# Sustained Oscillations in Hyperbolic-Parabolic Systems

Athanasios E. Tzavaras
Computer, Electrical and Mathematical Science and Engineering Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

We construct examples of oscillating solutions with persistent oscillations
for various hyperbolic-parabolic systems with singular diffusion matrices that
appear in mechanics. These include, an example for the equations of nonlinear
viscoelasticity of Kelvin-Voigt type with stored energy that violates rank-one
convexity, which amounts to a time-dependent variant of twinning solutions. An
example pertaining to the system of gas dynamics with thermal effects for a
viscous, adiabatic gas. Finally, an example for the compressible Navier-Stokes
system in one-space dimension with nonmonotone pressure function. We also
study the existence of oscillating solutions for linear hyperbolic-parabolic
systems with singular diffusion matrices.

###### Contents

  1. 1 Introduction
  2. 2 Oscillatory Solutions in Linear Hyperbolic-Parabolic Systems
    1. 2.1 Oscillations in a 1-d linear thermoviscoelasticity
    2. 2.2 An illustrating example
    3. 2.3 Oscillations in linear multi-dimensional systems
    4. 2.4 Linear adiabatic thermoviscoelasticity
  3. 3 Oscillations in quasi-linear dynamical models for phase transitions
    1. 3.1  Time-dependent twinning solutions in nonlinear viscoelasticity
      1. 3.1.1 Steady solutions with jumps in the deformation gradient in elasticity
      2. 3.1.2 Dynamic solutions with jumps in deformation gradient and strain rate in viscoelasticity
    2. 3.2 Longitudinal motions of a viscoelastic bar
    3. 3.3 A class of nonlinear models describing viscoplastic response
  4. 4  Quasilinear systems in gas dynamics for a viscous adiabatic gas
    1. 4.1 Uniform extension solutions
    2. 4.2 Discontinuous Solutions
    3. 4.3 Construction of an oscillatory solution
  5. 5 The compressible Navier-Stokes system in one space dimension
    1. 5.1 Langrangian to Eulerian
    2. 5.2  Sustained oscillations

##  1\. Introduction

The effectiveness of dissipation on smoothing of discontinuities was proposed
by Dafermos [2] as a yardstick for classifying the hyperbolic-parabolic
systems of thermomechanics. The properties of the Green function for the
linearized operator around a constant state capture the efficacy of
dissipation, as exemplified in the penetrating study of Liu and Zeng [12]. The
picture is less understood regarding propagation of oscillations. It is well-
known that systems of hyperbolic conservation laws can propagate oscillations
from the initial-data to solutions, and propagation and/or cancellations of
oscillations and its relation to entropy inequalities has been an extensively
studied subject. By contrast, fully-parabolic systems instantaneously smoothen
initial oscillations. Less is known for the intermediate case of hyperbolic-
parabolic systems which is the objective of the present work.

This question was brought to focus by an example, exhibiting sustained
oscillations induced by the initial data, [10, sec 7], for the hyperbolic-
parabolic system

$\displaystyle u_{t}$ $\displaystyle=v_{x}$ (1.1) $\displaystyle v_{t}$
$\displaystyle=\sigma(u)_{x}+v_{xx}\,,$

with $\sigma(u)$ non-monotone. The example was produced to highlight the gap
among two available existence theories for motions
$y:(0,T)\times\mathbb{T}^{d}\to\mathbb{R}^{d}$ on the torus $\mathbb{T}^{d}$,
$d=2,3$, describing the Cauchy problem of nonlinear viscoelastic materials

$\displaystyle\partial_{tt}y-\mathop{\mathrm{div}}\Big{(}\frac{\partial
W}{\partial F}(\nabla y)\Big{)}=\Delta\partial_{t}y\,,$ (1.2) $\displaystyle
y|_{t=0}=y_{0},\quad\partial_{t}y|_{t=0}=v_{0}\,.$

Introducing the velocity $v=\partial_{t}y$ and the deformation gradient
$F=\nabla y$, the second order evolution (1.2) is expressed as a hyperbolic-
parabolic system

$\displaystyle\partial_{t}v-\mathop{\mathrm{div}}(S(F))$
$\displaystyle=\Delta\,v$ (1.3) $\displaystyle\partial_{t}F-\nabla\,v$
$\displaystyle=0$ $\displaystyle\mathop{\mathrm{curl}}\,F$
$\displaystyle=0\,.$

There are two frameworks for global existence of weak solutions for (1.3): (a)
one with spatial regularity $v\in L^{2}$, $F\in L^{p}$ in [6, 4]; and (b) a
second with spatial regularity $v\in L^{2}$, $F\in H^{1}$ in [10]. A key and
perhaps unexpected difference among the two is the possibility of sustained
oscillations in the framework $v\in L^{2}$, $F\in L^{p}$ demonstrated by the
example in [10, sec 7].

A second motivation stems from the existence theory of the compressible
Navier-Stokes system

$\displaystyle\partial_{t}\rho+\mathop{\mathrm{div}}\rho u$ $\displaystyle=0$
(1.4) $\displaystyle\partial_{t}\rho u+\mathop{\mathrm{div}}\rho u\otimes
u+\nabla p(\rho)$ $\displaystyle=\mathop{\mathrm{div}}\Big{(}\mu(\nabla
u+\nabla u^{T})+\lambda(\mathop{\mathrm{div}}u)\,I\,\Big{)}$

where $\rho$ and $u$ are the density and velocity of the fluid while $\mu$ the
shear viscosity and $\lambda$ the second viscosity (or Lame constants) satisfy
$\mu>0$, $\lambda+\mu>0$. The existence theory of weak solutions of Lions [11]
and Feireisl [5] makes essential use of the propagation of compactness
property from the initial data. Lions conjectured that the compressible
Navier-Stokes system may exhibit propagation of oscillations and provided an
example for a variant of (1.4) that includes a forcing term, see [11, Rmk
5.8]. We will provide an example of propagation of oscillations for the
(unforced) compressible Navier-Stokes system in one space dimension. The
question of deriving the effective equations of the associated homogenization
problem is studied in [13, 7].

Our objective is to explore persistent oscillations in hyperbolic-parabolic
systems. The oscillations are related to the fact that systems of mechanics
are second order evolutions for the motion and, as the reader will observe,
this plays a fundamental role in the examples that we provide. We explore the
existence of sustained oscillations first for linear systems and then for
nonlinear systems. We start in section 2 with linear hyperbolic-parabolic
systems,

$\partial_{t}U+A\partial_{x}U=B\partial_{xx}U\,,$ (1.5)

in one or several space dimensions, where $B$ is singular. We focus on systems
originating in mechanics and explore a special structure of oscillations
natural to such problems, where oscillations of the deformation gradient
balance with oscillations of the strain-rate. Examples include the equations
of linear thermoviscoelasticity in one dimension (2.1) or several space
dimensions (2.27).

In section 3, we consider the system (1.3) of nonlinear viscoelasticity with a
stored energy violating rank-1 convexity; such models are employed in the
theory of phase transitions. We provide a novel class of dynamic oscillating
solutions where the deformation gradient oscillates between two time dependent
homogeneous states and exhibits jumps across steady interfaces. Such solutions
propagate oscillations of initial data to solutions and lead to Young measures
that are convex combinations of Dirac masses. In section 3.2 we study the
system of one-dimensional viscoelasticity (3.13) (or one-dimensional gas
dynamics for viscous gases) where the calculations can be made explicit.
Similar behavior appears in a class of one-dimensional models in
viscoplasticity studied in section 3.3.

The existence of oscillatory solutions is extended in section 4 to the system
describing gas dynamics with thermal effects for viscous but adiabatic gases,

$\displaystyle u_{t}-v_{x}$ $\displaystyle=0$ (1.6) $\displaystyle
v_{t}-\sigma(u,\theta)_{x}$ $\displaystyle=(\frac{\mu}{u}v_{x})_{x}$
$\displaystyle\big{(}\tfrac{1}{2}v^{2}+e(u,\theta)\big{)}_{t}-\Big{(}\sigma(u,\theta)\,v\Big{)}_{x}$
$\displaystyle=\Big{(}\frac{\mu}{u}\,v\,v_{x}\Big{)}_{x}\,.$

For a special constitutive class (CF) and a carefully selected nonmonotone
pressure weak solutions are constructed that exhibit persistent oscillations.
The construction combines a class of universal solutions describing uniform
extension (or compression) solutions alternating via jumps across steady
interfaces that satisfy the Rankine-Hugoniot jump condition.

Finally, in section 5, we consider the one-dimensional compressible Navier-
Stokes system

$\displaystyle\rho_{t}+(\rho u)_{y}$ $\displaystyle=0$ (1.7)
$\displaystyle(\rho u_{t}+(\rho u^{2}+p(\rho))_{y}$ $\displaystyle=\mu
u_{yy}\,,$

By exploring the transformation between Lagrangian and Eulerian descriptions,
we map the Lagrangian variable solutions obtained in section 3.2 to the
Eulerian domain. This leads to a special solution of the compressible Navier-
Stokes system (1.7),

$\displaystyle u(t,y)=\frac{y}{t}\,,\qquad\rho(t,y)$
$\displaystyle=\begin{cases}\;\frac{1}{ta}\quad&\quad
kv_{0}(\theta)<\frac{y}{t}<kv_{0}(\theta)+a\theta\\\\[5.0pt]
\;\frac{1}{tb}\quad&\;\;kv_{0}(\theta)+a\theta<\frac{y}{t}<(k+1)v_{0}(\theta)\end{cases}\quad
k\in\mathbb{Z}\,,$ (1.8)

where $v_{0}(\theta)=\theta a+(1-\theta)b$, $\theta\in(0,1)$, with persistent
oscillations. The function $(\rho,u)$ solves (1.7) provided the nonmonotone
pressure satisfies $p(\frac{1}{ta})=p(\frac{1}{tb})$ for $t\in[1.2]$. The
solution suggests propagation of oscillations from the initial data at $t=1$
to the domain $[1,2]\times\mathbb{R}$.

##  2\. Oscillatory Solutions in Linear Hyperbolic-Parabolic Systems

In this section we study linear systems that combine hyperbolic and parabolic
characters first in one-space dimension and then in several space dimensions.
The goal is to build a class of solutions for which initial oscillations
propagate into solutions despite the smoothing effect of parabolic characters,
exploiting degeneracies of the parabolic characters. The form of solutions is
motivated by examples that arise in the theory of thermoviscoelasticity and
relate to the mechanical interpretation of such models.

In the first two subsections we study the one-dimensional case, and then we
extend the results to the linear systems in several space dimensions.

###  2.1. Oscillations in a 1-d linear thermoviscoelasticity

Consider the system

$\displaystyle\frac{\partial^{2}u}{\partial t^{2}}$
$\displaystyle=\lambda\frac{\partial^{2}u}{\partial
x^{2}}+m\frac{\partial\theta}{\partial x}+\mu\frac{\partial^{2}}{\partial
x^{2}}\frac{\partial u}{\partial t}$ (2.1)
$\displaystyle\frac{\partial\theta}{\partial t}$
$\displaystyle=\kappa\frac{\partial^{2}\theta}{\partial
x^{2}}+m\frac{\partial^{2}u}{\partial x\partial t}$

where $(u(t,x),\theta(t,x))$ describe a thermomechanical process consisting of
the motion $u$ and the temperature $\theta$, while $\lambda>0$,
$m\in\mathbb{R}$, $\mu>0$, $\kappa>0$ are parameters describing the elasticity
modulus, thermoelastic coupling, viscosity and heat diffusivity, respectively.
The system (2.1) is a second order evolution but may be expressed as a first
order (in time) system by introducing the velocity $v=\frac{\partial
u}{\partial t}$ and the elastic strain $w=\frac{\partial u}{\partial x}$ and
writing it as

$\displaystyle\frac{\partial w}{\partial t}$ $\displaystyle=\;\frac{\partial
v}{\partial x}$ (2.2) $\displaystyle\frac{\partial v}{\partial t}$
$\displaystyle=\lambda\frac{\partial w}{\partial
x}+m\frac{\partial\theta}{\partial x}+\mu\frac{\partial^{2}v}{\partial x^{2}}$
$\displaystyle\frac{\partial\theta}{\partial t}$
$\displaystyle=m\frac{\partial v}{\partial
x}+\kappa\frac{\partial^{2}\theta}{\partial x^{2}}$

The form (2.2) indicates a hyperbolic-parabolic system, but we will be using
mostly the form (2.1) of a second-order evolution. Solutions of (2.1) satisfy
the energy identity

$\displaystyle\frac{\partial}{\partial
t}\left(\tfrac{1}{2}\Big{|}\frac{\partial u}{\partial
t}\Big{|}^{2}+\tfrac{\lambda}{2}\Big{|}\frac{\partial u}{\partial
x}\Big{|}^{2}+\tfrac{1}{2}|\theta|^{2}\right)$
$\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial
t}\big{(}\lambda\frac{\partial u}{\partial
x}+m\theta+\mu\frac{\partial^{2}u}{\partial x\partial t}\big{)}\right)$ (2.3)
$\displaystyle\quad+\frac{\partial}{\partial
x}\left(\kappa\theta\frac{\partial\theta}{\partial
x}\right)-\kappa\Big{|}\frac{\partial\theta}{\partial
x}\Big{|}^{2}-\mu\Big{|}\frac{\partial^{2}u}{\partial x\partial t}\Big{|}^{2}$

Under appropriate boundary conditions (either periodic or no mechanical work
and adiabatic at the boundaries) we obtain the energy dissipation identity

$\frac{d}{dt}\int\Big{(}\tfrac{1}{2}\Big{|}\frac{\partial u}{\partial
t}\Big{|}^{2}+\tfrac{\lambda}{2}\Big{|}\frac{\partial u}{\partial
x}\Big{|}^{2}+\tfrac{1}{2}|\theta|^{2}\Big{)}\,dx+\int\kappa\Big{|}\frac{\partial\theta}{\partial
x}\Big{|}^{2}+\mu\Big{|}\frac{\partial^{2}u}{\partial x\partial
t}\Big{|}^{2}=0\,.$ (2.4)

Next, consider the system (2.1) and apply the ansatz for solutions of the form

$\displaystyle u(t,x)$ $\displaystyle=\frac{1}{n}a(t)e^{inx}$ (2.5)
$\displaystyle\theta(t,x)$ $\displaystyle=-ib(t)e^{inx}$

where $i$ is the imaginary unit, $n$ an integer, and $(a(t),-ib(t))$ are
complex amplitude functions. Introducing (2.5) to (2.1) we deduce that
$(a(t),b(t))$ satisfies the system of ordinary differential equations

$\displaystyle\ddot{a}$ $\displaystyle=-\lambda n^{2}a+mn^{2}b-\mu
n^{2}\dot{a}$ (2.6) $\displaystyle\dot{b}$ $\displaystyle=-\kappa
n^{2}b-m\dot{a}$

The amplitude $(a,b)$ in (2.6) was introduced as complex-valued but it is also
consistent to select it real-valued, and we opt for that selection here.

In summary, given $(a,b)$ a real-valued solution of (2.6) then
$(u(t,x),\theta(t,x))$ selected via the ansatz (2.5) produces an oscillatory
solution of (2.1). Taking the real (or imaginary) part of (2.5), we obtain a
real-valued solution of (2.1), which for large $n$, might be highly
oscillatory, depending on the behavior of $(a,b)$ as a function of $n$. The
energy of solutions to (2.6) gets damped according to the identity

$\frac{d}{dt}\left(\tfrac{1}{2}\dot{a}^{2}+\tfrac{\lambda
n^{2}}{2}a^{2}+\tfrac{n^{2}}{2}b^{2}\right)+\mu n^{2}\dot{a}^{2}+\kappa
n^{4}b^{2}=0\,.$ (2.7)

The latter is an equation for the Fourier modes (2.5) inherited from (2.4).
Observe that for the ansatz (2.5) we have

$\displaystyle\frac{\partial u}{\partial x}$ $\displaystyle=ia(t)e^{inx}$
$\displaystyle\frac{\partial u}{\partial t}$
$\displaystyle=\frac{1}{n}\dot{a}(t)e^{inx}$
$\displaystyle\frac{\partial^{2}u}{\partial x\partial t}$
$\displaystyle=i\dot{a}(t)e^{inx}$

We expect oscillations of $u(t,x)$ and the velocity $v=\frac{\partial
u}{\partial t}(t,x)$ decay to zero (for $n$ large) while oscillations of the
strain $w=\frac{\partial u}{\partial x}(t,x)$, the strain rate $\frac{\partial
v}{\partial x}=\frac{\partial^{2}u}{\partial x\partial t}(t,x)$ and the
temperature $\theta(t,x)$ induced by the initial data might persist in time.
This has to be validated by studying the behavior of solutions of (2.6).

Setting $v=\dot{a}$, (2.6) is expressed as a first order system,

$\frac{d}{dt}\begin{bmatrix}a\\\ v\\\ b\end{bmatrix}=\begin{bmatrix}0&1&0\\\
-\lambda n^{2}&-\mu n^{2}&mn^{2}\\\ 0&-m&-\kappa
n^{2}\end{bmatrix}\begin{bmatrix}a\\\ v\\\ b\end{bmatrix}$ (2.8)

Let $A=A(n)$ be the coefficient matrix. The eigenvalues $\rho(n)$ of $A(n)$
are computed as roots of the cubic polynomial $\det(A-\rho I)=0$ which reads

$\rho^{3}+(\kappa+\mu)n^{2}\rho^{2}+(\kappa\mu n^{4}+\lambda
n^{2}+m^{2}n^{2})\rho+\kappa\lambda n^{4}=0\,.$ (2.9)

The three roots $\rho_{1},\rho_{2},\rho_{3}$ satisfy the Vieta relations

$\displaystyle\rho_{1}+\rho_{2}+\rho_{3}$ $\displaystyle=-(\kappa+\mu)n^{2}$
(2.10) $\displaystyle\rho_{1}\rho_{2}+\rho_{2}\rho_{3}+\rho_{3}\rho_{1}$
$\displaystyle=\kappa\mu n^{4}+(\lambda+m^{2})n^{2}$
$\displaystyle\rho_{1}\rho_{2}\rho_{3}$ $\displaystyle=-\kappa\lambda n^{4}$

While we cannot identify the explicit form of the eigenvalues
$\rho_{1},\rho_{2},\rho_{3}$, it is possible to do an asymptotic analysis for
large $n$. Setting $r:=\frac{\rho}{n^{2}}$, then $r$ are the roots of the
polynomial

$r^{3}+(\kappa+\mu)r^{2}+\Big{(}\kappa\mu+(\lambda+m^{2})\frac{1}{n^{2}}\Big{)}\rho+\frac{\kappa\lambda}{n^{2}}=0$
(2.11)

Introduce the asymptotic expansion
$r=r^{0}+\frac{1}{n^{2}}r^{1}+\frac{1}{n^{4}}r^{2}+...$ to the algebraic
equation (2.11); after equating the terms of the same orders we obtain

at $O(1)$ $\displaystyle(r^{0})^{3}+(\kappa+\mu)(r^{0})^{2}+\kappa\mu r^{0}=0$
(2.12) at $O(\tfrac{1}{n^{2}})$
$\displaystyle\big{(}3(r^{0})^{2}+2(\kappa+\mu)r^{0}+\kappa\mu\big{)}r^{1}=-(\lambda+m^{2})r^{0}-\kappa\lambda$
(2.13) at $O(\tfrac{1}{n^{4}})$
$\displaystyle\big{(}3(r^{0})^{2}+2(\kappa+\mu)r^{0}+\kappa\mu\big{)}r^{2}=-3r^{0}(r^{1})^{2}-(\kappa+\mu)(r^{1})^{2}-(\lambda+m^{2})r^{1}$
(2.14)

From (2.12) the leading terms in the expansion of the roots are $r^{0}_{1}=0$,
$r^{0}_{2}=-\kappa$, $r^{0}_{3}=-\mu$. Since these are distinct, for $n$ large
the roots are real and we proceed to compute their asymptotic expansions by
solving consecutively (2.13), (2.14) in the terms $r^{1}$, $r^{2}$ etc. For
instance, for $r^{0}_{1}=0$ we obtain $r_{1}^{1}=-\frac{\lambda}{\mu}$ and in
turn
$r_{1}^{2}=-\frac{\lambda}{\kappa\mu^{2}}\big{(}\frac{\kappa\lambda}{\mu}-m^{2}\big{)}$.
After a straightforward calculation, using $\rho=rn^{2}$, they produce the
following asymptotic expansions for the roots $\rho_{i}(n)$, $n=1,2,3$:

$\displaystyle\rho_{1}(n)$
$\displaystyle=\qquad\qquad-\frac{\lambda}{\mu}+\Big{(}-\frac{\lambda}{\kappa\mu^{2}}\big{(}\frac{\kappa\lambda}{\mu}-m^{2}\big{)}\Big{)}\frac{1}{n^{2}}+O\Big{(}\frac{1}{n^{4}}\Big{)}$
(2.15) $\displaystyle\rho_{2}(n)$ $\displaystyle=-\kappa
n^{2}-\frac{m^{2}}{\mu-\kappa}+\Big{(}-\frac{m^{2}}{\kappa(\mu-\kappa)^{2}}\big{(}\lambda+\kappa\frac{m^{2}}{\mu-\kappa}\big{)}\Big{)}\frac{1}{n^{2}}+O\Big{(}\frac{1}{n^{4}}\Big{)}$
(2.16) $\displaystyle\rho_{3}(n)$ $\displaystyle=-\mu
n^{2}+\Big{(}\frac{\lambda}{\mu}+\frac{m^{2}}{\mu-\kappa}\Big{)}+\Big{(}\big{(}\frac{\lambda}{\mu}+\frac{m^{2}}{\mu-\kappa}\big{)}\big{(}\frac{\lambda}{\mu^{2}}+\kappa\frac{m^{2}}{\mu-\kappa}\big{)}\Big{)}\frac{1}{n^{2}}+O\Big{(}\frac{1}{n^{4}}\Big{)}$
(2.17)

The reader can easily verify that the first two terms in the expansion of the
roots satisfy the Vieta relations within $O\Big{(}\frac{1}{n^{2}}\Big{)}$.

Next, we focus on the eigenvalue $\rho_{1}(n)$ in (2.15) and the associated
eigenvector $\xi_{1}(n)$ of the matrix $A(n)$ in (2.8). Introduce the
expansions

$\displaystyle\rho_{1}(n)$
$\displaystyle=r_{1}^{1}+\frac{1}{n^{2}}r_{1}^{2}+O\Big{(}\frac{1}{n^{4}}\Big{)}\quad\mbox{
where $r_{1}^{1}=-\frac{\lambda}{\mu}$,
$r_{1}^{2}=-\frac{\lambda}{\kappa\mu^{2}}\big{(}\frac{\kappa\lambda}{\mu}-m^{2}\big{)}$
}$ $\displaystyle\xi_{1}(n)$
$\displaystyle=\xi_{1}^{0}+\frac{1}{n^{2}}\xi_{1}^{1}+O\Big{(}\frac{1}{n^{4}}\Big{)}$

into the equation $A(n)\xi_{1}(n)=\rho_{1}(n)\xi_{1}(n)$ to obtain

$\Bigg{\\{}n^{2}\underbrace{\begin{bmatrix}0&0&0\\\ -\lambda&-\mu&m\\\
0&0&-\kappa\end{bmatrix}}_{A_{0}}+\underbrace{\begin{bmatrix}0&1&0\\\ 0&0&0\\\
0&-m&0\end{bmatrix}}_{A_{1}}-\big{(}r_{1}^{1}+\frac{1}{n^{2}}r_{1}^{2}+O\big{(}\frac{1}{n^{4}}\big{)}\big{)}I\Bigg{\\}}\left(\xi_{1}^{0}+\frac{1}{n^{2}}\xi_{1}^{1}+O\Big{(}\frac{1}{n^{4}}\Big{)}\right)=0$

Equating terms of the same order,

$\displaystyle A_{0}\xi_{1}^{0}$ $\displaystyle=0$ at order $O(n^{2})$
$\displaystyle A_{0}\xi_{1}^{1}$
$\displaystyle=-(A_{1}-r_{1}^{1}I)\xi_{1}^{0}$ at order $O(1)$ $\displaystyle
A_{0}\xi_{1}^{2}$
$\displaystyle=-(A_{1}-r_{1}^{1}I)\xi_{1}^{1}+r_{1}^{2}\xi_{1}^{0}\quad$
$\displaystyle\mbox{at order $O(n^{-2})$}\,,$

we compute

$\xi_{1}^{0}=\begin{bmatrix}\mu\\\ -\lambda\\\
0\end{bmatrix}\,,\quad\xi_{1}^{1}=\begin{bmatrix}-\frac{\lambda}{\mu}\\\\[5.0pt]
\frac{m^{2}\lambda}{\kappa\mu}\\\\[5.0pt]
\frac{m\lambda}{\kappa}\end{bmatrix}$

The associated solution of the system (2.8),

$\begin{bmatrix}a\\\ v\\\
b\end{bmatrix}(t)=\left(\xi_{1}^{0}+\frac{1}{n^{2}}\xi_{1}^{1}+O\big{(}\frac{1}{n^{4}}\big{)}\right)\exp\left\\{-\frac{\lambda}{\mu}t+\frac{1}{n^{2}}r_{1}^{2}t+O\big{(}\frac{1}{n^{4}}\big{)}\right\\}\,,$

gives rise using (2.5) to oscillating solutions for (2.1) which read

$\begin{bmatrix}u\\\\[5.0pt] \frac{\partial u}{\partial t}\\\\[5.0pt]
\theta\end{bmatrix}(t,x)=\begin{bmatrix}\tfrac{1}{n}a(t)\\\
\tfrac{1}{n}v(t)\\\
-ib(t)\end{bmatrix}e^{inx}=\begin{bmatrix}\frac{1}{n}\mu-\frac{\lambda}{\mu}\frac{1}{n^{3}}+O(\frac{1}{n^{5}})\\\\[5.0pt]
-\frac{1}{n}\lambda+\frac{m^{2}\lambda}{\kappa\mu}\frac{1}{n^{3}}+O(\frac{1}{n^{5}})\\\\[5.0pt]
-i\frac{m\lambda}{\kappa}\frac{1}{n^{2}}+O(\frac{1}{n^{4}})\end{bmatrix}\exp\Big{\\{}inx-\frac{\lambda}{\mu}t+O\big{(}\frac{1}{n^{2}}\Big{)}t)\Big{\\}}$
(2.18)

Considering (2.18) we see that $u$, $\frac{\partial u}{\partial t}$, $\theta$
and $\frac{\partial\theta}{\partial x}$ converge to zero for $n$ large, but
oscillations persist for the strain $w=\frac{\partial u}{\partial x}$, the
strain rate $\frac{\partial v}{\partial x}$ and
$\frac{\partial^{2}\theta}{\partial x^{2}}$. To leading order these solutions
read

$\begin{bmatrix}w\\\\[5.0pt] \frac{\partial v}{\partial x}\\\\[5.0pt]
\frac{\partial^{2}\theta}{\partial
x^{2}}\end{bmatrix}(t,x)=\begin{bmatrix}\frac{\partial u}{\partial
x}\\\\[5.0pt] \frac{\partial^{2}u}{\partial t\partial x}\\\\[5.0pt]
\frac{\partial^{2}\theta}{\partial
x^{2}}\end{bmatrix}(t,x)=i\,\begin{bmatrix}\mu+O(\frac{1}{n^{2}})\\\\[5.0pt]
-\lambda+O(\frac{1}{n^{2}})\\\\[5.0pt]
\frac{m\lambda}{\kappa}+O(\frac{1}{n^{2}})\end{bmatrix}\left(1+O\big{(}\frac{1}{n^{2}}\big{)}t\right)\exp\Big{\\{}inx-\frac{\lambda}{\mu}t\Big{\\}}$
(2.19)

######  Remark 2.1.

A special case of (2.35) is the system of adiabatic 1-$d$ linear
thermoviscoelasticity when $\kappa=0$,

$\displaystyle\frac{\partial w}{\partial t}$ $\displaystyle=\;\frac{\partial
v}{\partial x}$ (2.20) $\displaystyle\frac{\partial v}{\partial t}$
$\displaystyle=\lambda\frac{\partial w}{\partial
x}+m\frac{\partial\theta}{\partial x}+\mu\frac{\partial^{2}v}{\partial x^{2}}$
$\displaystyle\frac{\partial\theta}{\partial t}$
$\displaystyle=m\frac{\partial v}{\partial x}$

We will see in section 2.4 that one may construct solutions of (2.20) for
which oscillations appear to leading order for the vector function
$(w,v_{x},\theta)$. This should be contrasted to the behavior of (2.2) (which
has $\kappa\neq 0$) where oscillations appear to leading order for the vector
function $(w,v_{x},\theta_{xx})$, see (2.18), (2.19).

###  2.2. An illustrating example

The special case of linear viscoelasticity (in one and several space
dimensions) presents an example that can be worked out explicitly. For the
equation

$u_{tt}=\lambda u_{xx}+\mu u_{txx}\qquad x\in(-\pi,\pi)\,,t>0\,,$ (2.21)

where $u:(0,T)\times\mathbb{T}^{1}\to\mathbb{R}$, we consider solutions of the
form $u(t,x)=\frac{1}{n}\alpha(t)e^{inx}$. (For the multi-dimensional case,
see section 2.4). The amplitude $\alpha(t)$ satisfies

$\frac{d^{2}\alpha}{dt^{2}}+\mu
n^{2}\frac{d\alpha}{dt}+n^{2}\lambda\alpha=0\,.$ (2.22)

The associated characteristic polynomial $\rho^{2}+\mu n^{2}+\lambda n^{2}=0$
has roots

$\rho_{\pm}=\frac{\mu
n^{2}}{2}\left(-1\pm\sqrt{1-\frac{4\lambda}{\mu^{2}n^{2}}}\right)$

which are real for $n$ large, and enjoy the asymptotic expansions

$\displaystyle\rho_{-}$ $\displaystyle=-\mu
n^{2}+\frac{\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{3}n^{2}}+O\big{(}\tfrac{1}{n^{4}}\big{)}$
$\displaystyle\rho_{+}$
$\displaystyle=-\frac{\lambda}{\mu}-\frac{\lambda^{2}}{\mu^{3}n^{2}}+O\big{(}\tfrac{1}{n^{4}}\big{)}$

They give rise to solutions $\alpha_{\pm}=e^{\rho_{\pm}t}$ for (2.22) and
$u_{n,\pm}=\frac{1}{n}e^{inx+\rho_{\pm}t}$ for (2.21). Of interest here is the
solution associated to $\rho_{+}$ and we consider $(u_{n},v_{n})$ where

$u_{n}(t,x)=\frac{1}{n}\exp\left\\{inx-\frac{\lambda}{\mu}t-\frac{\lambda^{2}}{\mu^{3}n^{2}}t+O\big{(}\tfrac{1}{n^{4}}\big{)}t\right\\}$
(2.23)

while $v_{n}=\frac{\partial u_{n}}{\partial t}$. Note that $u_{n}\,,v_{n}\to
0$ as $n\to\infty$. By contrast,

$\displaystyle\frac{\partial u_{n}}{\partial x}$
$\displaystyle=i\left(1+O\big{(}\frac{1}{n^{2}}\big{)}\right)\exp\left\\{inx-\frac{\lambda}{\mu}t\right\\}$
(2.24) $\displaystyle\frac{\partial v_{n}}{\partial x}$
$\displaystyle=i\left(-\frac{\lambda}{\mu}+O\big{(}\frac{1}{n^{2}}\big{)}\right)\exp\left\\{inx-\frac{\lambda}{\mu}t\right\\}$

exhibit persistent oscillations induced by the initial data.

To put this behavior into perspective, note that (2.21) can be expressed as
the hyperbolic-parabolic system,

$\displaystyle\frac{\partial w}{\partial t}$ $\displaystyle=\;\frac{\partial
v}{\partial x}$ (2.25) $\displaystyle\frac{\partial v}{\partial t}$
$\displaystyle=\lambda\frac{\partial w}{\partial
x}+\mu\frac{\partial^{2}v}{\partial x^{2}}$

with $w=\frac{\partial u}{\partial x}$. The oscillation in (2.24) reflects a
balance between the terms $w$ and $\frac{\partial v}{\partial x}$ and is
stationary in time.

It should be contrasted with the usual oscillatory solutions of hyperbolic
systems. Recall, that the hyperbolic model

$\displaystyle\frac{\partial w}{\partial t}$ $\displaystyle=\;\frac{\partial
v}{\partial x}$ (2.26) $\displaystyle\frac{\partial v}{\partial t}$
$\displaystyle=\lambda\frac{\partial w}{\partial x}$

has oscillatory solutions propagating in the directions of the wave speeds

$\begin{bmatrix}w\\\\[5.0pt]
v\end{bmatrix}(t,x)=\xi_{\pm}\exp\big{\\{}in(x-\pm{\sqrt{\lambda}}t)\big{\\}}$

where $\xi_{\pm}$ are the eigenvectors associated to the wave speeds
$A\xi_{\pm}=\pm\sqrt{\lambda}\xi_{\pm}$. These propagate with speeds
$\pm\sqrt{\lambda}$ and are of different nature than (2.23), (2.24) which are
stationary in time.

###  2.3. Oscillations in linear multi-dimensional systems

Consider next the multi-dimensional system

$\displaystyle\frac{\partial^{2}u_{k}}{\partial t^{2}}$
$\displaystyle=\frac{\partial}{\partial
x_{\alpha}}\left(A_{kl\alpha\beta}\frac{\partial u_{l}}{\partial
x_{\beta}}+M_{k\alpha}\theta\right)+\mu\Delta\left(\frac{\partial
u_{k}}{\partial t}\right)$ (2.27) $\displaystyle\frac{\partial\theta}{\partial
t}$ $\displaystyle=\kappa\Delta\theta+M_{k\alpha}\frac{\partial}{\partial
x_{\alpha}}\left(\frac{\partial u_{k}}{\partial t}\right)$

describing the evolution of a thermomechanical process
$(u,\theta):(0,\infty)\times\mathbb{R}^{d}\to\mathbb{R}^{d}\times\mathbb{R}^{+}$.
Here, the summation convention is used, the parameters $\kappa>0,\mu>0$
describe the viscosity and thermal diffusivity, $A_{kl\alpha\beta}$ is a
fourth order rank-1 convex tensor which is symmetric in the sense
$A_{kl\alpha\beta}=A_{lk\alpha\beta}$, $A_{kl\alpha\beta}=A_{kl\beta\alpha}$
and describes the elastic response, while $M_{k\alpha}$ is a second order
tensor describing the thermoelastic coupling.

Consider the ansatz of oscillating solutions of the form

$\displaystyle u_{k}(t,x)$ $\displaystyle=\frac{1}{n}a_{k}(t)e^{in\nu\cdot
x}\qquad k=1,...,d$ (2.28) $\displaystyle\theta(t,x)$
$\displaystyle=-ib(t)e^{in\nu\cdot x}$

where $\nu=(\nu_{1},...,\nu_{d})$ is a unit vector, $|\nu|=1$, and n is a
natural number. Introducing the ansatz (2.28) to (2.27) we see that the
amplitudes $a_{k}(t)$, $\beta(t)$ satisfy the system of ordinary differential
equations

$\displaystyle\ddot{a}_{k}$
$\displaystyle=-n^{2}(A_{kl\alpha\beta}\nu_{\alpha}\nu_{\beta})a_{l}+n^{2}(M_{k\alpha}\nu_{\alpha})b-\mu
n^{2}\dot{a}_{k}$ (2.29) $\displaystyle\dot{b}$ $\displaystyle=-n^{2}\kappa
b-(M_{k\alpha}\nu_{\alpha})\dot{a}_{k}$

Note that (2.29) admits real valued solutions $(a_{1},...,a_{d},b)$ which by
taking real and imaginary parts in (2.28) give rise to real-valued oscillating
solutions for (2.27).

Recalling the summation convention is used, we proceed to solve the system
(2.29). As already noted, the fourth order tensor $A_{kl\alpha\beta}$ is
assumed symmetric and rank-1 convex:

$\displaystyle A_{kl\alpha\beta}=A_{lk\alpha\beta}\,,\quad$ $\displaystyle
A_{kl\alpha\beta}=A_{kl\beta\alpha}\,,$ (H1) $\displaystyle
A_{kl\alpha\beta}\nu_{\alpha}\nu_{\beta}\xi_{k}\xi_{l}>0\quad$
$\displaystyle\forall\xi\in\mathbb{R}^{d}-\\{0\\}\,,\nu\in S^{d-1}\,.$ (H2)

Hypothesis (H2) implies the acoustic tensor
$Q_{kl}:=A_{kl\alpha\beta}\nu_{\alpha}\nu_{\beta}$ is symmetric, positive
definite. It thus has positive eigenvalues $\lambda^{r}>0$, and a complete set
of linearly independent eigenvectors $\xi^{r}$, $r=1,...,d$, satisfying

$Q_{kl}\xi_{l}^{r}=\lambda^{r}\xi_{k}^{r}\quad k=1,...,d\,\;\;\mbox{
normalized so that $|\xi^{r}|=1$. }$ (2.30)

We now place an assumption on the thermoelastic interaction matrix
$M_{k\alpha}$. Given $\xi^{r}$ we assume there is $\nu\in S^{d-1}$ such that
for some $m\in\mathbb{R}$

$M_{k\alpha}\nu_{\alpha}=m\xi^{r}_{k}\quad k=1,...d.$ (H3)

Note that $|\xi^{r}|=1$ implies $m^{r}=\xi_{k}M_{k\alpha}\nu_{\alpha}$.
Hypothesis (H3) is automatically satisfied if $\text{rank}M=d$. Otherwise, it
is an assumption connecting pairs of eigenvectors $\xi^{r}$ to associated
propagation directions $\nu$.

With these in place, fix $r=1,...,d$ and proceed to solve (2.29). We make the
ansatz

$a_{k}(t)=\alpha(t)\xi_{k}^{r}\quad\mbox{ where $\xi^{r}\in\mathbb{R}^{d}$,
$|\xi^{r}|=1$ and $\alpha(t)$ scalar valued}.$ (2.31)

Then $(\alpha(t),b(t))$ satisfies the system of ordinary differential
equations

$\displaystyle\ddot{\alpha}$
$\displaystyle=-n^{2}\lambda^{r}\alpha+n^{2}m^{r}b-\mu n^{2}\dot{\alpha}$
(2.32) $\displaystyle\dot{b}$ $\displaystyle=-\kappa n^{2}b-m^{r}\dot{\alpha}$

where $m^{r}=\xi^{r}\cdot M\nu$, and $\lambda^{r}$, $\xi^{r}$ are an
eigenvalue and corresponding eigenvector of the acoustic tensor. Conversely,
(H3), (2.30) imply solutions of (2.32) generate via (2.31) solutions of
(2.29).

The system (2.32) is precisely (2.8) studied in section 2.1. Combining the
analysis of subsection 2.1 with (2.31) we conclude that if (H1), (H2) and (H3)
are satisfied for an eigenpair $(\lambda^{r},\xi^{r})$ and associated
direction $\nu$, there will be oscillating solutions for (2.27) of the form

$\displaystyle\begin{bmatrix}u\\\\[5.0pt] \frac{\partial u}{\partial
t}\\\\[5.0pt] \theta\end{bmatrix}(t,x)$
$\displaystyle=\begin{bmatrix}\tfrac{1}{n}\alpha(t)\xi^{r}\\\\[5.0pt]
\tfrac{1}{n}\dot{\alpha}(t)\xi^{r}\\\\[5.0pt] -ib(t)\end{bmatrix}e^{in\nu\cdot
x}$ (2.33)
$\displaystyle=\begin{bmatrix}\left(\frac{1}{n}\mu+O\big{(}\frac{1}{n^{3}}\big{)}\right)\xi^{r}\\\\[8.0pt]
\left(-\frac{1}{n}\lambda^{r}+O(\frac{1}{n^{3}})\right)\xi^{r}\\\\[8.0pt]
-i\frac{1}{n^{2}}\frac{m^{r}\lambda^{r}}{\kappa}+O(\frac{1}{n^{4}})\end{bmatrix}\exp\Big{\\{}in\nu\cdot
x-\frac{\lambda^{r}}{\mu}t+O\big{(}\frac{1}{n^{2}}\Big{)}t)\Big{\\}}$

where $(\alpha(t),b(t))$ satisfy (2.32). These provide oscillating progressive
waves in the direction $\nu$. Again $u$, $v=\frac{\partial u}{\partial t}$ and
$\theta$ converge to zero, $\nabla u$, $\nabla v$ and $\Delta\theta$ have
persistent oscillations

$\displaystyle\nabla u$
$\displaystyle=i\mu(\xi^{r}\otimes\nu)\left(1+O\big{(}\frac{1}{n^{2}}\Big{)}t\right)\exp\Big{\\{}in\nu\cdot
x-\frac{\lambda^{r}}{\mu}t\Big{\\}}$ (2.34) $\displaystyle\Delta\theta$
$\displaystyle=i\frac{m^{r}\lambda^{r}}{\kappa}\left(1+O\big{(}\frac{1}{n^{2}}\Big{)}t\right)\exp\Big{\\{}in\nu\cdot
x-\frac{\lambda^{r}}{\mu}t\Big{\\}}$

induced by oscillations in the initial data.

###  2.4. Linear adiabatic thermoviscoelasticity

Another system where oscillatory solutions may be constructed explicitly is
the case of adiabatic thermoviscoelastcity, namely the special case of (2.27)
when $\kappa=0$, which reads

$\displaystyle\frac{\partial^{2}u_{k}}{\partial t^{2}}$
$\displaystyle=\frac{\partial}{\partial
x_{\alpha}}\left(A_{kl\alpha\beta}\frac{\partial u_{l}}{\partial
x_{\beta}}+M_{k\alpha}\theta\right)+\mu\frac{\partial}{\partial
x_{\alpha}}\frac{\partial}{\partial x_{\alpha}}\left(\frac{\partial
u_{k}}{\partial t}\right)$ (2.35) $\displaystyle\frac{\partial\theta}{\partial
t}$ $\displaystyle=M_{k\alpha}\frac{\partial}{\partial
x_{\alpha}}\left(\frac{\partial u_{k}}{\partial t}\right)$

Introducing (2.28) leads to $(a_{1},...a_{d},b)$ solving the system of
ordinary differential equations (2.29) with $\kappa=0$, To this end, we look
for $a(t)=(a_{1},...a_{d})(t)$ solving

$\displaystyle\ddot{a}_{k}$
$\displaystyle=-n^{2}(A_{kl\alpha\beta}\nu_{\alpha}\nu_{\beta})a_{l}-n^{2}(M_{k\alpha}\nu_{\alpha})(M_{l\beta}\nu_{\beta})a_{l}-\mu
n^{2}\dot{a}_{k}$ (2.36)

and then select

$\displaystyle b=-(M_{k\alpha}\nu_{\alpha})a_{k}\,.$ (2.37)

The resulting $(a,b)$ will produce via (2.28) solutions of (2.35).

We impose the hypotheses (H1), (H2) of rank-1 convexity for
$A_{kl\alpha\beta}$ (but not hypothesis (H3)). Fix a direction $\nu\in
S^{d-1}$ and consider the modified acoustic tensor

$Q^{m}_{kl}=A_{kl\alpha\beta}\nu_{\alpha}\nu_{\beta}+(M_{k\alpha}\nu_{\alpha})(M_{l\beta}\nu_{\beta})$

By (H2) the matrix $Q^{m}_{kl}$ is positive definite. Let $\sigma^{r}>0$ be an
eigenvalue and $\zeta^{r}$ the corresponding eigenvector,

$Q^{m}\zeta^{r}=\big{(}Q+(M\nu)\otimes(M\nu)\big{)}\zeta^{r}=\sigma^{r}\zeta^{r}\,,\quad|\zeta^{r}|=1\,.$

Set $a_{k}=\alpha(t)\zeta^{r}_{k}$, then (2.36) is equivalent to solving the
equation for the amplitude $\alpha(t)$

$\frac{d^{2}\alpha}{dt^{2}}+\mu n^{2}\frac{d\alpha}{\partial
t}+n^{2}\sigma^{r}\alpha=0\,,$ (2.38)

which is precisely the example studied in section 2.2.

The analysis in section 2.2 indicates there are two solutions of (2.38) one
that decays fast, and a second denoted there by $\alpha_{+}(t)=e^{\rho_{+}t}$
which decays slowly. Utilizing (2.28), (2.37) the slowly decaying solution we
conclude that (2.35) has solutions of the form

$\displaystyle u_{k}=\frac{1}{n}\alpha(t)\zeta_{k}^{r}e^{in\nu\cdot x}$
$\displaystyle=\frac{1}{n}\zeta_{k}^{r}\exp\left\\{in\nu\cdot
x-\frac{\sigma^{r}}{\mu}t+O\big{(}\frac{1}{n^{2}}\big{)}t\right\\}$ (2.39)
$\displaystyle\theta=i(M_{k\alpha}\nu_{\alpha}\zeta^{r}_{k})\alpha(t)e^{in\nu\cdot
x}$
$\displaystyle=i(M_{k\alpha}\nu_{\alpha}\zeta^{r}_{k})\exp\left\\{in\nu\cdot
x-\frac{\sigma^{r}}{\mu}t+O\big{(}\frac{1}{n^{2}}\big{)}t\right\\}$
$\displaystyle=i(\zeta^{r}\cdot
M\nu)\Big{(}1+O\big{(}\frac{1}{n^{2}}\big{)}t\Big{)}\exp\left\\{in\nu\cdot
x-\frac{\sigma^{r}}{\mu}t\right\\}$ (2.40)

Note that $u_{n},v_{n}\to 0$ as $n\to\infty$ but oscillations persist for
$\theta$,

$F=\nabla
u=i(\zeta^{r}\otimes\nu)\Big{(}1+O\big{(}\frac{1}{n^{2}}\big{)}t\Big{)}\,\exp\left\\{in\nu\cdot
x-\frac{\sigma^{r}}{\mu}t\right\\}$ (2.41)

and $\nabla v$.

In the above example there are oscillations of $\theta$ to leading order, see
(2.40). This behavior should be contrasted to the example of section 2.3
where, due to the presence of heat diffusion, oscillations first appear in
$\Delta\theta$, see (2.34).

##  3\. Oscillations in quasi-linear dynamical models for phase transitions

We record some examples of solutions exhibiting sustained oscillations for
various quasi-linear hyperbolic-parabolic systems. The first class of models
concerns viscoelastic systems of the type used in modeling of phase
transitions.

###  3.1.  Time-dependent twinning solutions in nonlinear viscoelasticity

In preparation, we mention a well known result from continuum mechanics. Let
$\psi(x)$ be a displacement field defined on a reference configuration
$\mathcal{R}$ and suppose that $\mathcal{R}$ is split into two subdomains,
$\mathcal{R}=\mathcal{R}^{+}\cup\mathcal{R}^{-}$, by a smooth nonsingular
hypersurface $\mathcal{S}$ defined by

$f(x)=0\quad\mbox{with $\nabla f\neq 0$}.$

Suppose that $\psi$ enjoys the regularity $\psi$ is continuous on
$\mathcal{R}$, $\psi\in C^{1}\big{(}\overline{\mathcal{R}^{-}}\big{)}$,
$\psi\in C^{1}\big{(}\overline{\mathcal{R}^{+}}\big{)}$, and the limits

$\lim_{\scriptsize\begin{matrix}x\to x_{0}\\\
x>x_{0}\end{matrix}}\nabla\psi(x)=F^{+}\,,\quad\lim_{\scriptsize\begin{matrix}x\to
x_{0}\\\ x<x_{0}\end{matrix}}\nabla\psi(x)=F^{-}\,,$

exist for $x_{0}\in\mathcal{S}$ and are finite. Then the deformation gradients
satisfy across $\mathcal{S}$ the jump conditions

$F^{+}-F^{-}=a\otimes N$

where $N$ is the normal to the surface and $a$ is an amplitude capturing the
jump of the gradient in the normal direction at $\mathcal{S}$. This result is
the backbone in the construction of twinning solutions in elasticity, [9, 1].

Consider now the system of viscoelasticity of Kelvin-Voigt type,

$\displaystyle\partial_{tt}y-\mathop{\mathrm{div}}\Big{(}\frac{\partial
W}{\partial F}(\nabla y)+\nabla y_{t}\Big{)}=0\,.$ (3.1)

It is written as a system of conservation laws expressed in coordinate form,
for the quantities $v_{i}=\frac{\partial y_{i}}{\partial t}$,
$F_{i\alpha}=\frac{\partial y_{i}}{\partial x_{\alpha}}$, by

$\displaystyle\partial_{t}F_{i\alpha}$ $\displaystyle=\partial_{\alpha}v_{i}$
(3.2) $\displaystyle\partial_{t}v_{i}$
$\displaystyle=\partial_{\alpha}\Big{(}\frac{\partial W}{\partial
F_{i\alpha}}(F)+\partial_{\alpha}v_{i}\Big{)}$ $\displaystyle 0$
$\displaystyle=\partial_{\beta}F_{i\alpha}-\partial_{\alpha}F_{i\beta}\,.$

The last equation constrains $F$ to be a gradient. It is an involution, i.e. a
constraint that propagates from the initial data to solutions.

Our purpose is to construct oscillating solutions. This will result from two
ingredients, a special class of solutions combined with conditions that
guarantee joining these solutions across interfaces. We start with the
interface conditions.

Suppose that $\mathcal{S}=\\{(t,x):f(x,t)=0\\}$ is a space-time hypersurface
that splits a space time domain $Q_{T}=\mathcal{R}\times(0,T)$ into two parts,
$Q_{T}=\mathcal{R}^{+}\cup\mathcal{R}^{-}$. We assume that $\mathcal{S}$ is
smooth and depicts the motion of an interface $x=x(t)$ which is moving in the
direction of the normal $\nu=\frac{\nabla_{x}f}{|\nabla_{x}f|}$ with speed
$\dot{x}=s\nu$. Since the interface $x(t)$ moves on the surface, we have
$f(x(t),t)=0$, which easily yields for the interfacial speed that
$s=-\frac{f_{t}}{|\nabla_{x}f|}$. Suppose a solution $(v,F)$ of (3.1) that has
smoothness $v\in C(Q_{T})$, and such that $F,\nabla v\in
C\big{(}\overline{\mathcal{R}^{-}}\big{)}$, $F,\nabla v\in
C\big{(}\overline{\mathcal{R}^{+}}\big{)}$ and the right and left limits of
$F,\nabla v$ exist and are finite on both sides of the space-time surface
$\mathcal{S}$. The limits will satisfy the Rankine-Hugoniot conditions

$\displaystyle-s[F_{i\alpha}]$ $\displaystyle=\nu_{\alpha}[v_{i}]$ (3.3)
$\displaystyle-s[v_{i}]$ $\displaystyle=\nu_{\alpha}\left[\frac{\partial
W}{\partial F_{i\alpha}}(F)+\partial_{\alpha}v_{i}\right]$ (3.4)
$\displaystyle 0$
$\displaystyle=\nu_{\beta}[F_{i\alpha}]-\nu_{\alpha}[F_{i\beta}]$ (3.5)

where $[\cdot]$ denotes the jump across the interface. Condition (3.5) implies
that $[F]\tau=0$ for any vector $\tau\perp\nu$ and thus $[F]=a\otimes\nu$.
Together with (3.3) they lead to

$[F]=a\otimes\nu\,,\quad[v]=-sa\,.$ (3.6)

We are here interested in $[F]\neq 0$ and $[v]=0$. The latter precludes
$\nabla v$ from having delta masses and allows the jump to be computed via
(3.4). This suggests to consider standing shocks $s=0$ in conformance with the
considerations of the one-dimensional case studied in [8]. In summary, $s=0$
and $[F]\neq 0$ have to satisfy

$[F]=a\otimes\nu\,,\quad\big{[}T\nu\big{]}=\Big{[}\Big{(}\frac{\partial
W}{\partial F}(F)+\nabla v\Big{)}\nu\Big{]}=0$ (3.7)

where $T=\frac{\partial W}{\partial F}(F)+\nabla v$ is the total stress.
Observe that (3.7) requires that the steady interface is in equilibrium.

The second ingredient is special solutions of (3.1) of the particular form
$y(t,x)=F(t)x$ with $F(t)\in\mathbb{R}^{d\times d}$ a time dependent matrix.
One checks that $F(t)$ satisfies $\ddot{F}=0$, which implies that the solution
of (3.1) must be of the form

$y(t,x)=(tF_{0}+F_{1})x\,,\quad F_{0},F_{1}\in\mathbb{R}^{d\times d}\,,$ (3.8)

where $F_{0}$, $F_{1}$ are constant matrices. They describe uniform shear and
are universal in the sense that they are independent of the form of the stored
energy $W(F)$.

We use (3.6) and (3.7) to construct some special solutions of (3.2), which are
in turn used to construct oscillatory solutions for the same system.

####  3.1.1. Steady solutions with jumps in the deformation gradient in
elasticity

The first solution is known from work of Ball and James [1] and provides
solutions with discontinuities in the deformation gradient for the equations
of elasticity. The solutions are steady and solve (3.2). Namely, let $F_{-}$,
$F_{+}$ be such that $F_{+}-F_{-}=a\otimes\nu$ with $a\in\mathbb{R}^{d}$,
$\nu\in\mathcal{S}^{d-1}$. The hyperplane $\nu\cdot x=0$ passes through the
origin and is orthogonal to the unit vector $\nu$. The function

$y(t,x)=\begin{cases}F_{-}x&x\cdot\nu<0\\\ F_{+}x&x\cdot\nu>0\\\ \end{cases}$
(3.9)

satisfies $v(t,x)=0$ and is a steady solution of

$\displaystyle\partial_{t}F_{i\alpha}$ $\displaystyle=\partial_{\alpha}v_{i}$
(3.10) $\displaystyle\partial_{t}v_{i}$
$\displaystyle=\partial_{\alpha}\Big{(}\frac{\partial W}{\partial
F_{i\alpha}}(F)\Big{)}$ $\displaystyle 0$
$\displaystyle=\partial_{\beta}F_{i\alpha}-\partial_{\alpha}F_{i\beta}$

and also of (3.2) provided

$F_{+}-F_{-}=a\otimes\nu\,,\qquad[T\nu\big{]}=\Big{(}\frac{\partial
W}{\partial F}(F_{+})-\frac{\partial W}{\partial F}(F_{-})\Big{)}\nu=0\,.$

This solution has discontinuous deformation gradient across an interface, and
is related to twinning [9]. The position of the matrices $F_{-}$, $F_{+}$ can
be interchanged and leads ato a different weak solution. Combining these
together can lead to a persistent steady oscillatory structure that is related
to phase transitions, [1].

####  3.1.2. Dynamic solutions with jumps in deformation gradient and strain
rate in viscoelasticity

Next consider $F_{0}$ a baseline deformation gradient,
$F_{-}=F_{0}+a\otimes\nu$, $F_{+}=F_{0}+b\otimes\nu$, where $a$,
$b\in\mathbb{R}^{d}$, $\nu\in\mathcal{S}^{d-1}$. Then
$F_{+}-F_{-}=(b-a)\otimes\nu$. The hyperplane $x\cdot\nu=0$ passes through the
origin. Consider the motion

$\displaystyle y(t,x)$
$\displaystyle=\begin{cases}y_{-}(t,x)=t\big{(}F_{0}+a\otimes\nu\big{)}x&\quad
x\cdot\nu<0\\\ y_{+}(t,x)=t\big{(}F_{0}+b\otimes\nu\big{)}x&\quad
x\cdot\nu>0\\\ \end{cases}$ (3.11) $\displaystyle v(t,x)$
$\displaystyle=\begin{cases}\big{(}F_{0}+a\otimes\nu\big{)}x&\quad
x\cdot\nu<0\\\ \big{(}F_{0}+b\otimes\nu\big{)}x&\quad x\cdot\nu>0\\\
\end{cases}$

Note that $y$ and $v$ are both continuous across the interface $x\cdot\nu=0$
which is steady in time. The deformation gradient $\nabla y$ and stretch
tensor $\nabla v$ have discontinuities across that interface. Moreover, by
virtue of (3.7), $y$ will be a weak solution of (3.2) on the domain
$[1,2]\times\mathbb{R}^{d}$ provided

$\Big{(}\frac{\partial W}{\partial F}(tF_{-}+t(b-a)\otimes\nu)-\frac{\partial
W}{\partial F}(tF_{-})+(b-a)\otimes\nu\Big{)}\nu=0\,,\quad\mbox{ for
$t\in[1,2]$},$ (C)

is satisfied. The reader should note that under (C) the role of $a$ and $b$
(or $F_{-}$ and $F_{+}$) can be interchanged so as to obtain a second solution
with discontinuities in the deformation gradient $\nabla y$ and stretching
$\nabla v$.

$x_{1}$$x_{2}$$tF_{-}$$tF_{+}$$-2\mathstrut$ $-2{+}\theta\mathstrut$
$tF_{-}$$tF_{+}$$-1\mathstrut$ $-1{+}\theta\mathstrut$
$tF_{-}$$tF_{+}$$0\mathstrut$ $\theta\mathstrut$ $tF_{-}$$tF_{+}$$1\mathstrut$
$1{+}\theta\mathstrut$ $tF_{-}$$tF_{+}$$2\mathstrut$ $2{+}\theta\mathstrut$
$3\mathstrut$$x\cdot\nu=0$$x\cdot\nu=\theta$$x\cdot\nu=1$$x\cdot\nu=2$$\nu$
Figure 1. Deformation gradient for oscillating solutions,
$F_{+}=F_{0}+a\otimes\nu$, $F_{-}=F_{0}+b\otimes\nu$.

Suppose that the condition (C) is satisfied. By interlacing the solution
(3.11) with the corresponding solution with the position of $F_{-}$ and
$F_{+}$ interchanged, we construct an oscillating solution defined on
$[1,2]\times\mathbb{R}^{d}$ that jumps across the steady interfaces
$x\cdot\nu=k$ and $x\cdot\nu=k+\theta$, $k\in\mathbb{Z}$. The deformation
gradient and stretching are given respectively by

$\displaystyle\nabla y(t,x)$
$\displaystyle=\begin{cases}t(F_{0}+a\otimes\nu)&\quad k<x\cdot\nu<k+\theta\\\
t(F_{0}+b\otimes\nu)&\quad k+\theta<x\cdot\nu<k+1\\\ \end{cases}\qquad
k\in\mathbb{Z}$ (3.12) $\displaystyle\nabla v(t,x)$
$\displaystyle=\begin{cases}F_{0}+a\otimes\nu&\quad k<x\cdot\nu<k+\theta\\\
F_{0}+b\otimes\nu&\quad k+\theta<x\cdot\nu<k+1\\\ \end{cases}\qquad
k\in\mathbb{Z}$

and the deformation gradient is presented in Figure 1.

A commonly used notion in elasticity theory is rank-one convexity. Recall that
$W(F)$ is rank-one convex on the domain $D$ if for any $F\in D$

$\sum_{i,j,\alpha,\beta}\frac{\partial^{2}W}{\partial F_{i\alpha}\partial
F_{j\beta}}(F)\;\nu_{\alpha}\nu_{\beta}\;\xi_{i}\xi_{j}>0\,,\quad\xi\in\mathbb{R}^{d}-\\{0\\}\,,\,\nu\in\mathcal{S}^{d-1}\,.$
(ROC)

Observe that :

######  Lemma 3.1.

If condition (C) is satisfied then (ROC) is violated

###### Proof.

Indeed, using the summation convention, we express (C) as

$\displaystyle 0$ $\displaystyle=\Big{(}\frac{\partial W}{\partial
F_{i\alpha}}(tF_{-}+t(b-a)\otimes\nu)-\frac{\partial W}{\partial
F_{i\alpha}}(tF_{-})+(b-a)_{i}\nu_{\alpha}\Big{)}\nu_{\alpha}$
$\displaystyle=\left(\int_{0}^{1}\frac{d}{ds}\frac{\partial W}{\partial
F_{i\alpha}}(tF_{-}+st(b-a)\otimes\nu)\,ds\right)\nu_{\alpha}+(b-a)_{i}$
$\displaystyle=t\left(\int_{0}^{1}\frac{\partial^{2}W}{\partial
F_{i\alpha}\partial
F_{j\beta}}(tF_{-}+st(b-a)\otimes\nu)\,ds\right)(b-a)_{j}\nu_{\alpha}\nu_{\beta}+(b-a)_{i}$

IF $W(F)$ is rank-one convex on the domain $D$ containing
$tF_{-}+st(b-a)\otimes\nu$ then we obtain $0>|b-a|^{2}$ leading to a
contradiction. ∎

###  3.2. Longitudinal motions of a viscoelastic bar

We outline a one-dimensional example which is a variant of the example in [10,
sec 7] and concerns the system describing one-dimensional longitudinal motions
of a viscoelastic bar. The system

$\displaystyle u_{t}$ $\displaystyle=v_{x}$ (3.13) $\displaystyle v_{t}$
$\displaystyle=\sigma(u)_{x}+\partial_{x}\left(\frac{\mu}{u}v_{x}\right)$

describes longitudinal motions $y(t,x):(0,T)\times[0,1]\to\mathbb{R}$ for a
bar, where $u=y_{x}>0$ is the longitudinal strain and $v=y_{t}$ the velocity.
The total stress

$S=\sigma(y_{x})+\frac{\mu}{y_{x}}y_{tx}$

has an elastic and a viscous component, with the viscosity in Lagrangian
coordinates expressed as $\frac{\mu}{u}$. Here we take $\mu=1$ and the
function $\sigma(u)$ is smooth and non-monotone. Such models have been used
for modeling phase transitions.

Assume two positive states $0<a<b$ are fixed and suppose the stress function
$\sigma(u)$ satisfies

$\sigma(\tau a)=\sigma(\tau b)\qquad\mbox{ for $\tau\in[1,2]$}\,.$ (3.14)

Clearly, (3.14) requires that $\sigma(u)$ is nonmonotone. The example detailed
below is based on two properties of the system (3.13):

  * (i)

It admits special solutions of the form

$\bar{u}(t)=\kappa t\,,\quad\bar{v}(x)=\kappa x$ (3.15)

where $\kappa>0$ is a rate of stretching. These solutions are universal in the
sense that they satisfy (3.13) for any function $\sigma(u)$.

  * (ii)

(3.13) admits piecewise smooth solutions that are continuous in $v$ but
discontinuous in $u$ and $v_{x}$, see [8], provided the Rankine-Hugoniot
conditions

$\displaystyle-s[u]$ $\displaystyle=[v]$ $\displaystyle-s[v]$
$\displaystyle=\left[\sigma(u)+\frac{v_{x}}{u}\right]$

are satisfied, where $s$ is the shock speed and $[q]=q_{+}-q_{-}$ the jump of
the quantity $q$. As $v$ is continuous, the shocks are stationary $s=0$ and
$[u]\neq 0$ has to satisfy

$s=0\,,\quad\left[\sigma(u)+\frac{u_{t}}{u}\right]=0\,.$ (3.16)

We construct a family of periodic solutions to (3.13) defined for
$(t,x)\in[1,2]\times\mathbb{R}$. Fix states $a,b$ satisfying $0<a<2a<b<2b$ and
suppose that (3.14) is satisfied. We denote by $S(t)$ the common value

$S(t):=\frac{1}{ta}a+\sigma(ta)=\frac{1}{tb}b+\sigma(tb)\,,\qquad 1\leq t\leq
2\,.$

Next, fix $0<\theta<1$ and define the periodic function

$U(t,x):=\begin{cases}a\,t&\quad\;\;k<x<k+\theta\\\
b\,t&k+\theta<x<k+1\end{cases}\quad\;k\in\mathbb{Z}$ (3.17)

Finally, let $c_{\theta}=\theta a+(1-\theta)b$ and set

$\displaystyle Y(t,x)=\int_{0}^{x}U(t,y)dy$
$\displaystyle=\begin{cases}kc_{\theta}t+(x-k)at&\quad\;\;k<x<k+\theta\\\
kc_{\theta}t+\theta at+\big{(}x-k-\theta)bt&k+\theta<x<k+1\end{cases}\,,\quad
k\in\mathbb{Z}\,,$ (3.18) $\displaystyle V(t,x)=\partial_{t}Y(t,x)$
$\displaystyle=\begin{cases}kc_{\theta}+(x-k)a&\quad\;\;k<x<k+\theta\\\
kc_{\theta}+\theta a+\big{(}x-k-\theta)b&k+\theta<x<k+1\end{cases}\,,\quad
k\in\mathbb{Z}\,.$

Then $(U,V)$ is a weak solution of (3.13) on $[1,2]\times\mathbb{R}$. It
satisfies the equations in a classical sense on $(1,2)\times(k,k+\theta)$,
$(1,2)\times(k+\theta,k+1)$ and the Rankine-Hugoniot conditions (3.16) at the
interfaces $x=k$ and $x=k+\theta$ for $1\leq t\leq 2$.

The function $Y(t,x)$ is then rescaled and restricted to the interval
$(t,x)\in Q=(1,2)\times(-1,1)$ to define

$y_{n}(t,x)=\frac{1}{n}Y(t,nx)\,\quad
u_{n}=\partial_{x}y_{n}=U(t,nx)\,\quad{v_{n}}=\partial_{t}y_{n}=\frac{1}{n}V(t,nx)$
(3.19)

One checks that $(u_{n},v_{n})$ is a weak solution of (3.13) which is
stationary for the momentum equation. Moreover, one easily computes the limits

$\displaystyle u_{n}$
$\displaystyle\rightharpoonup(a\theta+b(1-\theta))t\quad\mbox{ weakly-$\star$
in $L^{\infty}\big{(}Q\big{)}$ }$ $\displaystyle\partial_{x}v_{n}$
$\displaystyle\rightharpoonup(a\theta+b(1-\theta))\quad\mbox{ weakly-$\star$
in $L^{\infty}\big{(}Q\big{)}$ }$ $\displaystyle v_{n}$
$\displaystyle\to(a\theta+b(1-\theta))x\quad\mbox{ strongly in
$L^{2}\big{(}Q\big{)}$ }$

and

$\sigma(u_{n})\rightharpoonup\theta\sigma(at)+(1-\theta)\sigma(bt)\neq\sigma\big{(}\theta
at+(1-\theta)bt)\,,$

weak-$\star$ in $L^{\infty}(Q)$. The oscillations in solutions are
characterized by the Young measure

$\nu=\theta\delta_{at}+(1-\theta)\delta_{bt}$

and are induced by oscillations in the initial data $u_{n}(1,x)$ and
$\partial_{x}v_{n}(1,x)$.

###  3.3. A class of nonlinear models describing viscoplastic response

Consider next the equation

$y_{tt}=\big{(}\sigma(y_{x},y_{xt})\big{)}_{x}$

which may be expressed as a system in the form

$\displaystyle u_{t}$ $\displaystyle=v_{x}$ (3.20) $\displaystyle v_{t}$
$\displaystyle=\partial_{x}\big{(}\sigma(u,v_{x})\big{)}$

Such systems have been used as models of viscoplasticity describing high
strain-rate deformations and are hyperbolic parabolic under the assumption
$\sigma_{q}(p,q)>0$.

This system has analogous properties to the example of section 3.2, namely:
(i) It admits uniform shearing solutions (3.15) for any $\kappa\in\mathbb{R}$
independently of the form of the function $\sigma(p,q)$. (ii) It can have
piecewise smooth solutions which satisfy the system in regions separated by
stationary interfaces, and across the interfaces they satisfy $[v]=0$,
$[u]\neq 0$ and $[v_{x}]\neq 0$ and the Rankine-Hugoniot jump conditions

$s=0\,,\quad\left[\sigma(u,v_{x})\right]=0\,.$ (3.21)

To continue, let $a,b$ be two fixed states satisfying $0<a<2a<b<2b$ and
suppose that

$\sigma(\tau a,a)=\sigma(\tau b,b)\quad\mbox{for $\tau\in[1,2]$}\,.$ (3.22)

We may then define periodic solutions $Y(t,x)$ and $(U,V)$ for (3.20) defined
by (3.17), (3.18) on the domain $(t,x)\in(1,2)\times\mathbb{R}$. Then
$\\{y_{n}\\}$ defined via the rescaling (3.19) produces an oscillatory
solution of (3.20) with sustained oscillations induced by the initial data.

To see what the hypothesis (3.22) signifies, consider the special case
$\sigma(p,q)=\varphi(p)q^{n}$ where $n>0$ exponent and $\varphi(u)>0$. This
satisfies $\sigma_{q}(p,q)>0$ so the system (3.20) falls within the class of
hyperbolic-parabolic systems. For this class (3.22) dictates

$\varphi(\tau a)=\varphi(\tau
b)\Big{(}\frac{b}{a}\Big{)}^{n}\quad\tau\in[1,2]$

and one can select $\varphi(u)$ that satisfies that for $a,b$ as above.

##  4\.  Quasilinear systems in gas dynamics for a viscous adiabatic gas

The system of compressible gas dynamics in one space dimension has the form

$\displaystyle u_{t}-v_{x}$ $\displaystyle=0$ (4.1) $\displaystyle
v_{t}-\sigma(u,\theta)_{x}$ $\displaystyle=0$
$\displaystyle\big{(}\tfrac{1}{2}v^{2}+e(u,\theta)\big{)}_{t}-(\sigma(u,\theta)\,v)_{x}$
$\displaystyle=0$

The constitutive theory is determined by a free energy function
$\psi=\psi(u,\theta)$ via the formulas

$\sigma=\frac{\partial\psi}{\partial
u}\,,\quad\eta=-\frac{\partial\psi}{\partial\theta}\,,\quad
e=\psi+\theta\,\eta$ (4.2)

and satisfies the Maxwell relations

$\sigma_{\theta}=-\eta_{u}\,,\quad e_{\theta}=\theta\eta_{\theta}\,,\quad
e_{u}=\sigma-\theta\sigma_{\theta}\,.$ (4.3)

The relations (4.2) are motivated from thermodynamics and (are designed to)
induce an additional conservation equation expressing conservation of the
entropy $\eta(u,\theta)$ for smooth solutions of (4.1),

$\partial_{t}\eta(u,\theta)=0\,.$

The system (4.1) may be written as a system of conservation laws

$\partial_{t}A(U)+\partial_{x}F(U)=0$

where

$\begin{split}U=\begin{pmatrix}u\\\ v\\\ \theta\end{pmatrix}\quad
A(U)=\begin{pmatrix}u\\\ v\\\ \tfrac{1}{2}v^{2}+e(u,\theta)\end{pmatrix}\quad
F(U)=-\begin{pmatrix}v\\\ \sigma(u,\theta)\\\
v\,\sigma(u,\theta)\end{pmatrix}\end{split}$

It is hyperbolic if

$e_{\theta}>0\,,\quad\sigma_{u}>0\,,\quad\eta_{\theta}>0\,.$ (H)

with the wave speeds computed by $\lambda_{0}=0$ and
$\lambda_{\pm}=\pm\sqrt{\sigma_{u}+\tfrac{\sigma_{\theta}^{2}}{\eta_{\theta}}}$.
The condition of hyperbolicity (H) is equivalent to the hypothesis
$\psi_{uu}>0$ and $\psi_{\theta\theta}<0$.

We will consider the one-dimensional hyperbolic-parabolic system

$\displaystyle u_{t}-v_{x}$ $\displaystyle=0$ (4.4) $\displaystyle
v_{t}-\sigma(u,\theta)_{x}$ $\displaystyle=(\frac{\mu}{u}v_{x})_{x}$
$\displaystyle\big{(}\tfrac{1}{2}v^{2}+e(u,\theta)\big{)}_{t}-\Big{(}\sigma(u,\theta)\,v\Big{)}_{x}$
$\displaystyle=\Big{(}\frac{\mu}{u}\,v\,v_{x}\Big{)}_{x}+\Big{(}\frac{\kappa}{u}\theta_{x}\Big{)}_{x}\,.$

In this model $u$ stands for the specific volume (the inverse of the density),
$v$ is the longitudinal velocity, and $\theta$ the temperature, while the
internal energy $e$ and the stress $\sigma$ are determined via constitutive
relations (4.2), (4.3); in this interpretation $u>0$ and $\theta>0$. In (4.4)
we adopted a Stokes law for the viscous stress

$\tau_{viscous}=\mu(u,\theta)\frac{1}{u}v_{x}\quad\mbox{ with
$\mu(u,\theta)\geq 0$},$

and a Fourier law for the heat conduction,

$Q=\kappa(u,\theta)\frac{1}{u}\theta_{x}\quad\mbox{ with $\kappa(u,\theta)\geq
0$}.$

The form of $\frac{\mu}{u}$ for the viscosity and $\frac{\kappa}{u}$ for the
heat conductivity is due to the interpretation of the model in Lagrangian
coordinates. The theory of gas dynamics for viscous and heat-conducting gases
satisfies the entropy production identity

$\partial_{t}\eta(u,\theta)-\partial_{x}\left(\frac{Q}{\theta}\right)=\frac{\mu}{u}\frac{v_{x}^{2}}{\theta}+\frac{\kappa}{u}\left(\frac{\theta_{x}}{\theta}\right)^{2}\geq
0\,.$ (4.5)

Another interpretation of (4.4) is to describe one dimensional shear motions
of a thermoviscoelastic materials. In this case $u$ will be the shear strain,
$v$ the velocity in the shear direction, $\theta>0$ the temperature; in this
interpretation $u$ does not obey a positivity constraint, while the viscous
stress will be $\tau=\mu v_{x}$, the heat conductivity $q=\kappa\theta_{x}$,
and (4.5) has to be adapted accordingly.

Our objective is to construct a periodic solution of (4.4), defined for $1\leq
t\leq 2$ and $x\in\mathbb{R}$ periodic with period $p=1$, in analogy to the
construction of section 3.2. This by rescaling will produce a weakly
convergent sequence of exact solutions to (4.4) that will converge weakly but
not strongly. The construction will be done for $\mu$ constant and for
$\kappa=0$, that is for a viscous adiabatic gas, and will be based on
combining uniform extension solutions with functions that exhibit jumps on the
temperature, strain, and strain-rate but with continuous velocities.

###  4.1. Uniform extension solutions

Consider (4.4) with $\mu$ and $\kappa$ both constant with $\mu>0$ and
$\kappa\geq 0$. We study a special class of exact solutions
$(\bar{y}(x,t),\bar{\theta}(t))$ which are required to be of the form

$\bar{y}(x,t)=axt\,,\qquad\bar{\theta}=\bar{\theta}(t)$ (4.6)

where $a>0$ is a constant and $\bar{\theta}(t)$ depends on time only. In this
case

$\bar{v}(x,t)=ax\,,\quad\bar{u}(x,t)=\bar{u}(t)=at$

Such a solution describes uniform extension of the gas when $at>1$ or
compression when $at<1$. One easily checks that (4.6) solves (4.4) if
$\bar{\theta}(t)=\bar{\theta}(t;a,\varphi)$ solves tha initial value problem

$\left\\{\begin{aligned}
\bar{\theta}\partial_{t}\eta(\bar{u},\bar{\theta})&=\frac{\mu}{\bar{u}}(\bar{u}_{t})^{2}\\\
\;\;\bar{\theta}(t=1;a,\varphi)&=\varphi\end{aligned}\right.$ (4.7)

where $\varphi>0$ depicts the temperature at time $t=1$ which is assumed
constant. The latter equation follows from (4.5) which is equivalent to (4.4)3
for smooth processes.

In the sequel we employ a constitutive theory for the gas

$\psi(u,\theta)=\theta\int_{1}^{u}\tau(s)\,ds-\big{(}\theta\ln\theta-\theta\big{)}$
(CF)

whence

$\sigma=\theta\tau(u)\,,\quad\eta(u,\theta)=\ln\theta-W(u)$

with $W(u)=\int_{1}^{u}\tau(s)\,ds$. Then, the initial value problem (4.7)
reduces to asking that $\bar{\theta}(t;a,\varphi)$ solves the problem

$\left\\{\begin{aligned}
\frac{d\bar{\theta}}{dt}-a\,\tau(at)\bar{\theta}&=\frac{\mu}{\bar{u}}(\bar{u}_{t})^{2}=\frac{\mu
a}{t}\\\ \;\;\bar{\theta}(t=1)&=\varphi\end{aligned}\right.$ (4.8)

The solution in the interval of interest $t\in[1,2]$ has the explicit form

$\bar{\theta}(t;a,\varphi)=\varphi
e^{W(at)-W(a)}+e^{W(at)}\int_{1}^{t}\frac{\mu a}{s}e^{-W(as)}\,ds$

Observe that since $\mu,a,\varphi>0$ the function $\bar{\theta}(t)>0$ as well.

###  4.2. Discontinuous Solutions

Next consider the problem (4.4) now under adiabatic conditions $\kappa=0$ and
we look for solutions which across an interface are continuous in velocity,
$[v]=0$ but have jump discontinuities $[\theta]\neq 0$, $[u]\neq 0$ and
$[v_{x}]\neq 0$. The Rankine-Hugoniot conditions,

$\displaystyle-s[u]$ $\displaystyle=[v]$ $\displaystyle-s[v]$
$\displaystyle=[\sigma(u,\theta)+\frac{\mu}{u}v_{x}]$
$\displaystyle-s[\tfrac{1}{2}v^{2}+e(u,\theta)]$
$\displaystyle=\big{[}(\sigma(u,\theta)+\frac{\mu}{u}v_{x})v]$

reduce to the requirement that

$s=0\,,\quad[v]=0\,,\quad\Big{[}\sigma(u,\theta)+\frac{\mu}{u}v_{x}\Big{]}=0$
(4.9)

If two solutions are smooth on the domains right and left of an interface
located at $x=x_{0}$ and satisfy across the interface the jump conditions
(4.9) then they give rise to a weak solution of the problem (4.4).

###  4.3. Construction of an oscillatory solution

We construct a weak solution consisting of two uniform extension solutions
connected via a jump discontinuity satisfying (4.9) along an interface at
$x=0$.

######  Lemma 4.1.

Let the constitutive theory be of the form (CF), and let $A,B$ satisfy
$0<A<2A<B$ and $\varphi_{A},\varphi_{B}>0$ be given and suppose that
$\tau(u)>0$ for $u\in[A,2A]$. Define for $t\in[1,2]$

$u_{A}=At\,,\quad\theta_{A}=\bar{\theta}(t;A,\varphi_{A})\,,\qquad
u_{B}=Bt\,,\quad\theta_{B}=\bar{\theta}(t;B,\varphi_{B})$

where $\bar{\theta}(t,a,\varphi)$ solves the initial value problem (4.8). Then
if $\tau(\cdot)$ is selected to satisfy

$\tau(At)\theta_{A}(t)=\tau(Bt)\theta_{B}(t)\quad t\in[1,2]$ (4.10)

then (4.9) is satisfied and

$(u,v,\theta)=\begin{cases}(At,Ax,\theta_{A}(t))&x<0\\\
(Bt,Bx,\theta_{B}(t))&x>0\end{cases}$ (4.11)

is a weak solution of (4.4) with $\kappa=0$ for $t\in[1,2]$.

###### Proof.

Let $\tau(\cdot)$ de defined in $[A,2A]$ and let $\theta_{A}(t)$ be defined by
(4.8) with $a=A$ and $\varphi=\varphi_{A}$. Let

$g(t)=\tau(At)\theta_{A}(t)$

and define $\hat{\theta}(t)$ to be the solution of the initial value problem

$\displaystyle\frac{d\hat{\theta}}{dt}$ $\displaystyle=\frac{\mu
B}{t}+B\tau(At)\theta_{A}(t)$ $\displaystyle\hat{\theta}(t=1)$
$\displaystyle=\varphi_{B}$

Note that $\hat{\theta}(t)>0$.

Given $\hat{\theta}(t)$ we select $\tau(\cdot)$ in the interval $[B,2B]$ so
that

$\tau(Bt)=\tau(At)\frac{\theta_{A}(t)}{\hat{\theta}(t)}$

Since $2A<B$ the function $\tau(\cdot)$ is well defined on $[B,2B]$. Moreover,
$\hat{\theta}$ solves (4.8) and thus $\hat{\theta}=\theta_{B}$. We conclude by
noticing that (4.10) implies the interface condition (4.9) across the
interface $x=0$ and thus (4.11) is a weak solution. ∎

The above argument is symmetric in the placement of $A$ and $B$ and thus

$(u,v,\theta)=\begin{cases}(Bt,Bx,\theta_{B}(t))&x<0\\\
(At,Ax,\theta_{A}(t))&x>0\end{cases}$ (4.12)

is again a weak solution of (4.4) with $\kappa=0$ for $t\in[1,2]$. We conclude
by iterating these two solutions to construct an oscillating weak solution
defined on $\mathbb{R}\times[1,2]$. We proceed along the lines of section 3.2.

Fix states $A,B$, $\theta_{A}(t)$, $\theta_{B}(t)$ as above and suppose that
(4.10) is satisfied. Let $0<\lambda<1$, let $c_{\lambda}=\lambda
A+(1-\lambda)B$ and define

$\displaystyle u(t,x)$
$\displaystyle:=\begin{cases}A\,t&\quad\;\;k<x<k+\lambda\\\
B\,t&k+\lambda<x<k+1\end{cases}\quad\;k\in\mathbb{Z}$ (4.13)
$\displaystyle\theta(t,x)$
$\displaystyle:=\begin{cases}\theta_{A}(t)&\quad\;\;k<x<k+\lambda\\\
\theta_{B}(t)&k+\lambda<x<k+1\end{cases}\quad\;k\in\mathbb{Z}$

Next, set

$\displaystyle y(t,x)=\int_{0}^{x}u(t,z)dz$
$\displaystyle=\begin{cases}kc_{\lambda}t+(x-k)At&\quad\;\;k<x<k+\lambda\\\
kc_{\lambda}t+\lambda
At+\big{(}x-k-\lambda)Bt&k+\lambda<x<k+1\end{cases}\,,\quad k\in\mathbb{Z}\,,$
(4.14) $\displaystyle y(t,x)=\partial_{t}y(t,x)$
$\displaystyle=\begin{cases}kc_{\lambda}+(x-k)A&\quad\;\;k<x<k+\lambda\\\
kc_{\lambda}+\lambda A+\big{(}x-k-\lambda)B&k+\lambda<x<k+1\end{cases}\,,\quad
k\in\mathbb{Z}\,.$

Then $(u,v,\theta)$ is a weak solution of (3.13) on $[1,2]\times\mathbb{R}$
satisfying the equations a classical sense on $(1,2)\times(k,k+\lambda)$,
$(1,2)\times(k+\lambda,k+1)$ and the Rankine-Hugoniot conditions at the
interfaces $x=k$, $x=k+\lambda$ for $k\in\mathbb{Z}$ and $1\leq t\leq 2$.

Finally, by rescaling

$y_{n}(t,x)=\frac{1}{n}y(t,nx)\,\quad
u_{n}=\partial_{x}y_{n}=u(t,nx)\,\quad{v_{n}}=\partial_{t}y_{n}=\frac{1}{n}v(t,nx)\,\quad\theta_{n}(t,x)=\theta(t,nx)$
(4.15)

and restricting to the interval $(t,x)\in Q=(1,2)\times(-1,1)$ we obtain a
solution with sustained oscillations propagating from the data at $t=1$ to the
solution in the interval $t\in[1,2]$. One checks the limits

$\displaystyle u_{n}$
$\displaystyle\rightharpoonup(A\lambda+B(1-\lambda))t\quad\mbox{
weakly-$\star$ in $L^{\infty}\big{(}Q\big{)}$ }$
$\displaystyle\partial_{x}v_{n}$
$\displaystyle\rightharpoonup(A\lambda+B(1-\lambda))\quad\mbox{ weakly-$\star$
in $L^{\infty}\big{(}Q\big{)}$ }$ $\displaystyle v_{n}$
$\displaystyle\to((A\lambda+B(1-\lambda))x\quad\mbox{ strongly in
$L^{2}\big{(}Q\big{)}$ }$ $\displaystyle\theta_{n}$
$\displaystyle\rightharpoonup\big{(}\theta_{A}(t)\lambda+\theta_{B}(t)(1-\lambda)\big{)}\quad\mbox{
weakly-$\star$ in $L^{\infty}\big{(}Q\big{)}$ }$

##  5\. The compressible Navier-Stokes system in one space dimension

In the final section we consider the one-dimensional version of (1.4) in
Eulerian coordinates:

$\displaystyle\rho_{t}+(\rho u)_{y}$ $\displaystyle=0$ (5.1)
$\displaystyle(\rho u)_{t}+(\rho u^{2}+p(\rho))_{y}$ $\displaystyle=\mu
u_{yy}$

where $\rho(t,y)$ and $u(t,y)$ are the density and velocity of the fluid
expressed in Eulerian coordinates $(t,y)$ while $\mu>0$ is the viscosity.

The Rankine-Hugoniot jump conditions across a jump discontinuity that moves
with speed $s$ have the form

$\displaystyle\big{[}\rho(u-s)\big{]}$ $\displaystyle=0$ (5.2)
$\displaystyle\big{[}\rho u(u-s)\big{]}$ $\displaystyle=\big{[}-p(\rho)+\mu
u_{y}\big{]}$

It follows that there can be solutions that are continuous in velocity $[u]=0$
but discontinuous in density, $[\rho]\neq 0$, across the interface moving with
speed $u=s$. Such solutions will satisfy the conservation of mass on the
interface (5.2)1. The conservation of momentum reduces to the condition

$\big{[}-p(\rho)+\mu u_{y}\big{]}=0$ (5.3)

Observe that since $[u]=0$ across the interface the jump $[u_{y}]$ is well
defined.

###  5.1. Langrangian to Eulerian

The system (3.13), which we now write in the notation

$\displaystyle w_{t}$ $\displaystyle=v_{x}$ (5.4) $\displaystyle v_{t}$
$\displaystyle=\sigma(w)_{x}+\partial_{x}\left(\frac{\mu}{w}v_{x}\right)$

is the Lagrangian form of the equation (5.1). The equivalence is effected via
the transformation

$w=\frac{\partial y}{\partial x}\,,\quad v=\frac{\partial y}{\partial t}$

and setting

$\frac{\partial y}{\partial
t}(t,x)=v(t,x)=u(t,y(t,x))\,,\quad\rho(t,y(t,x))=\frac{1}{\frac{\partial
y}{\partial x}(t,x)}=\frac{1}{w(t,x)}$ (5.5)

A direct calculation shows that for smooth solutions (5.4) transforms to
(5.1). The same result can be achieved by a transformation of the weak forms
provided that $y(t,x)$ is assumed to be a bi-Lipschitz homeomorphism, [3].

Consider the special periodic solution of (3.13) that has the form (3.17),
(3.18). Recall, that $a<2a<b$ and that for this solution

$\displaystyle y(t,x)$
$\displaystyle=\begin{cases}kv_{0}(\theta)t+(x-k)at&\quad\;\;k<x<k+\theta\\\
kv_{0}(\theta)t+\theta
at+\big{(}x-k-\theta)bt&k+\theta<x<k+1\end{cases}\,,\quad k\in\mathbb{Z}\,,$
(5.6) $\displaystyle v(t,x)$
$\displaystyle=\begin{cases}kv_{0}(\theta)+(x-k)a&\quad\;\;k<x<k+\theta\\\
kv_{0}(\theta)+\theta a+\big{(}x-k-\theta)b&k+\theta<x<k+1\end{cases}\,,\quad
k\in\mathbb{Z}\,.$

where $v_{0}(\theta)=\theta a+(1-\theta)b$ and $0<\theta<1$. Also, recall that

$\displaystyle w(t,x)$ $\displaystyle=\begin{cases}ta&\quad\;\;k<x<k+\theta\\\
tb&k+\theta<x<k+1\end{cases}\,,\quad k\in\mathbb{Z}\,,$ (5.7)

When we transform (5.6) and (5.7) to Eulerian coordinates via the
transformation (5.5) we obtain the following Eulerian form

$\displaystyle u(t,y)$ $\displaystyle=\frac{y}{t}$ (5.8)
$\displaystyle\rho(t,y)$ $\displaystyle=\begin{cases}\;\frac{1}{ta}\quad&\quad
0<\frac{y}{t}-kv_{0}(\theta)<a\theta\\\\[5.0pt]
\;\frac{1}{tb}\quad&\;\;a\theta<\frac{y}{t}-kv_{0}(\theta)<v_{0}(\theta)\end{cases}\quad
k\in\mathbb{Z}$ (5.9)

######  Remark 5.1.

The function

$\bar{\rho}(t,y)=\frac{1}{ta}\,,\quad\bar{u}(t,y)=\frac{y}{t}$

with $a>0$ is an exact solution of the system (5.1)

For the solution (5.8)-(5.9) we note that (i) it has discontinuities in $\rho$
on the lines $\xi_{k}(t)=ktv_{0}(\theta)$ and
$\xi_{k,\theta}=ktv_{0}(\theta)+ta\theta$. (ii) On the line $\xi_{k}(t)$ we
have

$\frac{d\xi_{k}}{dt}=kv_{0}(\theta)=u(t,y)\Big{|}_{y=\xi_{k}(t)}$

(iii) On the line $\xi_{k,\theta}(t)$ we have

$\frac{d\xi_{k}}{dt}=kv_{0}(\theta)+a\theta=u(t,y)\Big{|}_{y=\xi_{k,\theta}(t)}$

(iv) Let us denote

$I_{k}=\\{(t,y):0<y-ktv_{0}(\theta)<ta\theta\\}\,,\quad
J_{k}=\\{(t,y):ta\theta<y-ktv_{0}(\theta)<tv_{0}(\theta)\\}\,.$

Then we compute on each domain $I_{k}$ and $J_{k}$ that

$\partial_{t}\rho+\partial_{y}(\rho u)=0$

and

$\displaystyle\partial_{t}(\rho u)+\partial_{y}(\rho u^{2})$ $\displaystyle=0$
$\displaystyle\partial_{y}\Big{(}-p(\rho)+\mu\frac{\partial u}{\partial
y}\Big{)}=0$

that is the equations (5.1) are satisfied on each of $I_{k}$, $J_{k}$. (v)
Suppose that we impose the hypothesis on the pressure

$p\left(\frac{1}{at}\right)=p\left(\frac{1}{bt}\right)\quad\mbox{for \;
$t\in[1,2]$}$ (AP)

then on the interface $\xi_{k}(t)=ktv_{0}(\theta)$ we have

$[\rho u(u-\dot{\xi}_{k})]=0,[-p(\rho)+\mu u_{y}]=0$

The same relations also hold on the interface
$\xi_{k,\theta}(t)=ktv_{0}(\theta)+ta\theta$. We thus conclude

######  Proposition 5.2.

Let the pressure satisfy (AP). Then if $0<a<2a<b<2b$ then (5.8)-(5.9) define a
weak solution of the compressible Navier-Stokes system (5.1) defined for
$y\in\mathbb{R}$, $t\in[1,2]$.

For states $a$, $b$ satisfying $0<a<2a<b<2b$ the condition (AP) can be
fulfilled and results to nonmonotone equations for the pressure.

###  5.2.  Sustained oscillations

Next define the sequence of solutions $(\rho_{n},u_{n})$ by

$u_{n}(t,y)=\frac{1}{n}u(t,ny)\,,\quad\rho_{n}(t,y)=\rho(t,ny)\qquad
y\in(-1,1),\;t\in[1,2]\,,$ (5.10)

where $(\rho,u)$ is given by (5.8)-(5.9), $n\in\mathbb{N}$. Then we have

$u_{n}(t,y)=\frac{y}{t}$

and

$\rho_{n}(t,y)=\begin{cases}\;\frac{1}{ta}\quad&\qquad\quad\frac{k}{n}v_{0}(\theta)<\frac{y}{t}<\frac{k}{n}v_{0}(\theta)+\frac{1}{n}a\theta\\\\[5.0pt]
\;\frac{1}{tb}\quad&\frac{k}{n}v_{0}(\theta)+\frac{1}{n}a\theta<\frac{y}{t}<\frac{k+1}{n}v_{0}(\theta)\end{cases}\quad
k\in\mathbb{Z}$

It is easy to check that $(\rho_{n},u_{n})$ is a weak solution of (5.1).
Indeed, since as we saw $(\rho,u)$ is a weak solution of (5.1) on
$[1,2]\times\mathbb{R}$, we have for $\phi\in
C^{\infty}_{c}([1,2)\times\mathbb{R})$

$\displaystyle\iint\rho(t,y)\partial_{t}\phi(t,y)dtdy+\iint\rho(t,y)u(t,y)\partial_{y}\phi(t,y)dtdy+\int\rho(1,y)\phi(1,y)dy=0\,.$
(5.11)

We introduce the test function $\phi(t,y)=\frac{1}{n}\varphi(t,\frac{1}{n}y)$
in (5.11), perform the change of variables $y^{\prime}=\frac{1}{n}y$ and use
(5.10) to obtain

$\displaystyle\iint\rho_{n}(t,y)\partial_{t}\varphi(t,y)dtdy+\iint\rho_{n}(t,y)u_{n}(t,y)\partial_{y}\varphi(t,y)dtdy+\int\rho_{n}(1,y)\varphi(1,y)dy=0\,,$
(5.12)

that is $(\rho_{n},u_{n})$ satisfies the weak form of (5.1)1. A similar
calculation shows that $(\rho_{n},u_{n})$ also satisfies the weak form of
(5.1)2.

If the sequence $(\rho_{n},u_{n})$ is restricted on $Q=[1,2]\times(-1,1)$ we
see that

$\rho_{n}\rightharpoonup\rho=\theta\frac{1}{ta}+(1-\theta)\frac{1}{tb}\quad\mbox{
weakly-$\star$ in $L^{\infty}\big{(}Q\big{)}$ }$

Of course $u_{n}=\frac{y}{t}$ converges strongly.

######  Theorem 5.3.

Under the hypothesis (AP) there is a sequence of weakly converging initial
data so that the solution $(\rho_{n},u_{n})$ of (5.1) has the property that
$\rho_{n}\rightharpoonup\rho$ weakly while $u_{n}\to u$ strongly.

######  Remark 5.4.

Note that $(\rho_{n},u_{n})$ are also weak solutions of the pressureless Euler
equations, and the assumption (AP) guarantees that the total stress is a
function of time.

Statements and Declarations

Data availability The data supporting the findings of this study are available
within the paper,

Competing Interests The author has no relevant competing interests to declare.

Funding Research supported by KAUST baseline funds, No BAS/1/1652-01-01

## References

  * [1] J. M. Ball and R. D. James.  Fine phase mixtures as minimizers of energy.  Arch. Rational Mech. Anal., 100(1):13–52, 1987.
  * [2] C. M. Dafermos.  Can dissipation prevent the breaking of waves?  In Transactions of the Twenty-Sixth Conference of Army Mathematicians (Hanover, N.H., 1980), volume 1 of ARO Rep. 81, pages 187–198. U.S. Army Res. Office, Research Triangle Park, NC, 1981.
  * [3] C. M. Dafermos.  Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, fourth edition, 2016.
  * [4] S. Demoulini.  Weak solutions for a class of nonlinear systems of viscoelasticity.  Arch. Ration. Mech. Anal., 155(4):299–334, 2000.
  * [5] E. Feireisl.  Dynamics of viscous compressible fluids, volume 26 of Oxford Lecture Series in Mathematics and its Applications.  Oxford University Press, Oxford, 2004.
  * [6] G. Friesecke and G. Dolzmann.  Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy.  SIAM J. Math. Anal., 28(2):363–380, 1997.
  * [7] M. Hillairet.  Propagation of density-oscillations in solutions to the barotropic compressible Navier-Stokes system.  J. Math. Fluid Mech., 9(3):343–376, 2007.
  * [8] D. Hoff.  Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data.  Proc. Roy. Soc. Edinburgh Sect. A, 103(3-4):301–315, 1986.
  * [9] R. D. James.  Finite deformation by mechanical twinning.  Arch. Rational Mech. Anal., 77(2):143–176, 1981.
  * [10] K. Koumatos, C. Lattanzio, S. Spirito, and A. E. Tzavaras.  Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy.  J. Hyperbolic Differ. Equ., 20(2):433–474, 2023.
  * [11] P.-L. Lions.  Mathematical topics in fluid mechanics. Vol. 2, volume 10 of Oxford Lecture Series in Mathematics and its Applications.  The Clarendon Press, Oxford University Press, New York, 1998.  Compressible models, Oxford Science Publications.
  * [12] T.-P. Liu and Y. Zeng.  Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws.  Mem. Amer. Math. Soc., 125(599):viii+120, 1997.
  * [13] D. Serre.  Variations de grande amplitude pour la densité d’un fluide visqueux compressible.  Phys. D, 48(1):113–128, 1991.