\documentclass[reqno]{amsart}
\usepackage{graphicx}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 149, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/149\hfil Nonclassical shock waves]
{Nonclassical shock waves of conservation laws:
Flux function having two inflection points}
\author[H. D. Nghia, M. D. Thanh\hfil EJDE-2006/149\hfilneg]
{Ho Dac Nghia, Mai Duc Thanh} % in alphabetical order
\address{Ho Dac Nghia \newline
Horizon International Bilingual School,
02 Luong Huu hanh, Pham Ngu Lao Ward,
District 1, Ho Chi Minh City, Vietnam}
\email{dacnghia@yahoo.com}
\address{Mai Duc Thanh \newline
Department of Mathematics,
International University, Quarter 6,
Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam}
\email{mdthanh@hcmiu.edu.vn}
\thanks{Submitted September 7, 2006. Published December 5, 2006.}
\subjclass[2000]{35L65, 76N10, 76L05}
\keywords{Conservation law; non-genuine nonlinearity;
nonclassical solution; \hfill\break\indent kinetic relation}
\begin{abstract}
We consider the Riemann problem for non-genuinely nonlinear
conservation laws where the flux function admits two inflection
points. This is a simplification of van der Waals fluid pressure,
which can be seen as a function of the specific volume for a
specific entropy at which the system lacks the non-genuine
nonlinearity. Corresponding to each inflection point, A
nonclassical Riemann solver can be uniquely constructed.
Furthermore, two kinetic relations can be used to construct
nonclassical Riemann solutions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
The theory of nonclassical solutions of hyperbolic systems of
conservation laws has been introduced by LeFloch and has been
developed for many years. Nonclassical shocks may appear when the
system fails to be genuinely nonlinear. Briefly, nonclassical
shock waves violate the standard the Oleinik criterion
\cite{Oleinik} in the scalar case and the Lax shock inequalities
\cite{Lax71} and the Liu entropy conditions \cite{Liu74} for the
case of hyperbolic systems of conservation laws. To select
nonclassical shock waves, by a standard way, one follows the
strategy proposed by Abeyaratne-Knowles
\cite{AbeyaratneKnowles88,AbeyaratneKnowles91}, and by Hayes and
LeFloch \cite{HayesLeFloch97, HayesLeFloch00, LeFlochbook} to
describe the whole family of {\it nonclassical Riemann solutions}
and then to use a {\it kinetic relation} to determine the relevant
physical solution. For material on this subject, see the text book
\cite{LeFlochbook}. Related works can be found in
\cite{LeFloch93, HayesLeFloch97, HayesLeFloch00, LeFlochbook,
LeFlochThanh00, LeFlochThanh01, LeFlochThanh02, LeFlochThanh03,
James80, Shearer82, Slemrod83, SchulzeShearer, Truskinovsky87,
Truskinovsky93, ShearerYang95} and the references therein.
In this paper, we consider the Riemann problem for
conservation laws, where the flux-function has two inflection
points
\begin{equation}
\begin{gathered}
\partial_t u + \partial_x f(u) = 0,\\
u(x,0) = \begin{cases}
u_l & \text{for } x < 0, \\
u_r & \text{for } x > 0.
\end{cases}
\end{gathered} \label{1.1}
\end{equation}
Here, $u_l$ and $u_r$ are constants.
The flux function $f$ is a twice differentiable function of
$u\in\mathbb{R}$ and is assumed to satisfy the following hypotheses
\begin{equation}
\begin{gathered}
f''(u) > 0 \quad \text{for } u \in (-\infty,0)\cup (1,+\infty),\\
f''(u) < 0 \quad \text{for } u \in (0,1),\\
\lim_{u \to \pm\infty} f'(u) = +\infty, \quad \lim_{u \to
\pm\infty} f(u) = +\infty.
\end{gathered}\label{1.2}
\end{equation}
Thus the flux $f$ has two inflection points at $u=0$ and $u=1$.
The specification of these two values does not restrict the scope
of consideration of this paper. By assumption, the function $f$ is
clearly convex in each interval $(-\infty,0)$ and $(1,+\infty)$,
and is concave in the interval $(0,1)$. To specify these
intervals, we denote
\begin{equation}
\begin{gathered}
E_I := (-\infty, 0),\\
E_{II} := [0,1],\\
E_{III} := (1,+\infty),
\end{gathered} \label{1.3}
\end{equation}
and call each of them a {\it phase}.
In studying nonclassical shocks, one is concerned at the break of
the genuine nonlinearity of the system on a manifold. In many
situations, this manifold may be reduced to be simply an
inflection point of a flux function in appropriate coordinates. In
several models such as the Van der Waals fluids, non-genuine
nonlinearity may occur not only on one, but on two manifolds of
phase domains.
In the work of LeFloch-Thanh
\cite{LeFlochThanh00}, the presence of two inflection points in
the flux function was studied. The nonclassical Riemann solver was
constructed by restricting only on the first kinetic function,
though we may have several kinetic functions on a Hugoniot curve.
More clearly, following the strategy proposed by
Abeyaratne-Knowles \cite{AbeyaratneKnowles88,AbeyaratneKnowles91},
and by Hayes-LeFloch \cite{HayesLeFloch97, HayesLeFloch00,
LeFlochbook}, the authors define the entropy dissipation to
describe the whole set of nonclassical waves. It appears that the
entropy dissipation may vanish three times. And this would lead to
the definition of two kinetic functions. The domain as well as the
range of each of these two kinetic function contains one
inflection point and its values are symmetric to the variable
values with respect to the inflection point. The difficulty to use
the second kinetic function is that the shock speed involving the
second kinetic function may be less than that of the shock speed
using the first kinetic function. Consequently, the Riemann
solution may not be well-defined when two kinetic functions are to
be involved. In LeFloch-Thanh \cite{LeFlochThanh03}, phase
transitions were observed. All nonclassical shock waves satisfying
a single entropy condition that entropy should be nondecreasing in
time were also characterized.
This paper will deal with the case of two apparent kinetic
functions, continuing works in \cite{LeFlochThanh00,
LeFlochThanh03}. For simplicity, we restrict our attention to the
scalar case where we have a single conservation law. The flux
function will have the shape of the pressure of van der Waals
fluids in the region where it admits two inflection points.
Accordingly, we may have two kinetic functions, and we will
consider when we can use each of them, or both. Moreover, as the
entropy dissipation selects nonclassical waves like the rule of
equal areas, we will define the kinetic functions relying on the
rule of equal areas to set up their domains. To select a unique
solution, however, we have to restrict the range of kinetic
functions such that the chord connecting two states of a
nonclassical shock cuts the graph of the flux function at only one
point. The construction may be more visual in some sense. We note
that a similar way was constructed for classical shock waves by
Oleinik \cite{Oleinik}.
We note that the existence of nonclassical shock waves is related
to the existence of travelling waves of a regularized problem for
diffusive-dispersive models, when the diffusive and dispersive
coefficients tend to zero, see \cite{LeFlochbook}.
This paper is organized as follows. In Section 2 we will
investigate the properties of tangents to the graph of $f$, and
then we review the Oleinik construction of the entropy solution.
Section 3 will be devoted to selecting non-classical Riemann
solutions relying on one kinetic relation corresponding to each
inflection point. In Section 4 we will give a Riemann solver
which involves two kinetic relations.
\section{Basic Properties and Oleinik Construction}
In this section, first we describe several essential properties of
the flux function $f$. Tangents to the graph of $f$ will be used
to select nonclassical shocks instead of an entropy dissipation.
Second, we review the Oleinik construction for {\it classical
entropy solutions} of the problem \eqref{1.1}.
Recall that a discontinuity of \eqref{1.1} the form
\begin{equation}
u(x,t)=\begin{cases}
u_l & \text{for } x < st, \\
u_r & \text{for } x > st,
\end{cases} \label{2.1}
\end{equation}
connecting the left-hand state $u_l$ and the right-hand state
$u_r$ with shock speed $s$, is called a classical shock of
\eqref{1.1} if it satisfies the Rankine-Hugoniot relations
\begin{equation}
-s(u_r-u_l) + (f(u_r)-f(u_l)) =0, \label{2.2}
\end{equation}
and the Oleinik entropy criterion
\begin{equation}
\frac{f(u)-f(u_l)}{u-u_l}\ge\frac{f(u_r)-f(u_l)}{u_r-u_l},\quad
\text{for $u$ between $u_r$ and $u_l$}.
\label{2.3}
\end{equation}
The condition \eqref{2.3} means that the graph of $f$ is lying below
(above) the line connecting $u_l$ to $u_r$ when $u_r u_l$).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
%\centerline{\psfig{file=flux.eps,width=8truecm}}
\caption{Flux function having two inflection points}
\label{Flux}
\end{figure}
Under the hypotheses \eqref{1.2}, the tangents at $1$ and $0$ cut the
graph of the flux function $f$ at points $a$ and $b$,
respectively, with $a<0<1**d$, the solution is combined from three elementary waves:
a rarefaction wave from $u_l$ to $c$, followed by a shock from $c$ to $d$,
and then followed
by a rarefaction wave from $d$ to $u_r$.
Second, we deal with $u_l\in [c,0]$. If $u_r\in (-\infty,u_l)$,
the Riemann solution is a single shock. A single rarefaction wave
can connect $u_l$ with the states $u_r \in (u_l,0]$. If $u_r\in
[0,\phi^{-\natural}(u_l)]$, then ${\phi}^\natural(u_r)\in [u_l,0]$ and the Riemann
solution is composed by a rarefaction wave from $u_l$ to
${\phi}^\natural(u_r)$ followed by a shock from ${\phi}^\natural(u_r)$ to $u_r$. A single
shock from $u_l$ can reach $u_r\in (\phi^{-\natural}(u_l),{\psi}^\natural(u_l]$. Finally,
if $u_r> {\psi}^\natural(u_l)$, the solution is a composite of a shock from
$u_l$ to ${\psi}^\natural(u_l)$ followed with a rarefaction wave connecting
${\psi}^\natural(u_l)$ to $u_r$.
Third, $u_l\in (0,1)$. A single shock from $u_l$ can reach
$u_r\in (-\infty,\phi^{-\natural}(u_l)]\cup [u_l,{\psi}^\natural(u_l)]$.
A single rarefaction wave from $u_l$ can connect
to $u_r\in [0,u_l]$.
If $u_r\in (\phi^{-\natural}(u_l),0)$, then there exists a unique value $u^*\in (0,u_l)$ such
that $\phi^{-\natural}(u^*)=u_r$. That is $u^*={\phi}^\natural(u_r)$. In that case the Riemann
solution is a rarefaction wave
connecting $u_l$ to $u^*$ followed by a shock connecting $u^*$ to $u_r$.
Finally, if $u_r> {\psi}^\natural(u_l)$, the Riemann solution is a shock connecting
$u_l$ to ${\psi}^\natural(u_l)$
followed with a rarefaction wave from ${\psi}^\natural(u_l)$ to $u_r$.
Fourth, assume that $u_l\in [1,b]$. A single shock from $u_l$ can
reach
$$
u_r\in (-\infty,\phi^{-\natural}({\psi}^\natural(u_l))]\cup [{\psi}^\natural(u_l),u_l].
$$
A single rarefaction wave from $u_l$ can connect
to $u_r\in [u_l,+\infty)$.
If $u_r\in [0,{\psi}^\natural(u_l))$, the Riemann solution
is combined by a shock from $u_l$ to ${\psi}^\natural(u_l)$ followed by a rarefaction
from ${\psi}^\natural(u_l)$ to $u_r$.
If $u_r\in (\phi^{-\natural}({\psi}^\natural(u_l)),a)$, the solution contained three waves:
a shock from $u_l$ to ${\psi}^\natural(u_l)$, followed by a
rarefaction from ${\psi}^\natural(u_l)$ to ${\phi}^\natural(u_r)$, and followed by a shock connecting
${\phi}^\natural(u_r)$ to $u_r$.
Finally, if $u_l\in (b,+\infty)$, then the Riemann solution is
simply a shock if $u_r\phi(u)>{\phi}^\natural(u),\quad\forall u < 0,\\
\phi^{-\natural}(u)<\phi(u)<{\phi}^\natural(u),\quad\forall u > 0,\\
\phi(0)={\phi}^\natural(0)=\phi^{-\natural}(0)=0;
\end{gathered} \label{3.1}
\end{equation}
\item[(A2)] The {\it contraction property}
\begin{equation}
|\phi \circ \phi (u)| < |u|, \quad\forall u\in \Omega_0, \quad u\ne 0.
\label{3.2}
\end{equation}
\item[(A3)] Conditions at limits:
\begin{equation}
\phi(p)={\psi}^\natural(p),\quad \phi(1)=a. \label{3.3}
\end{equation}
\end{itemize}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
%\centerline{\psfig{file=kinetic_phi.eps,width=8truecm}}
\caption{Kinetic Function $\phi$} \label{Kinetic1} %fig 3.1
\end{figure}
For an arbitrary non-classical shock between a given left-hand
state $u_0$ and a given right-hand state $u_1$, {\it kinetic
relation} is the requirement that
\begin{equation}
u_1=\phi (u_0). \label{3.6}
\end{equation}
To select non-classical shock rather than classical ones, we
postulate that
\begin{itemize}
\item[(C)] Non-classical shocks are preferred whenever
available.
\end{itemize}
We now solve the Riemann problem relying on this condition .
The construction in this section is similar to the non-classical one
for the $1$-wave family in
\cite{LeFlochThanh00}, but we want to recall it here for
completeness.
Suppose first that $u_l \in (-\infty,p)$. Any point $u_r \in
(-\infty,u_l)$ can be achieved by a single classical shock. Any
point $u_r\in (u_l,0]$ is attainable by a single rarefaction wave.
If $u_r\in (0,\phi(p)]$, there exists a unique point $u_*\in
[p,0)$ such that $u_r=\phi(u_*)$. The solution is then the
composite of a rarefaction wave from $u_l$ to $u_*$ followed by a
nonclassical shock from $u_*$ to $u_r$. If $u_r\in
(\phi(p),+\infty)$, the solution consists of three parts: A
rarefaction wave from $u_l$ to $p$ followed by a nonclassical
shock from $p$ to $\phi(p)$, followed by a rarefaction wave from
$\phi(p)$ to $u_r$.
Second, suppose that $u_l\in [p,0)$. A point $u_r\in (-\infty,
u_l)$ can be attained by a single classical shock. A point
$u_r\in (u_l,0]$ is attainable by a single rarefaction wave.
If $u_r\in (0,\phi(u_l)]$, there exists a unique point
$u_*\in [u_l,a)$ such that $u_r=\phi(u_*)$. The solution is then
the composite of the rarefaction wave from $u_l$ to $u_*$
followed by a nonclassical
shock from $u_*$ to $u_r$. If $u_r\in (\phi(u_l),\phi(p)]$, there
exists a unique point $u^*\in [p,u_l)$ such that $u_r=\phi(u^*)$.
For this construction to make sense, one must here check whether
the classical shock from $u_l$ to $u^*$ is slower than the
nonclassical shock from $u^*$ to $u_r$. So, consider the function
\begin{equation}
\tilde f(v) := \begin{cases}
f(v),&\text{if } v\in (-\infty, u_l],\\
f(u_l)+f'(u_l)(v-u_l),&\text{if } v \in (u_l,+\infty).
\end{cases} \label{3.7}
\end{equation}
If $u_r\in (\phi(u_l), \eta)$, where
$$
\eta:=\min\{\phi(p),\phi^{-\natural}(u_l)\},
$$
the function $\tilde f$ is convex on $(-\infty,+\infty)$ and the
points $u^*$ and $u_r$ belong to its epigraph. Therefore, the
line segment connecting $u^*$ and $u_r$ should lie above the
graph of $\tilde f$ in the interval $(u^*,u_r)\ni u_l$. That is to
say
$$
{\tilde f(u_l)-\tilde f(u^*)\over u_l-u^*} <
{f(u_r)-f(u^*)\over u_r-u^*},
$$
i.e.,
\begin{equation}
s(u_l,u^*)**~~\psi(u)>{\psi}^\natural(u),\quad\forall u < 1,\\
\psi^{-\natural}(u)<\psi(u)<{\psi}^\natural(u),\quad\forall u > 1,\\
\psi(1)={\psi}^\natural(1)=\psi^{-\natural}(1)=1;
\end{gathered} \label{3.10}
\end{equation}
\item[(B2)] The {\it contraction property}
\begin{equation}
|\psi \circ \psi (u)-1| < |u-1|, \quad\forall u\in \Omega_1,\quad u\ne 1.
\label{3.11}
\end{equation}
\item[(B3)] Conditions at limits:
\begin{equation}
\psi(q)={\phi}^\natural(q),\quad {\phi}^\natural(\psi(\theta))=\theta. \label{3.12}
\end{equation}
\end{itemize}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3}
\end{center}
%\centerline{\psfig{file=kinetic_psi.eps,width=8truecm}}
\caption{Kinetic Function $\psi$}
\label{Kinetic2} %fig3.2}
\end{figure}
For any non-classical shock between a given left-hand state $u_0$
and a given right-hand state $u_1$, kinetic relation for the
coming construction is the requirement that
\begin{equation}
u_1=\psi (u_0). \label{3.15}
\end{equation}
So, we begin to construct the Riemann solver, postulating the condition
(C) in the previous subsection.
Assume first that $u_l\in (-\infty,\theta)$. A single classical
shock can jump to any $u_r\in (-\infty,u_l)$. A single rarefaction
wave can connect $u_l$ from the left to any $u_r\in [u_l,0]$ from
the right. If $u_r\in (0,\phi^{-\natural}(\theta)]$, then ${\phi}^\natural(u_r)\in
[\theta,0)$, the solution thus is a rarefaction wave from $u_l$ to
${\phi}^\natural(u_r)$ followed by a classical shock from ${\phi}^\natural(u_r$ to $u_r$.
If now $u_r\in (\phi^{-\natural}(\theta),\psi(\theta))$, the solution consists
of a rarefaction wave from $u_l$ to $\theta$, followed by a
non-classical shock from $\theta$ to $\psi(\theta)$, then followed
by a classical shock from $\psi(\theta)$ to $u_r$. If $u_r\in
[\psi(\theta),+\infty)$, the solution is a composite of a
rarefaction wave from $u_l$ to $\theta$, followed by a
non-classical shock from $\theta$ to $\psi(\theta)$, then followed
by a rareffaction wave from $\psi(\theta)$ to $u_r$.
Second, let $u_l\in [\theta,0]$. A single classical shock can jump
to any $u_r\in (-\infty,u_l)$. A single rarefaction wave can
connect $u_l$ to any $u_r\in [u_l,0]$. If $u_r\in (0,\phi^{-\natural}(u_l)]$,
then ${\phi}^\natural(u_r)\in [u_l,0)$, and therefore the solution is a
rarefaction wave from $u_l$ to ${\phi}^\natural(u_r)$ followed by a classical
shock from ${\phi}^\natural(u_r)$ to $u_r$. If now $u_r\in
(\phi^{-\natural}(u_l),\psi^\sharp(u_l,\psi(u_l))]$, the solution is a single
classical shock. If $u_r\in (\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, then
the solution is a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical shock from $\psi(u_l)$ to $u_r$. If
$u_r\in [\psi(u_l),+\infty)$, then the solution is composed from a
non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
rarefaction wave from $\psi(u_l)$ to $u_r$.
Third, let $u_l\in (0,1)$. A single classical shock can arrive at
any $u_r\in (-\infty,\newline \phi^{-\natural}(u_l)]$. If
$u_r\in (\phi^{-\natural}(u_l),0]$, then ${\phi}^\natural(u_r)\in [0,u_l)$.
The solution is thus a
rarefaction wave from $u_l$ to ${\phi}^\natural(u_r)$ attached by a classical
shock from ${\phi}^\natural(u_r)$ to $u_r$. If $u_r\in (0,u_l]$, the solution
is a single rarefaction wave. A single classical shock can arrive
at any $u_r\in (u_l,\psi^\sharp(u_l,\psi(u_l))]$. If $u_r\in
(\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, then the solution is a composite
of a non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
classical shock from $\psi(u_l)$ to $u_r$. If $u_r\in
[\psi(u_l),+\infty)$, then the solution is combined from a
non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
rarefaction wave from $\psi(u_l)$ to $u_r$.
Fourth, assume $u_l\in (1,\psi^{-1}(0))$. By the monotony, we have
$$
\psi(u_l)>0.
$$
If $u_r\in [u_l,+\infty)$, then the solution is a rarefaction
wave. A single classical shock can jump from $u_l$ to any $u_r\in
[\psi^\sharp(u_l,\psi(u_l)), u_l)$.\newline If $u_r\in
(\psi(u_l),\psi^\sharp(u_l,\psi(u_l)))$, then the solution is combined
from two shocks: a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical one from $\psi(u_l)$ to $u_r$. If $u_r\in
[0,\psi(u_l)]$, then the solution is a non-classical shock from
$u_l$ to $\psi(u_l)$ followed by a rarefaction wave from
$\psi(u_l)$ to $u_r$. If now $u_r\in (\phi^{-\natural}(\psi(u_l)), 0)$, then
${\phi}^\natural(u_r)\in (0,\psi(u_l))$. The solution is thus a non-classical
shock from $u_l$ to $\psi(u_l)$ followed by a rarefaction wave
from $\psi(u_l)$ to ${\phi}^\natural(u_r)$ attached by a classical shock from
${\phi}^\natural(u_r)$ to $u_r$. If $u_r\in
[\psi^\sharp(u_l,\psi(u_l)),\phi^{-\natural}(\psi(u_l))]$, then
the solution is a non-classical shock from $u_l$ to $\psi(u_l)$ followed by
a classical shock from $\psi(u_l)$ to $u_r$.
If $u_r\in (-\infty,\psi^\sharp(u_l,\psi(u_l)))$, then no non-classical
shocks can be involved in the construction. We thus use the
classical construction in Section 2 in this interval. The
discontinuity in this regime is
\begin{equation}
u_l\in (1,\psi^{-1}(0)),\quad u_r=\psi^\sharp(u_l,\psi(u_l)).
\label{3.16}
\end{equation}
Fifth, let $u_l\in [\psi^{-1}(0),q]$. The monotony of $\psi$
yields
$$
\psi(u_l)\le 0.
$$
A single rarefaction wave can connect $u_l$ to any $u_r\in
[u_l,+\infty)$. A single classical shock can jump from $u_l$ to
any $u_r\in [\psi^\sharp(u_l,\psi(u_l)), u_l)$.
If
$u_r\in [\phi^{-\natural}(\psi(u_l)),\psi^\sharp(u_l,\psi(u_l)))$,
then the solution is combined from two shocks: a non-classical
shock from $u_l$ to
$\psi(u_l)$ followed by a classical one from $\psi(u_l)$ to $u_r$.
If $u_r\in (0,\phi^{-\natural}(\psi(u_l)))$, then ${\phi}^\natural(u_r)\in (\psi(u_l),0)$.
The solution is therefore a composite of a non-classical shock
from $u_l$ to $\psi(u_l)$ followed by a rarefaction wave from
$\psi(u_l)$ to ${\phi}^\natural(u_r)$, then attached by a classical shock from
${\phi}^\natural(u_r)$ to $u_r$. If $u_r\in [\psi(u_l),0]$, then the solution
is a non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
rarefaction wave from $\psi(u_l)$ to $u_r$. If $u_r\in
[\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, then the solution is a
non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
classical one from $\psi(u_l)$ to $u_r$. If $u_r\in
(-\infty,\psi^\sharp(u_l,\psi(u_l)))$, then there are no non-classical
shocks and we have a situation similar to the previous one. In
this construction, we have a discontinuity sharing the same
formula for $u_r$ but $u_l\in [\psi^{-1}(0),q]$ instead. Combining
this argument and (3.16), we obtain the curve of discontinuity of
the construction
\begin{equation}
\{u_l\in (1,q],\quad u_r=\psi^\sharp(u_l,\psi(u_l))\}. \label{3.17}
\end{equation}
Finally, let $u_l\in (q,+\infty)$. In this case we have no
non-classical shocks and we use the classical construction as
well.
The above arguments can be summarized in the following theorem
\begin{theorem} \label{thm3.2}
Given the initial Riemann data $(u_l,u_r)$. Under the hypotheses
\eqref{1.2} and the condition (C), the Riemann problem (1.1)
admits a unique self-similar solution made of rarefaction waves,
classical shocks and non-classical shocks satisfying the kinetic
relation (3.15). The Riemann solver is described by
\noindent Case 1: $u_l\in (-\infty,\theta)$.
\begin{itemize}
\item
If $u_r\in (-\infty,u_l)$, the solution is
a single classical shock.
\item If $u_r\in [u_l,0]$, the solution
is a single rarefaction wave.
\item If $u_r\in (0,\phi^{-\natural}(\theta)]$, the solution
is a rarefaction wave from $u_l$ to ${\phi}^\natural(u_r)$ followed by a
classical shock from ${\phi}^\natural(u_r$ to $u_r$.
\item
If $u_r\in (\phi^{-\natural}(\theta),\psi(\theta))$, the solution is a composite of
a rarefaction wave from $u_l$ to $\theta$, followed by a non-classical
shock from $\theta$ to $\psi(\theta)$,
followed by a classical shock from $\psi(\theta)$ to $u_r$.
If $u_r\in [\psi(\theta),+\infty)$, the solution is a
rarefaction wave from $u_l$ to $\theta$, followed by a
non-classical shock from $\theta$ to $\psi(\theta)$, then followed
by a rarefaction wave from $\psi(\theta)$ to $u_r$.
\end{itemize}
\noindent Case 2: $u_l\in [\theta,0]$.
\begin{itemize}
\item
If $u_r\in (-\infty,u_l)$, then the solution
is a classical shock.
\item
If $u_r\in [u_l,0]$, the solution is a
single rarefaction wave.
\item
If $u_r\in (0,\phi^{-\natural}(u_l)]$, the
solution is a rarefaction wave from $u_l$ to ${\phi}^\natural(u_r)$ followed
by a classical shock from ${\phi}^\natural(u_r)$ to $u_r$.
\item
If $u_r\in (\phi^{-\natural}(u_l),\psi^\sharp(u_l,\psi(u_l))]$, the solution is a single classical shock.
\item If $u_r\in (\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, the solution
is a non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
classical shock from $\psi(u_l)$ to $u_r$.
\item
If $u_r\in [\psi(u_l),+\infty)$, the solution is a composite of a non-classical shock
from $u_l$ to $\psi(u_l)$ followed by a rarefaction wave from $\psi(u_l)$ to $u_r$.
\end{itemize}
\noindent Case 3: $u_l\in (0,1)$.
If $u_r\in (-\infty,\phi^{-\natural}(u_l)]$, the solution is
a classical shock.
\begin{itemize}
\item
If $u_r\in (\phi^{-\natural}(u_l),0]$, the solution
is a rarefaction wave from $u_l$ to ${\phi}^\natural(u_r)$ attached by a
classical shock from ${\phi}^\natural(u_r)$ to $u_r$.
\item If $u_r\in
(0,u_l]$, the solution is a single rarefaction wave.
\item If $u_r\in (u_l,\psi^\sharp(u_l,\psi(u_l))]$, the solution is a single
classical shock.
\item If $u_r\in (\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, the solution
is a composite of a non-classical shock
from $u_l$ to $\psi(u_l)$ followed by a classical shock from
$\psi(u_l)$ to $u_r$.
\item If $u_r\in [\psi(u_l),+\infty)$, then the solution is a composite
of a non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
rarefaction wave from $\psi(u_l)$ to $u_r$.
\end{itemize}
\noindent Case 4: $u_l\in (1,\psi^{-1}(0))$.
\begin{itemize}
\item If $u_r\in [u_l,+\infty)$, the solution is a
rarefaction wave.
\item If $u_r\in [\psi^\sharp(u_l,\psi(u_l)), u_l)$, the
solution is a single classical shock.
\item If $u_r\in (\psi(u_l),\psi^\sharp(u_l,\psi(u_l)))$,
the solution is a composite of
two shocks: a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical one from $\psi(u_l)$ to $u_r$.
\item If
$u_r\in [0,\psi(u_l)]$, the solution is a non-classical shock
from $u_l$ to $\psi(u_l)$ followed by a rarefaction wave from
$\psi(u_l)$ to $u_r$.
\item If $u_r\in (\phi^{-\natural}(\psi(u_l)), 0)$, the
solution is a composite of a non-classical shock
from $u_l$ to $\psi(u_l)$ followed by a rarefaction wave
from $\psi(u_l)$ to ${\phi}^\natural(u_r)$ attached by a classical
shock from ${\phi}^\natural(u_r)$ to $u_r$.
\item If $u_r\in [\psi^\sharp(u_l,\psi(u_l)),\phi^{-\natural}(\psi(u_l))]$,
the solution is a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical shock from $\psi(u_l)$ to $u_r$.
\item If $u_r\in (-\infty,\psi^\sharp(u_l,\psi(u_l)))$,
then the construction is classical (Section 2).
\end{itemize}
\noindent Case 5: $u_l\in [\psi^{-1}(0),q]$.
\begin{itemize}
\item If $u_r\in [u_l,+\infty)$, the solution is a
single rarefaction wave.
\item If $u_r\in [\psi^\sharp(u_l,\psi(u_l)), u_l)$, the solution is a
single classical shock.
\item If $u_r\in [\phi^{-\natural}(\psi(u_l)),\psi^\sharp(u_l,\psi(u_l)))$,
the solution is a composite of
two shocks: a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical one from $\psi(u_l)$ to $u_r$.
\item If $u_r\in (0,\phi^{-\natural}(\psi(u_l)))$, the solution is a composite
of three elementary waves:
a non-classical shock from $u_l$ to $\psi(u_l)$ followed by a
rarefaction wave from $\psi(u_l)$ to ${\phi}^\natural(u_r)$, then attached by
a classical shock from ${\phi}^\natural(u_r)$ to $u_r$.
\item If $u_r\in
[\psi(u_l),0]$, the solution is a non-classical shock from $u_l$
to $\psi(u_l)$ followed by a rarefaction wave from $\psi(u_l)$ to
$u_r$.
\item If $u_r\in [\psi^\sharp(u_l,\psi(u_l)),\psi(u_l))$, then the
solution is a non-classical shock from $u_l$ to $\psi(u_l)$
followed by a classical one from $\psi(u_l)$ to $u_r$.
\item If
$u_r\in (-\infty,\psi^\sharp(u_l,\psi(u_l)))$, then the construction is
classical.
\end{itemize}
\noindent Case 6: $u_l\in (q,+\infty)$, the construction is
classical.
The curve of discontinuity is
$$
\{u_l\in (1,q],\quad u_r=\psi^\sharp(u_l,\psi(u_l))\}\subset \mathbb{R}^2.
$$
\end{theorem}
\section{Non-Classical Riemann Solver Using two Kinetic
Relations}
In this section, we discuss the Riemann solver to the problem
\eqref{1.1} using two kinetic relations for non-classical
shock-waves between two phases. It turns out that even under the
condition (C), non-uniqueness appears. A stronger condition is
imposed to guarantee there is a unique choice of non-classical
shocks. As expected, the unique Riemann solution does not depend
continuously globally on the Riemann data.
Let us first point out several circumstances in which there are
distinct choices of non-classical solutions adaptable to the
condition (C). Firtly, assume that $u_l\in (1,\psi^{-1}(0))$,
then $\psi(u_l)\in (0,1)$. Therefore,
$$
\phi(\psi(u_l))\in (\phi^{-\natural}(\psi(u_l)),{\phi}^\natural(\psi(u_l)))\subset
(\phi^{-\natural}(\psi(u_l)),0).
$$
If we take
$$
u_r=\phi(\psi(u_l)),
$$
then we obtain a solution contains two nonclassical shocks: one
nonclassical shock corresponding to the kinetic function $\psi$
from $u_l$ to $\psi(u_l)$, followed by one nonclassical shock
corresponding to the kinetic function $\phi$ from $\psi(u_l)$ to
$\phi(\psi(u_l))$. However, as derived from the construction in
the subsection 3.2 that in this case, we obtained a non-classical
solution containing a non-classical shock corresponding to the
kinetic function $\psi$: one nonclassical shock from $u_l$ to
$\psi(u_l)$ followed by a rarefaction wave from $\psi(u_l)$ to
$\phi^{-\natural}(u_r)$, attached by a classical shock from $\phi^{-\natural}(u_r)$ to
$u_r$. This illustrates the co-existence of two nonclassical
solutions, one contains more nonclassical shocks than the other,
see Figure 4.1.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4}
\end{center}
%\centerline{\psfig{file=double_nshock.eps,width=8truecm}}
\caption{Two possible solutions: one
contains one and the other contains two nonclassical shocks}
\label{two_solution1} %fig4.1
\end{figure}
Secondly, assume now
$$
u_l\in [\theta, 0],\quad\text{and}\quad u_r\in (\psi^\sharp(u_l,\psi(u_l)),1).
$$
According to the description in Section 3, we could have two
nonclassical Riemann solutions, each of them contain one nonclassical shock.
Precisely, the solution would be
\begin{itemize}
\item either a classical shock from $u_l$ to $\phi^{-1}(u_r)$,
followed by a nonclassical shock corresponding to the kinetic
function $\phi$ from $\phi^{-1}(u_r)$ to $u_r$;
\item or a nonclassical shock corresponding to the kinetic function $\psi$
from $u_l$ to $\psi(u_l)$
followed by a classical shock from $\psi(u_l)$ to $u_r$.
\end{itemize}
This is an example of the co-existence of nonclassical solutions
including the same number of nonclassical shocks, see Figure
\ref{two_solution2}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5}
\end{center}
%\centerline{\psfig{file=single_nshock.eps,width=8truecm}}
\caption{Two distinct solutions: each contains a nonclassical shock}
\label{two_solution2} % fig4.2
\end{figure}
For short, in the sequel we will name a $0-shock$ (or a $1-shock$)
is a nonclasscial shock corresponding to the kinetic function
$\phi$ (the kinetic function $\psi$, resp.).
In order to select a unique solution, at least we must avoid the
above circumstances. In the following, we need a more restrictive
procedure than (C). That is the procedure
\begin{itemize}
\item[(P)]
-- A classical solution is understood to contain zero nonclassical shock.
-- Nonclassical shocks are preferred whenever available in the
extended sense that: If a solver $R_1(u_l,u_r)$ contains $m$
nonclassical shocks, and a solver $R_2(u_l,u_r)$ contains $n$
nonclassical shocks with $m>n$, then $R_1$ excludes $R_2$.
-- If the left-hand state belongs to the phase $E_I$, then the
$0$-shocks are preferred than the $1$-shocks in the sense that:
if $R_1(u_l,u_r)$ and $R_2(u_l,u_r)$ contain the same total number
of nonclassical shocks, and $R_1(u_l,u_r)$ contain $m$ $0$-shocks
and $R_2(u_l,u_r)$ contain $n$ $0$-shocks with $m>n$, then $R_1$
excludes $R_2$. Similarly, if the left-hand state belongs to the
phase $E_{III}$, then $1$-shocks are preferred than the
$0$-shocks.
\end{itemize}
For the construction, we first make it clear that a $1-shock$ can
not follow a $0-shock$.
\begin{proposition} \label{prop4.1}
In any Riemann solution, a $1-shock$ can
not follow a $0-shock$.
\end{proposition}
\begin{proof}
Let the states $u_0, u_1, u_2$ be given. Denote
$N_0(u_0,u_1)$ is the $0$-shock from $u_0$ to $u_1$ and
$N_1(u_1,u_2)$ is the $1$-shock from $u_1$ to $u_2$. That is to
say
$$
u_1=\phi(u_0),\quad\text{and}\quad u_2=\psi(u_1).
$$
In order to for $N_1$ to follow $N_0$ we must have the condition on
shock speeds:
\begin{equation}
s(u_1,u_2)>s(u_0,u_1).
\label{4.1}
\end{equation}
By the definition of kinetic functions, the shock speed
$s(u_1,u_2)$ has to be smaller than the slope of the tangent at
$u_1$, which is greater than the shock speed $s(u_0,u_1)$. This
contradicts with the condition \eqref{4.1}. The proposition is proved.
\end{proof}
Based on the procedure (P), we proceed now to construct the
Riemann solution. First, assume that $u_l\in
(-\infty,\min\{\theta, {\phi}^\natural(\phi(p))]$. Since $u_l$ is out of the
domain of the kinetic function $\psi$ and, as described in the
subsection 3.1, any $u_r\in (0,+\infty)$ can be arrived at by a
solution contain one $0$-shock. By virtue of the procedure (P), we
thus use the construction in the subsection 3.1 for this interval.
Second, let $u_l\in (\min\{\theta, {\phi}^\natural(\phi(p)), 0)$. The
construction of the subsection 3.1 is valid for $u_r<\phi^{-\natural}(u_l)$.
If $u_r\in [\phi^{-\natural}(u_l),\phi(p)]$, the solution can be a classical
shock connecting $u_l$ to $u^*$ followed by a nonclassical shock
from $u^*$ to $u_r$ provided \eqref{3.8} holds. If \eqref{3.8} fails, then the
construction in the subsection 3.2 can be applied here: if $u_l\le
\theta$ we have a rarefaction wave from $u_l$ to $\theta$ followed
by a $1$-shock from $\theta$ to $\psi(\theta)$, then followed by a
classical shock from $\psi(\theta)$ to $u_r$, if $u_l>\theta$,
then we have a $1$-shock from $u_l$ to $\psi(u_l)$ followed by a
classical one from $\psi(u_l)$ to $u_r$. For $u_r\in
(\phi(p),+\infty)$, if \eqref{3.9} holds then we use the construction in
the subsection 3.1 to cover $0$-shocks, else we use the one in the
subsection 3.2 to cover $1$-shocks or classical construction.
Third, let $u_l\in [0,1]$. We know from Proposition \ref{prop4.1} that
$1$-shocks can not follow $0$-shocks, so we need only find the
possibility of a $0$-shock following a $1$-shock. The interval
$[0,1]$ can be separated by two regions
\begin{equation}
\begin{gathered}
\mathcal{A} := \{u\in [0,1]: \psi^\sharp(u,\psi(u))< 1\}\text{ relatively open in }
[0,1],\\
\mathcal{A}^C = [0,1]\setminus \mathcal{A}.
\end{gathered}
\label{4.2}
\end{equation}
The relatively open set $\mathcal{A}$ is thus a union of certain
relatively open subintervals of the interval $[0,1]$. For any
$u\in \mathcal{A}$, there corresponds a set defined by
\begin{equation}
\mathcal{B }:= \{v \in (a,0): v <
\phi(\psi^\sharp(u,\psi(u))),\quad\text{and}\quad v >
\phi^\sharp(\psi(u),\phi^{-1}(v))\}.
\label{4.3}
\end{equation}
The set $\mathcal{B}$ is an open subset of $\mathbb{R}$. By definition, given
any left-hand state $u_r\in \mathcal{B}$, the Riemann solution for the
initial datum $(u_l,u_r)$ is a three-jump wave: first a $1$-shock
from $u_l$ to $\psi(u_l)$, followed by a classical jump from
$\psi(u_l)$ to $\phi^{-1}(u_r)$, then followed by a $0$-shock from
$\phi^{-1}(u_r)$ to $u_r$. For $u_r\in (-\infty,u_l]\setminus \mathcal{B}$, no $1$-shocks to be followed by a $0$-shock, so we use the
construction in the subsection 3.1. The states $u_r\in
(u_l,+\infty)$ can be reached by the construction in the
subsection 3.2, as no $0$-shocks are available.
Fourth, assume $u_l\in (1,\psi^{-1}(0)]$. By the monotony, we have
\begin{equation}
\psi(u_l)> 0. \label{4.4}
\end{equation}
Due to \eqref{4.4} the right-hand states $u_r\in [0,+\infty)$ should
be involved with $1$-shocks and the construction is the one of the
subsection 3.2. If $u_r\in (\phi(\psi(u_l)),0)$, then the solution
is a $1$-shock from $u_l$ to $\psi(u_l)$ followed by a rarefaction
wave from $\psi(u_l)$ to $\phi^{-1}(u_r)$, then followed by a
$0$-shock from $\phi^{-1}(u_r)$ to $u_r$. If $u_r\in
(\phi^\sharp(u_l,\psi(u_l)),\newline \phi(\psi(u_l))]$, then
$\phi^{-1}(u_r)\in (\psi(u_l),1)$. The solution is a $1$-shock
from $u_l$ to $\psi(u_l)$ followed by a classical shock from
$\psi(u_l)$ to $\phi^{-1}(u_r)$, then followed by a $0$-shock from
$\phi^{-1}(u_r)$ to $u_r$ iff
\begin{equation}
s(\psi(u_l),\phi^{-1}(u_r)) < s(\phi^{-1}(u_r),u_r). \label{4.5}
\end{equation}
If \eqref{4.5} fails, then no $0$-shocks are involved in the
construction and we use the one in the subsection 3.2. If now
$u_r\in (-\infty, \phi^\sharp(u_l,\psi(u_l))]$, then the classical
construction is invoked.
Fifth, let $u_l\in (\psi^{-1}(0),q]$, then
\begin{equation}
\psi(u_l)< 0.
\label{4.6}
\end{equation}
The right-hand states $u_r\in [\psi^\sharp(u_l,\psi(u_l)),+\infty)\cup
(-\infty,0]$ can be arrived at as in the construction of the
subsection 3.2. If $u_r\in (0,\phi(\psi(u_l)))$, then the solution
is a $1$-shock from $u_l$ to $\psi(u_l)$ followed by a rarefaction
wave from $\psi(u_l)$ to $\phi^{-1}(u_r)$ by virtue of (4.6), then
followed by a $0$-shock from $\phi^{-1}(u_r)$ to $u_r$. If $u_r\in
[\phi(\psi(u_l)),\psi^\sharp(u_l,\psi(u_l)))$, then $\phi^{-1}(u_r)\in
(p,\psi(u_l))$. The solution is a $1$-shock from $u_l$ to
$\psi(u_l)$ followed by a classical shock from $\psi(u_l)$ to
$\phi^{-1}(u_r)$, then followed by a $0$-shock from
$\phi^{-1}(u_r)$ to $u_r$ if and only if
\begin{equation}
s(\psi(u_l),\phi^{-1}(u_r)) < s(\phi^{-1}(u_r),u_r). \label{4.7}
\end{equation}
If \eqref{4.7} fails, then we use the one in the subsection 3.2.
Finally, if $u_l\in (q,+\infty)$, then the classical construction
is valid.
Summarizing the above arguments, we arrive at the following
theorem.
\begin{theorem} \label{thm4.2}
Given the initial Riemann data $(u_l,u_r)$. Under the hypotheses
\eqref{1.2}, There exists a unique Riemann solution made of
rarefaction waves, classical shocks and non-classical shocks
satisfying the kinetic relations \eqref{3.6} and \eqref{3.15}, and
the selective procedure (P).
\end{theorem}
\subsection*{Acknowledgements}
The authors would like to thank Professor P. G. LeFloch for his
helpful discussions.
\begin{thebibliography}{10}
\bibitem{AbeyaratneKnowles88}
R.~Abeyaratne and J. K. Knowles,
\newblock On the dissipative response due to discontinuous strains in bars of
unstable elastic materials.
\newblock {\em Int. J. Solids Structures}, 24:1021--1044, 1988.
\bibitem{AbeyaratneKnowles91}
R.~Abeyaratne and J. K. Knowles,
\newblock Kinetic relations and the propagation of phase boundaries in solids.
\newblock {\em Arch. Rational Mech. Anal.}, 114:119--154, 1991.
\bibitem{HayesLeFloch97}
B. T. Hayes and P. G. LeFloch,
\newblock {Non-classical shocks and kinetic relations: Scalar conservation
laws.}
\newblock {\em Arch. Ration. Mech. Anal.}, 139(1):1--56, 1997.
\bibitem{HayesLeFloch00}
B. T. Hayes and P. G. LeFloch,
\newblock Nonclassical shocks and kinetic relations: strictly hyperbolic
systems.
\newblock {\em SIAM J.~Math.~Anal.}, 31:941--991, 2000.
\bibitem{James80}
R. D. James,
\newblock The propagation of phase boundaries in elastic bars.
\newblock {\em Arch. Rational Mech. Anal.}, 73:125--158, 1980.
\bibitem{Lax71}
P. D. Lax,
\newblock {Shock waves and entropy, in: E.H. Zarantonello, Ed.,}.
\newblock {\em Contributions to Nonlinear Functional Analysis}, pages 603--634,
1971.
\bibitem{LeFloch93}
P. G. LeFloch,
\newblock {Propagating phase boundaries: Formulation of the problem and
existence via the Glimm method.}
\newblock {\em Arch. Rational Mech. Anal.}, 123:153--197, 1993.
\bibitem{LeFlochbook}
P. G. LeFloch,
\newblock {\em {Hyperbolic systems of conservation laws. The theory of
classical and nonclassical shock waves}}.
\newblock {Birkh\"auser}, 2002.
\bibitem{LeFlochThanh00}
P. G. LeFloch and M. D. Thanh,
\newblock {Nonclassical Riemann solvers and kinetic relations. III. A nonconvex
hyperbolic model for van der Waals fluids}.
\newblock {\em Electron. J. Differential Equations}, (72):19 pp, 2000.
\bibitem{LeFlochThanh01}
P. G. LeFloch and M. D. Thanh,
\newblock {Nonclassical Riemann solvers and kinetic relations. I. An hyperbolic
model of elastodynamics}.
\newblock {\em Z.A.M.P.}, 52:597--619, 2001.
\bibitem{LeFlochThanh02}
P. G. LeFloch and M. D. Thanh,
\newblock {Nonclassical Riemann solvers and kinetic relations. II. An
hyperbolic-elliptic model of phase transition dynamics}.
\newblock {\em Proc. Roy. Soc. Edinburgh Sect.}, 132 A(1):181--219, 2002.
\bibitem{LeFlochThanh03}
P. G. LeFloch and M. D. Thanh,
\newblock {Properties of Rankine-Hugoniot curves for Van der Waals fluid
flows}.
\newblock {\em Japan J. Indus. and Appl. Math.}, 20(2):211--238, 2003.
\bibitem{Liu74}
T.-P. Liu,
\newblock The Riemann problem for general $2\times 2$ conservation laws.
\newblock {\em Trans. Amer. Math. Soc.}, 199:89--112, 1974.
\bibitem{Oleinik}
O. A. Oleinik,
\newblock Construction of a generalized solution of the cauchy problem for a
quasi-linear equation of first order by the introduction of vanishing
viscosity.
\newblock {\em Amer. Math. Soc. Transl. Ser. 2}, 33:277--283, 1963.
\bibitem{SchulzeShearer}
S.~Schulze and M.~Shearer,
\newblock Undercompressive shocks for a system of hyperbolic conservation laws
with cubic nonlinearity.
\newblock {\em J. Math. Anal. Appl.}, 229:344--362, 1999.
\bibitem{Shearer82}
M.~Shearer,
\newblock The riemann problem for a class of conservation laws of mixed type.
\newblock {\em J. Differential Equations}, 46:426--443, 1982.
\bibitem{ShearerYang95}
M.~Shearer and Y.~Yang,
\newblock The riemann problem for a system of mixed type with a cubic
nonlinearity.
\newblock {\em Proc. Royal Soc. Edinburgh}, 125A:675--699, 1995.
\bibitem{Slemrod83}
M.~Slemrod,
\newblock {Admissibility criteria for propagating phase boundaries in a van der
Waals fluid.}
\newblock {\em Arch. Rational Mech. Anal.}, 81:301--315, 1983.
\bibitem{Truskinovsky93}
L.~Truskinovsky,
\newblock {\em {Kinks versus shocks, in ``Shock induced transitions and phase
structures in general media''}}.
\bibitem{Truskinovsky87}
L.~Truskinovsky,
\newblock {Dynamics of non-equilibrium phase boundaries in a heat conducting
nonlinear elastic medium.}
\newblock {\em J. Appl. Math. and Mech. (PMM)}, 51:777--784, 1987.
\end{thebibliography}
\end{document}
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