\documentclass[reqno]{amsart}
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\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 $\Omega=\{(x,t):-\infty0\}$) 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}),\; u0$, 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]}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.
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\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}