Conditional thermal entropy, stability, and finite propagation

For the global-evolution and forward-time clauses below, assume that the postulate's open time interval $I$ contains $0$. The local identities and telegrapher equation hold on any open interval on which a smooth solution is defined.

Let $C,\kappa ,\tau ,T_0,c,L>0$ and let $\mathrm{\Sigma }$ be ${\mathbb{R}}^3$ or ${\mathbb{T}}_L^3$. If $\mathrm{\Sigma }={\mathbb{T}}_L^3$, use the periodic-lift convention of Supported-rest effective Cattaneo thermal model: torus fields represent smooth spatially $L$-periodic fields on the rest slices of $M={\mathbb{R}}^4$. Adopt the supported-rest Cattaneo equations and effective current of Supported-rest effective Cattaneo thermal model. The equations are $C{\vartheta }_t+\mathrm{div}q=0$ and $\tau q_t+q=-\kappa \mathrm{\nabla }\vartheta$, with Euclidean spatial derivatives. For any smooth solution on an open time interval, the exact effective entropy-production and quadratic Lyapunov identities are

${\mathrm{\partial }}_t{s}_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{div}{j}_s=\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa T_0^2}\ge 0,{\mathrm{\partial }}_t{e}_L+\mathrm{div}(\vartheta q)=-\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa },e_L:=\frac{1}{2}(C{\vartheta }^2+\frac{\tau }{\kappa }\mathrm{\mid }q{\mathrm{\mid }}^2)\mathrm{.}$

The associated telegrapher equation is $\tau {\vartheta }_{tt}+{\vartheta }_t-(\kappa \mathrm{/}C)\mathrm{\Delta }\vartheta =0$, where $\mathrm{\Delta }:=\mathrm{div}\mathrm{\nabla }={\sum }_{i=1}^3{\mathrm{\partial }}_{x^i}^2$ on the Euclidean rest space, and its characteristic propagation speed is $v_h:=\sqrt{\kappa \mathrm{/}(C\tau )}$. Put $U_0=({\vartheta }_0,q_0):=(\vartheta (0,\cdot ),q(0,\cdot ))$. For Schwartz data on ${\mathbb{R}}^3$ or smooth periodic data on ${\mathbb{T}}_L^3$, the unique global smooth evolution, weighted-norm contraction, and finite propagation at $v_h$ follow conditionally from the exact mathematical interface in [F1]. Here Schwartz data mean componentwise $C^{\mathrm{\infty }}$ data for which every $x^{\alpha }{\mathrm{\partial }}^{\beta }{U}_0$ is bounded for multi-indices $\alpha ,\beta \in {\mathbb{N}}^3$, as specified in [F2]. Since the spacetime model uses light speed $c$, the heat subsystem is relativistically causal exactly when

$v_h\le c⟺\kappa \le C\tau c^2\mathrm{.}$

On ${\mathbb{T}}_L^3$, for each integer $r>3\mathrm{/}2$, the periodic solution supplied conditionally by [F1] obeys $T_0+\vartheta (t,x)>0$ for all $t\ge 0$ whenever $K_{r,L,C}\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^r({\mathbb{T}}_L^3)}<T_0$, with $K_{r,L,C}$ and $H_M^r$ exactly as defined there. This bound ensures positive temperature but does not by itself ensure that the solution stays in the selected domain $D$ of the postulate. The global linear PDE statement concerns the mathematical extension of the subsystem; its physical interpretation is restricted to times when $(\vartheta ,q)\in D$. No empirical coefficient or material-specific validity is asserted.