Causal stable bulk-relaxation linear response

Given: The physical time and spatial coordinates $(t,\mathbf{x})$, the adiabatic bulk-only branch, and the state and transport parameters and units specified in Specified small-perturbation material dynamics.

$h_0,T_0,c,a_s,{\zeta }_0,{\tau }_0>0$ and $a_s+{\zeta }_0\mathrm{/}({\tau }_0{h}_0)\le 1$. Positivity makes the displayed energy positive definite; the upper bound is the additional causal-branch assumption.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$: for every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a function $f$ with $f(n)\in X_n$ for every $n$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). This is the exact choice strength used by the mathematical supplier proof; no stronger choice principle is assumed.

Specified small-perturbation material dynamics: the bulk branch has $e\in C^2(X)$, $n_0,T_0,h_0>0$, $a_s=e_{vv}\mathrm{/}(n_0{h}_0)>0$, $\delta {\sigma }_p=0$, constant ${\zeta }_0,{\tau }_0>0$, and adopts the displayed three-equation system about $u_0=(1,0,0,0)$ and ${\mathrm{\Pi }}_0=0$. In particular $e_{vv}>0$ follows from the response identity and these strict inequalities; [A1] supplies the additional causal inequality. Put $U=(r,w,\widehat{\mathrm{\Pi }}):\mathbb{R}\times {\mathbb{R}}^3\to {\mathbb{R}}^5$.

Bulk linear relaxation energy, stability and causal speed (current batch-56 mathematical draft): under its exact state and adiabatic hypotheses, $n_0,T_0,h_0,e_{vv},c,{\zeta }_0,{\tau }_0>0$, $a_s=e_{vv}\mathrm{/}(n_0{h}_0)>0$, the corrected bulk-model interface, and [A2], its complete conditional proof gives the positive energy identity, unique global smooth solutions for $U_0\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^5)$ or $C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^5)$, and weighted Sobolev contraction $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{H_M^q}\le \mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^q}$ for every $q\ge 0$. Here $M=\mathrm{diag}(a_s\mathrm{/}{h}_0,h_0{I}_3,{\tau }_0\mathrm{/}{\zeta }_0)$, where $I_3$ is the $3\times 3$ identity; on ${\mathbb{R}}^3$ the Fourier transform is $\widehat{U}(\xi )={\int }_{{\mathbb{R}}^3}{e}^{-2\pi i\mathbf{x}\cdot \xi }U(\mathbf{x})d\mathbf{x}$ and $\mathrm{\parallel }U{\mathrm{\parallel }}_{H_M^q}^2={\int }_{{\mathbb{R}}^3}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^q\widehat{U}(\xi {)}^{\ast }M\widehat{U}(\xi )d\xi$. On the torus, ${\widehat{U}}_k=L^{-3}{\int }_{[0,L{]}^3}U(\mathbf{x})e^{-2\pi i\mathbf{k}\cdot \mathbf{x}\mathrm{/}L}d\mathbf{x}$ and $\mathrm{\parallel }U{\mathrm{\parallel }}_{H_M^q}^2={\sum }_{k\in {\mathbb{Z}}^3}(1+\mathrm{\mid }\mathbf{k}\mathrm{/}L{\mathrm{\mid }}^2{)}^q{\widehat{U}}_k^{\ast }M{\widehat{U}}_k$. It gives characteristic speeds $0,\pm v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ and the ball-shrinking finite-propagation property stated above. The positive-response, zero-reference-bulk-pressure, data-class and countable-choice hypotheses here match that exact mathematical interface.