Complete symmetrized constant-coefficient evolution

Assume countable choice. Let $m\ge 1$, let $U:\mathbb{R}\times {\mathbb{R}}^3\to {\mathbb{R}}^m$, and let $A_1,A_2,A_3,B,M\in {\mathbb{R}}^{m\times m}$ be constant matrices. Assume $M=M^{\mathsf{T}}$ is positive definite, each $M{A}_j$ is symmetric, and $S:=(MB+B^{\mathsf{T}}M)\mathrm{/}2$ is positive semidefinite. Consider

${\mathrm{\partial }}_{t}U+{\sum }_{j=1}^3{A}_j{\mathrm{\partial }}_{x_j}U+BU=0.$

Use Schwartz space and its seminorms: for $f\in C^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C})$ and multi-indices $\alpha ,\beta \in {\mathbb{N}}^3$, the scalar seminorm is $p_{\alpha \beta }(f):={\sup }_{x\in {\mathbb{R}}^3}\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }f(x)\mathrm{\mid }$, and $\mathcal{S}({\mathbb{R}}^3)$ consists of functions for which all these seminorms are finite. Here $\mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^m)$ means maps $U=(U_1,\dots ,U_m)$ with real-valued components such that each component, viewed as complex-valued with zero imaginary part, belongs to $\mathcal{S}({\mathbb{R}}^3)$; its vector seminorms are $p_{\alpha \beta }^{(m)}(U):={\max }_{1\le \mathrm{\ell }\le m}{p}_{\alpha \beta }(U_{\mathrm{\ell }})$. For $U_0\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^m)$ there is a unique global smooth solution, Schwartz in $x$ at each time and continuous in every $p_{\alpha \beta }^{(m)}$ on compact time intervals. With the negative-sign, $2\pi$-normalized Fourier transform, put $\widehat{U}(\xi )={\int }_{{\mathbb{R}}^3}{e}^{-2\pi ix\cdot \xi }U(x)dx$ and define, for $s\ge 0$,

$\mathrm{\parallel }U{\mathrm{\parallel }}_{H_M^s}^2:={\int }_{{\mathbb{R}}^3}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^s\widehat{U}(\xi {)}^{\ast }M\widehat{U}(\xi )d\xi \mathrm{.}$

For $t\ge 0$, $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{H_M^s}\le \mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^s}$ for every $s\ge 0$; hence the solution depends continuously on the data in each such norm. The same conclusions hold for smooth periodic data on the flat torus ${\mathbb{T}}_L^3:=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$, $L>0$, with Fourier coefficients ${\widehat{U}}_k=L^{-3}{\int }_{[0,L{]}^3}U(x)e^{-2\pi ik\cdot x\mathrm{/}L}dx$ and

$\mathrm{\parallel }U{\mathrm{\parallel }}_{H_M^s({\mathbb{T}}_L^3)}^2:={\sum }_{k\in {\mathbb{Z}}^3}(1+\mathrm{\mid }k\mathrm{/}L{\mathrm{\mid }}^2{)}^s{\widehat{U}}_k^{\ast }M{\widehat{U}}_k\mathrm{.}$

For periodic data, $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le C_{s,L,M}\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^s({\mathbb{T}}_L^3)}$ for each integer $s>3\mathrm{/}2$.

For a unit vector $\nu \in {\mathbb{R}}^3$, write $A(\nu )={\sum }_j{\nu }_j{A}_j$ and $\tilde{A}(\nu )=M^{1\mathrm{/}2}A(\nu )M^{-1\mathrm{/}2}$. The finite speed

$v_{\ast }:={\sup }_{\mathrm{\mid }\nu \mathrm{\mid }=1}\max \{\mathrm{\mid }\lambda \mathrm{\mid }:\lambda \in \mathrm{spec}(\tilde{A}(\nu ))\}$

is finite. If $v_{\ast }>0$, $R>v_{\ast }t$, and $U_0$ vanishes on $B_R(x_0)$, then $U(t)$ vanishes on $B_{R-v_{\ast }t}(x_0)$ for $0\le t<R\mathrm{/}{v}_{\ast }$. The same assertion holds on ${\mathbb{T}}_L^3$ using its geodesic distance. If $v_{\ast }=0$, every $A_j=0$ and the system is a pointwise linear ODE, so spatial support does not move. These are conclusions for the displayed linear systems; no variable-coefficient or nonlinear existence claim is included.

Fix $c>0$ and take $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,\mathbf{x})$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and future causal cone $\{V\in {\mathbb{R}}^4:V^0\ge 0,{\eta }_{ab}{V}^a{V}^b\le 0\}$. For $V=(V^0,\mathbf{V})\in {\mathbb{R}}^4$ with $V^0>0$, define its coordinate velocity by $\mathbf{v}(V):=c\mathbf{V}\mathrm{/}{V}^0$. Since $\eta (V,V)=(V^0{)}^2(\mathrm{\mid }\mathbf{v}(V){\mathrm{\mid }}^2\mathrm{/}{c}^2-1)$, membership in this cone 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. The constant rest observer is $\bar{u}=(1,0,0,0)$.

Let $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ be open, let $e\in C^2(X;\mathbb{R})$, and fix $(v_0,{\sigma }_{p0})\in X$. 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 $h_0:={ϵ}_0+p_0$. Assume $n_0,T_0,h_0,e_{vv}>0$, with $e_{vv}$ evaluated at fixed ${\sigma }_p$. A first variation is defined by a family of state maps $(v_{\lambda },{\sigma }_{p,\lambda }):M\to X$ and fields $u_{\lambda }:M\to {\mathbb{R}}^4$, jointly smooth in dimensionless $\lambda$ near $0$ and in $x$, with the displayed reference state and $\bar{u}$ as their constant $\lambda =0$ values and with $\eta (u_{\lambda },u_{\lambda })=-1$, $u_{\lambda }^0>0$. For any such field $f_{\lambda }$, write $\delta f:={{\mathrm{\partial }}_{\lambda }{f}_{\lambda }\mid }_{\lambda =0}$; put $r:=\delta ϵ$, $w:=(\delta u^1,\delta u^2,\delta u^3)$, and impose the adiabatic first-variation condition $\delta {\sigma }_p=0$. The linearized equations mean the first variations at $\lambda =0$ of the no-force balances ${\mathrm{\partial }}_a{T}_{\lambda }^{ab}=0$ and ${\mathrm{\partial }}_a{J}_{\lambda }^a=0$, with $n_{\lambda }=1\mathrm{/}{v}_{\lambda }$, ${ϵ}_{\lambda }=n_{\lambda }e(v_{\lambda },{\sigma }_{p,\lambda })$, $p_{\lambda }=-e_v(v_{\lambda },{\sigma }_{p,\lambda })$, ${\mathrm{\Delta }}_{\lambda }^{ab}={\eta }^{ab}+u_{\lambda }^a{u}_{\lambda }^b$, $T_{\lambda }^{ab}={ϵ}_{\lambda }{u}_{\lambda }^a{u}_{\lambda }^b+p_{\lambda }{\mathrm{\Delta }}_{\lambda }^{ab}$, and $J_{\lambda }^a=n_{\lambda }{u}_{\lambda }^a$; no member with $\lambda \mathrm{\ne }0$ is required to solve the nonlinear balances. Differentiating $\eta (u_{\lambda },u_{\lambda })=-1$ gives $\delta u^0=0$.

The local Gibbs identity of Local Gibbs, Euler, parcel first law and response identities gives

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

and assume additionally $a_s\le 1$. In the adopted units, $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $r$ and $h_0$ have units Pa, and $w$ and $a_s$ are dimensionless. Since $x^0=ct$ and $\bar{u}={\mathrm{\partial }}_{x^0}$, the first variations of the no-force perfect-fluid projection equations of Perfect stress and particle balances with entropy identity are

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

For initial data $U_0=(r_0,w_0)$ whose real components belong componentwise to $\mathcal{S}({\mathbb{R}}^3)$ as defined by Schwartz space and its seminorms, or for smooth periodic data on ${\mathbb{T}}_L^3$ with $L>0$, these equations have the unique global smooth solution supplied by Complete symmetrized constant-coefficient evolution. The positive quadratic energy density $\mathcal{E}=\frac{1}{2}[(a_s\mathrm{/}{h}_0)r^2+h_0\mathrm{\mid }w{\mathrm{\mid }}^2]$ has flux $c{a}_{s}rw$, and the solution is contractive in the corresponding weighted $L^2$ and Sobolev norms. Its characteristic speeds are $0$ and $\pm c\sqrt{a_s}$; for positive $a_s$, this propagation cone lies inside the Minkowski reference cone exactly when $a_s\le 1$. These are statements about this frozen linear system only; they do not assert a nonlinear or curved-spacetime evolution theorem.

For the formal coefficient comparison in [A3], if the same linear equations are assigned $a_s>1$, their plane-wave speed is $c\sqrt{a_s}>c$. If they are assigned $a_s<0$, smooth periodic modes grow as $e^{c\sqrt{-a_s}\mathrm{\mid }k\mathrm{\mid }t}$ at arbitrarily large wave numbers. The case $a_s=0$ is degenerate: the displayed positive energy matrix is no longer positive definite, so the stability proof above does not apply.

Use the smooth $(-,+,+,+)$ Lorentzian data, local state, and conventions of Perfect stress and particle balances with entropy identity and Frame-qualified dissipative entropy cancellation, with $n>0$, $T>0$, $c>0$, $x^0=ct$, and smooth future unit $u$. On the nonempty open spacetime domain $O$ of those suppliers, let $\zeta ,\tau \in C^1(O;(0,\mathrm{\infty }))$ be scalar coefficient fields, set $\beta :=\tau \mathrm{/}\zeta \in C^1(O;(0,\mathrm{\infty }))$, and let $\mathrm{\Pi }:O\to \mathbb{R}$ be smooth. In the bulk-only branch take

$T^{ab}=ϵu^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,{\mathrm{\nabla }}_a{T}^{ab}=0,{\mathrm{\nabla }}_a{J}^a=0,$

where ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a_{\mathrm{a}\mathrm{c}\mathrm{c}}^b=D{u}^b$. Adopt the corrected relaxation law

$\beta D\mathrm{\Pi }+\frac{\mathrm{\Pi }}{\zeta c}=-\theta -\frac{T\mathrm{\Pi }}{2}{\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\beta u^a}{T}\right)$

and the extended entropy current $S^a=[s-\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)]u^a$. Then the stress and particle equations imply

$Dϵ+(h+\mathrm{\Pi })\theta =0,(h+\mathrm{\Pi })a_{\mathrm{a}\mathrm{c}\mathrm{c}}^b+{\mathrm{\Delta }}^{ba}{\mathrm{\nabla }}_a(p+\mathrm{\Pi })=0,Dn+n\theta =0,$

and every actual smooth solution of the stated corrected law satisfies

${\mathrm{\nabla }}_a{S}^a=\frac{{\mathrm{\Pi }}^2}{\zeta cT}\ge 0.$

Here $\beta$ has units $\mathrm{P}{\mathrm{a}}^{-1}$, $\mathrm{\Pi },ϵ,p,h$ have units Pa, $D,\theta ,a_{\mathrm{a}\mathrm{c}\mathrm{c}}$ have inverse-length components, and the correction $\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)$ has the units of entropy density. The law is an explicit model assumption; dropping its derivative term does not yield the stated sign identity in general.

There is an exact homogeneous solution on the explicitly specified Minkowski spacetime $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,x^1,x^2,x^3)$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and future direction ${\mathrm{\partial }}_{x^0}$; its Levi-Civita derivative is ordinary coordinate differentiation. Fix a state point $(v_0,{\sigma }_{p0})$ in the supplied local state domain with $v_0>0$ and $T_0=e_{{\sigma }_p}(v_0,{\sigma }_{p0})>0$, and take the constant state map $(v,{\sigma }_p)(x)=(v_0,{\sigma }_{p0})$, so $n_0=1\mathrm{/}{v}_0$, ${ϵ}_0=n_{0}e(v_0,{\sigma }_{p0})$, and $p_0=-e_v(v_0,{\sigma }_{p0})$ are constant. For constant positive coefficients ${\zeta }_0,{\tau }_0$, let $u={\mathrm{\partial }}_{x^0}=(1,0,0,0)$, $T=T_0$, and define the smooth scalar field $\mathrm{\Pi }(x)={\mathrm{\Pi }}_0{e}^{-t\mathrm{/}{\tau }_0}$ on all of $M$, where $t=x^0\mathrm{/}c$. The stress and particle balances and the corrected law hold, with strict entropy production wherever $\mathrm{\Pi }(x)\mathrm{\ne }0$. This mathematical witness is not a general nonlinear existence theorem or a claim of physical applicability in any separately selected near-equilibrium regime.

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.