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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
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\def\rightheadline{EJDE--2000/72\hfil 
Riemann solvers for Van der Waals fluids \hfil\folio}
\def\leftheadline{\folio\hfil Philippe G. LeFloch \&  Mai Duc Thanh
 \hfil EJDE--2000/72}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.~{\eightbf 2000}(2000), No.~72, pp.~1--19.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title
Nonclassical Riemann solvers\\
and kinetic relations III:  
A nonconvex hyperbolic model for Van der Waals fluids
\endtitle

\thanks 
{\it 2000 Mathematics Subject Classifications:} 35L65, 76N10, 76L05.\hfil\break\indent
{\it Key words:} compressible fluid dynamics, phase transitions, 
Van der Waals, entropy inequality,  \hfil\break\indent
hyperbolic conservation law, kinetic relation, nonclassical solutions, Riemann solver. 
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted June 7, 2000. Published December 1, 2000. 
\hfil\break\indent
P.L.F. was supported in part by the Centre National de la Recherche 
Scientifique.
\hfil\break\indent
This work was done while M.D.T. was visiting the Ecole Polytechnique 
(1999-2000) and was
\hfil\break\indent
partially supported by the French-Vietnamese Institute ``ForMathVietnam".
\endthanks

\author Philippe G. LeFloch \&  Mai Duc Thanh  \endauthor

 \address 
 Philippe G. LeFloch \hfill\break
 Centre de Math\'ematiques Appliqu\'ees \& 
 Centre National de la Recherche Scientifique, U.M.R. 7641, 
 Ecole Polytechnique, 91128 Palaiseau Cedex, France. 
 \endaddress
 \email lefloch\@cmapx.polytechnique.fr
 \endemail

 \address 
 Mai Duc Thanh \hfill\break
 Hanoi Institute of Mathematics, P.O. Box 631 BoHo, 
 10.000 Hanoi, Vietnam.
 \endaddress
 \email mdthanh\@hanimath.ac.vn, 
 thanh\@cmapx.polytechnique.fr
 \endemail

\abstract
This paper deals with the so-called $p$-system 
describing the dynamics of isothermal and compressible 
fluids. The constitutive equation is assumed to have the 
typical convexity/concavity properties of the van der Waals equation.
We search for discontinuous solutions 
constrained by the associated mathematical entropy inequality. 
First, following a strategy proposed by Abeyaratne and Knowles
and by Hayes and LeFloch, we describe here the whole family of 
{\it nonclassical Riemann solutions\/} for this model. 
Second, we supplement the set of equations with a {\it kinetic relation\/} 
for the propagation of nonclassical undercompressive shocks, 
and we arrive at a uniquely defined solution of the Riemann problem. 
We also prove that the solutions depend $L^1$-continuously upon their data. 
The main novelty of the present paper is the presence of
{\it  two inflection points\/} in the constitutive equation.  
The Riemann solver constructed here is relevant 
for fluids in which viscosity and capillarity effects are kept 
in balance. 
\endabstract
\endtopmatter

\document

 
\heading{1. Introduction} \endheading

We consider the Riemann problem for a compressible and isothermal fluid
described by the following two conservation laws of mass and momentum: 
$$
\aligned 
&\partial _t u + \partial _x p(v) = 0, \\
&\partial _t v - \partial _x u = 0.
\endaligned
\tag 1.1
$$
Here $v>0$ and $u$ denote the specific volume and the velocity of the fluid, 
respectively, while the pressure $p=p(v)$ is a given smooth function depending
on the fluid under consideration. The initial datum has the form: 
$$
(u,v)(x,0)  = \cases
  (u_l, v_l) & \text{ for } x < 0, \\
  (u_r, v_r) & \text{ for } x > 0, \\
\endcases
\tag 1.2
$$
where $(u_l, v_l)$ and $(u_r, v_r)$ are constants. 
In typical models of (liquid-vapor) phase transformation, 
the pressure $p$ admits two inflection points and tends to $+\infty$ at $v=0$. 
That is, for some constants $a$ and $b$, we have 
$$ 
\aligned
& p''(v) > 0  \quad \text{ for } v \in (0,a)\cup (b,+\infty),\\ 
& p''(v) < 0  \quad \text{ for } v \in (a,b),\\
& p'(a) < 0,\\
& \lim_{v \to 0} p(v) = +\infty,\quad \lim_{v \to +\infty} p(v) \geq 0.
\endaligned
\tag 1.3
$$
As a consequence, the first derivative $p'$ attains a maximum value at $v=a$ and, 
since $p'(a)<0$, 
$$
p'(v) < 0 \quad \text{ for } v>0.
$$
Of course, the case where $v$ is restricted to remain above some threshold $v_*$
is covered also by the theory in this paper, provides one 
changes $v$ into $v-v_*$. 


The system (1.1) 
under consideration has the general form of a system of conservation laws, 
$$
\partial _t U + \partial _x F(U) = 0,  \qquad U: = (u,v), \quad F(U) = \bigl(p(v),-u\bigr). 
\tag 1.4
$$
Since $p'<0$, the matrix $DF(U)$ admits two real and distinct eigenvalues,  
depending only on $v$, 
$$
\lambda_1(v):= -\sqrt {-p'(v)} < 0 < \sqrt{-p'(v)}:= \lambda_2(v).
$$  
Therefore, (1.1) is strictly hyperbolic. 
Setting $c(v):= \sqrt{-p'(v)}$, which is called the sound speed, 
right-eigenvectors of $DF(U)$ may be chosen to be 
$r_1(v): = (c(v),1)$ and $r_2(v): = (-c(v),1)$ . 


As is customarily, all of the weak solutions of the system (1.1) are required 
to fullfil the following entropy inequality 
$$ 
\aligned 
&  \partial _t {\Cal U}(u,v) + \partial _x  {\Cal F}(u,v) \leq 0,\\
&  {\Cal U}(u,v):= {u^2\over 2} + \Sigma(v), \quad {\Cal F}(u,v)= u \, p(v), \\
&  \Sigma(v):= -\int_0^v p (w) \, dw,  
\endaligned 
\tag 1.5   
$$
where $(U,F)$ is a mathematical entropy-entropy flux pair for the system of conservation laws 
(1.1) (Lax \cite{11}). Under the assumption (1.3), the entropy $U$ is strictly convex in 
the conservative variables $(u,v)$. 

The present paper is based on recent work by Abeyaratne and Knowles \cite{1, 2}, 
LeFloch et al. \cite{8--10, 12--14}, 
and Shearer et al. \cite{18, 19} on nonclassical undercompressive shock waves 
of hyperbolic and hyperbolic-elliptic systems of conservation laws. 
We also rely on earlier contributions 
on propagating phase boundaries in van der Waals fluids, 
especially the pioneering work by Slemrod \cite{20--22} 
and the papers \cite{3--7}. 

First of all, in Section 2, we provide a precise description of the 
set of {\it all Riemann solutions\/} consistent with the two conservation laws (1.1) and 
the entropy inequality (1.5). In Section 3, 
we recall that the Riemann problem admits a unique (classical) solution 
characterized by the so-called Liu entropy criterion \cite{17}. 
This is the solution usually described in the engineering literature. 
However the solutions generated by zero viscosity-capillarity limits associated with 
the system (1.1) do not coincide with the (classical) Riemann solution. 
Therefore, in Section 4, we construct solutions that only satisfy the entropy ionequality (1.5). 
For the sake of uniqueness, it is known that the so-called {\it kinetic relation\/} should be added. 
Our main result (Theorem 4.3) establishes the existence and uniqueness of the weak 
solution of the Riemann problem (1.1)-(1.2)-(1.5) satisfying a prescribed kinetic relation. 
This represents an extension of previous results by the authors 
\cite{15-16} on nonclassical Riemann solvers 
and kinetic relation. Comparing with our earlier study \cite{15} 
of a nonconvex hyperbolic model of elastodynamics, 
the major novelty is the existence of two inflexion points in the equation 
of state (1.3), which significantly complicates the analysis of the Riemann 
problem. 


%==============================
\heading{2. Entropy Dissipation Function} 
\endheading

We are going to investigate the properties of the entropy dissipation function associated 
with the entropy inequality (1.5). First, we need to point out basic properties 
of the pressure function and introduce some useful notation. Virtually all of the properties 
stated in this section can be checked {\it geometrically\/}
from the graph of the function $p$. 
In the following we consider points on this graph and refer to them 
simply by their $v$-coordinate. 

We rely here on the assumptions (1.3) made on the pressure function. 
In the interval $(a,b)$, the function $p$ is concave, and thus remains 
above its tangent at the inflection point $b$. 
This tangent intersects the graph of $p$ at some other point, outside the interval 
$(a,b)$, whose coordinate will be denoted by $b^{-\natural} < a$. 
Similarly
the tangent to the curve at the other inflection point $a$ 
also intersects the graph of $p$ at some point $a^{-\natural} > b$. 
(This notation will become clear as we will introduce shortly 
some functions $\varphi^{-\natural}$ and $\psi^{-\natural}$.) 


%____________________________________________________________________________ 
\midinsert
\centerline{\psfig{file=fig21.eps,width=10.5truecm}}
\botcaption{Figure 2.1: Pressure function.}
\endcaption
\endinsert
%____________________________________________________________________________ 


Geometrically on the graph of $p$, we see that 
for any $v\in (b^{-\natural}, a^{-\natural})$ there exists exactly two
lines which are passing through the point with the coordinate $v$ and 
are also tangent to the graph. Call these two points $\psi^\natural(v)$ and $\varphi^\natural(v)$
with the convention that $\varphi^\natural(v) < \psi^\natural(v)$. In other words we have 
$$ 
\aligned
p'\big(\varphi^\natural(v)\big) &= {p(v) - p\big(\varphi^\natural(v)\big) \over v -\varphi^\natural(v)}, \\
p'\big(\psi^\natural(v)\big) &= {p(v) - p\big(\psi^\natural(v)\big) \over v -\psi^\natural(v)}. 
\endaligned
\tag 2.1
$$
The definition extends to the end points of the interval 
under consideration by setting  
$$
\varphi^\natural(b^{-\natural}) = \psi^\natural(b^{-\natural}) = b \quad \text{ and } \quad 
\varphi^\natural(a^{-\natural}) = \psi^\natural(a^{-\natural}) = a.
$$ 
No tangent can be draw from a point outside the interval $[b^{-\natural},a^{-\natural}]$ as the function 
$p$ ``resembles'' a convex function in that region. 
The two tangent functions $\varphi^\natural$ and $\psi^\natural: [b^{-\natural},a^{-\natural}]\to {\Bbb R}$
are going to play a central role in the forthcoming constructions in Sections 3 and 4. 

The following properties are elementary: 


\proclaim{Proposition 2.1} 
\roster
\item"(i)"
The values $v$ and $\psi^\natural(v)$ always lie on different sides with respect to $b$, 
and the values $v$ and $\varphi^\natural(v)$ always lie on different sides with respect to $a$, 
in the sense that: 
$$
\aligned 
(\varphi^\natural(v)-a)(v-a) < 0 \quad \text{ for } v \ne a, \qquad \varphi^\natural(a)=a,\\
(\psi^\natural(v)-b)(v-b) < 0 \quad \text{for } v\ne b, \qquad  \psi^\natural(b)=b.
\endaligned 
$$
\item"(ii)" Considering the convex hull of the epigraph of $p$, we see that 
there exist two points $c$ and $d$ such that (Figure 2.1) 
$$
b^{-\natural}<c<a<b<d<a^{-\natural} 
$$
and 
$$
\psi^\natural(c)= d \quad \text{ and } \quad \varphi^\natural(d) = c.
$$
\item"(iii)" 
The function $\psi^\natural$ is  increasing for $v \in [b^{-\natural},c]$ and 
decreasing for $v\in [c,a^{-\natural}]$. The function $\varphi^\natural$ is decreasing 
for $v\in [b^{-\natural}, d]$ and increasing for $v\in [d,a^{-\natural}]$. 
Moreover $\varphi^\natural$ maps $[b^{-\natural},a^{-\natural}]$ onto $[c,b]$, while  
$\psi^\natural$ maps $[b^{-\natural}, a^{-\natural}]$ onto $[a,d]$. 
\endroster
\endproclaim


Consider next the graph of $p$ from a somewhat different perspective. 
We are interested in the intersection points of any tangent line with the graph itself. 
Observe first that the convex hull of the epigraph of $p$  
coincides with the epigraph except in the interval $[c,d]$, defined in Proposition 2.1. 
Equivalently, the points $c$ and $d$ can be characterized by the conditions $c< a < d < d$
and  
$$
p'(c) = {p(d) - p(c) \over d - c } = p'(d). 
$$ 
We observe that the tangent at any point $v \notin [c,d]$ remains globally below the graph of $p$. 
So we focus on values $v \in [c,d]$.

For all $v \in (c,d)$, the tangent at the point with coordinate $v$ intersects 
the graph of $p$ at exactly two distinct points, say 
denoted by $\varphi^{-\natural}(v)$ and $\psi^{-\natural}(v)$ with the convention 
$$ 
\varphi^{-\natural}(v)<\psi^{-\natural}(v).
$$
The functions $\varphi^{-\natural}$ and $\psi^{-\natural}$ are not one-to-one, however  
one can check geometrically that, by 
restricting attention to the interval $[a,b]$, their inverses coincide with  
the functions $\varphi^\natural$ and $\psi^\natural$ defined above: 
$$
\varphi^\natural \circ \varphi^{-\natural} = \psi^\natural \circ \psi^{-\natural} = id \qquad \text{ on the interval } [a,b].
$$



We are now ready to investigate the sign of the entropy 
dissipation function associated with shock waves. 
Consider a shock wave solution of the hyperbolic system (1.1), 
connecting a left-hand state $(u_0,v_0)$ 
to a right-hand state $(u_1,v_1)$ and propagating with the speed $s \in {\Bbb R}$. 
For this shock wave to be a weak solution, the Rankine-Hugoniot relations 
must hold: 
$$ 
s \, (u_1 - u_0) - p(v_1) - p(v_0) = 0, \qquad 
s \, (v_1 - v_0) + u_1 - u_0 = 0, 
\tag 2.2
$$
which yield  
$$ 
s = {p(v_1) -p(v_0) \over u_1 - u_0} 
  = -{u_1 - u_0 \over v_1 - v_0}. 
$$
Therefore, whenever $(p(v_1) - p(v_0))/ ( v_1 - v_0) \leq 0$, 
the shock speed 
$$
s = \mp  \overline{c}(v_0, v_1) 
:= \mp \sqrt {- {p(v_1) - p(v_0) \over v_1 - v_0}} 
\tag 2.3
$$
is well-defined and 
independent of $u_0$ and $u_1$, so we simply set $s=s(v_0,v_1)$.
In (2.3), the $1$-- and $2$--shocks correspond to the $\mp$ signs, respectively. 

Similarly, the entropy inequality (1.5) holds for the shock wave 
provided the corresponding entropy dissipation function is negative: 
$$ 
\aligned 
E(u_0, v_0; u_1, v_1)
&:= -s(v_0,v_1) \, \Big({u_1^2 - u_0^2 \over 2} +  
       \Sigma(v_1) - \Sigma(v_0) \Big) + u_1 \, p(v_1) - u_0 \, p(v_0)\\
& \leq 0. 
\endaligned 
\tag 2.4
$$
An easy calculation based on the Rankine-Hugoniot relations 
leads to the simpler expression  
$$
E(v_0, v_1) 
        = - s(v_0,v_1) \, \Big( \Sigma(v_1) - \Sigma(v_0) 
        + {p(v_1) + p(v_0) \over 2} \, (v_1-v_0)   \Big). 
\tag 2.5
$$
In particular, $E=E(v_0,v_1)$ is independent of $u_0$ and $u_1$. 


It is not difficult to determine the sign of the function $E$ 
geometrically. Given some values $v_0$ and $v_1$, the straightline connecting 
the two corresponding points on the graph of $p$ may cut the 
graph at four points at most, and thus may determine at most three 
signed areas comprised between the line and the graph, say $A_1(v_0, v_1)$, $A_2(v_0, v_1)$, and $A_3(v_0, v_1)$.
By convention, an area is positive when the graph is above the straightline and negative otherwise. 
The notation can be trivially extended to the situations where only one or two areas are 
determined by the given line. In view of (2.5) we find that the entropy dissipation 
is essentially the sum of these three areas: 
$$ 
E(v_0, v_1)
        = - s(v_0,v_1) \, \Big( A_1(v_0, v_1) + A_2(v_0, v_1) + A_3(v_0, v_1)   \Big). 
$$  
In the following we will state various monotonicity properties for the functions 
associated with zeros of $E$. Those porperties can be checked immediately from this geometrical 
interpretation of the entropy dissipation. 


For definiteness, from now on,  we restrict attention to waves propagating with negative 
speed. A tedious calculation yields 
$$
{\partial  E(v_0,v_1) \over \partial  v_1} = 
{ 1 \over 2} \, \sqrt{-{v_1-v_0 \over p (v_1)-p (v_0)}}
\, {1\over (w-v_0)^2}\, M(v_0,v_1) \, N(v_0, v_1),
\tag 2.6
$$
where
$$ 
M(v_0,v_1):= -p (v_1) + p (v_0) + p'(v_1) \, (v_1-v_0) 
$$
and
$$
N(v_0,v_1):= 2\, \bigl(\Sigma(v_0) - \Sigma(v_1)\bigr) - (3 \, p(v_1)-p(v_0)) \, (v_1-v_0).
$$
Therefore, the sign of $E$ is given by the signs of $M$ and $N$, 
which we now investigate. 

The following properties of the function $M$ are immediately obtained geometrically: 
\roster
\item If $v_0\in (0,b^{-\natural})\cup (a^{-\natural},+\infty)$,
then 
$$
M(v_0,v_1) > 0\quad\text{for all } v_1\ne v_0. 
\tag 2.7a
$$
\item If $v_0\in [b^{-\natural},a^{-\natural}]$, then
$$
\aligned 
& M(v_0,v_1) < 0 \quad \text{ if } v_1 \in \bigl(\varphi^\natural(v_0), \psi^\natural(v_0)\bigr), \\
& M(v_0,v_1) = 0 \quad \text{ if } v_1 = v_0,\varphi^\natural(v_0) \, \text{ or } \psi^\natural(v_0),\\
& M(v_0,v_1) > 0 \quad \text{ otherwise.} 
\endaligned 
\tag 2.7b
$$
\endroster

On the other hand, the function $\Sigma$ being convex, 
$N$ is bounded away from zero. Namely we have 
$$
\aligned
N(v_0,v_1)&\ge 2p(v_1)(v_1-v_0) - (3p(v_1)-p(v_0))(v_1-v_0) \\
& = - (p(v_1)-p(v_0))(v_1-v_0) > 0,\quad\text{for all } v_1\ne v_0.
\endaligned
\tag 2.8
$$
We conclude that the functions $E$ and $M$ have the same sign. 


If $v_0\in (0,b^{-\natural}) \cup \bigl(a^{-\natural},+\infty\bigr)$, 
then, by (2.7a), the entropy dissipation function $E(v_0,.)$ 
is globally monotone increasing in $v_1>0$. 
If $v_0\in [b^{-\natural},a^{-\natural}]$, then, by (2.7b), it is monotone increasing in 
$\bigl(0, \varphi^\natural(v_0)\bigr]$ and in $\bigl[\psi^\natural(v_0), +\infty\bigr)$, 
but is monotone decreasing in $\bigl[\varphi^\natural(v_0), \psi^\natural(v_0)\bigr]$. 
Therefore, in this latter case, the entropy dissipation 
attains a maximal value $F(v_0):= E(v_0,\varphi^\natural(v_0))$ at
$v_1=\varphi^\natural(v_0)$ and a minimal value $G(v_0):= E(v_0,\psi^\natural(v_0))$ at
$v_1=\psi^\natural(v_0)$. To determine the sign of $E$, one must know the sign
of $F(v_0)$ and $G(v_0)$. 


Regarding $F$ and $G$ as functions of $v\in \bigl[b^{-\natural},a^{-\natural}\bigr]$,
we obtain
$$\aligned
{dF\over dv}(v) & =-(p(\varphi^\natural(v))-v)(\varphi^\natural(v)-v) < 0\quad\text{iff } v\in (c,d),\\
{dG\over dv}(v) & =-(p(\psi^\natural(v))-v)(\psi^\natural(v)-v) < 0 \quad\text{iff } v\in (c,d).
\endaligned
$$
Thus,  both functions $F$ and $G$ are decreasing in each of the intervals 
$(b^{-\natural},c)$ and $(d,a^{-\natural})$, and
are increasing in the interval $(c,d)$. {} Moreover we have 
$$
F(a) = G(b) = 0,
$$
which indecate that $F$ and $G$ are both negative at $v=c$ and
positive at $v=d$. Also it is not difficult to check that $F$ and $G$ 
are both positive at $v=b^{-\natural}$ and both negative at $v=a^{-\natural}$.
Geometrically, in the interval $(b^{-\natural},b)$, the graph of
$p$ remains below its tangent at $v=b$.
In the interval $(a,a^{-\natural})$, the graph remains 
below its tangent at $v=a$. 

For each of the functions $F$ and $G$, 
there exist two values denoted by $e<f$ and $e' < f'$, respectively,
which satisfy 
$$
e, e'\in (b^{-\natural},c), \qquad  f,f'\in (d,a^{-\natural})
$$
and 
$$
\aligned 
& F(v) < 0 \quad \text{ iff } v \in (e',a)\cup (f,a^{-\natural}), \\
& F(e') = F(a)= F(f) = 0,\\
& F(v) > 0 \quad \text{ otherwise,} 
\endaligned
\tag 2.9
$$
and 
$$
\aligned
& G(v) < 0 \quad \text{ iff } v \in (e, b) \cup (f',a^{-\natural}), \\
& G(e) = G(b) = G(f') = 0,\\
& G(v) > 0 \quad \text{ otherwise.} 
\endaligned 
\tag 2.10
$$
In view of (2.7a)--(2.10), we arrive to the following conclusions: 



\proclaim{Theorem 2.2} {\rm
(Fundamental properties of the entropy dissipation) }

For each $v_0\in (0,b^{-\natural})\cup (a^{-\natural},+\infty)$,
the entropy dissipation $E(v_0,v_1)$ is a globally monotone decreasing
function of $v_1>0$. 
For each $v_0\in [b^{-\natural},a^{-\natural}]$, the function $v_1 \mapsto E(v_0, v_1)$ 
is monotone increasing in the intervals  
$\bigl(0, \varphi^\natural(v_0)\bigr]$ and $\bigl[\psi^\natural(v_0), +\infty\bigr)$, 
but is monotone decreasing in the interval $\bigl[\varphi^\natural(v_0), \psi^\natural(v_0)\bigr]$. 

More precisely, the entropy inequality $(2.4)$ select the following 
admissible shock waves: 
\roster
\item"(i)"
If $v_0\in (0,e]\cup [f,+\infty)$, then the constraint $(2.4)$ is equivalent to 
$$
v_1\le v_0.
$$
\item"(ii)" If $v_0\in (e,a]$, then we have $E(v_0,\psi^\natural(v_0))=G(v_0)<0$
and the entropy dissipation admits three roots. 
Hence, there exist two values, distinct from $v_0$ and denoted by
$\varphi^\flat_\infty(v_0)$ and $\psi^\flat_\infty(v_0)$, such that
$$
v_0\le a\le \varphi^\natural(v_0)\le\varphi^\flat_\infty(v_0)<\psi^\natural(v_0)<\psi^\flat_\infty(v_0)
$$
and
$$
E(v_0,\varphi^\flat_\infty(v_0))=E(v_0,\psi^\flat_\infty(v_0))=E(v_0,v_0)=0.
$$
The inequality $(2.4)$ is equivalent to
$$
v_1\in (0,v_0]\cup [\varphi^\flat_\infty(v_0),\psi^\flat_\infty(v_0)].
$$
% 
% 
\item"(iii)"
If $v_0\in (a,b)$, there exist two values, distinct from $v_0$ and denoted by
$\varphi^\flat_\infty(v_0)$ and $\psi^\flat_\infty(v_0)$, such that
$$
\varphi^\flat_\infty(v_0)<\varphi^\natural(v_0)<a<v_0<b<\psi^\natural(v_0)<\psi^\flat_\infty(v_0)
$$ 
and
$$
E(v_0,\varphi^\flat_\infty(v_0))=E(v_0,\psi^\flat_\infty(v_0))=E(v_0,v_0)=0.
$$
The inequality $(2.4)$ is equivalent to
$$
v_1\in (0,\varphi^\flat_\infty(v_0)]\cup [v_0,\psi^\flat_\infty(v_0)].
$$
%
\item"(iv)"
If $v_0\in [b,f)$, then we have $E(v_0,\varphi^\natural(v_0))=F(v_0)>0$. 
There exist two values, distinct from $v_0$ and denoted by
$\varphi^\flat_\infty(v_0)$ and $\psi^\flat_\infty(v_0)$, such that
$$
\varphi^\flat_\infty(v_0)<\varphi^\natural(v_0))<\psi^\flat_\infty(v_0) \le\psi^\natural(v_0)\le b\le v_0
$$
and
$$
E(v_0,\varphi^\flat_\infty(v_0))=E(v_0,\psi^\flat_\infty(v_0))=E(v_0,v_0)=0.
$$
The inequality $(2.4)$ is equivalent to
$$
v_1\in (0,\varphi^\flat_\infty(v_0)]\cup [\psi^\flat_\infty(v_0),v_0].
$$
\endroster
\endproclaim 


The two functions $\varphi^\flat_\infty$ and $\psi^\flat_\infty: [e,f]\to{\Bbb R}$ introduced in Theorem 2.2
play a central role in the construction of the Riemann solutions. 
Indeed they determine some important boundaries of the set of right-hand 
states that can be reached by an (admissible) shock wave 
satisfying the entropy inequality (1.5).
Their monotonicity properties are summarized in the following proposition: 

\proclaim{Corollary 2.3}   
The function $\varphi^\flat_\infty$ is monotone decreasing in the interval $[e,\psi^\natural(e)]$ with
$$
\varphi^\flat_\infty(\varphi^\flat_\infty(v))=v,\quad v\in [e,\psi^\natural(e)],
\tag2.12a
$$
and is monotone increasing in the interval $[\psi^\natural(e),f]$ with
$$
\psi^\flat_\infty(\varphi^\flat_\infty(v))=v,\quad v\in [\psi^\natural(e),f].
\tag2.12b
$$
The function $\psi^\flat_\infty$  is monotone decreasing in the interval $[\varphi^\natural(f),f]$ with
$$
\psi^\flat_\infty(\psi^\flat_\infty(v))=v,\quad v\in [\varphi^\natural(f),f],
\tag2.13a
$$
and is monotone increasing in the interval 
$[e,\varphi^\natural(f)]$ with
$$
\varphi^\flat_\infty(\psi^\flat_\infty(v))=v,\quad v\in [e,\varphi^\natural(f)].
\tag2.13b
$$
Moreover,
$$
\varphi^\flat_\infty(e)=\psi^\flat_\infty(e)=\psi^\natural(e), \qquad \varphi^\flat_\infty(f)=\psi^\flat_\infty(f)=\varphi^\natural(f), 
$$
and 
$$
\varphi^\flat_\infty(a)=a, \qquad \psi^\flat_\infty(b)=b.
$$
\endproclaim


\demo{Proof} The last conclusion is an immediate consequence of the values $e, f$ in
(2.9) and (2.10). First of all, we claim that 
$$
\varphi^\flat_\infty(v)\le \psi^\natural(e),v\in [e,f].
\tag 2.14
$$ 
Actually, the values $v\ge a$ satisfy 
$$
\varphi^\flat_\infty(v)\le\varphi^\natural(v)\le a\le a<b<\psi^\natural(e).
$$
We are left to considering values $v\in (e,a)$. The line between $e$ and $\psi^\natural(e)$ crosses
the graph of $p$ at a middle point $e^*$. Since $\psi^\natural(e)$ lies on the convex part of
$p$, the line connecting any $v\in (e^*,\psi^\natural(\psi^\natural(e))), \psi^\natural(\psi^\natural(e)) > a$ and $\psi^\natural(e)$
cut the graph of $p$ at some middle point $v_1$ such that 
$$
p'(v_1)<{p(v)-p(\psi^\natural(e))\over v-\psi^\natural(e)} < p'(\psi^\natural(e)). 
$$
By a continuity argument, we deduce that there exists a (unique) point
$v^*\in (v_1,\psi^\natural(e))$ such that
$$
{p(v)-p(v^*)\over v-v^*}=p'(v^*),
$$
i.e.,
$$
\psi^\natural(e)>v^* = \psi^\natural(v)>\varphi^\flat_\infty(v),
$$ 
satisfying (2.14).
If $v\in (e,e^*)$, it is easy to see that the line connecting $v$ 
and $\psi^\natural(e)$ lies below the line connecting $e$ and $\psi^\natural(e)$. The convexity and 
concavity properties of the pressure function then guarantees that
$$
E(v,\psi^\natural(e))  <0. 
$$
In view of the item  (ii) of Lemma 2.2, we deduce (2.14). 
Besides, it is not difficult to check that
$$
\psi^{-\natural}(v) > \psi^\natural(v), \quad \text{ for all } v \in (b,d).
\tag 2.15
$$

Now, let $v\in [e,a)$,  so that $\varphi^\flat_\infty(v)> a$. If $\varphi^\flat_\infty(v)\in (a,b]$,
then
$$
\psi^\flat_\infty(\varphi^\flat_\infty(v))>b>v,
$$
which, by the skew-symmetry property of $E$, yields (2.12a). 
Assume that $\varphi^\flat_\infty(v)  \in (b,\psi^\natural(e)]$. We have, since $v\in (a,b)$
$$
v_1:=\varphi^\flat_\infty(v)<\varphi^{-\natural}(v):=v_2\in (b,d).
$$
In view of (2.15) and the monotonicity of the function $\psi^{-\natural}$ on the interval $(a,d)$,
it holds that
$$
\psi^\flat_\infty(\varphi^\flat_\infty(v))= \psi^\flat_\infty(v_1)>\psi^{-\natural}(v_1)>\psi^{-\natural}(v_2) > \varphi^\natural(v_2) = v.
$$
The last inequality and the skew-symmetry of $E$, yields (2.12a) as well.
Let $v\in (a,b)$, then $\varphi^\flat_\infty(v)<a$. The inequalities (ii) of Lemma 2.2 yield 
$$
\psi^\flat_\infty(\varphi^\flat_\infty(v)) > \psi^\natural(\varphi^\flat_\infty(v))\ge b>v,
$$
which again yields (2.12a).
Let $v\in (b,\psi^\natural(e))$, then $\varphi^\flat_\infty(v)<\varphi^\natural(v)<a<\psi^\flat_\infty(v)$. Therefore,
$$
E(v',\psi^\natural(e)) = -E(\psi^\natural(e),v')<0,\quad\text{for all } v'\in (e,a)\subset (e,\psi^\flat_\infty(\psi^\natural(e))),
$$
which, in particular for $v'=\varphi^\flat_\infty(v)$, leads us to 
$$
\psi^\flat_\infty(v') > \psi^\natural(e).
$$
Hence, (2.12a) is again a consequence of the skew-symmetry of $E$ and the last inequality.
Finally (2.14) yields (2.12b). 

The proof of (2.13) is entrirely similar.
The monotonicity properties are consequences of (2.12a)-(2.13b). 
The proof of Corollary 2.3 is complete.
\quad \qed
\enddemo

Recall finally that an arbitrary solution of the Riemann problem (1.1)-(1.2) 
may also contain rarefaction waves. Given a left-hand state $(u_0,v_0)$, 
the integral curve associated with the vector field $r_1(v)$ is: 
$$
{\Cal O}_1(u_0, v_0):= \bigl\{(u,v)/ u-u_0 = \int^v_{v_0} c(w) \, dw \bigr\}. 
\tag 2.16
$$ 
Based on the property that the characteristic speed be increasing in a rarefaction fan, 
we find easily: 

\proclaim{Lemma 2.4} {\rm ($1$--Rarefaction waves)}

Given some left-hand state $(u_0,v_0)$, 
the set of all right-hand states $(u_1,v_1)$ attainable through a $1$-rarefaction wave 
is the portion of the integral curve ${\Cal O}_1(w_0)$ determined by the following constrains: 
\roster
\item"(i)" If $v_0\in (0,a]$, then $v_1\in [v_0,a]$. 
\item"(ii)" If $v_0\in (a,b)$, then $v_1\in [a,v_0]$. 
\item"(iii)" If $v_0\in [b,+\infty)$, then $v_1\in [v_0,+\infty)$. 
\endroster
\endproclaim


%================================== 
\heading{3. Classical Riemann Solver} 
\endheading


To begin with the construction of Riemann solutions, in this section we restrict attention
to shock waves satisfying the so-called Liu entropy condition, which is much 
stronger than our condition (1.5). The solutions constructed now are 
referred to as the {\it classical Riemann solutions.\/} 
Recall that a {\it $1$--shock wave\/} connecting $(u_0, v_0)$ 
to $(u_1, v_1)$ satisfies the {\it Liu entropy condition\/} (see (2.3))
iff 
$$
-\overline{c}(v_0,v) \geq -\overline{c}(v_0,v_1) 
\qquad 
\text{ for all $v$ between $v_0$ and $v_1$.} 
\tag 3.1
$$ 
Note that the condition (3.1)
implies the {\it Lax shock inequalities\/}  
$$
\lambda_1(v_0) = -c(v_0) \geq -\bar c(v_0, v_1) \geq - c(v_1) = \lambda_1(v_1). 
\tag 3.2
$$ 
The Liu condition can be interpreted geometrically, since it is equivalent to 
$$
{p(v) - p(v_0) \over v - v_0} 
\geq 
{p(v_1) - p(v_0) \over v_1 - v_0} 
\quad 
\text{ for all } v \text{ between $v_0$ and $v_1$.} 
$$ 
In other words, for all $v$ between $v_0$ and $v_1$,
the graph of $p$ is below (respectively above) 
the line connecting $v_0$ to $v_1$ when $v_1 <v_0$ (resp. $v_1 > v_0$). 

Given some left-hand state $(u_0, v_0)$, we now determine
the {\it $1$--wave curve\/} made of all right-hand states that can be arrived at 
by combining one or several  elementary waves. 
That is, we try to combine together  rarefaction fans and 
shocks satisfying the Hugoniot relations and the Liu condition.  
Observe that, in view of (2.2)-(2.3), the Hugoniot curve 
for the first wave family is given by 
$$ 
{\Cal H}_1(u_0, v_0):= \Big\{(u,v) \, / \, u-u_0 = \overline{c}(v_0,v) \, (v-v_0) \Big\}. 
\tag 3.3
$$


The following lemma singles out those shock waves that are admissible for the Liu criterion. 
 

\proclaim{ Lemma 3.1} {\rm (Liu admissible shock waves)}

Given a left-hand state $(u_0,v_0)$, the set of right-hand states $(u_1,v_1)$ 
attainable by a $1$-shock satisfying the Liu entropy condition $(3.1)$ 
is characterized as follows: 
\roster
\item"(i)"
If $v_0\in (0,c)\cup (a^{-\natural},+\infty)$, then $v_1\in (0,v_0]$. 
\item"(ii)" 
If $v_0\in [c,a]$, then
$v_1\in (0,v_0]\cup [\varphi^{-\natural}(v_0), \psi^\natural(v_0)]$. 
\item"(iii)" 
If $v_0\in (a,b)$, then
$ v_1\in (0,\varphi^{-\natural}(v_0)]\cup [v_0,\psi^\natural(v_0)]$. 
\item"(iv)" 
If $v_0\in [b,a^{-\natural}]$, then
$v_1\in (0,\varphi^{-\natural}(\psi^\natural(v_0))]\cup [\psi^\natural(v_0),v_0]$. 
\endroster
\endproclaim


We are ready to construct the classical $1$--wave curve ${\Cal W}_1^c(u_l,v_l)$
consisting of all right-hand states $(u_m,v_m)$ that can be arrived at by a combination of 
of Liu admissible shocks and rarefactions. We rely here 
on Lemma 3.1 for the shocks and Lemma 2.4 for the rarefactions. 
The solution is actually directly determined from the convex hull and the concave hull 
of the graph of the function $p$. 


First, assume that $v_l \in (0,c)$. According to Lemma 3.1, 
all the states $(v_m,u_m)$ having $v_m\in (0,v_l)$ 
can be arrived at by a single Liu admissible $1$--shock. 
By Lemma 2.4, all of the points $(v_m,u_m)$ with $v_m\in (v_l,a]$ 
can be arrived at by a single $1$-rarefaction.
If now $v_m \in [a,d]$, we have $\varphi^\natural(v_m)\in [c,a]$. In that case, the solution is thus
a rarefaction wave from $v_l$ to $\varphi^\natural(v_m)$ followed by a shock from $\varphi^\natural(v_m)$
to $v_m$. Finally, if $v_m>d$, the solution is made of three elementary waves:
a rarefaction wave from $v_l$ to $c$, followed by a shock from $c$ to $d$, and followed
by a rarefaction wave from $d$ to $v_m$. 

Second, assume that $v_l\in [c,a]$. If $v_m\in (0,v_l)$, the Riemann solution
is a single Liu-admissible $1$--shock. The states $(v_m,u_m)$ with 
$v_m \in (v_l,a]$ can be arrived at by a single $1$-rarefaction.
If $v_m\in [a,\varphi^{-\natural}(v_l)]$, then $\varphi^\natural(v_m)\in [v_l,a]$ and the Riemann solution is 
a rarefaction wave from $v_l$ to $\varphi^\natural(v_m)$ followed by a shock from $\varphi^\natural(v_m)$
to $v_m$. If $v_m\in (\varphi^{-\natural}(v_l),\psi^\natural(v_l]$, the solution is a single shock.
Finally, if $v_m> \psi^\natural(v_l)$, the solution is a shock from $v_l$ to $\psi^\natural(v_l)$ followed with 
a rarefaction wave connecting $\psi^\natural(v_l)$ to $v_m$. 

Third, assume that $v_l\in (a,b)$. The points $(v_m,u_m)$ with 
$v_m\in (0,\varphi^{-\natural}(v_l)]\cup [v_l,\psi^\natural(v_l)]$ can be arrived at 
by a single shock. The points $w_m$ with 
$v_m\in [a,v_l]$ can be arrived at by a single rarefaction wave.
If $v_m\in (\varphi^{-\natural}(v_l),a)$, then there exists a unique value $v^*\in (a,v_l)$ such
that $\varphi^{-\natural}(v^*)=v_m$. That is $v^*=\varphi^\natural(v_m)$. In that case the Riemann solution is a rarefaction wave
connecting $v_l$ to $v^*$ followed by a shock connecting $v^*$ to $v_m$. 
Finally, if $v_m> \psi^\natural(v_l)$, the Riemann solution is a shock connecting $v_l$ to $\psi^\natural(v_l)$ 
followed with a rarefaction wave from $\psi^\natural(v_l)$ to $v_m$.

Fourth, assume that $v_l\in [b,a^{-\natural}]$.  The states $w_m$ with 
$v_m\in (0,\varphi^{-\natural}(\psi^\natural(v_l))]\cup 
[\psi^\natural(v_l),v_l]$ can be arrived at by a single shock. The states $w_m$ with 
$v_m\in [v_l,+\infty)$ can be reached by a single rarefaction wave. 
If $v_m\in [a,\psi^\natural(v_l))$, the Riemann solution
is a shock from $v_l$ to $\psi^\natural(v_l)$ followed by a rarefaction from $\psi^\natural(v_l)$ to $v_m$.
If $v_m\in (\varphi^{-\natural}(\psi^\natural(v_l)),a)$, the solution contained three waves: 
a shock from $v_l$ to $\psi^\natural(v_l)$, followed by a 
rarefaction from $\psi^\natural(v_l)$ to $\varphi^\natural(v_m)$, and followed by a shock connecting 
$\varphi^\natural(v_m)$ to $v_m$.

Finally, assume that $v_l\in (a^{-\natural},+\infty)$. In that case the Riemann solution is 
simply a shock if $v_m<v_l$ and a rarefaction wave otherwise.

>From now on, in addition to (1.3) we also assume that
$$
\int_b^{\infty}\sqrt{-p'(v)} dv =+\infty.
\tag3.4
$$
It is not difficult to check that the wave curve described above is smooth and 
monotone increasing and covers the whole range of values $u \in (-\infty, +\infty)$. 
A similar construction can be given for the $2$--wave curve ${\Cal W}^c_2(u_r, v_r)$ 
made of all left-hand states attainable through a combination of $2$--rarefaction fans or 
Liu-admissible $2$-shocks, starting from the right-hand state $(u_r, v_r)$. 
Additionally, it can be seen from the explicit formulas of the Hugoniot 
and rarefaction curves that 
the two wave curves are globally transverse and intersect at a single point. 

We arrive at the following main result in this section.

\proclaim{Theorem 3.2} {\rm (Classical Riemann solver)}

Under the assumption $(1.3)$, 
the Riemann problem $(1.1)$-$(1.2)$ admits a unique classical solution in the class 
of piecewise smooth self-similar functions 
made of rarefaction fans and shock waves satisfying the Liu entropy criterion. 
\endproclaim



%================================== 
\heading{4. Nonclassical Riemann Solvers} 
\endheading


We return to the general conditions in Theorem 2.2. A shock wave 
is said to be {\it nonclassical} if the entropy condition (1.5) holds 
but the Liu entropy condition (3.1) does not. Determining the set
of all right-hand states $(u_1,v_1)$ attainable through nonclassical 
shocks from a given left-hand state $(u_0,v_0)$ 
is immediate from Theorem 2.2 and Lemma 3.1.


\proclaim{Corollary 4.1} Given a left-hand state $(u_0,v_0)$, the set
of all right-hand states $(u_1,v_1)$ that can be connected to $w_0$
by a nonclassical shock wave is determined as follows:
\roster
\item"(i)" If $v_0\in (e,c]$, then $v_1\in [\varphi^\flat_\infty(v_0),\psi^\flat_\infty(v_0)]$.
\item"(ii)" If $v_0\in (c,a]$, then $v_1\in [\varphi^\flat_\infty(v_0),\varphi^{-\natural}(v_0))
\cup (\psi^\natural(v_0),\psi^\flat_\infty(v_0)]$.
\item"(iii)" If $v_0\in (a,b)$, then $v_1\in (\varphi^{-\natural}(v_0),\varphi^\flat_\infty(v_0)]
\cup (\psi^\natural(v_0),\psi^\flat_\infty(v_0)]$.
\item"(iv)" If $v_0\in [b,f)$, then $v_1\in (\varphi^{-\natural}(\psi^\natural(v_0)),\varphi^\flat_\infty(v_0)]
\cup [\psi^\flat_\infty(v_0),\psi^\natural(v_0))$.
\item"(v)" If $v_0\in [f,a^{-\natural} )$, then $v_1\in (\varphi^{-\natural}(\psi^\natural(v_0)),\psi^\natural(v_0))$.
\endroster
When $v_0\in (0,e) \cup (a^{-\natural}, \infty)$, no such shock exists. 
\endproclaim


Denote by ${\Cal N}(v_0)$ the closure of the set of all values 
attainable by nonclassical shocks, as described in Corollary 4.1. 
The {\it kinetic function} $\varphi^{\flat}$ is defined to be a decreasing function defined 
in the interval $[e,b]$ and such that 
$$
\varphi^{\flat}(v) \in {\Cal N}(v_0), \qquad v \in [e,b]. 
\tag 4.1a 
$$
We also impose the condition 
$$
\varphi^{\flat}(b) = b^{-\natural} 
\tag 4.1b 
$$
which, as we will see, 
guarantees the continuity of the Riemann solution with respect to its end states. 


%____________________________________________________________________________ 
\midinsert
\centerline{\psfig{file=fig41.eps,width=10.5truecm}}
\botcaption{Figure 4.1: Kinetic function.}
\endcaption
\endinsert
%____________________________________________________________________________ 


The graph of the kinetic function 
then intersects the one of the function $\psi^\natural$
at a unique point, denoted by $g\in [e,c]$. The straightline connecting
$v$ and $\varphi^{\flat}(v)$ intersects the graph of $p$ at three points when $v\in [g,b]$,
limiting two finite areas, and four points when $v\in [e,g)$, 
limiting three finite areas. 
Motivated by the derivation of the model made in phase transition dynamics 
(only the first inflection point is actually physically meaningful), 
we propose to restrict attention to the interval $[g,b]$, 
as far as nonclassical shocks are concerned. 
The {\it kinetic relation} is the requirement that, for any nonclassical shock
connecting some left-hand state $(u_0,v_0)$ to a right-hand state $(u_1,v_1)$,  
we have 
$$
v_1 = \varphi^{\flat}(v_0).
\tag 4.2
$$

By the results in Section 3, the Riemann problem (1.1)-(1.2)
always admits a solution satisfying the Liu entropy criterion (3.1). 
Since classical shocks are still admissible in the nonclassical construction
to be discussed in the present section, 
the classical solution is in principle admissible. 
We are going to allow nonclassical shocks as well 
and, therefore, to ensure uniqueness, it is clear that one must exclude 
the classical solution. We postulate here that 
$$
\text{Nonclassical shock waves are prefered, whenever available.}
\tag P
$$

We now proceed with the construction of the $1$-wave curve ${\Cal W}_1(u_l,v_l)$. 


Suppose first that $v_l \in (0,g)$. Any point $v_m \in (0,v_l)$ can be achieved 
by a single classical shock. Any point $v_m\in (v_l,a]$ is attainable 
by a single rarefaction wave. If $v_m\in (a,\varphi^{\flat}(g)]$, there exists a unique point
$v_*\in [g,a)$ such that $v_m=\varphi^{\flat}(v_*)$. The solution is then the composite 
of a rarefaction wave from $v_l$ to $v_*$ followed by a nonclassical shock 
from $v_*$ to $v_m$. If $v_m\in (\varphi^{\flat}(g),+\infty)$, the solution consists 
of three parts: A rarefaction wave from $v_l$ to $g$ followed by a 
nonclassical shock from $g$ to $\varphi^{\flat}(g)$, followed by a rarefaction wave 
from $\varphi^{\flat}(g)$ to $v_m$.


Second, suppose that $v_l\in [g,a)$. 
A point $v_m\in (0,v_l)$ can be attained by a 
single classical shock. A point $v_m\in (v_l,a]$ is attainable by a single
rarefaction wave. If $v_m\in (a,\varphi^{\flat}(v_l)]$, there exists a unique point
$v_*\in [v_l,a)$ such that $v_m=\varphi^{\flat}(v_*)$. The solution is then the composite
of the rarefaction wave from $v_l$ to $v_*$ followed by a nonclassical shock 
from $v_*$ to $v_m$. If $v_m\in (\varphi^{\flat}(v_l),\varphi^{\flat}(g)]$, there exists a 
unique point
$v^*\in [g,v_l)$ such that $v_m=\varphi^{\flat}(v^*)$. For this construction 
to make sense, one must here check whether
the classical shock from $v_l$ to $v^*$ is slower than the nonclassical shock 
from $v^*$ to $v_m$. So, consider the function
$$
\tilde p(v):= \cases p(v) \qquad\qquad&\text{if } v\in (0, v_l],\\
p(v_l)+p'(v_l)(v-v_l) &\text{if } v \in (v_l,+\infty).
\endcases
\tag 4.3 
$$

If $v_m\in (\varphi^{\flat}(v_l), h)$, where
$$
h:=\min\{\varphi^{\flat}(g),\varphi^{-\natural}(v_l)\},
$$
the function $\tilde p$ is convex on $(0,+\infty)$ and the points
$v^*$ and $v_m$ belong to its epigraph. Therefore, the straightline 
connecting $v^*$ and $v_m$ should lie above the graph of $\tilde p$ 
in the interval $(v^*,v_m)\ni v_l$. This is to say
$$
{\tilde p(v_l)-\tilde p(v^*)\over v_l-v^*} < 
{p(v_m)-p(v^*)\over v_m-v^*},
$$
i.e., 
$$
s(v_l,v^*)<s(v^*,v_m).
\tag 4.4
$$
The latter inequality means that the classical shock from $v_l$ to $v^*$ can be followed by the
nonclassical shock from $v^*$ to $v_m$. 
In the latter construction, if $v_l\in [g, \varphi^\natural(\varphi^{\flat}(g))$, then
$$
h = \varphi^{\flat}(g),
$$
and we have completed the argument when $v_m\in (\varphi^{\flat}(v_l),\varphi^{\flat}(g))$.


For $v_m\in (\varphi^{\flat}(g),+\infty)$, 
the Riemann solution consists of three parts: 
A classical shock from $v_l$ to $g$ followed by a nonclassical
shock from $g$ to $\varphi^{\flat}(g)$,
followed by a rarefaction wave from $\varphi^{\flat}(g)$ to $v_m$.

Suppose next that $v_l\in [\varphi^\natural(\varphi^{\flat}(g)),a)$, then 
$$
h=\varphi^{-\natural}(v_l).
$$
%           In this case, we are left with $v_m\in [\varphi^{-\natural}(v_l),\varphi^{\flat}(g)]$.
If $v_m\in [\varphi^{-\natural}(v_l),\psi^\natural(v_l)]$, the solution can be
a classical shock connecting $v_l$ to $v^*$ followed by a 
nonclassical shock from $v^*$ to $v_m$ provided (4.4) holds, or else 
a single classical shock.  
For $v_m\in (\psi^\natural(v_l),+\infty)$, 
the solution consists of a classical shock from $v_l$ to $\psi^\natural(v_l)$,  
followed by a rarefaction wave from $\psi^\natural(v_l)$ to $v_m$.


Third, suppose that $v_l \in [a,b)$. 
The points $v_m\in [a,+\infty)$ are reached by the classical 
construction described in Section 3. 
If $v_m\in [\varphi^{\flat}(v_l),a]$, there exists a unique point
$v^*\in [a,v_l]$ such that $v_m=\varphi^{\flat}(v^*)$. 
The solution then consists of a rarefaction wave 
connecting $v_l$ to $v^*$ followed by a nonclassical shock 
from $v^*$ to $v_m$. 
If $v_m\in [\varphi^{-\natural}(v_l),\varphi^{\flat}(v_l))$, then there exists a unique point
$v_*\in [v_l,b)$ such that $v_m=\varphi^{\flat}(v_*)$. Since both $v_l$ and $v_*$
belong to $[a,b]$ and the function $p$ is concave in this interval, 
we have
$$
{p(v_l)-p(v_*)\over v_l-v_*} 
< {p(\varphi^{\flat}(v_l))-p(v_*)\over \varphi^{\flat}(v_l)-v_*} 
< {p(v_m)-p(v_*)\over v_m-v_*}.
$$
This means the shock speed $s(v_l,v_*)$ is less  
than the shock speed $s(v_*,v_m)$. 
Therefore the Riemann solution can be a classical shock from $v_l$
 to $v_*$ followed by a nonclassical shock from $v_*$ to $v_m$.
If $v_m\in [\varphi^{-\natural}(v_l),b^{-\natural})$, there exists a unique point
$v^*\in [v_l,b)$ such that $v_m=\varphi^{\flat}(v^*)$. 
The solution then consists of a classical shock from $v_l$ to $v^*$ 
followed by a nonclassical shock from $v^*$ to $v_m$ provided
$$
- \bar c(v_l,v^*) < - \bar c(v^*,v_m),
\tag 4.5
$$
or else a single classical shock. 
The states $v_m\in (0,b^{-\natural}]$ are reached by single classical shocks.

Finally, when $v_l\in [b,+\infty)$, we also use the classical construction 
described in Section 3.

Denote by $\varphi^{-\flat }: [b^{-\natural},\varphi^{\flat}(g)]\to [g,b]$, the inverse of the kinetic function $\varphi^{\flat}$, 
which is also a monotone decreasing mapping. 
The arguments presented above are summarized as follows: 


\proclaim{Theorem 4.2} {\rm (Construction of the $1$-wave curve)}

Fix some left-hand state $(u_l,v_l)$. 
Under the assumptions $(1.3)$ and $(3.4)$, we have the following description 
of the $1$-wave curve ${\Cal W}_1(u_l, v_l)$ 
consisting of all of the right-hand states $(u_m,v_m)$ 
that can be reached by a combination of rarefaction fans and shock waves, 
satisfying the entropy inequality $(1.5)$, the kinetic relation $(4.2)$
(for nonclassical shocks), and the condition $(P)$:  

\vskip.1cm 

\noindent Case 1: $v_l\in (0,g)$.

\roster
\item 
If $v_m\in (0,v_l)$, the solution is a single classical shock.
\item
If $v_m\in (v_l,a]$, the solution is a single rarefaction wave.
\item
 If $v_m\in (a,\varphi^{\flat}(g)]$, the solution is
the composite of a rarefaction wave connecting 
$v_l$ to $v_*:=\varphi^{-\flat }(v_m)$ followed by a nonclassical shock from $v_*$ to $v_m$.
\item
If $v_m\in (\varphi^{\flat}(g),+\infty)$, 
the solution consists of three parts: A rarefaction wave from $v_l$ to $g$ 
followed by a nonclassical shock from $g$ to $\varphi^{\flat}(g)$, 
followed by a rarefaction wave from $\varphi^{\flat}(g)$ to $v_m$.
\endroster

\noindent Case 2: $v_l\in [g,a)$.

\roster
\item
If $v_m\in (0,v_l)$, the solution is a single classical shock.
\item
If $v_m\in (v_l,a]$, the solution is a single rarefaction wave. 
\item
If $v_m\in (a,\varphi^{\flat}(v_l)]$, the solution is  the composite of
a rarefaction wave from $v_l$ to $v_*:=\varphi^{-\flat }(v_m)$ followed by a
nonclassical shock from $v_*$ to $v_m$. 
\item
If $v_l\in [g,\varphi^\natural(\varphi^{\flat}(g)))$ and 
$v_m\in (\varphi^{\flat}(v_l),\varphi^{\flat}(g))$, then the solution consists of
a classical shock from $v_l$ to $v^*:=\varphi^{-\flat }(v_m)$ followed
by a nonclassical shock from $v^*$ to $v_m$. 
\item
If $v_l\in [g,\varphi^\natural(\varphi^{\flat}(g)))$ and 
$v_m\in (\varphi^{\flat}(g),+\infty)$, the solution consists of three
waves: A classical shock from $v_l$ to $g$ followed by a 
nonclassical shock from $g$ to $\varphi^{\flat}(g)$, followed by a rarefaction 
wave from $\varphi^{\flat}(g)$ to $v_m$.
\item
If $v_l\in [\varphi^\natural(\varphi^{\flat}(g)), a)$ and 
$v_m\in (\varphi^{\flat}(v_l),\varphi^{-\natural}(v_l))$, the solution consists of
the classical shock from $v_l$ to $v^*:=\varphi^{-\flat }(v_m)$ followed
by a nonclassical shock from $v^*$ to $v_m$. 
\item
If $v_l\in [\varphi^\natural(\varphi^{\flat}(g)), a)$ and $v_m\in [\varphi^{-\natural}(v_l),\psi^\natural(v_l)]$,
the solution is a classical shock from $v_l$ to $v^*$ 
followed by a nonclassical shock from $v^*$ to $v_m$ if (4.3) holds,
or else a single classical shock.  
\item 
If $v_l\in [\varphi^\natural(\varphi^{\flat}(g)), a)$ and
 $v_m\in (\psi^\natural(v_l),+\infty)$, the solution consists of 
a classical shock  from $v_l$ to $\psi^\natural(v_l)$ 
followed by a rarefaction wave from $\psi^\natural(v_l)$ to $v_m$.
\endroster


\noindent Case 3: $v_l \in [a,b)$.

\roster
\item
If $v_m\in [a,+\infty)$, the solution is classical (Section 3).
\item
 If $v_m\in [\varphi^{\flat}(v_l),a]$, the solution consists of
the rarefaction wave from $v_l$ to $v^*:=\varphi^{-\flat }(v_m)$ 
followed by a nonclassical shock from $v^*$ to $v_m$. 
\item
If $v_m\in [\varphi^{-\natural}(v_l),\varphi^{\flat}(v_l))$, 
the solution consists of a classical shock 
from $v_l$  to $v_*:=\varphi^{-\flat }(v_m)$ followed by a nonclassical shock 
from $v_*$ to $v_m$.
\item
If $v_m\in [\varphi^{-\natural}(v_l),b^{-\natural})$, the solution consists of
the classical shock wave from $v_l$ to $v^*:=\varphi^{\flat}(v_m)$ 
followed by a nonclassical shock from $v^*$ to $v_m$ provided (4.3) holds,
or else a single classical shock.
\item
The states $v_m\in (0,b^{-\natural}]$ are reached by a single classical shock.
\endroster

\noindent Case 4: $v_l\in [b,+\infty)$. 

The construction is classical (Section 3).
\endproclaim



%=================================

A similar result holds for the $2$-wave curve. 
We are now in the position to state the main result of this paper.


\proclaim{Theorem 4.3}
Under the assumptions $(1.3)$ and $(3.4)$, 
the Riemann problem $(1.1)$-$(1.2)$ admits a unique piecewise smooth, 
self-similar solution made of rarefaction fans and shock waves, satisfying 
the entropy inequality $(1.5)$, the kinetic relation $(4.2)$, and the condition $(P)$. 
Moreover, the Riemann solution depends $L^1_{\text{loc}}$ continuously
upon its data.
\endproclaim


\demo{Proof} We only need to check that the $1$-wave curve ${\Cal W}_1(u_l, v_l)$
constructed earlier is continuous, monotone increasing and extends from 
$(u_m,v_m) = (-\infty, 0)$ to $(u_m,v_m) = (+\infty, +\infty)$. 
To begin with, the continuity is easy checked from our construction.
For large values of $v_m$, any right-hand wave in the $1$-wave fan 
connecting $v_l$ and $v_m$ should be a rarefaction wave. The formulation (2.16) and
the assumption (3.4) yield
$$
u_m\to +\infty \quad \text{ as } \quad v_m \to +\infty.
$$
Any $1$-wave pattern connecting $v_l$ to $v_m$ with $v_m<
b^{-\natural}$ must be a single classical shock, by construction. The hypotheses (1.3) and the
formulation (3.3) then yield
$$
u_m \to -\infty \quad \text{ as } \quad v_m \to 0.
$$ 
Finally, since the shock speed $-\bar c(v_l,v_m)$ is a continuous
function in both variables $v_m$ and $v_l$, we conclude that the Riemann solution 
depends $L^1_{\text{loc}}$-continuously on the data. 

It remains only to check the monotonicity of the wave curve. The classical parts 
are easily seen to be monotone increasing, so we omit the details. 
We observe that, in the construction of Theorem 4.2, besides the classical ones, 
four distinct wave patterns can be distinguished:  
\roster
\item"(i)" A rarefaction wave followed by a nonclassical shock. This happens for instance when 
$v_l\in (0,g)$ and $v_m\in (a,\varphi^{\flat}(g))$.
\item"(ii)" A classical shock followed by a nonclassical one, say 
$v_l\in (g,\varphi^\natural(\varphi^{\flat}(g)))$ and $v_m\in (\varphi^{\flat}(v_l),\varphi^{\flat}(g))$.
\item"(iii)" In for instance the interval $v_l\in [a,b)$ and $v_m\in
[\varphi^{-\natural}(v_l),b^{-\natural})$, a classical shock followed by a nonclassical one if (4.3)
holds true, or a single classical shock elsewhere. 
 \endroster

Consider first the case (iii). For any fixed $v_l \in [a,b)$, the set of
$v_m\in [\varphi^{-\natural}(v_l),b^{-\natural})$ satisfying the condition (4.5) is open,
and therefore is a countable union of intervals. In each subinterval, 
we are back to the case (ii) or to the classical construction. Thus, we only 
need to treat Cases (i) and (ii). In the rest of the proof, we consider a 
specific situation arising in these cases, as other possibilities are similar. 
Recall that 
$$
c(v):=\sqrt{-p'(v)} 
$$ 
and
$$
\overline{c}(v_0, v_1):= \sqrt {- {p(v_1) - p(v_0) \over v_1 - v_0}}.  
$$

Consider the pattern (i). The  solution is made of a rarefaction wave followed by
a nonclassical shock. In other words, with the notation introduced earlier, 
$$
\aligned
u_m(v_m) - u_m(\varphi^{-\flat }(v_m)) & = \overline{c} (\varphi^{-\flat }(v_m),v_m) \, (v_m-\varphi^{-\flat }(v_m)),\\
u_m(\varphi^{-\flat }(v_m))-u_l & = \int_{v_l}^{\varphi^{-\flat }(v_m)} c(z) \, dz.
\endaligned
\tag 4.6 
$$
For $v_m$ in the interval $(a,\varphi^{\flat}(g))$, we deduce from (4.6) that 
$$  \aligned
{du_m \over dv_m} =& - {d \varphi^{-\flat }(v_m) \over 2dv_m}
\dfrac{\theta}{\overline{c}(\varphi^{-\flat }(v_m),v_m)}
\Big(c(\varphi^{-\flat }(v_m)) - \overline{c}(\varphi^{-\flat }(v_m),v_m)
\Big)^2\\
&+ c^2(v_m) + \overline{c}^2(\varphi^{-\flat }(v_m),v_m) > 0, 
\endaligned$$
which  yields the desired monotone property of the wave curve.

Consider next the pattern (ii). The solution is 
a composite of a classical shock connecting
$v_l$ to $\varphi^{-\flat }(v_m)$ followed by a nonclassical shock
connecting $\varphi^{-\flat }(v_m)$ with $v_m$.
>From (2.16) and (3.3) we deduce that 
$$
\aligned
u_m(v_m)-u_m(\varphi^{-\flat }(v_m)) & = \overline{c} (\varphi^{-\flat }(v_m),v_m)(v_m-\varphi^{-\flat }(v_m)),\\
u_m(\varphi^{-\flat }(v_m))-u_l & = \overline{c}(v_l,\varphi^{-\flat }(v_m))(\varphi^{-\flat }(v_m)-v_l). 
\endaligned
\tag 4.7 
$$
This yields 
$$
\aligned
{du_m\over dv_m} =  
 & - {d \varphi^{-\flat }(v_m)\over 2dv_m} \, 
\Big(\overline{c}(\varphi^{-\flat }(v_m),v_l)-\overline{c}
(\varphi^{-\flat }(v_m),v_m)\Big) \\
&\times \Big(\dfrac{c^2(\varphi^{-\flat }(v_m)}{\overline{c}(\varphi^{-\flat }(v_m),v_l)
\, \overline{c}(\varphi^{-\flat }(v_m),v_m)} - 1\Big) 
+ \dfrac{c^2(v_m) + 1}{2\overline{c}(\varphi^{-\flat }(v_m),v_m)}.
\endaligned
\tag 4.8 
$$
Since the function $p$ is convex in the interval $(0,a)\ni v_l,\varphi^{-\flat }(v_m)$ and
since $v_l>\varphi^{-\flat }(v_m)$, we have
$$
{{p(\varphi^{-\flat }(v_m))-p(v_l)\over \varphi^{-\flat }(v_m)-v_l}} > p'(\varphi^{-\flat }(v_m)).
$$
Hence we obtain 
$$
c(\varphi^{-\flat }(v_m)) > \overline{c}(\varphi^{-\flat }(v_m),v_l)>\overline{c}(\varphi^{-\flat }(v_m)v_m),
\tag4.9
$$
where the last inequality follows from the fact that the shock speed is
increasing and the classical shock is followed by the nonclassical one.
The inequalities (4.9) used in (4.8) yield 
$$
 {du_m\over dv_m} >0,
$$
which implies the monotonicity of the wave curve. 
The proof of Theorem 4.3 is complete.
\quad\qed
\enddemo


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\enddocument