Specified small-perturbation material dynamics

Use the selected local material state, perfect stress branch, bulk-only relaxation branch, and unit conventions of Smooth perfect local material balance model, Specified bulk extended entropy and relaxation law, and Thermodynamics-GR fields, currents, reference normalization and units. For this effective model take prescribed Minkowski spacetime ${\mathbb{R}}^4$ with coordinates $(x^0,\mathbf{x})=(ct,\mathbf{x})$, $t\in \mathbb{R}$, $\mathbf{x}\in {\mathbb{R}}^3$ measured in metres, and metric $\mathrm{diag}(-1,1,1,1)$. The reference material is a constant state $(v_0,{\sigma }_{p0})$ in the supplier's open state domain, with supplied $e\in C^2$, $u_0=(1,0,0,0)$, ${\mathrm{\Pi }}_0=0$, $n_0=1\mathrm{/}{v}_0>0$, ${\epsilon }_0=n_{0}e(v_0,{\sigma }_{p0})$, $p_0=-e_v(v_0,{\sigma }_{p0})$, $T_0=e_{{\sigma }_p}(v_0,{\sigma }_{p0})>0$, and enthalpy density $h_0={\epsilon }_0+p_0>0$. The local Gibbs supplier Local Gibbs, Euler, parcel first law and response identities gives the dimensionless adiabatic response

$a_s:={\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }\epsilon }\right)}_{{\sigma }_p}=\frac{e_{vv}(v_0,{\sigma }_{p0})}{n_0{h}_0};$

restrict this pair of effective branches to $a_s>0$. Here $e_{vv}$ is the second derivative of the supplied $C^2$ state function with respect to $v$ at fixed ${\sigma }_p$. The equality between the derivative and the state-function ratio is the exact interface stated by Local Gibbs, Euler, parcel first law and response identities under the reference-state hypotheses above. The derivative is the local fixed-composition response at the reference state; the equation does not assign a universal value to $a_s$.

For this postulate, a first variation means the derivative at $\lambda =0$ of a smooth one-parameter family of field configurations through the reference state, with dimensionless $\lambda$; no member with $\lambda \mathrm{\ne }0$ is required to solve a nonlinear equation. The fields $r,\widehat{\mathrm{\Pi }}:\mathbb{R}\times {\mathbb{R}}^3\to \mathbb{R}$ and $w:\mathbb{R}\times {\mathbb{R}}^3\to {\mathbb{R}}^3$ are smooth. Write $r=\delta \epsilon$, $w=(\delta u^1,\delta u^2,\delta u^3)$, and, in the bulk branch, $\widehat{\mathrm{\Pi }}=\delta \mathrm{\Pi }$. The adiabatic sector means $\delta {\sigma }_p=0$. These first variations have $r$ and $\widehat{\mathrm{\Pi }}$ in Pa and $w$ and $a_s$ dimensionless. Let ${\zeta }_0>0$ and ${\tau }_0>0$ be the bulk viscosity and relaxation time evaluated at the reference state, with units Pa s and s. Spatial derivatives have units ${\mathrm{m}}^{-1}$ and $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$.

Adopt the following two constant-coefficient effective systems on the stated background:

$r_t+c{h}_0\mathrm{div}w=0,w_t+\frac{c{a}_s}{h_0}\mathrm{\nabla }r=0$

for the perfect branch, and

$r_t+c{h}_0\mathrm{div}w=0,w_t+\frac{c}{h_0}\mathrm{\nabla }(a_{s}r+\widehat{\mathrm{\Pi }})=0,{\tau }_0{\widehat{\mathrm{\Pi }}}_t+\widehat{\mathrm{\Pi }}=-{\zeta }_{0}c\mathrm{div}w$

for the bulk-only branch. Derivatives $\mathrm{\nabla }$ and $\mathrm{div}$ act on $\mathbf{x}$; $t$ is physical time and $c$ is the speed parameter in Thermodynamics-GR fields, currents, reference normalization and units. The equations define the effective linear models. No uniform nonlinear error estimate, nonlinear solution lifespan, curved-spacetime evolution result, or existence claim for the nonlinear constitutive equations is postulated. The parameters and systems are adopted model inputs; they are not measurements or claims of universal material behavior.

For later solution claims, ${\mathbb{T}}_L^3:=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$ is the flat three-torus for $L>0$, and smooth periodic data mean $C^{\mathrm{\infty }}$ maps on that quotient. A multi-index is a triple in ${\mathbb{N}}^3$. A Schwartz map $U_0:{\mathbb{R}}^3\to {\mathbb{R}}^m$ means a $C^{\mathrm{\infty }}$ map such that $x^{\alpha }{\mathrm{\partial }}^{\beta }{U}_0(x)$ is bounded for every pair of multi-indices $\alpha ,\beta$. The systems have $m=4$ and $m=5$ components, respectively.

Assume the adiabatic perfect-material linear branch of Specified small-perturbation material dynamics on its prescribed Minkowski background, with $c>0$, $h_0>0$, and $0<a_s\le 1$, and assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. For every initial datum $U_0=(r_0,w_0)\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^4)$ or $U_0\in C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^4)$ with $L>0$, the Cauchy problem $U(0,\cdot )=U_0$ for the system

$r_t+c{h}_0\mathrm{div}w=0,w_t+\frac{c{a}_s}{h_0}\mathrm{\nabla }r=0$

has a unique global smooth solution on the corresponding space-time domain, contractive in the positive quadratic energy and its weighted Sobolev norms. Its energy density

${\mathcal{E}}_{\mathrm{p}\mathrm{f}}=\frac{1}{2}[\frac{a_s}{h_0}{r}^2+h_0\mathrm{\mid }w{\mathrm{\mid }}^2]$

satisfies ${\mathrm{\partial }}_t{\mathcal{E}}_{\mathrm{p}\mathrm{f}}+\mathrm{div}(c{a}_{s}rw)=0$. For each Sobolev exponent $q\ge 0$, its weighted Sobolev norm is nonincreasing for $t\ge 0$. The characteristic speeds, meaning eigenvalues of the principal matrix in a unit spatial direction, are $0$ and $\pm c\sqrt{a_s}$. Finite propagation means that if initial data vanish on a ball of radius $R$, then at time $t$ they vanish on the concentric ball of radius $R-c\sqrt{a_s}t$ whenever $0\le t<R\mathrm{/}(c\sqrt{a_s})$; on ${\mathbb{T}}_L^3$ use geodesic balls. Thus the exact linear propagation speed is $c\sqrt{a_s}\le c$. Equality is permitted at $a_s=1$. This is a conditional model deduction and makes no claim about nonlinear or curved-spacetime evolution, or about nature beyond the adopted model.

Assume the adiabatic bulk-only linear branch of Specified small-perturbation material dynamics on its prescribed Minkowski background, with $h_0,T_0,c,a_s,{\zeta }_0,{\tau }_0>0$, the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, and

$a_s+\frac{{\zeta }_0}{{\tau }_0{h}_0}\le 1.$

For every initial datum $U_0=(r_0,w_0,{\widehat{\mathrm{\Pi }}}_0)\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^5)$ or $U_0\in C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^5)$ with $L>0$, the Cauchy problem $U(0,\cdot )=U_0$ for the system

$r_t+c{h}_0\mathrm{div}w=0,w_t+\frac{c}{h_0}\mathrm{\nabla }(a_{s}r+\widehat{\mathrm{\Pi }})=0,{\tau }_0{\widehat{\mathrm{\Pi }}}_t+\widehat{\mathrm{\Pi }}=-{\zeta }_{0}c\mathrm{div}w$

has a unique global smooth solution on the corresponding space-time domain and is contractive in its positive quadratic energy and weighted Sobolev norms. Its energy density

${\mathcal{E}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=\frac{1}{2}[\frac{a_s}{h_0}{r}^2+h_0\mathrm{\mid }w{\mathrm{\mid }}^2+\frac{{\tau }_0}{{\zeta }_0}{\widehat{\mathrm{\Pi }}}^2]$

satisfies ${\mathrm{\partial }}_t{\mathcal{E}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}+\mathrm{div}[c(a_{s}r+\widehat{\mathrm{\Pi }})w]=-{\widehat{\mathrm{\Pi }}}^2\mathrm{/}{\zeta }_0$. For each Sobolev exponent $q\ge 0$, its weighted Sobolev norm is nonincreasing for $t\ge 0$. The characteristic speeds, meaning eigenvalues of the principal matrix in a unit spatial direction, are $0$ and $\pm c\sqrt{a_s+{\zeta }_0\mathrm{/}({\tau }_0{h}_0)}$. Finite propagation means that if initial data vanish on a ball of radius $R$, then at time $t$ they vanish on the concentric ball of radius $R-v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}t$ whenever $0\le t<R\mathrm{/}{v}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$, where $v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=c\sqrt{a_s+{\zeta }_0\mathrm{/}({\tau }_0{h}_0)}$; on ${\mathbb{T}}_L^3$ use geodesic balls. Hence propagation is bounded by $c$ under the stated inequality. Equality at the causal threshold is allowed. Energy contraction does not assert decay of every mode. The result is conditional on the adopted linear model; it is not a nonlinear or curved-spacetime existence claim or an empirical conclusion.