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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 58, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/58\hfil Singular solutions]
{Singular solutions for 2x2 systems in nonconservative form with
incomplete \\ set of eigenvectors}

\author[A. P. Choudhury \hfil EJDE-2013/58\hfilneg]
{Anupam Pal Choudhury}  % in alphabetical order

\address{Anupam Pal Choudhury \newline
TIFR Centre for Applicable Mathematics\\
Sharada Nagar, Chikkabommasandra, GKVK P.O.\\
Bangalore 560065, India}
\email{anupam@math.tifrbng.res.in}

\thanks{Submitted December 7, 2012. Published February 26, 2013.}
\subjclass[2000]{35L65, 35L67}
\keywords{Hyperbolic systems of conservation laws; $\delta$-shock
wave type solution; \hfill\break\indent weak asymptotic method}

\begin{abstract}
 In this article, we study the initial-value problem for two first-order
 systems in non-conservative form. The first system arises in elastodynamics
 and belongs to the class of strictly hyperbolic, genuinely nonlinear systems.
 The second system has repeated eigenvalues and an incomplete set of right
 eigenvectors. Solutions to such systems are expected to develop singular
 concentrations. Existence of singular solutions to both the systems have
 been shown using the method of weak asymptotics. The second system has
 been shown to develop singular concentrations even from Riemann-type initial
 data. The first system differing from the second in having an extra term
 containing a positive constant $k$, the solution constructed for the
 first system have been shown to converge to the solution of the second as
 $k$ tends to $0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

The initial-value problem for the first-order quasilinear hyperbolic system
\begin{equation}
\begin{gathered}
 \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}
  -\frac{\partial \sigma}{\partial x}=0, \\
 \frac{\partial \sigma}{\partial t}+u\frac{\partial \sigma}{\partial x}
  -k^2\frac{\partial u}{\partial x}=0
 \end{gathered} \label{e1.1}
\end{equation}
 (in the domain $\Omega=\{(x,t):-\infty<x<\infty,t>0\}$) arising in 
applications in elastodynamics, has been well studied 
(see \cite{c1,j1,j2,l1}). Here $k$ is a positive constant. 
It is a strictly hyperbolic system having two real distinct eigenvalues 
given by
$$
\lambda_{1}(u,\sigma)=u-k,\ \lambda_{2}(u,\sigma)=u+k
$$
with the corresponding right eigenvectors
$$
E_{1}(u,\sigma)=\begin{pmatrix}
1 \\
k \end{pmatrix},\quad
E_{2}(u,\sigma)=\begin{pmatrix}
                       1 \\
                       -k \end{pmatrix}.
$$
Now letting $k \to 0$, we see that the eigenvalues 
$\lambda_{1}(u,\sigma)$ and $\lambda_{2}(u,\sigma)$ tend to coincide. 
In particular, taking $k=0$ in \eqref{e1.1} we arrive at the system
\begin{equation}
\begin{gathered}
 \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}
 -\frac{\partial \sigma}{\partial x}=0, \\
 \frac{\partial \sigma}{\partial t}+u\frac{\partial \sigma}{\partial x}=0.
\end{gathered}
\label{e1.2}
\end{equation}
which has repeated eigenvalues $\lambda_{1}(u,\sigma)=\lambda_{2}(u,\sigma)=u$ 
and an incomplete set of right eigenvectors (we can take
$\begin{pmatrix} 1\\ 0\end{pmatrix}$ to be a right eigenvector).

In \cite{z1}, a class of 2x2 systems in conservative form having an 
incomplete set of eigenvectors everywhere has been considered. 
These systems exhibit development of singular concentrations. 
We expect a similar kind of development of singular concentration for 
the system \eqref{e1.2}. But the analysis in \cite{z1} cannot 
be applied directly due to the following two reasons:

1. The system \eqref{e1.2} is in nonconservative form and hence we need 
to give a suitable meaning to the nonconservative products which in general
 lead to different solutions depending upon the meaning attached 
(see \cite{d1},\cite{r1},\cite{v1}).

2. One of the assumptions that has been used in \cite{z1} is that the 
eigenvalue should have vanishing directional derivative along a right 
eigenvector. But the eigenvalue $u$ in this case has a nonvanishing 
directional derivative along any right eigenvector. 
Thus the assumption is not satisfied.

Another reason to expect singular solutions for the system \eqref{e1.2} 
comes from studying the behaviour as $k\to 0$ of the
shock and rarefaction curves obtained for the Riemann problem for the 
system \eqref{e1.1}. In \cite{j2}, the Riemann problem for the 
system \eqref{e1.1} has been studied using Volpert's product. 
Starting with Riemann type initial data 
$$
(u(x,0),\sigma(x,0))=\begin{cases}
 (u_{L},\sigma_{L}),& x<0\\
 (u_{R},\sigma_{R}),& x>0.
\end{cases}
$$
the shock curves $S_{1}(u_{L},\sigma_{L}),S_{2}(u_{L},\sigma_{L})$ 
and the rarefaction curves $R_{1}(u_{L},\sigma_{L}),$\\
$R_{2}(u_{L},\sigma_{L})$ can be written down in the $u-\sigma$
 plane as in \cite{j2}:
\begin{equation}
 \begin{gathered}
  R_{1}(u_{L},\sigma_{L}):\sigma=\sigma_{L}+k(u-u_{L}),\; u>u_{L},\\
  R_{2}(u_{L},\sigma_{L}):\sigma=\sigma_{L}-k(u-u_{L}),\; u>u_{L},\\
  S_{1}(u_{L},\sigma_{L}):\sigma=\sigma_{L}+k(u-u_{L}),\; u<u_{L},\\
  S_{2}(u_{L},\sigma_{L}):\sigma=\sigma_{L}-k(u-u_{L}),\; u<u_{L}.
 \end{gathered}
\label{e1.3}
\end{equation}
Now as $k \to 0$, it seems from the above curves that we cannot have a 
jump in $\sigma$ and therefore the Riemann problem might not be solvable
using Volpert's product. That this is indeed true when $k=0$, that is, 
for the system \eqref{e1.2} follows from the following:

For the solution of the Riemann problem using Volpert's product the following 
relations have to be satisfied (see \cite{d1})
\begin{gather*}
-s[u]+[\frac{u^2}{2}]-[\sigma]=0,\\
-s[\sigma]+\frac{1}{2}(u_{L}+u_{R})[\sigma]=0
\end{gather*}
where $s$ denotes the speed of the discontinuity and $[w]$ denotes 
the jump in the function $w$ across the discontinuity. 
It follows from the above relations that $[\sigma]=0$ thus proving what
 we expected.
This observation again makes us suspect that there might be development 
of singular concentrations even if we start with Riemann type initial data.

In what follows our main aim is to construct \textit{generalised 
$\delta$-shock wave solutions} for the systems \eqref{e1.1} and \eqref{e1.2} 
and examine the role played by the constant $k$.
We use the \textit{method of weak asymptotics} 
(see \cite{a1,d2,k1}) as the main tool. Depending upon the cases, 
we shall sometimes use the complex-valued \textit{weak asymptotic solutions} 
(see \cite{k1}) for our construction.

The plan of this article is as follows: 
In Section 2, the notions of the weak asymptotic solutions,
generalised $\delta$-shock wave type solutions are recalled and a brief sketch
of the method of weak asymptotics is given. 
In Section 3, a few weak asymptotic expansions are proved which are 
crucially used in the construction of singular solutions.
In Section 4, existence of generalised $\delta$-shock wave type 
solutions for the systems \eqref{e1.1} and \eqref{e1.2}
is proved. The role played by $k$ is examined and the development 
of singular solutions from Riemann type initial data is exhibited.

\section{A brief discussion on the method of weak asymptotics}

In this section, we recall the definitions of weak asymptotic solutions 
and generalised $\delta$-shock wave type solutions and discuss
the method of weak asymptotics; see \cite{a1,d2,k1}.

Let $\mathcal{D}$ and $\mathcal{D}'$ denote the space of smooth
functions of compact support and the space of distributions respectively.
Let $O_{\mathcal{D}'}(\epsilon^{\alpha})$ denote the collection
of distributions $f(x,t,\epsilon) \in \mathcal{D}'(\mathbb{R})$ 
such that for any test function
$\varphi(x) \in \mathcal{D}(\mathbb{R})$ the estimate 
$$
\langle f(x,t,\epsilon),\varphi(x)\rangle=O(\epsilon^{\alpha})
$$
holds and is uniform with respect to $t$. The relation 
$o_{\mathcal{D}'}(\epsilon^{\alpha})$ is interpreted similarly.

\begin{definition}[\cite{k1}] \label{def2.1}\rm
 A pair of smooth complex-valued (real-valued) functions \\
$(u(x,t,\epsilon),\sigma(x,t,\epsilon))$ is called a 
\textit{weak asymptotic solution}
of the system \eqref{e1.1} with the initial data $(u(x,0),\sigma(x,0))$ if
\begin{equation}
 \begin{gathered}
  \frac{\partial u(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial u(x,t,\epsilon)}{\partial x}-\frac{\partial \sigma(x,t,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1), \\
  \frac{\partial \sigma(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial \sigma(x,t,\epsilon)}{\partial x}-k^2\frac{\partial u(x,t,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\\
  u(x,0,\epsilon)-u(x,0)=o_{\mathcal{D}'}(1),\\
  \sigma(x,0,\epsilon)-\sigma(x,0)=o_{\mathcal{D}'}(1),\quad \epsilon \to 0.
   \end{gathered}
 \label{e2.1}
\end{equation}
\end{definition}
Similarly, for the system \eqref{e1.2} we have the following concept.

\begin{definition}[\cite{k1}] \label{def2.2} \rm
A pair of smooth complex-valued (real-valued) functions \\
$(u(x,t,\epsilon),\sigma(x,t,\epsilon))$ is called a 
\textit{weak asymptotic solution}
of the system \eqref{e1.2} with the initial data $(u(x,0),\sigma(x,0))$ if
\begin{equation}
 \begin{gathered}
  \frac{\partial u(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial u(x,t,\epsilon)}{\partial x}-\frac{\partial \sigma(x,t,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1), \\
  \frac{\partial \sigma(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial \sigma(x,t,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\\
  u(x,0,\epsilon)-u(x,0)=o_{\mathcal{D}'}(1),\\
  \sigma(x,0,\epsilon)-\sigma(x,0)=o_{\mathcal{D}'}(1),\quad \epsilon \to 0.
   \end{gathered}
 \label{e2.2}
\end{equation}
\end{definition}

\begin{definition} \label{def2.3} \rm
 A pair of real-valued distributions 
$(u(x,t),\sigma(x,t))$ in the space 
$C(\mathbb{R}_{+};\mathcal{D}'(\mathbb{R}))$
is called a \textit{generalised solution} of the systems \eqref{e1.1} 
or \eqref{e1.2} if it is the weak limit
(limit in the sense of distributions) of a weak asymptotic 
solution $(u(x,t,\epsilon),\sigma(x,t,\epsilon))$ as $\epsilon \to 0$.
\end{definition}

\subsection{The method of weak asymptotics} 

Instead of writing it down schematically (see \cite{a1}), 
here we shall try to illustrate the method of weak asymptotics by applying it to
the systems \eqref{e1.1} or \eqref{e1.2} with initial data of the form
\begin{equation}
 \begin{gathered}
  u(x,0)=u_{0}+u_{1}H(-x),\\
  \sigma(x,0)=\sigma_{0}+\sigma_{1}H(-x)+e^{0}\delta(x).
 \end{gathered}
 \label{e2.3}
 \end{equation}
where $u_{0},u_{1},\sigma_{0},\sigma_{1},e^{0}$ are constants. 
In this case we seek solutions in the form of the singular ansatz
\begin{equation}
 \begin{gathered}
  u(x,t)=u_{0}+u_{1}H(-x+\phi(t)),\\
  \sigma(x,t)=\sigma_{0}+\sigma_{1}H(-x+\phi(t))+e(t)\delta(x-\phi(t))
 \end{gathered}
\label{e2.4}
\end{equation}
Such a solution will be called a \textit{generalised $\delta$-shock wave type solution} for the systems \eqref{e1.1} or \eqref{e1.2}.\\
The first step is to regularize the Heaviside functions and delta distributions and add \textit{correction} terms so as to form a
smooth ansatz for the weak asymptotic solutions in the form:
\begin{equation}
 \begin{gathered}
  u(x,t,\epsilon)=u_{0}+u_{1}H_{u}(-x+\phi(t),\epsilon)+R_{u}(x,t,\epsilon),\\
  \sigma(x,t,\epsilon)=\sigma_{0}+\sigma_{1}H_{\sigma}(-x+\phi(t),\epsilon)+e(t)\delta(x-\phi(t),\epsilon)+R_{\sigma}(x,t,\epsilon).
 \end{gathered}
 \label{e2.5}
\end{equation}
Here $H_{u},H_{\sigma}$ are regularizations of the Heaviside function,
$\delta(x,\epsilon)$ is a regularization of the delta function and
$R_{u},R_{\sigma}$ are \textit{correction} terms chosen so as to satisfy 
the conditions:
$$
R_{i}(x,t,\epsilon)=o_{\mathcal{D}'}(1),\quad
\frac{\partial R_{i}(x,t,\epsilon)}{\partial t}=o_{\mathcal{D}'}(1),\quad
\epsilon \to 0,\; i=u,\sigma.
$$
The next step is to prove the existence of a weak asymptotic solution 
by substituting the smooth ansatz in place of u and $\sigma$ in the 
left hand side of the systems and finding out the functions $\phi(t)$ and $e(t)$.
Once this step is completed, we obtain a generalised $\delta$-shock wave 
type solution of the form \eqref{e2.4} as the distributional limit
of the weak asymptotic solution as $\epsilon \to 0$.

\section{The regularizations and weak asymptotic expansions}

In this section, we describe the regularizations and correction term 
to be used in the next section and prove a few weak asymptotic expansions for
various products containing them.
As mentioned in Section 2, we seek 
\textit{generalised $\delta$-shock wave type solutions} of the form
\begin{equation}
 \begin{gathered}
  u(x,t)=u_{0}+u_{1}H(-x+\phi(t)),\\
  \sigma(x,t)=\sigma_{0}+\sigma_{1}H(-x+\phi(t))+e(t)\delta(x-\phi(t))
 \end{gathered}
 \label{e3.1}
\end{equation}
 for the systems \eqref{e1.1} and \eqref{e1.2} with initial data of the form
 \begin{equation}
 \begin{gathered}
  u(x,0)=u_{0}+u_{1}H(-x),\\
  \sigma(x,0)=\sigma_{0}+\sigma_{1}H(-x)+e^{0}\delta(x).
 \end{gathered}
 \label{e3.2}
 \end{equation}
Here $\phi(t)$ and $e(t)$ are smooth functions to be determined.
To apply the method of weak asymptotics, we suggest a smooth ansatz of the form (we do not take any correction term in the expression for $\sigma(x,t,\epsilon)$)
\begin{equation}
 \begin{gathered}
 u(x,t,\epsilon)=u_{0}+u_{1}H_{u}(-x+\phi(t),\epsilon)
 +p(t)R(x-\phi(t),\epsilon),\\
 \sigma(x,t,\epsilon)=\sigma_{0}+\sigma_{1}H_{\sigma}(-x+\phi(t),\epsilon)
 +e(t)\delta(x-\phi(t),\epsilon).
 \end{gathered}
 \label{e3.3}
\end{equation}
Here $H_{u}(.,\epsilon),H_{\sigma}(.,\epsilon)$ are regularizations 
of the Heaviside function, $\delta(.,\epsilon)$ is the regularization
of the delta distribution and $p(t)R(.,\epsilon)$ is a correction 
term where $p(t)$ is a smooth function (complex-valued or real-valued)
to be chosen afterwards.

The regularizations of the delta distribution and the correction term 
used in the proofs are same as in \cite{k1}. The regularization of
the Heaviside function has a subtle change though.

Let $\omega:\mathbb{R}\to \mathbb{R}$ be a non-negative, smooth, 
even function with support in $(-1,1)$ and satisfying
$$
\int_{\mathbb{R}}\omega(x)dx=1.
$$
Let $\omega_{0}=\int_{\mathbb{R}}\omega^2(x)dx$ and let
\begin{equation}
 R(x,t,\epsilon)=\frac{1}{\sqrt{\epsilon}}\omega(\frac{x-2\epsilon}{\epsilon}),\quad
\delta(x,\epsilon)=\frac{1}{\epsilon}\omega(\frac{x+2\epsilon}{\epsilon}).
\label{e3.4}
\end{equation}
Since by definition, $R$ is independent of $t$, we henceforth denote 
it by $R(x,\epsilon)$.
An easy calculation using test functions then shows that 
$R(x,\epsilon)=o_{\mathcal{D}'}(\epsilon)$ and since $R$ is independent of $t$,
 we also have
$\frac{\partial R(x,\epsilon)}{\partial t}=o_{\mathcal{D}'}(\epsilon)$. 
Thus it satisfies the criteria for being a \textit{correction}
term as stated in Section 2.
Also it can be easily checked that 
\begin{gather*}
\frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(\epsilon),\quad
R^2(x,\epsilon)=\omega_{0}\delta(x)+o_{\mathcal{D}'}(\epsilon),\\
R(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x}
=\frac{1}{2}\omega_{0}\delta'(x)+o_{\mathcal{D}'}(\epsilon)
\end{gather*}
and that 
$$
\delta(x,\epsilon)=\delta(x)+o_{\mathcal{D}'}(\epsilon),\quad
\frac{\partial \delta(x,\epsilon)}{\partial x}
=\delta'(x)+o_{\mathcal{D}'}(\epsilon).
$$
An important observation we need to make at this point is that the supports 
of $R(x,\epsilon)$ and $\delta(x,\epsilon)$ are disjoint. Therefore
we have 
$$
R(x,\epsilon)\delta(x,\epsilon)=0,\ R(x,\epsilon)
\frac{\partial \delta(x,\epsilon)}{\partial x}=0.
$$
Let $c=(\frac{1}{2}-\frac{\sigma_{1}}{u_{1}^2})$.
Let us define the smooth function $H(x,\epsilon)$ as follows:
\begin{equation}
H(x,\epsilon)=\begin{cases} 
1,& x\leq -4\epsilon\\
c,& -3\epsilon\leq x \leq 3\epsilon\\
0,& x\geq 4\epsilon
\end{cases}
\label{e3.5}
\end{equation}
and is continued smoothly in the regions $(-4\epsilon,-3\epsilon)$ 
and $(3\epsilon,4\epsilon)$.
We take $H_{u}(x,\epsilon)=H_{\sigma}(x,\epsilon)=H(x,\epsilon)$.
Again a little bit of calculation shows that
\begin{gather*}
H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(\epsilon),\quad
\frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(\epsilon),\\
H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}
=\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(\epsilon)
\end{gather*}
Since the supports of $R(x,\epsilon)$ and $\delta(x,\epsilon)$ 
are contained in $(-3\epsilon,3\epsilon)$, it again follows that
 \begin{gather*}
 H(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x}=c\frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(\epsilon),\\
 R(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=0.R(x,t,\epsilon)=o_{\mathcal{D}'}(\epsilon),\\
 H(x,\epsilon)\frac{\partial \delta(x,\epsilon)}{\partial x}=c\delta'(x)+o_{\mathcal{D}'}(\epsilon).
  \end{gather*}
 From the above discussions we then have the following lemma.

\begin{lemma} \label{lem3.1}
Choosing the regularizations and corrections as in \eqref{e3.4} and
 \eqref{e3.5} we have the following weak asymptotic expansions:
 \begin{gather*}
 R(x,\epsilon)=o_{\mathcal{D}'}(1),\ \ \frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\\
 R^2(x,\epsilon)=\omega_{0}\delta(x)+o_{\mathcal{D}'}(1),\\
 R(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x}=\frac{1}{2}\omega_{0}\delta'(x)+o_{\mathcal{D}'}(1),\\
 \delta(x,\epsilon)=\delta(x)+o_{\mathcal{D}'}(1),\ \ \frac{\partial \delta(x,\epsilon)}{\partial x}=\delta'(x)+o_{\mathcal{D}'}(1),\\
 R(x,\epsilon)\delta(x,\epsilon)=0,\ R(x,\epsilon)\frac{\partial \delta(x,\epsilon)}{\partial x}=0,\\
 H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(1),\ \frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(1),\\
 H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(1),\\
 H(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\ R(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\\
 H(x,\epsilon)\frac{\partial \delta(x,\epsilon)}{\partial x}
=c\delta'(x)+o_{\mathcal{D}'}(1),\ \epsilon \to 0.
 \end{gather*}
\end{lemma}

\section{Generalised Delta-shock wave type solutions via construction 
of weak asymptotic solutions}

In this section, we construct $\delta$-shock wave type solutions for the 
systems \eqref{e1.1} and \eqref{e1.2} using the method of weak
asymptotics.
For the rest of the discussion, we use the convention $[v]=v_{L}-v_{R}$,
 where $v_{L},v_{R}$ respectively denote the left and right states of $v$ 
across the discontinuity.

\begin{theorem} \label{thm4.1}
 For $t\in [0,\infty)$, the Cauchy problem \eqref{e1.1},\eqref{e3.2} 
has a weak asymptotic solution \eqref{e3.3} with $\phi(t),e(t)\ and\ p(t)$
given by the relations
\begin{equation}
\begin{gathered}
\dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]},\  \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}}-k^2u_{1},\\
\frac{1}{2}p^2(t)\omega_{0}-e(t)=0,
\end{gathered}
\label{e4.1}
\end{equation}
where $\omega_{0}$ is a positive constant,  defined in Section 3.
\end{theorem}

\begin{proof}
 To begin, let us recall the form of the smooth ansatz:
\begin{gather*}
  u(x,t,\epsilon)=u_{0}+u_{1}H_{u}(-x+\phi(t),\epsilon)
+p(t)R(x-\phi(t),\epsilon), \\
  \sigma(x,t,\epsilon)=\sigma_{0}+\sigma_{1}H_{\sigma}(-x+\phi(t),\epsilon)
+e(t)\delta(x-\phi(t),\epsilon).
 \end{gather*}
Then the partial derivatives of $u(x,t,\epsilon)$ and 
$\sigma(x,t,\epsilon)$ are given by
\begin{gather*}
\begin{aligned}
  \frac{\partial u(x,t,\epsilon)}{\partial t}  
&=u_{1}\dot{\phi}(t)\frac{dH_{u}(-x+\phi(t),\epsilon)}{d\xi}
+\dot{p}(t)R(x-\phi(t),\epsilon)\\
&\quad -p(t)\dot{\phi}(t)\frac{dR(x-\phi(t),\epsilon)}{d\xi}, 
\end{aligned}\\
  \frac{\partial u(x,t,\epsilon)}{\partial x} 
=-u_{1}\frac{dH_{u}(-x+\phi(t),\epsilon)}{d\xi}+p(t)
\frac{dR(x-\phi(t),\epsilon)}{d\xi},  
\\
\begin{aligned}
\frac{\partial \sigma(x,t,\epsilon)}{\partial t} 
&=\sigma_{1}\dot{\phi}(t)\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d\xi}
+\dot{e}(t)\delta(x-\phi(t),\epsilon)\\
&\quad -e(t)\dot{\phi}(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi},
\end{aligned} \\
  \frac{\partial \sigma(x,t,\epsilon)}{\partial x} 
=-\sigma_{1}\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d\xi}
+e(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi},
 \end{gather*}
where $\frac{du(.,\epsilon)}{d\xi}$ denotes the derivative of 
$u(.,\epsilon)$ with respect to the first component.
Substituting these relations into the left-hand side of 
the system \eqref{e1.1}, we have 
\begin{equation}
 \begin{aligned}
  &\frac{\partial u(x,t,\epsilon)}{\partial t} 
+u(x,t,\epsilon)\frac{\partial u(x,t,\epsilon)}{\partial x}
 -\frac{\partial \sigma(x,t,\epsilon)}{\partial x}   \\
  &=\ u_{1}\dot{\phi}(t)\frac{dH_{u}(-x+\phi(t),\epsilon)}{d\xi} 
+\dot{p}(t)R(x-\phi(t),\epsilon)-p(t)\dot{\phi}(t)
 \frac{dR(x-\phi(t),\epsilon)}{d\xi}   \\
  &\quad -u_{0}u_{1}\frac{dH_{u}(-x+\phi(t),\epsilon)}{d\xi} -u_{1}^2H_{u}
 (-x+\phi(t),\epsilon)\frac{dH_{u}(-x+\phi(t),\epsilon)}{d \xi}   \\
  &\quad +u_{0} p(t)\frac{dR(x-\phi(t),\epsilon)}{d\xi} +u_{1}p(t)H_{u}
 (-x+\phi(t),\epsilon)\frac{dR(x-\phi(t),\epsilon)}{d \xi}    \\
  &\quad -p(t)u_{1}R(x-\phi(t),\epsilon)\frac{dH_{u}(-x+\phi(t),\epsilon)}{d \xi}
 +p^2(t)R(x-\phi(t),\epsilon)\frac{dR(x-\phi(t),\epsilon)}{d \xi}    \\
  &\quad +\sigma_{1}\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d \xi}
-e(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi}
 \end{aligned}
\label{e4.2}
\end{equation}
 and
 \begin{equation}
 \begin{aligned}
  &\frac{\partial \sigma(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial \sigma(x,t,\epsilon)}{\partial x}-k^2\frac{\partial u(x,t,\epsilon)}{\partial x}  \\
  &= \sigma_{1}\dot{\phi}(t)\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d\xi}+\dot{e}(t)\delta(x-\phi(t),\epsilon)
  -e(t)\dot{\phi}(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi} \\
  &\quad -u_{0}\sigma_{1}\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d \xi}-u_{1}\sigma_{1}H_{u}(-x+\phi(t),\epsilon)\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d \xi} \\
  &\quad +u_{0}e(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi}+u_{1}e(t)H_{u}(-x+\phi(t),\epsilon)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi}  \\
  &\quad -\sigma_{1}p(t)R(x-\phi(t),\epsilon)\frac{dH_{\sigma}(-x+\phi(t),\epsilon)}{d \xi}+p(t)e(t)R(x-\phi(t),\epsilon)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi}  \\
  &\quad +k^2u_{1}\frac{dH_{u}(-x+\phi(t),\epsilon)}{d\xi}-k^2p(t)
 \frac{dR(x-\phi(t),\epsilon)}{d\xi}.
\end{aligned}
\label{e4.3}
\end{equation}
Now using the weak asymptotics relations from Lemma 3.1 in 
 relation \eqref{e4.2}, we obtain
 \begin{align*}
  &\frac{\partial u(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)
 \frac{\partial u(x,t,\epsilon)}{\partial x}
 -\frac{\partial \sigma(x,t,\epsilon)}{\partial x} \\
  &= \{u_{1}\dot{\phi}(t)-u_{0}u_{1}-\frac{1}{2}u_{1}^2+\sigma_{1}\}
 \delta(x-\phi(t))+\{\frac{1}{2}p^2(t)\omega_{0}-e(t)\}\dot{\delta}(x-\phi(t))\\
  &\quad +o_{\mathcal{D}'}(1).
\end{align*}
Setting the coefficients of $\delta$ and $\dot{\delta}$ in the above 
relation to be zero, we obtain
 \begin{gather*}
  u_{1}\dot{\phi}(t)-u_{0}u_{1}-\frac{1}{2}u_{1}^2+\sigma_{1}=0, \\
 \frac{1}{2}p^2(t)\omega_{0}-e(t)=0.
   \end{gather*}
The first equation above, when rewritten, gives
\[
 \dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}.
\]
Substituting $\dot{\phi}(t)$ from above in the relation \eqref{e4.3} 
and observing that
\begin{align*}
&u_{1}e(t)H_{u}(-x+\phi(t),\epsilon)
 \frac{d\delta(x-\phi(t),\epsilon)}{d\xi}\\
&+u_{0}e(t)\frac{d\delta(x-\phi(t), \epsilon)}{d\xi}
 -e(t)\dot{\phi}(t)\frac{d\delta(x-\phi(t),\epsilon)}{d\xi}
=o_{\mathcal{D}'}(\epsilon),
\end{align*}
(the choice of $c$ as in Section 3 helps us in getting this asymptotics)
we have
 \begin{align*}
  &\frac{\partial \sigma(x,t,\epsilon)}{\partial t}+u(x,t,\epsilon)\frac{\partial \sigma(x,t,\epsilon)}{\partial x}-k^2\frac{\partial u(x,t,\epsilon)}{\partial x} \\
  &=\{\dot{e}(t)+\sigma_{1}\dot{\phi}(t)-\frac{1}{2}u_{1}\sigma_{1}-u_{0}\sigma_{1}+k^2u_{1}\}\delta(x-\phi(t))
  +o_{\mathcal{D}'}(1),\epsilon \to 0.
 \end{align*}
Setting the coefficient of $\delta$ in the above relation to be zero, we obtain
\[
  \dot{e}(t)+\sigma_{1}\dot{\phi}(t)-\frac{1}{2}u_{1}\sigma_{1}-u_{0}\sigma_{1}
+k^2u_{1}=0,
 \]
which on simplification gives
\[
  \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}}-k^2u_{1}.
\]
Therefore, we find that the smooth ansatz \eqref{e3.3} is a 
 weak  asymptotic solution
provided $p(t),\phi(t),e(t)$ can be solved from the following equations
\begin{equation}
 \begin{gathered}
  \dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}, \\
  \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}}-k^2u_{1},\\
  \frac{1}{2}p^2(t)\omega_{0}-e(t)=0.
   \end{gathered}
\label{e4.4}
\end{equation}
The above ordinary differential equations can be solved with the 
initial conditions $\phi(0)=0$ and $e(0)=e^{0}$ and we have
\begin{equation}
  \phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad
  e(t)=(\frac{\sigma_{1}^2}{u_{1}}-k^2u_{1})t+e^{0}.
\label{e4.5}
 \end{equation}
Next substituting $e(t)$ in the last equation of \eqref{e4.4}, 
we can solve for $p(t)$ taken in the form $p(t)=p_{1}(t)+ip_{2}(t)$ and hence
we have a weak asymptotic  solution of  \eqref{e1.1}.
\end{proof}

Since a  $\delta$-shock wave type solution is the distributional limit of
 a weak asymptotic solution,  from the previous theorem we 
have the following result.

\begin{theorem} \label{thm4.2}
 For $t\in [0,\infty)$, the Cauchy problem \eqref{e1.1},\eqref{e3.2}
 has a generalised $\delta$-shock wave type solution \eqref{e3.1} 
with $\phi(t)$ and $e(t)$ given by the relations
\begin{equation}
\phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad
e(t)=(\frac{\sigma_{1}^2}{u_{1}}-k^2u_{1})t+e^{0}.
\label{e4.6}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk4.3} \rm
 Let $e^{0}=0$. Now $\dot e(t)=0$ in \eqref{e4.1} would imply the existence 
of shock-wave solution (in the class of bounded variation functions)
 of the system \eqref{e1.1}.
But that implies $\sigma_{1}^2=k^2 u_{1}^2$. Imposing Lax's admissibility 
condition, which in this case is $u_{1}>0$, we get the shock curves
\begin{gather*}
S_{1}:[\sigma]=k[u],\\
S_{2}:[\sigma]=-k[u].
\end{gather*}
Also in this case, we have 
$u\frac{\partial \sigma}{\partial x}=\lim_{\epsilon \to 0}u(x,t,\epsilon)
\frac{\partial \sigma(x,t,\epsilon)}{\partial x}=-\sigma_{1}(u_{0}
+\frac{u_{1}}{2})\delta$,
which again is the Volpert's product (the negative sign arises because 
of the convention on $[\sigma]$). Thus we recover the results proved 
in \cite{j2} for the shock-wave case; see \eqref{e1.3}.
\end{remark}

\begin{remark}[Overcompressivity condition for $\delta$-shock wave solutions]\rm
 We recall that the overcompressivity condition 
(see \cite{a1,k1}) for the $\delta$-shock wave solutions for a
 $n\times n$ system is
$$
\lambda_{k}(v_{R})<\dot \phi(t)<\lambda_{k}(v_{L}),\quad k=1,\dots,n.
$$
Therefore, for the system \eqref{e1.1}, it takes the form
 \begin{gather*}
 u_{0}-k<\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}<u_{0}+u_{1}-k,\\
 u_{0}+k<\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}<u_{0}+u_{1}+k.
 \end{gather*}
The above relations on simplification yield the conditions:
\begin{gather*}
u_{1}>2k>0, \\
-(\frac{u_{1}}{2}-k)<\frac{\sigma_{1}}{u_{1}}<\frac{u_{1}}{2}-k.
\end{gather*}
\end{remark}

Next we prove the existence of a weak asymptotic solution for 
system \eqref{e1.2}.

\begin{theorem} 
For $t\in [0,\infty)$, the Cauchy problem \eqref{e1.2}, \eqref{e3.2} 
has a weak asymptotic solution \eqref{e3.3} with $\phi(t),e(t)\ and\ p(t)$
given by the relations
\begin{equation}
\begin{gathered}
\dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]},\  \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}},\\
\frac{1}{2}p^2(t)\omega_{0}-e(t)=0,
\end{gathered}
\label{e4.7}
\end{equation}
where $\omega_{0}$ is a positive constant $($defined in Section $3$ $)$.
\end{theorem}

\begin{proof}
 Proceeding as in the proof of the Theorem 4.1,
we find that the smooth ansatz \eqref{e3.3} is a  weak  asymptotic solution
provided $p(t),\phi(t),e(t)$ can be solved from the  equations
\begin{equation}
 \begin{gathered}
  \dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}, \\
  \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}}, \\
  \frac{1}{2}p^2(t)\omega_{0}-e(t)=0.
 \end{gathered}
\label{e4.8}
\end{equation}
The ordinary differential equations for $\phi(t)$ and $e(t)$ can be 
solved with the initial conditions $\phi(0)=0$ and $e(0)=e^{0}$ and we have
\[
  \phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad
  e(t)=\frac{\sigma_{1}^2}{u_{1}}t+e^{0}.
\]
Next substituting $e(t)$ in the last equation of \eqref{e4.8},
 we can solve for $p(t)$ taken in the form $p(t)=p_{1}(t)+ip_{2}(t)$ and hence
we have a  weak asymptotic  solution of  system \eqref{e1.2}.
\end{proof}

Therefore, from the previous theorem, we have the following result.

\begin{theorem}
For $t\in [0,\infty)$, the Cauchy problem \eqref{e1.2}, \eqref{e3.2}
 has a generalised $\delta$-shock wave type solution \eqref{e3.1} 
with $\phi(t)$ and $e(t)$
given by the relations
\begin{equation}
\phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad
e(t)=\frac{\sigma_{1}^2}{u_{1}}t+e^{0}.
\label{e4.9}
\end{equation}
\end{theorem}

\begin{remark} \rm
 The overcompressivity assumption for the system \eqref{e1.2} yields 
$u_{1}>0$ and $-\frac{u_{1}}{2}<\frac{\sigma_{1}}{u_{1}}<\frac{u_{1}}{2}$.
\end{remark}

\begin{remark} \rm
 If $u_{1}>0$ (follows from the overcompressivity condition above), 
then from equation \eqref{e4.7} it follows that $\dot{e}(t)>0$. If in addition
we have $e^{0}=0$, then $e(t)=\frac{\sigma_{1}^2}{u_{1}}t$ which is greater 
than zero for all t. In this case, it is sufficient to consider
$p(t)$ as a real-valued function only.
\end{remark}

\begin{remark} \rm
From Remark 4.8 it also follows that we might have a 
singular concentration developing in the solution of the system 
\eqref{e1.2} even if we start with Riemann type initial data.
\end{remark}

\begin{remark} \rm
 If we take $\sigma_{0}=\sigma_{1}=0$ in \eqref{e3.2}, then proceeding 
as in the proof of the Theorem 4.5 we obtain a generalised $\delta$-shock
 wave type solution for the system \eqref{e1.2} of the form:
 \begin{gather*}
  u(x,t)=u_{0}+u_{1}H(-x+\phi(t)),\\
  \sigma(x,t)=e^{0}\delta(x-\phi(t))
 \end{gather*}
where $\dot{\phi}(t)=\frac{[\frac{u^2}{2}]}{[u]}$.
\end{remark}

\begin{remark}[Dependence of the solutions on $k$] \rm
 From the structure of the generalised solutions for  systems the
\eqref{e1.1}, \eqref{e1.2} obtained from Theorem 4.2 and Theorem 4.6,
it is quite evident that as $k$ tends to $0$, the generalised solution 
obtained for the system \eqref{e1.1} actually converges (in distributional limit)
 to that obtained for the system \eqref{e1.2}.

This observation therefore justifies our motivation to study the 
system \eqref{e1.2} based upon the solutions of the system \eqref{e1.1}
(letting $k\to 0$).
\end{remark}

\subsection*{Acknowledgements} 
The author is grateful to the anonymous referee whose constructive 
criticism helped to improve the contents and presentation of this article.


\begin{thebibliography}{00}

\bibitem{a1}  Albeverio, S.; Shelkovich, V. M.; 
 On the delta-shock front problem.
\textit{Analytical approaches to multidimensional balance laws, Nova Sci. Publ.,
 New York} (2006), 45-87

\bibitem{c1}  Cauret, J. J.; Colombeau, J. F.; LeRoux, A. Y.;
Discontinuous generalized solutions of nonlinear nonconservative 
hyperbolic equation, \textit{J. Math. Anal. Appl.} 139 (1989), 552-573.

\bibitem{d1} Dal Maso, G.; Lefloch, P. G.; Murat, F.;
 Definition and weak stability of non-conservative products,
\textit{J. Math. Pures Appl.} 74 (1995), 483-548.

\bibitem{d2} Danilov, V.G.;  Mitrovic, D.;
 Weak asymptotic of shock wave formation process, \textit{Nonlinear Anal.} 
61 (2005), 613-635.

\bibitem{j1}  Joseph, K. T.;
 Generalized solutions to a Cauchy problem for a nonconservative hyperbolic
 system, \textit{J. Math. Anal. Appl.} 207  (1997), 361-389.

\bibitem{j2}  Joseph, K. T.;  Sachdev, P. L.;
Exact solutions for some non-conservative hyperbolic systems,
\textit{Internat. J. Non-Linear Mech.} 38 (2003), 1377-1386.

\bibitem{k1}  Kalisch, H.; Mitrovic, D.;
Singular solutions for the shallow-water equations,
\textit{IMA J. Appl. Math.} 77 (2012), 340-350.

\bibitem{l1}  Lefloch, P. G.; Liu, T.-P.;
 Existence theory for nonlinear hyperbolic systems in nonconservative form,
\textit{Forum Math.} 3 (1993), 261-280.

\bibitem{r1}  Raymond, J. P.;
 A new definition of nonconservative products and weak stability results,
\textit{Boll. Un. Mat. Ital. B (7)} 10 (1996), 681-699.

\bibitem{v1}  Volpert, A. I.; 
The space BV and quasilinear equations, \textit{Math. USSR Sb. 2} (1967), 
225-267.

\bibitem{z1}  Zheng, Y.; 
 Systems of conservation laws with incomplete sets of eigenvectors everywhere.
\textit{Advances in nonlinear partial differential equations and related
 areas (Beijing, 1997), World Sci. Publ., River Edge, NJ.} (1998), 399-426.

\end{thebibliography}

\section*{Addendum posted on July 26, 2013} 

The author would like to make a few minor corrections, and to 
mention that no changes in the main results take place.
Since all the corrections are on page 6, we rewrite this page
from the top to the beginning of Section 4.
\medskip

Let $c=(\frac{1}{2}-\frac{\sigma_{1}}{u_{1}^2})$.
Let us define the smooth function $H(x,\epsilon)$ as follows:
\begin{equation}
H(x,\epsilon)=\begin{cases} 
0,& x\leq -4\epsilon\\
c,& -3\epsilon\leq x \leq 3\epsilon\\
1,& x\geq 4\epsilon
\end{cases} \tag{3.5}
\end{equation}
and is continued smoothly in the regions $(-4\epsilon,-3\epsilon)$ 
and $(3\epsilon,4\epsilon)$.
We take $H_{u}(x,\epsilon)=H_{\sigma}(x,\epsilon)=H(x,\epsilon)$.
Again a little bit of calculation shows that
\begin{gather*}
H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(\epsilon),\quad
\frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(\epsilon),\\
H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}
=\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(\epsilon)
\end{gather*}
Since the supports of $R(x,\epsilon)$ and $\delta(x,\epsilon)$ 
are contained in $(-3\epsilon,3\epsilon)$, it again follows that
 \begin{gather*}
 H(x,\epsilon)\frac{\partial R(-x,\epsilon)}{\partial x}=c\frac{\partial R(-x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(\epsilon),\\
 R(-x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=0.R(-x,\epsilon)=0,\\
 H(x,\epsilon)\frac{\partial \delta(-x,\epsilon)}{\partial x}=c\delta'(-x)+o_{\mathcal{D}'}(\epsilon).
  \end{gather*}
 From the above discussions we then have the following lemma.
\medskip

\textbf{Lemma 3.1.} {\it
Choosing the regularizations and corrections as in $(3.4)$ and
 $(3.5)$ we have the following weak asymptotic expansions:
 \begin{gather*}
 R(x,\epsilon)=o_{\mathcal{D}'}(1),\quad
\frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),
\\
 R^2(x,\epsilon)=\omega_{0}\delta(x)+o_{\mathcal{D}'}(1),
\\
 R(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x}
=\frac{1}{2}\omega_{0}\delta'(x)+o_{\mathcal{D}'}(1),
\\
 \delta(x,\epsilon)=\delta(x)+o_{\mathcal{D}'}(1),\quad
\frac{\partial \delta(x,\epsilon)}{\partial x}=\delta'(x)+o_{\mathcal{D}'}(1),
\\
 R(x,\epsilon)\delta(x,\epsilon)=0,\quad
 R(x,\epsilon)\frac{\partial \delta(x,\epsilon)}{\partial x}=0,
\\
 H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(1),\quad
\frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(1),
\\
 H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}
=\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(1),
\\
 H(x,\epsilon)\frac{\partial R(-x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\quad
R(-x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=0,
\\
 H(x,\epsilon)\frac{\partial \delta(-x,\epsilon)}{\partial x}
=c\delta'(-x)+o_{\mathcal{D}'}(1),\quad \epsilon \to 0.
 \end{gather*}
}

End of addendum.

\textbf{Acknowledgement.} 
The author would like to thank Prof. Evgeniy Panov for 
bringing the corrections to the author's attention.



\end{document}