Bulk linear relaxation energy, stability and causal speed

Use $\mathcal{M}={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,\mathbf{x})$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, future direction ${\mathrm{\partial }}_{x^0}$, and reference causal cone $\{V\in {\mathbb{R}}^4:V^0\ge 0,{\eta }_{ab}{V}^a{V}^b\le 0\}$. For $V=(V^0,\mathbf{V})$ with $V^0>0$, define $\mathbf{v}(V):=c\mathbf{V}\mathrm{/}{V}^0$; then $\eta (V,V)=(V^0{)}^2(\mathrm{\mid }\mathbf{v}(V){\mathrm{\mid }}^2\mathrm{/}{c}^2-1)$, so cone membership is equivalent to $\mathrm{\mid }\mathbf{v}(V)\mathrm{\mid }\le c$, with equality on its nonzero null boundary. The zero vector belongs to the cone but has no coordinate velocity under this definition. Let $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ be open, $e\in C^2(X;\mathbb{R})$, and $(v_0,{\sigma }_{p0})\in X$. Define $n_0:=1\mathrm{/}{v}_0$, ${ϵ}_0:=n_{0}e(v_0,{\sigma }_{p0})$, $p_0:=-e_v(v_0,{\sigma }_{p0})$, $T_0:=e_{{\sigma }_p}(v_0,{\sigma }_{p0})$, and $h_0:={ϵ}_0+p_0$. The background fields are this constant state, $u_0={\mathrm{\partial }}_{x^0}=(1,0,0,0)$, and ${\mathrm{\Pi }}_0=0$. Assume $n_0,T_0,h_0,e_{vv},c>0$, with $e_{vv}$ evaluated at fixed ${\sigma }_p$.

A first variation is defined by a family $(v_{\lambda },{\sigma }_{p,\lambda },u_{\lambda },{\mathrm{\Pi }}_{\lambda })$ jointly smooth in dimensionless $\lambda$ near $0$ and in $x$, through the displayed background, with $u_{\lambda }$ future unit, meaning $\eta (u_{\lambda },u_{\lambda })=-1$ and $u_{\lambda }^0>0$. For any such field $f_{\lambda }$, set $\delta f:={{\mathrm{\partial }}_{\lambda }{f}_{\lambda }\mid }_{\lambda =0}$. At each $\lambda$, set $n_{\lambda }=1\mathrm{/}{v}_{\lambda }$, ${ϵ}_{\lambda }=n_{\lambda }e(v_{\lambda },{\sigma }_{p,\lambda })$, $p_{\lambda }=-e_v(v_{\lambda },{\sigma }_{p,\lambda })$, $T_{\lambda }=e_{{\sigma }_p}(v_{\lambda },{\sigma }_{p,\lambda })$, ${\mathrm{\Delta }}_{\lambda }^{ab}={\eta }^{ab}+u_{\lambda }^a{u}_{\lambda }^b$, $D_{\lambda }=u_{\lambda }^a{\mathrm{\partial }}_a$, and ${\theta }_{\lambda }={\mathrm{\partial }}_a{u}_{\lambda }^a$. The bulk stress and current are $T_{\lambda }^{ab}={ϵ}_{\lambda }{u}_{\lambda }^a{u}_{\lambda }^b+(p_{\lambda }+{\mathrm{\Pi }}_{\lambda }){\mathrm{\Delta }}_{\lambda }^{ab}$ and $J_{\lambda }^a=n_{\lambda }{u}_{\lambda }^a$, and the corrected law of Corrected bulk relaxation and exact extended entropy is $\beta D_{\lambda }{\mathrm{\Pi }}_{\lambda }+{\mathrm{\Pi }}_{\lambda }\mathrm{/}(\zeta c)=-{\theta }_{\lambda }-(T_{\lambda }{\mathrm{\Pi }}_{\lambda }\mathrm{/}2){\mathrm{\partial }}_a(\beta u_{\lambda }^a\mathrm{/}{T}_{\lambda })$, where $\beta =\tau \mathrm{/}\zeta$. The linearized equations mean the first variations at $\lambda =0$ of ${\mathrm{\partial }}_a{T}_{\lambda }^{ab}=0$, ${\mathrm{\partial }}_a{J}_{\lambda }^a=0$, and this corrected law, set equal to zero; no member with $\lambda \mathrm{\ne }0$ is required to solve the nonlinear system. Impose $\delta {\sigma }_p=0$, put $r:=\delta ϵ$, $w:=(\delta u^1,\delta u^2,\delta u^3)$, and denote $\delta \mathrm{\Pi }$ by $\mathrm{\Pi }$ in the linearized system below. Take constant coefficients $\zeta ,\tau >0$.

$a_s:={\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }ϵ}\right)}_{{\sigma }_p}=\frac{e_{vv}}{n_0{h}_0}>0$

as supplied by Local Gibbs, Euler, parcel first law and response identities. The resulting linearized system is

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

Its positive quadratic energy density

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

satisfies ${\mathrm{\partial }}_t{\mathcal{E}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}+\mathrm{div}[c(a_{s}r+\mathrm{\Pi })w]=-{\mathrm{\Pi }}^2\mathrm{/}\zeta$. The normal characteristic speeds are $0$ and $\pm v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$, where

$v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}=c\sqrt{a_s+\frac{\zeta }{\tau h_0}}\mathrm{.}$

By Complete symmetrized constant-coefficient evolution, initial data $U_0\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^5)$ with the componentwise real convention of Schwartz space and its seminorms, or smooth periodic data on ${\mathbb{T}}_L^3$ with $L>0$, have unique global smooth solutions, contractive in the corresponding energy and Sobolev norms, with finite propagation at $v_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$. The branch is inside the Minkowski reference cone exactly when $a_s+\zeta \mathrm{/}(\tau h_0)\le 1$. Constant density and transverse velocity modes can persist, so contraction does not mean every mode decays. No nonlinear or variable-coefficient existence theorem is claimed.