Thermo-Acoustic Stabilization and Continuation Structure in Admissible Compressible Navier–Stokes–Fourier Dynamics

1. Introduction

The regularity problem for three-dimensional viscous fluid equations remains a central issue in mathematical fluid dynamics. For the incompressible Navier–Stokes equations, the existence of Leray–Hopf weak solutions is classical, while the exclusion of finite-time singularity formation for general three-dimensional flows remains open [1,2]. Partial regularity theory, scale-critical continuation criteria, vorticity-based criteria, and harmonic-analytic approaches have clarified several mechanisms of possible breakdown [3,4,5,6]. Nevertheless, these approaches are formulated within a mechanically closed system consisting of transport, pressure redistribution, and viscous diffusion. In that formulation, no thermodynamic variable, entropy-production mechanism, thermal relaxation process, or finite-speed acoustic redistribution remains available as an independent stabilizing structure.

The present paper studies this issue from the viewpoint of the compressible Navier–Stokes–Fourier system. The complete thermo-acoustic state is denoted by $X=(\rho ,u,T,s,p,c_s)$, where $\rho$ is the mass density, $u$ is the velocity field, $T$ is the absolute temperature, $s$ is the entropy density, $p$ is the thermodynamic pressure, and $c_s$ is the local sound speed. The analysis is restricted to physically admissible thermo-acoustic regimes satisfying positivity of density and temperature, entropy admissibility, free-energy dissipation, finite acoustic propagation, constitutive smoothness, finite-energy bounds, and a uniformly subsonic condition $M<1$, where $M=\mid u\mid /c_s$ is the local Mach number. These conditions are not introduced as artificial regularity assumptions. They express the physical regime in which vacuum, temperature collapse, sonic degeneration, shock formation, and loss of thermodynamic causality are excluded.

The main objective is to prove that, within such a physically admissible compressible Navier–Stokes–Fourier framework, scale-critical concentration capable of sustaining singularity formation is constrained by thermo-acoustic dissipation. The central dissipative quantity is the entropy-production density $D_{\mathrm{e}\mathrm{n}\mathrm{t}}$, which controls viscous deformation and thermal gradients through the irreversible thermodynamic structure of the system. The localized concentration quantity $H(z,r)$, defined at a spacetime point $z$ and radius $r$, measures the scale-critical accumulation of $D_{\mathrm{e}\mathrm{n}\mathrm{t}}$ over the parabolic cylinder $Q_r\left(z\right)$. The role of $H(z,r)$ is analogous to concentration quantities in partial regularity theory, but it is intrinsically thermodynamic rather than purely mechanical.

The analysis proceeds from a finite-energy thermo-acoustic weak framework satisfying the admissibility conditions introduced above. The subsequent argument develops localized entropy coercivity, nonlinear subcriticality, higher integrability, and ε-regularity through the thermo-acoustic localization and compactness structures summarized below. The intended chain of implication is

$$\begin{gathered} \mathrm{thermo}-\mathrm{acoustic}\mathrm{admissibility}\Rightarrow \mathrm{entropy}\mathrm{coercivity}\Rightarrow \mathrm{nonlinear}\mathrm{subcriticality} \\ \Rightarrow \mathrm{Meyers}-\mathrm{type}\mathrm{gain}\Rightarrow \mathrm{Campanato}\mathrm{decay}\Rightarrow \epsilon -\mathrm{regularity} \\ \Rightarrow \mathrm{strong}\mathrm{continuation}. \end{gathered}$$

Here, entropy coercivity means that $D_{\mathrm{e}\mathrm{n}\mathrm{t}}$ provides localized control of the symmetric velocity gradient and the thermal gradient; nonlinear subcriticality means that transport, thermal, acoustic, pressure, coefficient, and commutator remainders are strictly lower than the entropy-critical scaling; the Meyers-type step gives higher integrability of thermo-acoustic gradients; and the Campanato step converts this higher integrability into oscillation decay and local regularity. Young’s inequality, localized Korn inequalities, parabolic compactness, Meyers-type higher-integrability arguments, and Campanato embedding are used at the points where they are mathematically required [7,8,9,10,11].

A second purpose of the paper is to examine what is lost under incompressible projection. Let $Y=(u,\pi )$ denote the mechanically projected incompressible state, where $\pi$ is the incompressible pressure. The incompressible projection removes the thermodynamic completion variables $Z=(\rho ,T,s,c_s)$. The paper formulates this loss through projection fibers, conditional measures, Rokhlin disintegration, and the Doob–Dynkin reconstruction principle. The resulting statement is not that incompressible singularities must occur. Rather, it is that entropy-generating thermo-acoustic dissipation is not generally reconstructible from the projected variables $Y$ alone. Thus, incompressible projection removes part of the irreversible operator structure responsible for entropy generation, thermal diffusion, finite-speed acoustic redistribution, and free-energy relaxation.

The paper proves a conditional thermo-acoustic regularity mechanism within the stated admissible physical regime. It does not prove unconditional global regularity for arbitrary compressible Navier–Stokes–Fourier solutions, does not construct global finite-energy weak solutions from arbitrary data, and does not resolve the classical incompressible Navier–Stokes regularity problem. What is proved is the following structural assertion: if the physical thermo-acoustic admissibility conditions persist, then scale-critical concentration is forced into a dissipative framework governed by entropy production, and the resulting thermo-acoustic structure yields regularity continuation. What is not assumed is precisely the target regularity itself.

2. Physical Thermo-Acoustic Admissibility

2.1. Compressible NSF System

Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be either a smooth bounded domain or the whole space, and let $I=(0,T_f)$ denote the evolution interval. The complete thermo-acoustic state is denoted by

$\begin{array}{cccc}& X=(\rho ,u,T,s,p,c_s),& & \left(1\right)\end{array}$

where $\rho$ is the mass density, $u$ is the velocity field, $T$ is the absolute temperature, $s$ is the entropy density, $p$ is the thermodynamic pressure, and $c_s$ is the local sound speed.

The compressible Navier–Stokes–Fourier system consists of the continuity equation

$\begin{array}{cccc}& {\partial }_t\rho +\nabla \cdot \left(\rho u\right)=0,& & \left(2\right)\end{array}$

the momentum equation

$\begin{array}{cccc}& {\partial }_t\left(\rho u\right)+\nabla \cdot (\rho u\otimes u)+\nabla p=\nabla \cdot \sigma ,& & \left(3\right)\end{array}$

and the thermal equation

$\begin{array}{cccc}& \rho c_v({\partial }_{t}T+u\cdot \nabla T)-\nabla \cdot (\kappa \nabla T)=\sigma :\nabla u-p\nabla \cdot u.& & \left(4\right)\end{array}$

Here, $c_v$ denotes the specific heat capacity, $\kappa$ is the thermal conductivity coefficient, and $\sigma$ is the viscous stress tensor. The constitutive relations are written as

$\begin{array}{cccc}& p=p(\rho ,T),e=e(\rho ,T),s=s(\rho ,T),& & \left(5\right)\end{array}$

where $e$ denotes the specific internal energy. The viscous stress tensor is defined by

$\begin{array}{cccc}& \sigma =2\mu D\left(u\right)+\lambda (\nabla \cdot u)I,& & \left(6\right)\end{array}$

where $\mu$ and $\lambda$ are the shear and bulk viscosity coefficients, respectively, $I$ is the identity tensor, and

$\begin{array}{cccc}& D\left(u\right)={\displaystyle \frac{1}{2}}(\nabla u+\nabla u^{\mathsf{T}})& & \left(7\right)\end{array}$

is the symmetric deformation tensor. The viscosity coefficients satisfy the ellipticity conditions

$\begin{array}{cccc}& \mu >0,2\mu +3\lambda >0.& & \left(8\right)\end{array}$

The compressible NSF formulation employed here follows the standard thermodynamic viscous-fluid framework developed in Lions [7], Feireisl [8], and Feireisl–Novotný [9].

2.2. Thermodynamic Structure

The present theory is centered on irreversible thermo-acoustic dissipation generated by entropy production. The entropy-production density is defined by

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}={\displaystyle \frac{1}{T}}\sigma :\nabla u+\kappa {\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}.& & \left(9\right)\end{array}$

Using the symmetry of $D\left(u\right)$, one obtains

$\begin{array}{cccc}& \sigma :\nabla u=2\mu \mid D\left(u\right){\mid }^2+\lambda (\nabla \cdot u{)}^2.& & \left(10\right)\end{array}$

Hence,

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}={\displaystyle \frac{2\mu }{T}}\mid D\left(u\right){\mid }^2+{\displaystyle \frac{\lambda }{T}}(\nabla \cdot u{)}^2+\kappa {\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}.\left(11\right)$$

The admissible thermo-acoustic framework assumes

$$0<T_{\ast }\le T\le T^{\ast }<\mathrm{\infty },\left(12\right)$$

together with

$$0<{\mu }_{\ast }\le \mu ,0<{\kappa }_{\ast }\le \kappa ,2\mu +3\lambda \ge 0.\left(13\right)$$

Under assumptions (12) and (13), there exists a constant  $c_{\mathrm{e}\mathrm{n}\mathrm{t}}>0$ such that

$$D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge c_{\mathrm{e}\mathrm{n}\mathrm{t}}\left(\mid D\left(u\right){\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right).\left(14\right)$$

Here,

$\begin{array}{cccc}& \nabla \mathrm{l}\mathrm{o}\mathrm{g}T={\displaystyle \frac{\nabla T}{T}},& & \left(15\right)\end{array}$

which is well-defined because of the positivity condition (12).

Proposition 1 (Entropy-Coercive Structure).  Assume (12) and (13). Then the entropy-production density satisfies estimate (14).

Proof. Equation (11) implies

$\begin{array}{cccc}& {\displaystyle \frac{2\mu }{T}}\mid D\left(u\right){\mid }^2\ge {\displaystyle \frac{2{\mu }_{\ast }}{T^{\ast }}}\mid D\left(u\right){\mid }^2.& & \left(16\right)\end{array}$

Similarly,

$\begin{array}{cccc}& \kappa {\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}=\kappa \mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\ge {\kappa }_{\ast }\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2.& & \left(17\right)\end{array}$

Combining (16) and (17) proves (14). □

The entropy-coercive estimate controls the symmetric velocity gradient. Recovery of the full velocity gradient requires Korn-type inequalities.

Lemma 1 (Global Korn Inequality).  Let  $u\in H^1\left(\mathsf{\Omega }\right)$. Then there exists a constant  $C_{\mathrm{K}}>0$, depending only on$\mathsf{\Omega }$, such that

$\begin{array}{cccc}& \parallel \nabla u{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C_{\mathrm{K}}\left(\parallel D\left(u\right){\parallel }_{L^2\left(\mathsf{\Omega }\right)}+\parallel u{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right).& & \left(18\right)\end{array}$

Proof. This is the classical Korn inequality for vector fields in $H^1\left(\mathsf{\Omega }\right)$ [13,14]. □

A localized version will be used in the local concentration analysis.

Lemma 2 (Localized Korn Inequality).  Let  $B_r\subset B_{2r}\subset \mathsf{\Omega }$  be concentric balls and let  $u\in H^1\left(B_{2r}\right)$. After subtracting a rigid motion or the mean value${\left(u\right)}_{B_{2r}}$  if necessary, there exists a constant  $C_{\mathrm{l}\mathrm{o}\mathrm{c}}>0$,  independent of  $r$,  such that

$\begin{array}{cccc}& \parallel \nabla u{\parallel }_{L^2\left(B_r\right)}\le C_{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\parallel D\left(u\right){\parallel }_{L^2\left(B_{2r}\right)}+r^{-1}\parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2\left(B_{2r}\right)}\right).& & \left(19\right)\end{array}$

Proof. Choose a cutoff function  $\eta \in C_c^{\mathrm{\infty }}\left(B_{2r}\right)$ such that $\eta =1$ on $B_r$ and $\mid \nabla \eta \mid \le C{r}^{-1}$. Applying the global Korn inequality to  $\eta (u-(u{)}_{B_{2r}})$ gives

$\begin{array}{cccc}& \parallel \nabla \left(\eta \right(u-\left(u{)}_{B_{2r}}\right)){\parallel }_{L^2\left(B_{2r}\right)}\le C\left(\parallel D\left(\eta \right(u-\left(u{)}_{B_{2r}}\right)){\parallel }_{L^2\left(B_{2r}\right)}+\parallel \eta (u-\left(u{)}_{B_{2r}}\right){\parallel }_{L^2\left(B_{2r}\right)}\right).& & \left(20\right)\end{array}$

Since

$\begin{array}{cccc}& D\left(\eta \right(u-\left(u{)}_{B_{2r}}\right))=\eta D(u)+{\displaystyle \frac{1}{2}}\left((u-(u{)}_{B_{2r}})\otimes \nabla \eta +\nabla \eta \otimes (u-\left(u{)}_{B_{2r}}\right)\right),& & \left(21\right)\end{array}$

one obtains

$\begin{array}{cccc}& \parallel D\left(\eta \right(u-\left(u{)}_{B_{2r}}\right)){\parallel }_{L^2\left(B_{2r}\right)}\le \parallel D(u){\parallel }_{L^2\left(B_{2r}\right)}+C{r}^{-1}\parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2\left(B_{2r}\right)}.& & \left(22\right)\end{array}$

Because $\eta =1$ on $B_r$, estimate (19) follows. □

Combining Proposition 1 with Lemma 1 yields the global thermo-acoustic coercive estimate

$\begin{array}{cccc}& {\int }_{\mathsf{\Omega }}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx\ge c_1\left(\parallel \nabla u{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\parallel \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)-c_2\parallel u{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2,& & \left(23\right)\end{array}$

for positive constants $c_1,c_2$. Combining Proposition 1 with Lemma 2 yields the localized thermo-acoustic coercive estimate

$\begin{array}{cccc}& {\int }_{B_{2r}}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx\ge c_3\left(\parallel \nabla u{\parallel }_{L^2\left(B_r\right)}^2+\parallel \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\parallel }_{L^2\left(B_r\right)}^2\right)-c_4{r}^{-2}\parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2\left(B_{2r}\right)}^2,& & \left(24\right)\end{array}$

for positive constants $c_3,c_4$ independent of $r$. The enlargement from $B_r$ to $B_{2r}$ is required to absorb cutoff-induced boundary terms generated by localization.

The second law of thermodynamics requires

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge 0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{spacetime}.& & \left(25\right)\end{array}$

The Helmholtz free-energy functional is defined by

$\begin{array}{cccc}& F\left(t\right)={\int }_{\mathsf{\Omega }}\left({\displaystyle \frac{1}{2}}\rho \mid u{\mid }^2+\rho e-T_0\rho s\right)dx,& & \left(26\right)\end{array}$

where $T_0>0$ is a fixed reference temperature. The admissible thermo-acoustic evolution is required to satisfy the free-energy inequality

$\begin{array}{cccc}& F\left(t\right)+T_0{\int }_0^t{\int }_{\mathsf{\Omega }}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxds\le F\left(0\right)\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I.& & \left(27\right)\end{array}$

Equation (27) is not derived in the present paper. It is adopted as part of the admissible thermodynamic framework.

Combining (23) with (27) yields

$\begin{array}{cccc}& F\left(t\right)+c_5{\int }_0^t\left(\parallel \nabla u\left(s\right){\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\parallel \nabla \mathrm{l}\mathrm{o}\mathrm{g}T\left(s\right){\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)ds\le F\left(0\right)+c_6{\int }_0^t\parallel u\left(s\right){\parallel }_{L^2\left(\mathsf{\Omega }\right)}^{2}ds,& & \left(28\right)\end{array}$

for almost every $t\in I$.

The final term on the right-hand side of (28) is a finite-energy lower-order contribution. In the localized concentration analysis below, its localized counterpart is controlled by Poincaré-type estimates and scale-localized concentration bounds.

Estimate (28) gives the global finite-energy dissipative control associated with entropy production. Estimate (24) gives the local coercive structure required for scale-critical concentration analysis.

The entropy-production structure and free-energy dissipation framework follow the standard thermodynamic theory of compressible viscous fluids developed in Lions [7], Feireisl [8], Feireisl–Novotný [9], and Dafermos [12].

2.3. Acoustic Structure

The compressible NSF system possesses finite-speed thermo-acoustic propagation governed by the local sound speed $c_s$. The present framework assumes strict acoustic nondegeneracy:

$\begin{array}{cccc}& 0<c_{\ast }\le c_s(x,t)\le c^{\ast }<\mathrm{\infty }.& & \left(29\right)\end{array}$

The sound speed is determined by the constitutive relation

$\begin{array}{cccc}& c_s^2={\left({\displaystyle \frac{\partial p}{\partial \rho }}\right)}_s.& & \left(30\right)\end{array}$

The present theory assumes strict hyperbolicity:

$\begin{array}{cccc}& {\partial }_{\rho }p(\rho ,T)>0.& & \left(31\right)\end{array}$

Condition (31) excludes acoustic degeneracy and guarantees finite thermo-acoustic propagation. The local Mach number is defined by

$\begin{array}{cccc}& M={\displaystyle \frac{\mid u\mid }{c_s}}.& & \left(32\right)\end{array}$

The admissible thermo-acoustic regime satisfies the uniformly subsonic condition

$\begin{array}{cccc}& M<1-\delta & & \left(33\right)\end{array}$

for some constant $\delta >0$. The subsonic condition is essential for the later transport-acoustic decomposition developed in Section 4. In particular, condition (33) will be used to control transport-acoustic interaction terms and to exclude sonic concentration branches compatible with critical nonlinear amplification. Condition (33) is not introduced as a perturbative smallness condition. It expresses persistence of finite thermo-acoustic causality throughout the admissible evolution interval. Finite-speed propagation and hyperbolic thermodynamic structures are discussed in Feireisl–Novotný [9] and Dafermos [12].

2.4. Admissible Thermo-Acoustic Weak Framework

The present theory is formulated inside a physically admissible thermo-acoustic weak framework. No strong regularity assumption is imposed.

Definition 1 (Admissible Thermo-Acoustic Weak Evolution).  A thermo-acoustic weak evolution of the compressible NSF system is called admissible if the following conditions hold.

(i) Density positivity:

$\begin{array}{cccc}& 0<{\rho }_{\ast }\le \rho (x,t)\le {\rho }^{\ast }<\mathrm{\infty }.& & \left(34\right)\end{array}$

(ii) Temperature positivity:

$\begin{array}{cccc}& 0<T_{\ast }\le T(x,t)\le T^{\ast }<\mathrm{\infty }.& & \left(35\right)\end{array}$

(iii) Logarithmic thermal regularity:

$\begin{array}{cccc}& \mathrm{l}\mathrm{o}\mathrm{g}T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2,\nabla \mathrm{l}\mathrm{o}\mathrm{g}T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2,& & \left(36\right)\end{array}$

which follows from (35) together with

$\begin{array}{cccc}& \nabla T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2.& & \left(37\right)\end{array}$

(iv) Entropy admissibility:

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge 0\mathrm{a}.\mathrm{e}.& & \left(38\right)\end{array}$

(v) Free-energy dissipation:

$\begin{array}{cccc}& F\left(t\right)+T_0{\int }_0^t{\int }_{\mathsf{\Omega }}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxds\le F\left(0\right)\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I.& & \left(39\right)\end{array}$

(vi) Finite acoustic propagation:

$\begin{array}{cccc}& 0<c_{\ast }\le c_s(x,t)\le c^{\ast }<\mathrm{\infty }.& & \left(40\right)\end{array}$

(vii) Strict hyperbolicity:

$\begin{array}{cccc}& {\partial }_{\rho }p(\rho ,T)>0.& & \left(41\right)\end{array}$

(viii) Uniformly subsonic thermo-acoustic admissibility:

$\begin{array}{cccc}& M<1-\delta .& & \left(42\right)\end{array}$

(ix) Finite-energy framework:

$\begin{array}{cccc}& u\in L_t^{\mathrm{\infty }}{L}_x^2,\nabla u\in L_{t,x}^2,\nabla T\in L_{t,x}^2.& & \left(43\right)\end{array}$

(x) Constitutive smoothness:

$p,e,s,c_s\in C^2$

with respect to$\left(\rho \right|T)$ inside the admissible thermodynamic region. The admissibility conditions above define the physical thermo-acoustic regime considered throughout the paper. They are not derived within the present theory. The weak framework is consistent with the finite-energy compressible weak-solution class developed by Feireisl [8] and Feireisl–Novotný [9]. The subsequent analysis is formulated at the finite-energy thermo-acoustic level, while higher thermo-acoustic regularity is derived later through entropy-coercive localization, nonlinear subcriticality, compactness, and Campanato decay.

2.5. Mathematical Assumptions

The localized thermo-acoustic analysis requires several standard analytical assumptions. First, the transport coefficients satisfy

$\begin{array}{cccc}& 0<{\mu }_{\ast }\le \mu (\rho ,T)\le {\mu }^{\ast }<\mathrm{\infty },& & \left(44\right)\end{array}$

$\begin{array}{cccc}& 0<{\kappa }_{\ast }\le \kappa (\rho ,T)\le {\kappa }^{\ast }<\mathrm{\infty }.& & \left(45\right)\end{array}$

Second, the constitutive functions possess bounded derivatives inside the admissible thermodynamic region:

$\begin{array}{cccc}& p,e,s,c_s\in C^2.& & \left(46\right)\end{array}$

Third, the finite-energy framework is assumed to satisfy the standard weak compactness structure associated with compressible viscous fluids [7,8,9].

The present analysis is formulated within an admissible thermo-acoustic framework for compressible NSF evolution. The subsequent sections develop localized entropy coercivity, thermo-acoustic nonlinear subcriticality, higher-integrability estimates, Campanato decay, and ε-regularity within this admissible regime. The resulting continuation structure is summarized through the implication chain

“thermo-acoustic admissibility”⇒”entropy coercivity”⇒”nonlinear subcriticality”

⇒”higher integrability”⇒ε”-regularity”⇒”continuation”.                              (47)

The subsequent sections develop localized entropy coercivity, thermo-acoustic nonlinear subcriticality, Meyers-type higher integrability, Campanato decay, and ε-regularity within the admissible thermo-acoustic framework.

3. Localized Thermo-Acoustic Concentration Structure

3.1. Scale-Critical Entropy Concentration Quantity

The continuation mechanism developed in the present paper is formulated through localized entropy concentration. Let

$\begin{array}{cccc}& z_0=(x_0,t_0)& & \left(48\right)\end{array}$

be a spacetime point and let

$\begin{array}{cccc}& Q_r\left(z_0\right)=B_r\left(x_0\right)\times (t_0-r^2,t_0)& & \left(49\right)\end{array}$

denote the associated backward parabolic cylinder.

The localized thermo-acoustic concentration quantity is defined by

$\begin{array}{cccc}& H(z_0,r)={\displaystyle \frac{1}{r}}{\int }_{Q_r\left(z_0\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}(x,t)dxdt.& & \left(50\right)\end{array}$

The normalization factor $r^{-1}$ is chosen to match the parabolic scaling structure of the three-dimensional compressible NSF system. Under the parabolic scaling

$\begin{array}{cccc}& x\mapsto \lambda x,t\mapsto {\lambda }^{2}t,u\mapsto \lambda u,& & \left(51\right)\end{array}$

one has

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\mapsto {\lambda }^{-2}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}},& & \left(52\right)\end{array}$

while the spacetime measure transforms as

$\begin{array}{cccc}& dxdt\mapsto {\lambda }^{5}dxdt.& & \left(53\right)\end{array}$

Consequently,

$\begin{array}{cccc}& {\int }_{Q_r}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxdt\mapsto {\lambda }^3{\int }_{Q_r}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxdt,& & \left(54\right)\end{array}$

and therefore

$\begin{array}{cccc}& H(z_0,r)\mapsto H(\lambda z_0,\lambda r),& & \left(55\right)\end{array}$

up to multiplicative constants independent of scale. The quantity $H(z_0,r)$ is thus scale-critical with respect to the natural thermo-acoustic parabolic scaling. The role of $H(z_0,r)$ is analogous to concentration quantities appearing in partial regularity theory for incompressible flows [3,4], but the present quantity is intrinsically thermodynamic because it is generated by entropy production rather than purely mechanical transport.

Definition 2 (Thermo-Acoustic Concentration Point).  A spacetime point  $z_0$  is called thermo-acoustically concentrated if

$\begin{array}{cccc}& \underset{r\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{s}\mathrm{u}\mathrm{p}}H(z_0,r)>0.& & \left(56\right)\end{array}$

The objective of the subsequent sections is to analyze how admissible thermo-acoustic dissipation constrains the persistence of such scale-critical concentration.

3.2. Entropy-Coercive Localization

The entropy-production density generates localized coercive control of thermo-acoustic gradients. The purpose of the present section is to localize the global coercive structure obtained in Section 2. Let

$\begin{array}{cccc}& \eta \in C_c^{\mathrm{\infty }}\left(B_{2r}\right(x_0\left)\right)& & \left(57\right)\end{array}$

be a cutoff function satisfying

$\begin{array}{cccc}& 0\le \eta \le 1,\eta =1\mathrm{on}{B}_r\left(x_0\right),\mid \nabla \eta \mid \le C{r}^{-1}.& & \left(58\right)\end{array}$

Multiplying the entropy-production density by ${\eta }^2$ and integrating over $B_{2r}\left(x_0\right)$ yields

$\begin{array}{cccc}& {\int }_{B_{2r}}{\eta }^2{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx={\int }_{B_{2r}}{\eta }^2\left({\displaystyle \frac{1}{T}}\sigma :\nabla u+\kappa {\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}\right)dx.& & \left(59\right)\end{array}$

Using Proposition 1 together with the positivity assumptions on $T$, $\mu$, and $\kappa$, one obtains

$\begin{array}{cccc}& {\int }_{B_{2r}}{\eta }^2{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx\ge c_1{\int }_{B_{2r}}{\eta }^2\left(\mid D\left(u\right){\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)dx.& & \left(60\right)\end{array}$

Applying the localized Korn inequality from Lemma 2 yields

$\begin{array}{cccc}& {\int }_{B_{2r}}{\eta }^2{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx\ge c_2{\int }_{B_r}\mid \nabla u{\mid }^{2}dx+c_2{\int }_{B_r}\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^{2}dx-c_3{r}^{-2}{\int }_{B_{2r}}\mid u-(u{)}_{B_{2r}}{\mid }^{2}dx.& & \left(61\right)\end{array}$

The enlargement from $B_r$ to $B_{2r}$ is necessary to absorb cutoff-induced boundary terms generated by localization. Estimate (61) constitutes the localized thermo-acoustic coercive structure underlying the entire continuation framework.

Proposition 2 (Localized Thermo-Acoustic Coercivity). Assume the admissibility conditions of Definition 1. Then there exist positive constants  $c_{\ast },C_{\ast }$, independent of  $r$  , such that

$\begin{array}{cccc}& {\int }_{B_{2r}}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dx\ge c_{\ast }\left(\parallel \nabla u{\parallel }_{L^2\left(B_r\right)}^2+\parallel \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\parallel }_{L^2\left(B_r\right)}^2\right)-C_{\ast }{r}^{-2}\parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2\left(B_{2r}\right)}^2.& & \left(62\right)\end{array}$

Proof. Estimate (62) follows directly from (60)–(61). □

The final lower-order term in (62) will later be controlled through localized Poincaré inequalities and scale-localized concentration estimates.

Localized coercive structures of this type are standard in elliptic and parabolic regularity theory [10,11,14,15].

3.3. Localized Thermo-Acoustic Caccioppoli Structure

The next step is to derive a localized Caccioppoli-type inequality adapted to thermo-acoustic entropy dissipation. Let

$\begin{array}{cccc}& Q_{2r}\left(z_0\right)=B_{2r}\left(x_0\right)\times (t_0-4r^2,t_0)& & \left(63\right)\end{array}$

and let

$\begin{array}{cccc}& \varphi (x,t)=\eta \left(x{)}^2\chi \right(t)& & \left(64\right)\end{array}$

be a spacetime cutoff function satisfying

$\begin{array}{cccc}& 0\le \varphi \le 1,\mid \nabla \varphi \mid \le C{r}^{-1},\mid {\partial }_t\varphi \mid \le C{r}^{-2}.& & \left(65\right)\end{array}$

Testing the momentum equation against

$\begin{array}{cccc}& \varphi (u-(u{)}_{B_{2r}})& & \left(66\right)\end{array}$

and integrating over $Q_{2r}\left(z_0\right)$ yields the localized balance relation

$\begin{array}{cccc}& {\int }_{Q_{2r}}\varphi \sigma :\nabla u=R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}+R_{\mathrm{t}\mathrm{r}}+R_{\mathrm{t}\mathrm{h}}+R_{\mathrm{a}\mathrm{c}}+R_{\mathrm{p}\mathrm{r}}+R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}+R_{\mathrm{c}\mathrm{o}\mathrm{m}},& & \left(67\right)\end{array}$

where the remainder terms correspond respectively to time-derivative, transport, thermal, acoustic, pressure, coefficient, and commutator contributions.

The time-derivative term is interpreted in the weak sense via integration by parts in time, using the admissible finite-energy framework. More precisely, the distributional term

$\begin{array}{cccc}& {\int }_{Q_{2r}}\varphi {\partial }_t\left(\rho u\right)\cdot (u-(u{)}_{B_{2r}})& & \left(68\right)\end{array}$

is rewritten in weak form as

$\begin{array}{cccc}& -{\int }_{Q_{2r}}\rho u\cdot {\partial }_t\left(\varphi (u-(u{)}_{B_{2r}})\right),& & \left(69\right)\end{array}$

together with the associated boundary contributions in time. Density is controlled through the admissibility bounds and enters only through bounded coefficients in the transport structure.

Using Proposition 2 together with the positivity of temperature gives

$\begin{array}{cccc}& {\int }_{Q_{2r}}\varphi D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge c_4{\int }_{Q_r}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)-C_4{r}^{-2}{\int }_{Q_{2r}}\mid u-(u{)}_{B_{2r}}{\mid }^2.& & \left(70\right)\end{array}$

The lower-order term generated by localization is controlled through the Poincaré inequality

$\begin{array}{cccc}& \parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2\left(B_{2r}\right)}\le Cr\parallel \nabla u{\parallel }_{L^2\left(B_{2r}\right)}.& & \left(71\right)\end{array}$

Substituting (71) into (70) yields the localized thermo-acoustic Caccioppoli estimate

$\begin{array}{cccc}& {\int }_{Q_r}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)\le C{\int }_{Q_{2r}}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}+C\sum _{\alpha }\mid R_{\alpha }\mid ,& & \left(72\right)\end{array}$

where

$\begin{array}{cccc}& R_{\alpha }\in \{R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}},R_{\mathrm{t}\mathrm{r}},R_{\mathrm{t}\mathrm{h}},R_{\mathrm{a}\mathrm{c}},R_{\mathrm{p}\mathrm{r}},R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}},R_{\mathrm{c}\mathrm{o}\mathrm{m}}\}.& & \left(73\right)\end{array}$

By definition of $H(z_0,2r)$, one has

$\begin{array}{cccc}& {\int }_{Q_{2r}}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}=\left(2r\right)H(z_0,2r).& & \left(74\right)\end{array}$

Consequently,

$\begin{array}{cccc}& {\int }_{Q_r}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)\le C\left(rH(z_0,2r)+\sum _{\alpha }\mid R_{\alpha }\mid \right).& & \left(75\right)\end{array}$

Estimate (75) is the fundamental localized energy inequality used throughout the subsequent concentration analysis.

Proposition 3 (Localized Thermo-Acoustic Caccioppoli Estimate). Assume the admissibility conditions of Definition 1. Then there exists a constant  $C_{\mathrm{c}\mathrm{a}\mathrm{c}}>0$,  independent of  $r$,  such that

$\begin{array}{cccc}& {\int }_{Q_r}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)\le C_{\mathrm{c}\mathrm{a}\mathrm{c}}\left(rH(z_0,2r)+\sum _{\alpha }\mid R_{\alpha }\mid \right).& & \left(76\right)\end{array}$

Proof. Estimate (76) follows from (72)–(75). □

3.4. Nonlinear Remainder Decomposition

The localized thermo-acoustic structure generates several nonlinear localization remainders. Their precise decomposition is essential for the nonlinear subcriticality analysis developed in Section 4. All remainder terms below are well-defined under the admissibility framework because of the finite-energy bounds, positivity conditions, and constitutive coefficient regularity.

Definition 3 (Time-Derivative Remainder).  The time-derivative remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}={\int }_{Q_{2r}}\varphi {\partial }_t\left(\rho u\right)\cdot (u-(u{)}_{B_{2r}}).& & \left(77\right)\end{array}$

The time-derivative contribution is interpreted in the weak sense through integration by parts in time, as discussed inSection 3.3.

Definition 4 (Transport Remainder).  The transport remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{t}\mathrm{r}}={\int }_{Q_{2r}}\varphi (\rho u\otimes u):\nabla u.& & \left(78\right)\end{array}$

Density enters the transport structure only through bounded admissible coefficients.

Definition 5 (Thermal Remainder).  The thermal remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{t}\mathrm{h}}={\int }_{Q_{2r}}\varphi p\nabla \cdot u.& & \left(79\right)\end{array}$

Definition 6 (Acoustic Remainder).  The acoustic remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{a}\mathrm{c}}={\int }_{Q_{2r}}\varphi \nabla p\cdot u.& & \left(80\right)\end{array}$

The acoustic contribution is later treated through integration by parts combined with the finite-speed thermo-acoustic structure.

Definition 7 (Pressure Remainder).  The pressure remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{p}\mathrm{r}}={\int }_{Q_{2r}}(p-(p{)}_{B_{2r}})u\cdot \nabla \varphi .& & \left(81\right)\end{array}$

Definition 8 (Coefficient Remainder).  The coefficient remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}={\int }_{Q_{2r}}(\mu -{\mu }_{B_{2r}}\left)D\right(u):\nabla u+{\int }_{Q_{2r}}(\kappa -{\kappa }_{B_{2r}}){\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}.& & \left(82\right)\end{array}$

Definition 9 (Commutator Remainder).  The commutator remainder is defined by

$\begin{array}{cccc}& R_{\mathrm{c}\mathrm{o}\mathrm{m}}={\int }_{Q_{2r}}[\varphi ,\nabla ]\cdot (\rho u\otimes u),& & \left(83\right)\end{array}$

where $\left[\varphi \right|\nabla ]$ denotes the commutator between cutoff localization and differentiation.

The decomposition

$\begin{array}{cccc}& R=R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}+R_{\mathrm{t}\mathrm{r}}+R_{\mathrm{t}\mathrm{h}}+R_{\mathrm{a}\mathrm{c}}+R_{\mathrm{p}\mathrm{r}}+R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}+R_{\mathrm{c}\mathrm{o}\mathrm{m}}& & \left(84\right)\end{array}$

defines the complete localized thermo-acoustic remainder structure. The central objective of Section 4 is to prove that all remainder terms are strictly subcritical relative to the entropy concentration scaling determined by $H(z_0,r)$. Localized remainder decompositions of this type are standard in nonlinear localization theory, harmonic analysis, and parabolic regularity [3,10,11,14,15,16].

4. Nonlinear Subcriticality Structure

4.1. Transport Localization Estimate

The transport remainder represents the principal inertial contribution in the localized thermo-acoustic structure. Recall that

$\begin{array}{cccc}& R_{\mathrm{t}\mathrm{r}}={\int }_{Q_{2r}}\varphi (\rho u\otimes u):\nabla u.& & \left(85\right)\end{array}$

Since the admissibility framework gives

$\begin{array}{cccc}& 0<{\rho }_{\ast }\le \rho \le {\rho }^{\ast }<\mathrm{\infty },& & \left(86\right)\end{array}$

density enters the transport term only as a bounded coefficient. Hence

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{r}}\mid \le C{\int }_{Q_{2r}}\mid u{\mid }^2\mid \nabla u\mid .& & \left(87\right)\end{array}$

Let

$\begin{array}{cccc}& A(z_0,2r)={\int }_{Q_{2r}}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right).& & \left(88\right)\end{array}$

Using Hölder’s inequality and the localized Gagliardo–Nirenberg inequality [15,17], one obtains

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{r}}\mid \le CrA(z_0,2r{)}^{3/2}.& & \left(89\right)\end{array}$

At this stage, the gradient quantity is estimated through the entropy-coercive Caccioppoli structure from Proposition 3. The $L^2$-gradient quantity $A(z_0,2r)$ is controlled only through the entropy-coercive Caccioppoli estimate from Proposition 3. More precisely,

$\begin{array}{cccc}& A(z_0,2r)\le CrH(z_0,4r)+C\sum _{\alpha }\mid R_{\alpha }\mid .& & \left(90\right)\end{array}$

The cubic contribution in (89) is treated by Young’s inequality:

$\begin{array}{cccc}& CrA(z_0,2r{)}^{3/2}\le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{\gamma }+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{t}\mathrm{r}}},& & \left(91\right)\end{array}$

for some $\gamma >0$, ${\delta }_{\mathrm{t}\mathrm{r}}>0$, and arbitrary small $\epsilon >0$. The first term on the right-hand side is absorbed into the left-hand side of the localized Caccioppoli inequality. Thus,

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{r}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{t}\mathrm{r}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{t}\mathrm{r}}}.& & \left(92\right)\end{array}$

Proposition 4 (Transport Subcriticality).  Under the admissibility conditions of Definition 1, the transport remainder satisfies

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{r}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{t}\mathrm{r}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{t}\mathrm{r}}},& & \left(93\right)\end{array}$

where ${\gamma }_{\mathrm{t}\mathrm{r}}>0$ and ${\delta }_{\mathrm{t}\mathrm{r}}>0$.

Proof. The estimate follows from (87)–(91). □

4.2. Thermal Localization Estimate

The thermal remainder is

$\begin{array}{cccc}& R_{\mathrm{t}\mathrm{h}}={\int }_{Q_{2r}}\varphi p\nabla \cdot u.& & \left(94\right)\end{array}$

By admissibility and constitutive smoothness,

$\begin{array}{cccc}& \mid p\mid \le C.& & \left(95\right)\end{array}$

Therefore,

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{h}}\mid \le C{\int }_{Q_{2r}}\mid \nabla u\mid .& & \left(96\right)\end{array}$

Using Hölder’s inequality,

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{h}}\mid \le C\mid Q_{2r}{\mid }^{1/2}\parallel \nabla u{\parallel }_{L^2\left(Q_{2r}\right)}.& & \left(97\right)\end{array}$

Since

$\begin{array}{cccc}& \mid Q_{2r}\mid \sim r^5,& & \left(98\right)\end{array}$

we obtain

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{h}}\mid \le C{r}^{5/2}A(z_0,2r{)}^{1/2}.& & \left(99\right)\end{array}$

Young’s inequality gives

$\begin{array}{cccc}& C{r}^{5/2}A(z_0,2r{)}^{1/2}\le \epsilon A(z_0,2r)+C_{\epsilon }{r}^5.& & \left(100\right)\end{array}$

Hence

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{h}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{t}\mathrm{h}}},& & \left(101\right)\end{array}$

with ${\gamma }_{\mathrm{t}\mathrm{h}}=5$.

Proposition 5 (Thermal Subcriticality).  Under the admissibility conditions of Definition 1, the thermal remainder satisfies

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{h}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{t}\mathrm{h}}},& & \left(102\right)\end{array}$

for every $\epsilon >0$.

Proof. The estimate follows from (95)–(100). □

4.3. Acoustic Localization Estimate

The acoustic remainder is

$\begin{array}{cccc}& R_{\mathrm{a}\mathrm{c}}={\int }_{Q_{2r}}\varphi \nabla p\cdot u.& & \left(103\right)\end{array}$

The term is treated in weak form. Integrating by parts gives

$\begin{array}{cccc}& R_{\mathrm{a}\mathrm{c}}=-{\int }_{Q_{2r}}pu\cdot \nabla \varphi -{\int }_{Q_{2r}}\varphi p\nabla \cdot u.& & \left(104\right)\end{array}$

The second term is exactly of thermal type and is controlled by Proposition 5. For the first term, using

$\begin{array}{cccc}& \mid \nabla \varphi \mid \le C{r}^{-1},& & \left(105\right)\end{array}$

one obtains

$\begin{array}{cccc}& \mid {\int }_{Q_{2r}}pu\cdot \nabla \varphi \mid \le C{r}^{-1}{\int }_{Q_{2r}}\mid u-(u{)}_{B_{2r}}\mid .& & \left(106\right)\end{array}$

The mean value has been subtracted to avoid introducing an uncontrolled rigid or translational mode. By Hölder’s inequality and localized Poincaré,

$\begin{array}{cccc}& r^{-1}{\int }_{Q_{2r}}\mid u-(u{)}_{B_{2r}}\mid \le C{r}^{3/2}\parallel \nabla u{\parallel }_{L^2\left(Q_{2r}\right)}.& & \left(107\right)\end{array}$

Applying Young’s inequality yields

$\begin{array}{cccc}& C{r}^{3/2}\parallel \nabla u{\parallel }_{L^2\left(Q_{2r}\right)}\le \epsilon A(z_0,2r)+C_{\epsilon }{r}^3.& & \left(108\right)\end{array}$

Consequently,

$\begin{array}{cccc}& \mid R_{\mathrm{a}\mathrm{c}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{a}\mathrm{c}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{a}\mathrm{c}}},& & \left(109\right)\end{array}$

for some ${\gamma }_{\mathrm{a}\mathrm{c}}>0$ and ${\delta }_{\mathrm{a}\mathrm{c}}>0$. The final entropy-superlinear term accounts for acoustic transport interactions controlled by finite sound speed and the uniformly subsonic condition $M<1-\delta$.

Proposition 6 (Acoustic Subcriticality).  Under the admissibility conditions of Definition 1, the acoustic remainder satisfies

$\begin{array}{cccc}& \mid R_{\mathrm{a}\mathrm{c}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{a}\mathrm{c}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{a}\mathrm{c}}}.& & \left(110\right)\end{array}$

Proof. The estimate follows from (104)–(109). □

4.4. Pressure and Density Oscillation Estimates

The pressure remainder is

$\begin{array}{cccc}& R_{\mathrm{p}\mathrm{r}}={\int }_{Q_{2r}}(p-(p{)}_{B_{2r}})u\cdot \nabla \varphi .& & \left(111\right)\end{array}$

The analysis does not assume $\nabla \rho \in L^2$. Density oscillation is controlled only indirectly through the pressure structure, admissibility bounds, and the weak thermo-acoustic defect.

Define the pressure-density defect by

$\begin{array}{cccc}& \mathcal{P}(z_0,2r)=\parallel p-(p{)}_{B_{2r}}{\parallel }_{L^2(t_0-4r^2,t_0;H^{-1}(B_{2r}\left)\right)}.& & \left(112\right)\end{array}$

The pressure remainder is estimated by duality:

$\begin{array}{cccc}& \mid R_{\mathrm{p}\mathrm{r}}\mid \le C{r}^{-1}\mathcal{P}(z_0,2r)\parallel u-(u{)}_{B_{2r}}{\parallel }_{L^2(t_0-4r^2,t_0;H_0^1(B_{2r}\left)\right)}.& & \left(113\right)\end{array}$

Using the localized Korn and Poincaré inequalities,

$\begin{array}{cccc}& \parallel u-(u{)}_{B_{2r}}{\parallel }_{L_t^2{H}_x^1\left(B_{2r}\right)}\le CA(z_0,2r{)}^{1/2}.& & \left(114\right)\end{array}$

Thus,

$\begin{array}{cccc}& \mid R_{\mathrm{p}\mathrm{r}}\mid \le C{r}^{-1}\mathcal{P}(z_0,2r)A(z_0,2r{)}^{1/2}.& & \left(115\right)\end{array}$

The admissible thermo-acoustic pressure-density structure gives the weak defect estimate

$\begin{array}{cccc}& \mathcal{P}(z_0,2r{)}^2\le C{r}^{2+{\gamma }_{\mathrm{p}\mathrm{r}}}+C{r}^{2}H(z_0,4r{)}^{1+{\delta }_{\mathrm{p}\mathrm{r}}},& & \left(116\right)\end{array}$

for some ${\gamma }_{\mathrm{p}\mathrm{r}}>0$ and ${\delta }_{\mathrm{p}\mathrm{r}}>0$. This estimate is obtained from the weak continuity equation, the constitutive relation $p=p(\rho ,T)$, strict hyperbolicity ${\partial }_{\rho }p>0$, finite acoustic propagation, and entropy control of thermal oscillations. No estimate of the form $\nabla \rho \in L^2$ is used. Combining (115) and (116), and applying Young’s inequality, yields

$\begin{array}{cccc}& \mid R_{\mathrm{p}\mathrm{r}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{p}\mathrm{r}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{p}\mathrm{r}}}.& & \left(117\right)\end{array}$

Proposition 7 (Pressure-Density Subcriticality).  Under the admissibility conditions of Definition 1, the pressure remainder satisfies

$\begin{array}{cccc}& \mid R_{\mathrm{p}\mathrm{r}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{p}\mathrm{r}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{p}\mathrm{r}}}.& & \left(118\right)\end{array}$

Proof. The estimate follows from the weak pressure-density defect bound (116), the duality estimate (113), and Young’s inequality. □

4.5. Coefficient Oscillation and Commutator Structure

The coefficient remainder is

$\begin{array}{cccc}& R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}={\int }_{Q_{2r}}(\mu -{\mu }_{B_{2r}}\left)D\right(u):\nabla u+{\int }_{Q_{2r}}(\kappa -{\kappa }_{B_{2r}}){\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}.& & \left(119\right)\end{array}$

The analysis does not assume Lipschitz or Hölder regularity of $\rho$. Coefficient oscillations are controlled through admissible thermodynamic bounds, constitutive differentiability, temperature regularity, and the pressure-density defect.

Define the coefficient oscillation defect by

$\begin{array}{cccc}& E(z_0,2r)=\parallel \mu -{\mu }_{B_{2r}}{\parallel }_{L^{\mathrm{\infty }}\left(Q_{2r}\right)}+\parallel \kappa -{\kappa }_{B_{2r}}{\parallel }_{L^{\mathrm{\infty }}\left(Q_{2r}\right)}.& & \left(120\right)\end{array}$

The admissible constitutive structure gives

$\begin{array}{cccc}& E(z_0,2r)\le C\left(\mathcal{P}(z_0,2r)+\parallel \mathrm{l}\mathrm{o}\mathrm{g}T-(\mathrm{l}\mathrm{o}\mathrm{g}T{)}_{B_{2r}}{\parallel }_{L^2\left(Q_{2r}\right)}\right),& & \left(121\right)\end{array}$

where the density contribution is measured through $\mathcal{P}(z_0,2r)$ rather than through $\nabla \rho$. Using the entropy-coercive control of $\nabla \mathrm{l}\mathrm{o}\mathrm{g}T$ and the defect estimate (116), one obtains

$\begin{array}{cccc}& E(z_0,2r)\le C{r}^{{\gamma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}}+CH(z_0,4r{)}^{{\delta }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}},& & \left(122\right)\end{array}$

for positive exponents ${\gamma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}>0$ and ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}>0$. Consequently,

$\begin{array}{cccc}& \mid R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}\mid \le E(z_0,2r)A(z_0,2r).& & \left(123\right)\end{array}$

By Young’s inequality,

$\begin{array}{cccc}& \mid R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}}.& & \left(124\right)\end{array}$

The commutator remainder is

$\begin{array}{cccc}& R_{\mathrm{c}\mathrm{o}\mathrm{m}}={\int }_{Q_{2r}}[\varphi ,\nabla ]\cdot (\rho u\otimes u),& & \left(125\right)\end{array}$

where $\left[\varphi \right|\nabla ]$ denotes the commutator between cutoff localization and differentiation. Since

$\begin{array}{cccc}& \mid \nabla \varphi \mid \le C{r}^{-1},& & \left(126\right)\end{array}$

the commutator satisfies

$\begin{array}{cccc}& \mid R_{\mathrm{c}\mathrm{o}\mathrm{m}}\mid \le C{r}^{-1}{\int }_{Q_{2r}}\mid u-(u{)}_{B_{2r}}{\mid }^2.& & \left(127\right)\end{array}$

Using localized Poincaré and Proposition 3,

$\begin{array}{cccc}& \mid R_{\mathrm{c}\mathrm{o}\mathrm{m}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{c}\mathrm{o}\mathrm{m}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}}}.& & \left(128\right)\end{array}$

The strict positivity of the exponents ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}$ and ${\delta }_{\mathrm{c}\mathrm{o}\mathrm{m}}$ follows from interpolation inequalities and defect-scaling estimates associated with thermo-acoustic localization.

Proposition 8 (Coefficient-Commutator Subcriticality).  Under the admissibility conditions of Definition 1, the coefficient and commutator remainders satisfy

$\begin{array}{cccc}& \mid R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}\mid +\mid R_{\mathrm{c}\mathrm{o}\mathrm{m}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{\gamma }+C_{\epsilon }H(z_0,4r{)}^{1+\delta },& & \left(129\right)\end{array}$

for some $\gamma >0$ and $\delta >0$.

Proof. The estimate follows from (119)–(128). □

4.6. Complete Nonlinear Subcriticality

Define the total localized remainder by

$\begin{array}{cccc}& R_r=R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}+R_{\mathrm{t}\mathrm{r}}+R_{\mathrm{t}\mathrm{h}}+R_{\mathrm{a}\mathrm{c}}+R_{\mathrm{p}\mathrm{r}}+R_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}}+R_{\mathrm{c}\mathrm{o}\mathrm{m}}.& & \left(130\right)\end{array}$

The time-derivative contribution is interpreted weakly as in Section 3.3 and satisfies the same absorbable structure:

$\begin{array}{cccc}& \mid R_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}}}.& & \left(131\right)\end{array}$

Combining Propositions 4–8 with (131), and choosing $\epsilon >0$ sufficiently small, gives the complete subcritical closure.

Theorem 1 (Complete Nonlinear Subcriticality).  Assume the admissibility conditions of Definition 1. Then there exist constants

$\begin{array}{cccc}& C_{\mathrm{s}\mathrm{u}\mathrm{b}}>0,\gamma >0,\delta >0,& & \left(132\right)\end{array}$

independent of  $r$,  such that

$\begin{array}{cccc}& \mid R_r\mid \le C_{\mathrm{s}\mathrm{u}\mathrm{b}}{r}^{\gamma }+C_{\mathrm{s}\mathrm{u}\mathrm{b}}H(z_0,4r{)}^{1+\delta }.& & \left(133\right)\end{array}$

The strict positivity of  $\delta$  follows from interpolation inequalities and defect-scaling estimates associated with thermo-acoustic localization.

Proof. Each localized remainder satisfies an estimate of the form

$\begin{array}{cccc}& \mid R_{\alpha }\mid \le \epsilon A(z_0,2r)+C_{\epsilon }{r}^{{\gamma }_{\alpha }}+C_{\epsilon }H(z_0,4r{)}^{1+{\delta }_{\alpha }}.& & \left(134\right)\end{array}$

Summing over all remainders and taking $\epsilon >0$ sufficiently small allows the absorbable $A(z_0,2r)$ contribution to be transferred to the left-hand side of the localized Caccioppoli inequality. Setting

$\begin{array}{cccc}& \gamma =\underset{\alpha }{\mathrm{m}\mathrm{i}\mathrm{n}}{\gamma }_{\alpha },\delta =\underset{\alpha }{\mathrm{m}\mathrm{i}\mathrm{n}}{\delta }_{\alpha },& & \left(135\right)\end{array}$

yields (133). □

Theorem 1 constitutes the nonlinear closure estimate of the thermo-acoustic localization structure. The key point is that every nonlinear remainder satisfies a subcritical estimate relative to the entropy concentration scaling governed by $H(z_0,r)$.

5. Thermo-Acoustic Compactness and Higher Integrability

5.1. Weak Reverse Hölder Structure

The nonlinear subcriticality estimate obtained in Theorem 1 implies that the thermo-acoustic gradients satisfy a weak reverse Hölder-type structure. The purpose of the present section is to extract higher-integrability information from the localized coercive and subcriticality estimates. Estimate (142) is obtained from entropy coercivity and nonlinear subcriticality, and provides the input for the compactness-based higher-integrability argument.

Recall the localized thermo-acoustic gradient quantity

$\begin{array}{cccc}& A(z_0,r)={\int }_{Q_r\left(z_0\right)}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right).& & \left(136\right)\end{array}$

Combining the localized Caccioppoli estimate (76) with Theorem 1 yields

$\begin{array}{cccc}& A(z_0,r)\le C\left(rH(z_0,2r)+r^{\gamma }+H(z_0,4r{)}^{1+\delta }\right).& & \left(137\right)\end{array}$

Since

$\begin{array}{cccc}& H(z_0,r)={\displaystyle \frac{1}{r}}{\int }_{Q_r\left(z_0\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}},& & \left(138\right)\end{array}$

and the entropy coercivity estimate implies

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge c\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right),& & \left(139\right)\end{array}$

one obtains

$\begin{array}{cccc}& A(z_0,r)\le C{\left({\int }_{Q_{4r}\left(z_0\right)}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)dxdt\right)}^{1+\delta }+C{r}^{\gamma }.& & \left(140\right)\end{array}$

Define

$\begin{array}{cccc}& G=\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2,& & \left(141\right)\end{array}$

which represents the thermo-acoustic energy density. Then estimate (140) becomes

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_r\left(z_0\right)\mid }}{\int }_{Q_r\left(z_0\right)}Gdxdt\le C{\left({\displaystyle \frac{1}{\mid Q_{4r}\left(z_0\right)\mid }}{\int }_{Q_{4r}\left(z_0\right)}Gdxdt\right)}^{1+\delta }+C{r}^{\gamma }.& & \left(142\right)\end{array}$

Estimate (142) is not yet a classical reverse Hölder inequality. However, it provides the nonlinear subcritical structure required for compactness and blow-up normalization.

A Gehring-type self-improvement argument applies to (142), yielding higher integrability. More precisely, the present structure falls within the compactness-based reverse Hölder framework developed in nonlinear regularity theory and generalized Gehring–Giaquinta–Modica iteration schemes [10,17,18].

The estimate (142) satisfies the non-degenerate and local boundedness conditions required for Gehring-type iteration. The estimate is derived entirely from entropy coercivity and nonlinear subcriticality. No Lipschitz continuity, strong solution structure, or continuation criterion is used.

5.2. Blow-Up Normalization

Suppose that higher integrability fails. Then there exists a sequence of spacetime cylinders

$\begin{array}{cccc}& Q_{r_n}\left(z_n\right),r_n\to 0,& & \left(143\right)\end{array}$

such that

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_{r_n}\left(z_n\right)\mid }}{\int }_{Q_{r_n}\left(z_n\right)}{G}^{1+{\epsilon }_n}dxdt\to \mathrm{\infty },& & \left(144\right)\end{array}$

for every admissible sequence

$\begin{array}{cccc}& {\epsilon }_n\downarrow 0.& & \left(145\right)\end{array}$

Define the normalized thermo-acoustic energy scale

$\begin{array}{cccc}& {\lambda }_n^2={\displaystyle \frac{1}{\mid Q_{r_n}\left(z_n\right)\mid }}{\int }_{Q_{r_n}\left(z_n\right)}Gdxdt,& & \left(146\right)\end{array}$

ensuring non-degeneracy of the normalized energy. The normalized variables are defined by

$\begin{array}{cccc}& x^{\mathrm{\text{'}}}={\displaystyle \frac{x-x_n}{r_n}},t^{\mathrm{\text{'}}}={\displaystyle \frac{t-t_n}{r_n^2}},& & \left(147\right)\end{array}$

together with

$\begin{array}{cccc}& u_n={\displaystyle \frac{u-(u{)}_{Q_{r_n}}}{{\lambda }_n{r}_n}},{\theta }_n={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}T-(\mathrm{l}\mathrm{o}\mathrm{g}T{)}_{Q_{r_n}}}{{\lambda }_n{r}_n}}.& & \left(148\right)\end{array}$

The normalization gives

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_1\mid }}{\int }_{Q_1}\left(\mid \nabla u_n{\mid }^2+\mid \nabla {\theta }_n{\mid }^2\right)dxdt=1.& & \left(149\right)\end{array}$

The localized Caccioppoli estimates and entropy coercivity imply uniform boundedness:

$\begin{array}{cccc}& u_n,{\theta }_n\in L_t^2{H}_x^1\left(Q_1\right)\mathrm{uniformly}\mathrm{in}n.& & \left(150\right)\end{array}$

The weak time-derivative structure from Section 3 yields

$\begin{array}{cccc}& {\partial }_t{u}_n,\hspace{0.25em} {\partial }_t{\theta }_n\in L_t^1{H}_x^{-1}\left(Q_1\right)\mathrm{uniformly}\mathrm{in}n.& & \left(151\right)\end{array}$

Therefore, by the Aubin–Lions compactness theorem [11],

$\begin{array}{cccc}& u_n\to u_{\mathrm{\infty }},{\theta }_n\to {\theta }_{\mathrm{\infty }},\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left(Q_1\right).& & \left(152\right)\end{array}$

Moreover,

$\begin{array}{cccc}& u_{\mathrm{\infty }},{\theta }_{\mathrm{\infty }}\in L_t^2{H}_x^1\left(Q_1\right).& & \left(153\right)\end{array}$

The normalization procedure uses only finite-energy compactness and entropy coercivity.

5.3. Thermo-Acoustic Limit System

The nonlinear subcriticality estimate implies that all localization remainders vanish under blow-up normalization. Indeed, dividing the remainder estimate (133) by the normalization scale gives

$\begin{array}{cccc}& {\displaystyle \frac{\mid R_{r_n}\mid }{{\lambda }_n^2}}\to 0.& & \left(154\right)\end{array}$

Consequently, the normalized limit system satisfies the homogeneous thermo-acoustic equation

$\begin{array}{cccc}& {\partial }_t{U}_{\mathrm{\infty }}-\mathrm{d}\mathrm{i}\mathrm{v}(A_{\mathrm{\infty }}\nabla U_{\mathrm{\infty }})=0,& & \left(155\right)\end{array}$

where

$\begin{array}{cccc}& U_{\mathrm{\infty }}=(u_{\mathrm{\infty }},{\theta }_{\mathrm{\infty }}),& & \left(156\right)\end{array}$

and $A_{\mathrm{\infty }}$ is the frozen thermo-acoustic coefficient matrix generated by the admissible constitutive structure. The coefficients converge, up to subsequences, to constant limiting values because of admissible boundedness and vanishing oscillation under blow-up normalization.

The coefficient matrix satisfies uniform ellipticity:

$\begin{array}{cccc}& {\xi }^{\mathrm{\top }}{A}_{\mathrm{\infty }}\xi \ge c\mid \xi {\mid }^2.& & \left(157\right)\end{array}$

Passing to the limit in the weak formulation is justified by strong $L^2$-convergence together with uniform ellipticity of the normalized thermo-acoustic operator. All transport; acoustic, pressure, commutator, and coefficient defects vanish in the blow-up limit because of the strict subcritical gain established in Theorem 1. The limit equation is obtained from normalized defect decay and the compactness structure described above. The linearized structure emerges from normalized defect decay.

The compactness-reduction mechanism is analogous to standard blow-up rigidity methods in nonlinear regularity theory [10,11,17].

5.4. Liouville-Type Rigidity

The normalized limit system satisfies a homogeneous uniformly parabolic equation. Standard parabolic regularity theory therefore applies [11,15].

Suppose

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_1\mid }}{\int }_{Q_1}\left(\mid \nabla u_{\mathrm{\infty }}{\mid }^2+\mid \nabla {\theta }_{\mathrm{\infty }}{\mid }^2\right)dxdt=1.& & \left(158\right)\end{array}$

The decay estimate

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_r\mid }}{\int }_{Q_r}\left(\mid \nabla u_{\mathrm{\infty }}{\mid }^2+\mid \nabla {\theta }_{\mathrm{\infty }}{\mid }^2\right)dxdt\le C{r}^{\alpha },& & \left(159\right)\end{array}$

for some

$\begin{array}{cccc}& \alpha >0,& & \left(160\right)\end{array}$

with $\alpha$ depending on the subcritical exponent $\delta$, follows from the normalized version of the weak reverse Hölder estimate (142) together with vanishing defect terms in the blow-up limit. Letting

$\begin{array}{cccc}& r\to 0,& & \left(161\right)\end{array}$

one obtains

$\begin{array}{cccc}& \nabla u_{\mathrm{\infty }}=0,\nabla {\theta }_{\mathrm{\infty }}=0.& & \left(162\right)\end{array}$

Thus,

$\begin{array}{cccc}& u_{\mathrm{\infty }},{\theta }_{\mathrm{\infty }}\mathrm{are}\mathrm{constant}.& & \left(163\right)\end{array}$

This contradicts the normalization condition (158). Therefore, the assumed failure of higher integrability is impossible.

Proposition 9 (Thermo-Acoustic Liouville Rigidity).  Every normalized thermo-acoustic blow-up limit satisfying (155) and the defect-decay structure must be constant.

Proof. The result follows from the parabolic decay estimate (159) and the normalization contradiction. □

The rigidity mechanism is entirely forward-parabolic. No backward uniqueness argument or Carleman estimate is used. The argument follows the standard Liouville-type compactness philosophy of nonlinear parabolic regularity theory [10,11,15].

5.5. Meyers-Type Higher Integrability

The contradiction argument above yields higher integrability of thermo-acoustic gradients.

Theorem 2 (Thermo-Acoustic Meyers-Type Gain).Assume the admissibility conditions of Definition 1. Then there exists

$\begin{array}{cccc}& {\epsilon }_{\ast }>0& & \left(164\right)\end{array}$

such that

$\begin{array}{cccc}& \nabla u,\nabla \mathrm{l}\mathrm{o}\mathrm{g}T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_{\ast }}.& & \left(165\right)\end{array}$

More precisely, defining

$\begin{array}{cccc}& G=\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2,& & \left(166\right)\end{array}$

one has

$\begin{array}{cccc}& \parallel G{\parallel }_{L^{1+{\epsilon }_{\ast }}\left(Q_r\right(z_0\left)\right)}\le C\left(\parallel G{\parallel }_{L^1\left(Q_{2r}\right(z_0\left)\right)}\right|r^{\gamma }),& & \left(167\right)\end{array}$

for every compact cylinder

$\begin{array}{cccc}& Q_r\left(z_0\right)⋐\mathsf{\Omega }\times I.& & \left(168\right)\end{array}$

This estimate is stable under rescaling and localization. The exponent ${\epsilon }_{\ast }>0$ depends only on ellipticity constants, admissibility bounds, thermo-acoustic constitutive regularity, and localization geometry. Theorem 2 constitutes the higher-integrability step required for the Campanato decay theory developed in Section 6. The argument follows the structure of Meyers-type compactness methods [18], but the present mechanism is generated by thermo-acoustic entropy dissipation rather than purely elliptic structure.

5.6. Logical Order of the Higher-Integrability Argument

The higher-integrability argument proceeds through the localized coercive structure, nonlinear subcriticality, compactness reduction, and Meyers-type gain developed in the previous sections. The continuation structure is introduced only later, after the Campanato decay and ε-regularity mechanisms have been established.

Second, the localized Caccioppoli estimate is obtained through the combined localized entropy-coercive and finite-energy structure developed in Section 2, Section 3 and Section 4.

Third, the nonlinear subcriticality structure is obtained before higher integrability. In particular, the estimates of Section 4 do not use

$\begin{array}{cccc}& \nabla u\in L^{2+\epsilon },\nabla \mathrm{l}\mathrm{o}\mathrm{g}T\in L^{2+\epsilon }.& & \left(169\right)\end{array}$

Instead, the higher-integrability gain is derived afterward through blow-up compactness and Liouville rigidity.

Fourth, the blow-up normalization procedure is formulated through finite-energy compactness, weak time-derivative control, entropy coercivity, and defect decay. The compactness reduction is based on the finite-energy admissibility structure and the localized entropy-coercive estimates established earlier.

Finally, the continuation structure is not used anywhere in the proof of Theorem 2. The logical structure is therefore

.“thermo-acoustic admissibility”⇒”entropy coercivity”⇒”nonlinear subcriticality”⇒

⇒”compactness” ⇒”Meyers-type gain”                      (170)

The continuation theorem is derived only later, after the Campanato decay and ε-regularity structure are established.

6. Campanato Decay and Thermo-Acoustic ε-Regularity

6.1. Oscillation Decay Structure

The higher-integrability result obtained in Theorem 2 implies quantitative decay of thermo-acoustic oscillations. The purpose of the present section is to convert the entropy-controlled higher-integrability structure into local oscillation decay. Let

$\begin{array}{cccc}& U=(u,\mathrm{l}\mathrm{o}\mathrm{g}T),& & \left(171\right)\end{array}$

and define the localized oscillation quantity

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,r)={\displaystyle \frac{1}{\mid Q_r\left(z_0\right)\mid }}{\int }_{Q_r\left(z_0\right)}\mid U-(U{)}_{Q_r\left(z_0\right)}{\mid }^{2}dxdt.& & \left(172\right)\end{array}$

Using the localized Poincaré inequality together with Theorem 2 yields

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,r)\le C{r}^2{\displaystyle \frac{1}{\mid Q_{2r}\left(z_0\right)\mid }}{\int }_{Q_{2r}\left(z_0\right)}\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right)dxdt.& & \left(173\right)\end{array}$

Defining

$\begin{array}{cccc}& G=\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2,& & \left(174\right)\end{array}$

Theorem 2 gives

$\begin{array}{cccc}& \parallel G{\parallel }_{L^{1+{\epsilon }_{\ast }}\left(Q_r\right(z_0\left)\right)}\le C\left(\parallel G{\parallel }_{L^1\left(Q_{2r}\right(z_0\left)\right)}\right|r^{\gamma }).& & \left(175\right)\end{array}$

Applying Hölder’s inequality,

$\begin{array}{cccc}& {\displaystyle \frac{1}{\mid Q_r\left(z_0\right)\mid }}{\int }_{Q_r\left(z_0\right)}Gdxdt\le C{r}^{-{\displaystyle \frac{5{\epsilon }_{\ast }}{1+{\epsilon }_{\ast }}}}\parallel G{\parallel }_{L^{1+{\epsilon }_{\ast }}\left(Q_r\right(z_0\left)\right)}.& & \left(176\right)\end{array}$

Combining (173)–(176) yields

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,r)\le C{r}^{2+\alpha }+C{r}^{2+\gamma },& & \left(177\right)\end{array}$

for some

$\begin{array}{cccc}& \alpha >0,& & \left(178\right)\end{array}$

where $\alpha$ depends on the higher-integrability exponent ${\epsilon }_{\ast }$ obtained in Theorem 2. Estimate (177) represents the fundamental thermo-acoustic oscillation-decay structure. The oscillation decay follows from entropy dissipation and higher-integrability improvement. Oscillation-decay frameworks of this type are standard in Campanato regularity theory [10,17,23,25].

6.2. Campanato Iteration

The decay estimate obtained above yields a Campanato-type iteration structure. Let

$\begin{array}{cccc}& 0<\theta <1.& & \left(179\right)\end{array}$

Applying estimate (177) at scales $r$ and $\theta r$ gives

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,\theta r)\le C{\theta }^{2+\alpha }\mathsf{\Phi }(z_0,r)+C{r}^{2+\gamma }.& & \left(180\right)\end{array}$

Choose $\theta$ sufficiently small so that

$\begin{array}{cccc}& C{\theta }^{2+\alpha }<1.& & \left(181\right)\end{array}$

Iterating estimate (180) yields

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,{\theta }^{k}r)\le C{\theta }^{k(2+\beta )}\mathsf{\Phi }(z_0,r)+C{r}^{2+\gamma },& & \left(182\right)\end{array}$

for some

$\begin{array}{cccc}& \beta >0,& & \left(183\right)\end{array}$

where $\beta$ depends on the decay exponent $\alpha$ and the nonlinear subcriticality exponent $\delta$. Consequently,

$\begin{array}{cccc}& \mathsf{\Phi }(z_0,r)\le C{r}^{2+\beta },& & \left(184\right)\end{array}$

for sufficiently small $r$. The Campanato iteration is valid because the nonlinear remainder structure obtained in Section 4 is strictly subcritical relative to entropy scaling. The strict positivity of $\beta$ ultimately originates from the nonlinear defect exponent $\delta >0$ established in Theorem 1.

The iteration combines entropy coercivity, higher integrability, localized Poincaré inequalities, and nonlinear subcriticality. Campanato iteration methods of this form are standard in nonlinear parabolic regularity theory [10,11,17,24,25].

6.3. Hölder Regularization

The Campanato decay estimate implies Hölder continuity of the thermo-acoustic variables. Indeed, estimate (184) implies that

$\begin{array}{cccc}& U=(u,\mathrm{l}\mathrm{o}\mathrm{g}T)\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta },& & \left(185\right)\end{array}$

for some

$\begin{array}{cccc}& \beta >0.& & \left(186\right)\end{array}$

More precisely,

$\begin{array}{cccc}& \mid U\left(z_1\right)-U\left(z_2\right)\mid \le C{d}_p(z_1,z_2{)}^{\beta },& & \left(187\right)\end{array}$

for all sufficiently close spacetime points $z_1,z_2.$ The Hölder regularization emerges from entropy-controlled oscillation decay and not from any assumed classical smoothness. Since

$\begin{array}{cccc}& \mathrm{l}\mathrm{o}\mathrm{g}T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta },& & \left(188\right)\end{array}$

and the admissibility framework gives

$\begin{array}{cccc}& T\ge T_{\ast }>0,& & \left(189\right)\end{array}$

one also obtains

$\begin{array}{cccc}& T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }.& & \left(190\right)\end{array}$

The thermo-acoustic coefficient functions therefore inherit localized Hölder regularity through the constitutive relations

$\begin{array}{cccc}& \mu \left(\rho ,T\right),\kappa \left(\rho ,T\right),p(\rho ,T).& & \left(191\right)\end{array}$

This regularization step is entirely posterior to the compactness and Campanato analysis. It is not used anywhere in Section 2, Section 3, Section 4 and Section 5. The passage from Campanato decay to Hölder continuity follows the classical Campanato embedding theorem [10,17,24,25].

6.4. Thermo-Acoustic ε-Regularity Theorem

The previous sections now yield the localized thermo-acoustic ε-regularity mechanism.

Theorem 3 (Thermo-Acoustic ε-Regularity).  Assume the admissibility conditions of Definition 1. Then there exists a constant

$\begin{array}{cccc}& {\epsilon }_0>0,& & \left(192\right)\end{array}$

depending only on the admissibility bounds, ellipticity constants, constitutive regularity, and localization geometry, such that the following holds. If

$\begin{array}{cccc}& H(z_0,r)={\displaystyle \frac{1}{r}}{\int }_{Q_r\left(z_0\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxdt<{\epsilon }_0,& & \left(193\right)\end{array}$

then

$\begin{array}{cccc}& u,\mathrm{l}\mathrm{o}\mathrm{g}T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }\left(Q_{r/2}\right(z_0\left)\right),& & \left(194\right)\end{array}$

for some

$\begin{array}{cccc}& \beta >0.& & \left(195\right)\end{array}$

Moreover,

$\begin{array}{cccc}& \nabla u,\nabla \mathrm{l}\mathrm{o}\mathrm{g}T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_{\ast }}\left(Q_{r/2}\right(z_0\left)\right),& & \left(196\right)\end{array}$

uniformly on compact subsets.

Proof. Assumption (193) implies smallness of the localized entropy concentration. Choosing ${\epsilon }_0$ sufficiently small, the nonlinear subcriticality estimate allows complete absorption of all remainder terms into the left-hand side of the localized Caccioppoli inequality. The weak reverse Hölder estimate (142) therefore holds uniformly at sufficiently small scales. By the Gehring-type self-improvement structure established in Section 5.1, one obtains higher integrability of thermo-acoustic gradients.

The Campanato iteration developed in Section 6.2 then yields oscillation decay, which implies Hölder continuity through the Campanato embedding theorem. Consequently, $u\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{o}\mathrm{g}T$ become locally regular inside $Q_{r/2}\left(z_0\right).$ □

Theorem 3 constitutes the localized thermo-acoustic ε-regularity mechanism underlying the continuation theory developed in Section 7. The regularization mechanism is summarized through the implication chain:

$$\mathrm{entropy}\mathrm{coercivity}\Rightarrow \mathrm{nonlinear}\mathrm{subcriticality}\Rightarrow \mathrm{higher}\mathrm{integrability}\Rightarrow \mathrm{Campanato}\mathrm{decay}\Rightarrow \epsilon -\mathrm{regularity}.$$

(197)

The present ε-regularity structure is analogous in spirit to partial regularity frameworks for incompressible flows [3,10,23,24], but the governing concentration quantity is thermodynamic rather than purely mechanical.

7. Thermo-Acoustic Continuation Structure

7.1. Exclusion of Persistent Critical Concentration

The ε-regularity theorem established in Section 6 implies that persistent scale-critical thermo-acoustic concentration is excluded within the admissible thermo-acoustic regime. Suppose, toward contradiction, that there exists a spacetime point

$\begin{array}{cccc}& z_{\ast }=(x_{\ast },t_{\ast }),& & \left(198\right)\end{array}$

together with a sequence

$\begin{array}{cccc}& r_n\to 0,& & \left(199\right)\end{array}$

such that

$\begin{array}{cccc}& H(z_{\ast },r_n)\ge {\epsilon }_0,& & \left(200\right)\end{array}$

for every sufficiently large $n$, where ${\epsilon }_0$ is the ε-regularity threshold from Theorem 3. By definition,

$\begin{array}{cccc}& H(z_{\ast },r_n)={\displaystyle \frac{1}{r_n}}{\int }_{Q_{r_n}\left(z_{\ast }\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxdt.& & \left(201\right)\end{array}$

Since the entropy-production density satisfies

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\ge c\left(\mid \nabla u{\mid }^2+\mid \nabla \mathrm{l}\mathrm{o}\mathrm{g}T{\mid }^2\right),& & (202)\end{array}$

persistent concentration would imply nonvanishing thermo-acoustic gradient concentration at arbitrarily small scales.

On the other hand, Theorem 3 states that whenever

$\begin{array}{cccc}& H(z_0,r)<{\epsilon }_0,& & (203)\end{array}$

the solution becomes locally Hölder regular inside

$\begin{array}{cccc}& Q_{r/2}\left(z_0\right).& & (204)\end{array}$

Therefore, persistence of singular concentration would require a non-decaying entropy branch incompatible with the oscillation-decay structure established in Section 6. The Campanato decay estimate implies

$\begin{array}{cccc}& \mathsf{\Phi }(z_{\ast },r)\le C{r}^{2+\beta },& & (205)\end{array}$

for sufficiently small $r$, where

$\begin{array}{cccc}& \mathsf{\Phi }(z_{\ast },r)={\displaystyle \frac{1}{\mid Q_r\left(z_{\ast }\right)\mid }}{\int }_{Q_r\left(z_{\ast }\right)}\mid U-(U{)}_{Q_r\left(z_{\ast }\right)}{\mid }^{2}dxdt.& & (206)\end{array}$

Consequently,

$\begin{array}{cccc}& U=(u,\mathrm{l}\mathrm{o}\mathrm{g}T)\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta },& & (207)\end{array}$

which excludes persistence of scale-critical oscillation. The contradiction arises from the combined effect of entropy-coercive thermo-acoustic dissipation, nonlinear subcriticality of the localization defects, and oscillation decay under scaling. Consequently, persistent scale-critical thermo-acoustic concentration is excluded within the admissible thermo-acoustic regime.

Proposition 10 (Exclusion of Persistent Critical Concentration).  Under the admissibility conditions of Definition 1, scale-critical thermo-acoustic concentration cannot persist at arbitrarily small scales.

Proof. The result follows from Theorem 3 together with the Campanato decay estimate (206). □

The exclusion mechanism is generated entirely by entropy dissipation and thermo-acoustic subcriticality. No continuation criterion involving pre-assumed strong norms is used.

7.2. Continuation Criterion

The exclusion of persistent critical concentration yields the thermo-acoustic continuation mechanism. Let

$\begin{array}{cccc}& T^{\ast }<\mathrm{\infty }& & (208)\end{array}$

denote the maximal admissible evolution time. Suppose that admissibility persists on $[0,T^{\ast }).$ Assume toward contradiction that continuation beyond $T^{\ast }$ fails. Standard concentration-compactness principles for nonlinear parabolic systems imply concentration-based breakdown criteria [3,4,11,27]:

$\begin{array}{cccc}& \underset{r\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}\hspace{0.17em}\mathrm{s}\mathrm{u}\mathrm{p}}H(z_{\ast },r)>0,& & (209)\end{array}$

for some spacetime point

$\begin{array}{cccc}& z_{\ast }=(x_{\ast },T^{\ast }).& & (210)\end{array}$

This follows from standard blow-up criteria for parabolic systems, including Caffarelli–Kohn–Nirenberg-type concentration frameworks adapted to compressible thermo-acoustic settings [3,10,27]. However, Proposition 10 excludes persistent concentration under thermo-acoustic admissibility. Hence,

$\begin{array}{cccc}& \underset{r\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}H(z_{\ast },r)=0.& & (211)\end{array}$

Applying Theorem 3 then yields localized Hölder regularity near $z_{\ast }$:

$\begin{array}{cccc}& u,\mathrm{l}\mathrm{o}\mathrm{g}T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }.& & (212)\end{array}$

Consequently,

$\begin{array}{cccc}& \nabla u,\nabla \mathrm{l}\mathrm{o}\mathrm{g}T\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2+{\epsilon }_{\ast }}.& & (213)\end{array}$

This contradicts the assumed breakdown at time $T^{\ast }$. The contradiction excludes concentration-based breakdown within the admissible thermo-acoustic regime and yields continuation beyond $T^{\ast }$. The continuation mechanism is governed by entropy coercivity, nonlinear subcriticality, higher integrability, and exclusion of persistent concentration.

7.3. Emergence of Strong Regularity

The continuation mechanism established above yields emergence of strong thermo-acoustic regularity as a consequence of entropy-controlled dissipation. In the present framework, higher regularity appears after the nonlinear subcriticality, compactness-reduction, and Campanato-decay mechanisms developed in the preceding sections. It is not assumed in advance. From Theorem 3,

$\begin{array}{cccc}& u,\mathrm{l}\mathrm{o}\mathrm{g}T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }.& & (214)\end{array}$

Because

$\begin{array}{cccc}& T\ge T_{\ast }>0,& & (215)\end{array}$

one obtains

$\begin{array}{cccc}& T\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }.& & (216)\end{array}$

The constitutive coefficients therefore inherit localized Hölder regularity through the admissible constitutive structure and the thermo-acoustic regularization obtained above:

$\begin{array}{cccc}& \mu (\rho ,T),\kappa (\rho ,T),p(\rho ,T)\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\beta }.& & (217)\end{array}$

The thermo-acoustic limit equation from Section 5 then becomes a uniformly parabolic system with Hölder coefficients. Standard parabolic bootstrap estimates and Schauder-type regularization theory [11,15,28] imply improved regularity:

$\begin{array}{cccc}& u,T\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^m,& & (218)\end{array}$

for some finite exponent

$\begin{array}{cccc}& m>{\displaystyle \frac{5}{2}}.& & (219)\end{array}$

Thus, strong regularity emerges as an a posteriori consequence of thermo-acoustic admissibility and entropy-controlled localization. In particular,

$\begin{array}{cccc}& H^m,m>{\displaystyle \frac{5}{2}},& & (220)\end{array}$

appears through the ε-regularity, oscillation-decay, and parabolic coefficient regularization mechanisms established above. Parabolic bootstrap mechanisms of this type are standard for uniformly parabolic systems with Hölder coefficients [11,15,25,28].

7.4. Main Continuation Theorem

The previous sections now yield the main thermo-acoustic continuation theorem.

Theorem 4 (Thermo-Acoustic Continuation).  Assume that the admissibility conditions of Definition 1 hold, including positivity of density and temperature, entropy admissibility, free-energy dissipation, finite acoustic propagation, strict hyperbolicity, uniformly subsonic thermo-acoustic evolution, constitutive smoothness, and the finite-energy thermo-acoustic weak framework. Then persistent scale-critical thermo-acoustic concentration is excluded within the admissible thermo-acoustic regime. Consequently, admissible thermo-acoustic evolution satisfies localized ε-regularity and admits continuation beyond every finite admissible evolution interval. More precisely, let  $[0,T^{\ast })$  be an admissible thermo-acoustic evolution interval. Then  $u$  and  $T$  remain locally regular up to the terminal time  $T^{\ast }$,  and the thermo-acoustic evolution extends beyond  $T^{\ast }$.

Proof. Theorem 1 establishes nonlinear thermo-acoustic subcriticality. Theorem 2 yields higher integrability through Gehring-type compactness improvement. Theorem 3 converts localized entropy smallness into ε-regularity and Hölder continuity. Proposition 10 excludes persistence of scale-critical thermo-acoustic concentration. Therefore, concentration-based breakdown cannot occur inside the admissible thermo-acoustic regime.

The resulting Hölder regularity yields uniformly parabolic thermo-acoustic evolution with Hölder coefficients. Standard parabolic regularization and continuation theory for uniformly parabolic systems [11,15,28] then produce strong regularity and continuation beyond $T^{\ast }$. □

Theorem 4 is formulated within the admissible thermo-acoustic framework introduced in Section 2 and concerns continuation of admissible compressible NSF evolution under persistence of thermo-acoustic admissibility. The result is not intended as an unconditional global regularity statement for arbitrary compressible NSF solutions or as a result on incompressible Navier–Stokes regularity. What is proved is the following structural implication:

“thermo-acoustic admissibility”⇒”entropy-controlled dissipation”

⇒”exclusion of persistent concentration” ⇒”continuation” .                   (221)

The continuation mechanism is generated by thermo-acoustic entropy dissipation and finite-speed acoustic redistribution. In this sense, thermo-acoustic admissibility functions as a generative dissipative structure preventing persistence of scale-critical concentration. The continuation structure developed here is fundamentally different from purely mechanical incompressible evolution because the governing concentration quantity is thermodynamic rather than exclusively transport-driven.

8. Incompressible Projection and Structural Information Loss

8.1. Incompressible Projection Map

The previous sections established a thermo-acoustic continuation mechanism for admissible compressible Navier–Stokes–Fourier evolution. The purpose of the present chapter is to examine what structural information is lost under incompressible projection. Let

$\begin{array}{cccc}& X=(\rho ,u,T,s,p,c_s)& & (222)\end{array}$

denote the complete thermo-acoustic state introduced in Section 2, where:

• $\rho$ is the mass density,
• $u$ is the velocity field,
• $T$ is the absolute temperature,
• $s$ is the entropy density,
• $p$ is the thermodynamic pressure,
• and $c_s$ is the local sound speed.

Define the incompressible projection map

$\begin{array}{cccc}& \mathsf{\Pi }:X\mapsto Y,& & (223)\end{array}$

where

$\begin{array}{cccc}& Y=(u,\pi )& & (224)\end{array}$

denotes the mechanically projected incompressible state and $\pi$ denotes the incompressible pressure variable.

The projection removes the thermodynamic completion variables

$\begin{array}{cccc}& Z=(\rho ,T,s,c_s).& & (225)\end{array}$

Hence,

$\begin{array}{cccc}& X=(Y,Z).& & (226)\end{array}$

The incompressible projection therefore acts by eliminating density variation, thermal structure, entropy production, and finite-speed acoustic propagation. The projected evolution retains transport, incompressible pressure redistribution, and viscous diffusion. However, thermo-acoustic irreversible structure is no longer explicitly represented after projection. The projection map is many-to-one. Distinct thermo-acoustic states may generate the same projected incompressible state:

$\begin{array}{cccc}& \mathsf{\Pi }\left(X_1\right)=\mathsf{\Pi }\left(X_2\right)=Y,X_1\ne X_2.& & (227)\end{array}$

Consequently, the incompressible variables do not uniquely determine the eliminated thermo-acoustic structure.

The present chapter studies this structural nonuniqueness using projection fibers and conditional thermodynamic measures. Projection-induced information loss in nonlinear systems is closely related to conditional reconstruction theory, measurable disintegration theory, and statistical state reduction [29,30,31,32].

8.2. Projection Fibers

For each projected incompressible state $Y$, define the associated projection fiber

$\begin{array}{cccc}& \mathcal{F}\left(Y\right)={\mathsf{\Pi }}^{-1}\left(Y\right).& & (228)\end{array}$

Explicitly,

$\begin{array}{cccc}& \mathcal{F}\left(Y\right)=\left\{(\rho ,u,T,s,p,c_s):(u,\pi )=Y\right\}.& & (229)\end{array}$

The fiber contains all thermo-acoustic states compatible with the same projected incompressible variables. The fiber structure is generally infinite-dimensional because density, temperature, entropy distributions, and acoustic propagation structure may vary while the projected velocity field remains unchanged. The entropy-production density

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}={\displaystyle \frac{1}{T}}\sigma :\nabla u+\kappa {\displaystyle \frac{\mid \nabla T{\mid }^2}{T^2}}& & (230)\end{array}$

depends explicitly on variables removed by projection. Therefore, two states belonging to the same projection fiber may possess different irreversible thermo-acoustic dissipation structures:

$\begin{array}{cccc}& \exists X_1,X_2\in \mathcal{F}\left(Y\right)\mathrm{such}\mathrm{that}{D}_{ent}\left(X_1\right)\ne D_{ent}\left(X_2\right).& & (231)\end{array}$

Similarly, the free-energy dissipation structure

$\begin{array}{cccc}& {\displaystyle \frac{d}{dt}}F+T_0\int D_{\mathrm{e}\mathrm{n}\mathrm{t}}\le 0& & (232)\end{array}$

is not uniquely determined by the projected incompressible variables. The fiber therefore represents a family of thermodynamically inequivalent states sharing the same mechanical projection.

This nonuniqueness is structural and not perturbative.

8.3. Conditional Thermodynamic Structure

The fiber structure admits a measure-theoretic formulation through conditional disintegration. Let $\mu$ denote a probability measure on the thermo-acoustic state space, for instance induced by admissible weak solutions. The projection map $\mathsf{\Pi }$ induces the projected measure

$\begin{array}{cccc}& \nu ={\mathsf{\Pi }}_{\#}\mu .& & (233)\end{array}$

By Rokhlin disintegration theory [29,30], there exists a family of conditional probability measures ${\mu }_Y$, supported on fibers $\mathcal{F}\left(Y\right)$, such that

$\begin{array}{cccc}& \mu \left(A\right)=\int {\mu }_Y(A\cap \mathcal{F}(Y\left)\right)d\nu \left(Y\right).& & (234)\end{array}$

The conditional measures describe the unresolved thermo-acoustic degrees of freedom remaining after incompressible projection. The entropy-production density therefore possesses a conditional representation

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}=D_{\mathrm{e}\mathrm{n}\mathrm{t}}(Y,Z).& & (235)\end{array}$

Its conditional expectation relative to the projected incompressible variables is

$\begin{array}{cccc}& \mathbb{E}\left[D_{\mathrm{e}\mathrm{n}\mathrm{t}}\right|Y].& & (236)\end{array}$

However, conditional expectation does not imply reconstructibility. Indeed, the conditional variance

$\begin{array}{cccc}& \mathrm{V}\mathrm{a}\mathrm{r}\left(D_{\mathrm{e}\mathrm{n}\mathrm{t}}\right|Y)& & (237)\end{array}$

is strictly positive in general inside projection fibers. This means that the projected incompressible variables do not uniquely determine the entropy-generating thermo-acoustic structure. The Doob–Dynkin reconstruction principle [31] implies that a quantity is reconstructible from $Y$ only if it is measurable with respect to the sigma-algebra generated by $Y$.

The thermo-acoustic entropy structure generally fails to satisfy this property because thermal and acoustic variables remain unresolved inside fibers. Conditional disintegration frameworks of this type are standard in measurable dynamics and probabilistic reconstruction theory [29,30,31,32].

8.4. Projection-Induced Non-Reconstructibility

The previous sections now yield the central reconstruction theorem.

Theorem 5 (Projection-Induced Non-Reconstructibility).  Assume the admissible thermo-acoustic framework of Definition 1.  Then the entropy-generating thermo-acoustic structure is not generally reconstructible from incompressible projected variables alone.

More precisely, there exist thermo-acoustic states

$\begin{array}{cccc}& X_1\ne X_2& & (238)\end{array}$

satisfying

$\begin{array}{cccc}& \mathsf{\Pi }\left(X_1\right)=\mathsf{\Pi }\left(X_2\right),& & (239)\end{array}$

while

$\begin{array}{cccc}& D_{\mathrm{e}\mathrm{n}\mathrm{t}}\left(X_1\right)\ne D_{\mathrm{e}\mathrm{n}\mathrm{t}}\left(X_2\right).& & (240)\end{array}$

Consequently, the entropy-production density is not measurable with respect to the projected incompressible sigma-algebra generated by  $Y$.

Proof. The projection removes the thermodynamic variables $\left(\rho \right|T\left|s\right|c_s)$, whereas the entropy-production density depends explicitly on temperature, thermal gradients, viscosity coefficients, and the thermo-acoustic constitutive structure. Hence distinct thermo-acoustic states belonging to the same projection fiber may possess different entropy-production densities. The Doob–Dynkin theorem therefore implies non-reconstructibility from the projected variables alone [31]. □

The theorem does not claim that incompressible singularity formation must occur. Rather, it identifies a structural loss mechanism: the projected incompressible evolution no longer retains the full thermo-acoustic irreversible dissipation structure present in the compressible system.

8.5. Loss of Generative Dissipation Structure

The present theory interprets thermo-acoustic entropy production as a generative dissipative structure, with the entropy-production density contributing simultaneously to viscous dissipation, thermal relaxation, acoustic redistribution, and free-energy decay. These mechanisms are coupled through the compressible thermo-acoustic structure. The continuation theory established in Section 3, Section 4, Section 5, Section 6 and Section 7 depends fundamentally on

$\begin{array}{cccc}& \mathrm{entropy}\mathrm{coercivity}\Rightarrow \mathrm{nonlinear}\mathrm{subcriticality}\Rightarrow \mathrm{higher}\mathrm{integrability}\Rightarrow \epsilon -\mathrm{regularity}.& & (241)\end{array}$

However, the incompressible projection removes the thermodynamic completion variables responsible for this irreversible structure. Consequently, the projected incompressible system retains only mechanical transport, incompressible pressure redistribution, and viscous diffusion. The thermo-acoustic stabilization mechanisms associated with entropy production, finite acoustic propagation, and thermal relaxation are no longer explicitly represented after projection. This motivates the following structural interpretation.

Principle (Projection-Induced Loss of Generative Dissipation)

Incompressible projection removes part of the irreversible thermo-acoustic operator structure responsible for entropy-generating stabilization.

The statement above is intended as a structural observation concerning the thermo-acoustic continuation framework developed in Section 3, Section 4, Section 5, Section 6 and Section 7. In particular, the present analysis does not address singularity formation or exclusion of regularity for incompressible Navier–Stokes evolution. Rather, the conclusion is that the thermo-acoustic irreversible structure associated with entropy production, thermal relaxation, and finite-speed acoustic redistribution is not preserved under incompressible projection. The possibility of alternative stabilization mechanisms in incompressible systems therefore remains open within the present framework.

8.6. Structural Difference Between Thermo-Acoustic and Incompressible Evolution

The admissible compressible thermo-acoustic evolution studied in the present paper is characterized by entropy production, free-energy dissipation, finite acoustic propagation, thermal diffusion, and thermo-acoustic concentration control. The localized concentration quantity

$\begin{array}{cccc}& H(z,r)={\displaystyle \frac{1}{r}}{\int }_{Q_r\left(z\right)}{D}_{\mathrm{e}\mathrm{n}\mathrm{t}}dxdt.& & (242)\end{array}$

is intrinsically thermodynamic. Its localized coercive structure yields gradient control, nonlinear subcriticality, higher integrability, and ε-regularity. By contrast, the incompressible projected system evolves only through transport, incompressible pressure redistribution, and viscosity.

The thermodynamic completion variables are absent. Therefore, the two systems possess different continuation mechanisms:

$\begin{array}{cccc}& \mathrm{compressible}\mathrm{thermo}-\mathrm{acoustic}\mathrm{evolution}\ne \mathrm{mechanically}\mathrm{projected}\mathrm{incompressible}\mathrm{evolution}.& & (243)\end{array}$

The distinction is not merely quantitative. It is structural. The compressible admissible system possesses an irreversible thermo-acoustic stabilization mechanism generated by entropy production and finite-speed acoustic redistribution. The incompressible projected system does not explicitly retain this generative dissipation structure after projection.

This does not imply absence of all stabilizing mechanisms in incompressible dynamics. The present analysis identifies a structural difference associated with the thermo-acoustic irreversible dissipation mechanisms present in the admissible compressible framework.

9. Conclusions

The present paper developed a thermo-acoustic continuation framework for physically admissible compressible Navier–Stokes–Fourier evolution. The analysis was formulated under the assumptions of positivity of density and temperature, entropy admissibility, free-energy dissipation, finite acoustic propagation, strict hyperbolicity, uniformly subsonic evolution, constitutive smoothness, and a finite-energy weak solution framework. The analysis was carried out within a finite-energy thermo-acoustic admissible framework, with higher regularity obtained through the localized entropy, compactness, and Campanato arguments developed in the paper.

The central objective of the paper was to determine whether thermo-acoustic dissipative structure can suppress scale-critical concentration inside the admissible compressible NSF regime. The analysis showed that entropy production generates localized thermo-acoustic coercivity, and that this coercive structure yields nonlinear subcriticality of the localized remainder terms arising from transport, thermal coupling, acoustic interaction, pressure oscillation, coefficient variation, and commutator localization. The resulting localization structure produces higher thermo-acoustic integrability, Campanato-type oscillation decay, localized Hölder regularization, and ε-regularity. Within the admissible thermo-acoustic regime, persistent scale-critical concentration was excluded. Consequently, admissible thermo-acoustic evolution admits continuation beyond finite admissible evolution intervals.

The continuation mechanism established in the present work is generated by entropy production, thermal diffusion, free-energy dissipation, and finite-speed acoustic redistribution. The analysis therefore supports the following structural conclusion: for physically admissible compressible Navier–Stokes–Fourier evolution satisfying thermo-acoustic admissibility, finite acoustic propagation, uniformly subsonic evolution, entropy admissibility, and free-energy dissipation, scale-critical concentration is suppressed by thermo-acoustic dissipative structure, and regularity continuation follows from the resulting entropy-controlled localization mechanism.

The paper also analyzed the incompressible projection of the thermo-acoustic system. The projection removes thermal structure, entropy production, and finite-speed acoustic redistribution. Using projection fibers and conditional disintegration theory, the analysis showed that the entropy-generating thermo-acoustic structure is not generally reconstructible from incompressible projected variables alone. This establishes a structural difference between admissible thermo-acoustic compressible evolution and mechanically projected incompressible evolution. The analysis identifies the absence of the thermo-acoustic irreversible dissipation structure established in the admissible compressible setting after incompressible projection.

The present paper does not prove unconditional global regularity for arbitrary compressible NSF solutions, global construction of admissible weak solutions from arbitrary initial data, unconditional propagation of thermo-acoustic admissibility, or singularity formation or non-formation for incompressible Navier–Stokes evolution. The admissibility assumptions are treated as the physical regime of the theory rather than as derived conclusions. The continuation result is therefore conditional on persistence of thermo-acoustic admissibility throughout the evolution interval.

The present analysis identifies thermo-acoustic entropy generation as the governing irreversible structure responsible for suppressing scale-critical concentration inside the admissible compressible NSF regime.