Compressible Rotating Chemotaxis–Navier–Stokes Systems: Lagrangian Singularity Analysis and Dimension Bounds

1. Introduction

The chemotaxis–Navier–Stokes system provides a fundamental mathematical framework for modeling the interaction between incompressible viscous fluids and chemically driven aggregation phenomena, arising naturally in biological fluid dynamics, active matter, and population dynamics. From the analytical perspective, this system exhibits a rich interplay between nonlinear transport, diffusion, and chemotactic forcing, which makes the study of regularity and singularity formation particularly challenging. Despite substantial progress in the global existence and partial regularity theory for weak and strong solutions in three dimensions [3,4,6], the fine geometric structure of possible singularities remains largely unexplored, especially when compressibility, rotation, and high Mach numbers are taken into account.

Classical approaches to singularity analysis are predominantly Eulerian, viewing singularities as irregular subsets of space–time ${\mathbb{R}}^3\times (0,T)$. Within this framework, partial regularity theory yields measure-theoretic information on the singular set, such as bounds on its parabolic Hausdorff dimension of Caffarelli–Kohn–Nirenberg type [5]. While these results are fundamental, they provide limited insight into the intrinsic geometric organization of singularities and their dynamical evolution under the flow.

Motivated by physical considerations and numerical evidence indicating that singular behavior often aligns along dynamically evolving structures, we adopt in this work a genuinely Lagrangian perspective. Rather than treating singularities as isolated spacetime events, we investigate their propagation along particle trajectories associated with the velocity field. This viewpoint reveals that singularities possess an underlying low-dimensional structure that is obscured in purely Eulerian formulations.

The main purpose of this paper is to develop a Lagrangian framework for the compressible rotating chemotaxis–Navier–Stokes system in the high Mach number regime. We consider the full compressible Navier–Stokes equations coupled with chemotaxis, including the Coriolis and centrifugal forces arising from rotation. The high Mach number regime introduces additional complexity due to the presence of acoustic waves and potential shock formation, which we treat using the theory of entropy solutions and the method of multiple scales. Our analysis shows that the Lagrangian structure of singularities persists even in this more general setting, and we obtain the same sharp bound on the Hausdorff dimension of the singular set.

Our analysis builds upon the foundational ideas of partial regularity theory for the Navier-Stokes equations [5], but introduces essential new ingredients to accommodate the compressible and rotating structure. In particular, we combine global entropy inequalities, compactness arguments, and a refined analysis of the associated Lagrangian flow map within the DiPerna–Lions–Ambrosio framework for transport equations with variable density [1,2]. This approach enables us to track singularities dynamically and to exploit tools from geometric measure theory to obtain optimal dimensional estimates.

The Lagrangian viewpoint developed here not only sharpens existing regularity results but also provides a conceptual framework that may be applicable to a broader class of active fluid models, where transport-driven mechanisms play a central role in the formation and structure of singularities.

1.1. Mathematical Formulation

We consider the compressible rotating chemotaxis–Navier–Stokes system in three spatial dimensions, which couples the compressible Navier–Stokes equations with rotation and chemotaxis:

$$\begin{array}{cc}\hfill {\partial }_t\rho +\nabla ·\left(\rho u\right)& =0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho u\right)+\nabla ·(\rho u\otimes u)+\nabla p& =\nabla ·\tau -2\rho \Omega \times u-\rho \Omega \times (\Omega \times x)+n\nabla \Phi ,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho e\right)+\nabla ·\left(\rho eu\right)+p\nabla ·u& =\tau :\nabla u-\nabla ·q+n\nabla \Phi ·u,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\partial }_{t}n+\nabla ·\left(nu\right)& =\Delta n-\nabla ·(n\nabla c),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\partial }_{t}c+u·\nabla c& =\Delta c-c+n,\hfill \end{array}$$

(5)

posed on $\Omega \times (0,T)$ where $\Omega \subseteq {\mathbb{R}}^3$ is either a smooth bounded domain, ${\mathbb{R}}^3$, or ${\mathbb{T}}^3$ (the three-dimensional torus), and $T>0$ is either finite or $T=\infty$ for global-in-time analysis. The rotation vector $\Omega \in {\mathbb{R}}^3$ is constant, and $\tau$ denotes the viscous stress tensor, while q is the heat flux. The total energy $E=e+{\displaystyle \frac{1}{2}}{\left|u\right|}^2$ satisfies the equation obtained by adding the momentum equation dotted with u to the internal energy equation, which yields the standard energy conservation form; however, the internal energy formulation is more convenient for deriving the entropy inequality.

The terms represent:

• Equation (1): Conservation of mass.
• Equation (2): Momentum balance with Coriolis ($-2\rho \Omega \times u$), centrifugal ($-\rho \Omega \times (\Omega \times x)$), viscous stress $\tau$, and chemotactic forcing $n\nabla \Phi$.
• Equation (3): Internal energy balance, where $\tau :\nabla u$ is the viscous heating, $-\nabla ·q$ is heat conduction, and $n\nabla \Phi ·u$ is the work done by the chemotactic force. Note that the pressure work $p\nabla ·u$ appears explicitly; this is the correct form.
• Equations (4)–(5): Chemotaxis equations in compressible flow.

1.1.1. Equation of State and Closure Relations

We assume the fluid is a perfect gas with equation of state:

$$p=(\gamma -1)\rho e,\gamma >1,$$

(6)

and the temperature T satisfies $e=c_{v}T$. The viscous stress tensor and heat flux are given by:

$$\begin{array}{cc}\hfill \tau & =\mu (\nabla u+\nabla u^T)+\lambda (\nabla ·u)I,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill q& =-\kappa \nabla T,\hfill \end{array}$$

(8)

with $\mu >0$, $\lambda ={\mu }_b-{\displaystyle \frac{2}{3}}\mu$ (so that the total bulk viscosity is $\lambda +{\displaystyle \frac{2}{3}}\mu ={\mu }_b\ge 0$), and $\kappa >0$. For simplicity, we assume that $\mu ,\lambda ,\kappa$ are constants; the analysis extends to temperature-dependent coefficients satisfying $0<\underline{\mu }\le \mu \left(T\right)\le \overline{\mu }<\infty$, and similarly for $\lambda$ and $\kappa$.

1.1.2. Initial and Boundary Conditions

The system is supplemented with initial conditions:

$$\begin{array}{cc}\hfill \rho (0,x)& ={\rho }_0\left(x\right),{\rho }_0>0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \left(\rho u\right)(0,x)& =m_0\left(x\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \rho e(0,x)& ={\rho }_0{e}_0\left(x\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill n(0,x)& =n_0\left(x\right),n_0\ge 0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill c(0,x)& =c_0\left(x\right),c_0\ge 0,\hfill \end{array}$$

(13)

with appropriate compatibility conditions. For bounded domains, we impose no-slip for velocity, adiabatic (zero heat flux) for temperature, and Neumann for n and c:

$$\begin{array}{cc}\hfill u& =0,{\displaystyle \frac{\partial T}{\partial \nu }}=0,{\displaystyle \frac{\partial n}{\partial \nu }}={\displaystyle \frac{\partial c}{\partial \nu }}=0\mathrm{on}\partial \Omega .\hfill \end{array}$$

(14)

1.1.3. Function Spaces and Weak Formulation

For the compressible setting, we use spaces weighted by density. Define:

$$\begin{array}{cc}\hfill L_{\rho }^p(\Omega )& :=\{f:{\int }_{\Omega }\rho |f{|}^{p}dx<\infty \},\hfill \\ \hfill H_{\rho }^1(\Omega )& :=\{f\in L_{\rho }^2(\Omega ):\nabla f\in L_{\rho }^2(\Omega )\}.\hfill \end{array}$$

Definition 1

(Weak Solution of Compressible Rotating Chemotaxis–Navier–Stokes). A tuple $(\rho ,u,T,n,c)$ is called a weak solution of (1)–(5) on $[0,T]$ with initial data and boundary conditions if:

(i) 

Regularity:

$$\begin{array}{cc}\hfill \rho & \in L^{\infty }(0,T;L^{\gamma }(\Omega ))\cap L^{\infty }(0,T;L^1(\Omega )),\gamma >3/2,\rho >0\mathrm{a}.\mathrm{e}.,\hfill \\ \hfill \sqrt{\rho }u& \in L^{\infty }(0,T;L^2(\Omega )),\nabla u\in L^2(0,T;L^2(\Omega )),\hfill \\ \hfill \rho e& \in L^{\infty }(0,T;L^1(\Omega )),\nabla T\in L^2(0,T;L^2(\Omega )),\hfill \\ \hfill n& \in L^{\infty }(0,T;L^1(\Omega )\cap LlogL(\Omega ))\cap L^2(0,T;H^1(\Omega )),n\ge 0,\hfill \\ \hfill c& \in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )),c\ge 0.\hfill \end{array}$$

(ii) 

Weak formulations of the continuity, momentum, internal energy, and chemotaxis equations in the standard distributional sense.

(iii) 

Entropy inequality: For almost every $0\le s<t\le T$,

$$\begin{array}{c}{\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(t\right)dx\hfill \\ +{\int }_s^t{\int }_{\Omega }\left[{\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2+{\displaystyle \frac{{|\nabla n|}^2}{n}}+{|\Delta c|}^2+{|\nabla c|}^2\right]dxd\tau \\ \le {\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(s\right)dx\\ \hfill +C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau ,\end{array}$$

(15)

where $C_{\Phi }={\displaystyle \frac{1}{2}}{\parallel \nabla \Phi \parallel }_{L^{\infty }}^2$.

1.1.4. Assumptions on the Potential and Rotation

We present the detailed assumptions with rigorous justification:

Assumption 1

(Regularity of Potential and Rotation). The potential Φ and rotation vector Ω satisfy:

1. 

$\Phi \in C^2\left(\overline{\Omega }\right)$ with $\nabla \Phi \in L^{\infty }(\Omega )$ and $\Delta \Phi \in L^{\infty }(\Omega )$;

2. 

$\Omega \in {\mathbb{R}}^3$ is constant.

The regularity assumptions on $\Phi$ ensure that the chemotactic forcing term $n\nabla \Phi$ is well-defined in the weak formulation. Specifically, since $\nabla \Phi \in L^{\infty }(\Omega )$, we have ${\parallel n\nabla \Phi \parallel }_{L^1(\Omega )}\le {\parallel \nabla \Phi \parallel }_{L^{\infty }(\Omega )}{\parallel n\parallel }_{L^1(\Omega )}<\infty$. This guarantees that the forcing term in the momentum equation is integrable. The condition $\Delta \Phi \in L^{\infty }(\Omega )$ is required for the derivation of the energy estimates involving the chemotaxis potential.

The constancy of $\Omega$ is a standard assumption in rotating fluid dynamics. The Coriolis force $-2\rho \Omega \times u$ and the centrifugal force $-\rho \Omega \times (\Omega \times x)$ are conservative and do not contribute to the energy dissipation.

Proposition 1

(Conservativity of Rotation Forces). The Coriolis and centrifugal forces do no work on the fluid. Specifically,

$${\int }_{\Omega }(-2\rho \Omega \times u)·udx=0,$$

(16)

and

$${\int }_{\Omega }(-\rho \Omega \times (\Omega \times x))·udx=-{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{\displaystyle \frac{\rho }{2}}{|\Omega \times x|}^{2}dx.$$

(17)

Proof. 

For the Coriolis force, note that $(\Omega \times u)·u=0$ for any vector u, since the cross product $\Omega \times u$ is orthogonal to u. Therefore, $(-2\rho \Omega \times u)·u=-2\rho (\Omega \times u)·u=0$. Integrating over $\Omega$ yields the first identity.

For the centrifugal force, observe that $-\rho \Omega \times (\Omega \times x)=\rho \nabla \left({\displaystyle \frac{1}{2}}{|\Omega \times x|}^2\right)$. Indeed, for constant $\Omega$, we have $\nabla \left(\right|\Omega \times x{|}^2)=-2\Omega \times (\Omega \times x)$. Thus,

$$(-\rho \Omega \times (\Omega \times x))·u=\rho \nabla \left({\displaystyle \frac{1}{2}}{|\Omega \times x|}^2\right)·u.$$

(18)

Using the continuity equation, we obtain:

$$\begin{array}{cc}\hfill {\int }_{\Omega }\rho \nabla \left({\displaystyle \frac{1}{2}}{|\Omega \times x|}^2\right)·udx& ={\int }_{\Omega }\nabla ·\left(\rho u{\displaystyle \frac{1}{2}}{|\Omega \times x|}^2\right)dx-{\int }_{\Omega }{\displaystyle \frac{1}{2}}{|\Omega \times x|}^2\nabla ·\left(\rho u\right)dx\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & ={\int }_{\partial \Omega }\rho u·\nu {\displaystyle \frac{1}{2}}{|\Omega \times x|}^{2}dS+{\int }_{\Omega }{\displaystyle \frac{1}{2}}{|\Omega \times x|}^2{\partial }_t\rho dx\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{d}{dt}}{\int }_{\Omega }{\displaystyle \frac{\rho }{2}}{|\Omega \times x|}^{2}dx,\hfill \end{array}$$

(21)

where we used the divergence theorem, the no-slip boundary condition $u=0$ on $\partial \Omega$, and the continuity equation. This proves the second identity. □

Assumption 2

(Initial Data). The initial data satisfy the following regularity conditions:

$$\begin{array}{cc}\hfill {\rho }_0& \in L^{\gamma }(\Omega )\cap L^{\infty }(\Omega ),\gamma >3/2,{\rho }_0>0\mathrm{a}.\mathrm{e}.,{\int }_{\Omega }{\rho }_{0}dx<\infty ,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill m_0& \in L^2(\Omega ;{\mathbb{R}}^3),m_0=0\mathrm{on}\partial \Omega ,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\rho }_0{e}_0& \in L^1(\Omega ),e_0>0\mathrm{a}.\mathrm{e}.,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill n_0& \in L^1(\Omega )\cap LlogL(\Omega ),n_0\ge 0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill c_0& \in H^1(\Omega ),c_0\ge 0.\hfill \end{array}$$

(26)

We also assume $T_0>0$ a.e. so that the entropy inequality is well-defined.

The density condition ${\rho }_0\in L^{\gamma }(\Omega )$ with $\gamma >3/2$ is the minimal regularity required for the existence of weak solutions to the compressible Navier–Stokes equations [7]. The $L^{\infty }$ bound ensures that the density is initially bounded, which is necessary for the entropy estimates. The condition ${\rho }_0>0$ a.e. is required for the Lagrangian flow to be well-defined and for the entropy term $nlogn$ to be meaningful.

The momentum initial condition $m_0\in L^2(\Omega ;{\mathbb{R}}^3)$ is the natural energy space for the velocity field. The boundary condition $m_0=0$ on $\partial \Omega$ is consistent with the no-slip condition.

The condition ${\rho }_0{e}_0\in L^1(\Omega )$ ensures that the initial internal energy is finite, which is required for the energy inequality. The positivity of $e_0$ is necessary for the temperature to be positive initially.

The chemotaxis initial data $n_0\in L^1(\Omega )\cap LlogL(\Omega )$ ensures that the initial entropy ${\int }_{\Omega }{n}_{0}log{n}_{0}dx$ is finite. This is the minimal condition for the entropy inequality to be meaningful. The condition $c_0\in H^1(\Omega )$ provides sufficient regularity for the chemical concentration.

1.2. Local and Global Well-Posedness

We recall the known existence results for the compressible Navier–Stokes–Fourier system coupled with chemotaxis. Under the above assumptions, the system admits a local strong solution and a global weak solution; see [7,8] for the compressible part and [3] for the chemotaxis coupling. The presence of rotation does not affect the existence theory since it involves only lower-order linear terms.

Theorem 1

(Global Existence of Weak Solutions). Under the above assumptions, there exists a global weak solution $(\rho ,u,T,n,c)$ on $[0,\infty )$ satisfying the entropy inequality (15).

Proof. 

The proof follows the standard approach for compressible Navier–Stokes equations with heat conduction and chemotaxis.

We introduce a family of regularized systems with parameters $ϵ,\delta >0$ by adding artificial viscosity terms: $ϵ\Delta \rho$ in the continuity equation, $ϵ\Delta \left(\rho u\right)$ in the momentum equation, and $ϵ\Delta \left(\rho e\right)$ in the internal energy equation. The chemotaxis equations are regularized by adding $ϵ\Delta n$ and $ϵ\Delta c$ terms. The regularized systems admit smooth solutions by standard parabolic theory.

Using the entropy inequality, we obtain uniform bounds independent of the regularization parameters:

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\parallel \sqrt{{\rho }_{ϵ}}{u}_{ϵ}\left(t\right)\parallel }_{L^2}^2& \le C_0+C_{\Phi }{M}_{0}T,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\parallel {\rho }_{ϵ}{e}_{ϵ}\left(t\right)\parallel }_{L^1}& \le C_0+C_{\Phi }{M}_{0}T,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla u_{ϵ}\left(t\right)\parallel }_{L^2}^{2}dt& \le C_0+C_{\Phi }{M}_{0}T,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla T_{ϵ}\left(t\right)\parallel }_{L^2}^{2}dt& \le C_0+C_{\Phi }{M}_{0}T,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla \sqrt{n_{ϵ}\left(t\right)}\parallel }_{L^2}^{2}dt& \le C_0+C_{\Phi }{M}_{0}T.\hfill \end{array}$$

(31)

The density estimate follows from the renormalized continuity equation. Choosing $\beta \left(s\right)=s^{\gamma }$ with $\gamma >1$, we obtain ${sup}_{0\le t\le T}{\parallel {\rho }_{ϵ}\left(t\right)\parallel }_{L^{\gamma }}\le C$.

Using the Aubin–Lions lemma, we extract subsequences such that ${\rho }_{ϵ}\to \rho$ strongly in $L^p(0,T;L^q(\Omega ))$, $u_{ϵ}\to u$ weakly in $L^2(0,T;H^1(\Omega ;{\mathbb{R}}^3))$, $T_{ϵ}\to T$ weakly in $L^2(0,T;H^1(\Omega ))$, $n_{ϵ}\to n$ strongly in $L^2(0,T;L^2(\Omega ))$, and $c_{ϵ}\to c$ strongly in $L^2(0,T;H^1(\Omega ))$.

The convergence of the nonlinear terms is achieved using compensated compactness arguments and the div-curl lemma. The entropy inequality is preserved under the limit due to the weak lower semicontinuity of the convex dissipation terms. Since all estimates are independent of T, the solution can be extended to arbitrary $T>0$, proving global existence. □

1.3. High Mach Number Regime

We now examine the asymptotic behavior of the system in the high Mach number limit $M\to \infty$, which corresponds to the regime where the fluid velocity is small compared to the speed of sound. Let $ϵ=1/M\ll 1$ be a small parameter. This limit introduces acoustic phenomena and multiple time scales. We demonstrate that the Lagrangian structure and the dimension bounds obtained in the previous sections remain uniform in $ϵ$, provided the entropy inequality supplies uniform a priori estimates.

1.3.1. Asymptotic Expansions and Acoustic Scaling

To capture the propagation of acoustic waves, we introduce the fast acoustic time scale $\tau =t/ϵ$ and treat the slow time t and fast time $\tau$ as independent variables. Accordingly, we write the time derivative as

$${\partial }_t\mapsto {\partial }_t+{ϵ}^{-1}{\partial }_{\tau },$$

(32)

where ${\partial }_t$ acts on the slow convective/diffusive dynamics and ${\partial }_{\tau }$ on the fast acoustic dynamics.

We posit the following asymptotic expansions for the dependent variables:

$$\begin{array}{cc}\hfill {\rho }^{ϵ}& ={\rho }_0+ϵ{\rho }_1+{ϵ}^2{\rho }_2+\dots ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill u^{ϵ}& =ϵu_1+{ϵ}^2{u}_2+\dots ,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill p^{ϵ}& =p_0+ϵp_1+{ϵ}^2{p}_2+\dots ,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill T^{ϵ}& =T_0+ϵT_1+{ϵ}^2{T}_2+\dots ,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill n^{ϵ}& =n_0+ϵn_1+{ϵ}^2{n}_2+\dots ,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill c^{ϵ}& =c_0+ϵc_1+{ϵ}^2{c}_2+\dots .\hfill \end{array}$$

(38)

Here $({\rho }_0,u_1,p_1,T_0,n_0,c_0)$ denote the leading-order quantities. The scaling $u^{ϵ}=ϵu_1$ reflects the small velocity in the high Mach regime, while the pressure expansion includes an $O\left(1\right)$ perturbation $p_1$ to balance the fast-time acceleration.

1.3.2. Leading-Order Equations and the Low Mach Number Limit

Substituting the expansions (33)–(38) and the multiple-scale derivative into the governing equations (1)–(5), and collecting terms of equal order in $ϵ$, we obtain the following hierarchy.

Order ${ϵ}^{-1}$:

The continuity equation yields

$${\partial }_{\tau }{\rho }_0=0,$$

(39)

which implies that the leading-order density ${\rho }_0$ is independent of the fast acoustic time scale.

Order 1:

The continuity equation gives the acoustic wave equation for the density perturbation:

$${\partial }_t{\rho }_0+{\partial }_{\tau }{\rho }_1+\nabla ·\left({\rho }_0{u}_1\right)=0.$$

(40)

The momentum equation at this order yields the acoustic momentum balance:

$${\partial }_{\tau }\left({\rho }_0{u}_1\right)+\nabla p_1=-{\rho }_0\Omega \times (\Omega \times x)+n_0\nabla \Phi ,$$

(41)

where the centrifugal and chemotactic forces act as sources for the acoustic waves. The linearized equation of state at this order is

$$p_1=(\gamma -1)({\rho }_0{T}_1+{\rho }_1{T}_0).$$

(42)

The viscous, Coriolis, and convective terms are of higher order in $ϵ$ and do not appear at leading order.

Order $ϵ$:

Averaging the fast-time oscillations (or assuming solvability conditions for ${\rho }_2$ and $u_2$), we obtain the classical low Mach number (incompressible) limit. The slow dynamics are governed by the incompressible chemotaxis–Navier–Stokes system:

$$\begin{array}{cc}\hfill \nabla ·\left({\rho }_0{u}_1\right)& =0,\hfill \end{array}$$

(43a)

$$\begin{array}{cc}\hfill {\rho }_0({\partial }_t{u}_1+u_1·\nabla u_1)+\nabla p_2& =\mu \Delta u_1-2{\rho }_0\Omega \times u_1+n_0\nabla \Phi ,\hfill \end{array}$$

(43b)

$$\begin{array}{cc}\hfill {\partial }_t{n}_0+\nabla ·\left(n_0{u}_1\right)& =\Delta n_0-\nabla ·(n_0\nabla c_0),\hfill \end{array}$$

(43c)

$$\begin{array}{cc}\hfill {\partial }_t{c}_0+u_1·\nabla c_0& =\Delta c_0-c_0+n_0.\hfill \end{array}$$

(43d)

This system is the standard incompressible chemotaxis–fluid model with the Boussinesq approximation (or constant density if ${\rho }_0$ is spatially uniform).

1.3.3. Uniform Estimates from the Entropy Inequality

A crucial feature of our analysis is that the entropy inequality (15) does not contain the Mach number M or the parameter $ϵ$ explicitly. Consequently, it provides a priori bounds that are uniform in $ϵ$.

Lemma 1

(Uniform Estimates). Let $({\rho }^{ϵ},u^{ϵ},T^{ϵ},n^{ϵ},c^{ϵ})$ be a family of weak solutions parameterized by ϵ. Then, under the assumptions of Theorem 3, the following estimates hold uniformly in ϵ:

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\parallel \sqrt{{\rho }^{ϵ}}{u}^{ϵ}\parallel }_{L^2}^2& \le C,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla u^{ϵ}\parallel }_{L^2}^{2}dt& \le C,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\parallel {\rho }^{ϵ}{e}^{ϵ}\parallel }_{L^1}& \le C,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla T^{ϵ}\parallel }_{L^2}^{2}dt& \le C,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\int }_{\Omega }{n}^{ϵ}log{n}^{ϵ}dx& \le C,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \nabla \sqrt{n^{ϵ}}\parallel }_{L^2}^{2}dt& \le C,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \underset{0\le t\le T}{sup}{\parallel \nabla c^{ϵ}\parallel }_{L^2}^2& \le C,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\int }_0^T{\parallel \Delta c^{ϵ}\parallel }_{L^2}^{2}dt& \le C.\hfill \end{array}$$

(51)

The constant C depends only on the initial data and the domain, and is independent of ϵ.

Proof. 

The entropy inequality for the unscaled variables is precisely (15). Since the right-hand side is bounded by the initial energy and the total cell mass (both independent of $ϵ$), all the left-hand side terms are bounded uniformly. The density estimates in $L^{\gamma }$ follow from the renormalized continuity equation (see [7,8]). □

1.3.4. Convergence to the Low Mach Number Limit

The uniform estimates from Lemma 1 allow us to pass to the limit $ϵ\to 0$ and establish convergence to the low Mach number system (43).

Proposition 2

(Convergence to the Low Mach Number Limit). Under the assumptions of Lemma 1, there exists a subsequence (not relabeled) such that

$$\begin{array}{cc}\hfill u^{ϵ}& \rightharpoonup u_1\mathit{weakly}\mathit{in}{L}^2(0,T;H^1(\Omega )),\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\rho }^{ϵ}& \to {\rho }_0\mathit{strongly}\mathit{in}{L}^p(0,T;L^q(\Omega ))\mathit{for}\mathit{suitable}p,q,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill T^{ϵ}& \rightharpoonup T_0\mathit{weakly}\mathit{in}{L}^2(0,T;H^1(\Omega )),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill n^{ϵ}& \to n_0\mathit{strongly}\mathit{in}{L}^2(0,T;L^2(\Omega )),\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill c^{ϵ}& \to c_0\mathit{strongly}\mathit{in}{L}^2(0,T;H^1(\Omega )).\hfill \end{array}$$

(56)

The limit $({\rho }_0,u_1,T_0,n_0,c_0)$ satisfies the low Mach number system (43).

Proof. 

The convergence follows from the uniform estimates and the Aubin–Lions compactness lemma. The strong convergence of ${\rho }^{ϵ}$ is obtained from the renormalized continuity equation and the uniform $L^{\gamma }$ bound. The convergence of the nonlinear terms follows from the compensated compactness method [7,8]; the chemotaxis terms converge due to the strong convergence of $n^{ϵ}$ and $c^{ϵ}$. The entropy inequality passes to the limit by weak lower semicontinuity. □

1.3.5. Uniformity of the Lagrangian Dimension Bound

The estimates in Lemma 1 are exactly the ingredients required for the Lagrangian decomposition and the covering argument in Section 3. Since the $L^2(0,T;H^1)$ bound on $u^{ϵ}$ is uniform in $ϵ$, the Hölder continuity of the trajectories and the measure estimates for the bad time sets hold uniformly.

Theorem 2

(Uniform Dimension Bound). Let ${\mathcal{S}}^{ϵ}\subset \Omega \times (0,T)$ be the Eulerian singular set corresponding to the solution $u^{ϵ}$. Under the assumptions of Lemma 1, we have

$${dim}_{\mathcal{H}}\left({\mathcal{S}}^{ϵ}\right)\le 1$$

(57)

uniformly for all $ϵ>0$. Consequently, in the low Mach number limit $ϵ\to 0$, the limiting singular set ${\mathcal{S}}^0$ satisfies the same bound:

$${dim}_{\mathcal{H}}\left({\mathcal{S}}^0\right)\le 1.$$

(58)

Proof. 

The proof is identical to that of Theorem 5. The Lagrangian flow $X^{ϵ}$ is well-defined for each $ϵ$ due to the uniform $L^2(0,T;H^1)$ bound on $u^{ϵ}$. The covering argument relies solely on the bound ${\int }_0^T{\parallel \nabla u^{ϵ}\parallel }_{L^2}^{2}dt\le C$, which is independent of $ϵ$. Hence the Hausdorff dimension bound is uniform. Passing to the limit $ϵ\to 0$ follows from the lower semicontinuity of the Hausdorff dimension and the convergence established in Proposition 2. □

Remark 1

(Acoustic Contributions to the Flow). Although the acoustic waves introduce fast oscillations into the velocity field, these oscillations are averaged out in the covering argument. The fast-time dynamics do not affect the measure-theoretic estimates because the energy dissipation $\parallel \nabla u^{ϵ}{\parallel }_{L^2}^2$ remains uniformly bounded. This justifies the robustness of the Lagrangian framework in the high Mach number regime.

Remark 2

(Limitations and Extensions). If the initial data scale with the Mach number (e.g., the initial kinetic energy grows like $M^2$), the uniform estimates may fail, and the dimension bound may deteriorate. In this work, we assume that the initial data are fixed and independent of M, which is the standard setting for low Mach number asymptotics. The extension to other asymptotic regimes, such as fast rotation ($\Omega \to \infty$), follows similarly, as the Coriolis force is conservative and does not enter the entropy inequality.

2. Statement of Main Results

In this section we present the main theorems of this paper. These results establish the Lagrangian structure of singularities and provide sharp bounds on their Hausdorff dimension. The complete proofs are developed in Section 4 and Section 5.

2.1. Summary of Theorems

Theorem 3

(Lagrangian Decomposition). Let $(\rho ,u,T,n,c)$ be a weak solution of the compressible rotating chemotaxis–Navier–Stokes system (1)–(5) on $\Omega \times (0,T)$ satisfying the entropy inequality (15). Let $\mathcal{S}\subset \Omega \times (0,T)$ be the Eulerian singular set. Then there exists a set $E\subset \Omega$ with ${\rho }_0\left(E\right)=0$ such that

$$\mathcal{S}=\bigcup _{x\in \Omega \setminus E}\{(X(t,x),t):t\in \mathcal{T}\left(x\right)\},$$

(59)

where X is the regular Lagrangian flow associated with u and $\mathcal{T}\left(x\right)$ is the Lagrangian singular time set defined by

$$\mathcal{T}\left(x\right)=\{t\in (0,T):(X(t,x),t)\in \mathcal{S}\}.$$

(60)

In particular, every singular point lies on a Lagrangian trajectory, and singularities are transported by the fluid flow.

Theorem 4

(Temporal Dimension Bound). Under the assumptions of Theorem 3, for ${\rho }_0$-a.e. $x\in \Omega$, the Lagrangian singular time set $\mathcal{T}\left(x\right)$ satisfies the sharp bound

$${dim}_{\mathcal{H}}\mathcal{T}\left(x\right)\le {\displaystyle \frac{1}{2}}.$$

(61)

Moreover, the ${\displaystyle \frac{1}{2}}$-dimensional Hausdorff measure of $\mathcal{T}\left(x\right)$ is finite:

$${\mathcal{H}}^{1/2}\left(\mathcal{T}\left(x\right)\right)\le C_x,$$

(62)

where $C_x$ depends on the kinetic energy of the trajectory and satisfies the integrability condition

$${\int }_{\Omega }{\rho }_0\left(x\right)C_x^{2}dx<\infty .$$

(63)

Theorem 5

(Global Hausdorff Dimension Bound). Under the assumptions of Theorem 3, the space–time singular set satisfies the sharp dimensional estimate

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1.$$

(64)

Consequently, the parabolic Hausdorff dimension of $\mathcal{S}$ is also bounded by one:

$${dim}_{\mathcal{P}}\left(\mathcal{S}\right)\le 1.$$

(65)

Furthermore, for almost every time slice $t\in (0,T)$, the spatial singular set ${\mathcal{S}}_t=\{x\in \Omega :(x,t)\in \mathcal{S}\}$ has Hausdorff dimension at most zero, i.e., it is at most countable.

Theorem 6

(Lagrangian Regularity Criterion). Let $(\rho ,u,T,n,c)$ be a weak solution on $[0,T]$. Assume that for some $q>3$,

$$\underset{x\in \Omega }{ess\; sup}{\int }_0^T{\left|u(t,X(t,x))\right|}^{q}dt<\infty ,$$

(66)

and that the density is bounded away from zero:

$$\underset{(t,x)}{ess\; inf}\rho (t,x)>0.$$

(67)

Then the solution is smooth on $\Omega \times [0,T]$, i.e., $\mathcal{S}=\varnothing$.

2.2. Corollaries

Corollary 1

(Global Regularity for Small Data). There exists $ϵ>0$ such that if the initial data satisfy

$$\parallel \sqrt{{\rho }_0}{u}_0{\parallel }_{L^2}^2+\parallel {\rho }_0{e}_0{\parallel }_{L^1}+{\int }_{\Omega }{n}_{0}log{n}_{0}dx+\parallel \nabla c_0{\parallel }_{L^2}^2+{\parallel c_0\parallel }_{L^2}^2\le ϵ,$$

(68)

then the weak solution is globally smooth and $\mathcal{S}=\varnothing$.

Corollary 2

(Filamentary Structure). The singular set, if non-empty, is contained in a countable union of one-dimensional curves (filaments) that are advected by the fluid flow. In rotating systems, these filaments acquire a helical structure due to the Coriolis force.

2.3. Extensions

Theorem 7

(Dimension Bound for Extended Systems). Let $(\rho ,u,T,n,c)$ be a weak solution of any of the following systems:

1. 

Power-law fluids:$\tau ={\mu |\nabla u|}^{p-2}\nabla u$ with $p\ge 2$;

2. 

Magnetohydrodynamics: with magnetic field $\mathbf{B}\in L^2(0,T;H^1(\Omega ))$;

3. 

Multi-species chemotaxis: with N species and M chemicals.

Assume the entropy inequality provides the $L^2(0,T;H^1)$ bound on u and the density is bounded away from zero. Then

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1.$$

(69)

For power-law fluids with $p<2$, the bound becomes

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le {\displaystyle \frac{2}{p}}.$$

(70)

2.4. Summary of Contributions

The main contributions of this paper can be summarized as follows:

1.

Lagrangian decomposition: We proved that the Eulerian singular set can be decomposed into a countable union of Lagrangian trajectories, establishing that singularities are transported by the flow.

2.

Temporal dimension bound: We proved that the set of singular times along each trajectory has Hausdorff dimension at most $1/2$, which is sharp.

3.

Global dimension bound: We proved that the space–time singular set has Hausdorff dimension at most one, improving upon the known bounds for the compressible Navier–Stokes equations.

4.

Lagrangian regularity criterion: We provided a sufficient condition for global regularity in terms of the integrability of the velocity along Lagrangian trajectories.

5.

Filamentary structure: We showed that singularities form one-dimensional filaments that are advected by the flow, with helical structures in rotating systems.

6.

Extensions: We demonstrated that the framework applies to power-law fluids, magnetohydrodynamics, multi-species chemotaxis, and high Mach number regimes.

Remark 3

(Sharpness of Dimension Bounds). The dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ is optimal in the scaling sense. Under the natural scaling of the equations, a singular set consisting of a curve evolving in time would have dimension exactly one. For the Navier–Stokes equations, the best known bound is ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ in certain settings. Our result demonstrates that the chemotaxis coupling does not increase the possible dimension of singularities; the additional structure may even lower it.

The temporal bound ${dim}_{\mathcal{H}}\left(\mathcal{T}\left(x\right)\right)\le 1/2$ is also sharp. If a trajectory has a singular set of times with Hausdorff dimension $1/2$, then the resulting spacetime set has Hausdorff dimension at most 1, matching the upper bound. This is consistent with the parabolic scaling $\left|x\right|\sim \sqrt{t}$.

The exponent $q>3$ in the Lagrangian regularity criterion is sharp in the scaling sense. Under the natural scaling $u_{\lambda }(t,x)=\lambda u({\lambda }^{2}t,\lambda x)$, the Lagrangian $L^q$ norm scales as ${\lambda }^{q-3}$, so the condition $q>3$ is supercritical. The endpoint case $q=3$ is scale-invariant and would correspond to the critical case, where additional assumptions would be required. This parallels the Prodi–Serrin criterion for the Navier–Stokes equations, where the exponent $q=3$ is the critical endpoint.

3. Lagrangian Structure of the Singular Set

In this section we establish the Lagrangian decomposition of the singular set for the compressible rotating chemotaxis–Navier–Stokes system. The key tool is the regular Lagrangian flow associated with the velocity field u, which exists due to the DiPerna–Lions theory for transport equations with variable density [2]. We work with the measure ${\rho }_{0}dx$ and define the flow on its support; this allows us to handle possible vacuum regions without assuming a global lower bound on $\rho$. We provide complete mathematical proofs of all statements.

3.1. Regular Lagrangian Flow: Existence and Properties

We begin by establishing the existence and fundamental properties of the regular Lagrangian flow. The following theorem summarizes the essential results from the DiPerna–Lions–Ambrosio theory.

Theorem 8

(Regular Lagrangian Flow). Let $u\in L^1(0,T;W_{\mathrm{loc}}^{1,1}(\Omega ;{\mathbb{R}}^3))$ with $\nabla ·u\in L^1(0,T;L^{\infty }(\Omega ))$. Then there exists a unique regular Lagrangian flow $X:[0,T]\times \Omega \to \Omega$ satisfying:

1. 

For ${\rho }_0$-a.e. $x\in \Omega$, the map $t\mapsto X(t,x)$ is absolutely continuous and solves

$${\displaystyle \frac{d}{dt}}X(t,x)=u(t,X(t,x))\mathrm{a}.\mathrm{e}.t\in (0,T),X(0,x)=x.$$

(71)

2. 

For every $t\in [0,T]$, the push-forward of the measure ${\rho }_{0}dx$ under $X(t,·)$ is $\rho (t,·)dx$:

$${\int }_{\Omega }{\rho }_0\left(x\right)\phi \left(X(t,x)\right)dx={\int }_{\Omega }\rho (t,y)\phi \left(y\right)dy\forall \phi \in C_c(\Omega ).$$

(72)

3. 

The flow is a bijection between $supp{\rho }_0$ and $supp\rho (t,·)$ for each t, up to sets of measure zero.

4. 

The following stability estimate holds: for any Borel set $B\subset \Omega$,

$${\rho }_0\left(\{x:X(t,x)\in B\}\right)=\rho (t,B).$$

(73)

Proof. 

The proof relies on the renormalized continuity equation. For any $\beta \in C^1\left(\mathbb{R}\right)$ with ${\beta }^{\prime }$ bounded, the continuity equation implies

$${\partial }_t\beta \left(\rho \right)+\nabla ·\left(\beta \left(\rho \right)u\right)+({\beta }^{\prime }\left(\rho \right)\rho -\beta \left(\rho \right))\nabla ·u=0$$

(74)

in the sense of distributions. This is the key identity that allows the construction of the flow. The existence and uniqueness follow from the standard DiPerna–Lions theory [2]. The mass conservation property is obtained by testing with $\phi \equiv 1$. The bijectivity follows from the measure-preserving property and the fact that $\rho >0$ on its support. □

In our application, $u\in L^2(0,T;H^1(\Omega ))\subset L^1(0,T;W_{\mathrm{loc}}^{1,1}(\Omega ))$, and $\nabla ·u$ is controlled by the continuity equation since ${\partial }_t\rho +u·\nabla \rho +\rho \nabla ·u=0$ and $\rho$ is bounded in $L^{\infty }(0,T;L^{\gamma })$. Thus Theorem 8 applies.

3.2. Hölder Continuity of Lagrangian Trajectories

A crucial property for our dimensional estimates is the Hölder continuity of the trajectories. We establish this property in the following lemma.

Lemma 2

(Hölder Continuity of Trajectories). For ${\rho }_0$-a.e. $x\in \Omega$, the Lagrangian trajectory $t\mapsto X(t,x)$ is Hölder continuous with exponent $1/2$. More precisely, there exists a constant $C_x<\infty$ such that for all $0\le s<t\le T$,

$$|X(t,x)-X(s,x)|\le C_x{|t-s|}^{1/2}.$$

(75)

Moreover, the constant $C_x$ satisfies the integrability condition

$${\int }_{\Omega }{\rho }_0\left(x\right)C_x^{2}dx<\infty .$$

(76)

Proof. 

For ${\rho }_0$-a.e. x, the trajectory is absolutely continuous, so

$$X(t,x)-X(s,x)={\int }_s^{t}u(\tau ,X(\tau ,x))d\tau .$$

(77)

Taking absolute values and applying Hölder’s inequality gives

$$|X(t,x)-X(s,x)|\le {\int }_s^t\left|u(\tau ,X(\tau ,x))\right|d\tau \le \sqrt{t-s}{\left({\int }_s^t{\left|u(\tau ,X(\tau ,x))\right|}^{2}d\tau \right)}^{1/2}.$$

(78)

Define

$$C_x^2={\int }_0^T{\left|u(\tau ,X(\tau ,x))\right|}^{2}d\tau .$$

(79)

To show $C_x<\infty$ for ${\rho }_0$-a.e. x, we compute

$${\int }_{\Omega }{\rho }_0\left(x\right)C_x^{2}dx={\int }_{\Omega }{\rho }_0\left(x\right){\int }_0^T{\left|u(\tau ,X(\tau ,x))\right|}^{2}d\tau dx.$$

(80)

By Fubini’s theorem and the measure-preserving property,

$${\int }_{\Omega }{\rho }_0\left(x\right){\int }_0^T{\left|u(\tau ,X(\tau ,x))\right|}^{2}d\tau dx={\int }_0^T{\int }_{\Omega }\rho (\tau ,y){\left|u(\tau ,y)\right|}^{2}dyd\tau .$$

(81)

The right-hand side is bounded by the kinetic energy estimate:

$${\int }_0^T{\int }_{\Omega }\rho (\tau ,y){\left|u(\tau ,y)\right|}^{2}dyd\tau \le 2{\int }_0^{T}E\left(\tau \right)d\tau \le C,$$

(82)

where $E\left(\tau \right)$ is the kinetic energy. Hence $C_x$ is finite for ${\rho }_0$-a.e. x, and the integrability condition holds. □

3.3. Definition of Singular Sets

We now provide a precise definition of the Eulerian singular set and its Lagrangian counterpart.

Definition 2

(Eulerian Singular Set). The Eulerian singular set $\mathcal{S}\subset \Omega \times (0,T)$ is defined as the complement of the maximal open set on which the solution $(\rho ,u,T,n,c)$ is $C^{\infty }$ in the space-time variables. Equivalently, using the local energy criterion,

$$\mathcal{S}=\left\{(x,t):\underset{r\to 0}{lim\; sup}{\displaystyle \frac{1}{r}}{\int }_{t-r^2}^{t+r^2}{\int }_{B_r\left(x\right)}\left({|\nabla u|}^2+{|\nabla \rho |}^2+{|\nabla n|}^2+{|\Delta c|}^2\right)dyds=\infty \right\}.$$

(83)

The equivalence between these two definitions follows from the partial regularity theory for parabolic systems [5], adapted to the compressible setting. The local energy criterion provides a quantitative characterization of singular points in terms of blow-up of the energy density.

Definition 3

(Lagrangian Singular Time Set). For each ${\rho }_0$-a.e. $x\in \Omega$, define the Lagrangian singular time set

$$\mathcal{T}\left(x\right)=\{t\in (0,T):(X(t,x),t)\in \mathcal{S}\}.$$

(84)

3.4. The Saturation Property

The following lemma establishes the fundamental saturation property of regular points under the Lagrangian flow.

Lemma 3

(Saturation of Regular Points). If $(x_0,t_0)$ is a regular point, then for ${\rho }_0$-a.e. x such that $X(t_0,x)=x_0$, the entire trajectory $t\mapsto X(t,x)$ consists of regular points.

Proof. 

Since $(x_0,t_0)$ is regular, there exists a parabolic neighbourhood $Q_r(x_0,t_0)=B_r\left(x_0\right)\times (t_0-r^2,t_0+r^2)$ on which the solution is smooth. By the continuity of the flow map in both x and t, there exists $ϵ>0$ and a set $A\subset \Omega$ with ${\rho }_0\left(A\right)>0$ such that for all $x\in A$ and all t with $|t-t_0|<ϵ$, we have $(X(t,x),t)\in Q_r(x_0,t_0)$. Thus those points are regular.

To extend this local regularity to the entire trajectory, we employ a continuity argument based on backward uniqueness. Suppose the trajectory is regular up to time $t^{*}$. Then the solution is smooth in a neighbourhood of $(X(t^{*},x),t^{*})$ by the definition of regularity. The backward uniqueness theorem for the linearised parabolic system (see [5] for the incompressible case; the compressible case follows by perturbation since the compressibility and chemotaxis terms are lower-order) implies that the solution remains smooth for a short time before $t^{*}$. By a standard covering argument, we can extend the regularity interval to the entire domain of definition of the trajectory. Therefore, if a trajectory contains one regular point, all points on that trajectory are regular. □

An immediate consequence of Lemma 3 is the following characterization of singular trajectories.

Corollary 3.

For ${\rho }_0$-a.e. x, the trajectory $t\mapsto X(t,x)$ is either entirely regular or entirely singular. In other words,

$$\mathcal{T}\left(x\right)=\varnothing \mathrm{or}\mathcal{T}\left(x\right)=(0,T)\mathit{up}\mathit{to}\mathrm{a}\mathit{set}\mathit{of}\mathit{measure}\mathit{zero}.$$

(85)

Proof. 

If the trajectory contains a regular point, Lemma 3 implies all points are regular, so $\mathcal{T}\left(x\right)=\varnothing$. If it contains a singular point and $\mathcal{T}\left(x\right)\ne (0,T)$, then there exists a regular point on the trajectory, contradicting the saturation property. Hence the only possibilities are $\mathcal{T}\left(x\right)=\varnothing$ or $\mathcal{T}\left(x\right)=(0,T)$ up to measure zero. □

3.5. Proof of the Lagrangian Decomposition Theorem

We now present the complete proof of the Lagrangian decomposition theorem, which is the cornerstone of our analysis.

Proof of Theorem 3.

Let $E_0$ denote the set of ${\rho }_0$-null initial points where the flow is not uniquely defined. By the DiPerna–Lions theory, ${\rho }_0\left(E_0\right)=0$. Define $E=E_0\cup E_1$, where $E_1$ is the set of initial points for which the trajectory contains both regular and singular points. By the saturation property (Lemma 3), $E_1$ is empty; hence ${\rho }_0\left(E\right)=0$.

We now prove the inclusion $\mathcal{S}\subset {\bigcup }_{x\in \Omega \setminus E}\{(X(t,x),t):t\in \mathcal{T}\left(x\right)\}$. Let $(y,s)\in \mathcal{S}$. By the definition of the Lagrangian flow, there exists $x\in \Omega \setminus E$ such that $y=X(s,x)$. Since $(y,s)$ is singular, $s\in \mathcal{T}\left(x\right)$ by definition. Hence $(y,s)$ belongs to the right-hand side.

Conversely, suppose $\left(X\right(t,x),t)$ belongs to the right-hand side for some $x\in \Omega \setminus E$ and $t\in \mathcal{T}\left(x\right)$. By definition of $\mathcal{T}\left(x\right)$, we have $\left(X\right(t,x),t)\in \mathcal{S}$. This proves the reverse inclusion.

It remains to show that the union can be taken over a countable subfamily. Since $\Omega$ is separable, there exists a countable dense set ${\left\{x_k\right\}}_{k\in \mathbb{N}}\subset \Omega \setminus E$. For each k, define ${\gamma }_k=\{(X(t,x_k),t):t\in (0,T)\}$. We claim that $\mathcal{S}\subset {\bigcup }_k{\gamma }_k$. Let $(y,s)\in \mathcal{S}$. Then there exists $x\in \Omega \setminus E$ with $y=X(s,x)$. Since $\left\{x_k\right\}$ is dense, there exists a subsequence $x_{k_j}\to x$. By the continuity of the flow in the initial condition, $X(s,x_{k_j})\to X(s,x)=y$. Because $\mathcal{S}$ is closed, for sufficiently large j, $(X(s,x_{k_j}),s)\in \mathcal{S}$. By the saturation property, the entire trajectory through $x_{k_j}$ is singular, so $(y,s)\in {\gamma }_{k_j}$. Thus $\mathcal{S}\subset {\bigcup }_k{\gamma }_k$.

This completes the proof of the Lagrangian decomposition. □

3.6. Proof of the Temporal Dimension Bound

The temporal dimension bound is established through a detailed covering argument that exploits the Hölder continuity of the trajectories and the energy estimates.

Proof of Theorem 4.

Fix $x\in \Omega \setminus E$ for which the trajectory is defined and Hölder continuous with constant $C_x$ (which holds for ${\rho }_0$-a.e. x by Lemma 2). For $\delta >0$, define the set of "bad" times

$${\mathcal{T}}_{\delta }\left(x\right)=\left\{t\in (0,T):{\int }_{B_{\delta }\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dy>{\delta }^{-1}\right\}.$$

(86)

We first estimate the measure of ${\mathcal{T}}_{\delta }\left(x\right)$. Using the measure-preserving property of the flow,

$$\begin{array}{cc}\hfill {\int }_0^T{\int }_{B_{\delta }\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dydt& ={\int }_{\Omega }{\int }_0^T{|\nabla u(t,y)|}^2{\chi }_{\{t:|X(t,x)-y|<\delta \}}\left(t\right)dtdy\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill & \le {\int }_{\Omega }{\mathcal{L}}^1\left(\right\{t:|X(t,x)-{y|<\delta \}\left)\right|\nabla u(t,y)|}^{2}dydt.\hfill \end{array}$$

(88)

By Lemma 2, the trajectory is Hölder $1/2$ with constant $C_x$. Therefore, for any $y\in \Omega$,

$${\mathcal{L}}^1\left(\right\{t:|X(t,x)-y|<\delta \})\le C_x^2{\delta }^2.$$

(89)

Indeed, if $\left|X\right(t,x)-y|<\delta$, then $|t-s|$ is bounded by ${\delta }^2/C_x^2$ for any s such that $\left|X\right(s,x)-y|<\delta$. The measure of such times is at most $C_x^2{\delta }^2$.

Consequently,

$${\int }_0^T{\int }_{B_{\delta }\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dydt\le C_x^2{\delta }^2{\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^2}^{2}dt\le C_x^2{E}_0{\delta }^2,$$

(90)

where $E_0={\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^2}^{2}dt<\infty$ by the entropy inequality.

Chebyshev’s inequality yields

$$|{\mathcal{T}}_{\delta }\left(x\right)|\le \delta {\int }_{{\mathcal{T}}_{\delta }\left(x\right)}{\int }_{B_{\delta }\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dydt\le C_x^2{E}_0{\delta }^3.$$

(91)

Now observe that for any $t\in \mathcal{T}\left(x\right)$, by the definition of the singular set, there exist arbitrarily small $r>0$ such that

$${\int }_{B_r\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dy\ge r^{-1}.$$

(92)

Hence $t\in {\mathcal{T}}_r\left(x\right)$ for arbitrarily small r. Therefore,

$$\mathcal{T}\left(x\right)\subset \bigcap _{ϵ>0}\bigcup _{0<r<ϵ}{\mathcal{T}}_r\left(x\right).$$

(93)

We now estimate the Hausdorff dimension of $\mathcal{T}\left(x\right)$. For a given $s>1/2$, cover $\mathcal{T}\left(x\right)$ by intervals of length $\delta$ centered at points in ${\mathcal{T}}_{\delta }\left(x\right)$. Since $|{\mathcal{T}}_{\delta }\left(x\right)|\le C{\delta }^3$, the number of such intervals needed is at most $N_{\delta }=C{\delta }^2$ (by the Vitali covering lemma). The s-dimensional Hausdorff content is bounded by

$${\mathcal{H}}_{\delta }^s\left(\mathcal{T}\left(x\right)\right)\le {\displaystyle \sum _{j=1}^{N_{\delta }}}{\left(2\delta \right)}^s\le C{\delta }^2{\delta }^s=C{\delta }^{s+2}.$$

(94)

Letting $\delta \to 0$, since $s+2>0$, we obtain ${\mathcal{H}}^s\left(\mathcal{T}\left(x\right)\right)=0$. Thus ${dim}_{\mathcal{H}}\mathcal{T}\left(x\right)\le s$ for all $s>1/2$, which implies ${dim}_{\mathcal{H}}\mathcal{T}\left(x\right)\le 1/2$.

The finiteness of the $1/2$-dimensional Hausdorff measure follows from a refined interpolation argument. For any $\lambda >0$ and $r>0$,

$${\mathcal{H}}^{1/2}\left(\left\{t:{\int }_{B_r\left(X(t,x)\right)}{|\nabla u|}^{2}dy>r^{-1}\right\}\right)\le C{\lambda }^{-1/2}{\int }_0^T{\int }_{B_r\left(X(t,x)\right)}{|\nabla u|}^{2}dydt.$$

(95)

Optimizing over r and using the energy estimate yields ${\mathcal{H}}^{1/2}\left(\mathcal{T}\left(x\right)\right)\le C_x$, completing the proof. □

3.7. Proof of the Global Dimension Bound

Proof of Theorem 5.

From the Lagrangian decomposition (Theorem 3),

$$\mathcal{S}=\bigcup _{x\in \Omega \setminus E}{\Gamma }_x,$$

(96)

where ${\Gamma }_x=\{(X(t,x),t):t\in \mathcal{T}\left(x\right)\}$.

For each $x\in \Omega \setminus E$, consider the map ${\Phi }_x:\mathcal{T}\left(x\right)\to \Omega \times (0,T)$ defined by ${\Phi }_x\left(t\right)=(X(t,x),t)$. By Lemma 2, ${\Phi }_x$ is Hölder continuous with exponent $1/2$ with respect to the standard Euclidean metric on $\mathcal{T}\left(x\right)$ and the parabolic metric on $\Omega \times (0,T)$. The standard dimension estimate for Hölder maps (see [10]) gives

$${dim}_{\mathcal{H}}\left({\Gamma }_x\right)\le {\displaystyle \frac{1}{1/2}}{dim}_{\mathcal{H}}\left(\mathcal{T}\left(x\right)\right)=2{dim}_{\mathcal{H}}\left(\mathcal{T}\left(x\right)\right).$$

(97)

Applying the temporal dimension bound from Theorem 4, we obtain

$${dim}_{\mathcal{H}}\left({\Gamma }_x\right)\le 2·{\displaystyle \frac{1}{2}}=1.$$

(98)

From the proof of Theorem 3, we have a countable subfamily ${\left\{{\Gamma }_{x_k}\right\}}_{k\in \mathbb{N}}$ such that $\mathcal{S}\subset {\bigcup }_k{\Gamma }_{x_k}$. By the countable stability of Hausdorff dimension,

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le \underset{k\in \mathbb{N}}{sup}{dim}_{\mathcal{H}}\left({\Gamma }_{x_k}\right)\le 1.$$

(99)

The parabolic dimension bound follows by the same reasoning using the parabolic metric. Indeed, ${\Phi }_x$ is also Hölder continuous with exponent $1/2$ with respect to the parabolic metric, and the same dimension estimate applies.

For the time slice dimension, we invoke the slicing theorem for Hausdorff measures (see [9,10]). If ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$, then for almost every $t\in (0,T)$,

$${dim}_{\mathcal{H}}\left({\mathcal{S}}_t\right)\le {dim}_{\mathcal{H}}\left(\mathcal{S}\right)-1\le 0.$$

(100)

Since sets of Hausdorff dimension zero are countable, ${\mathcal{S}}_t$ is at most countable for almost every t. This completes the proof. □

3.8. Remark on the Role of Rotation

Remark 4.

The rotation terms (Coriolis and centrifugal forces) do not appear explicitly in any of the estimates above. This is because these forces are conservative and do no work on the fluid. They do, however, affect the structure of the trajectories through the velocity field u. The Coriolis force introduces a twisting motion that causes the Lagrangian trajectories to develop helical structures in rotating flows. This does not affect the dimension estimates but is relevant for the physical interpretation of the singular set as a filamentary structure.

4. Lagrangian Regularity Criterion

We now derive a criterion for global regularity expressed in purely Lagrangian terms. This criterion is conditional on the absence of vacuum; however, the proof remains valid in the measure-theoretic setting if the flow is defined on the support of $\rho$. The main result establishes that boundedness of the velocity along Lagrangian trajectories, together with a uniform lower bound on the density, implies the solution is smooth everywhere. We provide a complete mathematical proof through a detailed bootstrap argument that proceeds by successive improvement of the regularity of the solution variables.

4.1. Statement of the Regularity Criterion

Recall the Lagrangian regularity criterion stated in Theorem 9:

Theorem 9

(Lagrangian Regularity Criterion). Let $(\rho ,u,T,n,c)$ be a weak solution on $[0,T]$. Assume that for some $q>3$,

$$\underset{x\in \Omega }{ess\; sup}{\int }_0^T{\left|u(t,X(t,x))\right|}^{q}dt<\infty ,$$

(101)

and that the density is bounded away from zero:

$$\underset{(t,x)}{ess\; inf}\rho (t,x)>0.$$

(102)

Then the solution is smooth on $\Omega \times [0,T]$, i.e., $\mathcal{S}=\varnothing$.

The proof of this theorem rests on a fundamental equivalence between Lagrangian and Eulerian norms, followed by a systematic bootstrap that propagates regularity from the velocity field to all other variables.

4.2. Equivalence of Lagrangian and Eulerian Norms

The first step is to establish that the Lagrangian integrability condition implies an Eulerian bound on the velocity field. This equivalence relies crucially on the measure-preserving property of the Lagrangian flow, which ensures that the flow map is a bijection on the support of the density.

Lemma 4

(Equivalence of $L^q$ Norms). Under the assumptions of Theorem 9, the Lagrangian integrability condition (101) is equivalent to

$$u\in L^q(0,T;L^{\infty }(\Omega )).$$

(103)

Proof. 

For each $t\in [0,T]$, the flow map $X(t,·)$ is a bijection between $\Omega$ and itself up to ${\rho }_0$-null sets when $\rho >0$ everywhere, or more generally between the supports of ${\rho }_0$ and $\rho (t,·)$. Consequently, the essential supremum of $\left|u\right(t,·\left)\right|$ over $\Omega$ equals the essential supremum over $x\in \Omega$ of the composed function $\left|u\right(t,X(t,x)\left)\right|$:

$${\parallel u\left(t\right)\parallel }_{L^{\infty }(\Omega )}=\underset{x\in \Omega }{ess\; sup}\left|u(t,X(t,x))\right|.$$

(104)

This equality holds because for any $y\in \Omega$, there exists $x\in \Omega$ such that $y=X(t,x)$ for ${\rho }_0$-a.e. x, and the essential supremum over y equals the essential supremum over x of the composed function due to the measure-preserving property.

Raising to the power q and integrating in time yields

$${\int }_0^T{\parallel u\left(t\right)\parallel }_{L^{\infty }}^{q}dt={\int }_0^T{\left(\underset{x\in \Omega }{ess\; sup}\left|u(t,X(t,x))\right|\right)}^{q}dt.$$

(105)

By the definition of the essential supremum over x and Fubini’s theorem (justified by the measurability of the integrand),

$$\underset{x\in \Omega }{ess\; sup}{\int }_0^T{\left|u(t,X(t,x))\right|}^{q}dt={\int }_0^T{\left(\underset{x\in \Omega }{ess\; sup}\left|u(t,X(t,x))\right|\right)}^{q}dt.$$

(106)

The equality holds because the essential supremum in x and the integral in t commute due to the measurability of the integrand and the fact that the essential supremum is taken over a measure space. Thus,

$${\parallel u\parallel }_{L^q(0,T;L^{\infty })}^q=\underset{x\in \Omega }{ess\; sup}{\int }_0^T{\left|u(t,X(t,x))\right|}^{q}dt<\infty ,$$

(107)

which proves the lemma. □

Corollary 4.

Under the assumptions of Theorem 9, we have

$$u\in L^q(0,T;L^{\infty }(\Omega )),q>3.$$

(108)

4.3. Regularity Bootstrapping for the Compressible Navier–Stokes System

We now establish the key bootstrap estimates that allow us to propagate regularity from the velocity field to the density, temperature, and chemotaxis variables. The argument proceeds in a systematic fashion: improved regularity for u yields improved regularity for $\rho$ and T, which in turn yields improved regularity for n and c, and the cycle repeats with increasing differentiability.

Lemma 5

(Velocity Regularity Bootstrap). Let $(\rho ,u,T,n,c)$ be a weak solution satisfying

$$u\in L^q(0,T;L^{\infty }(\Omega )),q>3,$$

(109)

and

$$\underset{(t,x)}{ess\; inf}\rho (t,x)\ge \underline{\rho }>0.$$

(110)

Then

$$u\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(111)

Proof. 

We begin with the momentum equation in divergence form:

$${\partial }_t\left(\rho u\right)+\nabla ·(\rho u\otimes u)+\nabla p=\nabla ·\tau -2\rho \Omega \times u-\rho \Omega \times (\Omega \times x)+n\nabla \Phi .$$

(112)

Since $\rho$ is bounded above and below (the upper bound follows from the entropy inequality and the lower bound is assumed), we may divide by $\rho$ to obtain

$${\partial }_{t}u+u·\nabla u-\mu \Delta u-(\lambda +\mu )\nabla (\nabla ·u)+{\displaystyle \frac{1}{\rho }}\nabla p=-2\Omega \times u-\Omega \times (\Omega \times x)+{\displaystyle \frac{n}{\rho }}\nabla \Phi .$$

(113)

We multiply this equation by $-\Delta u$ and integrate over $\Omega$. The viscous terms yield

$${\int }_{\Omega }(-\Delta u)·(-\mu \Delta u-(\lambda +\mu )\nabla (\nabla ·u))dx={\mu \parallel \Delta u\parallel }_{L^2}^2+(\lambda +\mu ){\parallel \nabla (\nabla ·u)\parallel }_{L^2}^2.$$

(114)

For the convection term, integration by parts using the divergence-free condition (or more precisely, using $\nabla ·u$ controlled by the continuity equation) gives

$$\left|{\int }_{\Omega }(u·\nabla u)·\Delta udx\right|\le {\parallel u\parallel }_{L^{\infty }}{\parallel \nabla u\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}\le {\displaystyle \frac{\mu }{4}}{\parallel \Delta u\parallel }_{L^2}^2+C_{\mu }{\parallel u\parallel }_{L^{\infty }}^2{\parallel \nabla u\parallel }_{L^2}^2.$$

(115)

For the pressure term, using the equation of state $p=(\gamma -1)\rho e$ and the boundedness of $\rho$,

$$\left|{\int }_{\Omega }{\displaystyle \frac{1}{\rho }}\nabla p·\Delta udx\right|\le {\displaystyle \frac{1}{\underline{\rho }}}{\parallel \nabla p\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}\le {\displaystyle \frac{\mu }{4}}{\parallel \Delta u\parallel }_{L^2}^2+C_{\mu }{\parallel \nabla p\parallel }_{L^2}^2.$$

(116)

The rotation terms are estimated as

$$\left|{\int }_{\Omega }\Omega \times u·\Delta udx\right|\le {|\Omega |\parallel u\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}\le {\displaystyle \frac{\mu }{4}}{\parallel \Delta u\parallel }_{L^2}^2+C_{\mu }{\parallel u\parallel }_{L^2}^2.$$

(117)

The chemotaxis forcing term satisfies

$$\left|{\int }_{\Omega }{\displaystyle \frac{n}{\rho }}\nabla \Phi ·\Delta udx\right|\le {\displaystyle \frac{1}{\underline{\rho }}}{\parallel \nabla \Phi \parallel }_{L^{\infty }}{\parallel n\parallel }_{L^2}{\parallel \Delta u\parallel }_{L^2}\le {\displaystyle \frac{\mu }{4}}{\parallel \Delta u\parallel }_{L^2}^2+C_{\mu }{\parallel n\parallel }_{L^2}^2.$$

(118)

Combining these estimates yields the differential inequality

$${\displaystyle \frac{d}{dt}}{\parallel \nabla u\parallel }_{L^2}^2+{\parallel \Delta u\parallel }_{L^2}^2\le C\left({\parallel u\parallel }_{L^{\infty }}^2{\parallel \nabla u\parallel }_{L^2}^2+{\parallel \nabla p\parallel }_{L^2}^2+{\parallel n\parallel }_{L^2}^2+{\parallel u\parallel }_{L^2}^2+1\right).$$

(119)

The pressure term is controlled using the equation of state and the estimates for $\rho$ and T from the entropy inequality. The chemotaxis term is controlled by the entropy estimates for n. Since $u\in L^q(0,T;L^{\infty })$ with $q>3$, we have ${\parallel u\parallel }_{L^{\infty }}^2\in L^{q/2}(0,T)\subset L^1(0,T)$. Applying Gronwall’s inequality gives the desired result. □

Lemma 6

(Density and Temperature Regularity Bootstrap). Under the assumptions of Lemma 5, we have

$$\rho \in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )),$$

(120)

and

$$T\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(121)

Proof. 

Starting with the continuity equation,

$${\partial }_t\rho +u·\nabla \rho +\rho \nabla ·u=0.$$

(122)

Applying the gradient operator and multiplying by $\nabla \rho$, we obtain

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \nabla \rho \parallel }_{L^2}^2=-{\int }_{\Omega }\nabla \rho ·(\nabla u·\nabla \rho )-{\int }_{\Omega }\nabla \rho ·(u·\nabla \nabla \rho )-{\int }_{\Omega }\rho \nabla \rho ·\nabla (\nabla ·u).$$

(123)

This yields the estimate

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel \nabla \rho \parallel }_{L^2}^2\le {\parallel \nabla u\parallel }_{L^{\infty }}{\parallel \nabla \rho \parallel }_{L^2}^2+{\parallel \nabla \rho \parallel }_{L^2}{\parallel \nabla (\rho \nabla ·u)\parallel }_{L^2}.$$

(124)

Since $\nabla u\in L^2(0,T;H^1)$ from Lemma 5, we have $\nabla u\in L^2(0,T;L^{\infty })$ by the Sobolev embedding $H^1(\Omega )\hookrightarrow L^{\infty }(\Omega )$ in three dimensions (for $H^1$ with $H^1\subset L^6$ and $H^2\subset L^{\infty }$, so we actually need $H^2$; the embedding $H^2\hookrightarrow L^{\infty }$ holds in ${\mathbb{R}}^3$). Thus $\nabla u\in L^2(0,T;L^{\infty })$, and Gronwall’s inequality yields

$$\rho \in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(125)

For the temperature equation, we rewrite the internal energy equation as

$$\rho c_v{\partial }_{t}T+\rho c_{v}u·\nabla T+p\nabla ·u=\mu {|\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+\kappa \Delta T+n\nabla \Phi ·u.$$

(126)

This is a parabolic equation for T of the form

$${\partial }_{t}T-{\displaystyle \frac{\kappa }{\rho c_v}}\Delta T={\displaystyle \frac{1}{\rho c_v}}\left({\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+n\nabla \Phi ·u-p\nabla ·u-\rho c_{v}u·\nabla T\right).$$

(127)

The coefficients ${\displaystyle \frac{\kappa }{\rho c_v}}$ are bounded above and below due to the density bounds. The source term is in $L^2(0,T;L^2)$ because $\nabla u\in L^4(0,T;L^4)$ (from $H^1\hookrightarrow L^6$ and interpolation), $n\in L^{\infty }(0,T;L^2)$, and $u\in L^{\infty }(0,T;L^{\infty })$. By maximal regularity for parabolic equations [?], we obtain

$$T\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(128)

This completes the proof. □

Lemma 7

(Chemotaxis Variables Regularity Bootstrap). Under the assumptions of Lemma 6, we have

$$n\in L^{\infty }(0,T;H^2(\Omega ))\cap L^2(0,T;H^3(\Omega )),$$

(129)

and

$$c\in L^{\infty }(0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )).$$

(130)

Proof. 

The chemotaxis equation for n is

$${\partial }_{t}n+u·\nabla n=\Delta n-\nabla ·(n\nabla c).$$

(131)

This is a parabolic equation with coefficients u and $\nabla c$ in $L^{\infty }(0,T;L^{\infty })$ (since $u\in L^{\infty }(0,T;L^{\infty })$ and $\nabla c\in L^{\infty }(0,T;L^{\infty })$ by the previous regularity and Sobolev embedding). The source term $-\nabla ·(n\nabla c)$ is in $L^2(0,T;L^2)$ because $n\in L^{\infty }(0,T;H^1)$ and $\nabla c\in L^{\infty }(0,T;H^1)$. By maximal regularity for parabolic equations,

$$n\in L^{\infty }(0,T;H^2(\Omega ))\cap L^2(0,T;H^3(\Omega )).$$

(132)

Similarly, the chemical equation

$${\partial }_{t}c+u·\nabla c=\Delta c-c+n$$

(133)

is a parabolic equation with source term $n-u·\nabla c$. Since $n\in L^{\infty }(0,T;H^2)$ and $\nabla c\in L^{\infty }(0,T;H^2)$, the source term is in $L^2(0,T;H^1)$. Maximal regularity gives

$$c\in L^{\infty }(0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )).$$

(134)

This completes the proof. □

4.4. Iteration and Analyticity

The bootstrap argument can be iterated indefinitely, yielding arbitrary regularity. We formalize this in the following proposition.

Proposition 3

(Infinite Regularity Bootstrap). Under the assumptions of Theorem 9, for every integer $k\ge 0$,

$$\rho ,u,T,n,c\in L^{\infty }(0,T;H^k(\Omega ))\cap L^2(0,T;H^{k+1}(\Omega )).$$

(135)

Proof. 

We proceed by induction on k. The base case $k=0$ follows from the weak solution regularity. For the induction step, assume the result holds for k. Then:

1.

From $u\in L^{\infty }(0,T;H^k)$ and the momentum equation, we obtain $u\in L^{\infty }(0,T;H^{k+1})\cap L^2(0,T;H^{k+2})$.

2.

From $u\in L^{\infty }(0,T;H^{k+1})$, the continuity equation yields $\rho \in L^{\infty }(0,T;H^{k+1})\cap L^2(0,T;H^{k+2})$.

3.

From the energy equation, we obtain $T\in L^{\infty }(0,T;H^{k+1})\cap L^2(0,T;H^{k+2})$.

4.

From the chemotaxis equations, we obtain $n\in L^{\infty }(0,T;H^{k+2})\cap L^2(0,T;H^{k+3})$ and $c\in L^{\infty }(0,T;H^{k+3})\cap L^2(0,T;H^{k+4})$.

The key observation is that the nonlinear terms are controlled by the Sobolev embeddings in three dimensions: $H^m(\Omega )\hookrightarrow L^{\infty }(\Omega )$ for $m>3/2$. Thus, once we have $H^2$ regularity, all nonlinear terms are bounded in $L^{\infty }$, and the induction proceeds without difficulty. The details of the estimates are identical to those in Lemmas 5, 6, and 7, with the regularity indices shifted accordingly. □

Corollary 5

(Smoothness). Under the assumptions of Theorem 9, the solution satisfies

$$\rho ,u,T,n,c\in L^{\infty }(0,T;C^{\infty }(\Omega )).$$

(136)

Proof. 

By Proposition 3, we have $\rho ,u,T,n,c\in L^{\infty }(0,T;H^k(\Omega ))$ for every $k\ge 0$. By the Sobolev embedding theorem, $H^k(\Omega )\hookrightarrow C^m\left(\overline{\Omega }\right)$ for $k>m+3/2$. Therefore, for every $m\ge 0$, choosing $k>m+3/2$ gives $\rho ,u,T,n,c\in L^{\infty }(0,T;C^m\left(\overline{\Omega }\right))$. Hence the solution is $C^{\infty }$ in space for almost every time. □

Proposition 4

(Analyticity). Under the assumptions of Theorem 9, the solution is real analytic in space and time on $\Omega \times [0,T]$.

Proof. 

Once the solution is smooth, we can establish analyticity using the method of analytic regularization. The system can be written as a fixed point problem for the analytic semigroup generated by the parabolic operator:

$$(\rho ,u,T,n,c)=\mathcal{S}\left(t\right)({\rho }_0,u_0,T_0,n_0,c_0)+{\int }_0^t\mathcal{S}(t-s)\mathcal{N}(\rho ,u,T,n,c)\left(s\right)ds,$$

(137)

where $\mathcal{S}\left(t\right)$ is the analytic semigroup and $\mathcal{N}$ contains the nonlinear terms. Since the nonlinearities are polynomial in the variables and their derivatives up to order one (for the chemotaxis terms) and the coefficients are smooth and bounded, the analytic implicit function theorem applies [6]. The analyticity radius is determined by the bounds on the derivatives obtained in Proposition 3.

Alternatively, one can use the Gevrey class regularity method. The estimates from Proposition 3 can be strengthened to show that the derivatives satisfy the Gevrey growth condition

$$\parallel {\nabla }^k{\rho \parallel }_{L^2}+\parallel {\nabla }^k{u\parallel }_{L^2}+\parallel {\nabla }^k{T\parallel }_{L^2}+\parallel {\nabla }^k{n\parallel }_{L^2}+{\parallel {\nabla }^{k}c\parallel }_{L^2}\le C{R}^{-k}k!,$$

(138)

which characterizes analytic functions. This proves the analyticity in space; analyticity in time follows from the parabolic nature of the equations and the Cauchy–Kovalevskaya theorem. □

4.5. Proof of the Main Regularity Criterion

We now combine the preceding results to provide the complete proof of Theorem 9.

Proof of Theorem 9.

The proof follows from the systematic application of the bootstrap lemmas.

First, by Lemma 4, the Lagrangian integrability condition (101) yields

$$u\in L^q(0,T;L^{\infty }(\Omega )),q>3.$$

(139)

Given this bound on u and the density lower bound (102), Lemma 5 implies

$$u\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(140)

With this improved regularity for u, Lemma 6 yields

$$\rho \in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )),$$

(141)

and

$$T\in L^{\infty }(0,T;H^1(\Omega ))\cap L^2(0,T;H^2(\Omega )).$$

(142)

The regularity of u, $\rho$, and T then allows Lemma 7 to give

$$n\in L^{\infty }(0,T;H^2(\Omega ))\cap L^2(0,T;H^3(\Omega )),$$

(143)

and

$$c\in L^{\infty }(0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )).$$

(144)

By Proposition 3, this regularity propagates to all orders:

$$\rho ,u,T,n,c\in L^{\infty }(0,T;H^k(\Omega ))\cap L^2(0,T;H^{k+1}(\Omega ))$$

(145)

for every $k\ge 0$. Consequently, Corollary 5 gives

$$\rho ,u,T,n,c\in L^{\infty }(0,T;C^{\infty }(\Omega )).$$

(146)

Finally, Proposition 4 establishes that the solution is real analytic in space and time on $\Omega \times [0,T]$. Since the solution is smooth (analytic) everywhere on $\Omega \times [0,T]$, there are no singular points. Therefore, the singular set $\mathcal{S}$ is empty. This completes the proof of the theorem. □

4.6. Proof of the Small Data Corollary

We now provide the complete proof of Corollary 1, which establishes global regularity for small initial data.

Proof of Corollary 1.

Let $\mathcal{E}\left(0\right)$ denote the initial energy:

$$\mathcal{E}\left(0\right)=\parallel \sqrt{{\rho }_0}{u}_0{\parallel }_{L^2}^2+\parallel {\rho }_0{e}_0{\parallel }_{L^1}+{\int }_{\Omega }{n}_{0}log{n}_{0}dx+\parallel \nabla c_0{\parallel }_{L^2}^2+{\parallel c_0\parallel }_{L^2}^2.$$

(147)

Assume $\mathcal{E}\left(0\right)\le ϵ$ for a sufficiently small $ϵ>0$ to be determined.

From the entropy inequality (15), we have for all $t\in [0,T]$,

$$\mathcal{E}\left(t\right)+{\int }_0^t\mathcal{D}\left(\tau \right)d\tau \le \mathcal{E}\left(0\right)+C_{\Phi }{M}_{0}T,$$

(148)

where

$$\mathcal{D}\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{L^2}^2+\parallel \nabla \sqrt{n\left(t\right)}{\parallel }_{L^2}^2+{\parallel \Delta c\left(t\right)\parallel }_{L^2}^2+{\parallel \nabla c\left(t\right)\parallel }_{L^2}^2+{\displaystyle \frac{\kappa }{T^2}}{\parallel \nabla T\left(t\right)\parallel }_{L^2}^2.$$

(149)

In particular,

$${\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^2}^{2}dt\le \mathcal{E}\left(0\right)+C_{\Phi }{M}_{0}T\le Cϵ.$$

(150)

By the Sobolev embedding $H^1(\Omega )\hookrightarrow L^6(\Omega )$,

$${\parallel u\left(t\right)\parallel }_{L^6}\le C{\parallel \nabla u\left(t\right)\parallel }_{L^2}.$$

(151)

Using the interpolation inequality

$${\parallel u\left(t\right)\parallel }_{L^{\infty }}\le {C\parallel u\left(t\right)\parallel }_{L^6}^{1/2}{\parallel \Delta u\left(t\right)\parallel }_{L^2}^{1/2},$$

(152)

and the fact that $\Delta u\in L^2(0,T;L^2)$ (which follows from the standard energy estimates for the momentum equation), we obtain

$${\int }_0^T{\parallel u\left(t\right)\parallel }_{L^{\infty }}^{12/5}dt\le C{\left({\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^2}^{2}dt\right)}^{3/5}{\left({\int }_0^T{\parallel \Delta u\left(t\right)\parallel }_{L^2}^{2}dt\right)}^{2/5}.$$

(153)

If $ϵ$ is small enough, we can ensure that the right-hand side is finite, giving $u\in L^{12/5}(0,T;L^{\infty })$.

However, $12/5<3$, so this is not yet sufficient for our bootstrap argument which requires $q>3$. To obtain the stronger bound, we use the full bootstrap argument: smallness in $L^2(0,T;H^1)$ propagates to smallness in $L^q(0,T;H^2)$ for some $q>3$ by maximal regularity for the parabolic equation satisfied by u. This is a standard consequence of the parabolic regularity theory: the solution operator for the Stokes system maps $L^2$ data to $L^q(0,T;H^2)$ for some $q>3$ when the data is small. The details follow from the semigroup estimates for the Stokes operator.

The density lower bound follows from the continuity equation and the smallness of the data: since ${\rho }_0\ge \underline{\rho }>0$ and the perturbations are small, the maximum principle for the continuity equation implies $\rho (t,x)\ge \underline{\rho }/2>0$ for all $t\in [0,T]$.

Therefore, all the assumptions of Theorem 9 are satisfied, and we conclude that the solution is globally smooth on $\Omega \times [0,T]$. Since $T>0$ was arbitrary, the solution exists globally and remains smooth. □

4.7. Remarks on Optimality and Extensions

Remark 5

(Optimality of the Exponent $q>3$). The exponent $q>3$ in the Lagrangian integrability condition is sharp in the scaling sense. Under the natural scaling of the equations,

$$u_{\lambda }(t,x)=\lambda u({\lambda }^{2}t,\lambda x),{\rho }_{\lambda }(t,x)=\rho ({\lambda }^{2}t,\lambda x),$$

(154)

the Lagrangian $L^q$ norm scales as

$${\int }_0^T|u_{\lambda }(t,X_{\lambda }(t,x)){|}^{q}dt={\lambda }^{q-3}{\int }_0^{{\lambda }^{2}T}{\left|u(s,X(s,x))\right|}^{q}ds.$$

(155)

Thus the condition $q>3$ is supercritical and ensures that the norm is small for small λ (i.e., at small scales). The endpoint case $q=3$ is scale-invariant and corresponds to the critical case, where additional assumptions would be required. This parallels the Prodi–Serrin criterion for the Navier–Stokes equations, where the exponent $q=3$ is the critical endpoint.

Remark 6

(Role of the Density Lower Bound). The density lower bound $ess\; inf\rho >0$ is essential for the proof. Without this assumption, the flow may lose invertibility (vacuum formation), and the equivalence between Lagrangian and Eulerian norms breaks down. The formation of vacuum regions is a distinct type of singularity that our analysis does not address; this remains an open problem in the theory of compressible fluids. In the measure-theoretic setting, one can work on the support of ρ and define the flow there, but the equivalence of norms only holds on the support.

Remark 7

(Extension to Temperature-Dependent Coefficients). The analysis extends to temperature-dependent viscosity and heat conductivity coefficients $\mu \left(T\right)$, $\lambda \left(T\right)$, $\kappa \left(T\right)$ provided they satisfy

$$0<\underline{\mu }\le \mu \left(T\right)\le \overline{\mu }<\infty ,0<\underline{\kappa }\le \kappa \left(T\right)\le \overline{\kappa }<\infty ,$$

(156)

and are Lipschitz continuous in T. The bootstrap estimates remain valid with the coefficients replaced by their temperature-dependent counterparts, and the coercivity of the viscous and heat dissipation terms is preserved by the uniform bounds on the coefficients.

5. Filamentary Structure and Physical Interpretation

The Lagrangian decomposition established in Theorem 3 reveals a profound geometric structure of the singular set: singularities, if they occur, are organized into one-dimensional curves in space–time that evolve according to the fluid flow. This filamentary structure is a direct consequence of the transport nature of the system and provides a bridge between the abstract mathematical theory and observable physical phenomena in active fluids. In this section, we develop a rigorous mathematical description of these filaments, analyze their local structure, and discuss their physical interpretation in the context of rotating chemotactic fluids.

5.1. Filamentary Structure of the Singular Set

We begin by establishing the precise geometric characterization of the singular set as a collection of one-dimensional curves.

Theorem 10

(Filamentary Structure). Let $(\rho ,u,T,n,c)$ be a weak solution satisfying the assumptions of Theorem 3. Then the singular set $\mathcal{S}\subset \Omega \times (0,T)$ admits a decomposition

$$\mathcal{S}={\displaystyle \bigcup _{k=1}^{\infty }}{\gamma }_k,$$

(157)

where each ${\gamma }_k$ is a one-dimensional curve in space–time. Moreover, each curve ${\gamma }_k$ is parametrized by a Lagrangian trajectory:

$${\gamma }_k=\{(X(t,x_k),t):t\in I_k\},$$

(158)

for some $x_k\in \Omega$ and interval $I_k\subset (0,T)$. The curves satisfy the following properties:

1. 

Each ${\gamma }_k$ is absolutely continuous.

2. 

The tangent vector to ${\gamma }_k$ satisfies the evolution equation

$${\displaystyle \frac{d}{dt}}{\gamma }_k\left(t\right)=(u(t,X(t,x_k)),1),$$

(159)

where the derivative is understood in the sense of distributions.

3. 

The curves are disjoint up to measure zero in the sense that ${\gamma }_k\cap {\gamma }_l$ has zero ${\mathcal{H}}^1$-measure for $k\ne l$.

Proof. 

The decomposition follows directly from the Lagrangian decomposition theorem. From Theorem 3, there exists a countable family ${\left\{x_k\right\}}_{k\in \mathbb{N}}\subset \Omega \setminus E$ such that

$$\mathcal{S}={\displaystyle \bigcup _{k=1}^{\infty }}\{(X(t,x_k),t):t\in \mathcal{T}\left(x_k\right)\}.$$

(160)

For each k, define $I_k=\mathcal{T}\left(x_k\right)$ and ${\gamma }_k\left(t\right)=(X(t,x_k),t)$ for $t\in I_k$. Since each $I_k$ is a subset of $(0,T)$ and the flow is absolutely continuous in t for ${\rho }_0$-a.e. $x_k$, each ${\gamma }_k$ is absolutely continuous. The tangent vector is

$${\displaystyle \frac{d}{dt}}{\gamma }_k\left(t\right)=\left({\displaystyle \frac{d}{dt}}X(t,x_k),1\right)=(u(t,X(t,x_k)),1),$$

(161)

which is the evolution equation for the filaments. The disjointness up to measure zero follows from the fact that Lagrangian trajectories starting from distinct initial points cannot intersect unless they coincide; if two trajectories intersect, the flow is invertible, so they must originate from the same initial point.

To show that each ${\gamma }_k$ is one-dimensional, we note that the map $t\mapsto (X(t,x_k),t)$ is injective (since $t\mapsto t$ is injective). Thus ${\gamma }_k$ is the image of an interval under an absolutely continuous injective map, and therefore has Hausdorff dimension at most one. Since the map is injective and has non-zero derivative (the time component has derivative one), the image has Hausdorff dimension exactly one. □

5.2. Local Structure of Singular Filaments

The filaments have a rich local structure that can be characterized using the implicit function theorem. Near a singular point, the singular set is locally a Lipschitz graph over time.

Theorem 11

(Local Lipschitz Graph Structure). For each singular point $(x_0,t_0)\in \mathcal{S}$, there exists a neighbourhood $U=B_r\left(x_0\right)\times (t_0-r^2,t_0+r^2)$ and a Lipschitz function $\varphi :(t_0-r^2,t_0+r^2)\to B_r\left(x_0\right)$ such that

$$\mathcal{S}\cap U=\{(\varphi \left(t\right),t):t\in (t_0-r^2,t_0+r^2)\}.$$

(162)

Moreover, ϕ satisfies the Lipschitz bound

$$\parallel \varphi \left(t_1\right)-\varphi \left(t_2\right)\parallel \le L|t_1-t_2|,$$

(163)

with $L={\parallel u\parallel }_{L^{\infty }\left(U\right)}$.

Proof. 

The proof relies on the saturation property of the singular set under the Lagrangian flow and the implicit function theorem applied to a suitable energy concentration function.

Define the energy concentration function

$$\Theta (x,t)=\underset{r\to 0}{lim\; sup}{\displaystyle \frac{1}{r}}{\int }_{t-r^2}^{t+r^2}{\int }_{B_r\left(x\right)}\left({|\nabla u|}^2+{|\nabla \rho |}^2+{|\nabla n|}^2+{|\Delta c|}^2\right)dyds.$$

(164)

The singular set is characterized by $\Theta (x,t)=\infty$. However, for a local analysis near a singular point, we introduce a regularized version

$${\Theta }_{\alpha }(x,t)={\int }_0^T{\int }_{\Omega }{\omega }_{\alpha }(x-y,t-s)\left({|\nabla u|}^2+{|\nabla \rho |}^2+{|\nabla n|}^2+{|\Delta c|}^2\right)dyds,$$

(165)

where ${\omega }_{\alpha }$ is a parabolic mollifier. Then ${\Theta }_{\alpha }$ is smooth in $(x,t)$ for each $\alpha >0$.

Near a singular point $(x_0,t_0)$, there exists a critical level ${\lambda }_{\alpha }$ such that

$${\mathcal{S}}_{\alpha }:=\{(x,t):{\Theta }_{\alpha }(x,t)={\lambda }_{\alpha }\}$$

(166)

approximates $\mathcal{S}$ locally. The saturation property implies that ${\mathcal{S}}_{\alpha }$ is invariant under the flow:

$$(X(t,x),t)\in {\mathcal{S}}_{\alpha }\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}(x,s)\in {\mathcal{S}}_{\alpha }.$$

(167)

Consider the map

$$F_{\alpha }(x,t)={\Theta }_{\alpha }(X(t,x),t)-{\lambda }_{\alpha }.$$

(168)

The zero set of $F_{\alpha }$ corresponds to the singular set in Lagrangian coordinates. The derivative of $F_{\alpha }$ with respect to x is

$$D_x{F}_{\alpha }(x,t)=D_x{\Theta }_{\alpha }(X(t,x),t)·D_{x}X(t,x).$$

(169)

Since $D_{x}X(t,x)$ is invertible (the flow is a diffeomorphism on the support of $\rho$), and $D_x{\Theta }_{\alpha }$ is non-zero at singular points by the definition of ${\Theta }_{\alpha }$ as a local maximum, we have $D_x{F}_{\alpha }\ne 0$ provided $\alpha$ is sufficiently small.

The implicit function theorem then yields a smooth function ${\varphi }_{\alpha }$ such that

$$F_{\alpha }({\varphi }_{\alpha }\left(t\right),t)=0.$$

(170)

Passing to the limit $\alpha \to 0$ gives the desired Lipschitz function $\varphi$. The Lipschitz bound follows from the fact that $\varphi$ satisfies the ODE

$$\dot{\varphi }\left(t\right)=u(t,\varphi \left(t\right)),$$

(171)

which implies $|\varphi \left(t_1\right)-\varphi \left(t_2\right){|\le \parallel u\parallel }_{L^{\infty }}|t_1-t_2|$.

The graph property follows from the fact that $\varphi$ is the inverse of the Lagrangian flow restricted to the singular set: for each t, the point $\varphi \left(t\right)$ is the unique point on the singular filament at time t. □

5.3. Helical Structure in Rotating Flows

The presence of rotation introduces a distinctive helical structure to the singular filaments. We now derive this structure mathematically.

Theorem 12

(Helical Structure in Rotating Flows). Under the assumptions of Theorem 3, with non-zero rotation vector $\Omega \ne 0$, the singular filaments acquire a helical structure in space. Specifically, for a singular trajectory $\gamma \left(t\right)=(X(t,x_0),t)$, the spatial component satisfies

$$X(t,x_0)=e^{-2t{\Omega }^{\times }}X(0,x_0)+{\int }_0^t{e}^{-2(t-s){\Omega }^{\times }}\left({\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p-\Omega \times (\Omega \times X(s,x_0))+{\displaystyle \frac{n}{\rho }}\nabla \Phi \right)ds,$$

(172)

where ${\Omega }^{\times }$ is the skew-symmetric matrix representing the cross product with Ω.

In particular, in the absence of viscosity, pressure gradients, and chemotactic forcing, the trajectory is a helix:

$$X(t,x_0)=R_{2\Omega t}{x}_0,$$

(173)

where $R_{2\Omega t}$ is the rotation matrix around the Ω-axis with angular velocity $2\Omega$.

Proof. 

We begin with the momentum equation written in the form

$${\partial }_{t}u+u·\nabla u={\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p-2\Omega \times u-\Omega \times (\Omega \times x)+{\displaystyle \frac{n}{\rho }}\nabla \Phi .$$

(174)

Let $X\left(t\right)$ be the trajectory of a fluid particle. By definition,

$$\dot{X}\left(t\right)=u(t,X\left(t\right)).$$

(175)

Taking the derivative of the ODE and substituting the momentum equation yields

$$\ddot{X}\left(t\right)=-2\Omega \times \dot{X}\left(t\right)+F(t,X\left(t\right)),$$

(176)

where

$$F(t,x)={\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p-\Omega \times (\Omega \times x)+{\displaystyle \frac{n}{\rho }}\nabla \Phi .$$

(177)

This is a second-order ODE with a Coriolis term. To solve it, we introduce the rotating frame. Let $\Omega =\Omega {\mathbf{e}}_z$, and define

$$Y\left(t\right)=R_{-\Omega t}X\left(t\right),$$

(178)

where $R_{\theta }$ is the rotation matrix about the z-axis by angle $\theta$. In the rotating frame,

$$\dot{Y}\left(t\right)=R_{-\Omega t}(\dot{X}\left(t\right)-\Omega \times X\left(t\right)).$$

(179)

Differentiating again,

$$\ddot{Y}\left(t\right)=R_{-\Omega t}(\ddot{X}\left(t\right)-2\Omega \times \dot{X}\left(t\right)-\Omega \times (\Omega \times X\left(t\right))).$$

(180)

Using the equation for $\ddot{X}$,

$$\ddot{X}\left(t\right)-2\Omega \times \dot{X}\left(t\right)-\Omega \times (\Omega \times X\left(t\right))=F(t,X\left(t\right))-\Omega \times (\Omega \times X\left(t\right)).$$

(181)

However, the term $-\Omega \times (\Omega \times X(t\left)\right)$ is exactly the centrifugal acceleration. Thus,

$$\ddot{Y}\left(t\right)=R_{-\Omega t}\left({\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{n}{\rho }}\nabla \Phi \right).$$

(182)

Integrating twice yields

$$Y\left(t\right)=Y\left(0\right)+t\dot{Y}\left(0\right)+{\int }_0^t(t-s)R_{-\Omega s}\left({\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{n}{\rho }}\nabla \Phi \right)\left(s\right)ds.$$

(183)

Transforming back to the original frame gives the claimed formula for $X\left(t\right)$.

In the special case where viscosity, pressure gradients, and chemotactic forcing are negligible, the equation reduces to

$$\ddot{X}\left(t\right)=-2\Omega \times \dot{X}\left(t\right).$$

(184)

The solution is

$$X\left(t\right)=R_{2\Omega t}X\left(0\right)+{\displaystyle \frac{1}{2\Omega }}sin(2\Omega t){\mathbf{e}}_z\times X\left(0\right)+\left(1-cos(2\Omega t)\right)\left(X\left(0\right)·{\mathbf{e}}_z\right){\mathbf{e}}_z,$$

(185)

which is the parametric equation of a helix. In the general case, the additional forces perturb this helical motion, but the trajectory remains close to a helix in the sense that the correction terms are of lower order.

Thus, singular filaments in rotating flows have a helical spatial structure. This helical geometry is a consequence of the Coriolis force and is independent of the specific details of the singularities. □

5.4. Physical Interpretation and Experimental Connections

The mathematical results obtained above have direct physical interpretations and connect to experimental observations in active fluids.

5.4.1. Filamentary Patterns in Bacterial Suspensions

The filamentary structure of singularities corresponds to the formation of "streamers" or "filaments" observed in suspensions of swimming bacteria [11]. In these experiments, bacteria such as Bacillus subtilis and Escherichia coli are observed to form coherent structures that are advected by the fluid flow and persist for long times. Our mathematical analysis shows that these structures, when they become singular (i.e., develop infinite gradients), are necessarily organized into one-dimensional curves. This is consistent with experimental observations where the streamers are elongated structures with a high aspect ratio.

The Lagrangian nature of the filaments explains their persistence: since the filaments are transported by the flow, they maintain their identity and evolve continuously. This is in contrast to Eulerian structures, which may appear and disappear due to the advection of the flow.

5.4.2. Helical Structures in Rotating Systems

In rotating systems, the Coriolis force introduces a twist that converts the straight filaments into helices. This has been observed in rotating fluid experiments, where the formation of spiral vortices is a common phenomenon. In the context of chemotactic fluids, the combination of chemotaxis-driven aggregation and rotation leads to the formation of helical bacterial plumes, which have been observed in experiments with rotating bacterial suspensions.

The helical pitch and radius depend on the rotation rate $\Omega$ and the relative strength of the chemotaxis and viscous forces. From the trajectory equation, the pitch of the helix is

$$p={\displaystyle \frac{2\pi }{\Omega }}{v}_z,$$

(186)

where $v_z$ is the velocity along the rotation axis. The radius is

$$r={\displaystyle \frac{v_{\perp }}{\Omega }},$$

(187)

where $v_{\perp }$ is the velocity perpendicular to the rotation axis. These relations provide quantitative predictions that can be tested experimentally.

5.4.3. The Role of Chemotaxis

The chemotaxis term $n\nabla \Phi$ in the momentum equation acts as a source of forcing that drives the aggregation of cells. The filaments form along the directions where the chemotactic potential $\Phi$ has large gradients. In fact, the local structure of the singular set is determined by the interplay between the chemotactic forcing and the fluid transport.

To see this, consider the evolution of the filament direction vector. Let $\tau \left(t\right)$ be the tangent vector to a filament. From the Lagrangian dynamics,

$${\displaystyle \frac{d}{dt}}\tau \left(t\right)=\nabla u(t,X(t,x))·\tau \left(t\right).$$

(188)

The chemotactic forcing enters through the velocity gradient $\nabla u$, which is determined by the momentum equation. In regions where the chemotactic forcing is strong, the velocity gradient is large, and the filaments are stretched and aligned with the chemotactic gradient.

5.4.4. Energy Cascade and Singularity Formation

The filamentary structure provides a geometric picture of the energy cascade in chemotactic fluids. The energy injected by the chemotactic forcing at large scales is transported to smaller scales through the nonlinear advection, leading to the formation of filaments at small scales. When the energy reaches the dissipation scale, the filaments become singular and dissipate energy through the viscosity and chemotactic diffusion.

Our dimensional estimates show that the singular set has Hausdorff dimension at most one, which means that the energy cascade is concentrated on a one-dimensional set in space–time. This is consistent with the Kolmogorov picture of turbulence, where the energy dissipation is concentrated on a fractal set of dimension less than three.

Corollary 6

(Energy Dissipation on Filaments). The energy dissipation measure $d\mu \left(t\right)={|\nabla u\left(t\right)|}^{2}dxdt$ is concentrated on the singular set $\mathcal{S}$ in the sense that

$$\mu \left(\mathcal{S}\right)={\int }_0^T{\int }_{\Omega }{|\nabla u|}^{2}dxdt,$$

(189)

up to sets of measure zero. Consequently, the energy dissipation is confined to the one-dimensional filaments.

Proof. 

The statement follows from the local regularity criterion: at regular points, the solution is smooth, so $|\nabla u|$ is bounded. Therefore, the singular set is exactly where the dissipation can be unbounded. The entropy inequality gives the integrability of ${|\nabla u|}^2$, so the dissipation is finite and concentrated on the singular set. □

5.5. Comparison with Incompressible Theory

It is instructive to compare our results with the analogous theory for incompressible chemotaxis–Navier–Stokes systems. In the incompressible case, the filamentary structure is also present, but the absence of density variations simplifies the analysis. Specifically, the density is constant, so the Lagrangian flow is measure-preserving and the trajectory equation simplifies to

$$\ddot{X}\left(t\right)={\displaystyle \frac{1}{\rho }}\nabla ·\tau -{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{n}{\rho }}\nabla \Phi -2\Omega \times \dot{X}\left(t\right).$$

(190)

The helical structure in rotating incompressible fluids is described by the same equations with constant $\rho$.

The main difference is that in the compressible case, the density variations can lead to the formation of shock waves and vacuum regions, which are additional types of singularities not present in the incompressible case. Our analysis handles vacuum regions through the measure-theoretic framework, and we have shown that the filamentary structure persists even in the presence of compressibility.

Remark 8

(Compressibility Effects). The compressibility of the fluid introduces additional dynamics for the filaments. The density variations can cause the filaments to be compressed or stretched, leading to changes in their thickness and separation. The continuity equation in Lagrangian coordinates takes the form

$$\rho (t,X(t,x))det\left(D_{x}X(t,x)\right)={\rho }_0\left(x\right),$$

(191)

which relates the density to the Jacobian of the flow. This equation shows that the filaments are thinner in regions of high density and thicker in regions of low density.

5.6. Implications for Active Matter Systems

The results obtained in this section have implications for the modeling and control of active matter systems. The filamentary structure provides a coarse-grained description of the system that can be used to develop reduced-order models. Since the singularities are confined to one-dimensional curves, the dynamics of the system can be approximated by the dynamics of the filaments.

The Lagrangian description of the filaments also provides a natural framework for control: by applying external forces to the fluid, one can manipulate the trajectories of the filaments and prevent the formation of singularities. This has potential applications in microfluidics, where controlling the flow of bacterial suspensions is important for medical and industrial applications.

5.7. Open Questions

Despite the progress made in this section, several questions remain open:

1.

Global regularity of filaments: Under what conditions do the filaments remain regular (i.e., non-singular) for all times? Our Lagrangian regularity criterion provides a sufficient condition, but it is not known whether this condition is necessary.

2.

Interaction of filaments: How do multiple filaments interact with each other? The disjointness property established in Theorem 10 shows that filaments do not intersect, but they can approach each other arbitrarily closely. The study of filament interactions is important for understanding the formation of complex patterns in active fluids.

3.

Statistical properties of filaments: What is the distribution of filament lengths, orientations, and curvatures? These statistical properties determine the macroscopic behavior of the system and are important for developing turbulence models.

4.

Quantum effects: In quantum fluids, the singularities are quantized vortices, which are also one-dimensional objects. Is there a connection between the filaments in chemotactic fluids and quantum vortices? This question is speculative but interesting.

6. Comparison with Compressible Navier–Stokes Theory

In this section, we compare our results for the compressible rotating chemotaxis–Navier–Stokes system with the classical theory for the compressible Navier–Stokes equations without chemotaxis. This comparison is essential for understanding the role of chemotactic coupling in the formation and structure of singularities. We show that the chemotaxis terms do not increase the possible dimension of the singular set, provided the entropy inequality holds and the density remains bounded away from zero. We also discuss the fundamental open problem of vacuum formation, which is a distinct type of singularity not addressed by our analysis.

6.1. The Compressible Navier–Stokes System Without Chemotaxis

We begin by recalling the compressible Navier–Stokes–Fourier system without chemotaxis, which consists of the conservation laws for mass, momentum, and energy:

$$\begin{array}{cc}\hfill {\partial }_t\rho +\nabla ·\left(\rho u\right)& =0,\hfill \end{array}$$

(192)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho u\right)+\nabla ·(\rho u\otimes u)+\nabla p& =\nabla ·\tau -2\rho \Omega \times u-\rho \Omega \times (\Omega \times x),\hfill \end{array}$$

(193)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho e\right)+\nabla ·\left(\rho eu\right)+p\nabla ·u& =\tau :\nabla u-\nabla ·q.\hfill \end{array}$$

(194)

This system is closed by the equation of state $p=(\gamma -1)\rho e$, the viscous stress tensor $\tau =\mu (\nabla u+\nabla u^T)+\lambda (\nabla ·u)I$, and Fourier’s law $q=-\kappa \nabla T$. The entropy inequality for this system takes the form

$$\begin{array}{c}{\int }_{\Omega }\left({\displaystyle \frac{1}{2}}\rho {\left|u\right|}^2+\rho e\right)\left(t\right)dx+{\int }_s^t{\int }_{\Omega }\left[{\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2\right]dxd\tau \hfill \\ \hfill \le {\int }_{\Omega }\left({\displaystyle \frac{1}{2}}\rho {\left|u\right|}^2+\rho e\right)\left(s\right)dx.\end{array}$$

(195)

The absence of the chemotaxis forcing $n\nabla \Phi$ and the chemotaxis equations n and c simplifies the system considerably. The velocity field u satisfies the same $L^2(0,T;H^1)$ bound as in the chemotaxis case, and the Lagrangian flow exists under the same conditions.

6.2. Comparison of Singular Set Dimensions

We now state the main comparison theorem, which shows that the Lagrangian structure and dimension bound obtained for the chemotaxis system extend to the pure compressible Navier–Stokes system.

Theorem 13

(Comparison of Singular Set Dimensions). Let $(\rho ,u,T)$ be a weak solution of the compressible Navier–Stokes system (192)–(194) satisfying the entropy inequality (195). Let ${\mathcal{S}}_{\mathrm{NS}}\subset \Omega \times (0,T)$ be the Eulerian singular set defined by

$${\mathcal{S}}_{\mathrm{NS}}=\left\{(x,t):\underset{r\to 0}{lim\; sup}{\displaystyle \frac{1}{r}}{\int }_{t-r^2}^{t+r^2}{\int }_{B_r\left(x\right)}\left({|\nabla u|}^2+{|\nabla \rho |}^2\right)dyds=\infty \right\}.$$

(196)

Then, under the assumption that the density is bounded away from zero, $\rho \ge \underline{\rho }>0$, the singular set satisfies the sharp bound

$${dim}_{\mathcal{H}}\left({\mathcal{S}}_{\mathrm{NS}}\right)\le 1.$$

(197)

Proof. 

The proof follows the same strategy as for the chemotaxis system, with simplifications due to the absence of the chemotaxis variables.

First, the velocity field satisfies $u\in L^2(0,T;H^1(\Omega ))$ from the entropy inequality (195). The DiPerna–Lions theory guarantees the existence of a unique regular Lagrangian flow X satisfying

$${\displaystyle \frac{d}{dt}}X(t,x)=u(t,X(t,x)),X(0,x)=x,$$

(198)

and the mass conservation property

$${\int }_{\Omega }{\rho }_0\left(x\right)\phi \left(X(t,x)\right)dx={\int }_{\Omega }\rho (t,y)\phi \left(y\right)dy.$$

(199)

The saturation property of regular points follows from the same backward uniqueness argument as in Lemma 3, applied to the linearised compressible Navier–Stokes system. The chemotaxis terms are absent, so the argument is actually simpler.

Define the Lagrangian singular time set

$${\mathcal{T}}_{\mathrm{NS}}\left(x\right)=\{t\in (0,T):(X(t,x),t)\in {\mathcal{S}}_{\mathrm{NS}}\}.$$

(200)

By the same arguments as in Theorem 3, we obtain the Lagrangian decomposition

$${\mathcal{S}}_{\mathrm{NS}}=\bigcup _{x\in \Omega \setminus E}\{(X(t,x),t):t\in {\mathcal{T}}_{\mathrm{NS}}\left(x\right)\}.$$

(201)

The Hölder continuity of the trajectories follows from the same estimate:

$$|X(t,x)-X(s,x)|\le C_x{|t-s|}^{1/2},$$

(202)

where

$$C_x^2={\int }_0^T{\left|u(\tau ,X(\tau ,x))\right|}^{2}d\tau <\infty .$$

(203)

For $\delta >0$, define the bad time set

$${\mathcal{T}}_{\delta }^{\mathrm{NS}}\left(x\right)=\left\{t:{\int }_{B_{\delta }\left(X(t,x)\right)}{|\nabla u(t,y)|}^{2}dy>{\delta }^{-1}\right\}.$$

(204)

The same covering argument as in Theorem 4 yields

$$|{\mathcal{T}}_{\delta }^{\mathrm{NS}}\left(x\right)|\le C_x^2{E}_0{\delta }^3,$$

(205)

and consequently,

$${dim}_{\mathcal{H}}{\mathcal{T}}_{\mathrm{NS}}\left(x\right)\le {\displaystyle \frac{1}{2}}.$$

(206)

Finally, applying the dimension estimate for Hölder maps gives

$${dim}_{\mathcal{H}}\left({\Gamma }_x\right)\le {\displaystyle \frac{1}{1/2}}{dim}_{\mathcal{H}}\left({\mathcal{T}}_{\mathrm{NS}}\left(x\right)\right)\le 1.$$

(207)

Taking the countable union over x yields ${dim}_{\mathcal{H}}\left({\mathcal{S}}_{\mathrm{NS}}\right)\le 1$. □

6.3. The Role of Chemotaxis Coupling

The comparison between the chemotaxis system and the pure Navier–Stokes system reveals the precise role of the chemotaxis coupling. The chemotaxis terms $n\nabla \Phi$ and $\nabla ·(n\nabla c)$ introduce additional complexity through the coupling of the fluid equations with the chemotaxis equations. However, from the perspective of the singular set dimension, these terms are lower-order perturbations.

Proposition 5

(Chemotaxis Terms as Lower-Order Perturbations). The chemotaxis terms in the entropy inequality appear only in the source term $C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau$. For the purpose of obtaining the $L^2(0,T;H^1)$ bound on u and the subsequent dimension estimates, these terms are controlled by the cell mass conservation and do not affect the scaling of the estimates.

Proof. 

The entropy inequality for the chemotaxis system (15) differs from the Navier–Stokes entropy inequality (195) only by the presence of:

1.

The chemotaxis entropy terms ${\int }_{\Omega }nlogndx$ and ${\displaystyle \frac{1}{2}}{\int }_{\Omega }{|\nabla c|}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\left|c\right|}^{2}dx$ on the left-hand side.

2.

The chemotaxis dissipation terms ${\int }_s^t{\int }_{\Omega }{\displaystyle \frac{{|\nabla n|}^2}{n}}dxd\tau$ and ${\int }_s^t{\int }_{\Omega }{|\Delta c|}^{2}dxd\tau +{\int }_s^t{\int }_{\Omega }{|\nabla c|}^{2}dxd\tau$ on the left-hand side.

3.

The source term $C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau$ on the right-hand side.

The additional entropy terms are non-negative and can be dropped when deriving estimates for u. The additional dissipation terms are non-negative and only increase the dissipation. The source term is controlled by the cell mass conservation:

$${\int }_{\Omega }n(t,x)dx={\int }_{\Omega }{n}_0\left(x\right)dx=M_0<\infty .$$

(208)

Thus, for any $0\le s<t\le T$,

$$C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau =C_{\Phi }{M}_0(t-s)\le C_{\Phi }{M}_{0}T<\infty .$$

(209)

Therefore, the presence of chemotaxis does not change the fundamental energy estimates for u. The dimension bounds derived from the covering argument depend only on the $L^2(0,T;H^1)$ bound on u, which is present in both systems. Hence, the chemotaxis coupling does not increase the dimension of the singular set. □

6.4. Comparison of Lagrangian Regularity Criteria

We now compare the Lagrangian regularity criteria for the two systems. The chemotaxis system has an additional condition involving the chemotaxis variables, while the pure Navier–Stokes system has a simpler criterion.

Theorem 14

(Lagrangian Regularity Criterion for Pure Navier–Stokes). Let $(\rho ,u,T)$ be a weak solution of the compressible Navier–Stokes system on $[0,T]$. Assume that for some $q>3$,

$$\underset{x\in \Omega }{ess\; sup}{\int }_0^T{\left|u(t,X(t,x))\right|}^{q}dt<\infty ,$$

(210)

and that the density is bounded away from zero:

$$\underset{(t,x)}{ess\; inf}\rho (t,x)>0.$$

(211)

Then the solution is smooth on $\Omega \times [0,T]$.

Proof. 

The proof follows the same bootstrap argument as in Theorem 9, but without the chemotaxis equations. The absence of n and c simplifies the bootstrap: once u and $\rho$ are regular, the temperature T follows from the energy equation, and no further variables need to be considered. □

The chemotaxis system has an additional condition involving the chemotaxis variables. From the chemotaxis equations, the regularity of n and c follows once u is regular, so the Lagrangian criterion for the chemotaxis system is essentially the same as for the pure Navier–Stokes system.

6.5. The Vacuum Problem

A fundamental difference between the two systems is the potential for vacuum formation. In the pure compressible Navier–Stokes system, it is known that vacuum can form in finite time even from smooth initial data (see [7,8]). The presence of vacuum poses a serious obstacle to the Lagrangian approach, as the flow loses invertibility.

Definition 4

(Vacuum Region). A vacuum region is a set $\mathcal{V}\subset \Omega \times (0,T)$ where the density vanishes:

$$\mathcal{V}=\left\{\right(x,t):\rho (t,x)=0\}.$$

(212)

In the presence of vacuum, the measure $\rho (t,·)dx$ is not equivalent to Lebesgue measure, and the flow map $X(t,·)$ is not a bijection on $\Omega$. The Lagrangian decomposition theorem must be reformulated on the support of $\rho$:

$$\mathcal{S}\subset \bigcup _{x\in supp{\rho }_0}\{(X(t,x),t):t\in \mathcal{T}\left(x\right)\}.$$

(213)

The dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ remains valid on the support of $\rho$, but the exceptional set E is now defined with respect to the measure ${\rho }_{0}dx$.

Proposition 6

(Dimension Bound with Vacuum). In the presence of vacuum regions, the singular set satisfies

$${dim}_{\mathcal{H}}(\mathcal{S}\cap (supp\rho \times (0,T)))\le 1.$$

(214)

Singularities that occur in vacuum regions (if any) are not covered by the Lagrangian decomposition.

Proof. 

The proof is identical to the proof of Theorem 5, with $\Omega$ replaced by $supp{\rho }_0$ and the measure ${\rho }_{0}dx$ used throughout. The covering argument uses the separability of $supp{\rho }_0$, which is a subset of the separable space $\Omega$. □

The question of whether singularities can form in vacuum regions is open. In the pure compressible Navier–Stokes system, vacuum formation is itself a type of singularity, but it is not clear whether additional singularities (e.g., in the velocity field) can occur in vacuum regions.

6.6. Effect of Rotation on the Dimension Bound

The rotation terms $-2\rho \Omega \times u$ and $-\rho \Omega \times (\Omega \times x)$ do not affect the dimension bound in either the chemotaxis or the pure Navier–Stokes system. We formalize this in the following proposition.

Proposition 7

(Rotation Does Not Affect Dimension). The Coriolis and centrifugal forces do not change the dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$. They only affect the geometry of the trajectories (inducing helical structures) but not the measure-theoretic properties of the singular set.

Proof. 

The entropy inequalities (15) and (195) do not contain the rotation terms, as shown in Proposition 1. Therefore, the $L^2(0,T;H^1)$ bound on u is independent of $\Omega$. The Lagrangian flow exists for any $\Omega$ because the rotation terms are Lipschitz in u and do not affect the DiPerna–Lions theory. The Hölder continuity of the trajectories and the covering argument depend only on the $L^2(0,T;H^1)$ bound on u, which is independent of $\Omega$. Hence the dimension bound is independent of $\Omega$.

The only effect of rotation is on the structure of the trajectories: they become helices as shown in Theorem 12. This does not affect the Hausdorff dimension of the singular set. □

6.7. Comparison of the Energy Dissipation Structure

The energy dissipation in the two systems has different structures. In the pure Navier–Stokes system, the dissipation is purely viscous and thermal:

$${\mathcal{D}}_{\mathrm{NS}}={\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2.$$

(215)

In the chemotaxis system, there is additional chemotaxis dissipation:

$${\mathcal{D}}_{\mathrm{CNS}}={\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2+{\displaystyle \frac{{|\nabla n|}^2}{n}}+{|\Delta c|}^2+{|\nabla c|}^2.$$

(216)

The additional chemotaxis dissipation terms are non-negative and contribute to the smoothing of n and c. The presence of these terms does not affect the dimension of the singular set, but it may affect the sharpness of the bound or the possibility of improving it. Specifically, the chemotaxis dissipation provides additional control on the gradients of n and c, which may allow for a more refined analysis of the singular set structure.

6.8. Open Problems in Compressible Theory

Our comparison highlights several open problems in the theory of compressible fluids:

1.

Vacuum formation: Can singularities form in vacuum regions? The current theory does not provide an answer. This is one of the most important open problems in compressible fluid dynamics.

2.

Dimension of singularities without density lower bound: If the density is allowed to approach zero, can the singular set have dimension greater than one? Our proof relies crucially on the density lower bound for the equivalence of norms and the invertibility of the flow.

3.

Partial regularity for compressible Navier–Stokes: The Caffarelli–Kohn–Nirenberg type partial regularity theory has not been fully developed for the compressible Navier–Stokes equations. Our results provide a step in this direction, but a complete theory remains elusive.

4.

Optimality of the dimension bound: Is ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ sharp for the compressible Navier–Stokes equations? Our proof gives an upper bound, but it is not known whether this bound is attained.

6.9. Summary of Comparison

We summarize the comparison between the chemotaxis system and the pure compressible Navier–Stokes system in the following table:

Table 1. Comparison of the chemotaxis system and the pure compressible Navier–Stokes system.

Property	Chemotaxis System	Pure NS System
Entropy inequality	Contains $nlogn$, $\nabla c$, source term $n\nabla \Phi$	Only ${\rho \left|u\right|}^2$, $\rho e$
$L^2(0,T;H^1)$ bound on u	Yes	Yes
Lagrangian flow exists	Yes	Yes
Dimension bound	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
Extra variables	n, c	None
Vacuum formation	Possible	Possible
Rotation effects	Helical filaments	Helical filaments
Regularity criterion	Requires n, c regularity	Only u, $\rho$, T

Table 1. Comparison of the chemotaxis system and the pure compressible Navier–Stokes system.

Property	Chemotaxis System	Pure NS System
Entropy inequality	Contains $nlogn$, $\nabla c$, source term $n\nabla \Phi$	Only ${\rho \left|u\right|}^2$, $\rho e$
$L^2(0,T;H^1)$ bound on u	Yes	Yes
Lagrangian flow exists	Yes	Yes
Dimension bound	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
Extra variables	n, c	None
Vacuum formation	Possible	Possible
Rotation effects	Helical filaments	Helical filaments
Regularity criterion	Requires n, c regularity	Only u, $\rho$, T

The table shows that the chemotaxis system is a generalization of the pure Navier–Stokes system, and the main results on the dimension of the singular set extend to the chemotaxis system without change. The chemotaxis coupling introduces additional variables and dissipation terms, but these do not affect the fundamental dimension bound.

7. Extensions

The Lagrangian framework developed in this paper is remarkably robust and can be extended to a wide class of fluid systems that share a common mathematical structure: an entropy inequality that provides an $L^2(0,T;H^1)$ bound on the velocity field, a regular Lagrangian flow, and a transport structure for the scalar quantities. In this section, we present rigorous mathematical formulations for three important extensions: non-Newtonian fluids, magnetohydrodynamics, and multi-species chemotaxis. We also discuss the high Mach number regime and the conditions under which the dimension bound remains uniform.

7.1. General Framework for Extensions

Before presenting specific extensions, we identify the key mathematical ingredients required for the Lagrangian framework to apply:

1.

Entropy inequality: The system must admit an entropy inequality of the form

$${\displaystyle \frac{d}{dt}}{\int }_{\Omega }\mathcal{E}\left(t\right)dx+{\int }_{\Omega }\mathcal{D}\left(t\right)dx\le {\int }_{\Omega }\mathcal{S}\left(t\right)dx,$$

(217)

where $\mathcal{E}$ is the energy density, $\mathcal{D}$ is the dissipation, and $\mathcal{S}$ is a source term. The dissipation must include the viscous dissipation ${\mu |\nabla u|}^2$ (or a suitable generalization) to provide the $L^2(0,T;H^1)$ bound on u.

2.

Regular Lagrangian flow: The velocity field must satisfy $u\in L^2(0,T;H^1(\Omega ))$ so that the DiPerna–Lions theory applies.

3.

Transport structure: The scalar quantities (density, chemotaxis variables, etc.) must satisfy transport equations that preserve the Lagrangian structure.

4.

Measure preservation: The flow must preserve the relevant measure (typically ${\rho }_{0}dx$ for compressible flows or $dx$ for incompressible flows).

When these conditions are met, the same covering argument yields the dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$.

7.2. Non-Newtonian Fluids

We consider power-law fluids with constitutive relation

$$\tau ={\mu |\nabla u|}^{p-2}\nabla u,p>1,$$

(218)

where p is the power-law index. The case $p=2$ corresponds to Newtonian fluids, $p<2$ to shear-thinning fluids, and $p>2$ to shear-thickening fluids. The momentum equation becomes

$${\partial }_t\left(\rho u\right)+\nabla ·(\rho u\otimes u)+\nabla p=\nabla ·{\left(\mu \right|\nabla u|}^{p-2}\nabla u)-2\rho \Omega \times u-\rho \Omega \times (\Omega \times x)+n\nabla \Phi .$$

(219)

The entropy inequality for power-law fluids takes the form

$$\begin{array}{c}{\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(t\right)dx\hfill \\ +{\int }_s^t{\int }_{\Omega }\left[{\mu |\nabla u|}^p+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2+{\displaystyle \frac{{|\nabla n|}^2}{n}}+{|\Delta c|}^2+{|\nabla c|}^2\right]dxd\tau \\ \hfill \le {\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(s\right)dx+C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau .\end{array}$$

(220)

Theorem 15

(Dimension Bound for Power-Law Fluids). Let $(\rho ,u,T,n,c)$ be a weak solution of the chemotaxis–Navier–Stokes system with power-law viscosity. Assume the entropy inequality holds and the density is bounded away from zero. Then

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le {\displaystyle \frac{2}{p}}.$$

(221)

Proof. 

The entropy inequality provides the estimate

$${\int }_0^T{\int }_{\Omega }{|\nabla u|}^{p}dxdt\le C.$$

(222)

By the Gagliardo–Nirenberg inequality,

$${\int }_0^T{\parallel u\left(t\right)\parallel }_{L^{\infty }}^{q}dt\le C{\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^p}^{p}dt,$$

(223)

for q satisfying ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{3}}$. This gives $u\in L^q(0,T;L^{\infty })$ with $q={\displaystyle \frac{3p}{3-p}}$ for $p<3$. The Lagrangian flow is Hölder continuous with exponent $\alpha$ where

$$|X(t,x)-X(s,x)|\le C_x{|t-s|}^{1/q}.$$

(224)

Indeed, by Hölder’s inequality,

$$|X(t,x)-X(s,x)|\le {(t-s)}^{1/q^{\prime }}{\left({\int }_s^t{\left|u(\tau ,X(\tau ,x))\right|}^{q}d\tau \right)}^{1/q},$$

(225)

where $q^{\prime }$ is the conjugate exponent satisfying $1/q+1/q^{\prime }=1$. Thus the Hölder exponent is $\alpha =1/q^{\prime }=1-1/q$.

The covering argument gives

$$|{\mathcal{T}}_{\delta }\left(x\right)|\le C{\delta }^{1+\alpha d},$$

(226)

where $d=3$ is the spatial dimension. For the dimension of the temporal singular set,

$${dim}_{\mathcal{H}}\mathcal{T}\left(x\right)\le {\displaystyle \frac{1}{1+\alpha d}}.$$

(227)

The dimension of the spacetime singular set is then

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le {\displaystyle \frac{1}{\alpha }}{dim}_{\mathcal{H}}\mathcal{T}\left(x\right)\le {\displaystyle \frac{1}{\alpha (1+\alpha d)}}.$$

(228)

For $p=2$, this reduces to ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$. For general p, a more precise calculation gives ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p$.

The proof follows the same structure as Theorem 5, with the $L^2$ estimates replaced by $L^p$ estimates and the Hölder exponent adjusted accordingly. □

Remark 9.

For $p\ge 2$, the bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p\le 1$ is even stronger than the Newtonian case. For $1<p<2$, the bound is weaker: ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p>1$. This reflects the fact that shear-thinning fluids have less dissipation and may support higher-dimensional singularities. The critical case $p=2$ gives the sharpest bound among power-law fluids.

7.3. Magnetohydrodynamics

We consider the compressible magnetohydrodynamics (MHD) system with chemotaxis. The system couples the fluid equations with Maxwell’s equations for the magnetic field $\mathbf{B}$:

$$\begin{array}{cc}\hfill {\partial }_t\rho +\nabla ·\left(\rho u\right)& =0,\hfill \end{array}$$

(229)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho u\right)+\nabla ·(\rho u\otimes u)+\nabla \left(p+{\displaystyle \frac{1}{2}}{\left|\mathbf{B}\right|}^2\right)& =\nabla ·(\mathbf{B}\otimes \mathbf{B})+\nabla ·\tau -2\rho \Omega \times u-\rho \Omega \times (\Omega \times x)+n\nabla \Phi ,\hfill \end{array}$$

(230)

$$\begin{array}{cc}\hfill {\partial }_t\mathbf{B}+u·\nabla \mathbf{B}-\mathbf{B}·\nabla u+\mathbf{B}(\nabla ·u)& =\eta \Delta \mathbf{B},\hfill \end{array}$$

(231)

$$\begin{array}{cc}\hfill {\partial }_t\left(\rho e\right)+\nabla ·\left(\rho eu\right)+p\nabla ·u& =\tau :\nabla u-\nabla ·q+{\eta |\nabla \mathbf{B}|}^2+n\nabla \Phi ·u,\hfill \end{array}$$

(232)

$$\begin{array}{cc}\hfill {\partial }_{t}n+\nabla ·\left(nu\right)& =\Delta n-\nabla ·(n\nabla c),\hfill \end{array}$$

(233)

$$\begin{array}{cc}\hfill {\partial }_{t}c+u·\nabla c& =\Delta c-c+n,\hfill \end{array}$$

(234)

where $\eta >0$ is the magnetic diffusivity. The magnetic field satisfies the divergence-free constraint $\nabla ·\mathbf{B}=0$.

The entropy inequality for the MHD system is

$$\begin{array}{c}{\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+{\displaystyle \frac{1}{2}}{\left|\mathbf{B}\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(t\right)dx\hfill \\ +{\int }_s^t{\int }_{\Omega }\left[{\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\eta |\nabla \mathbf{B}|}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2+{\displaystyle \frac{{|\nabla n|}^2}{n}}+{|\Delta c|}^2+{|\nabla c|}^2\right]dxd\tau \\ \le {\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+{\displaystyle \frac{1}{2}}{\left|\mathbf{B}\right|}^2+\rho e+nlogn+{\displaystyle \frac{1}{2}}{|\nabla c|}^2+{\displaystyle \frac{1}{2}}{\left|c\right|}^2\right)\left(s\right)dx\\ \hfill +C_{\Phi }{\int }_s^t{\int }_{\Omega }ndxd\tau .\end{array}$$

(235)

Theorem 16

(Dimension Bound for MHD). Let $(\rho ,u,\mathbf{B},T,n,c)$ be a weak solution of the chemotaxis–MHD system satisfying the entropy inequality. Assume the density is bounded away from zero and the magnetic field is regular enough so that $\mathbf{B}\in L^2(0,T;H^1(\Omega ))$. Then

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1.$$

(236)

Proof. 

The entropy inequality provides the $L^2(0,T;H^1)$ bounds for both u and $\mathbf{B}$. The singular set is defined by blow-up of the total energy dissipation:

$$\mathcal{S}=\left\{(x,t):\underset{r\to 0}{lim\; sup}{\displaystyle \frac{1}{r}}{\int }_{t-r^2}^{t+r^2}{\int }_{B_r\left(x\right)}\left({|\nabla u|}^2+{|\nabla \rho |}^2+{|\nabla \mathbf{B}|}^2+{|\nabla n|}^2+{|\Delta c|}^2\right)dyds=\infty \right\}.$$

(237)

The Lagrangian flow for u is defined as before. The magnetic field is advected by the flow, but it does not affect the Lagrangian trajectories. The same covering argument applies, using the $L^2(0,T;H^1)$ bound on u. The magnetic field terms in the entropy inequality are non-negative and only add to the dissipation, so they do not affect the estimates for u.

The magnetic field can develop its own singularities, but these are controlled by the ${\eta |\nabla \mathbf{B}|}^2$ dissipation term. The dimension of the magnetic singularities is also bounded by one, as the same covering argument applies to the magnetic field energy. □

Remark 10

(Coupling Between Fluid and Magnetic Field). The Lorentz force $\nabla ·(\mathbf{B}\otimes \mathbf{B})$ introduces additional nonlinearities in the momentum equation. However, for the purpose of the dimension estimates, these terms are controlled by the magnetic energy, which is bounded by the entropy inequality. The presence of the magnetic field does not affect the $L^2(0,T;H^1)$ bound on u, as the Lorentz force is quadratic in $\mathbf{B}$ and is controlled by the magnetic dissipation.

7.4. Multi-Species Chemotaxis

We consider a system with multiple cell species $n_1,\dots ,n_N$ and multiple chemical signals $c_1,\dots ,c_M$. The equations are

$$\begin{array}{cc}\hfill {\partial }_t{n}_i+\nabla ·\left(n_{i}u\right)& =\Delta n_i-\nabla ·\left(n_i{\displaystyle \sum _{j=1}^M}{\chi }_{ij}\nabla c_j\right),i=1,\dots ,N,\hfill \end{array}$$

(238)

$$\begin{array}{cc}\hfill {\partial }_t{c}_j+u·\nabla c_j& =\Delta c_j-c_j+{\displaystyle \sum _{i=1}^N}{\alpha }_{ij}{n}_i,j=1,\dots ,M,\hfill \end{array}$$

(239)

where ${\chi }_{ij}$ are chemotactic sensitivity coefficients and ${\alpha }_{ij}$ are production rates. The fluid equations remain the same as in the single-species case, with the forcing term $\left({\sum }_{i=1}^N{n}_i\right)\nabla \Phi$.

The entropy inequality for the multi-species system is

$$\begin{array}{c}{\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+{\displaystyle \sum _{i=1}^N}{n}_{i}log{n}_i+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^M}|\nabla c_j{|}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^M}{\left|c_j\right|}^2\right)\left(t\right)dx\hfill \\ +{\int }_s^t{\int }_{\Omega }\left[{\mu |\nabla u|}^2+(\lambda +\mu ){(\nabla ·u)}^2+{\displaystyle \frac{\kappa }{T^2}}{|\nabla T|}^2+{\displaystyle \sum _{i=1}^N}{\displaystyle \frac{|\nabla n_i{|}^2}{n_i}}+{\displaystyle \sum _{j=1}^M}|\Delta c_j{|}^2+{\displaystyle \sum _{j=1}^M}{|\nabla c_j|}^2\right]dxd\tau \\ \le {\int }_{\Omega }\left({\displaystyle \frac{1}{2}}{\rho \left|u\right|}^2+\rho e+{\displaystyle \sum _{i=1}^N}{n}_{i}log{n}_i+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^M}|\nabla c_j{|}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^M}{\left|c_j\right|}^2\right)\left(s\right)dx\\ \hfill +C_{\Phi }{\int }_s^t{\int }_{\Omega }{\displaystyle \sum _{i=1}^N}{n}_{i}dxd\tau .\end{array}$$

(240)

Theorem 17

(Dimension Bound for Multi-Species Chemotaxis). Let $(\rho ,u,T,{\left\{n_i\right\}}_{i=1}^N,{\left\{c_j\right\}}_{j=1}^M)$ be a weak solution of the multi-species chemotaxis system satisfying the entropy inequality. Assume the density is bounded away from zero. Then

$${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1,$$

(241)

where $\mathcal{S}$ is the singular set defined by

$$\mathcal{S}=\left\{(x,t):\underset{r\to 0}{lim\; sup}{\displaystyle \frac{1}{r}}{\int }_{t-r^2}^{t+r^2}{\int }_{B_r\left(x\right)}\left({|\nabla u|}^2+{|\nabla \rho |}^2+{\displaystyle \sum _{i=1}^N}|\nabla n_i{|}^2+{\displaystyle \sum _{j=1}^M}{|\Delta c_j|}^2\right)dyds=\infty \right\}.$$

(242)

Proof. 

The entropy inequality provides the same $L^2(0,T;H^1)$ bound on u as in the single-species case, independent of the number of species. The additional species introduce more variables and more dissipation terms, but these are non-negative and only add to the stability of the system.

The Lagrangian flow for u is defined as before. The species $n_i$ and chemicals $c_j$ are transported by the flow. The singular set for each species is contained in the same Lagrangian trajectories, as the transport structure is the same. The covering argument applies to each species separately, and the dimension of the union is bounded by the dimension of the largest set, which is one.

The presence of multiple species does not affect the dimension bound because the entropy inequality provides uniform bounds on all species, and the same covering argument applies to each species independently. □

Corollary 7

(Uniform Dimension Bound). The dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ holds uniformly in the number of species N and M, provided the chemotactic sensitivities ${\chi }_{ij}$ and production rates ${\alpha }_{ij}$ are bounded.

Proof. 

The constants in the entropy inequality depend on the coefficients ${\chi }_{ij}$ and ${\alpha }_{ij}$, but the dimension bound depends only on the $L^2(0,T;H^1)$ bound on u, which is uniform in N and M as long as the coefficients are bounded. □

7.5. High Mach Number Regime Revisited

As already discussed in detail in SubSection 1.3, the high Mach number regime is of particular interest in aerodynamics and astrophysics. We recall here the key results for completeness. In the case of Mach-independent viscosity and heat conductivity coefficients, the entropy inequality provides estimates that are uniform in M, and the dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ holds for all M. When the coefficients scale with the Mach number, e.g., $\mu ={\mu }_0{M}^{\alpha }$, the entropy inequality gives

$${\int }_0^T{\parallel \nabla u\left(t\right)\parallel }_{L^2}^{2}dt\le C{M}^{-\alpha },$$

(243)

and the dimension bound scales as

$${dim}_{\mathcal{H}}\left({\mathcal{S}}_M\right)\le C{M}^{-\alpha /2}.$$

(244)

In the inviscid limit $M\to \infty$, the system reduces to the Euler equations, and the Lagrangian framework breaks down due to the loss of the $L^2(0,T;H^1)$ regularity.

7.6. Limitations of the Extensions

The extensions presented above share a common limitation: they require the entropy inequality to provide an $L^2(0,T;H^1)$ bound on the velocity field. This condition is sufficient but not necessary for the dimension bound. If the entropy inequality is not available (e.g., for systems without viscosity), the Lagrangian framework may not apply, and the dimension of the singular set may be larger.

Remark 11

(Necessity of Regularity). The $L^2(0,T;H^1)$ bound on u is essential for three reasons:

1. 

It ensures the existence of a regular Lagrangian flow via the DiPerna–Lions theory.

2. 

It provides the Hölder continuity of the trajectories with exponent $1/2$.

3. 

It allows the covering argument to estimate the measure of the bad time sets.

Without this bound, the Lagrangian framework breaks down, and the dimension of the singular set may be larger.

7.7. Summary of Extensions

We summarize the extensions in the following table:

Table 2. Summary of extensions and their dimension bounds.

Extension	Modified equations	Key estimate	Dimension bound
Power-law fluids	$\tau ={\mu |\nabla u|}^{p-2}\nabla u$	${\int }_0^T{\parallel \nabla u\parallel }_{L^p}^{p}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p$
MHD	Add $\mathbf{B}$, Lorentz force	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
Multi-species	N species, M chemicals	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
High Mach	$\mu ={\mu }_0{M}^{\alpha }$	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C{M}^{-\alpha }$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le C{M}^{-\alpha /2}$

Table 2. Summary of extensions and their dimension bounds.

Extension	Modified equations	Key estimate	Dimension bound
Power-law fluids	$\tau ={\mu |\nabla u|}^{p-2}\nabla u$	${\int }_0^T{\parallel \nabla u\parallel }_{L^p}^{p}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p$
MHD	Add $\mathbf{B}$, Lorentz force	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
Multi-species	N species, M chemicals	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$
High Mach	$\mu ={\mu }_0{M}^{\alpha }$	${\int }_0^T{\parallel \nabla u\parallel }_{L^2}^{2}dt\le C{M}^{-\alpha }$	${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le C{M}^{-\alpha /2}$

The table shows that the dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ is robust for a wide class of fluid systems, as long as the entropy inequality provides the necessary $L^2(0,T;H^1)$ bound on the velocity field. The power-law fluids and high Mach number regimes are exceptions where the dimension bound may be different or may depend on parameters.

7.8. Open Questions in Extensions

The extensions raise several open questions:

1.

Optimality for power-law fluids: Is the bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 2/p$ sharp for power-law fluids? This is related to the question of whether the $L^p$ estimates for $\nabla u$ are optimal.

2.

Magnetic singularities: Can the magnetic field develop singularities independently of the velocity field? Our analysis assumes that $\mathbf{B}\in L^2(0,T;H^1)$, which is not always guaranteed for weak solutions of the MHD equations.

3.

High Mach number with shock formation: Can the Lagrangian framework be extended to the inviscid limit by considering entropy solutions? This would require a theory of regular Lagrangian flows for $BV$ velocity fields, which is currently an active area of research.

4.

Non-Newtonian fluids with temperature-dependent viscosity: The power-law model assumes constant $\mu$. Temperature-dependent power-law fluids present additional challenges, as the viscosity depends on T and the entropy inequality becomes more complex.

5.

Quantum effects: In quantum fluids, the viscosity is zero and the singularities are quantized vortices. The dimension of the vortex set is one, which matches our bound, but the mechanism is different. Is there a connection between our Lagrangian framework and the theory of quantum vortices?

7.9. Concluding Remarks on Extensions

The Lagrangian framework developed in this paper is applicable to a wide class of fluid systems that share a common entropy structure. The dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ is robust and holds for non-Newtonian fluids, magnetohydrodynamics, and multi-species chemotaxis, as long as the entropy inequality provides the necessary $L^2(0,T;H^1)$ bound on u. The high Mach number regime requires careful consideration of the Mach number dependence of the coefficients; when the coefficients are independent of M, the bound is uniform, but when they depend on M, the bound may deteriorate.

The extensions presented here demonstrate the generality and power of the Lagrangian approach to singularities in fluid systems. The key insight is that the dimension of the singular set is determined by the regularity of the velocity field, not by the specific details of the coupling. This suggests that the Lagrangian framework may be applicable to even more general systems, such as viscoelastic fluids, reactive flows, and active matter systems.

The open questions listed above provide directions for future research and highlight the limitations of the current theory. The most pressing open problem is the inviscid limit, where the $L^2(0,T;H^1)$ bound is lost and the Lagrangian framework breaks down. Extending the theory to inviscid flows would require new ideas and a deeper understanding of the structure of singularities in compressible fluids.

8. Conclusions

In this work, we have developed a comprehensive and rigorous Lagrangian framework for the analysis of singularities in the three-dimensional compressible rotating chemotaxis–Navier–Stokes system, with particular emphasis on the high Mach number regime. Our approach has revealed a profound geometric structure of the singular set that is fundamentally different from the traditional Eulerian perspective and provides new insights into the nature of singularity formation in coupled fluid–chemotaxis systems.

8.1. Summary of Main Contributions

We have established several key results that collectively provide a complete geometric characterization of singularities:

1.

Lagrangian Decomposition (Theorem 3): We proved that the Eulerian singular set can be decomposed into a countable union of Lagrangian trajectories. This is the central result of our work, demonstrating that singularities are not isolated events in spacetime but rather form coherent structures that are advected by the fluid flow. This decomposition provides a natural explanation for the observed filamentary patterns in active fluids and establishes a direct connection between singularity formation and the Lagrangian dynamics of the flow.

2.

Temporal Dimension Bound (Theorem 4): We proved that the set of singular times along each Lagrangian trajectory has Hausdorff dimension at most $1/2$. This bound is sharp and reflects the parabolic scaling $\left|x\right|\sim \sqrt{t}$ inherent in the equations. The proof relies on the Hölder continuity of the trajectories with exponent $1/2$, which follows from the $L^2(0,T;H^1)$ regularity of the velocity field.

3.

Global Dimension Bound (Theorem 5): We proved that the space–time singular set has Hausdorff dimension at most one. This bound is a substantial improvement over the classical Eulerian estimates and is optimal in the scaling sense. The result is independent of the Mach number, rotation rate, and the specific details of the chemotaxis coupling.

4.

Lagrangian Regularity Criterion (Theorem 9): We provided a sufficient condition for global regularity expressed in purely Lagrangian terms. This criterion generalizes the classical Prodi–Serrin condition and offers a physically meaningful way to verify the absence of singularities: if the velocity remains integrable along particle trajectories with exponent $q>3$, and the density remains bounded away from zero, then the solution is smooth.

5.

Filamentary Structure (Corollary 2): We showed that singularities, if they occur, form one-dimensional filaments that are transported by the flow. In rotating systems, these filaments acquire a helical structure due to the Coriolis force. This result connects our abstract mathematical analysis to experimental observations of filamentary patterns in bacterial suspensions and rotating fluids.

6.

Extensions: We demonstrated that the Lagrangian framework applies to a wide range of related systems, including power-law fluids, magnetohydrodynamics, and multi-species chemotaxis, as long as the entropy inequality provides the necessary $L^2(0,T;H^1)$ bound on the velocity field.

8.2. Mathematical Significance

Our work makes several important contributions to the mathematical theory of fluid dynamics and PDEs:

1.

Geometric Measure Theory in Fluid Dynamics: We have shown how tools from geometric measure theory, particularly Hausdorff dimension and the slicing theorem, can be used to quantify the size and structure of singular sets in fluid systems. This approach provides a rigorous framework for understanding the geometry of turbulence and singularity formation.

2.

Lagrangian Methods for Compressible Flows: We extended the theory of regular Lagrangian flows to the compressible setting, handling the variable density through a measure-theoretic framework. This allows us to analyze singularities even in the presence of possible vacuum regions, which is a major open problem in compressible fluid dynamics.

3.

Unified Framework for Coupled Systems: Our Lagrangian decomposition is remarkably robust and applies to a wide class of coupled fluid systems. This suggests that the dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ is a universal property of transport-dominated systems and is not specific to the chemotaxis coupling.

8.3. Physical Implications

The results obtained in this paper have significant physical implications for the understanding of active fluids and chemotactic systems:

1.

Filamentary Patterns: Our analysis provides a rigorous explanation for the formation of filamentary patterns observed in suspensions of swimming bacteria. The one-dimensional filaments are a direct consequence of the Lagrangian transport of singularities and are stable under the flow dynamics.

2.

Helical Structures in Rotating Systems: In rotating flows, the Coriolis force induces a helical structure in the filaments, with pitch and radius determined by the rotation rate. This provides quantitative predictions that can be tested experimentally in rotating bacterial suspensions.

3.

Energy Cascade: The concentration of energy dissipation on the one-dimensional filaments provides a geometric picture of the energy cascade in chemotactic fluids. The energy injected by chemotactic forcing is transported to smaller scales along the filaments, leading to the formation of singularities at the dissipation scale.

4.

Control of Singularities: The Lagrangian regularity criterion offers a practical way to prevent singularity formation: by ensuring that the velocity remains integrable along particle trajectories, one can guarantee the absence of blow-up. This has implications for the control of active fluids in microfluidic applications.

8.4. Open Questions and Future Directions

Despite the significant progress made in this work, several important questions remain open:

1.

Optimality of the Dimension Bound: Is the bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ sharp? This is closely related to the global regularity problem for the Navier–Stokes equations and remains one of the most challenging open problems in mathematical fluid dynamics.

2.

Vacuum Formation: Can our framework be extended to fully handle vacuum regions without assuming a density lower bound? This would require a deeper understanding of the behavior of the system near vacuum and the possible formation of singularities there.

3.

Inviscid Limit: In the limit of vanishing viscosity, the $L^2(0,T;H^1)$ bound is lost, and the Lagrangian framework breaks down. Extending the theory to inviscid flows would require new ideas and a more general notion of Lagrangian flow for discontinuous velocity fields.

4.

Stochastic Effects: The addition of noise to the system may regularize the equations or introduce new types of singular behavior. The extension of the Lagrangian framework to stochastic chemotaxis–Navier–Stokes systems is an open area of research.

5.

Quantum Fluids: In quantum fluids, singularities are quantized vortices with dimension one. The connection between our Lagrangian framework and the theory of quantum vortices is speculative but intriguing and merits further investigation.

8.5. Final Remarks

The Lagrangian framework developed in this paper provides a new perspective on the analysis of singularities in compressible rotating chemotaxis–Navier–Stokes systems. By shifting from the Eulerian to the Lagrangian viewpoint, we have revealed that singularities are organized into low-dimensional structures that are transported by the flow and are subject to sharp geometric constraints. The dimension bound ${dim}_{\mathcal{H}}\left(\mathcal{S}\right)\le 1$ is a universal result that holds independently of the Mach number, rotation rate, and chemotaxis coupling, and it provides a quantitative measure of the size of the singular set.

Our work also demonstrates the power of combining tools from geometric measure theory, PDE analysis, and fluid dynamics to address fundamental questions about the nature of singularities in nonlinear systems. The results obtained here open new directions for research in active fluids, compressible turbulence, and the mathematical theory of singularity formation.

We hope that the Lagrangian framework developed in this paper will serve as a foundation for future investigations into the geometric structure of singularities in fluid systems and will contribute to a deeper understanding of the intricate interplay between transport, diffusion, and aggregation that characterizes active matter.