# A Galilei Invariant Version of Yong’s Model

Heinrich Freistühler

(June 30, 2021)

In his pioneering paper [31], Yong has described the dynamics of a
compressible fluid with Maxwell delayed viscosity as a symmetric hyperbolic
system of balance laws, and shown that the solutions of this system tend to
solutions of the Navier-Stokes equations.

The original main purpose of the present note was to propose a Galilei
invariant version of Yong’s model, as a system of balance laws (while also
communicating some simple observations on its shock waves). This has changed
in the course of time.

The model. For a barotropic fluid, $e=e(v),v=1/\rho,$ and
$p=p(v)=-e^{\prime}(v)$, with $-p^{\prime}(v)=e^{\prime\prime}(v)>0$ consider
the equations of motion

$\displaystyle\begin{aligned} \rho_{t}+(\rho u^{j})_{,j}&=0\\\ (\rho
u^{i})_{t}+(\rho
u^{j}u^{i}+p\delta^{ij})_{,j}+\frac{1}{\epsilon}(\sigma^{ij}+\tau\delta^{ij})_{,j}&=0\\\
(\rho\sigma^{kl})_{t}+(\rho
u^{j}\sigma^{kl})_{,j}+\frac{1}{\epsilon}c^{klj}_{i}u^{i}_{,j}&=-\frac{\sigma^{kl}}{\mu\epsilon^{2}}\\\
(\rho\tau)_{t}+(\rho
u^{j}\tau)_{,j}+\frac{1}{\epsilon}u^{j}_{,j}&=-\frac{\tau}{\nu\epsilon^{2}}\end{aligned}$
(1)

with

$c^{klj}_{i}=\frac{1}{2}(\delta^{k}_{i}\delta^{lj}+\delta^{l}_{i}\delta^{kj})-\frac{1}{3}\delta^{r}_{i}\delta^{j}_{r}\delta^{kl},$

and define the total energy and its flux as

$\displaystyle
E=\rho(e+\frac{1}{2}u_{i}u^{i}+\frac{1}{2}\sigma_{kl}\sigma^{kl}+\frac{1}{2}\tau^{2}),\quad
F^{j}=((E+p)\delta^{ij}+\frac{1}{\epsilon}(\sigma^{ij}+\tau\delta^{ij}))u_{i}.$
(2)

######  Theorem 1.

(i) Using $(E,F^{j})$ from (2) as an (“entropy”, “entropy” flux) pair, system
(1) can be written as a symmetric hyperbolic system of balance laws. All its
solutions dissipate energy,

$\displaystyle E_{t}+F^{j}_{,j}\leq 0.$ (3)

(ii) For any fixed values of $\epsilon>0$ and $\rho_{*}>0$ and any data
$({{}^{0}\rho}_{0},{{}^{0}u}^{i},{{}^{0}\sigma}^{kl},{{}^{0}\tau})$ such that
$({{}^{0}\rho}_{0}-\rho_{*},{{}^{0}u}^{i},{{}^{0}\sigma}^{kl},{{}^{0}\tau})\in
H^{s}(\mathbb{R}^{d}),s>d/2+1$ and ${{}^{0}\rho}_{0}>0$ is uniformly bounded
away from $0$, there exists a $\bar{t}>0$ such that system (1) has a solution
in $C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ that assumes these data.

(iii) If for certain initial data, the Navier-Stokes equations

$\displaystyle\begin{aligned} \rho_{t}+(\rho u^{j})_{,j}&=0\\\ (\rho
u^{i})_{t}+(\rho u^{j}u^{i}+p\delta^{ij})_{,j}&=(\mu
c^{ijl}_{k}u^{k}_{,l}+\nu\delta^{ij}u^{k}_{,k})_{,j}\end{aligned}$ (4)

have a solution in $C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ for some
$\bar{t}>0$, then there exists an $\bar{\epsilon}>0$ such that for every
$\epsilon\in(0,\bar{\epsilon})$ system (1) has a unique solution in
$C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ for the same data with
$\sigma^{kl}=\epsilon\mu c_{i}^{klj}u^{i}_{,j},\tau=\epsilon\nu u^{j}_{,j}$,
and the $(\rho,u)$-parts of these solutions converge, as $\epsilon\searrow 0$,
to the solution of (4).

###### Proof.

(i) Noting that

$\displaystyle\rho(u^{i}_{t}+u^{j}u^{i}_{,j})+\delta^{ij}p_{,j}+\frac{1}{\epsilon}(\sigma^{ij}+\tau\delta^{ij})_{,j}$
$\displaystyle=0,$
$\displaystyle\rho(\sigma^{kl}_{t}+u^{j}\sigma^{kl}_{,j})+\frac{1}{\epsilon}c^{klj}_{i}u^{i}_{,j}$
$\displaystyle=-\frac{\sigma^{kl}}{\mu\epsilon^{2}},$
$\displaystyle\rho(\tau_{t}+u^{j}\tau)_{,j}+\frac{1}{\epsilon}u^{j}_{,j}$
$\displaystyle=-\frac{\tau}{\nu\epsilon^{2}},$

and

$\sigma_{kl}c_{i}^{klj}=\sigma_{kl}(\frac{1}{2}(\delta^{k}_{i}\delta^{lj}+\delta^{l}_{i}\delta^{kj})-\frac{1}{3}\delta^{r}_{i}\delta^{j}_{r}\delta^{kl})=\sigma_{i}^{j},$

one finds

$\displaystyle E_{t}+F^{j}_{,j}$
$\displaystyle=\rho(e+\frac{1}{2}u_{i}u^{i}+\frac{1}{2}\sigma_{kl}\sigma^{kl}+\tau^{2})_{t}+\rho
u^{j}(e+\frac{1}{2}u_{i}u^{i}+\frac{1}{2}\sigma_{kl}\sigma^{kl}+\tau^{2})_{,j}+(pu^{j})_{,j}+\frac{1}{\epsilon}((\sigma^{ij}+\tau\delta^{ij})u_{i})_{,j}$
$\displaystyle=\frac{p}{\rho}(\rho_{t}+u^{j}\rho_{,j})+\rho
u_{i}(u^{i}_{t}+u^{j}u^{i}_{,j})+\rho\sigma_{kl}(\sigma^{kl}_{t}+u^{j}\sigma^{kl}_{,j})+\rho\tau(\tau_{t}+u^{j}\tau_{,j})+(pu^{j})_{,j}+\frac{1}{\epsilon}((\sigma^{ij}+\tau\delta^{ij})u_{i})_{,j}$
$\displaystyle=\\!-pu^{j}_{,j}\\!\\!-u_{i}(\delta^{ij}p_{,j}+\frac{1}{\epsilon}(\sigma^{ij}\\!+\\!\tau\delta^{ij})_{,j})-\sigma_{kl}(\frac{1}{\epsilon}c^{klj}_{i}u^{i}_{,j}\\!+\\!\frac{\sigma^{kl}}{\mu\epsilon^{2}})-\tau(\frac{1}{\epsilon}u^{j}_{,j}\\!+\\!\frac{\tau}{\nu\epsilon^{2}})+(pu^{j})_{,j}+\frac{1}{\epsilon}((\sigma^{ij}\\!+\\!\tau\delta^{ij})u_{i})_{,j}$
$\displaystyle=-u_{i}(\frac{1}{\epsilon}\sigma^{ij}_{,j})-\sigma_{kl}(\frac{1}{\epsilon}c^{klj}_{i}u^{i}_{,j})+\frac{1}{\epsilon}(\sigma^{ij}u_{i})_{,j}-\sigma_{kl}\frac{\sigma^{kl}}{\mu\epsilon^{2}}-u_{i}(\frac{1}{\epsilon}(\tau\delta^{ij})_{,j})-\tau(\frac{1}{\epsilon}u^{j}_{,j}+\frac{\tau}{\nu\epsilon^{2}})+\frac{1}{\epsilon}(\tau\delta^{ij}u_{i})_{,j}$
$\displaystyle=\frac{1}{\epsilon}(-u_{i}\sigma^{ij}_{,j}-\sigma^{j}_{i}u^{i}_{,j}+(\sigma^{ij}u_{i})_{,j})-\sigma_{kl}\frac{\sigma^{kl}}{\mu\epsilon^{2}}+\frac{1}{\epsilon}(-u_{i}(\tau\delta^{ij})_{,j}-\tau
u^{j}_{,j}+(\tau\delta^{ij}u_{i})_{,j})-\frac{\tau^{2}}{\nu\epsilon^{2}}$
$\displaystyle=-\frac{\sigma_{kl}\sigma^{kl}}{\mu\epsilon^{2}}-\frac{\tau^{2}}{\nu\epsilon^{2}}.$

This proves (3).

For solutions to the homogeneous system associated with (1), the same
calculation yields the corresponding identity

$\displaystyle E_{t}+F^{j}_{,j}=0.$ (5)

Since $E$ as a function of the conserved quantities $\rho,m^{j}=\rho
u^{j},S^{kl}=\rho\sigma^{kl},{\mathcal{T}}=\rho\tau$,

$\displaystyle E=\rho
e+\frac{m^{2}+S^{2}+{\mathcal{T}}^{2}}{2\rho}\quad\text{with }\quad
m^{2}={m_{i}m^{i}},\ S^{2}={S_{kl}S^{kl}},$ (6)

is convex,

$D^{2}E={\begin{pmatrix}\displaystyle{\frac{p^{\prime}(\rho)}{\rho}+\frac{m^{2}+S^{2}+{\mathcal{T}}^{2}}{\rho^{3}}}&\displaystyle{-\frac{m^{j}}{\rho^{2}}}&\displaystyle{-\frac{S^{rs}}{\rho^{2}}}\displaystyle{-\frac{{\mathcal{T}}}{\rho^{2}}}&0\\\
\displaystyle{-\frac{m_{i}}{\rho^{2}}}&\displaystyle{\frac{1}{\rho}\delta_{i}^{j}}&0&0\\\
\displaystyle{\frac{S_{kl}}{\rho^{2}}}&0&\displaystyle{\frac{1}{\rho}\delta_{kl}^{rs}}&0\\\
\displaystyle{-\frac{{\mathcal{T}}}{\rho^{2}}}&0&0&\displaystyle{\frac{1}{\rho}}\end{pmatrix}>0},$

the remaining part of the assertion follows from [4].

Assertions (ii) and (iii) follow in direct analogy with Yong’s beautiful
argument for his Theorem 3.1 in [31]. ∎

Very similar considerations hold for nonbarotropic fluids, $e=e(v,\varsigma)$
with

$\displaystyle D^{2}e(v,\varsigma)>0,\ p=-e_{v}(v,\varsigma),$ (7)

the equations of motion now reading

$\displaystyle\begin{aligned} \rho_{t}+(\rho u^{j})_{,j}&=0\\\ (\rho
u^{i})_{t}+(\rho
u^{j}u^{i}+p\delta^{ij})_{,j}+\frac{1}{\epsilon}(\sigma^{ij}+\tau\delta^{ij})_{,j}&=0\\\
E_{t}+F^{j}_{,j}&=0\\\ (\rho\sigma^{kl})_{t}+(\rho
u^{j}\sigma^{kl})_{,j}+\frac{1}{\epsilon}c^{klj}_{i}u^{i}_{,j}&=-\frac{\sigma^{kl}}{\mu\epsilon^{2}}\\\
(\rho\tau)_{t}+(\rho
u^{j}\tau)_{,j}+\frac{1}{\epsilon}u^{j}_{,j}&=-\frac{\tau}{\nu\epsilon^{2}}.\end{aligned}$
(8)

The same calculation as above, now taking into account that in this situation
$e$ and $p$ depend also on the specific entropy $\varsigma$, leads to the
identity

$\displaystyle\rho\theta(\varsigma_{t}+u^{j}\varsigma_{,j})=\frac{\sigma_{kl}\sigma^{kl}}{\mu\epsilon^{2}}+\frac{\tau^{2}}{\nu\epsilon^{2}}$
(9)

where $\theta=e_{s}(v,\varsigma)$ is the temperature. Equation (9) implies
that $\eta_{t}+\zeta^{j}_{,j}\geq 0$, i.e., the second law of thermodynamics
holds with

$\displaystyle\eta=\rho\varsigma,\quad\zeta^{j}=\rho\varsigma u^{j}.$ (10)

For solutions to the asscociated homogeneous system, the calculation shows the
corresponding identity

$\displaystyle\eta_{t}+\zeta^{j}_{,j}=0,$ (11)

which means that $(\eta,\zeta^{j})$ is an (entropy, entropy flux) pair for it.
Assumption (7) implying that $E$ is a convex function of the conserved
quantities $\rho,m^{i},S^{kl},{\mathcal{T}},\eta$, we see that $-\eta$ is a
convex function of the conserved quantities
$\rho,m^{i},E,S^{kl},{\mathcal{T}}$.

We have thus proven

######  Theorem 2.

Using the (entropy, entropy flux) pair $(\eta,\zeta^{j})$, system (8) can be
written as a symmetric hyperbolic system of balance laws. All its solutions
generate entropy,

$\displaystyle\eta_{t}+\zeta^{j}_{,j}\geq 0.$ (12)

(ii) For any fixed values of $\epsilon>0$ and $\rho_{*},\theta_{*}>0$ and any
data
$({{}^{0}\rho},{{}^{0}\theta},{{}^{0}u}^{i},{{}^{0}\sigma}^{kl},{{}^{0}\tau})$
such that
$({{}^{0}\rho}_{0}-\rho_{*},{{}^{0}\theta}-\theta_{*},{{}^{0}u}^{i},{{}^{0}\sigma}^{kl},{{}^{0}\tau})\in
H^{s}(\mathbb{R}^{d}),s>d/2+1,$ and ${{}^{0}\rho},{{}^{0}\theta}>0$ are
uniformly bounded away from $0$, there exists a $\bar{t}>0$ such that system
(8) has a solution in $C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ that assumes
these data.
(iii) If for certain initial data, the nonbarotropic Navier-Stokes equations

$\displaystyle\begin{aligned} \rho_{t}+(\rho u^{j})_{,j}&=0\\\ (\rho
u^{i})_{t}+(\rho u^{j}u^{i}+p\delta^{ij})_{,j}&=(\mu
c^{ijl}_{k}u^{k}_{,l}+\nu\delta^{ij}u^{k}_{,k})_{,j}\\\
\hat{E}_{t}+\hat{F}^{j}_{,j}&=((\mu
c^{ijl}_{k}u^{k}_{,l}+\nu\delta^{ij}u^{k}_{,k})u_{i})_{,j}\end{aligned}$ (13)

with

$\displaystyle\hat{E}=\rho(e+\frac{1}{2}u^{2}),\quad\hat{F}^{j}=(\hat{E}+p)u^{j}$
(14)

have a solution in $C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ for some
$\bar{t}>0$, then there exists an $\bar{\epsilon}>0$ such that for every
$\epsilon\in(0,\bar{\epsilon})$ system (8) has a unique solution in
$C^{1}([0,\bar{t}],H^{s}(\mathbb{R}^{d}))$ for the same data, augmented via
$\sigma^{kl}=\epsilon\mu c_{i}^{klj}u^{i}_{,j},\tau=\epsilon\nu u^{j}_{,j}$,
and the $(\rho,u,s)$-parts of these solutions converge, as $\epsilon\searrow
0$, to the solution of (13).

Shock waves. For general fluids with convex internal energy,
$e^{\prime\prime}(v)>0$ or $D^{2}e(v,\varsigma)>0$ as considered already so
far, both the barotropic and the nonbarotropic Navier-Stokes equations support
shock waves in the sense of heteroclinic traveling wave solutions

$\displaystyle(\rho(t,x),u(t,x))=(R(x\cdot n-ct),U(x\cdot n-ct))$ (15)

or

$\displaystyle(\rho(t,x),u(t,x),\theta(t,x))=(R(x\cdot n-ct),U(x\cdot
n-ct),\Theta(x\cdot n-ct)),$ (16)

respectively [9, 28, 2], where $n\in S^{d-1}$ is the direction and $c$ the
speed of propagation and $(R,U)$, or $(R,U,\Theta)$ respectively, is the
“profile” of the shock, a heteroclinic solution of the associated ODE, with
limits

$\displaystyle(R,U)(\pm\infty)=(\rho^{\pm},u^{\pm})\\!\text{ or
}\\!(R,U,\Theta)(\pm\infty)=(\rho^{\pm},u^{\pm},\theta^{\pm}),\ \\!\text{
respectively,}$ (17)

that, as the respective pairs of downstream and upstream states, fulfil the
Rankine-Hugoniot jump conditions. The time-asymptotic stability of these
viscous shock waves against planar (“longitudinal“, “onedimensional”)
perturbations has early on been studied in [11, 15, 16, 29] and then, Zumbrun
et al. have shown [33, 22, 21, 20] that the nonlinear stability reduces
completely to the so-called Evans function condition, that is, the requirement
that the Evans function $D=D^{v}$, a certain analytic function defined on an
open superset of the closed right half
$\mathbb{H}=\\{\lambda:\text{Re}(\lambda)\geq 0\\}$ of $\mathbb{C}$ (cf., e.
g., [7]) has precisely one zero on $\mathbb{H}$, namely at $0$, and this zero
is simple.
Zumbrun and co-workers have also studied shock profiles in systems of balance
laws (“relaxed conservation laws“) [23, 25, 19, 18], again with the result
that the nonlinear stability reduces to the Evans function condition, now
applied to $D=D^{r}$, the relaxation counterpart of the viscous Evans function
$D^{v}$.

Now, (1) and (8) are relaxed systems of conservation laws and as such obvious
candidates for the application of Zumbrun’s stability theory. We observe,
first in one space dimension:

######  Theorem 3.

Consider any viscous shock wave solution $(R,U)$ or $(R,U,\Theta)$ of (4) or
(13), respectively. Then the following hold: (i) For sufficiently small
$\epsilon>0$, also the respective relaxed system, i.e., (1) or (8), admits a
shock wave, $(R_{\epsilon},U_{\epsilon},\Sigma_{\epsilon},T_{\epsilon})$ or
$(R_{\epsilon},U_{\epsilon},\Theta_{\epsilon},\Sigma_{\epsilon},T_{\epsilon}),$
with

$(R_{\epsilon},U_{\epsilon})(\pm\infty)=(R,U)(\pm\infty)\quad\text{or}\quad(R_{\epsilon},U_{\epsilon},\Theta_{\epsilon})(\pm\infty)=(R,U,\Theta)(\pm\infty),\
\ \text{respectively.}$

and

$\Sigma_{\epsilon}(\pm\infty)=T_{\epsilon}(\pm\infty)=0;$

these relaxation profiles are regular perturbations of the Navier-Stokes
profiles,

$\lim_{\epsilon\to
0}(R_{\epsilon},U_{\epsilon},\Sigma_{\epsilon}+T_{\epsilon})=(R,U,-{\tilde{\mu}}U^{\prime})\text{
or }\lim_{\epsilon\to
0}(R_{\epsilon},U_{\epsilon},\Theta_{\epsilon},\Sigma_{\epsilon}+T_{\epsilon})=(R,U,\Theta,-{\tilde{\mu}}U^{\prime}),\text{respectively.}$

(ii) There exist an $\bar{\epsilon}>0$ and an open superset $\Lambda$ of
$\mathbb{H}$ in $\mathbb{C}$ such that the Evans function $D^{v}$ of the
Navier-Stokes shock profile and the Evans functions
$D^{r}_{\epsilon},0<\epsilon<\bar{\epsilon},$ of the relaxation profiles are
all defined on $\Lambda$ and satisfy there

$\displaystyle\lim_{\epsilon\to 0}D^{r}_{\epsilon}=D^{v},\quad\text{uniformly
on compacta.}$ (18)

###### Proof.

We discuss only the barotropic case (as the nonbarotropic case can be treated
in complete analogy). Rescaling (and noticing that in 1D, setting
$\tilde{\mu}=(2/3)\mu+\nu$, $\tau$ is redundant) we see that system (1) reads

$\displaystyle\begin{aligned} \rho_{t}+(\rho u)_{x}&=0\\\ (\rho u)_{t}+(\rho
u^{2}+p(1/\rho))+\sigma)_{x}&=0\\\ \epsilon((\rho\sigma)_{t}+(\rho
u\sigma)_{x})+u_{x}&=-\frac{\sigma}{\tilde{\mu}}.\end{aligned}$ (19)

(i) Choosing, due to Galilean invariance w. l. o. g., $n=1$ and $c=0$, the ODE
governing the relaxation shock profiles
$(R_{\epsilon},U_{\epsilon},\Sigma_{\epsilon})$ is

$\displaystyle\begin{aligned} (RU)^{\prime}&=0\\\
(RU^{2}+p(1/R)+\Sigma)^{\prime}&=0\\\
\epsilon(RU\Sigma)^{\prime}+U^{\prime}&=-\frac{\Sigma}{\tilde{\mu}}.\end{aligned}$
(20)

This system is a regular perturbation of its $\epsilon\to 0$ limit

$\displaystyle\begin{aligned} (RU)^{\prime}&=0\\\
(RU^{2}+p(1/R)+\Sigma)^{\prime}&=0\\\
U^{\prime}&=-\frac{\Sigma}{\tilde{\mu}},\end{aligned}$ (21)

which has the same rest points. But (21) is the system governing the Navier-
Stokes shock profiles. In both ODEs, $\Sigma$ vanishes at the rest points; in
the latter one, $\Sigma=-{\tilde{\mu}}U^{\prime}$ identically along the
profile. Going up from (21) to (20), any heteroclinic solution persists
uniquely for small $\epsilon>0$, as for $\epsilon=0$ its orbit is a
_transverse_ intersection of the unstable manifold of one with the stable
manifold of the other hyperbolic fixed point (cf., e.g., [7, 34, 6] for the
idea).

(ii) The Evans function for a shock profile $(R,U,\Sigma)$ of the relaxed
system (19) is a particular Wronskian of solutions of the eigenvalue problem
for the linearization of (19) around the profile,

$\displaystyle\begin{aligned} \lambda r+Ur^{\prime}+R\upsilon^{\prime}&=0\\\
\lambda(Ur+R\upsilon)+U^{2}r^{\prime}+2m\upsilon^{\prime}-R^{-2}p^{\prime}(1/R)r^{\prime}+s^{\prime}&=0\\\
\epsilon\lambda Rs+\epsilon
ms^{\prime}+\upsilon^{\prime}&=\frac{-s}{\tilde{\mu}};\end{aligned}$ (22)

in (22), $m$($\neq 0$) is the constant value that $RU$ takes along the profile
according to the first line of (21); note that the mass transfer $m$ is $\neq
0$ for any genuine shock, and thus $U\neq 0$ uniformly along the shock
profile. System (22) again is a regular perturbation of its $\epsilon\to 0$
limit

$\displaystyle\begin{aligned} \lambda r+Ur^{\prime}+R\upsilon^{\prime}&=0\\\
\lambda(Ur+R\upsilon)+U^{2}r^{\prime}+2m\upsilon^{\prime}-R^{-2}p^{\prime}(1/R)r^{\prime}+s^{\prime}&=0\\\
\upsilon^{\prime}&=\frac{-s}{\tilde{\mu}},\end{aligned}$ (23)

and this immediately implies that for any $C>0$, there exist
$\delta(C),\bar{\epsilon}>0$ such that the Evans functions $D^{v}$ and
$D^{r}_{\epsilon},0<\epsilon<\bar{\epsilon},$ of (23) and (22) respectively,
are well-defined on the domain

$\Lambda_{C}=\\{\lambda\in\mathbb{C}:|\lambda|\leq C\text{ and
Re}(\lambda)\geq-\delta(C)\\}$

and satisfy $\lim_{\epsilon\to 0}D^{r}_{\epsilon}=D^{r}$ uniformly on
$\Lambda_{C}$. If now $D^{v}$ has only one, simple, zero on $\Lambda_{C}$, the
same thus obviously holds for $D^{r}_{\epsilon}$ as soon as $\epsilon$ is
sufficiently small. (That this zero is $0$ follows from the general fact that
$0$ always is an eigenvalue (namely the one corresponding to shifting the
profile; cf., e.g., [7]) or [6].) Finally, the standard arguments (cf., e.g.,
again [7]) showing the absence of Evans function zeroes outside a circle
$|\lambda|\geq C$ of sufficiently large radius $C$ here trivially apply
_uniformly_ to (23) and (22), so that altogether, for sufficiently small
$\epsilon>0$, $D^{r}_{\epsilon}$ satisfies the complete Evans function
condition when $D^{v}$ does. ∎

The cited works of Zumbrun and coworkers now readily imply the following.

######  Corollary 1.

For any Navier-Stokes shock that satisfies Majda’s condition on the Lopatinski
determinant [17] and Zumbrun’s Evans function condition, all corresponding
relaxation shocks as identified in Assertion (i) of Theorem 3 with sufficently
small $\epsilon>0$ are nonlinearly stable with respect to Sobolev norms chosen
appropriately according to Zumbrun theory.

Note that the corollary holds also regarding non-planar perturbations, i.e.,
as a fully multidimensional result. This is obvious from the fact that the
above reasoning carries over verbatim to the multidimensional Evans function
$D=D(\lambda,\omega)$ whose second argument is a transverse Fourier frequency
(cf. [32, 34, 5]).

The use of the corollary lies in the fact the two said conditions have been
verified for large families of Navier-Stokes shocks, of small and large
amplitude, cf. [14, 1, 13, 12]. —

Ruggeri’s model. Models like the above belong to the realm of rational
extended thermodynamics [24]. Exactly that model was indeed proposed almost 40
years ago by Ruggeri [26].

######  Definition 1.

(i) One calls $Y=(Y_{0},\ldots,Y_{n})$ _Godunov variables_ and a vector
$(X^{\alpha}(Y))_{\alpha=0,....,3}$ a _4-potential_ for a symmetric system of
conservation laws

$\displaystyle\partial_{\alpha}F^{\alpha b}(Y)=I^{\alpha b}(Y),\quad
b=0,\ldots,n,$ (24)

if the spatiotemporal fluxes $F^{\alpha b}$ derive from $X^{\alpha}$ as

$\displaystyle F^{\alpha b}(Y)=\frac{\partial X^{\alpha}(Y)}{\partial
Y_{b}},\quad\alpha=0,...,3,\ b=0,...,n.$ (25)

(ii) In that case, system (24) is called _symmetric-hyperbolic_ if

$\displaystyle\left(\frac{\partial^{2}X^{0}(Y)}{\partial Y_{b}\partial
Y_{c}}\right)_{b,c=0,\ldots,n}\quad\text{is definite.}$ (26)

(iii) In the situation of (i), a scalar function $X(Y)$ is called a
_protopotential_ for system (24) if

$X^{\alpha}(Y)=\frac{\partial X(Y)}{\partial Y_{\alpha}}.$

(iv) If a system as in (i) is Lorentz invariant, it is called _causal_ if

$\displaystyle\left(\frac{\partial^{2}X^{\alpha}(Y)}{\partial Y_{b}\partial
Y_{c}}T_{\alpha}\right)_{b,c=0,\ldots,n}\quad\text{is definite for all
}T_{\alpha}\text{ with }T_{\alpha}T^{\alpha}<0.$ (27)

For the Euler equations of classical inviscid fluid dynamics, the Godunov
variables

$\displaystyle\tilde{\psi}\equiv\psi-\frac{u^{2}}{2\theta},\
\tilde{u}^{i}\equiv\frac{u^{i}}{\theta},\
\tilde{\theta}\equiv\frac{1}{\theta}$ (28)

(where $\psi=g/\theta$ with $g=e+pv-\theta\varsigma$ the chemical potential)
and the corresponding 4-potential

$\displaystyle X_{E}^{0}=\frac{p}{\theta},\quad
X_{E}^{i}=\frac{p}{\theta}u^{i}$ (29)

were proposed by Godunov [10] (while the production term is $I_{E}^{\alpha
b}=0$ in that case). In [26], Ruggeri augmented the list (28) as

$\displaystyle\tilde{\psi},\ \tilde{u}^{i},\ \tilde{\theta},\
\tilde{\sigma}^{ij}\equiv\frac{\sigma^{ij}}{\theta},\
\tilde{\tau}\equiv\frac{\tau}{\theta},\
\tilde{q}^{i}\equiv\frac{q^{i}}{\theta^{2}},$ (30)

and showed two things: (a) With Godunov variables (30), the 4-potential

$\displaystyle X^{0}=\frac{p}{\theta},\quad
X^{i}=\frac{1}{\theta}\left(\left(p\delta^{ij}-(\sigma^{ij}+\tau\delta^{ij)}\right)u_{j}+q^{i}\right),$
(31)

and the production term

$\displaystyle(I^{b})_{b=0,...,13}=(0_{1},0_{3},0_{1},-\frac{1}{\theta\eta_{Q}}q,\frac{1}{\theta\eta_{S}}\sigma,\frac{1}{\theta\eta_{B}}\tau),$
(32)

system (24), (25) coincides with the classical Navier-Stokes-Fourier (NSF)
equations. In (29) and (31) the fluid can be identified through an equation of
state

$\displaystyle p=\hat{p}(\theta,\psi).$ (33)

(b) With the modified 4-potential

$\displaystyle\begin{aligned} X_{N}^{0}=\frac{p_{N}}{\theta},\quad
X_{N}^{i}=\frac{1}{\theta}\left(\left(p_{N}\delta^{ij}-(\sigma^{ij}+\tau\delta^{ij)}\right)u_{j}+q^{i}\right),\qquad\quad\\\
p_{N}=\hat{p}(\theta,\psi-\mathcal{S}),\quad\mathcal{S}=\mathcal{S}(\tilde{\sigma},\tilde{\tau},\tilde{q})=\frac{1}{2}\left(\epsilon_{S}\|\tilde{\sigma}\|^{2}+\epsilon_{B}\tilde{\tau}^{2}+\epsilon_{Q}|\tilde{q}|^{2}\right),\end{aligned}$
(34)

and the unmodified production term $I^{\alpha b}_{N}=I^{\alpha b}$, the
resulting system is, as he called it, “a symmetric-hyperbolic system of
conservative equations for a viscous heat conducting fluid”. The latter is
indeed the model discussed above!

A 4-potential comes with a protopotential whenever the rank-2 tensor

$\left(F^{\alpha\beta}\right)_{\alpha,\beta=0,...,3}$

is symmetric. This has been used by Geroch and Lindblom in the relativistic
context with $F^{\alpha\beta}=T^{\alpha\beta}$ the Lorentz invariant energy-
momentum 4-tensor [8]; it also applies to non-relativistic fluid dynamics with
$F^{\alpha\beta}=M^{\alpha\beta}$ the Galilei invariant mass-momentum
4-tensor. We note that the classical Euler equations, Navier-Stokes-Fourier
equations, and Ruggeri model have

$\displaystyle X_{E}(\tilde{\psi},\tilde{u},\tilde{\theta})$
$\displaystyle=\hat{X}(\theta,\psi)$ $\displaystyle
X(\tilde{\psi},\tilde{u},\tilde{\theta},\tilde{\sigma},\tilde{\tau},\tilde{q})$
$\displaystyle=\hat{X}(\theta,\psi)+\theta\left(-(1/2)(\tilde{\sigma}_{ij}+\tilde{\tau}\delta_{ij})\tilde{u}^{i}\tilde{u}^{j}+\tilde{q}_{i}\tilde{u}^{i}\right)$
$\displaystyle
X_{N}(\tilde{\psi},\tilde{u},\tilde{\theta},\tilde{\sigma},\tilde{\tau},\tilde{q})$
$\displaystyle=\hat{X}(\theta,\psi-\mathcal{S}(\tilde{\sigma},\tilde{\tau},\tilde{q}))+\theta\left(-(1/2)(\tilde{\sigma}_{ij}+\tilde{\tau}\delta_{ij})\tilde{u}^{i}\tilde{u}^{j}+\tilde{q}_{i}\tilde{u}^{i}\right)\
$

as protopotentials, respectively, where $\hat{X}$ relates to (33) as

$\hat{X}(\theta,\psi)=\int\hat{p}(\theta,\psi)d\psi.$

From the original perspective of the present paper we switch to a new one, now
proposing a

Lorentz invariant version of Ruggeri’s model. The relativistic Euler
equations,

$\displaystyle\partial_{\alpha}T^{\alpha\beta}_{E}$ $\displaystyle=0,$
$\displaystyle\partial_{\alpha}N^{\alpha}_{E}$ $\displaystyle=0$

admit the Godunov variables [27]

$\displaystyle\psi=\frac{g}{\theta},\quad\Upsilon^{\alpha}=\frac{u^{\alpha}}{\theta}$
(35)

and the protopotential [8]

$\displaystyle\bar{X}_{E}(\Upsilon,\psi)=\bar{\hat{X}}(\theta,\psi)$ (36)

with now

$\bar{\hat{X}}(\theta,\psi)=\int\hat{p}(\theta,\psi)d\theta;$

indeed, with $\theta^{-2}=-\Upsilon_{\alpha}\Upsilon^{\alpha}$,

$\displaystyle\begin{aligned}
T^{\alpha\beta}_{E}&=\frac{\partial^{2}\bar{X}_{E}(\Upsilon,\psi)}{\partial\Upsilon_{\alpha}\partial\Upsilon_{\beta}}=\theta^{3}\frac{\partial
p(\theta,\psi)}{\partial\theta}\Upsilon^{\alpha}\Upsilon^{\beta}+p(\theta,\psi)g^{\alpha\beta},\\\
N^{\alpha}_{E}&=\frac{\partial^{2}\bar{X}_{E}(\Upsilon,\psi)}{\partial\Upsilon_{\alpha}\partial\psi}=\frac{\partial
p(\theta,\psi)}{\partial\psi}\Upsilon^{\alpha}.\end{aligned}$ (37)

To prepare for including dissipative effects, we decompose symmetric rank-2
tensors via

$\displaystyle{{}^{S}\\!P}:V_{\alpha\beta}\mapsto{{}^{S}\\!P}_{\alpha\beta}^{\gamma\delta}V_{\gamma\delta}$
$\displaystyle\quad\text{with
}{{}^{S}\\!P}_{\alpha\beta}^{\gamma\delta}=\Pi_{(\alpha}^{\gamma}\Pi_{\beta)}^{\delta}-\frac{1}{3}\Pi_{\alpha\beta}\Pi^{\gamma\delta},$
$\displaystyle{{}^{S}\\!P}:V_{\alpha\beta}\mapsto{{}^{B}\\!P}_{\alpha\beta}^{\gamma\delta}V_{\gamma\delta}$
$\displaystyle\quad\text{with
}{{}^{B}\\!P}_{\alpha\beta}^{\gamma\delta}=\frac{1}{3}\Pi_{\alpha\beta}\Pi^{\gamma\delta},$
$\displaystyle{{}^{Q}\\!P}:V_{\alpha\beta}\mapsto{{}^{Q}\\!P}_{\alpha\beta}^{\gamma\delta}V_{\gamma\delta}$
$\displaystyle\quad\text{with
}{{}^{Q}\\!P}_{\alpha\beta}^{\gamma\delta}=-\left(\Pi_{\alpha}^{(\gamma}u_{\beta}+\Pi_{\beta}^{(\gamma}u_{\alpha}\right)u^{\delta)},$
$\displaystyle{{}^{H}\\!P}:V_{\alpha\beta}\mapsto{{}^{H}\\!P}_{\alpha\beta}^{\gamma\delta}V_{\gamma\delta}$
$\displaystyle\quad\text{with
}{{}^{H}\\!P}_{\alpha\beta}^{\gamma\delta}=u_{\alpha}u_{\beta}u^{\gamma}u^{\delta};$

the mappings ${{}^{S}\\!P},{{}^{B}\\!P},{{}^{Q}\\!P},{{}^{H}\\!P}$ are
complementing projectors, i.e.,

${{{}^{C}\\!P}}{{}^{D}\\!P}=\delta^{CD}\ {{}^{C}\\!P},\quad
C,D\in\\{S,B,Q,H\\},\quad\text{and}\quad{{}^{S}\\!P}+{{}^{B}\\!P}+{{}^{Q}\\!P}+{{}^{H}\\!P}=\text{id}.$

Now, slightly varying upon [8], we augment, as a first step, the list (35) as

$\displaystyle\psi,\quad\Upsilon^{\alpha},\quad\Sigma^{\alpha\beta}$ (38)

with a single symmetric rank-2 tensor $\Sigma^{\alpha\beta}$, and find that

$\displaystyle\bar{X}(\Upsilon,\psi,\Sigma)=\bar{\hat{X}}(\theta,\psi)-\frac{1}{2}\Sigma_{\alpha\beta}\Upsilon^{\alpha}\Upsilon^{\beta}$
(39)

as a protopotential and the production term
$(\bar{I}^{b})_{b}=(0_{5},(I^{\alpha\beta})_{\alpha\beta})$ with

$I_{\alpha\beta}=-\frac{1}{\theta}\left(\frac{1}{\eta_{S}}{{}^{S}\\!P}_{\alpha\beta}^{\gamma\delta}-\frac{1}{\eta_{B}}{{}^{B}\\!P}_{\alpha\beta}^{\gamma\delta}-\frac{1}{\eta_{Q}}{{}^{Q}\\!P}_{\alpha\beta}^{\gamma\delta}-\frac{1}{\eta_{H}}{{}^{H}\\!P}_{\alpha\beta}^{\gamma\delta}\right)\Sigma_{\gamma\delta}$

together induce Eckart’s counterpart [3] of the non-relativistic NSF
equations, here in the form

$\displaystyle\begin{aligned}
\partial_{\alpha}\left(T^{\alpha\beta}_{E}-\Sigma^{\alpha\beta}\right)&=0\\\
\partial_{\alpha}N^{\alpha}_{E}&=0\\\
\partial_{\alpha}\left(\frac{1}{2}\left(g^{\alpha\beta}\Upsilon^{\gamma}+g^{\alpha\gamma}\Upsilon^{\beta}\right)\right)&=I^{\beta\gamma}.\\\
\end{aligned}$ (40)

The last line of (40) is to be read as

$\displaystyle\Sigma_{\alpha\beta}=-\theta\left({\eta_{S}}{{}^{S}\\!P}_{\alpha\beta}^{\gamma\delta}+{\eta_{B}}{{}^{B}\\!P}_{\alpha\beta}^{\gamma\delta}+{\eta_{Q}}{{}^{Q}\\!P}_{\alpha\beta}^{\gamma\delta}+{\eta_{H}}{{}^{H}\\!P}_{\alpha\beta}^{\gamma\delta}\right)\left(\Upsilon_{\gamma,\delta}+\Upsilon_{\delta,\gamma}\right)/2,$
(41)

which essentially111with the exception of the last term, which could be there
mathematically (if $\eta_{H}\neq 0$), but to the author’s knowledge has not
been reported in the physics literature expresses the classically known
composition of dissipative stresses (cf., e.g., [30], p. 56), with
$\eta_{S},\eta_{B},\eta_{Q}$ coefficients of shear viscosity, bulk viscosity,
and heat conduction. Representation (41) and equations (40) remain meaningful
if one or several of the dissipation coefficients
$\eta_{S},\eta_{B},\eta_{Q},\eta_{H}$ vanish identically;222While it seems
generally reasonable to assume that the ‘heating dissipation’ coefficient
$\eta_{H}$ vanishes, many fluids also have $\eta_{B}=0$, no bulk viscosity.
mathematically such situations imply a reduction of the state space to a
corresponding equilibrium manifold.
We finally vary the protopotential $\bar{X}$ as

$\displaystyle\bar{X}_{N}(\Upsilon,\psi,\Sigma)=\bar{\hat{X}}(\theta,\psi-\bar{\mathcal{S}}(\Sigma,\Upsilon))-\frac{1}{2}\Sigma_{\alpha\beta}\Upsilon^{\alpha}\Upsilon^{\beta}$
(42)

with

$\bar{\mathcal{S}}(\Sigma,\Upsilon)=\frac{1}{2}\left(\epsilon_{S}{{}^{S}\\!P}_{\alpha\beta}^{\gamma\delta}{\Sigma}^{\alpha\beta}{\Sigma}_{\gamma\delta}+\epsilon_{B}{{}^{B}\\!P}_{\alpha\beta}^{\gamma\delta}{\Sigma}^{\alpha\beta}{\Sigma}_{\gamma\delta}+\epsilon_{Q}{{}^{Q}\\!P}_{\alpha\beta}^{\gamma\delta}{\Sigma}^{\alpha\beta}{\Sigma}_{\gamma\delta}+\epsilon_{H}{{}^{H}\\!P}_{\alpha\beta}^{\gamma\delta}{\Sigma}^{\alpha\beta}{\Sigma}_{\gamma\delta}\right).$

It appears that the resulting symmetric system

$\displaystyle\begin{aligned}
\partial_{\alpha}\left(\frac{\partial\bar{X}_{N}(\Upsilon,\psi,\Sigma)}{\partial\Upsilon_{\alpha}\partial\Upsilon_{\beta}}\right)&=0,\\\
\partial_{\alpha}\left(\frac{\partial\bar{X}_{N}(\Upsilon,\psi,\Sigma)}{\partial\Upsilon_{\alpha}\partial\psi}\right)&=0,\\\
\partial_{\alpha}\left(\frac{\partial\bar{X}_{N}(\Upsilon,\psi,\Sigma)}{\partial\Upsilon_{\alpha}\partial\Sigma_{\beta\gamma}}\right)&=0\end{aligned}$
(43)

is well-behaved.

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