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

\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_{-}), u<u_{-}\\
R_2(u_{-},\sigma_{-}): \sigma=\sigma_{-}-k(u-u_{-}), u>u_{-}\\
S_2(u_{-},\sigma_{-}): \sigma=\sigma_{-}-k(u-u_{-}), u<u_{-}.
\end{gathered}\label{e2.4}
\end{equation}
Further $j$-shock speed $s_j$ is given by
\begin{equation}
 s_j =\frac{u_{+}+u_{-}}{2} +(-1)^j k,\quad j=1,2 \label{e2.5}
\end{equation}
and  the Lax entropy condition requires that the $j$- shock speed satisfies 
inequality
\begin{equation}
 \lambda_j(u_{+},\sigma_{+}) \leq s_j \leq \lambda_j(u_{-},\sigma_{-}). \label{e2.6}
\end{equation}
This curves fill in the $u-\sigma$ plane and the Riemann problem can be solved 
uniquely for arbitrary initial states
$(u_{-},\sigma_{-})$ and $(u_{_+},\sigma_{+})$
in the class of self similar functions consisting of solutions of shock waves 
and rarefaction waves separated by constant states. These constant states 
are obtained from the shock curves and rarefaction curves
corresponding to the two families of the characteristic fields.

The nonconservative product and the selection criteria is associated with a 
regularization. In this paper we analyze shock wave solution of \eqref{e1.1}
with respect to the Volpert product and Lax's shock inequalities and its relation
to parabolic approximation.
To analyze this connection, first we ask the question, does there exists a 
traveling wave profile solution
$(u(\xi),\sigma(\xi))$ with  $\xi=\frac{x-s_jt}{\epsilon}$, of the system
the corresponding parabolic approximation
 \begin{equation}
\begin{gathered}
u_t + u u_x - \sigma_x = \epsilon u_{xx}, \\
\sigma_t + u \sigma_x - k^2  u_x = \epsilon \sigma_{xx}.
\end{gathered}
\label{e2.7}
\end{equation}
connecting $(u_{-},\sigma_{-})$ to $(u_{+},\sigma_{+})$
when  $(u_{+},\sigma_{+})$ lies on the shock curve $S_j, j=1,2$ 
passing through $(u_{-},\sigma_{-})$.

 This amounts to solving the boundary-value problem, for a
system of nonlinear ordinary differential equations,
\begin{equation}
\begin{gathered}
 -s_j \frac {du}{d\xi} + u \frac {du}{d\xi} -\frac {d\sigma}{d\xi} 
= \frac {d^2u}{d\xi^2}, \quad
-s_j  \frac {d\sigma}{d \xi} +u \frac {d\sigma}{d \xi} - k^2
\frac {du}{d \xi} =\frac {d^2\sigma}{d \xi^2}
\end{gathered} \label{e2.8}
\end{equation}
for $\xi  \in (-\infty,\infty)$ with boundary
conditions
\begin{equation}
u(-\infty) = u_{-},\quad  u(\infty)=u_{+},\quad 
\sigma(-\infty) = \sigma_{-}, \quad
\sigma(\infty)=\sigma_{+}. \label{e2.9}
\end{equation}


 \begin{theorem} \label{thm2.1}
If $(u_{+},\sigma_{+}) \in S_j(u_{-},\sigma_{-})$, then the viscous shock 
profile  $(u(\xi),\sigma(\xi)$ of \eqref{e2.8}-\eqref{e2.9}  with speed
 $s_j=\frac{u_{+}+u_{-}}{2} +(-1)^j k$ is given by
\begin{equation}
(u(\xi),\sigma(\xi))=(\phi(\xi),(-1)^{j+1} k \phi(\xi)+\sigma_{-} 
+ (-1)^{j} k u_{-}), \label{e2.10}
\end{equation}
where
\begin{equation}
\phi(\xi)=u_{+}+\frac{(u_{-}-u_{+})}{1+\exp(\frac{u_{-}-u_{+}}{2} \xi -\xi_0)},
\label{e2.11}
\end{equation}
where $\xi_0$ is a constant.
\end{theorem}

\begin{proof} 
Writing  \eqref{e2.8} in Riemann invariants
\begin{equation}
\begin{gathered}
\frac{d^2 w_1}{d\xi^2} = (\lambda_1(r,s) - s) \frac{dw_1}{d\xi},\\
\frac{d^2 w_2}{d\xi^2} = (\lambda_2(r,s) - s) \frac{dw_2}{d\xi}.
\end{gathered} \label{e2.12}
\end{equation}
It follows from the uniqueness of solutions of ODE, any solution $w_j$ 
of \eqref{e2.12} with first derivative is zero at
a point $\xi_0$ must be a constant equal to $w_j(\xi_0)$. 
So either solutions $w_j$ are constants or strictly monotone. 
Suppose $u_{+} \in S_1(u_{-})$, then
$w_1(u_{-},\sigma_{-})=w_1(u_{+},\sigma_{+})$ and so
 $w_1(u(\xi),\sigma(\xi))$ is a constant.
Thus $u$ and $\sigma$ are related by
\begin{equation}
\sigma(\xi)= ku +(\sigma_{-}- ku_{-}). \label{e2.13}
\end{equation}
Substituting this relation in any one of the equations in \eqref{e2.8}, we 
get the same single equation for $u$. Indeed
for $1$-shock, $u$ is given by the equation
\[
 -s_1 u' +uu' -k u' = u" ,\quad u(-\infty)=u_{-}, u(\infty)=u_{+}.
\]
Once $u$ is known, $\sigma$ is obtained from \eqref{e2.13}.
In terms of the new variable $v=u-k$, this problem reduces to
\[
 -s_1 v' +vv' = v" ,\quad v(-\infty)=u_{-}-k, u(\infty)=u_{+}-k,
\]
whose solution is
\[
 v(\xi)= u_{+}+\frac{(u_{-}-u_{+})}{1+\exp(\frac{u_{-}-u_{+}}{2} \xi -\xi_0)}-k .
\]
Since $u=v+k$, and $\sigma(\xi)= ku +(\sigma_{-}- ku_{-})$, 
we have traveling wave corresponding to $1$-shock wave is given by
\[
  u(\xi)=u_{+}+\frac{(u_{-}-u_{+})}{1+\exp(\frac{u_{-}-u_{+}}{2} \xi 
-\xi_0)},\quad \sigma(\xi)=k u(\xi) +(\sigma_{-}-k u_{-})
\]
where $\xi=\frac{x-s_1 t}{\epsilon}$,
and  the formula \eqref{e2.10}-\eqref{e2.11} follows for the case $j=1$. 
The analysis for the case $j=2$ is similar and is omitted.
\end{proof}


\section{Explicit formula for initial data lying on level sets of Riemann invariants}

Now we consider initial value problem for the viscous system
\begin{equation}
\begin{gathered}
u_t + u u_x - \sigma_x = \epsilon u_{xx}, \\
\sigma_t + u \sigma_x - k^2  u_x = \epsilon \sigma_{xx}.
\end{gathered}\label{e3.1}
\end{equation}
with initial data
\begin{equation}
 u(x,0)=u_0(x),\quad \sigma_0(x,0)=\sigma_0(x) \label{e3.2}
\end{equation}
When $u_0(x)$ and $\sigma_0(x)$ are functions of bounded variation and 
continuously differentiable, existence of classical  solution satisfying 
initial data was shown in \cite{j1}.
Additionally if we assume $(u_0(x),\sigma_0(x))$
lies on the level set on one of the $j$-Riemann invariants, 
$w_j(u,\sigma)=\sigma +(-1)^j k u$, $j=1, 2$, the system can be reduced to 
the Burgers equation and an explicit formula can be derived for the corresponding 
initial value problem.

\begin{theorem} \label{thm3.1}
 Assume that the initial data $(u_0,\sigma_0)$ is function of bounded variation 
and there exists $c$ a constant such that $w_j(u_0(x),\sigma_0(x))=c$ for all $x$, 
for fixed $j=1$ or $j=2$.

\noindent (a). Then the viscous system \eqref{e3.1}-\eqref{e3.2}, has a solution 
of the form
\begin{equation}
\begin{gathered}
u^\epsilon(x,t)=\frac{\int_{-\infty}^\infty \frac{x-y}{t}
e^{-\frac{1}{2\epsilon}\theta(x,y,t) dy}}{\int_{-\infty}^\infty e^{-\frac{1}{2\epsilon}\theta(x,y,t)} dy} +(-1)^{j+1} k\\
\sigma^\epsilon(x,t)=(-1)^{j+1}k[\frac{\int_{-\infty}^\infty \frac{x-y}{t}
e^{-\frac{1}{2\epsilon}\theta(x,y,t)}}{\int_{-\infty}^\infty e^{\theta(x,y,t)}}+(-1)^{j}k] +c,
\end{gathered}
\label{e3.3}
 \end{equation}
where 
 \begin{equation}
 \theta(x,y,t)=\frac{(x-y-(-1)^j kt)^2}{2 t} +\int_0^y u_0(z) dz.\label{e3.4}
 \end{equation}

\noindent (b). For each fixed $t>0$, 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}