\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 123, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/123\hfil The Riemann problem in gasdynamics]
{The Riemann problem in gasdynamics}
\author[E. D. Conway, S. I. Rosencrans\hfil EJDE-2009/123\hfilneg]
{Edward Daire Conway, Steven I. Rosencrans} % in alphabetical order
\address{Steven I. Rosencrans \newline
Mathematics Department\\
Tulane University\\
New Orleans, LA 70118, USA}
\email{srosenc@tulane.edu}
\thanks{Submitted August 2, 2009. Published September 29, 2009.}
\thanks{Supported in part by grant GP9019 from the National
Science Foundation}
\subjclass[2000]{35L03, 35L65, 35L67, 76L05}
\keywords{Riemann problem; shock wave; rarefaction wave}
\begin{abstract}
In this note we give a proof of the existence of a solution to the
Riemann problem in one-dimensional gasdynamics. Lax's 1957 paper on
conservation laws leaves no
doubt that such a solution exists, but it seems to us that there
may be interest in a brief and explicit proof favorable to
numerical computations. Our procedure also allows us to give a
simple characterization of those problems in which a given wave is
a shock or a rarefaction wave. In the final section we prove
a result of Von Neumann's concerning
the overtaking of two shocks.
This paper was written in 1969 and is being published now at the
suggestion of Jerry Goldstein, whose editorial note is included.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
Editorial note by Jerry Goldstein: This work was done in 1969 and
presented in the PDE Seminar at Tulane, where the two authors and
I were colleagues. I was struck by the beauty and simplicity of
the result, although Ed and Steve felt that the experts understood
that the result was implicitly contained in Peter Lax's earlier
paper \cite{lax}. Later these results were included in Joel
Smoller's book \cite{smoller1983}, but I still felt that the
Conway-Rosencrans global in time analysis of the Riemann problem
in gasdynamics deserved a wider audience. Ed Conway died in 1985.
In March 2009, I met Steve Rosencrans in New Orleans and urged him
again to publish this lovely short note. I am very pleased that
he finally agreed to do so.
\section{Introduction}
In this note we give a proof of the existence of a solution to the
Riemann problem in gasdynamics. Lax's paper \cite{lax} leaves no
doubt that such a solution exists, but it seems to us that there
may be interest in a brief and explicit proof favorable to
numerical computations. Our procedure also allows us to give a
simple characterization of those problems in which a given wave is
a shock or a rarefaction wave. In the final section we prove
a result of Von Neumann's concerning
the overtaking of two shocks.
The proof we give follows the general method given by Lax although
a familiarity with that work will not be needed. In the general
case treated by Lax existence is proved only for data close to a
constant state (see below) but in the special case of gasdynamics,
it is possible to prove a global result.
The Riemann problem is of current interest because of several
recent papers (e.g., \cite{glimm}, \cite{smoller}) considering
weak solutions of conservation laws, in which existence is proved
by approximating the given Cauchy problem by a sequence of Riemann
problems. The Riemann problem is the Cauchy problem for
one-dimensional gasdynamics with the initial data
\[
\mathbf{v}(0,x)=\begin{cases}
\mathbf{v}_r&x\ge0,\\
\mathbf{v}_l&x<0,
\end{cases}
\]
where $\mathbf{v}_r$ and $\mathbf{v}_l$ are any 3D vectors and
generally
\[
\mathbf{v}(t,x) = \begin{pmatrix}
\rho(t,x)\\ p(t,x)\\ u(t,x) \end{pmatrix}
\]
($t\ge0$ and $-\infty 0$)
\item $\mathbf{v}_l$ can be joined to $C(y)\mathbf{v}_l$ on the
right by a contact discontinuity.
\item $\mathbf{v}_l$ can be joined to $F(y)\mathbf{v}_l$ on the
right by a forward-facing shock (if $y < 0$) or a centered
forward-facing rarefaction wave (if $y > 0$).
\end{itemize}
We solve the Riemann problem by finding $y_1,y_2,y_3$ such that
\[
F(y_3)C(y_2)B(y_1)\mathbf{v}_l= \mathbf{v}_r.
\]
The solution then consists of constant states separated by
rarefaction waves, shocks, or contact discontinuities
(see \cite{cf1948}).
In the following sections we determine the maps $B,C,F$. Actually,
they are given almost explicitly in \cite{cf1948},
and we use these results. The choice of the logarithm of the
pressure ratio as a parameter makes the formulas explicit except for
the inversion of one scalar function.
\section{One-parameter family of forward-facing waves}
First we shall derive the set of all states reached from
$\mathbf{v}_l$ by a centered forward-facing rarefaction wave. In
this case $p_r>p_l$, see \cite[Section 81]{cf1948} so we can
write
\[
\frac{p_r}{p_l}=e^y, \quad y\ge0.
\]
Then
\[
\frac{p_r}{p_l}=\Big(\frac{\rho_r}{\rho_l}\Big)^{1/\gamma}
=e^{y/\gamma}
\]
(see \cite[Eqn. 40.10]{cf1948}). Finally \cite[Eqn. 40.09]{cf1948}
implies
\[
\frac{u_r-u_l}{c_l}=\frac{2}{\gamma-1}(e^{\tau y}-1),
\]
where
\[
\tau=\frac{\gamma-1}{2\gamma}
\]
and $c_l$ is the left-hand value of the sound speed.
Now we compute the set of states reached from $\mathbf{v}_l$
by a forward-facing shock. We need the following two equations,
\cite[Eqn. 67.02]{cf1948}
\begin{equation} \label{67.02}
\frac{\rho_1}{\rho_0}=\frac{p_1+\mu^2p_0}{p_0+\mu^2p_1},\quad
\mu^2=\frac{\gamma-1}{\gamma+1}
\end{equation}
and \cite[Eqn. 71.05]{cf1948}
\begin{equation} \label{71.05}
\frac{|u_1-u_0|}{c_0}=\frac{1-\mu^2}{\sqrt{1+\mu^2}}
\frac{p_1-p_0}{p_0}\sqrt{\frac{p_0}{p_1+\mu^2p_0}}
\end{equation}
where the subscript $0$ stands for the state in front of
(i.e., ``before") the shock, and $1$ stands for the state in
back of (i.e., ``after") the shock. In a forward-facing wave,
particles cross the shock from right to left, so $1 = l$ and $0 = r$.
Furthermore in a forward-facing shock
$p_r\le p_l$, so we can write
\[
\frac{p_r}{p_l}=e^x\quad (x\le0).
\]
Then \eqref{67.02} implies
\[
\frac{\rho_r}{\rho_l}=\frac{e^x+\mu^2}{1+\mu^2e^x}\,.
\]
In a forward-facing shock $u_1 \ge u_0$ so \eqref{71.05} implies
\[
\frac{u_r-u_l}{c_l}=-\frac{1-\mu^2}{\sqrt{1+\mu^2}}
\frac{1-e^x}{\sqrt{\mu^2+e^x}}.
\]
The above calculations show that
\[
F(y)\mathbf{v}=\begin{pmatrix}
f_3(y)v_1\\e^yv_2\\v_3+\sqrt{\frac{\gamma
v_2}{v_1}}h_3(y)\end{pmatrix},
\]
where
\[
h_3(y)=
\begin{cases}
\frac{2}{\gamma-1}(e^{\tau y}-1)&y\ge0\\[4pt]
\sqrt{\frac{2}{\gamma(\gamma+1)}}\frac{e^y-1}{\sqrt{\mu^2+e^y}}&y<0
\end{cases}
\]
and
\[
f_3(y)=\begin{cases}
e^{y/\gamma}& y\ge0\\
\frac{\mu^2+e^y}{1+\mu^2e^y}& y<0.
\end{cases}
\]
\section{One-parameter family of backward-facing waves}
The procedure is as in the previous section. The results are
\[
B(y)\mathbf{v}=\begin{pmatrix}f_1(y)v_1\\e^{-y}v_2\\
v_3+\sqrt{\frac{\gamma v_2}{v_1}}h_1(y)\end{pmatrix},
\]
where
\[
f_1(y)=1/f_3(y)\quad -\infty0$,
which by \eqref{e2} is equivalent to
\[
\sqrt{\frac{A_p}{A_\rho}}h_1(\log A_p)< A_u<\frac{2}{\gamma-1}
\Big(1+\sqrt{A_p/A_\rho}\Big)
\]
and is a shock otherwise. Similarly, we find that the forward-facing
wave is a rarefaction wave if and only if
$y_3>0$, which by \eqref{e2} is equivalent to
\[
h_1(-\log A_p) < A_u <\frac{2}{\gamma-1}
\Big(1+\sqrt{A_p/A_\rho}\Big)
\]
and is a shock otherwise. These explicit criteria seem to be new.
\section{Overtaking of two weak forward-facing shock waves}
We now apply the above results to the interaction of two forward-facing
shock waves. This problem has been treated by von Neumann (unpublished).
(However, see related material in \cite{vn}.) He showed that
if $\gamma \le 5/3$, then the resulting configuration after the
waves meet consists of a backward-facing rarefaction wave,
a contact discontinuity, and a forward-facing transmitted shock.
The proof is included in the report \cite{cf1943}.
The criteria derived at the end of Section 6 immediately enable
us to formulate the problem as an inequality. As a simple
application, we prove von Neumann's result for sufficiently
\textbf{weak} interacting shocks: the resulting backward wave
\textbf{is} a rarefaction wave if $\gamma<5/3$ and \textbf{is not}
a rarefaction wave if $\gamma >5/3$.
The problem is formulated as follows. We are given three states,
$\mathbf{v}_l$, $\mathbf{v}_m$, $\mathbf{v}_r$ such that
\begin{gather*}
\mathbf{v}_m=F_s(\mathbf{v}_l)\quad s<0\\
\mathbf{v}_r=F_t(\mathbf{v}_m)\quad t<0
\end{gather*}
and we wish to solve the Riemann problem with initial states
$\mathbf{v}_l$, $\mathbf{v}_r$. From these equations we see
\[
\mathbf{v}_r=F_t(F_s(\mathbf{v}_l))
\]
which implies
\begin{gather*}
A_\rho =f_3(t)f_3(s)\\
A_p =e^{t+s}\\
A_u =h_3(s)+h_3(t)\sqrt{e^s/f_3(s)}.
\end{gather*}
From Section 6 we know that (assuming \eqref{e3}) the backward-facing
wave is a rarefaction wave if and only if
\[
A_u>\sqrt{\frac{A_p}{A_\rho}}h_1(\log A_p).
\]
Using \eqref{e1} we see that this is equivalent to
\begin{equation} \label{e4}
h_1(t+s)