\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 116, pp. 1--8.\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-2006/116\hfil A Riemann problem with viscosity and dispersion]
{A Riemann problem with small  viscosity and dispersion}

\author[K. T. Joseph\hfil EJDE-2006/116\hfilneg]
{Kayyunnapara Thomas Joseph}

\address{Kayyunnapara Thomas Joseph \newline
School of Mathematics \\
Tata Institute of Fundamental Research \\
Homi Bhabha Road \\
Mumbai 400005, India}
\email{ktj@math.tifr.res.in}

\date{}
\thanks{Submitted July 11, 2006. Published September 26, 2006.}
\subjclass[2000]{35B40, 35L65}
\keywords{Elastodynamics equation; viscosity; dispersion; Riemann problem}

\begin{abstract}
 In this paper we prove existence of  global solutions to
 a hyperbolic system in elastodynamics, with small viscosity
 and dispersion terms and derive estimates uniform in the
 viscosity-dispersion parameters. By passing to the limit,
 we prove the existence of solution the Riemann problem
 for the hyperbolic system with arbitrary Riemann data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

In this paper first we consider the boundary-value problem, for a
system of nonlinear ordinary differential equations,
\begin{equation}
\begin{gathered}
 -\xi \frac {du}{d\xi} + u \frac {du}{d\xi} -\frac {d\sigma}{d\xi} =
\epsilon \frac {d^2u}{d\xi^2} + \gamma \epsilon^2 \frac{d^3u}{d\xi^3},  \\
-\xi  \frac {d\sigma}{d \xi} +u \frac {d\sigma}{d \xi} - k^2
\frac {du}{d \xi} = \epsilon \frac {d^2\sigma}{d \xi^2}
+ \gamma \epsilon^2  \frac{d^3\sigma}{d\xi^3}
\end{gathered}\label{e1.1}
\end{equation}
for $\xi \in [a,b]$ with boundary
conditions
\begin{equation}
\begin{gathered}
u(a) = u_L, u(b)=u_R, \\
\sigma(a) = \sigma_L,
\sigma(b)=\sigma_R.
\end{gathered} \label{e1.2}
\end{equation}
This system is the self-similar vanishing
diffusion-dispersion approximation of initial value problem
for the system of
 equations which comes in elastodynamics:
\begin{equation}
\begin{gathered}
u_t + u u_x - \sigma_x = 0,\\
\sigma_t + u \sigma_x - k^2  u_x = 0,
\end{gathered} \label{e1.3}
\end{equation}
with Riemann initial data
\begin{equation}
(u(x,0),\sigma(x,0))=\begin{cases}
(u_L,\sigma_L), & x <0 \\
(u_R,\sigma_R) & x >0.
\end{cases} \label{e1.4}
\end{equation}
Here $u$ is the velocity, $\sigma$ is the stress and $k>0$ is the speed
of propagation of the elastic waves. The system \eqref{e1.3} is
nonconservative, strictly
hyperbolic system with characteristic speeds
\begin{equation}
\lambda_1(u,\sigma) = u - k,   \lambda_2(u,\sigma) = u + k
\label{e1.5}
\end{equation}
and Riemann invariants
\begin{equation}
r_1(u,\sigma)=\sigma + k u, r_2(u,\sigma)= \sigma - k u \label{e1.6}
\end{equation}
and was studied by
many authors \cite{c1,j1,j2,j3,j5}, with initial datas under various
conditions using differece schemes or diffusion approximations. In this
paper we analyse diffusion-dispersion approximations for the Riemann
problem \eqref{e1.3} and \eqref{e1.4}.

 First we show the existence of
smooth solutions $(u^\epsilon,\sigma^\epsilon)$of the problem
\eqref{e1.1}-\eqref{e1.2} and derive
estimates in the space of bounded variation, uniformly
in $\epsilon>0$. We do not give any restrictions on the
size of the initial data.

Next we study $(u^\epsilon,\sigma^\epsilon)$ as
$\epsilon$ tends to $0$ and show the limit function is a weak solution
to \eqref{e1.3} with the Riemann initial data \eqref{e1.4}
The nonconservative product which appear
in the equation \eqref{e1.3} is justified
by the work of LeFloch and Tzavaras \cite{l2} on nonconservative
products.

\section{ Self-similar vanishing diffusion-dispersion approximation}

 In this section, we consider the system \eqref{e1.1} and \eqref{e1.2} and
prove the existence of smooth solutions. It is more convenient to work
with Riemann invariants \eqref{e1.5}.
Given the data $(u_L,\sigma_L), (u_R,\sigma_R)$, we define
\begin{equation}
\begin{gathered}
r_{1L} = \sigma_L + k u_L, r_{1R} = \sigma_R + k u_R,\\
 r_{2L} = \sigma_L -k u_L, r_{2R} = \sigma_R - k u_R.
\end{gathered}
\label{e2.1}
\end{equation}
The characteristic speeds \eqref{e1.5} in terms of the Riemann invariants
 take the form
\[
\lambda_1(r_1,r_2) = \frac{r_1-r_2}{2k}-k,\quad
\lambda_2(r_1,r_2) = \frac{r_1-r_2}{2k}+k.
\]
Consider the rectangle
\[
D = [\min(r_{1L},r_{1R}),\max(r_{1L},r_{1R})]\times
[\min(r_{2L},r_{2R}),\max(r_{2L},r_{2R})],
\]
and consider the minimum and maximum of the eigenvalues on this square
\begin{equation}
\lambda_j^m = \min_{D}\lambda_j(r_1,r_2), \quad
\lambda_j^M = \max_{D}\lambda_j(r_1,r_2),\quad j=1,2.
 \label{e2.2}
\end{equation}

We choose $\gamma>0$, small and the boundary points $a,b$ such that
\begin{equation}
0<\gamma<\frac{1}{4(\lambda_2^M - \lambda_1^m)}, \quad
 \lambda_1^m -\frac{1}{\gamma}<a<\lambda_1^m<\lambda_2^M<b
\label{e2.3}
\end{equation}

The choice of $a,b$, in this fashion is to ensure that there is
no boundary effect in the limit, that is, for $\xi < \lambda_1^m$ and
$\xi>\lambda_2^M$ the limiting values of $(u,\sigma)$ are $(u_L,\sigma_L)$
$(u_R,\sigma_R)$ respectively.

\begin{theorem} \label{thm2.1}
 Under the assumptions \eqref{e2.3}, for each fixed $\epsilon>0$
there exits a smooth solution $(u^\epsilon(\xi),\sigma^\epsilon(\xi))$
for \eqref{e1.1} and \eqref{e1.2} satisfying the estimates
\begin{gather}
|u^\epsilon(\xi)| + |\sigma^\epsilon(\xi)| \leq C, \quad
\int_{a}^{b}| \frac {du^\epsilon}{d \xi}|d \xi +
\int_{a}^{b}| \frac {d\sigma^\epsilon}{d \xi}(\xi)|d \xi \leq C,
\label{e2.4}
\\
|u^\epsilon(\xi) - u_L| + |\sigma^\epsilon(\xi) -\sigma_L| \leq
\frac{C}{\delta} e^\frac{-(\xi -\lambda_1^m)^2}{2\epsilon} , \quad
a \leq \xi \leq\lambda_1^m-\delta \label{e2.5}
\\
|u^\epsilon(\xi) - u_R| + |\sigma^\epsilon(\xi) -\sigma_R| \leq
\frac{C}{\delta} e^\frac{-(\xi -\lambda_2^M)^2}{2\epsilon} , \quad
\lambda_2^M+\delta \leq \xi \leq b, \label{e2.6}
\end{gather}
for some constant $C>0$ independent of $\epsilon>0$ and for
$\delta>0$, small.
\end{theorem}

\begin{proof}
For convenience of notation, in the rest of this section we drop
the dependence of $\epsilon$ and write $u,\sigma,r_1,r_2$ ect.
In terms of the Riemann
invariants \eqref{e1.5}, the problem \eqref{e1.1} and \eqref{e1.2}
takes the form
\begin{equation}
\begin{gathered}
-\xi \frac {dr_1}{d \xi} + \lambda_1(r_1,r_2) \frac {dr}{d \xi}
 = \epsilon^2 \frac {d^2 r_1}{d \xi^2} +
\gamma \epsilon \frac {d^3 r_1}{d\xi^3},\\
 -\xi \frac {dr_2}{d \xi} + \lambda_2(r_1,r_2) \frac {dr_2}{d \xi} =
\epsilon \frac {d^2 r_2}{d \xi^2}+
\gamma \epsilon^2 \frac {d^3 r_2}{d \xi^3}
\end{gathered}
\label{e2.7}
\end{equation}
on $[a,b]$ with boundary conditions
\begin{equation}
r_1(a) = r_{1L} ,\quad  r_1(b) = r_{1R} ,\quad r_2(a) = r_{2L},\quad
r_2(b) = r_{2R} \label{e2.8}
\end{equation}
where $r_{1L}$, $r_{1R}$, $r_{2L}$ and $r_{2R}$ are given by \eqref{e2.1}.

 From \eqref{e1.6}, the definition of $r_1(u,\sigma), r_2(u,\sigma)$, we
have
$$
u=\frac{r_1(u,\sigma) - r_2(u,\sigma)}{2k},\quad
\sigma = \frac{r_1(u,\sigma) + r_2(u,\sigma)}{2}.
$$
Then to prove the theorem, it is sufficient to prove the the existence of
$r_1,r_2$
solution of \eqref{e2.7} and \eqref{e2.8}, with following  estimates
\begin{equation}
\begin{gathered}
r_1(\xi) \in [\min(r_{1L},
r_{1R}),\; \max(r_{1L},r_{1R})], \quad  \xi \in [a,b],\\
r_2(\xi) \in [\min(r_{2L},
r_{2R}), \;\max(r_{2L},r_{2R})], \quad \xi \in [a,b];
\end{gathered}
\label{e2.9}
\end{equation}
\begin{equation}
\begin{gathered}
|r_1(\xi) - r_{1L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_1^m)^2}{2\epsilon}\big) ,\quad a\leq \xi \leq \lambda_1^m - \delta,\\
|r_2(\xi) - r_{2L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_2^m)^2}{2\epsilon}\big) ,\quad a \leq \xi \leq \lambda_2^m - \delta;
\end{gathered}
\label{e2.10}
\end{equation}
\begin{equation}
\begin{gathered}
|r_1(\xi) - r_{1R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_1^M)^2}{2\epsilon}\big) , \quad \lambda_1^M + \delta \leq \xi \leq
b,\\
|r_2(\xi) - r_{2R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_2^M)^2}{2\epsilon} \big), \quad \lambda_2^M + \delta \leq \xi \leq b;
\end{gathered}
\label{e2.11}
\end{equation}
\begin{equation}
\int_{a}^{b}| \frac {dr_1}{d \xi}|d \xi \leq
|r_{1R}-r_{1L}|,\quad \int_{a}^{b}| \frac {dr_2}{d \xi}|d
\xi \leq |r_{2R}-r_{2L}|. \label{e2.12}
\end{equation}
To prove this  we reduce \eqref{e2.7} and \eqref{e2.8} to an
integral equation using some ideas of LeFloch and Rohde  \cite{l1} and
Joseph and LeFloch \cite{j4} and use a fixed point argument.
First note that \eqref{e2.7} can be
written in the form
\begin{equation}
\begin{gathered}
\gamma \epsilon^2 \frac{d^3r_1}{d\xi^3} + \epsilon\frac{d^2r_1}{dx^2}
-(\lambda_1(r_1,r_2) - \xi) \frac{dr_1}{d\xi} =0,\\
\gamma \epsilon^2 \frac{d^2r_2}{d\xi^2} + \epsilon\frac{d^2r_2}{dx^2}
-(\lambda_2(r_1,r_2) - \xi)\frac{dr_2}{d\xi} = 0.
\end{gathered}
\label{e2.13}
\end{equation}
 For $j=1,2$, let
\begin{equation}
\varphi_1(\xi) = \frac{dr_1}{d\xi},\quad
\varphi_2(x)=\frac{dr_2}{d\xi}
\label{e2.14}
\end{equation}
Then from \eqref{e2.13} we get
\begin{equation}
\gamma \epsilon^2 \varphi_i''+ \epsilon \varphi_i' -
(\lambda_i(r_1(\xi),r_2(\xi))-\xi) \,
\varphi_i = 0.
\label{e2.15}
\end{equation}
Suppose we are given $r_1,r_2$ smooth functions, taking
values in the rectangle $D$ and of finite total variation
independent of $\epsilon$,
\eqref{e2.15} is a second order linear
ordinary differential equation for $\varphi_i$. Under the
transformation,
\[
H_i=e^{\frac{-1}{2\epsilon\gamma}\xi} \varphi_i
\]
\eqref{e2.15} reduces to
\begin{equation}
H_i''= \frac{\mu_i(y)}{\gamma\epsilon^2}H
\label{e2.16}
\end{equation}
where
\begin{equation}
\mu_i(\xi)=\frac{1}{4\gamma} +(\lambda_i(r_1(\xi),r_2(\xi))-\xi).
\label{e2.17}
\end{equation}
By taking $\gamma>0$ small, we have $\mu_i(y)>0$ and we can use
the theorem of Olver \cite{o1} to construct solutions to \eqref{e2.15}.
Indeed LeFloch and Rohde \cite{l1} showed the existence of
$\varphi_i(\xi), i=1,2$ satisfying the following properties:
\begin{equation}
0<\varphi_i(\xi)\leq C/\epsilon, \int_a^b \varphi_i(\xi) d\xi =1
\label{e2.18}
\end{equation}
and
\begin{equation}
\varphi_i(\xi)\leq \begin{cases}
\frac{C}{\epsilon} \exp\big(\frac{-c(x-\lambda_1^m)^2}{\epsilon}\big), & a \leq \xi
\leq \lambda_1^m,
\\
\frac{C}{\epsilon} \exp\big(\frac{-c(x-\lambda_1^m)^2}{\epsilon}\big), &
\lambda_2^M \leq \xi \leq b,
\end{cases}
\label{e2.19}
\end{equation}
Integrating once \eqref{e2.14} and using \eqref{e2.18} and the boundary
conditions, we get,
\begin{equation}
\begin{gathered}
r_1^\epsilon(\xi) = r_{1L} + (r_{1R} -
r_{1L})\int_{a}^{\xi} \varphi_1(y) dy,\\
r_2^\epsilon(\xi) = r_{2L} + (r_{2R} -
r_{2L})\int_{a}^{\xi} \varphi_2(y) dy.
\end{gathered}
\label{e2.20}
\end{equation}
It follows that to solve \eqref{e2.7} and \eqref{e2.8} with estimates
 \eqref{e2.4}--\eqref{e2.6},
it is enough to solve \eqref{e2.20}. To solve \eqref{e2.20}, we use the
Schauder fixed point theorem applied to the function
\[
F(r_1,r_2)(\xi) = (F_1(r_1,r_2)(\xi),F_2(r_1,r_2)(\xi))
\]
where
\begin{equation}
\begin{gathered}
F_1(r_1,r_2)(\xi)  = r_{1L} + (r_{1R} -
r_{1L})\int_{a}^{\xi} \varphi_1(y) dy,\\
F_2(r_1,r_2)(\xi) = r_{2L} + (r_{2R} -
r_{2L})\int_{a}^{\xi} \varphi_2(y) dy.
\end{gathered}
\label{e2.21}
\end{equation}
From \eqref{e2.18} and \eqref{e2.21}, it is clear that
$F_1(r,s)$ is a convex combination of $r_{1L}$ and $r_{1R}$ and
$F_2(r,s)$ is a convex combination of $r_{2L}$ and $r_{2R}$. So the
estimate
\begin{equation}
\begin{gathered}
F_1(r_1,r_2)(\xi) \in  [\min(r_{1L},r_{1R}),\max(r_{1L},r_{1R})],\\
F_2(r_1,r_2)(\xi) \in  [\min(r_{2L},r_{2R}),\max(r_{2L},r_{2R})]
\end{gathered}
\label{e2.22}
\end{equation}
easily follows. Also from \eqref{e2.18} and \eqref{e2.21}, we get for
$j=1,2$
\begin{equation}
|\frac {d F_j(r,s)}{d \xi}(\xi)| \leq \frac{C}{\epsilon}.
\label{e2.23}
\end{equation}
Further, from \eqref{e2.19}, we get:
\[
|F_1(r,s)(\xi) - r_{1L}| \leq \frac{C}{\epsilon}\int_{a}^\xi
\exp\big(\frac{-(s -\lambda_1^m)^2}{2\epsilon}\big) ds
= \frac{C \sqrt{2 \epsilon}}{\epsilon}
\int_\frac{a-\lambda_1^m}{\sqrt{2\epsilon}}^\frac{(\xi-\lambda_1^m)}
{\sqrt{2 \epsilon}}e^{-s^2 } ds,
\]
for $\quad a \leq \xi \leq \lambda_1^m$;
\[
|F_2(r,s)(\xi) - r_{2L}| \leq \frac{C}{\epsilon}\int_{0}^\xi
\exp\big(\frac{-(s -\lambda_2^m)^2}{2\epsilon}\big) ds
= \frac{C \sqrt{2 \epsilon}}
{\epsilon}\int_\frac{a -\lambda_2^m}{\sqrt{2 \epsilon}}^{
\frac{(\xi-\lambda_2^m)}{\sqrt{2 \epsilon}}}
e^{-s^2} ds,
\]
for $a \leq \xi \leq \lambda_2^m$;
\[
|F_1(r,s)(\xi) - r_{1R}| \leq \frac{C}{\epsilon}\int_\xi^{b}
\exp\big(\frac{-(s -\lambda_k^M)^2}{2\epsilon}\big) ds
= \frac{C \sqrt{2 \epsilon}}
{\epsilon}\int_{\frac{(\xi-\lambda_1^M)}{\sqrt{2 \epsilon}}}^{\frac
{b-\lambda_1^M}{\sqrt{2 \epsilon}}}
e^{-s^2} ds,
\]
for $\lambda_1^M \leq \xi \leq b$;
\[
|F_2(r_1,r_2)(\xi) - r_{2R}| \leq \frac{C}{\epsilon}\int_\xi^{b}
\exp\big(\frac{-(s -\lambda_k^M)^2}{2\epsilon}\big) ds
= \frac{C \sqrt{2 \epsilon}}
{\epsilon}\int_{\frac{(\xi-\lambda_2^M)}{\sqrt{2 \epsilon}}}^{\frac
{b-\lambda_2^M}{\sqrt{2 \epsilon}}}
e^{-s^2} ds,
\]
for $\lambda_2^M \leq \xi \leq b$.

Now using the asymptotic expansion
\[
\int_y^\infty e^{-y^2} dy = (\frac{1}{2y} -O(\frac{1}{y^2}))e^{-y^2}, \quad
y \to \infty
\]
in the above inequalities, we get
\begin{equation}
\begin{gathered}
|F_1(r_1,r_2)(\xi) - r_{1L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_1^m)^2}{2\epsilon}\big) ,\quad a\leq \xi \leq \lambda_1^m - \delta,\\
|F_2(r_1,r_2)(\xi) - r_{2L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_2^m)^2}{2\epsilon}\big) , \quad a\leq \xi \leq \lambda_2^m - \delta;
\end{gathered}
\label{e2.24}
\end{equation}
\begin{equation}
\begin{gathered}
|F_1(r_1,r_2)(\xi) - r_{1R}| \leq \frac{C}{\delta}\exp\big(\frac{-(\xi
-\lambda_1^M)^2}{2\epsilon}\big) , \quad \lambda_1^M + \delta \leq \xi \leq
b,\\
|F_2(r_1,r_2)(\xi) - r_{2R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi
-\lambda_2^M)^2}{2\epsilon}\big) ,\quad  \lambda_2^M + \delta \leq \xi \leq b.
\end{gathered}
\label{e2.25}
\end{equation}

The estimates \eqref{e2.22} and \eqref{e2.23}
show that $F=(F_1,F_2)$ is  compact and maps the convex set
$\{(r_1,r_2)\in C[a,b]\times C[a,b] : (r_1(\xi),r_2(\xi))\in D\}$
into itself, where D is the rectangle
$D=[\min(r_{B},r_{R}),\max(r_{B},r_{R})]\times [\min(s_{B},s_{R}),
\max(s_{B},s_{R})]$,
and $C[a,b]$ is the space of continuous functions with uniform norm.
So by Schauder fixed point theorem there exists $(r_1,r_2)$ such that
$F(r_1,r_2)=(r_1,r_2)$, satisfies \eqref{e2.20} and hence is a smooth
solution
to \eqref{e2.7} with boundary conditions \eqref{e2.8}. Further the
estimates \eqref{e2.9}-\eqref{e2.12} follows from
\eqref{e2.20}, \eqref{e2.22}, \eqref{e2.24}, \eqref{e2.25} and the
fact that $F(r_1,r_2)=(r_1,r_2)$.
The proof of the theorem is complete.
\end{proof}


\section {Passage to the limit as $\epsilon \to 0$; the Riemann
problem.}

Here we construct solution of the Riemann problem
\begin{equation}
\begin{gathered}
u_t + u u_x - \sigma_x = 0,\\
\sigma_t + u \sigma_x - k^2  u_x = 0,
\end{gathered}
\label{e3.1}
\end{equation}
with Riemann type initial data
\begin{equation}
(u(x,0),\sigma(x,0))=\begin{cases}
(u_L,\sigma_L) & x <0
\\
(u_R,\sigma_R) & x >0.
\end{cases}
\label{e3.2}
\end{equation}

Since the Riemann problem is invariant under scaling, solution
is sought in the form $(u(\xi,\sigma(\xi))$ with $\xi=x/t$.
Then \eqref{e3.1} and \eqref{e3.2} takes the form
\begin{equation}
\begin{gathered}
 -\xi \frac {du}{d\xi} + u \frac {du}{d\xi} -\frac {d\sigma}{d\xi} = 0
\\
-\xi  \frac {d\sigma}{d \xi} +u \frac {d\sigma}{d \xi} - k^2
\frac {du}{d \xi} = 0
\end{gathered}
\label{e3.3}
\end{equation}
for $\xi \in (-\infty,\infty)$ with boundary
conditions
\begin{equation}
\begin{gathered}
u(-\infty) = u_L, \quad u(\infty)=u_R, \\
\sigma(-\infty) = \sigma_L, \quad \sigma(\infty)=\sigma_R.
\end{gathered}
\label{e3.4}
\end{equation}

The smooth solution $(u^\epsilon,\sigma^\epsilon)$
of \eqref{e1.1} and \eqref{e1.2} constructed in
the previous section is an approximation to the problem
\eqref{e3.3} and \eqref{e3.4}. Because of the estimates \eqref{e2.4},
by compactnes, there exists a subsequence
$(u^{\epsilon_n},\sigma^{\epsilon_n})$
converges to a BV function $(u,\sigma)$ as $\epsilon_n \to 0$.
This limit function is not in general continuous and hence the
nonconservative product which appear in the equation \eqref{e3.3} does not
make sense in the theory of distribution. So we use the theory developed
by LeFloch and Tzavaras \cite{l2} for nonconservative products.
For completeness we briefly
describe in short their results that we use.

Let $u_n : [a,b] \to R^n$ be a sequence of continuous functions
uniformly bounded total variation:
\begin{equation}
\sup|u_n| +TV(u_n) \leq C.
\label{e3.5}
\end{equation}
where $TV(u)$ denotes the total variation of $u$ on $[a,b]$.
Define the Radon measure
\[
\langle \mu_n,g \rangle = \int_{[a,b]} g(u_n) du_n , g \in C(R^n)
\]
We have
\[
|\langle \mu_n,g \rangle | \leq TV(u_n).\sup_{|\lambda|\leq C}|g(\lambda)|.
\]
So by weak* compactness of $\mu_n$, there exists a subsequence $n_k$ and
a
measure $\mu \in M(R^n)$ such that
\[
\mu_{nk} \to \mu
\]
in weak* $M(R^n)$. To characterize $\mu$, we need the notion of
generalized graph of $u$.


\begin{definition} \label{def3.1} \rm
 Generalized graph of $u$ is defined as
a Lipschitz continuous map
\[
({X}, {U}):[0,1]\to [a,b]\times R^n
\]
such that
\begin{itemize}
\item[(a)] $({X}(0),{U}(0))=(a,u(a)),({X}(1),{U}(1))=(b,u(b))$
\item[(b)] ${X}$ is increasing :$s_1<s_2$ implies ${X}(s_1)\leq
{X}(s_2)$
\item[(c)] given $y\in [a,b]$, there exists $s \in [0,1]$ such that
${X}(s)=y$ and ${U}(s)=u(y)$.
\end{itemize}
\end{definition}

Generalized graph for a continuous BV function $u$ can be easily defined.
Let
\[
\sigma(x) = \frac{x-a-TV_{[a,x]}(u)}{b-a + TV_{[a,b]}(u)}
\]
$\sigma$ is strictly increasing , surjective and Lipschitz continuous. Let
${X}:[0,1]\to [a,b]$ is inverse of $\sigma$ and
${U}:[0,1]\to
R^n$ be defined as $uo{X}$, then
$({X},{U}):[0,1]\to [a,b]\times R^n$ is
the generalized graph of $u$. LeFloch and Tzavaras \cite{l2} proved
the following result.

\begin{theorem}[LeFloch and Tzavaras \cite{l2}] \label{Leflochthm}
If $u_n$ is a sequence of continuous functions satisfying a uniform bound
\eqref{e3.5}
Then there exists a subsequence $u_{nk}$ and associated
generalized graph $({X},{U})$ such that for any
continuous function $g$,
\[
\int_{[a,b]} \theta(x) g(u_{nk}(x) du_{nk}(x) \to \langle \mu_g,\theta \rangle
\]
where  $\mu:C_0(R^n) \to M[a,b]$ is defined by
\[
\langle \mu_g,\theta \rangle = \int_0^1(\theta({X}(s))g({U}(s))d{U}(s),
\quad \theta \in C[a,b]
\]
\end{theorem}

The product $g(u).u_x$ is defined as $\mu_g$ and is denoted by
$[g(u).u_x]_{({X},{U})}$.

LeFloch and Tzavaras \cite{l2} considered the Riemann
problem for a general nonconservative hyperbolic system
\begin{equation}
-\xi \frac{du}{d\xi} + A(u)\frac{du}{d\xi} = 0, \quad
u(-\infty)=u_L, \quad
u(\infty)=u_R.
\label{e3.6}
\end{equation}
They introduced the following definition.

\begin{definition} \label{def3.2}\rm
 A vector function $u(\xi)$ defined on $(-\infty, \infty)$
and of bounded variation  is a solution for the system \eqref{e3.6}
if there exists $({X}, {U})$, a generalised graph of $u$ such that as a
Borel measure
\[
[(-\xi + A(u))\frac{du}{d\xi}]_{({X},{U})} = 0, \quad u(-\infty)=u_L,
\quad u(\infty)=u_R
\]
\end{definition}

Under the condition that $|u_L-u_R|$ sufficiently small, the Riemann
problem \eqref{e3.6} was
solved in by LeFloch and Tzavaras \cite{l2} using vanishing diffusion
approximation. The Riemann
problem for the general systems with diffusion - dispersion
approximations were treated by LeFloch and Rohde \cite{l1} when
$|u_l-u_R|$
is small. Joseph \cite{j1} treated \eqref{e3.3} with only the diffusion
terms but
with large data. Present paper treats \eqref{e3.3} with diffusion and
dispersion terms and with arbitrary data \eqref{e3.4}. We shall prove the
following result.

\begin{theorem} \label{thm3.1}
There exits a function of bounded variation
$V=(u,\sigma)$ and
an associated generalized graph $({X},{U})$ solving the the
Riemann problem \eqref{e3.3} and \eqref{e3.4} in the sense of definition
\ref{def3.2}.
\end{theorem}

\begin{proof} As we have the estimate \eqref{e2.4}, we know
$\gamma \epsilon^2 u'''+\epsilon u''$ and
$\gamma \epsilon^2 \sigma'''+\epsilon \sigma''$
goes to zero in distribution as $\epsilon$ goes to zero. So by the
Theorem of LeFloch and Tzavaras \cite{l2} stated before, there exists a bounded
measurable
function $V=(u,\sigma)$ and associated generalized graph $(X,U)$
such that
\[
[-\xi + A(V) \frac{dV}{d\xi}]_{(X,U)} =0.
\]
where $A(V)= (A_{ij}(u,\sigma))$ is a $2\times2$ matrix with
$A_{11}(u,\sigma)=u, A_{12}(u,\sigma)=-1,A_{21}(u,\sigma)=-k^2,
A_{22}(u,\sigma)=u$.
We also have $(u(\xi),\sigma(\xi)=(u_L,\sigma_L)$ on $\xi \in
[a,\lambda_1^m]$ and so can be extended as $(u_L,\sigma_L)$ on
$(-\infty,a]$
and $(u(\xi),\sigma(\xi)=(u_R,\sigma_R)$ for $\xi \in
[\lambda_2^M,b]$ and so can be extended as $(u_R,\sigma_R)$ on
$[b,\infty)$. The proof of the theorem is
complete.
\end{proof}

\subsection*{Remarks} We note that the hyperbolic system \eqref{e3.1} and
\eqref{e3.2} has finite speed of propagation with minimum speed
$\lambda_1^m$ and maximum speed $\lambda_2^M$ defined by \eqref{e2.2}
which depends only on Riemann data $u_L,u_R,\sigma_L,\sigma_R$ and so
the solution $(u(\xi),\sigma(\xi))$ is $(u_L,\sigma_L)$ on
$(-\infty,\lambda_1^m]$ and $(u_R,\sigma_R)$ on
$[\lambda_2^M,\infty)$.
This is the meaning of the estimates \eqref{e2.5} and \eqref{e2.6}.

\subsection*{Acknowledgements}
This work is supported by a grant 2601-2 from the Indo-French
Centre for the promotion of advanced Research, IFCPAR
(Centre Franco-Indien pour la promotion de la Recherche Avancee, CEFIPRA),
New Delhi.

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\end{document}