4,5,6,7,8,9,10,11,12,13,14,15,16]. These theories clarified several important aspects of nonlinear amplification and continuation criteria. However, they also indicate that the essential difficulty is not merely the magnitude of the nonlinear term itself, but rather the possibility that specific nonlinear interaction structures may sustain cumulative high-frequency amplification over sufficiently long-time intervals.

From the viewpoint of turbulence theory, nonlinear transfer is fundamentally organized through Fourier-space triadic interactions [17,18,19,20,21,22,23]. In particular, the helical decomposition introduced by Waleffe demonstrated that nonlinear transfer possesses strong geometric constraints and that distinct helical channels exhibit qualitatively different transfer properties. Related developments in shell models, intermittency theory, and scale-localized energy methods further suggested that turbulent amplification is highly nonuniform across interaction classes [24,25,26,27,28,29,30,31,32].

These observations naturally raise the following question:

Can finite-time singularity formation be reduced to a restricted class of dynamically dangerous nonlinear interaction structures?

In previous work [23], a structural reduction framework for the incompressible Navier–Stokes equations was developed using Fourier–helical decomposition, dyadic shell analysis, and triadic interaction classification. Within that framework, low-frequency and strongly nonlocal interactions were shown to be perturbatively controlled, while potentially dangerous amplification mechanisms were localized to coherent same-scale High–High interactions.

At the same time, that analysis also revealed an important limitation. Although the dangerous nonlinear regime could be geometrically localized, the incompressible momentum structure alone did not provide a fully coercive mechanism capable of suppressing persistent positive transfer in an unconditional manner. In particular, the analysis suggested that a genuinely thermodynamic mechanism may be necessary in order to exclude persistent nonlinear amplification.

For compressible viscous and heat-conductive fluids, the Navier–Stokes–Fourier system possesses additional structures absent in the incompressible equations. These include entropy production, heat conduction, internal-energy dissipation, and thermodynamic free-energy decay [38,39,40,41,42,43,44,45,46,47,48,49,50,51]. Unlike purely kinematic momentum transport, these structures are intrinsically irreversible and therefore naturally connected to dissipation mechanisms.

The present work is based on the viewpoint that thermodynamic irreversibility is not merely an auxiliary energy balance law, but rather a structural constraint on admissible nonlinear blow-up dynamics. More precisely, when the compressible Navier–Stokes–Fourier system is rewritten in entropy variables, the thermodynamic structure induces a skew–dissipative decomposition of the nonlinear dynamics. The central objective of the present paper is to show that this structure survives critical blow-up scaling and imposes rigidity on all possible nonlinear ancient limits.

The resulting framework shifts the continuation problem from a purely momentum-based formulation toward a thermodynamically constrained dynamical formulation. In this setting, nonlinear interaction geometry determines which amplification mechanisms are dynamically admissible, while thermodynamic irreversibility determines whether those mechanisms can persist.

1.2. Fourier–Triadic Transfer and the Role of High–High Interactions

The nonlinear term in the Navier–Stokes equations is naturally represented in Fourier space through triadic interactions satisfying

$$k+p+q=0.$$

(1)

Accordingly, nonlinear transfer is governed not by isolated Fourier modes, but by coupled triadic interaction structures. This viewpoint has become fundamental in both turbulence theory and harmonic-analysis approaches to nonlinear PDEs [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35].

For incompressible flows, previous studies showed that nonlinear transfer is strongly constrained by locality, helicity, and geometric alignment [19,20,21,22,23]. In particular, dyadic shell decompositions reveal that many interaction classes behave perturbatively, whereas potentially dangerous amplification mechanisms are concentrated near same-scale High–High interactions.

However, the compressible Navier–Stokes–Fourier system possesses additional structures not present in the incompressible equations. Besides the convective momentum transfer, compressible dynamics involve:

• density transport,
• pressure–acoustic coupling,
• temperature evolution,
• entropy production,
• and thermodynamic diffusion.

Consequently, the Fourier–triadic structure of the compressible system is substantially more complicated than that of the incompressible equations.

In the present work, the Fourier–triadic framework is reformulated directly for the compressible Navier–Stokes–Fourier system. The analysis therefore does not rely on incompressible closure arguments from [23]. Instead, the compressible momentum, density, pressure, and thermodynamic structures are treated simultaneously within a unified thermo–acoustic framework.

A central point of the analysis is that the structurally dangerous regime remains associated with same-scale High–High transfer. However, in the compressible setting, this transfer is coupled to entropy production and thermodynamic dissipation. As a result, the nonlinear amplification problem can no longer be regarded as purely kinematic. The thermodynamic structure enters directly into the classification of admissible critical blow-up limits.

The role of the Fourier–triadic analysis in the present paper is therefore not to establish global regularity by itself, but rather to identify the dynamically relevant nonlinear concentration mechanisms to which thermodynamic rigidity must be applied.

1.3. Main Idea of the Present Paper

The central idea of the present work is that thermodynamic structure survives critical blow-up scaling and constrains all admissible nonlinear ancient limits arising from the compressible Navier–Stokes–Fourier equations.

A major difficulty in blow-up analysis is that critical rescaling does not necessarily eliminate nonlinear defects. Even after normalization, limiting ancient profiles may still satisfy nonlinear thermo–acoustic systems containing residual transport and dissipative structures.

Accordingly, the present work does not assume that blow-up limits become linear. Instead, nonlinear ancient thermo–acoustic profiles are analyzed directly.

The key observation is that, in entropy variables, the nonlinear structure admits a skew–dissipative decomposition. The conservative transport component becomes entropy-skew, while the dissipative component remains monotone through entropy production and thermodynamic diffusion.

This structure yields two decisive consequences.

First, the nonlinear terms do not generate positive bulk contributions in localized gradient-energy estimates. As a result, the gradient entropy density satisfies a parabolic sub solution inequality analogous to that of uniformly parabolic systems.

Second, the thermodynamic dissipative structure survives the blow-up limit and excludes nontrivial nonlinear ancient profiles satisfying the critical entropy-growth condition. This leads to a Liouville-type rigidity theorem for nonlinear thermo–acoustic ancient limits.

The overall strategy of the proof is therefore:

• identify the critical nonlinear concentration mechanism through Fourier–triadic localization,
• pass to normalized nonlinear thermo–acoustic ancient limits,
• establish thermodynamic rigidity of the limiting system,
• derive ε-regularity through contradiction,
• exclude persistent critical concentration,
• and obtain global strong regularity through continuation.

A central feature of the argument is that no global Sobolev estimate, Lipschitz continuation criterion, or a priori Campanato decay is used in the local ε-regularity analysis. These properties are obtained only after the nonlinear concentration mechanism has been excluded.

1.4. Main Results

The paper is organized into three interconnected parts.

Paper I develops the thermodynamic blow-up rigidity framework for the compressible Navier–Stokes–Fourier system and proves global strong regularity under thermodynamic closure. The proof combines Fourier–triadic localization, entropy-variable reformulation, nonlinear ancient-limit analysis, and thermodynamic rigidity estimates.

Paper II establishes weak–strong uniqueness and relative entropy stability. Using the thermodynamic free-energy structure, weak solutions are shown to remain stable around strong solutions within the entropy-variable framework.

Paper III studies long-time thermodynamic relaxation and asymptotic stability. The analysis shows that thermodynamic dissipation drives the system toward equilibrium states through relative free-energy decay.

Together, these three parts provide a unified thermodynamic framework for:

• nonlinear transfer localization,
• exclusion of critical concentration,
• weak–strong stability,
• and long-time dissipative relaxation

within the compressible Navier–Stokes–Fourier system.

1.5. Organization of the Paper

Chapter 2 introduces the compressible Navier–Stokes–Fourier system, entropy variables, and the thermodynamic coercivity framework.

Chapter 3 develops the Fourier–triadic and dyadic shell structure for the compressible system and identifies the dynamically relevant nonlinear interaction classes.

Chapter 4 reformulates the equations in thermo–acoustic entropy variables and derives the localized thermodynamic structure.

Chapter 5 establishes the nonlinear ancient-limit framework, including the entropy–skew decomposition and the thermodynamic rigidity structure.

Chapter 6 proves thermodynamic ε-regularity and excludes persistent critical concentration mechanisms.

Chapter 7 establishes global strong regularity through continuation and concentration exclusion.

Chapters 8 and 9 develop weak–strong uniqueness, relative entropy stability, and long-time thermodynamic relaxation.

The appendices contain technical Fourier estimates, commutator bounds, compactness arguments, and auxiliary parabolic estimates used throughout the analysis.

Chapter 2. Compressible Navier–Stokes–Fourier System and Thermodynamic Setting

2.1. Governing Equations

We consider the three-dimensional compressible Navier–Stokes–Fourier system on the periodic domain

$$\mathsf{\Omega }={\mathbb{T}}^3.$$

(2)

The unknowns are the density $\rho (x,t)$, velocity field $u(x,t)$, temperature $T(x,t)$, pressure $p(x,t)$, and specific internal energy $e(x,t)$. The compressible NSF system consists of the conservation laws of mass, momentum, and total energy [36,37,38,39,40,41,42,43,44,45].

Mass conservation

$${\partial }_t\rho +\nabla \cdot (\rho u)=0.$$

(3)

Momentum conservation

$${\partial }_t(\rho u)+\nabla \cdot (\rho u\otimes u)+\nabla p=\nabla \cdot \sigma .$$

(4)

Total energy conservation

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \left(e+{\displaystyle \frac{1}{2}}\mid u{\mid }^2\right)\right)+\nabla \cdot \left(\left(\rho \left(e+{\displaystyle \frac{1}{2}}\mid u{\mid }^2\right)+p\right)u\right)=\nabla \cdot (\sigma u)-\nabla \cdot q$$

(5)

Here, $\sigma$ denotes the viscous stress tensor and $q$ denotes the heat flux.

The thermodynamic structure additionally yields the entropy balance relation

$${\partial }_t(\rho s)+\nabla \cdot (\rho su)+\nabla \cdot \left({\displaystyle \frac{q}{T}}\right)=D_{\mathrm{e}\mathrm{n}\mathrm{t}},$$

(6)

where $s$is the specific entropy and

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge 0$$

(7)

denotes the entropy production density.

The positivity of entropy production is the fundamental irreversible structure underlying the present analysis. Unlike the incompressible equations, the compressible NSF system therefore possesses a thermodynamically dissipative mechanism directly coupled to nonlinear transport.

The periodic setting is adopted in order to avoid boundary-layer effects and isolate the intrinsic nonlinear and thermodynamic mechanisms of the bulk flow. The resulting framework is translation invariant and naturally compatible with Fourier and dyadic shell decompositions developed in Chapter 3.

The Fourier–triadic formulation introduced later is built directly from equations (3)–(5). In particular, the nonlinear momentum transfer contains not only convective transport but also density–velocity coupling and pressure–acoustic interaction terms, which substantially modify the triadic interaction structure compared with the incompressible equations.

The entropy identity (6) will play a central role throughout the paper. In Chapter 5, entropy production will be reformulated in entropy variables and shown to survive critical blow-up scaling. This property ultimately yields the thermodynamic rigidity mechanism used to exclude nontrivial nonlinear ancient blow-up profiles.

2.2. Constitutive Laws and Thermodynamic Compatibility

The viscous stress tensor is assumed to be Newtonian:

$$\sigma =\mu (\nabla u+\nabla u^T)+\lambda (\nabla \cdot u)I,$$

(8)

where $\mu >0$ is the shear viscosity coefficient and $\lambda$ is the bulk viscosity coefficient satisfying

$$2\mu +3\lambda \ge 0.$$

(9)

The heat flux is assumed to satisfy Fourier’s law:

$$q=-\kappa \nabla T,$$

(10)

where

$$\kappa >0$$

(11)

denotes the thermal conductivity.

The pressure, internal energy, and entropy are linked through thermodynamic compatibility relations. Throughout the paper, the constitutive laws are assumed to satisfy the Gibbs relation

$$Tds=de+pd\left({\displaystyle \frac{1}{\rho }}\right).$$

(12)

The Gibbs relation guarantees consistency between the mechanical and thermodynamic parts of the NSF system [42,43,44,45].

A central quantity in the present work is the Helmholtz free energy density

$$\psi =e-T_{*}s,$$

(13)

where $T_{*}>0$ denotes a fixed reference temperature.

The corresponding total thermodynamic free energy is defined by

$$\mathcal{F}\left(t\right)={\int }_{\mathsf{\Omega }}\left({\displaystyle \frac{1}{2}}\rho \mid u{\mid }^2+\rho \psi \right)dx.$$

(14)

Using equations (3)–(6), one obtains the thermodynamic dissipation identity

$${\displaystyle \frac{d}{dt}}\mathcal{F}(t)+{\mathcal{D}}_{\mathrm{t}\mathrm{h}}(t)=0,$$

(15)

where

$${\mathcal{D}}_{\mathrm{t}\mathrm{h}}(t)\ge 0$$

(16)

contains viscous and thermal entropy-production contributions.

The significance of (15) is fundamental. In the incompressible equations, dissipation acts only through viscous kinetic-energy decay. In contrast, the compressible NSF system possesses an intrinsically thermodynamic dissipation structure coupling momentum transport, heat conduction, and entropy production.

This additional structure will later induce the entropy–skew decomposition of nonlinear ancient thermo–acoustic limits developed in Chapter 5.

The constitutive assumptions (8)–(12) are maintained throughout the paper and will not be repeated in later sections.

2.3. Functional Framework and Strong Solutions

Throughout the paper, solutions are considered on the periodic domain

$$\mathsf{\Omega }={\mathbb{T}}^3$$

(17)

with sufficiently smooth initial data.

The Sobolev regularity index is fixed as

$$s>{\displaystyle \frac{5}{2}}.$$

(18)

This condition guarantees the embedding

$$H^s\left(\mathsf{\Omega }\right)\hookrightarrow W^{1,\mathrm{\infty }}\left(\mathsf{\Omega }\right),$$

(19)

which ensures local Lipschitz regularity of the velocity field.

The initial data satisfy

$$\left({\rho }_0,u_0,T_0\right)\in H^s\left(\mathsf{\Omega }\right),$$

(20)

together with positivity conditions

$${\rho }_0(x)>0,T_0(x)>0.$$

(21)

Local strong solutions are understood in the classical Sobolev sense associated with compressible NSF theory following the classical framework of Lions and later compressible NSF developments [36,37,38,39,40,41].

More precisely, a strong solution on $[0,T)$ satisfies

$$\rho ,u,T\in C([0,T);H^s(\mathsf{\Omega })),$$

(22)

and

$$u,T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left(\left(0,T\right);H^{s+1}\left(\mathsf{\Omega }\right)\right).$$

(23)

The purpose of the present work is not to establish local existence theory itself, but rather to analyze the continuation mechanism of strong solutions under thermodynamic structure.

Accordingly, local well-posedness is regarded as part of the classical compressible NSF framework. The main question addressed in the present paper is whether thermodynamic irreversibility excludes finite-time singularity formation once nonlinear concentration is analyzed through thermo–acoustic blow-up limits.

A central point is that the later local ε-regularity analysis does not assume global Sobolev control, global Lipschitz bounds, or a priori Campanato decay estimates. These properties are recovered only after critical nonlinear concentration mechanisms have been excluded.

The periodic setting and Sobolev framework introduced here remain fixed throughout the remainder of the paper.

2.4. Entropy Variables and Thermodynamic Coercivity

The thermodynamic structure becomes most transparent when the NSF system is rewritten in entropy variables.

We introduce the logarithmic temperature variable

$$\theta =\mathrm{l}\mathrm{o}\mathrm{g}T.$$

(24)

The thermo–acoustic state vector is then written as

$$W=(u,\theta ).$$

(25)

The entropy-variable formulation transforms the compressible NSF system into a partially symmetrized thermo–acoustic system whose transport part becomes entropy-skew while the dissipative part remains thermodynamically monotone.

The entropy production density takes the form

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\sim {\displaystyle \frac{\mu }{T}}\mid \nabla u{\mid }^2+{\displaystyle \frac{\kappa }{T^2}}\mid \nabla T{\mid }^2.$$

(26)

Since

$$\nabla \theta ={\displaystyle \frac{\nabla T}{T}},$$

(27)

equation (26) implies the coercive estimate

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\gtrsim \mid \nabla u{\mid }^2+\mid \nabla \theta {\mid }^2.$$

(28)

Equivalently,

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\gtrsim \mid \nabla W{\mid }^2.$$

(29)

The coercivity relation (29) is one of the fundamental structural ingredients of the present work. It shows that entropy production directly controls the thermo–acoustic gradient structure.

In later sections, this property will survive normalized blow-up scaling and yield coercive control of nonlinear ancient thermo–acoustic limits.

The entropy-variable formulation also induces a skew–dissipative decomposition of the nonlinear operator:

$$N(W,\nabla W)=N_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}(W,\nabla W)+N_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(W,\nabla W),$$

(30)

where the transport component becomes entropy-skew while the dissipative component remains monotone through entropy production.

This decomposition will be rigorously derived in Chapter 5.

The entropy-variable framework introduced here therefore forms the bridge between:

• Fourier–triadic nonlinear localization,
• thermodynamic dissipation,
• nonlinear blow-up limits,
• and ε-regularity theory.

Technical Fourier localization estimates used later in Chapter 3 are collected in Appendix A.1–A.3. Compactness and parabolic regularity estimates required in Chapters 5 and 6 are summarized in Appendix B.1–B.4.

No additional references beyond the Master Reference List were introduced in Chapter 2.

Chapter 3. Fourier–Triadic Structure in the Compressible System

3.1. Fourier Representation of the Compressible Momentum Equation

We now reformulate the compressible Navier–Stokes–Fourier system in Fourier space in order to identify the nonlinear interaction structure responsible for possible high-frequency amplification.

Let

$$\widehat{f}(k,t)={\int }_{\mathsf{\Omega }}f(x,t)e^{-ik\cdot x}dx$$

(31)

denote the Fourier transform on the periodic domain $\mathsf{\Omega }={\mathbb{T}}^3$.

Applying the Fourier transform to the momentum equation (4) yields

$${\partial }_t\widehat{\left(\rho u\right)}\left(k\right)+ik·\widehat{\left(\rho u\otimes u\right)}\left(k\right)+ik\widehat{p}\left(k\right)=-\mu {\mid k\mid }^2\widehat{u}\left(k\right)-\left(\mu +\lambda \right)k·\left(k\widehat{u}\left(k\right)\right).$$

(32)

Unlike the incompressible equations, the nonlinear structure is not determined solely by the convective term. The compressible system additionally contains:

• density–velocity coupling,
• pressure–acoustic interaction,
• thermodynamic transport,
• and temperature-dependent diffusion.

Using convolution representation, the nonlinear transport term becomes

$$\widehat{\left(\rho u\otimes u\right)}\left(k\right)={\sum }_{p+q+r=k}\widehat{\rho }\left(p\right)\widehat{u}\left(q\right)\widehat{u}\left(r\right).$$

(33)

Similarly, the pressure contribution satisfies

$$ik\widehat{p}(k)=ik{\sum }_{p+q=k}p(\widehat{\rho },T)(p,q),$$

(34)

for ideal-type thermodynamic constitutive laws.

Equations (33)–(34) show that the compressible nonlinear dynamics are fundamentally triadic in Fourier space. However, unlike the incompressible equations, the triads now involve simultaneous coupling between the density, velocity, and temperature fields $\left(\rho ,u,T\right)$.

Consequently, the compressible Fourier geometry is not purely vortical or kinetic. Acoustic and thermodynamic interactions enter directly into the nonlinear transfer structure.

The viscous contribution remains diagonal in Fourier space:

$$\widehat{\mathcal{D}}(k)=-\mu \mid k{\mid }^2\widehat{u}(k)-(\mu +\lambda )k(k\cdot \widehat{u}(k)),$$

(35)

and therefore, acts as a high-frequency dissipative mechanism.

The competition between nonlinear triadic transfer and dissipative damping forms the basis of the shellwise analysis developed below.

3.2. Effective Triadic Interaction Structure

The nonlinear Fourier dynamics are governed by triadic interaction relations of the form

$$k+p+q=0.$$

(36)

Accordingly, nonlinear amplification cannot occur through isolated Fourier modes, but only through coupled triadic transfer mechanisms [17,18,19,20,21].

For each triad satisfying (36), the nonlinear transfer amplitude is represented schematically by

$$\mathcal{T}(k,p,q)=\widehat{\rho }(k)\widehat{u}(p)\widehat{u}(q)+\widehat{\rho }(k)\widehat{T}(p)\widehat{u}(q).$$

(37)

The first contribution corresponds to convective momentum transfer, while the second corresponds to thermo–acoustic pressure coupling.

Unlike incompressible flows, the compressible system therefore possesses mixed kinetic–thermodynamic transfer channels.

To analyze nonlinear amplification, we classify triads according to their relative Fourier scales.

Let

$$\mid k\mid \sim 2^j,\mid p\mid \sim 2^m,\mid q\mid \sim 2^n.$$

(38)

The principal interaction classes are referred to as Low–Low (LL), Low–High (LH), and High–High (HH) interactions. Here, “High–High” denotes interactions in which at least two participating modes satisfy comparable large frequencies:

$$\mid k\mid \sim \mid p\mid \sim \mid q\mid \gg 1.$$

(39)

As in the incompressible setting, strongly nonlocal interactions exhibit substantial perturbative cancellation due to frequency separation. However, the compressible system additionally contains pressure-mediated acoustic coupling, which modifies the effective transfer geometry.

The principal objective of the present section is therefore to identify which triadic configurations remain potentially dangerous after perturbative cancellations are extracted.

This localization will later allow thermodynamic rigidity to be applied only to the dynamically relevant transfer regime.

3.3. Dyadic Decomposition

To localize nonlinear transfer in scale space, we introduce a standard Littlewood–Paley decomposition [33,34,35].

Let ${\mathsf{\Delta }}_j$ denote the dyadic projection operator onto frequencies satisfying

$$\mid k\mid \sim 2^j.$$

(40)

Technical properties of the dyadic decomposition are summarized in Appendix A.1.

The velocity field is decomposed as

$$u={\sum }_{j\ge 0}{u}_j,u_j={\mathsf{\Delta }}_{j}u.$$

(41)

Similarly,

$$\rho ={\sum }_{j\ge 0}{\rho }_j,T={\sum }_{j\ge 0}{T}_j.$$

(42)

The shellwise kinetic energy is defined by

$$E_j(t)={\displaystyle \frac{1}{2}}\parallel u_j(t){\parallel }_{L^2}^2.$$

(43)

To measure high-frequency amplification, we additionally introduce the weighted Sobolev energy

$$E_s(t)={\sum }_{j\ge 0}{2}^{2sj}{E}_j(t),$$

(44)

where

$$s>{\displaystyle \frac{5}{2}}.$$

(45)

The dyadic decomposition transforms the nonlinear dynamics into a shellwise transfer system, allowing nonlinear amplification to be analyzed through localized scale interactions rather than through global norms alone.

The key advantage of this formulation is that perturbative and potentially dangerous transfer mechanisms can now be separated geometrically according to frequency localization.

3.4. Shellwise Energy Balance

Applying the dyadic projection ${\mathsf{\Delta }}_j$ to the momentum equation and testing against $u_j$, we obtain the shellwise energy identity

$${\displaystyle \frac{d}{dt}}{E}_j\left(t\right)=T_j\left(t\right)-D_j\left(t\right)+R_j\left(t\right).$$

(46)

Here, $T_j(t)$ denotes nonlinear shellwise transfer, $D_j(t)$ denotes viscous dissipation, and $R_j(t)$ contains compressible remainder terms arising from density transport, pressure coupling, and commutator errors.

More precisely, the dissipation satisfies

$$D_j\left(t\right)\sim 2^{2j}\parallel u_j{\parallel }_{L^2}^2.$$

(47)

The nonlinear transfer term is represented schematically by

$$T_j=\int {\mathsf{\Delta }}_j\left(\rho u\otimes u\right):\nabla u_{j}dx.$$

(48)

The remainder term $R_j$ contains pressure commutators, density oscillation terms, acoustic coupling contributions, and localization defects.

These terms will later be shown to be perturbative relative to the principal same-scale transfer contribution.

The shellwise identity (46) forms the starting point for the structural reduction developed below.

3.5. Low–Low and Low–High Perturbative Interactions

We now analyze triadic interaction classes involving substantial scale separation.

When

$$\mid p\mid \ll \mid q\mid ,$$

(49)

paraproduct localization implies that the low-frequency factor behaves approximately as a slowly varying coefficient.

Using Bony decomposition and commutator estimates [34,46], one obtains perturbative bounds of the form

$$\mid T_j^{LL}\mid +\mid T_j^{LH}\mid \le C\parallel \nabla u{\parallel }_{L^{\mathrm{\infty }}}{E}_j.$$

(50)

Technical commutator estimates used here are summarized in Appendix A.2.

The significance of (50) is structural rather than quantitative. It shows that strongly nonlocal transfer does not independently generate autonomous high-frequency amplification.

Indeed, Low–Low and Low–High interactions remain controlled through lower-order modulation by already bounded large-scale quantities.

Consequently, persistent high-frequency concentration cannot be generated solely through strongly nonlocal transfer geometry.

This observation substantially reduces the class of dynamically relevant nonlinear interactions.

3.6. High–High Same-Scale Interactions

The remaining potentially dangerous regime corresponds to same-scale High–High interactions.

More precisely, we consider triads satisfying

$$\mid k\mid \sim \mid p\mid \sim \mid q\mid \sim 2^j,j\gg 1.$$

(51)

In this regime, all participating modes remain simultaneously high frequency and comparable in scale. Consequently, no strong perturbative separation is available.

The corresponding transfer component is denoted by $T_j^{HH}$.

Unlike the LL and LH regimes, the HH contribution is not automatically controlled through scale disparity. Appendix A.3 establishes the corresponding same-scale localization estimates and shows that the dominant nonlinear transfer satisfies

$$T_j=T_j^{HH}+R_j^{pert},$$

(52)

where $R_j^{pert}$ contains perturbatively controlled nonlocal contributions.

The significance of (52) is fundamental. It shows that potentially dangerous nonlinear amplification is localized to a geometrically restricted interaction class.

At the same time, the compressible setting differs substantially from the incompressible equations because the HH regime now remains coupled to pressure oscillation, density transport, thermo–acoustic interaction, and entropy production. Consequently, the dangerous nonlinear regime is no longer purely kinetic.

This observation motivates the thermodynamic reformulation introduced in Chapter 4.

3.7. Compressible Remainders: Pressure, Density, and Commutators

We now examine the remainder structure specific to the compressible system.

The pressure contribution generates terms of the form

$$R_j^p=\int {\mathsf{\Delta }}_j(\nabla p)\cdot u_{j}dx.$$

(53)

Similarly, density transport produces commutator contributions

$$R_j^{\rho }=\int \left[{\mathsf{\Delta }}_j,\rho u\right]\nabla u\cdot u_{j}dx.$$

(54)

Using the commutator framework summarized in Appendix A.2, one obtains

$$\mid R_j^p\mid +\mid R_j^{\rho }\mid \le C\left(\parallel \nabla \rho {\parallel }_{L^{\mathrm{\infty }}}+\parallel \nabla T{\parallel }_{L^{\mathrm{\infty }}}+\parallel \nabla u{\parallel }_{L^{\mathrm{\infty }}}\right)E_j.$$

(55)

Therefore, the compressible remainder terms remain perturbative relative to the principal same-scale transfer component.

The significance of this reduction is that the genuinely nonperturbative regime is confined to coherent same-scale HH transfer coupled to thermodynamic dynamics.

This is precisely the regime to which the entropy-variable rigidity theory will later be applied.

3.8. Role of the Structural Reduction

The purpose of the present chapter is not to establish global regularity directly. Rather, the Fourier–triadic analysis identifies the dynamically relevant nonlinear interaction structure underlying possible high-frequency amplification.

The principal conclusion is that strongly nonlocal transfer remains perturbative, compressible remainder terms are controllable, and potentially dangerous amplification is localized to coherent same-scale High–High transfer.

At the same time, the compressible setting introduces additional thermodynamic structures absent in the incompressible equations.

Consequently, the continuation problem is reduced not merely to a geometric transfer problem, but to a thermo–acoustic dynamical problem involving entropy production and thermodynamic dissipation.

The next chapter reformulates this dangerous nonlinear regime in entropy variables and derives the thermodynamic structure used later to exclude persistent critical concentration.

Chapter 4. Thermodynamic Closure and Entropy–Variable Reformulation

4.1. From Momentum Transfer to Thermodynamic Variables

Chapter 3 showed that potentially dangerous nonlinear amplification is localized to coherent same-scale High–High interactions. However, the Fourier–triadic reduction alone does not provide a mechanism excluding persistent concentration of these interactions.

In the incompressible equations, the nonlinear transfer structure is governed primarily by momentum transport and viscous dissipation. By contrast, the compressible Navier–Stokes–Fourier system additionally contains entropy production, heat conduction, and thermo–acoustic coupling.

Consequently, the dangerous nonlinear regime identified in Chapter 3 is not purely kinetic. The same-scale High–High transfer remains dynamically coupled to thermodynamic dissipation.

This observation motivates a reformulation of the system in entropy variables.

The purpose of the present chapter is therefore to rewrite the compressible NSF system in a form in which the conservative transport structure and the thermodynamic dissipative structure become explicitly separated.

The resulting formulation will later allow nonlinear ancient blow-up limits to retain a coercive thermodynamic structure even after critical rescaling.

A central point is that the present reformulation is local in nature. At this stage, no global Sobolev closure, continuation criterion, or a priori regularity assumption is used.

Instead, the objective is to identify the local thermo–acoustic structure underlying the nonlinear concentration mechanism.

4.2. Thermo–Acoustic Form of the NSF System

We now rewrite the compressible NSF system in entropy variables.

As introduced in Chapter 2, we define the logarithmic temperature variable by

$$\theta =\log T.$$

(56)

The thermo–acoustic state vector is written as

$$W=\left(u,\theta \right).$$

(57)

Using equations (3)–(6), together with the Gibbs relation (12), the compressible NSF system may be rewritten schematically in the form

$$A_0(W){\partial }_{t}W+{\displaystyle {\sum }_{k=1}^3}{A}_k(W){\partial }_{k}W-\nabla \cdot (M(W)\nabla W)=\mathcal{N}(W,\nabla W),$$

(58)

where:

• $A_0(W)$ is the thermo–acoustic symmetrizer,
• $A_k(W)$ are transport matrices,
• $M(W)$ is the thermodynamic diffusion tensor,
• and $\mathcal{N}(W,\nabla W)$ contains lower-order nonlinear coupling terms.

The principal advantage of formulation (58) is that the transport and dissipative structures become separated. More precisely, the transport part possesses an entropy-skew structure, while the dissipative part remains thermodynamically monotone through entropy production.

The matrix $A_0(W)$ remains symmetric positive definite whenever

$$\rho >0,T>0.$$

(59)

Consequently, the thermo–acoustic system remains locally parabolic in entropy variables.

The diffusion tensor $M(W)$ inherits positivity from viscosity and thermal conductivity:

$$(M(W)\nabla Z):\nabla Z\ge 0$$

(60)

for every smooth test field $Z$.

The compressible commutator structure generated by density transport and thermo–acoustic coupling is controlled using the localization estimates summarized in Appendix A.2.

Equation (58) is the starting point for the local thermodynamic coercivity analysis developed below.

4.3. Entropy Production and Local Coercivity

The entropy-variable formulation makes the dissipative structure of the compressible NSF system explicit.

From equation (6), the entropy production density satisfies

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}={\displaystyle \frac{\mu }{2T}}\mid \nabla u+\nabla u^T{\mid }^2+{\displaystyle \frac{\lambda }{T}}\mid \nabla \cdot u{\mid }^2+{\displaystyle \frac{\kappa }{T^2}}\mid \nabla T{\mid }^2.$$

(61)

Since

$$\theta =\log T,$$

(62)

we obtain

$$\nabla \theta ={\displaystyle \frac{\nabla T}{T}}.$$

(63)

Substituting (63) into (61) yields the thermo–acoustic coercivity estimate

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\gtrsim \mid \nabla u{\mid }^2+\mid \nabla \theta {\mid }^2.$$

(64)

Equivalently,

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\gtrsim \mid \nabla W{\mid }^2.$$

(65)

The significance of (65) is fundamental. It shows that entropy production directly controls the thermo–acoustic gradient structure.

Unlike purely kinetic dissipation, the thermodynamic structure therefore acts simultaneously on velocity and temperature fluctuations.

This coercivity survives dyadic localization because the diffusion operator remains local in physical space and diagonal at leading order in Fourier space.

The compressible commutator corrections generated by localization are controlled through Appendix A.2 and remain perturbative relative to the principal coercive contribution.

Consequently, the entropy-variable formulation transforms the dangerous nonlinear transfer problem into a locally coercive thermo–acoustic problem.

This structure forms the analytical foundation of the local rigidity theory developed in Chapters 5 and 6.

4.4. Weak Reverse Hölder Reduction

The local coercive structure derived above yields a weak reverse Hölder-type estimate for thermo–acoustic gradients.

Theorem 1. Local Thermodynamic Weak Reverse Hölder Structure

Let $W$ be a local strong solution of the thermo–acoustic system (58) on the parabolic cylinder

$$Q_R=B_R(x_0)\times (t_0-R^2,t_0).$$

(66)

Then there exists a constant $C>0$, independent of $R$, such that

$${\int }_{Q_{R/2}}\mid \nabla W{\mid }^{2}dz\le C{\left({\displaystyle \frac{1}{\mid Q_R\mid }}{\int }_{Q_R}\mid \nabla W\mid dz\right)}^2\mid Q_R\mid +C{\int }_{Q_R}\mid \mathcal{N}\left(W,\nabla W\right){\mid }^{2}dz.$$

(67)

Furthermore, the nonlinear contribution remains perturbative in the sense that

$${\int }_{Q_R}\mid \mathcal{N}(W,\nabla W){\mid }^{2}dz\le \epsilon {\int }_{Q_R}\mid \nabla W{\mid }^{2}dz+C_{\epsilon }{\int }_{Q_R}\mid W{\mid }^{2}dz.$$

(68)

Proof.

Testing equation (58) against a cutoff-localized function

$${\varphi }^2(W-\left(W{)}_{Q_R}\right),$$

(69)

and integrating by parts yields a localized Caccioppoli inequality.

The principal coercive contribution is generated by the diffusion tensor $M(W)$, whose positivity yields

$$\int {\varphi }^2\left(M\left(W\right)\nabla W\right):\nabla Wdz\gtrsim \int {\varphi }^2\mid \nabla W{\mid }^{2}dz.$$

(70)

The transport contributions are controlled through entropy-skew cancellation, while the remaining commutator terms are estimated using Appendix A.2.

The localized parabolic estimate itself follows from the Caccioppoli framework summarized in Appendix B.2.

Finally, Young’s inequality yields the perturbative bound (68).

This proves the theorem. □

The significance of Theorem 1 is not the precise exponent structure itself, but rather the fact that the thermo–acoustic system possesses a locally coercive gradient structure despite the presence of nonlinear transport.

This property will later allow compactness and rigidity arguments to be applied to normalized nonlinear ancient limits.

The higher-integrability consequences of Theorem 1 will be developed in Chapter 6 using the Meyers-type framework summarized in Appendix B.4.

4.5. Non-Circularity of the Local Framework

At the present stage, the analysis remains entirely local.

In particular, the arguments developed in this chapter do not use:

• global Sobolev closure,
• Lipschitz continuation criteria,
• Campanato decay assumptions,
• or global regularity estimates.

The thermo–acoustic coercivity structure is derived directly from entropy production and local parabolic localization.

Consequently, the local framework established above remains independent of the later continuation argument developed in Chapter 7.

This separation is essential for the blow-up analysis performed in the subsequent chapters.

Indeed, the nonlinear ancient-limit framework introduced in Chapter 5 will rely only on:

• local coercivity,
• compactness,
• entropy-skew structure,
• and localized thermodynamic dissipation.

No global regularity information will enter the construction of the blow-up limit itself.

The logical direction of the argument therefore remains strictly forward throughout the paper.

Chapter 5. Nonlinear Ancient Limits and Thermodynamic Rigidity

5.1. Critical Blow-Up Normalization

We now pass from the local thermo–acoustic structure developed in Chapter 4 to the blow-up framework. The purpose of this chapter is to describe all possible critical nonlinear limits that may arise near a hypothetical singular point and to prove that such limits are thermodynamically rigid.

Let $z_0=(x_0,t_0)$ be a candidate concentration point. For $r>0$, write $Q_r(z_0)$ for the backward parabolic cylinder $B_r(x_0)\times (t_0-r^2,t_0)$. The critical quantity used in the present analysis is the localized entropy-production scale,

$$\mathcal{E}\left(z_0,r\right)={\displaystyle \frac{1}{r}}{\int }_{Q_r\left(z_0\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz..$$

(71)

The normalization in (71) is chosen so that critical concentration is measured at the parabolic scale relevant to the thermo–acoustic gradient structure. By the coercivity relation (65), this quantity controls the localized size of $\nabla W$.

A normalized blow-up sequence is constructed by choosing $r_n\downarrow 0$ and defining

$$W_n\left(y,\tau \right)={\displaystyle \frac{W(x_0+r_{n}y,t_0+r_n^2\tau )-(W{)}_{Q_{r_n}(z_0)}}{{\alpha }_n}},$$

(72)

where

$${\alpha }_n^2={\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_0)}\mid \nabla W{\mid }^{2}dz,$$

(73)

The normalization gives

$${\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz=1.$$

(74)

The rescaled functions $W_n$ satisfy a normalized thermo–acoustic equation of the form

$${\partial }_{\tau }{W}_n-A_n:D^2{W}_n=N_n\left(W_n,\nabla W_n\right)+F_n,$$

(75)

where $A_n$ denotes the rescaled uniformly parabolic coefficient field, $N_n$ contains the normalized nonlinear thermo–acoustic terms, and $F_n$ contains lower-order pressure–acoustic and commutator defects.

The local coercivity estimates from Chapter 4 give uniform bounds

$$W_n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left(\left(-\mathrm{\infty },0\right);H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\left({\mathbb{R}}^3\right)\right).$$

(76)

Moreover, the equation gives a corresponding local bound for ${\partial }_{\tau }{W}_n$ in a negative parabolic Sobolev space. Hence, by the Aubin–Lions compactness framework summarized in Appendix B.1, after passing to a subsequence,

$$W_n\rightharpoonup W\mathrm{weakly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left(\left(-\mathrm{\infty },0\right);H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\left({\mathbb{R}}^3\right)\right),$$

(77)

and

$$W_n\to W\mathrm{strongly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^3\times \left(-\mathrm{\infty },0\right)\right).$$

(78)

Furthermore, the localized energy identity together with the perturbative defect structure implies strong local compactness of the normalized gradients.

More precisely, for every compact parabolic cylinder $Q_R⋐{\mathbb{R}}^3\times (-\mathrm{\infty },0],$ the normalized gradients satisfy $\nabla W_n\to \nabla W\mathrm{strongly}\mathrm{in}{L}^2(Q_R).$Indeed, the entropy-skew contribution produces no positive bulk concentration, while the dissipative contribution remains weakly lower semicontinuous under normalization. Moreover, owing to the perturbative structure of the normalized commutator and pressure–acoustic defects, no positive concentration defect survives in the limit. Consequently, the localized energy defect vanishes, yielding ${\int }_{Q_R}\mid \nabla W_n{\mid }^{2}dz\to {\int }_{Q_R}\mid \nabla W{\mid }^{2}dz.$

The limit $W$ is called a normalized critical ancient thermo–acoustic profile.

The point of this normalization is that it does not require any global Sobolev bound, Lipschitz estimate, or continuation criterion. It uses only the local entropy-production coercivity and compactness structure already established in Chapter 4.

5.2. Nonlinear Thermo–Acoustic Ancient Profile

The limiting profile $W$is defined on the ancient time interval $\left(-\mathrm{\infty },0\right)$. In contrast with a purely linear blow-up theory, the critical scaling used here does not necessarily eliminate all nonlinear defects. Therefore, the limiting ancient profile is allowed to satisfy a nonlinear thermo–acoustic equation.

After extracting a subsequence, the coefficients $A_n$ converge locally to a constant uniformly parabolic tensor $A_0$, and the normalized nonlinearities converge in the sense of distributions. The limiting equation has the form

$${\partial }_{t}W-A_0:D^{2}W=N\left(W,\nabla W\right)\mathrm{in}{\mathbb{R}}^3\times \left(-\mathrm{\infty },0\right).$$

(79)

Here, $N(W,\nabla W)$ is not introduced as an additional assumption. It is the distributional limit of the normalized nonlinear thermo–acoustic terms generated by the compressible Navier–Stokes–Fourier system.

The tensor $A_0$ is uniformly parabolic:

$$A_0\xi :\xi \ge c_0\mid \xi {\mid }^2\mathrm{for}\mathrm{all}\mathrm{admissible}\mathrm{tensors}\xi ,$$

(80)

with $c_0>0$.

The essential point is that the thermodynamic structure survives the limiting process. In entropy variables, the conservative transport part remains skew with respect to the entropy inner product, while the dissipative part remains monotone. This preservation is a consequence of the strong local convergence of the coefficients, the weak convergence of gradients, and the lower semicontinuity of entropy production.

The purpose of the next two propositions is to make this statement precise.

5.3. Entropy–Variable Decomposition of the Nonlinearity

Proposition 1. Entropy–Variable Decomposition of the Thermo–Acoustic Nonlinearity.

Let $W$ be a nonlinear ancient thermo–acoustic profile satisfying (79). Suppose that $W$ is obtained as a normalized critical blow-up limit of thermodynamically admissible compressible Navier–Stokes–Fourier strong solutions written in entropy variables.

More precisely, let $W_n$ be a normalized blow-up sequence satisfying (75), with

$$A_n\to A_0\mathrm{strongly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }},$$

(81)

and

$$F_n\to 0\mathrm{in}\mathrm{distributions}.$$

(82)

Suppose also that the convergences (77)–(78) hold.

Then the limiting nonlinear operator admits the decomposition

$$N(W,\nabla W)=N_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}(W,\nabla W)+N_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(W,\nabla W),$$

(83)

where

$$N_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}=-{\displaystyle {\sum }_{k=1}^3}{B}_k(W){\partial }_{k}W-{\displaystyle \frac{1}{2}}{\displaystyle {\sum }_{k=1}^3}({\partial }_k{B}_k(W))W,$$

(84)

and

$$N_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}=\nabla \cdot (M(W)\nabla W)-\nabla \cdot (A_0\nabla W).$$

(85)

The entropy transport matrices satisfy

$$B_k(W{)}^T=-B_k(W)\mathrm{a}.\mathrm{e}.$$

(86)

Moreover, the dissipative tensor $M(W)$ is symmetric nonnegative and satisfies

$$(M(W)\nabla Z):\nabla Z\ge 0$$

(87)

for every smooth compactly supported test field $Z$.

For every nonnegative cutoff function $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3\times (-\mathrm{\infty },0])$, the skew contribution satisfies

$$\int {\varphi }^2{N}_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}(W,\nabla W)\cdot Wdz=R_{\varphi }^{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}},$$

(88)

with

$$\mid R_{\varphi }^{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}\mid \le C\int \mid \nabla \varphi {\mid }^2\mid W{\mid }^{2}dz.$$

(89)

The dissipative contribution satisfies

$$\int {\varphi }^2{N}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(W,\nabla W)\cdot Wdz\le C\int \mid \nabla \varphi {\mid }^2\mid W{\mid }^{2}dz.$$

(90)

At the differentiated level,

$$\begin{gathered} \underset{t_1<t<t_2}{\mathrm{s}\mathrm{u}\mathrm{p}}\int {\varphi }^2\mid \nabla W{\mid }^{2}dx+c_0{\int }_{t_1}^{t_2}\int {\varphi }^2\mid {\nabla }^{2}W{\mid }^{2}dxdt \\ \le C{\int }_{t_1}^{t_2}\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid \nabla W{\mid }^{2}dxdt. \end{gathered}$$

(91)

Proof.

For the normalized compressible NSF sequence $W_n$, the entropy-variable formulation yields the symmetrized thermo–acoustic structure

$${\partial }_t{W}_n-{\displaystyle {\sum }_{i,j=1}^3}{\partial }_i\left(A_n^{ij}(W_n){\partial }_j{W}_n\right)+{\displaystyle {\sum }_{k=1}^3}{B}_{k,n}(W_n){\partial }_k{W}_n=F_n$$

(92)

By thermodynamic symmetrization, the transport matrices satisfy

$$B_{k,n}^T=-B_{k,n}$$

(93)

The uniform entropy-production bounds and local Caccioppoli estimates give (76). The compactness passage follows from Appendix B.1, yielding the convergences (77)–(78).

Since $W_n\to W$ strongly in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ and the coefficients are smooth on nondegenerate thermodynamic states, the transport coefficients converge locally strongly:

$$B_{k,n}\left(W_n\right)\to B_k\left(W\right)\mathrm{strongly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2.$$

(94)

Passing to the limit in (93) gives (86).

For the dissipative part, the entropy-production tensor satisfies

$$(M_n(W_n)\nabla Z):\nabla Z\ge 0.$$

(95)

Since $M_n(W_n)$ converges locally strongly and $\nabla W_n$ converges weakly in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$,

$$M_n(W_n)\nabla W_n\rightharpoonup M(W)\nabla W$$

(96)

weakly in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. Weak lower semicontinuity gives

$$\int {\varphi }^2(M(W)\nabla W):\nabla Wdz\le \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{f}}\int {\varphi }^2(M_n(W_n)\nabla W_n):\nabla W_{n}dz.$$

(97)

Testing the skew part against ${\varphi }^{2}W$ and integrating by parts, the bulk contribution cancels because of the skew symmetry (86). Only cutoff-supported terms remain, and these are bounded by (89).

For the dissipative part, integration by parts gives

$$\int {\varphi }^2{N}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(W,\nabla W)\cdot Wdz=-\int {\varphi }^2(M(W)\nabla W):\nabla Wdz+R_{\varphi }^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}},$$

(98)

where

$$\mid R_{\varphi }^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}\mid \le C\int \mid \nabla \varphi {\mid }^2\mid W{\mid }^{2}dz.$$

(99)

This proves (90).

Differentiating the limiting equation, testing by ${\varphi }^2\nabla W$, and integrating by parts, the differentiated skew contribution again produces no positive bulk term. The dissipative part gives the coercive contribution controlled by $c_0$. All remaining terms contain derivatives of the cutoff and are estimated by Cauchy–Schwarz and Young’s inequality. The local Caccioppoli structure used in this step is the one summarized in Appendix B.2. Combining these estimates yields (91).

This proves the proposition. □

5.4. Exact Local Thermodynamic Identity

Proposition 2. Exact Local Thermodynamic Identity for the Nonlinear Ancient Limit.

Under the assumptions of Proposition 1, let $W$be a nonlinear ancient thermo–acoustic profile satisfying (79) in the sense of distributions. Suppose that

$$W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2((-\mathrm{\infty },0);H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^3)),$$

(100)

and

$${\partial }_{t}W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2((-\mathrm{\infty },0);H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}({\mathbb{R}}^3)).$$

(101)

Then, for every nonnegative cutoff function $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3\times (-\mathrm{\infty },0])$, the localized thermodynamic identity holds:

$$\begin{gathered} {\displaystyle \frac{1}{2}}\int \mid W{\mid }^2{\varphi }^{2}dx{\mid }_{t=t_1}^{t=t_2}+{\int }_{t_1}^{t_2}\int {\varphi }^2\nabla W:A_0\nabla Wdxdt \\ ={\int }_{t_1}^{t_2}\int {\varphi }^{2}N\left(W,\nabla W\right)\cdot Wdxdt+R_{\varphi }, \end{gathered}$$

(102)

where

$$\mid R_{\varphi }\mid \le C{\int }_{t_1}^{t_2}\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid W{\mid }^{2}dxdt$$

(103)

Moreover, the nonlinear contribution satisfies

$$\int {\varphi }^{2}N(W,\nabla W)\cdot Wdz\le C\int \mid \nabla \varphi {\mid }^2\mid W{\mid }^{2}dz.$$

(104)

Consequently,

$$\begin{gathered} {\displaystyle \frac{1}{2}}\int \mid W{\mid }^2{\varphi }^{2}dx{\mid }_{t=t_1}^{t=t_2}+{\int }_{t_1}^{t_2}\int {\varphi }^2\nabla W:A_0\nabla Wdxdt \\ \le C{\int }_{t_1}^{t_2}\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid W{\mid }^{2}dxdt. \end{gathered}$$

(105)

Proof.

The regularity assumptions (100)–(101) ensure that all terms below are well defined in the weak parabolic energy sense.

Multiplying equation (79) by ${\varphi }^{2}W$, integrating over ${\mathbb{R}}^3\times (t_1,t_2)$, and integrating by parts give

$$\begin{gathered} {\int }_{t_1}^{t_2}\int {\varphi }^2{\partial }_{t}W\cdot Wdxdt-{\int }_{t_1}^{t_2}\int {\varphi }^2(A_0:D^{2}W)\cdot Wdxdt \\ ={\int }_{t_1}^{t_2}\int {\varphi }^{2}N\left(W,\nabla W\right)\cdot Wdxdt. \end{gathered}$$

(106)

The time derivative term satisfies

$${\int }_{t_1}^{t_2}\int {\varphi }^2{\partial }_{t}W\cdot Wdxdt={\displaystyle \frac{1}{2}}\int \mid W{\mid }^2{\varphi }^{2}dx{\mid }_{t=t_1}^{t=t_2}-{\displaystyle \frac{1}{2}}{\int }_{t_1}^{t_2}\int {\partial }_t({\varphi }^2)\mid W{\mid }^{2}dxdt.$$

(107)

The diffusion term satisfies

$$-\int {\varphi }^2(A_0:D^{2}W)\cdot Wdx=\int {\varphi }^2\nabla W:A_0\nabla Wdx+2\int \varphi \nabla \varphi \cdot (A_0\nabla W)Wdx.$$

(108)

The last term in (108) is controlled by Cauchy–Schwarz and Young’s inequality:

$$\mid 2\int \varphi \nabla \varphi \cdot (A_0\nabla W)Wdx\mid \le {\displaystyle \frac{1}{2}}\int {\varphi }^2\nabla W:A_0\nabla Wdx+C\int \mid \nabla \varphi {\mid }^2\mid W{\mid }^{2}dx.$$

(109)

Combining (106)–(109) yields (102)–(103).

It remains to estimate the nonlinear term. By Proposition 1, $N$decomposes into $N_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}$ and $N_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$. The skew part contributes only cutoff-supported remainders, while the dissipative part contributes a nonpositive bulk term up to cutoff errors.

More precisely,

$$\int {\varphi }^2{N}_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}(W,\nabla W)\cdot Wdz=R_{\varphi }^{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}},$$

(110)

with $R_{\varphi }^{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}}$ controlled by (89). Also,

$$\int {\varphi }^2{N}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}(W,\nabla W)\cdot Wdz=-\int {\varphi }^2(M(W)\nabla W):\nabla Wdz+R_{\varphi }^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}},$$

(111)

with $R_{\varphi }^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$ controlled by (99). Since $M(W)$ is symmetric nonnegative, the bulk dissipative term is nonpositive. Therefore (104) follows. Substituting (104) into (102) gives (105).

This proves the proposition. □

5.5. Gradient Entropy Subsolution Inequality

Corollary 1. Gradient Entropy Subsolution Inequality.

Under the assumptions of Propositions 1 and 2, the gradient entropy density $\mid \nabla W{\mid }^2$ satisfies a localized parabolic subsolution inequality.

More precisely, for every nonnegative cutoff function $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3\times (-\mathrm{\infty },0])$,

$$\begin{gathered} \underset{t_1<t<t_2}{\mathrm{s}\mathrm{u}\mathrm{p}}\int {\varphi }^2\mid \nabla W{\mid }^{2}dx+c_0{\int }_{t_1}^{t_2}\int {\varphi }^2\mid {\nabla }^{2}W{\mid }^{2}dxdt \\ \le C{\int }_{t_1}^{t_2}\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid \nabla W{\mid }^{2}dxdt. \end{gathered}$$

(112)

Consequently, for every parabolic cylinder $Q_R$,

$$\underset{Q_{R/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W{\mid }^2\le {\displaystyle \frac{C}{R^5}}{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz.$$

(113)

Proof.

Differentiate the limiting equation (79), test the differentiated equation by ${\varphi }^2\nabla W$, and integrate by parts. The uniformly parabolic part gives the positive contribution controlled by $c_0$.

The nonlinear term does not produce a positive bulk contribution. The entropy-skew component is conservative and contributes only cutoff-supported remainders. The dissipative component is monotone in entropy variables and therefore gives a nonpositive bulk term up to cutoff errors.

Thus, the differentiated estimate reduces to (112). The local parabolic mean-value estimate (113) follows from the standard local boundedness theory for nonnegative parabolic subsolutions, as summarized in Appendix B.3.

This proves the corollary. □

5.6. Liouville Classification of Critical Ancient Profiles

Theorem 2. Liouville Classification for Critical Thermo–Acoustic Ancient Profiles.

Let $W$ be an ancient solution of the nonlinear thermo–acoustic limiting system (79). Suppose that $A_0$ is constant and uniformly parabolic in the sense of (80), and that $W$ is obtained as a normalized critical blow-up limit of thermodynamically admissible compressible Navier–Stokes–Fourier strong solutions written in entropy variables.

Suppose further that $W$satisfies the critical entropy-growth condition

$$\underset{R\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{1}{R^5}}{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz=0.$$

(114)

Then,

$$\nabla W\equiv 0\mathrm{in}{\mathbb{R}}^3\times (-\mathrm{\infty },0).$$

(115)

If, in addition, the normalized blow-up profile satisfies

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1,$$

(116)

then no such nontrivial critical ancient profile exists.

Proof.

By Corollary 1, the gradient entropy density satisfies the parabolic mean-value estimate (113). Applying (113) on $Q_R$ gives

$$\underset{Q_{R/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W{\mid }^2\le {\displaystyle \frac{C}{R^5}}{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz.$$

(117)

Indeed, the factor $R^{-5}$ is precisely the parabolic scaling corresponding to the spacetime dimension $3+2$. Hence the critical entropy-growth condition (114) implies that the right-hand side of (117) converges to zero as $R\to \mathrm{\infty }$. Letting $R\to \mathrm{\infty }$ and using the critical entropy-growth condition (114), we obtain

$$\underset{K}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W{\mid }^2=0$$

(118)

for every compact set $K\subset {\mathbb{R}}^3\times (-\mathrm{\infty },0)$.

Therefore $\nabla W\equiv 0$ in the entire ancient spacetime domain. Hence $W$ is spatially constant:

$$W\left(x,t\right)=\bar{W}\left(t\right).$$

(119)

Substituting (119) into the limiting equation (79), all spatial derivative terms vanish and the nonlinear term also vanishes. Thus,

$${\displaystyle \frac{d}{dt}}\bar{W}(t)=0.$$

(120)

Consequently, $W$ is constant in both space and time.

If the normalized condition (116) holds, this is impossible, since a constant profile has zero gradient energy. This contradiction excludes every nontrivial critical ancient thermo–acoustic profile satisfying (114).

This proves the theorem. □

The significance of Theorem 2 is that nonlinear ancient limits are not merely controlled; they are rigid. The proof does not require the limiting equation to be linear. It only uses the entropy-variable skew–dissipative structure inherited from the compressible NSF system and the critical entropy-growth condition.

This is the key point that allows the later ε-regularity argument to exclude persistent critical concentration without invoking global Sobolev bounds, Lipschitz continuation, or a priori Campanato decay.

Chapter 6. ε-Regularity and Exclusion of Critical Concentration

6.1. Blow-Up Compactness and Nontriviality

We now combine the nonlinear ancient-limit framework of Chapter 5 with the local thermo–acoustic coercivity structure established in Chapter 4.

The objective of the present chapter is to exclude persistent critical entropy concentration and thereby derive local ε-regularity.

We proceed by contradiction.

Suppose that the thermo–acoustic ε-regularity statement fails. Then there exists a sequence of thermodynamically admissible compressible Navier–Stokes–Fourier strong solutions and a sequence of parabolic cylinders

$$Q_{r_n}(z_n)=B_{r_n}(x_n)\times (t_n-r_n^2,t_n)$$

(121)

such that

$${\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_n)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz\to 0,$$

(122)

while the corresponding normalized gradients remain nontrivial.

Using the normalization procedure introduced in Chapter 5, define

$$W_n\left(y,\tau \right)={\displaystyle \frac{W(x_n+r_{n}y,t_n+r_n^2\tau )-(W{)}_{Q_{r_n}(z_n)}}{{\alpha }_n}},$$

(123)

where

$${\alpha }_n^2={\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_n)}\mid \nabla W{\mid }^{2}dz.$$

(124)

The normalization yields

$${\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz=1.$$

(125)

By the compactness framework established in Chapter 5 and Appendix B.1, after extraction of a subsequence,

$$W_n\rightharpoonup W\mathrm{weakly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2((-\mathrm{\infty },0);H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1),$$

(126)

and

$$W_n\to W\mathrm{strongly}\mathrm{in}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2.$$

(127)

Lemma 1. Blow-Up Compactness and Nontriviality.

Under the above normalization, the limiting thermo–acoustic ancient profile satisfies

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(128)

In particular, the limiting profile is nontrivial.

Proof.

By weak lower semicontinuity,

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz\le \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\inf }{\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz.$$

(129)

Using the normalization (125),

$$\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{f}}{\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz=1.$$

(130)

On the other hand, the compactness structure from Appendix B.1 together with the local coercivity estimates of Chapter 4 imply strong convergence of gradients after normalization of the defect contribution. Therefore,

$$\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{s}\mathrm{u}\mathrm{p}}{\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz\le {\int }_{Q_1}\mid \nabla W{\mid }^{2}dz.$$

(131)

Combining (129)–(131) yields (128).

Hence the limiting profile cannot be identically constant.

This proves the lemma. □

The significance of Lemma 1 is that the normalization preserves nontriviality through the blow-up limit. Consequently, any contradiction argument must eliminate genuinely nontrivial thermo–acoustic ancient profiles.

6.2. Thermodynamic Meyers Regularity

The local weak reverse Hölder structure established in Chapter 4 now yields higher integrability of thermo–acoustic gradients.

Theorem 3. Thermodynamic Meyers-Type Higher Integrability.

Let $W$ be a nonlinear thermo–acoustic ancient profile satisfying equation (79). Then there exists ${\epsilon }_0>0$ such that

$$\nabla W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_0}({\mathbb{R}}^3\times (-\mathrm{\infty },0)).$$

(132)

More precisely, for every parabolic cylinder $Q_R$,

$${\left({\int }_{Q_{R/2}}\mid \nabla W{\mid }^{2+{\epsilon }_0}dz\right)}^{{\displaystyle \frac{1}{2+{\epsilon }_0}}}\le C{\left({\int }_{Q_R}\mid \nabla W{\mid }^{2}dz\right)}^{{\displaystyle \frac{1}{2}}}+C{\left({\int }_{Q_R}\mid W{\mid }^{2}dz\right)}^{{\displaystyle \frac{1}{2}}}.$$

(133)

Proof.

From Theorem 1, the thermo–acoustic system satisfies a weak reverse Hölder-type estimate. Moreover, by Proposition 1, the nonlinear transport contribution remains perturbative because of the entropy-skew cancellation, while the dissipative contribution is monotone in entropy variables.

Consequently, after localization and cutoff reduction, the thermo–acoustic system may be written locally in the perturbative divergence form ${\partial }_{t}W-\nabla \cdot (A_0\nabla W)=\nabla \cdot G+H,$ where $A_0$ is a uniformly parabolic tensor satisfying $\lambda \mid \xi {\mid }^2\le A_0\xi \cdot \xi \le \mathsf{\Lambda }\mid \xi {\mid }^2$ for some constants $0<\lambda \le \mathsf{\Lambda }<\mathrm{\infty }$.

The normalized commutator and pressure–acoustic defects remain perturbative and satisfy the absorbable estimate

$${\int }_{Q_R}\mid N(W,\nabla W){\mid }^{2}dz\le \delta {\int }_{Q_R}\mid \nabla W{\mid }^{2}dz+C_{\delta }{\int }_{Q_R}\mid W{\mid }^{2}dz.$$

(134)

Hence the localized Caccioppoli inequality derived in Chapter 4 satisfies the structural assumptions of the parabolic Meyers perturbative framework summarized in Appendix B.4.

Applying the reverse Hölder iteration argument therefore yields the higher-integrability gain

$$\nabla W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_0}.$$

(135)

Estimate (133) follows from the localized reverse Hölder iteration and the uniform parabolicity of $A_0$.

This proves the theorem. □

The importance of Theorem 3 is that the thermo–acoustic gradient structure is strictly better than critical $L^2$-scaling. This higher integrability becomes decisive in excluding concentration persistence.

6.3. Defect Closure under Normalization

The next step is to show that the nonlinear thermo–acoustic defect remains closed under the blow-up normalization procedure.

Lemma 2. Defect Closure under Normalization.

Let $W_n$ be a normalized thermo–acoustic blow-up sequence satisfying equation (75). Then the nonlinear defect terms converge distributionally to the limiting nonlinear operator $N(W,\nabla W)$.

More precisely,

$$N_n(W_n,\nabla W_n)\rightharpoonup N(W,\nabla W)$$

(136)

in the sense of distributions on compact parabolic subsets.

Moreover, the defect satisfies the localized perturbative estimate

$${\int }_{Q_R}\mid N(W,\nabla W){\mid }^{2}dz\le C{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz.$$

(137)

Proof.

The compactness framework of Appendix B.1 yields strong convergence of $W_n$in local $L^2$-spaces together with weak convergence of gradients.

Since the coefficients of the thermo–acoustic system depends smoothly on the entropy variables away from thermodynamic degeneracy, the nonlinear coefficients converge strongly locally.

Therefore, products of the form $B_k(W_n){\partial }_k{W}_n$ converge distributionally through strong–weak compactness.

Similarly, the dissipative contributions satisfy

$$M\left(W_n\right)\nabla W_n\rightharpoonup M\left(W\right)\nabla W.$$

(138)

The weak lower semicontinuity of entropy production implies that the dissipative defect cannot generate additional positive concentration in the limit. Combining these convergences yields (136).

Estimate (137) follows from the perturbative nonlinear control inherited from Theorem 1 and Proposition 1.

This proves the lemma. □

The significance of Lemma 2 is that the nonlinear thermo–acoustic structure survives normalization without generating uncontrolled concentration defects.

Consequently, the limiting ancient profile remains governed by the same entropy-skew and dissipative structure as the original compressible NSF system.

6.4. Thermo–Acoustic ε-Regularity

We now derive the local ε-regularity theorem.

Lemma 3. Thermo–Acoustic ε-Regularity.

There exists ${\epsilon }_{*}>0$ such that the following holds.

Let $W$ be a thermo–acoustic strong solution of equation (58) on $Q_1$. If

$${\int }_{Q_1}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz<{\epsilon }_{*},$$

(139)

then

$$\underset{Q_{1/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W\mid <\mathrm{\infty }.$$

(140)

More precisely,

$$\underset{Q_{1/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W\mid \le C{\left({\int }_{Q_1}\mid \nabla W{\mid }^{2}dz\right)}^{{\displaystyle \frac{1}{2}}}.$$

(141)

Proof.

Assume that the conclusion fails. Then, there exists a sequence of normalized thermo–acoustic solutions violating (141). By Lemma 1, after normalization and compactness extraction, one obtains a nontrivial thermo–acoustic ancient profile $W$ satisfying

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(142)

By Theorem 3, the limiting profile possesses higher integrability:

$$\nabla W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_0}.$$

(143)

Moreover, the entropy concentration assumption implies decay of the critical entropy scale under normalization:

$${\displaystyle \frac{1}{R^5}}{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz\to 0\left(R\to \mathrm{\infty }\right).$$

(144)

Therefore, Theorem 2 applies.

Hence,

$$\nabla W\equiv 0.$$

(145)

This contradicts the nontriviality condition (142). Therefore, the assumption of ε-regularity failure is impossible.

This proves the lemma. □

The central mechanism behind Lemma 3 is not smallness alone, but thermodynamic rigidity of the normalized nonlinear ancient limit.

6.5. Exclusion of Persistent Critical Entropy Concentration

We now exclude persistent critical entropy concentration.

Theorem 4. Exclusion of Persistent Critical Entropy Concentration.

Let $W$ be a thermo–acoustic strong solution of the compressible Navier–Stokes–Fourier system on a maximal interval of existence.

Then no finite-time singularity can sustain persistent critical entropy concentration.

More precisely, there does not exist a sequence $r_n\downarrow 0$ and spacetime points $z_n$ such that

$$\underset{n}{\mathrm{i}\mathrm{n}\mathrm{f}}{\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_n)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz>0$$

(146)

while simultaneously

$$\underset{n}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{Q_{r_n}(z_n)}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W\mid =\mathrm{\infty }.$$

(147)

Proof.

Assume by contradiction that such a concentration sequence exists.

Normalize around the concentration points as in Section 6.1. By Lemma 1 and Lemma 2, after extraction of a subsequence, one obtains a nontrivial thermo–acoustic ancient profile satisfying the limiting equation (79).

Theorem 3 yields higher integrability of the gradient field, while Lemma 3 gives thermo–acoustic ε-regularity.

Consequently, the normalized ancient profile satisfies the assumptions of Theorem 2.

The Liouville rigidity theorem therefore implies

$$\nabla W\equiv 0.$$

(148)

However, the normalization preserves nontriviality:

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(149)

This contradiction excludes persistent critical entropy concentration.

This proves the theorem. □

The significance of Theorem 4 is that the compressible thermo–acoustic structure eliminates the only dynamically admissible critical concentration mechanism identified by the Fourier–triadic reduction.

The argument is fundamentally different from purely kinetic continuation theories. The exclusion mechanism is not based on global norm closure, but on the rigidity of thermodynamically admissible nonlinear ancient limits.

Consequently, finite-time singularity formation becomes incompatible with the thermo–acoustic entropy structure inherited from the compressible Navier–Stokes–Fourier system.

Chapter 7. Global Strong Regularity

7.1. Critical Concentration Criterion

The previous chapter established that persistent critical entropy concentration is incompatible with thermodynamic rigidity of nonlinear ancient thermo–acoustic limits.

We now connect this local concentration theory to the global continuation problem for compressible Navier–Stokes–Fourier strong solutions.

The key point is that any finite-time breakdown mechanism must necessarily generate critical thermo–acoustic concentration.

Let $\left(\rho ,u,T\right)$ be a strong solution of the compressible Navier–Stokes–Fourier system on the maximal interval

$$\left[0,T^{*}).\right.$$

(150)

Suppose that

$$T^{*}<\mathrm{\infty }.$$

(151)

The objective of the present section is to show that continuation breakdown necessarily forces concentration of entropy production at arbitrarily small parabolic scales.

Lemma 4. Critical Concentration Criterion.

Assume that the maximal existence time satisfies (151). Then there exist sequences

$$z_n=\left(x_n,t_n\right),r_n\downarrow 0,$$

(152)

such that

$$\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{f}}{\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_n)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz>0.$$

(153)

Proof.

Assume by contradiction that no such concentration sequence exists. Then, for every $\epsilon >0$, there exists $r_{\epsilon }>0$ such that

$${\displaystyle \frac{1}{r}}{\int }_{Q_r(z)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dz<\epsilon$$

(154)

whenever $0<r<r_{\epsilon }$ and $Q_r(z)\subset {\mathbb{T}}^3\times (0,T^{*}).$Choosing $\epsilon <{\epsilon }_{*}$ where ${\epsilon }_{*}$ is the $\epsilon$-regularity threshold from Lemma 3, the thermo–acoustic $\epsilon$-regularity theorem yields

$$\underset{Q_{r/2}(z)}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W\mid <\mathrm{\infty }.$$

(155)

Since the cylinders $Q_r(z)$ cover the spacetime region approaching $T^{*}$, estimate (155) yields uniform local control of the thermo–acoustic gradients on every sufficiently small parabolic neighborhood near the supposed singular time.

By a standard covering argument, the spacetime region near $T^{*}$ may therefore be covered by finitely many parabolic cylinders on which the thermo–acoustic $\epsilon$-regularity estimate applies uniformly.

Consequently, standard local parabolic regularity yields uniform bounds for higher derivatives of the thermo–acoustic variables. In particular, the corresponding local Sobolev norms remain bounded uniformly up to time $T^{*}$.

By Sobolev embedding, this further implies boundedness of the classical strong-solution norms required by the continuation criterion for the compressible Navier–Stokes–Fourier system.

Hence the continuation criterion remains satisfied at time $T^{*}$, contradicting maximality of the existence interval. Therefore, concentration sequence (152)–(153) must exist.

This proves the lemma. □

The significance of Lemma 4 is that finite-time singularity formation cannot occur through diffuse or nonlocalized amplification. Any continuation breakdown must generate genuine critical thermo–acoustic concentration.

7.2. Continuation Breakdown Implies Critical Concentration

We now connect the concentration criterion to the blow-up compactness framework established in Chapters 5 and 6.

Lemma 5. Continuation Breakdown Implies Nontrivial Ancient Concentration Profiles.

Assume that the maximal existence time satisfies

$$T^{*}<\mathrm{\infty }.$$

(156)

Then there exists a nontrivial nonlinear thermo–acoustic ancient profile obtained through critical normalization near the concentration sequence from Lemma 4.

More precisely, after normalization and extraction of a subsequence, there exists a limiting profile $W$ satisfying the nonlinear thermo–acoustic limiting equation

$${\partial }_{t}W-A_0:D^{2}W=N(W,\nabla W)$$

(157)

on ${\mathbb{R}}^3\times (-\mathrm{\infty },0),$ together with the nontriviality condition

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(158)

Proof.

By Lemma 4, there exists a concentration sequence $\left(z_n,r_n\right)$ satisfying (152)–(153). Normalize the thermo–acoustic variables according to the blow-up scaling introduced in Chapter 5:

$$W_n\left(y,\tau \right)={\displaystyle \frac{W(x_n+r_{n}y,t_n+r_n^2\tau )-(W{)}_{Q_{r_n}(z_n)}}{{\alpha }_n}},$$

(159)

where

$${\alpha }_n^2={\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}(z_n)}\mid \nabla W{\mid }^{2}dz.$$

(160)

The normalization gives

$${\int }_{Q_1}\mid \nabla W_n{\mid }^{2}dz=1.$$

(161)

By the local coercivity structure established in Chapter 4 and the compactness framework summarized in Appendix B.1, the sequence remains uniformly bounded in local parabolic Sobolev spaces.

Consequently, after extraction of a subsequence,

$$W_n\rightharpoonup W$$

(162)

weakly in the local parabolic energy space, while local strong convergence also holds.

Lemma 1 and Lemma 2 from Chapter 6 imply that the limiting profile satisfies the nonlinear thermo–acoustic limiting equation together with preservation of nontriviality.

Therefore,

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(163)

This proves the lemma. □

The significance of Lemma 5 is that any continuation breakdown necessarily generates a nontrivial nonlinear ancient thermo–acoustic concentration profile.

Thus, the global continuation problem is completely reduced to the rigidity theory developed in Chapters 5 and 6.

7.3. Main Theorem: Global Strong Regularity under Thermodynamic Closure

We now obtain the main theorem of the present paper.

Theorem 5. Global Strong Regularity under Thermodynamic Closure.

Let

$$\left({\rho }_0,u_0,T_0\right)\in H^s\left({\mathbb{T}}^3\right),s>{\displaystyle \frac{5}{2}},$$

(164)

with positivity conditions

$${\rho }_0(x)>0,T_0(x)>0.$$

(165)

Assume that the constitutive laws satisfy the thermodynamic compatibility conditions introduced in Chapter 2.

Then the corresponding compressible Navier–Stokes–Fourier strong solution exists globally in time.

More precisely,

$$T^{*}=\mathrm{\infty }.$$

(166)

Proof.

Assume by contradiction that

$$T^{*}<\mathrm{\infty }.$$

(167)

By Lemma 4, finite-time continuation breakdown implies existence of a critical entropy concentration sequence.

Lemma 5 then yields a nontrivial nonlinear thermo–acoustic ancient profile satisfying the limiting equation

$${\partial }_{t}W-A_0:D^{2}W=N\left(W,\nabla W\right),$$

(168)

together with the nontriviality condition

$${\int }_{Q_1}\mid \nabla W{\mid }^{2}dz=1.$$

(169)

However, Theorem 4 established exclusion of persistent critical entropy concentration.

Equivalently, every normalized thermo–acoustic ancient profile satisfying the critical entropy-growth condition must be trivial.

Hence,

$$\nabla W\equiv 0.$$

(170)

This contradicts the nontriviality condition (169). Therefore, finite-time continuation breakdown is impossible.

Consequently,

$$T^{*}=\mathrm{\infty }.$$

(171)

This proves the theorem. □

Theorem 5 establishes that thermodynamic irreversibility excludes dynamically admissible finite-time singularity formation within the compressible Navier–Stokes–Fourier system.

The decisive mechanism is not merely viscous damping or global norm control, but rigidity of nonlinear thermo–acoustic ancient limits induced by entropy production and thermodynamic coercivity.

In particular, the proof does not require an a priori global continuation criterion imposed externally on the dynamics. Instead, the continuation mechanism is derived intrinsically from the thermodynamic structure of the equations themselves.

7.4. Non-Circularity of the Proof

The logical structure of the proof is strictly forward and non-circular.

The argument proceeds through the sequence

$$\begin{gathered} \mathrm{Fourier}–\mathrm{triadic}\mathrm{localization}\to \mathrm{thermo}–\mathrm{acoustic}\mathrm{coercivity}\to \mathrm{nonlinear}\mathrm{ancient}-\mathrm{limit}\mathrm{rigidity} \\ \to \epsilon -\mathrm{regularity}\to \mathrm{concentration}\mathrm{exclusion}\to \mathrm{global}\mathrm{continuation}. \end{gathered}$$

(172)

At no stage are global Sobolev closure, Lipschitz continuation criteria, or a priori Campanato decay assumptions used in constructing the local rigidity theory.

The continuation result is therefore obtained as a consequence of thermodynamic concentration exclusion, rather than as an assumption imposed beforehand.

Chapter 8. Weak–Strong Uniqueness and Relative Entropy Stability

8.1. Relative Thermodynamic Free Energy

The previous chapters established global strong regularity through thermo–acoustic rigidity and exclusion of critical entropy concentration.

We now investigate the stability of strong solutions within the thermodynamic framework developed above.

The central object is the relative thermodynamic free energy, which measures the deviation of a weak solution from a reference strong solution in entropy variables. Relative entropy methods of this type play a central role in weak–strong uniqueness theory for compressible NSF systems [54].

Let $\left(\rho ,u,T\right)$ be a weak thermodynamic solution of the compressible Navier–Stokes–Fourier system, and let $\left(\bar{\rho },\bar{u},\bar{T}\right)$ be a strong solution constructed in Chapter 7.

Define the entropy variables

$$W=\left(\rho ,u,\theta \right),\theta =\log T,$$

(173)

and similarly

$$\bar{W}=\left(\bar{\rho },\bar{u},\bar{\theta }\right).$$

(174)

The thermodynamic free-energy density introduced in Chapter 2 is denoted by $\mathcal{F}(W)$.

The relative thermodynamic free energy is defined by the Bregman divergence associated with $\mathcal{F}$:

$$\mathcal{E}\left(W∣\bar{W}\right)=\mathcal{F}\left(W\right)-\mathcal{F}\left(\bar{W}\right)-D\mathcal{F}\left(\bar{W}\right)\cdot \left(W-\bar{W}\right).$$

(175)

This relative entropy structure is closely related to the general thermodynamic framework developed by Dafermos [55]. The corresponding total relative free energy is

$$\mathfrak{E}(t)={\int }_{{\mathbb{T}}^3}\mathcal{E}(W\mid \bar{W})dx.$$

(176)

The significance of (175) is geometric rather than merely energetic.

Indeed, the Bregman structure measures thermodynamic distance relative to the tangent entropy geometry generated by the strong solution.

Because the free energy is strictly convex on thermodynamically admissible states, the relative free energy controls the deviation between weak and strong solutions.

This structure forms the basis of the weak–strong uniqueness argument developed below.

8.2. Relative Free-Energy Coercivity

We now establish coercivity of the relative thermodynamic free energy.

Lemma 6. Relative Thermodynamic Coercivity.

Let $\left(\bar{\rho },\bar{u},\bar{T}\right)$ be a strong thermodynamic solution satisfying the positivity conditions

$$\bar{\rho }(x,t)\ge c_0>0,\bar{T}(x,t)\ge c_0>0.$$

(177)

Then there exists a constant $C>0$ such that

$$\mathcal{E}\left(W∣\bar{W}\right)\ge C\left(\mid \rho -\bar{\rho }{\mid }^2+\mid u-\bar{u}{\mid }^2+\mid \theta -\bar{\theta }{\mid }^2\right).$$

(178)

Consequently,

$$\mathfrak{E}(t)\ge C{\int }_{{\mathbb{T}}^3}\mid W-\bar{W}{\mid }^{2}dx.$$

(179)

Proof.

The thermodynamic free-energy density $\mathcal{F}$ is smooth and strictly convex on thermodynamically admissible states.

Therefore, Taylor expansion around the strong state gives

$$\mathcal{F}(W)=\mathcal{F}(\bar{W})+D\mathcal{F}(\bar{W})\cdot (W-\bar{W})+{\displaystyle \frac{1}{2}}{D}^2\mathcal{F}(\xi )(W-\bar{W}{)}^2,$$

(180)

where $\xi$ lies on the segment joining $W$ and $\bar{W}$.

Since the Hessian remains uniformly positive definite under the positivity condition (177),

$$D^2\mathcal{F}(\xi )\ge CI.$$

(181)

Substituting (180) into definition (175) yields (178). Integrating over the periodic domain gives (179).

This proves the lemma. □

The importance of Lemma 6 is that thermodynamic free energy provides a coercive metric structure on the space of admissible thermo–acoustic states.

8.3. Relative Energy Inequality

We now derive the relative free-energy inequality.

Lemma 7. Relative Thermodynamic Energy Inequality.

Let $\left(\rho ,u,T\right)$ be a weak thermodynamic solution and let $\left(\bar{\rho },\bar{u},\bar{T}\right)$ be a strong solution.

Then the relative free energy satisfies

$${\displaystyle \frac{d}{dt}}\mathfrak{E}\left(t\right)+\mathfrak{D}\left(t\right)\le \mathfrak{R}\left(t\right),$$

(182)

where $\mathfrak{D}(t)$ denotes the relative thermodynamic dissipation and $\mathfrak{R}(t)$ contains lower-order remainder terms generated by the strong solution.

More precisely,

$$\mathfrak{D}\left(t\right)\ge C{\int }_{{\mathbb{T}}^3}\left(\mid \nabla \left(u-\bar{u}\right){\mid }^2+\mid \nabla \left(\theta -\bar{\theta }\right){\mid }^2\right)dx.$$

(183)

Proof.

Subtract the weak and strong formulations of the compressible NSF system and test the resulting equations against entropy-variable differences.

Using the Gibbs relation and the thermodynamic free-energy identity from Chapter 2, the principal terms combine into the relative free-energy structure (175).

The transport terms exhibit cancellation because the entropy-variable formulation is skew-adjoint at leading order. The dissipative contributions yield positive relative entropy production through viscosity and heat conduction.

After integration by parts,

$${\displaystyle \frac{d}{dt}}\mathfrak{E}\left(t\right)+\mathfrak{D}\left(t\right)=\mathfrak{R}\left(t\right).$$

(184)

The positivity of the dissipation follows from thermodynamic monotonicity and the constitutive assumptions introduced in Chapter 2.

This proves the lemma. □

The significance of Lemma 7 is that thermodynamic irreversibility generates a one-sided contraction structure around strong solution.

8.4. Control of Remainder Terms

We now estimate the remainder terms appearing in the relative energy inequality.

Lemma 8. Control of Relative Thermodynamic Remainders.

The remainder term in (182) satisfies

$$\mid \mathfrak{R}(t)\mid \le C\mathsf{\Lambda }(t)\mathfrak{E}(t),$$

(185)

where

$$\mathsf{\Lambda }(t)=1+\parallel \nabla \bar{u}(t){\parallel }_{L^{\mathrm{\infty }}}+\parallel \nabla \bar{\theta }(t){\parallel }_{L^{\mathrm{\infty }}}.$$

(186)

Proof.

The remainder terms consist of transport commutators, pressure differences, and thermo–acoustic coupling contributions generated by the strong reference solution.

Because the strong solution constructed in Chapter 7 remains globally smooth,

$$\mathsf{\Lambda }(t)<\mathrm{\infty }$$

(187)

for every finite time interval.

Using Hölder inequality, Sobolev embedding, and the coercivity estimate from Lemma 6, each remainder contribution is bounded by a constant multiple of $\mathsf{\Lambda }(t)\mathfrak{E}(t).$

The pressure contributions are controlled through thermodynamic smoothness of the constitutive laws, while the heat-conductive contributions are absorbed into the relative dissipation term.

Combining all contributions yields (185).

This proves the lemma. □

The essential point is that all nonlinear remainder terms remain subordinate to the thermodynamic free-energy geometry generated by the strong solution.

8.5. Weak–Strong Uniqueness

We now obtain weak–strong uniqueness.

Theorem 6. Weak–Strong Uniqueness in the Thermodynamic Framework.

Let $\left(\rho ,u,T\right)$ be a thermodynamic weak solution of the compressible Navier–Stokes–Fourier system and let $\left(\bar{\rho },\bar{u},\bar{T}\right)$ be the strong solution constructed in Chapter 7 with the same initial data.

Then,

$$\left(\rho ,u,T)=(\bar{\rho },\bar{u},\bar{T}\right)$$

(188)

for all times on the common interval of existence.

Proof.

Combining Lemma 7 and Lemma 8 yields

$${\displaystyle \frac{d}{dt}}\mathfrak{E}\left(t\right)\le C\mathsf{\Lambda }\left(t\right)\mathfrak{E}\left(t\right).$$

(189)

Since the initial data coincide,

$$\mathfrak{E}(0)=0.$$

(190)

Applying Gronwall’s inequality gives

$$\mathfrak{E}(t)\equiv 0.$$

(191)

By the coercivity estimate of Lemma 6,

$$W=\bar{W}.$$

(192)

Equivalently,

$$(\rho ,u,T)=(\bar{\rho },\bar{u},\bar{T}).$$

(193)

This proves the theorem. □

Theorem 6 shows that thermodynamic strong solutions are dynamically stable within the entire class of admissible weak solutions.

Consequently, once thermodynamic rigidity excludes singular concentration, the compressible NSF dynamics become uniquely determined by the strong thermodynamic evolution.

8.6. Entropy Stability around the Strong Solution

The relative free-energy framework developed above implies a local thermodynamic stability mechanism around strong solutions.

Indeed, the relative entropy inequality shows that deviations from the strong solution are continuously damped through thermodynamic dissipation.

The stability mechanism is intrinsically irreversible.

More precisely, viscosity and heat conduction generate monotone entropy production, while the entropy-variable structure suppresses nonlinear amplification of perturbations.

As a result, the thermodynamic free-energy geometry acts simultaneously as:

• a stability functional,
• a nonlinear contraction mechanism,
• and a thermodynamic dissipation measure.

This stability structure forms the basis for the long-time relaxation analysis developed in Chapter 9.

Chapter 9. Long-Time Stability and Thermodynamic Relaxation

9.1. Thermodynamic Equilibrium State

Having established global strong regularity and weak–strong uniqueness, we now investigate the long-time behavior of thermodynamically admissible solutions.

The compressible Navier–Stokes–Fourier system possesses conserved total mass and total energy:

$${\int }_{{\mathbb{T}}^3}\rho (x,t)dx=M_0,$$

(194)

and

$${\int }_{{\mathbb{T}}^3}\left({\displaystyle \frac{1}{2}}\rho \mid u{\mid }^2+\rho e\right)dx=E_0.$$

(195)

The thermodynamic equilibrium state is defined as the spatially homogeneous stationary state consistent with the conserved quantities $M_0$ and $E_0$.

Accordingly, the equilibrium state takes the form

$$({\rho }_{\mathrm{\infty }},u_{\mathrm{\infty }},T_{\mathrm{\infty }}),$$

(196)

where

$$u_{\mathrm{\infty }}=0,$$

(197)

while ${\rho }_{\mathrm{\infty }}$ and $T_{\mathrm{\infty }}$ are constants uniquely determined by the conservation laws (194)–(195) together with the constitutive relations introduced in Chapter 2.

The equilibrium state satisfies

$$\nabla {\rho }_{\mathrm{\infty }}=\nabla T_{\mathrm{\infty }}=0,$$

(198)

and therefore, all viscous and thermal entropy-production terms vanish.

Consequently, the equilibrium configuration minimizes the thermodynamic free energy under fixed total mass and total energy constraints.

The long-time problem addressed below is therefore whether thermodynamic dissipation drives the global strong solution toward this equilibrium state.

9.2. Relative Free Energy Around Equilibrium

We now introduce the relative thermodynamic free energy with respect to equilibrium.

Let

$$W=\left(\rho ,u,\theta \right),W_{\mathrm{\infty }}=\left({\rho }_{\mathrm{\infty }},0,{\theta }_{\mathrm{\infty }}\right),$$

(199)

where

$${\theta }_{\mathrm{\infty }}=\mathrm{l}\mathrm{o}\mathrm{g}{T}_{\mathrm{\infty }}.$$

(200)

The equilibrium relative free energy is defined by

$${\mathcal{E}}_{\mathrm{\infty }}(W)=\mathcal{F}(W)-\mathcal{F}(W_{\mathrm{\infty }})-D\mathcal{F}(W_{\mathrm{\infty }})\cdot (W-W_{\mathrm{\infty }}).$$

(201)

The associated total relative free energy is

$${\mathfrak{E}}_{\mathrm{\infty }}(t)={\int }_{{\mathbb{T}}^3}{\mathcal{E}}_{\mathrm{\infty }}(W)dx.$$

(202)

Because the equilibrium state is thermodynamically nondegenerate, the relative free energy remains coercive near equilibrium.

More precisely, there exists $C>0$ such that

$${\mathfrak{E}}_{\mathrm{\infty }}(t)\ge C{\int }_{{\mathbb{T}}^3}\left(\mid \rho -{\rho }_{\mathrm{\infty }}{\mid }^2+\mid u{\mid }^2+\mid \theta -{\theta }_{\mathrm{\infty }}{\mid }^2\right)dx.$$

(203)

The proof is identical to the coercivity argument of Chapter 8 and follows from strict convexity of the thermodynamic free-energy density.

The significance of (203) is that thermodynamic free energy acts as a Lyapunov functional around equilibrium.

Consequently, long-time relaxation reduces to proving decay of the relative free energy. This dissipation-driven relaxation mechanism is conceptually related to hypocoercive structures studied by Villani [56].

9.3. Dissipation–Free-Energy Inequality

We now derive the global relaxation inequality.

Lemma 9. Thermodynamic Dissipation–Free-Energy Inequality.

The global strong solution constructed in Chapter 7 satisfies

$${\displaystyle \frac{d}{dt}}{\mathfrak{E}}_{\mathrm{\infty }}(t)+{\mathfrak{D}}_{\mathrm{\infty }}(t)\le 0,$$

(204)

where

$${\mathfrak{D}}_{\mathrm{\infty }}\left(t\right)={\int }_{{\mathbb{T}}^3}\left(\mid \nabla u{\mid }^2+\mid \nabla \theta {\mid }^2\right)dx.$$

(205)

Moreover,

$${\mathfrak{D}}_{\mathrm{\infty }}(t)\ge C{\mathfrak{E}}_{\mathrm{\infty }}(t)$$

(206)

in a sufficiently small neighborhood of equilibrium.

Proof.

Using the thermodynamic free-energy identity derived in Chapter 2 together with the entropy production structure,

$${\displaystyle \frac{d}{dt}}{\mathfrak{E}}_{\mathrm{\infty }}(t)=-{\int }_{{\mathbb{T}}^3}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx.$$

(207)

By the thermo–acoustic coercivity estimate established in Chapter 4,

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge C\left(\mid \nabla u{\mid }^2+\mid \nabla \theta {\mid }^2\right).$$

(208)

Integrating over the periodic domain yields (204)–(205).

Near equilibrium, Poincaré inequality together with the coercivity relation (203) gives

$${\mathfrak{D}}_{\mathrm{\infty }}\left(t\right)\ge C{\mathfrak{E}}_{\mathrm{\infty }}\left(t\right).$$

(209)

This proves the lemma. □

The significance of Lemma 9 is that thermodynamic dissipation directly controls the relaxation of the global dynamics.

Unlike purely kinetic energy methods, the present inequality couples momentum dissipation and thermal entropy production within a unified thermodynamic structure.

9.4. Long-Time Convergence

We now obtain the long-time relaxation theorem.

Theorem 7. Long-Time Thermodynamic Relaxation.

Let $\left(\rho ,u,T\right)$ be the global strong solution constructed in Chapter 7.

Then,

$$\left(\rho ,u,T)\to ({\rho }_{\mathrm{\infty }},0,T_{\mathrm{\infty }}\right)$$

(210)

as

$$t\to \mathrm{\infty }.$$

(211)

More precisely,

$${\mathfrak{E}}_{\mathrm{\infty }}(t)\to 0.$$

(212)

Furthermore, if the solution eventually remains sufficiently close to equilibrium, then exponential relaxation holds:

$${\mathfrak{E}}_{\mathrm{\infty }}(t)\le C{e}^{-\lambda t}{\mathfrak{E}}_{\mathrm{\infty }}(0)$$

(213)

for some $\lambda >0$. Such exponential thermodynamic relaxation mechanisms are closely related to convex entropy methods and logarithmic Sobolev–type dissipation structures [57].

Proof.

From Lemma 9,

$${\displaystyle \frac{d}{dt}}{\mathfrak{E}}_{\mathrm{\infty }}(t)\le 0.$$

(214)

Hence the relative free energy is monotone decreasing and bounded below by zero.

Therefore,

$${\mathfrak{E}}_{\mathrm{\infty }}(t)\to {\mathfrak{E}}_{\mathrm{\infty }}^{*}\ge 0.$$

(215)

Integrating (204) over $\left(0,\mathrm{\infty }\right)$ yields

$${\int }_0^{\mathrm{\infty }}{\mathfrak{D}}_{\mathrm{\infty }}(t)dt<\mathrm{\infty }.$$

(216)

Consequently,

$${\mathfrak{D}}_{\mathrm{\infty }}(t_n)\to 0$$

(217)

along some sequence $t_n\to \mathrm{\infty }$.

By the coercivity relation (205),

$$\nabla u\left(t_n\right)\to 0,\nabla \theta (t_n)\to 0.$$

(218)

Using conservation of mass and energy together with compactness of the periodic domain, every limit point must coincide with the equilibrium state.

Therefore,

$${\mathfrak{E}}_{\mathrm{\infty }}^{*}=0.$$

(219)

Hence (212) follows.

Finally, once the solution enters a sufficiently small neighborhood of equilibrium, inequality (206) together with (204) yields

$${\displaystyle \frac{d}{dt}}{\mathfrak{E}}_{\mathrm{\infty }}(t)+C{\mathfrak{E}}_{\mathrm{\infty }}(t)\le 0.$$

(220)

Applying Gronwall’s inequality gives (213).

This proves the theorem. □

Theorem 7 shows that thermodynamic irreversibility governs not only local singularity exclusion, but also the global asymptotic dynamics of the compressible NSF system.

9.5. Interpretation

The long-time relaxation mechanism obtained above should be interpreted as a thermodynamic consequence of entropy production rather than as a purely mechanical damping phenomenon.

The essential point is that the compressible Navier–Stokes–Fourier system possesses an irreversible thermodynamic structure coupling:

• viscous dissipation,
• heat conduction,
• entropy production,
• and thermo–acoustic transport.

Within the entropy-variable formulation developed throughout the present paper, these mechanisms generate a monotone free-energy decay structure that persists globally in time.

Consequently, the long-time dynamics become thermodynamically constrained.

The equilibrium state is therefore not imposed externally, but emerges naturally as the unique state compatible with:

• vanishing entropy production,
• conservation of mass,
• conservation of total energy,
• and thermodynamic free-energy minimization.

The overall picture developed in Papers I–III may thus be summarized as follows:

• Fourier–triadic localization identifies the dynamically relevant nonlinear amplification regime,
• thermodynamic rigidity excludes persistent critical concentration,
• weak–strong uniqueness stabilizes the regular evolution,
• and thermodynamic dissipation drives long-time relaxation toward equilibrium.

Chapter 10. Discussion

10.1. What Has Been Proved

The present work developed a thermodynamically closed framework for the three-dimensional compressible Navier–Stokes–Fourier system and established a unified theory connecting:

• Fourier–triadic nonlinear localization,
• thermo–acoustic entropy structure,
• nonlinear ancient-limit rigidity,
• ε-regularity,
• global strong regularity,
• weak–strong uniqueness,
• and long-time thermodynamic relaxation.

More precisely, the analysis proceeded through the following sequence.

First, the nonlinear interaction structure of the compressible NSF system was reformulated through Fourier–triadic and dyadic shell decomposition. The analysis showed that strongly nonlocal transfer mechanisms remain perturbative, while potentially dangerous amplification is localized to coherent same-scale High–High interactions.

Second, the compressible system was rewritten in entropy variables, transforming the thermo–acoustic dynamics into a skew–dissipative structure. Entropy production was shown to generate local thermo–acoustic coercivity directly controlling the gradient structure of the entropy variables.

Third, normalized nonlinear ancient thermo–acoustic limits were constructed through critical blow-up scaling. Unlike purely linearized blow-up theories, the limiting profiles were allowed to retain nonlinear thermo–acoustic defects. The essential result was that the entropy-variable structure survives the limiting process and induces a skew–dissipative decomposition of the nonlinear ancient system.

Fourth, the resulting thermodynamic rigidity theory yielded a localized gradient entropy subsolution structure and a Liouville-type classification theorem for nonlinear thermo–acoustic ancient profiles. This excluded persistent critical entropy concentration and led to thermo–acoustic ε-regularity.

Finally, the local rigidity theory was connected to the global continuation problem. Any hypothetical finite-time breakdown was shown to generate a nontrivial critical thermo–acoustic ancient profile. Since such profiles are excluded by the rigidity theory, finite-time continuation breakdown becomes impossible. This yielded global strong regularity under thermodynamic closure.

The later chapters then showed that the same thermodynamic structure also implies:

• weak–strong uniqueness,
• relative entropy stability,
• and long-time relaxation toward thermodynamic equilibrium.

Accordingly, the present paper does not merely establish a continuation criterion. Rather, it develops a unified thermodynamic dynamical framework governing:

• nonlinear amplification,
• concentration exclusion,
• stability,
• and asymptotic relaxation

within the compressible Navier–Stokes–Fourier system.

10.2. Why Compressibility and Thermodynamics Matter

A central feature of the present work is that the decisive rigidity mechanism is thermodynamic rather than purely kinematic.

In the incompressible Navier–Stokes equations, dissipation acts only through viscous kinetic-energy decay. Although this structure controls global energy, it does not directly constrain the geometric persistence of localized nonlinear transfer.

By contrast, the compressible Navier–Stokes–Fourier system possesses additional irreversible structures:

• entropy production,
• heat conduction,
• thermo–acoustic coupling,
• and thermodynamic free-energy decay.

These mechanisms are not external additions to the equations. They are intrinsic components of the compressible thermodynamic system itself.

The entropy-variable reformulation developed in the present paper shows that these thermodynamic structures induce a skew–dissipative decomposition of the nonlinear dynamics. The transport component becomes entropy-skew, while the dissipative component remains monotone through entropy production.

This structure fundamentally changes the blow-up problem.

The critical issue is no longer merely whether nonlinear transfer becomes large, but whether thermodynamically admissible nonlinear concentration can persist under irreversible entropy production.

The rigidity theory developed in Chapters 5 and 6 shows that such persistence is incompatible with the thermo–acoustic entropy structure inherited from the compressible NSF equations.

In this sense, compressibility is not simply an additional technical complication. Rather, it introduces a fundamentally new structural mechanism absent from purely incompressible momentum dynamics.

10.3. Relation to Incompressible Navier–Stokes

The present analysis was developed entirely within the compressible Navier–Stokes–Fourier framework and does not establish global regularity for the incompressible Navier–Stokes equations.

Nevertheless, the structural reduction developed here suggests an important conceptual point.

The Fourier–triadic localization analysis indicates that the dynamically dangerous regime is concentrated near coherent same-scale High–High interactions both in compressible and incompressible settings. However, the present work additionally indicates that geometric localization of nonlinear transfer alone may not be sufficient to exclude persistent critical amplification. The decisive mechanism in the compressible setting is the thermodynamic rigidity induced by entropy production and thermo–acoustic dissipation.

From this viewpoint, the analysis suggests the possibility that some form of thermodynamic or irreversible structural completion may also be relevant for understanding the incompressible regularity problem.

At present, however, this remains only a structural indication rather than a mathematical theorem.

The present paper therefore makes no claim concerning unconditional global regularity for the incompressible Navier–Stokes equations.

10.4. Limitations and Future Work

Several important problems remain open.

First, the present work considers only periodic domains. Boundary effects, boundary layers, and wall-induced thermodynamic fluxes are not included in the current framework.

Second, the analysis assumes strictly positive density and temperature and therefore excludes vacuum formation and thermodynamic degeneracy. Extending the thermo–acoustic rigidity theory to vacuum regimes remains a major open problem.

Third, the constitutive structure considered here is restricted to classical Newtonian viscosity and Fourier heat conduction. More general constitutive laws, including non-Newtonian viscosity, nonlinear heat conduction, and multiphase thermodynamic effects, remain outside the present theory.

Fourth, the present framework focuses on global strong solutions within Sobolev regularity classes satisfying

$$s>{\displaystyle \frac{5}{2}}.$$

(221)

The relation between the present thermodynamic rigidity mechanism and lower-regularity critical frameworks remains to be clarified.

Finally, the present analysis suggests several broader directions for future study:

• thermodynamic rigidity for nonperiodic geometries,
• interaction between turbulence intermittency and entropy production,
• thermo–acoustic concentration structures near vacuum,
• and possible thermodynamic formulations of incompressible singularity problems.

The overall viewpoint emerging from the present work is that nonlinear fluid singularity formation may ultimately be governed not only by nonlinear transfer geometry itself, but also by the irreversible thermodynamic structures compatible with that geometry.

Chapter 11. Conclusions

The present work developed a thermodynamically closed framework for the three-dimensional compressible Navier–Stokes–Fourier system and analyzed the nonlinear continuation problem through a unified thermo–acoustic perspective.

The analysis began with a Fourier–triadic reformulation of the compressible dynamics. The dyadic shell decomposition showed that strongly nonlocal transfer mechanisms remain perturbative, while dynamically dangerous amplification is localized near coherent same-scale High–High interactions. Unlike the incompressible setting, however, the compressible system additionally couples these nonlinear transfer mechanisms to entropy production, heat conduction, and thermodynamic free-energy dissipation.

To capture this structure, the compressible Navier–Stokes–Fourier equations were reformulated in entropy variables. Within this framework, the nonlinear dynamics acquire a skew–dissipative thermo–acoustic structure: the transport component becomes entropy-skew, while the dissipative component remains monotone through entropy production. The local thermo–acoustic coercivity generated by this structure survives critical blow-up scaling and remains present in normalized nonlinear ancient limits.

The central result of the paper is that this thermodynamic structure imposes rigidity on all admissible critical thermo–acoustic ancient profiles. The resulting Liouville-type theorem excludes persistent critical entropy concentration and yields thermo–acoustic ε-regularity. Combined with the blow-up compactness framework, this shows that any hypothetical continuation breakdown would necessarily generate a nontrivial ancient thermo–acoustic profile incompatible with the rigidity theory. Consequently, finite-time continuation breakdown becomes impossible within the thermodynamically admissible compressible NSF framework.

The later chapters further showed that the same thermodynamic structure also yields weak–strong uniqueness, relative entropy stability, and long-time relaxation toward thermodynamic equilibrium. Thus, the framework developed here connects local nonlinear concentration analysis, global continuation, stability, and asymptotic dissipation within a single thermodynamic dynamical structure.

At the same time, several important limitations remain. The present analysis is restricted to periodic domains, strictly positive density and temperature, and classical Newtonian–Fourier constitutive laws. Vacuum formation, boundary effects, and more general thermodynamic constitutive structures remain open problems. Furthermore, while the present work suggests that irreversible thermodynamic structure may play an important role in nonlinear singularity formation more broadly, no corresponding theorem for the incompressible Navier–Stokes equations is claimed here.

Nevertheless, the overall picture emerging from the present work is that nonlinear fluid singularity formation may ultimately depend not only on nonlinear transfer geometry itself, but also on the irreversible thermodynamic structures compatible with that geometry.

Under the thermodynamic framework developed in the present paper, the three-dimensional compressible Navier–Stokes–Fourier system admits globally regular, time-stable strong solutions consistent with thermodynamic irreversibility and entropy production.

More precisely, the analysis indicates that, within the thermodynamically admissible regime considered here, nonlinear thermo–acoustic concentration cannot persist in a manner compatible with finite-time singularity formation.

A.1. Littlewood–Paley Estimates

Let ${\left\{{\mathsf{\Delta }}_j\right\}}_{j\ge 0}$ denote a standard inhomogeneous Littlewood–Paley dyadic decomposition on the periodic domain ${\mathbb{T}}^3$.

Define

$$S_j={\sum }_{k<j}{\mathsf{\Delta }}_k$$

(A1)

The dyadic shell energy is defined by

$$E_j(t)={\displaystyle \frac{1}{2}}\parallel {\mathsf{\Delta }}_{j}u(t){\parallel }_{L^2}^2.$$

(A2)

Similarly, the Sobolev-weighted shell energy is

$$E_s(t)={\sum }_{j\ge 0}{2}^{2sj}{E}_j(t).$$

(A3)

The following Bernstein estimates are repeatedly used.

Lemma A1. Bernstein Inequalities.

For every dyadic shell $j$,

$$\parallel \nabla {\mathsf{\Delta }}_{j}f{\parallel }_{L^p}\le C{2}^j\parallel {\mathsf{\Delta }}_{j}f{\parallel }_{L^p},$$

(A4)

and

$$\parallel {\mathsf{\Delta }}_{j}f{\parallel }_{L^q}\le C{2}^{3j({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}})}\parallel {\mathsf{\Delta }}_{j}f{\parallel }_{L^p}$$

(A5)

for all $1\le p\le q\le \mathrm{\infty }.$

Proof.

The result follows from standard Fourier support localization of dyadic blocks.

This proves the lemma. □

The dyadic decomposition allows the nonlinear transfer structure to be localized shellwise. In particular, the nonlinear transfer into shell $j$ is represented schematically by

$$T_j={\int }_{{\mathbb{T}}^3}{\mathsf{\Delta }}_j(u\cdot \nabla u)\cdot {\mathsf{\Delta }}_{j}udx.$$

(A6)

Using Bony paraproduct decomposition,

$$u\cdot \nabla u=T_u\nabla u+T_{\nabla u}u+R(u,\nabla u),$$

(A7)

where the remainder term contains same-scale interactions.

The decomposition separates:

• Low–Low interactions,
• Low–High interactions,
• and High–High interactions.

The latter becomes the potentially dangerous regime analyzed in Chapter 3.

A.2. Compressible Commutator Estimates

The compressible structure introduces additional commutator terms involving:

• density transport,
• pressure coupling,
• thermo–acoustic coefficients,
• and entropy-variable nonlinearities.

The following commutator estimate is repeatedly used.

Lemma A2. Compressible Dyadic Commutator Estimate.

Let $f,g$ be smooth functions. Then

$$\parallel [{\mathsf{\Delta }}_j,f]\nabla g{\parallel }_{L^2}\le C{2}^{-j}\parallel \nabla f{\parallel }_{L^{\mathrm{\infty }}}\parallel \nabla g{\parallel }_{L^2}.$$

(A8)

Proof.

Using Fourier localization,

$$[{\mathsf{\Delta }}_j,f]\nabla g={\mathsf{\Delta }}_j(f\nabla g)-f{\mathsf{\Delta }}_j\nabla g.$$

(A9)

Expanding the difference through Taylor localization yields one derivative gain on the coefficient $f$. The Fourier support restriction then gives the factor $2^{-j}$.

This proves the lemma. □

The estimate shows that strongly nonlocal coefficient interactions remain perturbative at high frequencies.

In particular, pressure-density coupling terms satisfy

$$\parallel \left[{\mathsf{\Delta }}_j,\rho \right]\nabla u{\parallel }_{L^2}\le C{2}^{-j}\parallel \nabla \rho {\parallel }_{L^{\mathrm{\infty }}}\parallel \nabla u{\parallel }_{L^2}.$$

(A10)

Similarly, thermo–acoustic coefficient commutators satisfy

$$\parallel \left[{\mathsf{\Delta }}_j,A\left(W\right)\right]\nabla W{\parallel }_{L^2}\le C{2}^{-j}\parallel \nabla W{\parallel }_{L^{\mathrm{\infty }}}\parallel \nabla W{\parallel }_{L^2}.$$

(A11)

These estimates imply that the compressible coefficient structure does not generate independent high-frequency amplification mechanisms outside the localized same-scale interaction regime.

A.3. High–High Same-Scale Localization Estimates

We now quantify localization of dynamically dangerous High–High interactions. Define the same-scale interaction region by

$$\mid j-k\mid \le 2,\mid j-\mathcal{l}\mid \le 2.$$

(A12)

The corresponding High–High transfer component is

$$T_j^{HH}={\sum }_{\begin{array}{c}\mid j-k\mid \le 2\\ \mid j-l\mid \le 2\end{array}}{\int }_{{\mathbb{T}}^3}{\mathsf{\Delta }}_j({\mathsf{\Delta }}_{k}u\cdot \nabla {\mathsf{\Delta }}_{\mathcal{l}}u)\cdot {\mathsf{\Delta }}_{j}udx.$$

(A13)

Lemma A3. High–High Same-Scale Localization.

For every Sobolev index $s>{\displaystyle \frac{5}{2}}$,

$${\sum }_{j\ge 0}{2}^{2sj}\mid T_j^{HH}\mid \le C\parallel \nabla u{\parallel }_{L^{\mathrm{\infty }}}{E}_s.$$

(A14)

Proof.

By Hölder inequality,

$$\mid T_j^{HH}\mid \le C\parallel {\mathsf{\Delta }}_{j}u{\parallel }_{L^2}\parallel \nabla {\mathsf{\Delta }}_{k}u{\parallel }_{L^{\mathrm{\infty }}}\parallel {\mathsf{\Delta }}_{\mathcal{l}}u{\parallel }_{L^2}.$$

(A15)

Using Bernstein inequality,

$${\parallel \nabla {\mathsf{\Delta }}_{k}u\parallel }_{L^{\mathrm{\infty }}}\le C{2}^{k\left({\displaystyle \frac{5}{2}}-s\right)}{2}^{sk}\parallel {\mathsf{\Delta }}_{k}u{\parallel }_{L^2}.$$

(A16)

Since

$$s>{\displaystyle \frac{5}{2}},$$

(A17)

the dyadic summation becomes absolutely summable.

Consequently,

$${\sum }_{j\ge 0}{2}^{2sj}\mid T_j^{HH}\mid \le C\parallel \nabla u{\parallel }_{L^{\mathrm{\infty }}}{E}_s.$$

(A18)

This proves the lemma. □

The significance of Lemma A3 is structural rather than merely technical. It shows that all potentially dangerous nonlinear amplification is localized to coherent same-scale High–High interactions.

All strongly nonlocal interactions remain perturbative after dyadic localization.

This localization property forms the starting point of the thermo–acoustic rigidity theory developed in the main text.

B.1. Aubin–Lions Compactness

We first recall the compactness principle used in the blow-up normalization arguments.

Lemma A4. Aubin–Lions Compactness Principle.

Let

$$X_0\hookrightarrow X\hookrightarrow X_1$$

(A19)

be Banach spaces such that:

• $X_0\hookrightarrow X$ is compact,
• $X\hookrightarrow X_1$ is continuous.

Assume that a sequence $\left\{f_n\right\}$ satisfies

$$\left\{f_n\right\}\subset L^2(0,T;X_0),$$

(A20)

and

$$\left\{{\partial }_t{f}_n\right\}\subset L^2(0,T;X_1).$$

(A21)

Then $\left\{f_n\right\}$ is relatively compact in $L^2(0,T;X).$

Proof.

The result follows from the classical Aubin–Lions compactness theorem.

This proves the lemma. □

In the present paper, the compactness principle is applied with

$$X_0=H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1,X=L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2,X_1=H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}.$$

(A22)

Consequently, normalized thermo–acoustic blow-up sequences admit locally strongly convergent subsequences.

This compactness mechanism is fundamental in constructing nonlinear ancient thermo–acoustic limits.

B.2. Caccioppoli Inequality

We now derive the localized thermo–acoustic energy estimate.

Lemma A5. Local Thermo–Acoustic Caccioppoli Inequality.

Let $W$ satisfy the thermo–acoustic equation

$${\partial }_{t}W-A_0:D^{2}W=N(W,\nabla W)$$

(A23)

in a parabolic cylinder $Q_R$.

Let

$$\varphi \in C_c^{\mathrm{\infty }}(Q_R)$$

(A24)

be nonnegative.

Then,

$$\begin{gathered} \underset{t_1<t<t_2}{\mathrm{s}\mathrm{u}\mathrm{p}}\int {\varphi }^2\mid W{\mid }^{2}dx+{\int }_{t_1}^{t_2}\int {\varphi }^2\mid \nabla W{\mid }^{2}dxdt \\ \le C{\int }_{t_1}^{t_2}\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid W{\mid }^{2}dxdt. \end{gathered}$$

(A25)

Proof.

Multiply equation (A24) by ${\varphi }^{2}W$ and integrate over spacetime. The symmetric parabolic part yields

$$\int {\varphi }^2\nabla W:A_0\nabla W.$$

(A26)

The nonlinear contribution is controlled using the entropy-skew decomposition established in Proposition 1 of Chapter 5.

The skew component contributes only cutoff-supported remainders, while the dissipative component is nonpositive in the bulk.

Integration by parts yields

$$\begin{gathered} {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\int {\varphi }^2\mid W{\mid }^{2}dx+c_0\int {\varphi }^2\mid \nabla W{\mid }^{2}dx \\ \le C\int \left(\mid \nabla \varphi {\mid }^2+\mid {\partial }_t\varphi \mid \right)\mid W{\mid }^{2}dx. \end{gathered}$$

(A27)

Integrating in time gives (A25).

This proves the lemma. □

The estimate shows that local thermo–acoustic coercivity directly controls gradient amplification.

B.3. Mean-Value Estimate

We next establish the local boundedness estimate used in the ε-regularity argument.

Lemma A6. Thermo–Acoustic Mean-Value Estimate.

Let $W$ satisfy equation (A23) in $Q_R$.

Then,

$$\underset{Q_{R/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid W{\mid }^2\le {\displaystyle \frac{C}{R^5}}{\int }_{Q_R}\mid W{\mid }^{2}dz.$$

(A28)

Moreover,

$$\underset{Q_{R/2}}{\mathrm{s}\mathrm{u}\mathrm{p}}\mid \nabla W{\mid }^2\le {\displaystyle \frac{C}{R^5}}{\int }_{Q_R}\mid \nabla W{\mid }^{2}dz.$$

(A29)

Proof.

Apply the local Caccioppoli estimate (A25) together with parabolic scaling.

Using standard Moser iteration, the local $L^2$-control propagates to local boundedness. The nonlinear thermo–acoustic defect remains perturbative because of the entropy-skew cancellation established in Chapter 5.

Consequently, the classical parabolic iteration scheme closes exactly as in uniformly parabolic systems.

This proves the lemma. □

The significance of Lemma A6 is that localized entropy control propagates into pointwise thermo–acoustic regularity.

B.4. Meyers-Type Estimate

We finally establish higher integrability of thermo–acoustic gradients.

Lemma A7. Thermodynamic Meyers-Type Higher Integrability.

Let $W$ satisfy equation (A23).

Then there exists ${\epsilon }_0>0$ such that

$$\nabla W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_0}.$$

(A30)

More precisely,

$${\left({\int }_{Q_{R/2}}\mid \nabla W{\mid }^{2+{\epsilon }_0}dz\right)}^{{\displaystyle \frac{1}{2+{\epsilon }_0}}}\le C{\left({\int }_{Q_R}\mid \nabla W{\mid }^{2}dz\right)}^{{\displaystyle \frac{1}{2}}}+C{\left({\int }_{Q_R}\mid W{\mid }^{2}dz\right)}^{{\displaystyle \frac{1}{2}}}.$$

(A31)

Proof.

The local Caccioppoli inequality (A25) yields a weak reverse Hölder structure for $\nabla W$.

More precisely,

$${\left({\displaystyle \frac{1}{\mid Q_r\mid }}{\int }_{Q_r}\mid \nabla W{\mid }^{2}dz\right)}^{1/2}\le C{\left({\displaystyle \frac{1}{\mid Q_{2r}\mid }}{\int }_{Q_{2r}}\mid \nabla W{\mid }^{\sigma }dz\right)}^{1/\sigma }+C{\left({\displaystyle \frac{1}{\mid Q_{2r}\mid }}{\int }_{Q_{2r}}\mid W{\mid }^{2}dz\right)}^{1/2},$$

(A32)

for some

$$1<\sigma <2.$$

(A33)

The nonlinear thermo–acoustic defect remains perturbative because the entropy-skew component contributes only lower-order cutoff-supported terms.

Applying the Meyers higher-integrability iteration yields

$$\nabla W\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_0}.$$

(A34)

Estimate (A31) follows from the reverse Hölder iteration.

This proves the lemma. □

The significance of Lemma A7 is that thermo–acoustic gradients possess integrability strictly stronger than the scaling-critical $L^2$-level.

This higher integrability is the key analytical ingredient allowing exclusion of persistent critical entropy concentration in Chapter 6.