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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\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)
$$
b^{-\natural} 0\quad\text{for all } v_1\ne v_0.
\tag 2.7a
$$
\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 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)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
$$
\varphi^\flat_\infty(v)\le\varphi^\natural(v)\le a\le a** a$ and $\psi^\natural(e)$
cut the graph of $p$ at some middle point $v_1$ such that
$$
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.
$$
The last inequality and the skew-symmetry of $E$, yields (2.12a) as well.
Let $v\in (a,b)$, then $\varphi^\flat_\infty(v) \psi^\natural(\varphi^\flat_\infty(v))\ge b>v,
$$
which again yields (2.12a).
Let $v\in (b,\psi^\natural(e))$, then $\varphi^\flat_\infty(v)<\varphi^\natural(v) \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 $v_1$.}
$$
In other words, for all $v$ between $v_0$ and $v_1$,
the graph of $p$ is below (respectively above)
the line connecting $v_0$ to $v_1$ when $v_1 v_0$).
Given some left-hand state $(u_0, v_0)$, we now determine
the {\it $1$--wave curve\/} made of all right-hand states that can be arrived at
by combining one or several elementary waves.
That is, we try to combine together rarefaction fans and
shocks satisfying the Hugoniot relations and the Liu condition.
Observe that, in view of (2.2)-(2.3), the Hugoniot curve
for the first wave family is given by
$$
{\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_mFrom 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^*)**~~ 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
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