\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 34, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/34\hfil Remarks on vacuum state]
{Remarks on vacuum state and uniqueness of concentration process}
\author[V. G. Danilov\hfil EJDE-2008/34\hfilneg]
{Vladimir G. Danilov}
\address{Moscow Technical University of Communications
and Informatics, Aviamotornaya, 8a, 111024, Moscow, Russia}
\email{danilov@miem.edu.ru}
\thanks{Submitted December 12, 2007. Published March 12, 2008.}
\thanks{Supported by grants 05-01-00912 from RFBR and
436 RUS 13/895/0-1 from DFG Project}
\subjclass[2000]{35L65, 35L67}
\keywords{Zero-pressure gas dynamics system; vacuum state; \hfill\break\indent
delta-shock solution}
\begin{abstract}
We give two examples of nonuniqueness of generalized solutions
of pressureless gas dynamics systems.
In both of these examples, the presence of the Dirac $\delta$-function
leads to nonuniqueness.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction}
In this note, we present two examples of
nonuniqueness of the solution of the pressureless gas dynamics
system. This system has the form
\begin{equation} \label{e1}
\begin{gathered}
\partial_t\rho+\partial_x(\rho u)=0, \\
\partial_t(\rho u)+\partial_x (\rho u^2)=0,
\end{gathered}
\end{equation}
and, as is well known, in the domain, where the solution belongs
to $C^1$, it is equivalent to the system
\begin{equation} \label{e2}
\begin{gathered}
\partial\rho_t + \partial_x(\rho u)=0,\\
\partial_t u +\frac12 \partial_x u^2=0.
\end{gathered}
\end{equation}
However, these systems are quite different if
one considers generalized solutions.
In both cases, the nonuniqueness of the solution originates from
the fact that the initial condition contains the Dirac
$\delta$-function. We also note that the nonuniqueness of the
solution of system \eqref{e1} in the examples presented here
arises because of a quite different mechanism than that found in
\cite[p.~145]{h1}.
This type of nonuniqueness for system \eqref{e1} arises because of
the properties of the conditions posed on the discontinuity curve,
which are analogs of the Rankine--Hugoniot conditions for shock
waves. (These conditions are usually implicit conditions in works
concerning the study of \eqref{e1}, \eqref{e2} in general
functional spaces).
Moreover, our solutions for system \eqref{e1} is an entropy
solution in the sense of \cite[Definition 2]{h1}. Such a solution
must be unique due to the result of \cite{h1}.
Apparently, this can be explained by the fact that there are
different definitions of the generalized solution. In this paper,
we use the definitions given in \cite{d1}. These definitions are direct
analogies of the definitions (in the form of integral identities)
of generalized solutions of conservation laws belonging to
$L^1\cap L^\infty$.
A definition of $\delta$-shock wave type generalized solutions
in the form of integral identities was given in \cite{d1},
where the integral identities were obtained as the limits
of the results obtained by substituting approximate solutions
(weak asymptotic solutions in \cite{d1})
into the original equations. In what follows, we present another
(heuristic) method for obtaining these integral identities.
All of this shows that these definitions based on integral
identities are quite natural. We note that, in the literature,
another approach is well known in the definition of $\delta$-shock
wave type solutions for systems \eqref{e1} and \eqref{e2}, see,
e.g., \cite{b1,c1,h1,y1}.
In these works, the generalized solution of the continuity
equation in systems \eqref{e1} and \eqref{e2} is determined in the
form of an integral identity over the measure determined by the
function $\rho$. Since the Dirac $\delta$-function on the
trajectory of the discontinuity is considered as a term in the
density of this measure and the functions depending on the
velocity $u$ must be integrated, it follows from formal
considerations that the value of $u$ must be determined on the
trajectory of the discontinuity. It is clear that such an approach
cannot be uniquely possible. For example, in system \eqref{e2},
the definition of the generalized solution for the second equation
(for $u$) is in no way related to $\rho$ and it is defined by a
general definition of the $L'\cap L^\infty$ solution of the
conservation law. But it turns out that this definition must take
account of the second equation (about which the first equation
does not know anything).
Apparently, all this originates from the attempts to define
the product $\delta(z)H(z)$ (of the Dirac $\delta$-function
by the Heaviside function), which formally appears
in substituting the $\delta$-shock wave type solution into
the system of equations.
Nevertheless, it is well known that such a definition is not unique.
In \cite{d4}, a new method for constructing integral identities
determining the $\delta$-shock wave type solutions
was proposed.
In this method, we priorly do not assume that there is some fixed
definition of the product $\delta(z)H(z)$. We only assume that the
equation holds in the sense of the space
$\mathcal{D}'(\mathbb{R}^{n+1})$. As a rule, such an assumption in
the case of conservation laws implies a definition of the usual
generalized solution in the form of an integral identity.
Absolutely the same was obtained in \cite{d4}. Moreover, we present
these considerations and construct the corresponding definition
for the system of equations \eqref{e1}, see Sec.~2. Thus, our
remark about the nonuniqueness of the $\delta$-shock wave type
solution to system \eqref{e1} has one more explanation: this
nonuniqueness is related to the nonuniqueness of the definition of
the product of generalized functions. If we fix such a definition,
then, of course, the nonuniqueness disappears. We describe this in
more detail in Sec.~2.
The nonuniqueness for system \eqref{e2} arises when we consider an
unstable step in the initial data for $u$ and the
$\delta$-function for $\rho$ at the point of jump of $u$. This
means that the mass concentrated at the origin of the rarefaction
domain fills the vacuum ``nonuniquely.''
Of course, the examples given below cast a shadow on the physical
consistency of the models related to systems \eqref{e1} and
\eqref{e2}. In any case, one must attentively examine the
conclusions about the real processes obtained using these models.
\section{Results}
\subsubsection*{Definitions of generalized $\delta$-shock wave
type solutions to systems \eqref{e1} and \eqref{e2}}
\begin{definition} \label{def1} \rm
Let $\Gamma=\{\gamma_i,i\in I\}$ be a graph in the
half-plane $\{ x\in \mathbb{R}^1, t\geq0\}$ containing $C^1$ arcs
$\gamma_i$, and let $I$ be a finite set. By $I_0\subset I$ we
denoted the arcs starting from the point $x^0_k\in\mathbb{R}^1$.
A distribution $\rho(x,t)$ and a graph $\Gamma$,
where
\begin{equation} \label{e3}
\begin{gathered}
\rho(x,t)=R(x,t)+E(t)\delta(\Gamma),\quad
E(t)\delta(\Gamma)=\sum_{i\in I} e_i(t)\delta(\gamma_i),
\\
e_i(t)\in C^1(\gamma_i),\quad \gamma_i=\{x=\varphi_i(t)\},
\end{gathered}
\end{equation}
$R(x,t)\in C^1((\mathbb{R}^1\times\mathbb{R}^+)\setminus \Gamma)$
and a function $u=u(x,t)\in L^\infty(\mathbb{R}^1\times\mathbb{R}^+)
\cap C^1((\mathbb{R}^1\times\mathbb{R}^+)\setminus \Gamma)$.
is called a {\it generalized $\delta$-shock wave type solution to}
\eqref{e2} if the integral identities
\begin{gather} \label{e4}
\int^\infty_0\int_{\mathbb{R}^1} (u\zeta_t +\frac12 u^2 \zeta_x)\,dx dt
+\int_{\mathbb{R}^1} (u\zeta)\Big|_{t=0}dx=0,
\\
\label{e5}
\begin{aligned}
&\int^\infty_0 \int_{\mathbb{R}^1} (R\zeta_t+u R\zeta_x)\,dx dt
+\sum_{i\in I} \int_{\gamma_i} e_i(t)\frac{d\zeta}{dt_i}\,dt
\\
&+\int_{\mathbb{R}^1} R\zeta\Big|_{t=0}\,dx +\sum_{k\in
I_0} e_k(0)\zeta(\varphi_k(0),0)=0,
\end{aligned}
\end{gather}
hold for all test functions $\zeta(x,t)\in
\mathcal{D}(\mathbb{R}^1\times\mathbb{R}^1_+)$ and
$\frac{d}{dt_i}=\frac{\partial}{\partial t}+\varphi_{it}
\frac{\partial}{\partial x}$.
\end{definition}
The appearance of the summand
$$
\sum_{i\in I}\int_{\gamma_i}e_i(t)\frac{d\zeta}{dt_i}\,dt
$$
in \eqref{e5} can easily be explained. Indeed, let $\rho$ have the form \eqref{e3}, then
differentiating in $t$, we obtain (see \cite{d4})
$$
\rho_t=\sum_{i\in I}e_i(t)(-\varphi_{it})\delta'(\gamma_i)
+ \text{smoother summands}.
$$
Hence it is clear that we must have
\begin{equation} \label{e6}
(\rho u)_x=-\sum_{i\in I}e_i(-\varphi_{it})\delta'(\gamma_i)
+ \text{smoother summands}.
\end{equation}
Now, for any test function $\zeta(x,t)$ such that $\zeta(x,0)=0$, we have
\begin{align*}
\langle\rho_t+(u\rho)_x,\zeta\rangle
&=-\langle\rho,\zeta_t(x,t)\rangle-\langle\rho u,\zeta_x(x,t)\rangle
\\
&=-\langle R,\zeta_t(x,t)\rangle
-\langle E(t)\delta(\Gamma),\zeta_t(x,t)\rangle
-\langle Ru,\zeta_x\rangle
\\
&\quad
+\sum_{i\in I}\langle e_i(t)(-\varphi_{it})\delta(\gamma_i),\zeta_x(x,t)\rangle
\\
&=-\langle R,\zeta_t\rangle-\langle Ru,\zeta_x\rangle
-\sum_{i\in I}\int_{\gamma_i} e_i(t)(\zeta_t+\varphi_{it}\zeta_x)\,dt.
\end{align*}
Here $\langle,\zeta\rangle$ denotes the action
of a generalized function on a test function~$\zeta$.
Of course, these calculations are not a proof, this is only a motivation.
Definition~1 gives a method for calculating the functions contained
in \eqref{e3}.
Suppose that
\begin{equation} \label{e7}
u=u_0+\sum_{i\in I}H(x-\varphi_i) u_i,
\end{equation}
where $\varphi_i(t)$, $u_0(x,t)$, and $u_i(x,t)$ are smooth functions
(i.e., the velocities have jumps on the curves $x=\varphi_i$).
Then, integrating by parts, we obtain
\begin{align*}
&\int^\infty_0 \int_{\mathbb{R}^1\setminus \bigcup\{x=\varphi_i\}}
(R_t+(uR)_x)\zeta\,dx dt
\\
&-\sum_{i\in I}\int_{x=\varphi_i}
\{[R]\varphi_{it}-[uR]\}\zeta\,dt +\sum_{i\in I}\int_{\gamma^i}
e_{it}\zeta\,dt=0,
\end{align*}
where $[g]$ is a jump of the function $g$ across
the discontinuity curve $x=\varphi_i(t)$,
$[g]=g(\varphi_i+0)-g(\varphi_i-0)$.
This and \eqref{e4} imply the system of equations
\begin{gather} \label{e8}
\begin{gathered}
u_t+\frac12 (u^2)_x=0, \\
R_t+(uR)_x=0\quad (x,t)\in \mathbb{R}^1\times
\mathbb{R}^1_+\setminus \bigcup_{i\in I}\{x=\varphi_i\},
\end{gathered}
\\ \label{e9}
\begin{gathered}
\varphi_{it}=u(\varphi_i+0,t)+u(\varphi_i-0,t)
=\frac12\frac{[u^2]}{[u]}\Big|_{x=\varphi_i}, \\
e_{it}=\varphi_{it}[R]|_{x=\varphi_{i}}-[uR]|_{x=\varphi_i},\quad
i\in I.
\end{gathered}
\end{gather}
The signs of the summands in \eqref{e9} differ from the signs
of the similar summands in \cite{d1}, since the jumps on the curves
$x=\varphi_i$ are defined in different ways.
Systems \eqref{e8}, \eqref{e9} and \eqref{e15}, \eqref{e16} are,
in fact, known, see \cite{b1,y1}.
In this case, at the nodes of the graph $\Gamma$ lying above the
axis $\{t=0\}$, the following ``Kirchhoff laws'' must be
satisfied:
\begin{equation} \label{e10}
\sum_{i\in \mathop{\rm In}
A_k}e_i(t^*_k-0)=\sum_{i\in\mathop{\rm Out} A_k}e_i(t^*_k+0),
\end{equation}
where $\mathop{\rm In}$ and $\mathop{\rm Out}$ are the respective sets of incoming and
outgoing arcs associated with a certain node $A_k=(x_k,t^*_k)$.
For system \eqref{e1}, we have the definition of the solution in
the following form.
\begin{definition} \label{def2} \rm
Let $\Gamma=\{\gamma_i,i\in I\}$ be a graph in the half-plane
$\{x\in\mathbb{R}^1, t\geq0\}$ containing $C^1$ arcs $\gamma_i$,
and let $I$ be a finite set.
By $I_0\subset I$ we denoted the arcs starting from the point
$x^0_k\in\mathbb{R}^1$. A functions
$u=u(x,t)\in L^\infty(\mathbb{R}^1\times\mathbb{R}^+)
\cap C^1((\mathbb{R}^1\times\mathbb{R}^+)\setminus \Gamma)$,
a distribution $\rho(x,t)$, and a graph $\Gamma$, where
\begin{equation} \label{e11}
\begin{gathered}
\rho(x,t)=R(x,t)+E(t)\delta(\Gamma),\quad
E(t)\delta(\Gamma)=\sum_{i\in I} e_i(t)\delta(\gamma_i), \\
e_i(t)\in C^1(\gamma_i),\quad \gamma_i=\{x=\varphi_i(t)\}\,.
\end{gathered}
\end{equation}
The function
$R(x,t)\in C^1((\mathbb{R}^1\times\mathbb{R}^+)\setminus \Gamma)$
is called a {\it generalized $\delta$-shock wave type solution}
of \eqref{e1} if the integral equalities
\begin{gather} \label{e12}
\begin{aligned}
&\int^\infty_0 \int_{\mathbb{R}^1} (Ru\zeta_t+Ru^2\zeta_x)\,dx dt
+\sum_{i\in I} \int_{\gamma_i}\varphi_{it} e_i(t)\frac{d\zeta}{dt_i}\,dt
\\
& +\int_{\mathbb{R}^1} (R u\zeta)\Big|_{t=0}\,dx +\sum_{k\in
I_0}\varphi_{kt}(0) e_k(0)\zeta(\varphi_k(0),0)=0,
\end{aligned}
\\ \label{e13}
\begin{aligned}
&\int^\infty_0 \int_{\mathbb{R}^1} (R\zeta_t+uR\zeta_x)\,dx dt +\sum_{i\in
I} \int_{\gamma_i} e_i(t)\frac{d\zeta}{dt_i}\,dt
\\
&+\int_{\mathbb{R}^1} R\zeta\Big|_{t=0}\,dx +\sum_{k\in
I_0}e_k(0)\zeta(\varphi_k(0),0)=0,
\end{aligned}
\end{gather}
hold for all test functions
$\zeta(x,t)\in \mathcal{D}(\mathbb{R}^1\times\mathbb{R}^1_+)$
and $\frac{d}{dt_i}=\frac{\partial}{\partial t}+\varphi_{it}
\frac{\partial}{\partial x}$.
\end{definition}
Relation \eqref{e13} coincides exactly with relation \eqref{e5}.
The second summand in the left-hand side of \eqref{e12} can also
be easily explained
as well as the corresponding summand in \eqref{e11}. Indeed, in view of
\eqref{e6}, we have
\begin{equation} \label{e14}
(\rho u^2)_x=\sum_{i\in I}
e_i\varphi^{2}_{it}\delta'(\gamma_i) + \text{smoother summands}.
\end{equation}
Now, just as above, for any test function $\zeta(x,t)$ such that
$\zeta(x,0)=0$, we have
\begin{align*}
&\langle (\rho u)_t+(\rho u^2)_x,\zeta\rangle =-\langle \rho u,\zeta_t
\rangle-\langle \rho u^2,\zeta_x\rangle
\\
& =-\langle Ru,\zeta_t\rangle -\sum_{i\in I}e_i\varphi_{it}
\langle\delta(\gamma_i),\zeta_t\rangle -\langle
Ru^2,\zeta_x\rangle -\sum_{i\in I} e_i(\varphi_{it})^2
\langle\delta(\gamma_i),\zeta_x\rangle
\\
& =-\langle Ru,\zeta_t\rangle-\langle Ru^2,\zeta_x\rangle
-\sum_{i\in I} \int_{\gamma_i} e_i \varphi_{it}
(\zeta_t+\varphi_{it}\zeta_x)\,dt.
\end{align*}
As in Definition \ref{def1}, Definition \ref{def2} leads to a
system of equations for the
unknown functions $u$, $R$, $e_i$, $\varphi_i$ contained
in \eqref{e12}, \eqref{e13}:
\begin{gather}
\begin{gathered}
(R u)_t+(Ru^2)_x=0, \\
R_t+(Ru)_x=0,\quad (x,t)\in (R^1\times R^+_1)\setminus
\bigcup\{x=\varphi_i\},
\end{gathered} \label{e15}
\\
\begin{gathered}
(e_i\varphi_{it})'_t =\varphi_{it}[R u]\Big|_{x=\varphi_i}
-[R u^2]\Big|_{x=\varphi_i}, \\
e_{it}=\varphi_{it}[R]\Big|_{x=\varphi_i} -[R
u]\Big|_{x=\varphi_i},\quad i\in I.
\end{gathered} \label{e16}
\end{gather}
Here an analog of the two ``Kirchhoff laws'' is given by the equations
\begin{equation} \label{e17}
\begin{gathered}
\sum_{i\in \mathop{\rm In} A_k} e_i(t^*_k-0) = \sum_{i\in \mathop{\rm Out} A_k} e_i(t^*_k+0),
\\
\sum_{i\in \mathop{\rm In} A_k} e_i(t^*_k-0)\varphi_{it}(t^*_k-0)
=\sum_{i\in \mathop{\rm Out} A_k} e_i(t^*_k+0)\varphi_{it}(t^*_k+0).
\end{gathered}
\end{equation}
Obviously, a significant distinction of system \eqref{e15} from Eqs.~\eqref{e9} is that
system \eqref{e15} consists of second-order equations and, formally, to solve this
system, it is required to know the values $e_i(0)$, $\varphi_i(0)$, and
$\varphi_{it}(0)$ (!). Obviously, these values cannot be found from the
initial conditions for the original problem (we discuss this later in more
detail).
Thus, we have the following theorem.
\begin{theorem} \label{thm1}
Suppose that system \eqref{e8}--\eqref{e10} (\eqref{e15}--\eqref{e17})
for $t\in[0,T]$ has a classical solution. Then system \eqref{e2}
(respectively, \eqref{e1}) has a generalized
$\delta$-shock wave type solution in the sense of
Definition \ref{def1} (Definition \ref{def2}).
\end{theorem}
Thus, just as in the case of classical shock waves, constructing
generalized $\delta$-shock type solutions is reduced to solving
a system of ordinary differential equations \cite{m1,m2,m3}.
The existence of the solution, for example, in the case of piecewise
constant initial functions $u|_{t=0}$ and $\rho|_{t=0}$,
can be proved easily.
As is easy to see, the solutions to systems \eqref{e1} and
\eqref{e2} in the sense of the definitions given above satisfy the
conservation laws in the following form.
Suppose that there exists a number $A$ such that
$$
\langle \rho(x,t),\eta(x)\rangle=0,\quad t\in [0,T],
$$
for any test function $\eta(x)$, $\sup
\eta(x)\in\mathbb{R}^1\setminus[-A,A]$, where
$\langle\rho,\eta\rangle$ denotes the action of a generalized
function $\rho(x,t)$ on the the test function $\eta(x)$, $t$ is a
parameter, and $\rho(x,t)$ is a component of the solution to
system \eqref{e1} or \eqref{e2} in the sense of the above
definitions constructed using the solutions of systems
\eqref{e15}--\eqref{e17} or \eqref{e8}--\eqref{e10}.
\begin{lemma} \label{lem1}
For any test function $\zeta(x)$, $\zeta(x)=1$ for $x\in[-A,A]$,
$t\in[0,T]$, the following relation holds:
$$
\langle \rho(x,t),\zeta(x)\rangle
=\langle\rho(x,0),\zeta(x)\rangle.
$$
\end{lemma}
For system \eqref{e1}, we can formulate one more conservation law.
\begin{lemma} \label{lem2}
The following relation holds:
$$
\langle \rho u,\zeta(x)\rangle
=\langle\rho u|_{t=0},\zeta(x)\rangle.
$$
where $t\in[0,T]$, $\zeta$ is a test function satisfying the
assumption of Lemma~{\rm1}, $\rho$ and $u$ are solutions of
system \eqref{e1} in the sense of Definition \ref{def2}
constructed using solutions of system \eqref{e15}--\eqref{e17}.
\end{lemma}
The proof of Lemma~1 can be found in \cite{d2}.
Here we only prove Lemma~2 whose proof is similar to
the proof of Lemma~1.
For simplicity, we consider the case in which the graph $\Gamma$
contains a single arc $x=\varphi(t)$.
Then we have
\begin{align*}
\frac{d}{dt}\langle \rho u,\zeta\rangle
&=\frac{d}{dt}\int_{\mathbb{R}^1} Ru\,dt +(e\varphi_t)_t
\\
&=-\varphi_t[Ru]\Big|_{x=\varphi}
+\int^{\varphi}_{-\infty}(Ru)_t\,dx
+\int^{\infty}_{\varphi}(Ru)_t\,dx
+(e\varphi'_t)'_t
\\
&=-\int^{\varphi}_{-\infty}(Ru^2)_x\,dx
-\int_{\varphi}^{\infty}(Ru^2)_x\,dx
+(e\varphi_t)_t
-\varphi_t[Ru]\Big|_{x=\varphi}
\\
&=[Ru^2]\Big|_{x=\varphi} -\varphi_t[Ru]\Big|_{x=\varphi}
+(e\varphi_t)_t=0.
\end{align*}
The last equality is precisely the first equation in \eqref{e16}.
Now it is natural to pose the problem of the uniqueness of the solution.
\subsubsection*{Examples of nonuniqueness.}
In what follows, we give an answer to this question
about the uniqueness in the form of examples. The
general answer is the following: the solution may be nonunique if
the initial conditions for $\rho$ contain an atomic measure
(the Dirac $\delta$-function).
In particular, the solution of the Cauchy problem to system
\eqref{e2} is constructed in \cite{c1} in the case where the initial
profile of velocity is an unstable step function. To construct the
second component $\rho$ of the solution, i.e., to solve the
continuity equation, the authors \cite{c1} choose a class of
functions invariant under the scaling transformation
$$
x\to kx,\quad t\to kt.
$$
Indeed, the group of scaling transformations acts on
the solution of the system considered.
But these are particular solutions. For example, in \cite{r1},
such solutions are considered in a quite different context.
In \cite{c1},
the statement that a vacuum domain exists is derived
from the assumption that such invariant solutions
are unique. Such a statement
cannot be made based only on the consideration
of {\it particular} solutions.
The following natural question arises:
Can solutions
that are not contained in the class of solutions
invariant under the action of the scaling group
help to fill a vacuum?
More generally, the question can be formulated as follows:
Do there exist any natural conditions ensuring the uniqueness
of the Goursat problem solutions considered in \cite{c1}.
In this small note, we give an affirmative answer to this
question. Namely, for any initial regular distribution $\rho$
with compact support (perhaps, with a first kind discontinuity),
the solution of the Goursat problem is zero
in the rarefaction domain.
In our considerations, we do not use the regularization procedure,
which uniformly approximates the solution of the Cauchy problem
for~\eqref{e1}. This can be done using the simple formulas
from \cite{d3}, but the problem is very simple and does not
require any special {\it technical} methods.
So the solution of the Cauchy problem for \eqref{e2} with the
initial conditions
\begin{equation} \label{e18}%18
\begin{aligned}
u|_{t=0}&=\begin{cases}
u_l, &xx_0,\end{cases}
\quad u_{l,r}=\mbox{const},\quad u_l< u_r,
\\
\rho|_{t=0}&=\begin{cases}
\rho_l, &xx_0,\end{cases}
\quad \rho_{l,r}\geq 0,
\end{aligned}
\end{equation}
for $t>0$ has the form
\begin{equation} \label{e19}%3
u=\begin{cases}
u_l, &x x_0+u_rt,
\end{cases}
\end{equation}
and, respectively,
\begin{equation} \label{e20}%4
\rho=\begin{cases}
\rho_l(x-u_lt), &x x_0+u_rt,
\end{cases}
\end{equation}
where $\rho_0=\rho_0(z)$ is an arbitrary $C^1$-function.
Formulas \eqref{e19}, \eqref{e20} can be verified by a direct
substitution.
We only note that since the function $u$ in \eqref{e19}
is continuous for $t>0$,
the Rankine-Hugoniot type conditions are identically satisfied
on the lines $x=x_0+u_j t$, $j=l,r$, because the equation
for $\rho$ is linear in~$\rho$.
Calculating the integral $\int_{R^1}\rho(x,t)\,dt$
for $t>0$ ($\rho(x,t)$ is defined in \eqref{e20}), we obtain
$$
\int_{R^1} \rho(x,t)\,dx=\int_{xx_0} \rho_r\,dx+\int^{u_r}_{u_l} \rho_0(z)\,dz.
$$
Hence, since $\rho|_{t=0}$ is nonnegative and
$\langle\rho,\zeta\rangle$ is preserved by Lemma~1, we have
$$
\int^{u_r}_{u_l} \rho_0(z)\,dz=0.
$$
Otherwise, we come to a contradiction, because from \eqref{e18} and the
mass conservation law we must have
$$
\langle \rho|_{t=0},\zeta\rangle =\int_{xx_0}\rho_r\,dx
=\int_{xx_0} \rho_r\,dx+\int^{u_r}_{u_l} \rho_0(z)\,dz.
$$
We point out that we derived this relation without any assumptions
on the properties of {\it particular} solutions to
system~\eqref{e2}. We also note that a (more general than that
in \cite{c1}) assumption ensuring the uniqueness of the solution of
the Goursat problem in the case under study could be the
assumption that $\rho$ is bounded. However, if simultaneously with
the rarefaction wave we consider shock waves in the $u$-component,
then $\delta$-shock type solutions arise, which is prohibited by
the boundedness condition.
But if the initial condition for $\rho$ is replaced
by the condition
$$
\rho|_{t=0}=\rho_l H(x_0-x)+\rho_r H(x-x_0)
+\hat\rho \delta(x-x_0),
$$
then the choice of the function $\rho_0$ in \eqref{e20} is restricted
only by the condition
$$
\int^{u_r}_{u_l}\rho_0(z)\,dz=\hat\rho
$$
and the solution of this ``singular'' Goursat problem
is not unique.
\subsubsection*{Nonuniqueness of the Cauchy problem solution
in the case of system~\eqref{e1}}
It is proved in \cite{d1} that for system \eqref{e1} to have a
solution of the form
\begin{equation} \label{e21}
\begin{gathered}
u=u_0(x,t)+u_1(x,t) H(\varphi(t)-x),\\
\rho =\rho_0(x,t)+\rho_1(x,t)
H(\varphi(t)-x)+e(t)\delta(x-\varphi(t))
\end{gathered}
\end{equation}
in the sense of the integral identity introduced in \cite{d1}
(also see Definition~2 at the beginning),
it is necessary that besides of other relations
the following equations must be satisfied
\begin{equation} \label{e22}
e_t(t)=-([u\rho]-[\rho]\varphi_t)|_{x=\varphi}, \quad
\frac{d}{dt}(e\varphi_t)+([u^2\rho]-[u\rho]\varphi_t)|_{x=\varphi}=0,
\end{equation}
where $[f]|_{x=\varphi}=f(\varphi(t)+0)-f(\varphi(t)-0)$
as above.
We restrict ourselves to considering the case
\begin{equation} \label{e23}
u_0=u_1=\text{const},\quad u_0<0,\quad u_0+u_1>0,\quad
\rho_0=\rho_1=\text{const}\geq0.
\end{equation}
Obviously, to construct the solution in this case, it suffices to
construct the solution to system \eqref{e2}, which in this case is
a system with constant coefficients.
We note that if conditions \eqref{e22} are satisfied,
then the inequality
$$
\frac{u(x_2,t)-u(x_1,t)}{x_2-x_1}\leq 0
$$
holds for any small $x_1,x_2$ and hence the condition
(see \cite[p.~119]{h1})
$$
\frac{u(x_2,t)-u(x_1,t)}{x_2-x_1}\leq \frac1t
$$
that the solution is an entropy solution
is also satisfied.
Next, it follows from Definition \ref{def2} that
$$
\rho u^2=\rho_r u^2_r
+H(\varphi(t)-x)(\rho_1 u^2_l-\rho_r u^2_r)
+e(\varphi_t)^2\delta(x-\varphi(t))
$$
and $\rho u^2$ weakly converges to its initial value
under the assumption that the functions $e(t)$ and $\varphi_t(t)$
are continuous for $t\geq0$.
This readily follows from the formulas for the solution of
system \eqref{e22} given below.
It is easy to see that \eqref{e22} form
a second-order system of equations for $e,\,\varphi$.
The original system is a first-order system, hence the value
$\varphi_t(0)$ remains undetermined.
In \cite{y1}, it is shown that, in the case of constant
$u_i,\,\rho_i$, $i=1,2$, system \eqref{e5} has a unique solution
if $e(0)=0$. In this case, the solution is independent
of $\varphi_t(0)$.
The formula for $\varphi(t)$ obtained in
\cite[Theorem 4.3]{y1} has the form
$$
\varphi(t)= \frac{e(0)+[u\rho]t
-\sqrt{e(0)^2+2e(0)e_t(0)t+\rho_r\rho_l(u_r -u_l)^2t^2}}{[\rho]},
\quad [\rho]\ne0,
$$
It follows from the first equation in \eqref{e22} that
the values of the constants $e_t(0)$ and $\varphi_t(0)$
can be expressed linearly in terms of each other
$$
e_t(0)=[u\rho]^0-[\rho]^0\varphi_t(0),
$$
where $[\,\,]^0 = [\,\,]\big|_{t=0}$ and hence the quantities
$[\,\,]^0$ are functions of the argument $\varphi(0)$.
From these formulas it is easily seen that,
in the case $e(0)=0$, the expression $\varphi_t(0)$ ($e_t(0)$)
is not contained in the formula for $\varphi(t)$.
Indeed, for $e(0)=0$ and $[\rho]\ne0$, relations~\eqref{e22} imply
the following equation for $\varphi_t(0)$:
$$
\varphi_t(0)^2 [\rho]^0-2\varphi_t(0) [u\rho]^0+[u^2\rho]^0=0.
$$
Solving this equation under the additional condition
$u_r|_{t=0}<\dot\varphi(0)< u_l|_{t=0}$,
which is necessary for the existence of the desired
$\delta$-shock type solutions, we obtain
$$
\varphi_t(0)=\Big([u\rho]^0
-\sqrt{([u\rho]^0)^2-[\rho]^0[u^2\rho]^0}\Big) ([\rho]^0)^{-1}
=: G(\varphi(0)).
$$
Thus, in the case $e(0)=0$, the missing constant
is determined by the natural initial data of the problem.
It is also easy to verify that $e_t(0)>0$
in this case.
Hence from the second equation in (21) we obtain
$|\frac{d^2\varphi}{dt^2}(0)|<\infty$.
Therefore, although the coefficient
of the second derivative $\frac{d^2\varphi}{dt^2}$
vanishes for $t=0$,
system \eqref{e22} has a smooth solution at least in the small
in~$t$. This can be proved as usual, by reducing
the problem to an integral equation.
The case $\rho_1|_{t=0}=-[\rho]^0=0$ is considered similarly
(see \cite{y1}) for $u_i,\rho_i=\text{const}$.
In this case,
$$
e(t)=e(0)-t[\rho u]=e(0)-t\rho[u]
$$
and the problem is reduced to solving the ordinary differential equation
$$
e_t(t)\varphi_t +e\varphi_{tt} =\varphi_t\rho[u]-\rho[u^2].
$$
Hence we obtain
$$
\varphi_t=\frac{[u^2]}{2[u]}
+\Big(\varphi_t(0)-\frac{[u^2]}{2[u]}\Big)
\frac{e(0)}{(e(0)-t\rho[u])^2}.
$$
It is clear that for $e(0)=0$,
the solution is independent of the constant $\varphi_t(0)$,
which cannot be determined from the Cauchy conditions.
We note once more that (23) implies that $[u]=-u_1<0$.
We note that if we use the construction of the nonconservative
Volpert-Khudaev product and the definition of the measure
solution that follows from this construction, then we, as was
already noted in the Introduction, fix the definition of the
product $\delta(z)H(z)$. In this case, this means that we set
$u|_{x=\varphi}=\varphi_t$ and, in particular,
$\varphi_t(0)=u|_{\substack{x=\varphi\\ t=0}}$. Thus, the last
term in the first integral identity in Definition~3 seems to be
determined by specifying the initial velocity and the solution of
the entire system \eqref{e1} is determined by specifying two
initial conditions (for the velocity and density) in the form
(21). But, as was already noted, this is a delusion: such a method
for specifying the initial conditions is not necessary (unique).
Thus, we see that the ``singular'' Cauchy problem for
system \eqref{e1} does not have the property that the solution is
unique and the problem whose initial conditions do not contain the
Dirac function has a unique solution.
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\end{document}