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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 259, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/259\hfil A nonconservative system in elastodynamics]
{Exact solutions of a nonconservative system in elastodynamics}
\author[K. T. Joseph \hfil EJDE-2015/259\hfilneg]
{Kayyunnapara Thomas Joseph}
\dedicatory{In memory of Professor S. Raghavan}
\address{Kayyunapara Thomas Joseph \newline
Centre for applicable Mathematics,
Tata Institute of Fundamental Research, \newline
Sharadanagar,Post Bag no. 6503,
GKVK Post Office,
Bangalore 560065, India}
\email{ktj@math.tifrbng.res.in}
\thanks{Submitted September 4, 2015. Published October 7, 2015.}
\subjclass[2010]{35A20, 35L45, 35B25}
\keywords{Elastodynamics; viscous shocks}
\begin{abstract}
In this article we find an explicit formula for solutions of a
nonconservative system when the initial data lies in the level set
of one of the Riemann invariants. Also for nonconservative shock waves
in the sense of Volpert we derive an explicit formula for the viscous
shock profile.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks
\section{Introduction}
One of the systems of equations that comes in modelling propagation of elastic waves,
is the nonconservative system
\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{e1.1}
\end{equation}
which was introduced in \cite{c2}.
Here $u$ is the velocity, $\sigma$ is the stress and $k>0$ is the speed
of propagation of the elastic waves. The system \eqref{e1.1} is strictly
hyperbolic with characteristic speeds
\begin{equation}
\lambda_1(u,\sigma) = u - k, \quad \lambda_2(u,\sigma) = u + k
\label{e1.2}
\end{equation}
and corresponding Riemann invariants
\begin{equation}
\it{w_1}(u,\sigma)=\sigma - k u,\quad \it{w_2}(u,\sigma)= \sigma +k u.
\label{e1.3}
\end{equation}
It is well known that smooth global in time solutions do not exist even if
the initial data is smooth, then the term $u \sigma_x$ appearing in equations,
does not make sense in the theory of distributions,
and classical theory of Lax \cite{la1} does not work.
There are many approaches starting with Volpert \cite{v1}, and subsequently
by Colombeau \cite{ca1,c1,c2}, Dal Maso, LeFloch and Murat \cite{le1}
and LeFloch and Tzavaras \cite{le2} to define such products.
They are not equivalent but are related and have some common features.
They consider systems of $N$ equations of the form
\begin{equation}
U_t +A(U)U_x =0,\label{e1.4}
\end{equation}
where $A(U)U_x$, is not in conservative form $F(U)_x$.
Here $A(U)$ an $N \times N$ matrix, depending smoothly on $U \in \Omega$,
and $\Omega$ is an open connected set in $\mathbb{R}^N$.
Assume that $U$ has a discontinuity along $x=s t$ and of the form
\begin{equation}
U(x,t) = \begin{cases}
U_{-},&\text{if } x < s t,\\
U_{+},&\text{if } x > s t.
\end{cases}\label{e1.5}
\end{equation}
where $U_{-}$ and $U_{+}$ are constant vectors in $\Omega$.
Volpert \cite{v1} defined $A(U) U_x$ as a measure
\begin{equation}
A(U)U_x=\frac{1}{2}(A(U_{+}+A(U_{-})(U_{+}-U_{-})\delta_{x=s t}.
\label{e1.6}
\end{equation}
As this definition is inadequate for many applications, Dal Maso,LeFloch, Murat \cite{le1} generalized this definition by
\begin{equation}
A(U(x,t))U_x(x,t)
=\Big(\int_0^1 A(\phi(s,U_{-},U_{+}))\partial_s\phi(s;U_{-},U_{+})\Big)
\delta_{x=s t}
\label{e1.7}
\end{equation}
where $\phi$ is a family of Lipschitz paths,
$\phi:[0,1] \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}^N$, with
$\phi(0,U_{-},U_{+})\\ =U_{-}$ and $\phi(1,U_{-},U_{+})=U_{+}$,
with some natural conditions.
Volpert product corresponds to taking $\phi$ the straight line path connecting
$U_{-}$ and $U_{+}$. Further they solved Riemann problem for \eqref{e1.4}
with Riemann data
\begin{equation}
U(x,t) = \begin{cases}
U_{-},&\text{if } x < 0,\\
U_{+},,&\text{if } x > 0.
\end{cases}\label{e1.8}
\end{equation}
when the system is strictly hyperbolic and $|U_{+}-U_{-}|$ is small.
Choudhury \cite{ch1} has recently shown that Riemann problem for \eqref{e1.1}
with $k=0$, in which case the system is not strictly hyperbolic, do have
a solution in the class of shock waves and rarefaction waves if one uses
the product in \cite{le1}, with special choice of paths but not for straight
line paths. This example shows advantages of the product in \cite{le1} over
the Volpert product.
Different paths give different solutions. So as pointed out in \cite{le1, le2, le3}
any discussion of well-posedness of solution for nonconservative systems,
should be based on a given nonconservative product in addition to admissibility
criterion for shock discontinuities.
As the system of the type \eqref{e1.4}, is an approximation and is obtained
when one ignores higher order derivative terms, which give smoothing effects with
small parameters as coefficients . So a natural way to construct the physical
solution is, by the limit of a given regularization as these small parameters
goes to zero. Different regularizations correspond to different nonconservative
product and admissibility condition, see \cite{le1,le2} and the references there
more details.
Another method to handle the nonconservative product is using Colombeau algebra.
Initial value problem for the system \eqref{e1.1} was solved in this space
first in \cite{ca1,c2} using numerical approximation for a restricted class
of initial data. More general class of initial data including the $L^\infty$
space was treated in \cite{j1} by parabolic approximations with out any
conditions on the smallness of data. Dafermos regularization and the approach
of \cite{le2} was used \cite{j3,j4} to study Riemann problem.
In this paper we take a parabolic regularization and explain its connection with
the Volpert nonconservative product and Lax admissibility conditions.
Also we give explicit formula for the solution
when the initial data lie in the level set of one of the Riemann invariants
of the system \eqref{e1.1}.
\section{Viscous shocks profile for Volpert shock}
First we recall some known facts about the Riemann problem for \eqref{e1.1}.
Here the initial data takes the form
\begin{equation}
(u(x,0),\sigma(x,0)) = \begin{cases}
(u_{-},\sigma_{-}),&\text{if } x< 0,\\
(u_{+},\sigma_{+}),&\text{if } x>0.
\end{cases}\label{e2.1}
\end{equation}
A shock wave is a weak solution of \eqref{e1.1}, with speed $s$ is of the form
\begin{equation}
(u(x,t),\sigma(x,0)) = \begin{cases}
(u_{-},\sigma_{-}),&\text{if } x < s t,\\
(u_{+},\sigma_{+}),&\text{if } x > s t.
\end{cases} \label{e2.2}
\end{equation}
When Volpert product is used
the Rankine Hugoniot condition takes the form
\begin{equation}
\begin{gathered}
-s(u_{+}-u_{-})+\frac{u_{+}^2 - u_{-}^2}{2} -(\sigma_{+}-\sigma_{-})=0\\
-s(\sigma_{+}-\sigma_{-})+\frac{u_{+}+u_{-}}{2}(\sigma_{+}
-\sigma_{-})-k^2(u_{+}-u_{-})
\end{gathered} \label{e2.3}
\end{equation}
In \cite{j2}, it was shown that the Riemann problem can be solved without
any smallness assumptions on the Riemann data when the nonconservative product is
understood in the sense of Volpert \cite{v1} with Lax's admissibility conditions
for shock speed. Indeed, corresponding to each characteristic family
$\lambda_j, j=1,2$ we can define shock waves and rarefaction waves.
Fix a state $(u_{-},\sigma_{-})$ the set of states $(u_{+},\sigma_{+})$
which can be connected by a single $j$-shock wave is a straight line called
$j$-shock curve and is denoted by $S_j$ and the states which can be
connected by a single $j$-rarefaction wave is a straight line is called
$j$-rarefaction curve and is denoted by $R_j$. These wave curves are given by
\begin{equation}
\begin{gathered}
R_1(u_{-},\sigma_{-}): \sigma=\sigma_{-}+k(u-u_{-}), u>u_{-}\\
S_1(u_{-},\sigma_{-}): \sigma=\sigma_{-}+k(u-u_{-}), uu_{-}\\
S_2(u_{-},\sigma_{-}): \sigma=\sigma_{-}-k(u-u_{-}), u0$, except on a countable points $x \in \mathbb{R}^1$,
there exits a unique minimizer $y(x,t)$ for
\begin{equation}
\min_{y\in \mathbb{R}^1} [\frac{(x-y-(-1)^j kt)^2}{2t} + \int_0^y u_0(z) dz].
\label{e3.5}
\end{equation}
At these points the point wise limit
$\lim_{\epsilon \to 0}(u^\epsilon(x,t),\sigma^\epsilon(x,t))
= (u(x,t),\sigma(x,t))$
exits and is given by
\begin{equation}
\begin{gathered}
u(x,t)=(-1)^{j+1}k + \frac{(x-y(x,t))}{t},\\
\sigma(x,t)=(-1)^{j+1}k[(-1)^{j}k + \frac{(x-y(x,t))}{t}]+c.
\end{gathered}
\label{e3.6}
\end{equation}
Further $(u, \sigma)$ given by \eqref{e3.6} is a weak solution to \eqref{e1.1}
with initial condition \eqref{e3.2}.
\end{theorem}
\begin{proof}
Since the initial data in the level set of $j$-Riemann invariant, we seek a
solution lying in the same invariant set. So we seek $(u,\sigma)$ satisfying
\begin{equation}
\sigma=(-1)^{j+1} k u +c.
\label{e3.7}
\end{equation}
The an easy computation shows that the system become a single Burgers
equation for $u$,
\[
u_t +u u_x - (-1)^{j+1}ku_x =\epsilon u_{xx}
\]
Once $u$, is known then formula for $\sigma$ follows.
To find $u$ we make a substitution
\begin{equation}
v=u-(-1)^{j+} k \label{e3.8}
\end{equation}
and then the equation for $v$ can be written as
\[
v_t +v v_x =\epsilon v_{xx}
\]
with initial conditions
\[
v(x,0)=u_0(x) -(-1)^{j+1} k.
\]
Applying Hopf-Cole transformation \cite{h1}
\begin{equation}
v=-2\epsilon \frac {w_x}{w}
\label{e3.9}
\end{equation}
the problem is reduced to
\[
w_t=\epsilon w_{xx}
\]
with initial conditions
\[
w(x,0)=e^{\frac{-1}{2\epsilon}(\int_0^x u_0(z)dz -(-1)^{j+1}kx)}.
\]
Solving this system, we get
\begin{equation}
w(x,t)=\frac{1}{(4 \pi t \epsilon)^{1/2}}\int_{-\infty}^{\infty}
e^{\frac{-1}{2 \epsilon}[\frac{(x-y)^2}{2 t }+ \int_0^y u_0(z) dz
-(-1)^{j+1} ky]} dy.
\label{e3.10}
\end{equation}
An easy computation shows that
\begin{equation}
w_x(x,t)=\frac{-1}{2\epsilon }.\frac{1}{(4 \pi t \epsilon)^{1/2}}
\int_{-\infty}^{\infty} \frac{(x-y)}{t}e^{\frac{-1}{2 \epsilon}
[\frac{(x-y)^2}{2 t}+\int_0^y u_0(z) dz -(-1)^{j+1} ky]} dy.
\label{e3.11}
\end{equation}
Notice that
\begin{equation}
(x-y)^2 - (-1)^{j+1} 2 t k y =(x-y-(-1)^j kt)^2 +(-1)^j 2 t kx -t^2 k^2.
\label{e3.12}
\end{equation}
Using \eqref{e3.12} in \eqref{e3.10} and \eqref{e3.11}, substituting the
resulting expressions in \eqref{e3.9}, and using $u=v+(-1)^{j+1} k$,
from \eqref{e3.8} we get the formula for $u$ in \eqref{e3.3}.
Then the formula for $\sigma$ is obtained from the relation \eqref{e3.7}.
The formula for vanishing viscosity limit follows
from analysis of Hopf \cite{h1} and Lax \cite{la1}.
Indeed for each fixed $(x,t)$, there is at least one minimizer
for \eqref{e3.5}. There may be many minimizers, take $y(x,t)_{-}$ is the smallest
such minimizer and $y(x,t)_{+}$ is the largest one. Hopf has proved that,
for each fixed $t>0$, $y(x,t)_{\pm}$ is a nondecreasing function of $x$
and so has at most countable points of discontinuities and except these points,
these minimizer is unique and $y(x,t)=y(x,t)_{-}=y(x,t)_{+}$.
Then formula \eqref{e3.6} holds at these points $(x,t)$.
Now to show that the limit satisfies \eqref{e1.1}, we just notice that
\begin{equation}
\begin{gathered}
u_t +uu_x -\sigma_x-\epsilon u_{xx}=u_t +\frac{(u^2)_x}{2} -(-1)^{j+1} k u_x
-\epsilon u_{xx}\\
\sigma_t +u\sigma_x-k^2 u_x -\epsilon \sigma_{xx}=(-1)^{j+1}
k [u_t +\frac{(u^2)_x}{2}-(-1)^{j+1} k u_x -\epsilon u_{xx}].
\end{gathered}
\label{e3.13}
\end{equation}
which is conservative, and by standard theory of conservation laws works
\cite{h1,la1}, and we can pass to the limit in the equation in the weak sense.
Also from \cite {h1,la1} it follows that the solution satisfies the initial
data in weak sense.
\end{proof}
In the above theorem the solution of the inviscid
system \eqref{e1.1}, that we have constructed lie in the level set of a
Riemann invariant. Assume that the solution is on the $j$-Riemann invariant.
Then $\sigma$ and $u$ are related by \eqref{e3.7} and then
$u\sigma_x =(-1)^{j+1} k (\frac{u^2}{2})_x$, a conservative product.
A computation as in \eqref{e3.13} show that the system \eqref{e1.1}
becomes a single equation in conservation form for $u$, namely
\[
u_t +(\frac{(u^2)}{2} -(-1)^{j+1} k u)_x =0.
\]
Then all paths give the same Rankine- Hugoniot conditions for the shocks,
see \cite{le1}.
\subsection*{Acknowledgements}
I am very grateful to the anonymous referee for the corrections and suggestions
which improved the presentation of the paper.
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\end{document}