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\markboth{\hfil Non-classical phase transitions at a sonic point \hfil EJDE--2003/22}
{EJDE--2003/22\hfil Monique Sabl\'e-Tougeron \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 22, pp. 1--28. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Non-classical phase transitions at a sonic point 
 %
\thanks{ {\em Mathematics Subject Classifications:} 
35L65, 35L67, 74B20, 76L05, 80A32.
\hfil\break\indent
{\em Key words:} Hyperbolic, phase transition, Chapman-Jouguet regime, 
kinetic relation.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted September 5, 2002. Published March 7, 2003.} }
\date{}
%
\author{Monique Sabl\'e-Tougeron}
\maketitle

\begin{abstract} 
 The relevant mathematical features of phase transition for a general
 hyperbolic nonlinear system near a sonic discontinuity are clarified.
 A well-posed Riemann's problem is obtained, including non-classical
 undercompressive shocks, defined by a geometrical kinetic relation.
 A counterpart is the geometrical rejection of some compressive shocks.
 The result is consistent with the structure profiles of the elasticity
 model of Shearer-Yang and the combustion model of Majda.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Hyperbolic phase transitions occur in important physical contexts as
elasticity and combustion. In
the so-called Chapman-Jouguet regime, the waves are sonic on one side of
their phase-transition
discontinuity; here we call these waves semi-characteristic transition waves. Uniqueness fails in
the Riemann problem near a sonic phase transition and the geometric-compressibility
criterion of Lax \cite{L}, is not appropriate to select such large amplitude
transition discontinuities.

Some admissibility criteria for particular physical systems have been
suggested in many papers. Most of them require a simple phase
transition to be asymptotic to travelling waves of
appropriate augmented systems. The problem of  existing travelling waves is that of heteroclinic
connections of rest points for ordinary differential equations. At a sonic transition, one of the
rest point is not hyperbolic. In addition to the non-planar general context obstruction, these are all
reasons why there are so few responses to the wave structure problems.

Nevertheless, complete resolution is
achieved near a sonic point in case of two significant models. In elasticity, for a cubic stress,
Shearer and Yang \cite{S-Y}, characterize the simple phase transition waves which satisfy the
viscosity-capillarity criterion of Slemrod \cite{S}. In combustion, for a
qualitative model Majda \cite{Ma}, characterize the simple phase transition waves which satisfy a
viscosity-reacting criterion with ignition temperature kinetics. These
two models not only underline undercompressive admissible transition
shocks but also compressive non-admissible ones.
The undercompressive admissible shocks enter into the so-called kinetic relation of
Abeyaratne and Knowles \cite{A-K} and their set is called the kinetic set.
The selection in each of the two models of non-compressive shocks as
either kinetic or sonic appears as a mixed sonic-kinetic geometrical selection.


In the present paper, we carry out the local-mathematical analysis at a
semi-characteristic hyperbolic nonlinear phase transition, up to the
well-posedness of the mixed sonic-kinetic Riemann problem.
Under a purely sonic-stability
assumption, the local Riemann problem is well-posed and BV-stable.
One could use it to get a  global-existence result by a Glimm's type
scheme, in the way of \cite{C-ST}.

Section \ref{semichar} provides the mathematical justification for
the claims made in this paper. There, we identify the
relevant phase boundary variables in regards to Hugoniot's large-jump equation, near a sonic phase
transition, the speed of which is genuinely nonlinear. In this space, we underline the fundamental part
of what we call a Hugoniot stationary vector field, as well as the
Hugoniot nonlinear dipoles. The analysis leads to a geometrical
admissibility criterion which  explains in a natural way the
rejection of compressible transition shocks as a counterpart
of the admissibility of undercompressive ones.

In section \ref{SY-M} we show how the results of Shearer-Yang or Majda
 enter the sonic-kinetic general case.

Section \ref{entropy} analyzes the entropy, so that as in Abeyaratne and Knowles \cite{A-K},
sonic-kinetic geometries  can be defined
when physical examples are not available.

Section \ref{C-J} applies the analysis to the perturbations of
Chapman-Jouguet detonations, for the complete compressible Euler system.
At a Chapman-Jouguet detonation, the system is characteristic to the side
of the burnt gas for a genuinely nonlinear mode but  non-characteristic to the side of the unburnt
gas. In a geometrical admissibility criterion, kinetic phase boundaries are weak detonations, and
compressive non-admissible ones are strong detonations. The mixed sonic-kinetic Riemann problem
is solved in the case of a particular geometry in order to show how the
mathematical analysis applies in the absence of an explicit physical example.


\section{Semi-characteristic hyperbolic transition \\ waves}\label{semichar}

Let $\Omega^\pm$ be two disjoint open subsets of
$\mathbb{R}^N$, $\underline u^-,\underline u^+$ two constant states in $\Omega^-, \Omega^+$, $f$ a
smooth
function defined in $\Omega :=
\Omega ^- \cup \Omega ^+$ with values in $\mathbb{R}^N$; we are
concerned with the system of conservation laws in one space dimension
\begin{equation}\label{lc}
\partial_tu+\partial_xf(u)=0.
\end{equation}
We assume that $f$ satisfies the Lax conditions \cite{L}, in each open
set $\Omega^\pm$:
\begin{itemize}
\item The eigenvalues
$\lambda_1(u)<\dots <\lambda_N(u)$ of the differential $Df(u)$ are
real and distinct (strict hyperbolicity),
\item For  $i=1,\ldots,N$,
either $D\lambda_i(u)\cdot r_i(u)\neq 0$ for every $u$
(genuine nonlinearity), or
$D\lambda_i\cdot r_i=0$ for  every $u$ (linear degeneracy),
 where $r_i(u)$   denotes a right eigenvector of $Df(u)$ associated
to the eigenvalue $\lambda_i(u)$.
\end{itemize}
%
Let $s$ be a fixed  integer between $1$ and $N$; we assume that
\begin{equation}\label{nch}
\lambda_{s}(\underline u^-)\neq \lambda_i(\underline u^+)\quad
\forall i
\end{equation}
 and that the eigenvalue $\lambda_{s}$ is genuinely nonlinear at $\underline u^-$
\begin{equation}\label{vnl}
D\lambda_{s}(\underline u^-)\cdot r_{s}(\underline u^-)=1.
\end{equation}
We also assume that
the open sets $\Omega^\pm$ are small enough to conserve these properties
uniformly.

 A \textit{simple phase wave}  with speed
$\sigma \in\mathbb{R}$ is a solution $u$ of the system  (\ref{lc})
which is constant on each side of the linear curve $x=\sigma t$ and
satisfies $u := u^{\pm}\in \Omega^\pm$ in $\pm(x-\sigma t)>0$;
we denote by $( u^-, \sigma , u^+)$ such a phase wave. Let
$(\underline u^-,\underline
\sigma ,\underline u^+)$ be an \textit{$s$-sonic  phase wave}, that is
$\lambda_{s}(\underline u^-)=\underline \sigma$.
Since by (\ref{nch}) $\underline \sigma$ is not an
eigenvalue at $\underline u^+$, we also call this phase transition wave
a \textit{semi-characteristic}.

\subsection{Hugoniot's stationary vector field}

Let $(u^-, \sigma, u^+)$ be a phase wave close to the $s$-sonic background
phase wave $(\underline u^-,\underline \sigma, \underline u^+)$.
Across the linear curve $x=\sigma t$ the Rankine
Hugoniot equation is
\begin{equation}\label{RH}
J(u^-,\sigma ,u^+) :=-\sigma  [u]+[f(u)]=0
\end{equation}
where $[u]=u^+-u^-$. Note that the function $J$ satisfies
\begin{gather*}
D_{u^+}J(u^-,\sigma ,u^+) = Df(u^+)-\sigma I\,,
\quad D_{\sigma }J(u^-,\sigma,u^+) = -[u]\,, \\
D_{u^-}J(u^-,\sigma ,u^+) = -Df(u^-)+\sigma I\,.
\end{gather*}
 From (\ref{nch}),  $D_{u^+}J(\underline u^-,\underline \sigma, \underline u^+)$ is not singular. By
the implicit function theorem, the jump condition (\ref{RH}) is locally
equivalent to $u^+=H(\sigma ,u^-)$ for a smooth \textit{Hugoniot function } $H$. A simple phase
wave is thus defined by the phase-boundary  variables $(\sigma , u^-)$. In
that  phase-boundary space, we call \textit{sonic manifold} the hypersurface
$$
\Sigma_s:= \{(\sigma , u^-):  \sigma  = \lambda_{s}(u^-)\}\,.
$$
The sonic hypersurface keeps apart the subsonic phase
boundaries, which are \textit{determinate} (as waves in the $(t,x)$
independent variables):
$$
\mathcal{P}\mathcal{B}_{\rm det}:= \{(\sigma , u^-):  \sigma  \leq \lambda_{s}(u^-)\}\,,$$
and the supersonic phase
boundaries, which are \textit{indeterminate} :
$$
\mathcal{P}\mathcal{B}_{\rm ind}:= \{(\sigma , u^-):  \sigma  > \lambda_{s}(u^-)\}\,,
$$
(because some $s$-shock or $s$-rarefaction small waves may appear at the left side of the phase
discontinuity).

 At the sonic manifold $\Sigma_s$, the eigenvalue $(\lambda_{s}(u^-)-\sigma )$\
of the  partial
differential
$D_{u^-}J(u^-,\sigma ,u^+)$, (which does not depend on $u^+$), vanishes and the associated
eigenspace is  $\mathbb{R}.r_{s}(u^-)$; we  denote by
$$\partial_{r_s}:=  r_{s,1}(u^-)\partial _{u^-_1}+\dots +
r_{s,N}(u^-)\partial
_{u^-_N}\,,$$
the vector field defined by the eigenvector $r_{s}(u^-)$, as a vector field in the $u^-$-space as
well as in the
$(\sigma,u^-)$-space. By (\ref{vnl}),
$\partial_{r_s}$ is transversal to
$\Sigma_s$. For $y \in \{\sigma , u^-_1,\dots , u^-_N\}$,
the solution $u^+=H(\sigma ,u^-)$ of (\ref{RH}), satisfies
\begin{multline*}
D_{u^+}J(u^-,\sigma ,H(\sigma ,u^-)) \partial _y H(\sigma ,u^-)\\
=-(D_{u^-}J)(u^-,\sigma ,H(\sigma ,u^-)) \partial _y u^- - (D_\sigma
J)(u^-,\sigma ,H(\sigma ,u^-)) \partial _y \sigma \,.
\end{multline*}
In particular, we have
\begin{gather*}
(\partial_{r_s} H)(\lambda_{s}(u^-) ,u^-)=0\,,\\
 \partial_\sigma H(\sigma ,u^-)=
(D_{u^+}J(u^-,\sigma ,H(\sigma ,u^-)))^{-1}(H(\sigma ,u^-)-u^-)\neq 0.
\end{gather*}

For applications, a \textit{general Hugoniot function} is
a smooth family of smooth functions
$H(\alpha,\cdot):\Omega^-\to \Omega^+$, with parameter
$\alpha\in \omega$, such as:\\ 
0 is a simple eigenvalue of $D_{u^-}H(\underline \alpha,\underline u^-)$,\\ 
the set $\{det  (D_{u^-}H(\alpha,u^-))=0\}$ is a graph
$\Sigma_s:=\{(\alpha,u^-)\in \omega\times \Omega^-: \alpha =\alpha_s(u^-)\}$.

A \textit{Hugoniot stationary vector field} is a smooth family
of smooth fields $X=X(\alpha,u^-,\partial_{u^-})$ on $\Omega^-$,
 with parameter $\alpha$, which satisfies
$X H=0$  on  $\Sigma_s$.

A \textit{Hugoniot stationary manifold} is a smooth hypersurface
$\Sigma:= \{(\alpha , u^-): \Lambda(\alpha ,u^-)=0\}$ which is transversal to
$\Sigma_s$ and satisfies $X\Lambda(\alpha  ,u^-)=0$ along
$\Sigma \cap \Sigma_s$.

\subsection{Hugoniot's nonlinear dipoles}

In the  phase boundary space, near $(\underline\sigma ,\underline u^-)$, let us consider
the equation
\begin{equation}\label{H}
H(\sigma ,v^-)-H(\sigma ,u^-)=0\,,
\end{equation}
where $v^-$ is the unknown, and $v^-=u^-$ is an obvious solution.
 This equation can be written as
\begin{equation}\label{La}
f(v^-) -f(u^-)-\sigma (v^--u^-)=0\,,
\end{equation}
which means that $v^-$ is connected to $u^-$ by a
small amplitude $s$-shock
with speed $\sigma$ (because $\sigma$ is close to $\lambda_{s}(\underline u^-) $). To define $v^-$
as a function of the phase boundary variable $(\sigma , u^-)$, we review Lax's analysis.  The
equation (\ref{La}) is equivalent to,
$$
A(\sigma ,u^-,v^-) (v^--u^-):= \int_0^1 (Df(u^-+t(v^--u^-))-\sigma I) dt \
(v^--u^-) =0\,,
$$
which means that $ (v^--u^-)$ belongs to the kernel of $A(\sigma ,u^-,v^-)$.
The eigenvalues of $A(\underline \sigma ,\underline u^-,\underline u^-)$
are $(\lambda _i(\underline u^-)-\lambda _{s}(\underline u^-))$. Near
$(\underline \sigma ,\underline u^-,\underline u^-)$, let $\mu _{s}(\sigma ,u^-,v^-)$ denote the
continuous eigenvalue of
$A(\sigma ,u^-,v^-)$ which satisfies
$\mu _{s}(\sigma ,u^-,u^-)= \lambda _{s}(u^-)-\sigma $, 
$\{r_i(\sigma,u^-,v^-): 1\leq i\leq N\}$  a basis of right eigenvectors of
$A(\sigma ,u^-,v^-)$,
$\{\ell_i(\sigma,u^-,v^-)\ :\  1\leq i\leq N\}$ the dual basis, and
$M_{s}(\sigma ,u^-,v^-)$
the matrix composed of rows, with the left eigenvectors $\ell_i(\sigma
,u^-,v^-)$, $i\neq s$. From the symmetry of $A(\sigma ,u^-,v^-)$ in
$(u^-,v^-)$ we get
$$D_{v^-}\mu_{s}( \sigma , u^-, u^-)= \frac{1}{2}
D_{u^-}(\mu_{s}( \sigma ,u^-, u^-))= \frac{1}{2}D\lambda_{s}( u^-)\,.
$$
As we have $D_{v^-}(M_{s}(\sigma , u^-,\cdot)(\cdot - u^-))(
u^-) =
M_{s}(\sigma , u^-, u^-)$, by the genuine non linearity property
(\ref{vnl}), the
classical inverse function theorem applies, near $(\underline\sigma ,\underline u^-,\underline u^-)$,
to the equation
\begin{equation}\label{B}
B(\sigma ,u^-,v^-):= \begin{pmatrix}
2 \mu_{s}(\sigma ,u^-,v^-) \\
M_{s}(\sigma , u^-,v^-)(v^--u^-)
\end{pmatrix}= 0\,.
\end{equation}
The smooth solution $v^-=v^-(\sigma , u^-)$ satisfies (\ref{H}) and also $v^-(\lambda_s(u^-) , u^-)=u^-$;
if $(\sigma ,u^-)$ is sonic, that is, $\sigma  =
\lambda_{s}(u^-)$, then $(\sigma,v^-(\sigma,u^-))$ also is sonic, and coincides with $(\lambda_{s}(u^-)
,u^-)$.   We may assume that $r_{i}( \sigma, u^-, u^-)=r_{i}(
u^-)$. Then the
first column of the matrix $(D_{v^-}B( \sigma, u^-, u^-))^{-1}$ is
$r_{s}(
u^-)$ and we have
 \begin{equation}\label{dv-}
D\lambda_{s}( u^-) \partial_{\sigma }v^-(\lambda_{s}(u^-) , u^-)=2\,,
\end{equation}

%%%%%%%%%%%%%%%%

%\begin{equation} \label{Dv-}
%\begin{aligned}
%&D_{u^-}v^-(\lambda_{s}(u^-) , u^-)r_{s}( u^-)\\
%&= (D_{v^-}B( \lambda_{s}(u^-),u^-,u^-))^{-1}
%\begin{pmatrix}
%-D\lambda_{s}( u^-)\\
%M_{s}(\lambda_{s}(u^-) , u^-, u^-)
%\end{pmatrix} r_{s}( u^-) \\
%&= (D_{v^-}B(\lambda_{s}(u^-), u^-, u^-))^{-1}
%\begin{pmatrix} -1\\  0 \end{pmatrix} = - r_{s}( u^-),
%\end{aligned}
%\end{equation}

%%%%%%%%%%%%% MY COMPUTER DO NOT HAVE {aligned}, but maths above are good. Below,
%%%%%%% same maths:

$$D_{u^-}v^-(\lambda_{s}(u^-) , u^-)r_{s}( u^-)= \qquad \qquad \qquad
\qquad\qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad
 $$ 
$$\qquad \qquad  (D_{v^-}B( \lambda_{s}(u^-),
u^-, 
u^-))^{-1}
\left( \begin{array}{ll} 
-D\lambda_{s}( u^-)\\
M_{s}(\lambda_{s}(u^-) , u^-, u^-)
\end{array} \right) r_{s}( u^-)$$
\begin{equation}\label{Dv-}
\qquad \qquad \qquad  = (D_{v^-}B(\lambda_{s}(u^-), u^-, 
u^-))^{-1}
\left( \begin{array}{ll} 
-1\\
0
\end{array} \right) = -\ r_{s}( u^-),
\end{equation}

%%%%%%%%%%%%%%

\noindent from which follows
 \begin{equation}\label{lr}
 \ell _{s}( u^-) (\partial_{r_s}v^-)(\lambda_{s}(u^-) , u^-)\equiv
 \ell _{s}( u^-) (D_{u^-}v^-)( \lambda_{s}(u^-) , u^-) r_{s}(
u^-)=-1\,.
 \end{equation}
Now,  differentiating the equation
\begin{equation}\label{rh-}
-\sigma  (v^-(\sigma,u^-)-u^-)+(f(v^-(\sigma,u^-)-f(u^-))=0\,,
\end{equation}
we get
$$ (\lambda_{s}(v^-)-\sigma ) \ell _{s}(v^-) D_{u^-}v^-( \sigma ,
u^-) r_{s}(u^-)=(\lambda_{s}(u^-)-\sigma ) \ell _{s}(v^-) r_{s}( u^-)
$$
 then,  (\ref{lr}) implies that near $(\underline \sigma ,\underline u^-)$, $
(\lambda_{s}(v^-)-\sigma
)$ and
$(\lambda_{s}(u^-)-\sigma )$ have opposite signs (which is a classical
result of Lax).
As a consequence, if $(\sigma ,u^-)$ is subsonic, that is,
$\sigma  < \lambda_{s}(u^-)$, then
$(\sigma,v^-(\sigma,u^-))$ is supersonic, that is $ \sigma >
\lambda_{s}(v^-)$. The sonic
twin solution of (\ref {H}),
$( (\lambda_{s}(u^-),u^-)\,, (\lambda_{s}(u^-),u^-) )$,
leaves the sonic manifold as a
\textit{subsonic-supersonic dipole}
$((\sigma ,u^-)_{sub}\,, (\sigma,v^-(\sigma,u^-))_{sup} )$.

Denoting by $\epsilon _{s}\mapsto S _{s}(\epsilon _{s},u^-)$ a smooth
parametrization
of the $s$-shock curve
near $\underline u^-$, we can write
\begin{equation}\label{S}
v^-(\sigma,u^-)= S_{s}(\epsilon_{s}^d(\sigma,u^-),u^-)\,,
\end{equation}
for a smooth function $\epsilon_{s}^d(\sigma,u^-)$. For a classical
parametrization 
\cite{L, G-R}
$$
S _{s}(\epsilon _{s},u^-)= u^-+ \epsilon _{s} r_{s}(
u^-)+\frac{\epsilon _{s}^2}{2} Dr_{s}( u^-).r_{s}( u^-)+O(\epsilon_{s}^3)
,$$
 the speed $s _{s}(\epsilon _{s},u^-)$ of the ${s}$-shock is
$$s _{s}(\epsilon _{s},u^-)=\lambda_{s}(u^-)+\frac{\epsilon
_{s}}{2} D\lambda_{s}(u^-).r_{s}( u^-)+O(\epsilon
_{s}^2)\,.$$
When $(\sigma ,u^-)$ is subsonic, $\epsilon_{s}^d(\sigma,u^-)<0$ is the strength of an
admissible shock, called \textit{(micro)-detonating strength}.
Moreover we have, from (\ref{dv-}) and (\ref{Dv-}),
\begin{equation}\label{deps}
\begin{gathered}
\partial_{\sigma }\epsilon_{s}^d(\lambda_{s}(u^-), u^-)= 2\,,\\
(\partial_{r_s}\epsilon_{s}^d)(\lambda_{s}(u^-),u^-)\equiv (\partial_{u^-}\epsilon_{s}^d)(\lambda_{s}(u^-),
u^-).  r_{s}( u^-)= - 2 \,.
\end{gathered}\end{equation}

\subsection{The non-classical admissibility criterion}\label{microdet}

Let us consider a hypersurface $\Sigma $ in the phase
boundary space, near $(\underline \sigma, \underline u^-)$,
 parametrized by $u^-$, 
$$\Sigma := \{(\sigma,u^-): \sigma = \sigma_k (u^-)\}$$
where
\begin {equation}\label{transs}
\partial_{r_s}(\sigma_k -\lambda_{s})(\underline u^-))\neq 0\,.
\end{equation}
 Then $\Sigma$ is transversal
to $\Sigma_s $ along the smooth manifold $\Sigma \cap\Sigma_s:={\mathcal{C}}_s$
$$
{\mathcal{C}}_s = \{(\sigma,u^-): \sigma_k (u^-)=\sigma
= \lambda_{s}(u^-)\}\,.
$$
Moreover, the $u^-$-projection $\Pi {\mathcal{C}}_s$
of ${\mathcal{C}}_s$,
$$\Pi{\mathcal{C}}_s = \{u^-: \sigma_k (u^-) = \lambda_{s}(u^-)\}\,,
$$
is a smooth hypersurface of ${\bf R}^N$ near $\underline u^-$.
Let $\Sigma_k$ be the indeterminate part  (here supersonic) of $\Sigma $
$$
\Sigma_k = \{(\sigma_k (u^-),u^-):  \sigma_k (u^-)> \lambda_{s}(u^-)\}
.$$
Referring to \cite{A-K},  we call  $\Sigma_k$  the
\textit{kinetic hypersurface}.
Let $\Sigma _d$ be the (micro)-detonating set
(determinate, here subsonic) associated to $\Sigma_k$
$$
\Sigma _d :=  \{(\sigma ,v^-(\sigma,u^-)): (\sigma,u^-)\in \Sigma_k\}\,,
$$
where $v^-(\sigma,u^-)$ is the solution of (\ref{B}), or equivalently,
\begin{align*}
\Sigma _d &=  \{(\sigma, u^-):
(\sigma , S_{s}(\epsilon_{s}^d(\sigma,u^-),u^-))\in \Sigma_k \}\\
&=\{(\sigma, u^-):  \sigma _k(S_{s}(\epsilon_{s}^d(\sigma,u^-),u^-))
=\sigma \}\,.
\end{align*}
 From (\ref{deps}), if
\begin{equation}\label{hyp}
\partial_{r_s}\sigma_k(\underline u^-)\equiv D\sigma_k(\underline u^-)\,.\,r_{s}(\underline u^-)\neq \frac{1}{2}\,,
\end{equation}
then $\Sigma _d $ is an hypersurface with boundary ${\mathcal{C}}_s$,
parametrized by $u^-$,
$$\Sigma_d := \{(\sigma,u^-): \sigma = \sigma_d (u^-)\}\,.
$$
In this case, we call  $\Sigma_d$  the \textit{(micro)-detonating hypersurface}.
From (\ref{deps}) again, the smooth function $\sigma_d$ satisfies
\begin{equation}\label{dk1}
\partial_{r_s}\sigma_d( u^-)\equiv D\sigma_d( u^-)\,.\,r_{s}( u^-)
=\frac {\partial_{r_s}\sigma_k( u^-)}{2 \partial_{r_s}\sigma_k( u^-)-1}\,,
\quad \mbox{along }  \tilde {\mathcal{C}}_s \,.
\end{equation}
In particular, $\Sigma_d $ is transversal to
$\Sigma_s$ along ${\mathcal{C}}_s$.

In this geometry, we define the \textit{admissible phase boundaries}
by cutting up the  supersonic phase boundaries but
$\Sigma_k$,
and deleting the subsonic ones belonging to
$$
\mathcal{D}:= \{(\sigma, S_{s}(\epsilon_{s},u^-)):  (\sigma,u^-)\in \Sigma_d , \
\epsilon_{s}^d(\sigma,u^-)<\epsilon_{s}\leq 0 \}\,.
$$
Near the sonic phase boundary $(\underline \sigma,\underline u^-)$, the set $\mathcal{P}\mathcal{B}_{ad}$ of the admissible
phase boundaries writes
$$\mathcal{P}\mathcal{B}_{ad}=\Sigma_k \cup ( \mathcal{P}\mathcal{B}_{\rm det}\setminus \mathcal{D} )\,.
$$
From (\ref{S})
the $S_{s}$ function pastes the hypersurfaces $\Sigma _k$ and $\Sigma _d$ as
 $$
 \Sigma _d \ni (\sigma,u^-)\longmapsto (\sigma ,
 S_{s}(\epsilon_{s}^d(\sigma,u^-),u^-))\in \Sigma_k\,,
 $$
and the Hugoniot function $H$ is defined on the pasted set,
\begin{equation}\label{paste}
H(\sigma,S_{s}(\epsilon_{s}^d(\sigma,u^-),u^-)=H(\sigma , u^-)\,.
\end{equation}

\subsection{A geometrical well-posedness criterion, with BV stability}

Here we assume that $\Sigma_k$ is not an Hugoniot stationary manifold :
\begin{equation}\label{neqH}
(\partial_{r_s}\sigma_k)(u^-) \neq 0 \,, \quad \forall  u^-\in \Pi {\mathcal{C}}_s\,.
\end{equation}
$\Sigma_k$ is located above one side of
$\Pi {\mathcal{C}}_s$; on that side we define $\tilde \sigma_k (u^-)= \sigma_k (u^-)$, on the
other side we
define $\tilde \sigma_k (u^-)= \lambda_{s} (u^-)$. The function $\tilde \sigma_k$ is continuous
near $\underline
u^-$ and piecewise smooth. Its graph $\tilde \Sigma_k$ is the union of $\Sigma_k$ and of one part
of $\Sigma_s$,
$$\tilde\Sigma_k = \Sigma_k \cup\{(\lambda_{s}(u^-),u^-):  \sigma_k (u^-)\leq
\lambda_{s}(u^-)\}\,.
$$
Denoting by  $L_{s}(\epsilon_{s},u^-)$ the $s$-Lax function which
restriction to the shocks is $S_{s}(\epsilon_{s},u^-)$, for a given
$(\sigma_0,u^-_0)$ close to $(\lambda_{s} (\underline u^-),\underline u^-)$,
we obtain that the curve $\epsilon_{s}\mapsto (\sigma_0, L_{s}(\epsilon_{s},u^-_0))$
is transversal to $\tilde \Sigma_k$ according to (\ref{neqH}),
and contained in the hyperplane $\{\sigma =\sigma_0\}$.
It reaches the Lipschitz-continuous hypersurface $\tilde \Sigma_k$ for
a defined $\epsilon_{s} =\tilde \epsilon_{s}(\sigma_0, u^-_0)$. The
function
$(\sigma, u^-)\mapsto \tilde \epsilon_{s}(\sigma, u^-)$ is Lipschitz-continuous and piecewise smooth; thus, the
composition
$H(\sigma, L_{s}(\tilde \epsilon_{s}(\sigma, u^-),u^-))$ has the same
property

\begin{figure}[th]
\begin{center}
\begin{picture}(214,170)(-50,-48) %(-112,-50)
\setlength{\unitlength}{1.4pt}
\put(-7,30){\makebox(0,0)[l]{{ ${\mathcal{C}}_s$}}}
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\put(120,20){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(15,-45){\qbezier(-20,50)(20,55)(40,40)}
\put(40,8){\makebox(0,0)[l]{{ $\Sigma_s$}}}
\put(0,50){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(60,30){\qbezier(-80,20)(-70,0)(-65,-25)}
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\put(0,0){\qbezier(-75,10)(-60,22)(-55,24)}
\put(-55,-18){\makebox(0,0)[l]{{ $\Sigma_d$}}}
\put(0,0){\qbezier(-75,10)(-60,-5)(-55,-30)}
\put(0,0){\qbezier(-65,26)(-30,20)(-5,5)}
\put(-8,41){\qbezier(-70,28)(-30,21)(-12,8)}
\put(-70,58){\makebox(0,0)[l]{{ $\Sigma_k$}}}
\put(80,55){\qbezier(-60,20)(-67,-5)(-65,-20)}
\put(10,58){\makebox(0,0)[l]{{$s$-rarefaction}}}
\put(12,82){\makebox(0,0)[l]{{$ _{(\sigma, u^-) \in\
\mathcal{P}\mathcal{B}_{\rm ind}\setminus \mathcal{D}}$}}}
\put(18.5,75){\makebox(0,0)[l]{{$\bullet$}}}
\put(13.5,32){\makebox(0,0)[l]{{$\downarrow$}}}
\put(6,25){\makebox(0,0)[l]{{
$ _{(\sigma,L_s(\tilde\epsilon_s(\sigma,u^-),u^-))}$
}}}
\put(15,30){\qbezier(-58,5)(-70,-8)(-65,-25)}
\put(-60,15){\makebox(0,0)[l]{{$s$-shock}}}
\put(-65,-0){\makebox(0,0)[l]{{$ _{(\sigma, u^-)\
\in  \mathcal{D}}$}}}
\put(-51.5,5){\makebox(0,0)[l]{{$\bullet$}}}
\put(-46,35){\makebox(0,0)[l]{{$\nearrow$}}}
\put(-86,42){\makebox(0,0)[l]{{
$ _{(\sigma,L_s(\tilde\epsilon_s(\sigma,u^-),u^-))}$
}}}
\put(-10,-15){\makebox(0,0)[l]{{$\bullet$}}}
\put(-9,-15){\makebox(0,0)[l]{{
$ _{(\sigma,u^-)=(\sigma,L_s(\epsilon_s(\sigma,u^-),u^-)) \in \
\mathcal{P}\mathcal{B}_{\rm det}\setminus \mathcal{D}}$ }}}
\put(110,45){\makebox(0,0)[t]
{\hbox{$\epsilon_s\mapsto (\sigma, L_s(\epsilon_s,u^-))$ curves :
}}}
\put(110,35){\makebox(0,0)[t]
{\hbox{ $s$-shock is $\tilde\epsilon_s(\sigma,u^-)<0$}}}
\put(110,25){\makebox(0,0)[t]
{\hbox{$s$-rarefaction is $\tilde\epsilon_s(\sigma,u^-)>0$}}}
\end{picture}
\end{center}
\caption{Mixed sonic-kinetic geometry: how to reach admissible
  phase boundaries}
\end{figure}

In regard to the pasting property (\ref{paste}), we already have
a definition of Lipschitz-continuous  for the
phase boundary part in a mixed sonic-kinetic Riemann problem
$$
u^+=H(\sigma,L_{s}(\epsilon_{s}(\sigma,u^-),u^-))
$$
where the discontinuous function
\begin{equation}\label{epss}
\epsilon_{s}(\sigma,u^-):=  \begin{cases}
\tilde \epsilon_{s}(\sigma, u^-)\,, & \mbox{if }
(\sigma, u^-)\in \mathcal{D} \mbox{ or }   \sigma \geq \lambda_{s}(u^-)
\\
0   \,, & \mbox{if }   (\sigma,u^-)\notin \mathcal{D} \mbox{ and }
 \sigma \leq \lambda_{s}(u^-)
\end{cases}
\end{equation}
is defined in such a way that $(\sigma,L_s(\epsilon_s(\sigma,u^-),u^-))$ is an
admissible phase boundary.
 To consider at most one indetermination in the Riemann
problems, we assume that
\begin{equation}\label{sub1}
\lambda_{s }(\underline u^+)<\lambda_{s}(\underline u^-)< \lambda_{s+1}(\underline u^+)\,.
\end{equation}
We use classical Lax functions $L^\pm_i(\epsilon_i,u^\pm)$ to describe
the small simple waves on the
left $(-)$, or on the right
$(+)$, of the phase transition curve $x=\sigma t$ in the space of the independent
variables $(t,x)$. We denote by
$\epsilon^-_{nc}:= (\epsilon_1,\dots ,\epsilon_{s-1})$ the strength of  the waves on the left,
called \textit{not causal}, that (surely) leave the discontinuity $x=\sigma t$; in the
same way
$\epsilon^+_{nc}:= (\epsilon_{s+1},\dots ,\epsilon_{N})$. We  write also
\begin{gather*}
L^-_{nc}(\epsilon^-_{nc},u^-)
:= L^-_{s-1}(\epsilon_{s-1},\dots ,L^-_1(\epsilon_1,u^-)\dots ),\\
L^+_{nc}(\epsilon^+_{nc},u^+)
:= L^+_{N}(\epsilon_{N},\dots ,L^+_{s+1}(\epsilon_{s+1},u^+)\dots ),
\end{gather*}
and $r^\pm_{nc}(u^\pm) := \partial_{\epsilon^\pm_{nc}}L^\pm_{nc}(0,u^\pm)$. \\
The  mixed sonic-kinetic Riemann problem reads
\begin{equation}\label{RP}
\begin{gathered}
u^+=L^+_{nc}(\epsilon^+_{nc},H(\sigma, L^-_s(\epsilon_s(\sigma, u^\#), L^-_{nc}(\epsilon^-_{nc},u^-))))\,,
\\
u^\# := L^-_{nc}(\epsilon^-_{nc},u^-)\,,
\end{gathered}
\end{equation}
where $\epsilon_s(\sigma, u^\#)$, defined by (\ref{epss}), forces the extra $s-$wave to
reach  transversally the mixed sonic-kinetic
geometry when the phase boundary $(\sigma, u^\#)$ is not admissible. Its well-posedness may be
obtained from the Lipschitz-continuous inverse function theorem of
Clarke \cite{C}.

\begin{figure}[th]
\begin{center}
\begin{picture}(120,110)(-40,-25) %(-140,-40)
\setlength{\unitlength}{1.4pt}
\put(-50,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\thicklines\line(1,3){17}}
\put(25,55){\makebox(0,0)[tr]
{\hbox{ $ _{x=\sigma t}$}}}
\put(0,35){\makebox(0,0)[b]
{\hbox{ $u^{\#}$}}}
\put(0,0){\line(-3,1){50}}
\put(-30,20){\makebox(0,0)[br]
{\hbox{$\epsilon _{nc}^-$}}}
\put(0,0){\line(2,1){45}}
\put(45,25){\makebox(0,0)[br]
{\hbox{$\epsilon_{nc}^+$}}}
\put(-40,-6){\makebox{$u^-$}}
\put(30,-6){\makebox{$u^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{$(\sigma,u^\#)$ admissible}}}
%RIGHT:
\put(120,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\line(1,3){16}}
\put(0,0){\thicklines\line(2,3){33}}
\put(43,55){\makebox(0,0)[tr]
{\hbox{$ _{x=\sigma  t}$}}}
\put(0,0){\line(-3,1){60}}
\put(-30,18){\makebox(0,0)[br]
{\hbox{$\epsilon _{nc}^-$}}}
\put(0,0){\line(2,1){55}}
\put(55,18){\makebox(0,0)[br]
{\hbox{$\epsilon_{nc}^+$}}}
\put(5,30){\makebox(0,0)[b]
{\hbox{$u^{\#}$}}}
\put(15,27){\makebox(0,0)[b]
{\hbox{$u^b$}}}
\put(12,50){\makebox(0,0)[b]
{\hbox{$ _{\epsilon_s(\sigma,u^{\#})}$}}}
\put(-40,-6){\makebox{$u^-$}}
\put(30,-6){\makebox{$u^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{$(\sigma,u^\#)$ non admissible, $(\sigma,u^b)\in \tilde \Sigma_k$}}} }
}
\end{picture}
\end{center}
\caption{}
\end{figure}


 A necessary condition for the well-posedness is the stability condition
\begin{equation}\label{stab}
\det \big(r^-_{nc}(\underline u^-)\,, \partial_\sigma H(\underline
\sigma, \underline u^-)\,, r^+_{nc}(\underline
u^+)\big) :=\underline D\neq 0,
\end{equation}
where
$$
\partial_\sigma H(\underline \sigma, \underline u^-)
= (Df(\underline u^+)-\lambda_s(\underline u^-)I)^{-1}[\,\underline u\,]\,.
$$
To complete the proof of well-posedness we need to compute
$$
D_{\epsilon^-_{nc}}(H(\underline \sigma, L^-_s(\tilde \epsilon_s(\underline \sigma,L^-_{nc}(\epsilon^-_{nc},\underline u^-)),
L^-_{nc}(\epsilon^-_{nc},\underline u^-))))(0)
$$
for the two smooth forms of $\tilde \epsilon_s(\sigma,u^-)$ associated to the impact of the
$s-$Lax curves, starting from
$(\sigma,u^-)$,
 with $\Sigma_s$ or $\Sigma_k$. This differential contains the term $\partial_{r_s}H(\underline \sigma,
\underline  u^-) =0$ as a
factor at each time where $\tilde \epsilon_s$ occurs genuinely. Thus, at point
$(\epsilon^-_{nc},\sigma, u^-)=(0,\underline \sigma,\underline u^-)$ the determinant of the
differential of the composition associated to these two forms in (\ref{RP}) is still
$\underline D$. Clarke's local inverse function theorem applies and the solution
$(\epsilon^-_{nc},\sigma, \epsilon^+_{nc})(u^-,u^+)$ is
Lipschitz-continuous.

To control the variation of the wave solution in the $(t,x)$ independent variables, we need to
estimate the strength
$$
\tilde \epsilon_s(u^-,u^+):= \tilde \epsilon_s(\sigma(u^-,u^+),L^-_{nc}
(\epsilon^-_{nc}(u^-,u^+),u^-))
$$
of the possible $s$-wave at the left of the phase boundary $x=\sigma t$.
Since, from (\ref{neqH}), the function $\tilde\epsilon_s$ is
Lipschitz-continuous we have
\begin{align*}
\tilde \epsilon_s(u^-,u^+)
&=O(1)(|\sigma(u^-,u^+)-\underline \sigma |+|\epsilon^-_{nc}(u^-,u^+)|
+|u^--\underline u^-|)\\
&= O(1)(|u^--\underline u^-|+|u^+-\underline u^+|)\,.
\end{align*}
We have proved the following theorem.

\begin{theorem}[mixed sonic-kinetic Riemann problem]\label{skrp} \quad \\
Let $(\underline u^-,\underline \sigma,\underline u^+)$ be a sonic simple phase wave satisfying (\ref{nch}),
(\ref{vnl}),  (\ref{sub1}),
(\ref{stab}). Let $\Sigma$ a smooth hypersurface of ${\bf R}^{1+N}$ containing
$(\underline \sigma, \underline u^-)$ and satisfying (\ref{transs}), (\ref{hyp}), (\ref{neqH}). There exists a
neighbourhood
$\Omega^-\times\Omega^+$ of $(\underline u^-,\underline u^+)$, a neighbourhood
$\omega^-_{nc}\times\omega\times \omega^+_{nc}$ of $(0,\underline \sigma ,0)$, such that for every
$(u^-,  u^+)\in
\Omega^-\times\Omega^+$, there exists a unique
$(\epsilon^-_{nc},\sigma,\epsilon^+_{nc})\in \omega^-_{nc}\times\omega\times \omega^+_{nc}$
satisfying
$$
u^+=L^+_{nc}(\epsilon^+_{nc},H(\sigma, L^-_s(\epsilon_s(\sigma, u^\#), L^-_{nc}(\epsilon^-_{nc},u^-))))\,,
 \quad  u^\# := L^-_{nc}(\epsilon^-_{nc},u^-)\,,
$$
where $\epsilon_s(\sigma, u^-)$ is defined by (\ref{epss}),  in such a way that
$(\sigma, L^-_s(\epsilon_s(\sigma, u^\#))\in \mathcal{P}\mathcal{B}_{ad}$. \\
The function
$(u^-,u^+)\mapsto  (\epsilon^-_{nc},\sigma,\epsilon^+_{nc})$ is
Lipschitz-continuous and the variation of the phase wave solution
is estimated by
$$
|\epsilon^-_{nc}|+|\epsilon_s(\sigma, u^\#)|+
|\sigma-\underline \sigma |+|\epsilon^+_{nc}|
=O(1) (|u^--\underline u^-| +|u^+-\underline u^+|)\,.
$$
\end{theorem}


\paragraph{Remark}
When $\Sigma$ is an Hugoniot stationary manifold,
$$(\partial_{r_s}\sigma_k)(u^-) =0 \,, \quad \forall  u^-\in \Pi {\mathcal{C}}_s\,,$$
which is not degenerate,
$(\partial_{r_s}^2\sigma_k)(u^-) \neq 0 $, for all
$u^-\in \Pi {\mathcal{C}}_s$,
$\Sigma$ is located at one side of the hypersurface drawn from
${\mathcal{C}}_s$ by the integral curves of the Hugoniot stationary field $\partial_{r_s}$, with no
intersection  but ${\mathcal{C}}_s$, where they are tangent. Moreover, from (\ref{dk1}), $\Sigma_d$ also is an Hugoniot's
stationary manifold. This
geometry leads to an amplification of the variation of the solution of the mixed
sonic-kinetic Riemann problem. The
amplification concerns the polarization of $u^-$ in the $r_s(u^-)$ direction. A simple example in
the elasticity setting is
studied in \cite{ST}.

\section{Two non-classical phase transition examples}\label{SY-M}


\subsection{Shearer-Yang  model for dynamic elasticity}
In \cite{S-Y}, Shearer and Yang prove the existence of phase boundary profiles with
finite viscosity and finite capillarity for a qualitative model
in elasticity. The system
\begin{equation}\label{VC}
\begin{gathered}
\partial_t u-\partial_x v=0 \\
\partial_t v-\partial_x f(v)
=\epsilon \partial_x^2v-A \epsilon^2 \partial_x^3u
\end{gathered}
\end{equation}
depends on a viscosity parameter $\epsilon >0$ and a balance viscosity-capillarity fixed
parameter
$A>0$. The stress $f$ is cubic, $f(u) = u^3-u$.
Travelling waves solutions of (\ref{VC}), $(u,v)(t,x)=(U,V)(\frac{x-\sigma t}{\epsilon})$, with
asymptotic conditions
$$
(U,V)(-\infty)=(u_L,v_L)\,, \quad (U,V)(+\infty)=(u_R,v_R)\,,\quad
(U',U'')(\pm \infty)=0\,,
$$
 are defined by profiles $(U,V)(\xi)$, which satisfy $V(\xi)=v_L-\sigma (U(\xi)-U_L)$, and the
ordinary differential equation
$$
AU''=-\sigma U'+ f(U)-f(u_L)-\sigma^2(U-U_L)\,.
$$
The state $u_R$ is a rest point for this equation if $(u_L,\sigma,u_R)$ satisfies the
Rankine-Hugoniot condition
$$ f(u_R)-f(u_L)-\sigma ^2 (u_R-u_L)=0\,.
$$
This is one of the jump equation for the discontinuous solution
$(u,v)=(u_L,v_L)$ if $x<\sigma t$, $(u,v)=(u_R,v_R)$ if
$x>\sigma t$, of the elasticity equation
\begin{equation}\label{S-Y}
\begin{gathered}
\partial_t u-\partial_x v=0\\
\partial_t v-\partial_x f(v)=0
\end{gathered}
\end{equation}
The other jump condition, $(v_R-v_L)+\sigma (u_R-u_L)=0$, is explicit in $v_R$; so, the relevant phase
boundary variables are $(\sigma,u_L)$. Phase transitions occur from $\{u_L<
-1/\sqrt 3\}$ to $\{u_R>
1/\sqrt 3\}$ across a discontinuity with speed $\sigma <0$. The boundary phase variables for
the hyperbolic equation (\ref{S-Y}) are $(\sigma,u_-)$, $\sigma<0$, $u_-<-1/\sqrt 3$. In this
space the sonic manifold is the curve
$\sigma=-\sqrt{3u_-^2-1}$.

For every $A$ in $]1/3,4/9[$, near the sonic phase boundary,
$(\underline \sigma,\underline u_-)
=\big(-\sqrt {3\frac{2+A}{2-3A}},-\frac{2}{3}\sqrt\frac{6}{2-3A}\big)$,
Shearer and Yang obtain simple phase waves as limit
$\epsilon\to 0$ of viscosity-capillarity profiles
$(U,V)(\frac{x-\sigma t}{\epsilon})$ for the following cases:
\\
Case 1: $\sigma =\sigma_k(u_-):=-3\sqrt\frac{A}{
2} \frac{u_--\sqrt{3(6A-1)u_-^2-2(9A-2)}}{2-9A}
$ in the domain
$\sigma>-\sqrt{3u_-^2-1}$, this is the curve $\Sigma_k$,
\\
Case 2: $\sigma <\sigma_d(u_-)$ in the domain $\sigma\leq -\sqrt{3u_-^2-1}$,
$ u_-\geq \underline u_-$,
\\
Case 3: $\sigma \leq -\sqrt{3u_-^2-1}$ in the domain $u_-\leq \underline u_-$,
\\
where $\sigma_d$ is defined from the Hugoniot dipoles $((\sigma,\sigma_k(u_-)),(\sigma,\sigma_d(u_-)))$. The
borderline of the second domain is the detonation curve $\Sigma_d$, which is not admissible, and
the borderline of the third domain is the admissible part of the sonic
curve $\Sigma_s$.

\begin{figure}[th]
\begin{center}
\begin{picture}(266,180)(60,-75)
\setlength{\unitlength}{1.4pt}
%\put(0,0){
\put(145,0){
\put(0,55){\vector(1,0){60}}
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\put(39,57){\makebox{$ _0$}}
\put(48,33){\makebox{$\frac{-1}{\sqrt 3}$}}
\put(46,-40){\makebox{$ _{-1}$}}
\put(0,55){\line(-1,0){90}}
\put(62,54){\makebox{$\sigma$}}
\put(42,72){\makebox{$u_-$}}
\put(5,45){
%s
\qbezier(-72,-90)(-29,-10)(40,-10)
\put(-77,-80){\makebox{$\Sigma_s$}}
%d
\qbezier(-45,-50)(-35,-23)(-15,-5)
\put(-14,-2){\makebox{$\Sigma_d\not\subset \mathcal{P}\mathcal{B}_{ad}$}}
%k
\qbezier(-45,-50)(-10,-40)(0,-40)
\put(-0,-42){\makebox{$\Sigma_k\subset \mathcal{P}\mathcal{B}_{ad}$}}
}
}
\put(130,15){\makebox{$\mathcal{D}$}}
\put(112,10){\makebox{$ _{\rm no connection}$}}
\put(67,11){\makebox{$ _{\rm connections}$}}
\put(108,-15){\makebox{$ _{\rm no connection}$}}
\put(85,-7){\makebox{$ _{(\underline \sigma,\underline u_-)}$}}
\put(103,53){\makebox{$|$}}
\put(85,67){\makebox{$\
_{\underline \sigma_-=-\sqrt{3\frac{2+A}{2-3A}}}$}}
\put(192,0){\makebox{$\
_{\underline u_-=-\frac{2}{3}\sqrt\frac{6}{2-3A}}$}}
\put(189,-7){\makebox{-}}
\put(189,-42){\makebox{-}}
\end{picture}
\end{center}
\caption{Viscosity-capillarity admissible phase boundaries
of Shearer-Yang's model}
\end{figure}


\paragraph{Remark}  The viscosity-capillarity admissibility criterion, in the elasticity
setting, has been introduced by Slemrod \cite{S}.

\subsection{Majda's model for dynamic combustion}\label{Majda}

In \cite{Ma}, Majda proves the existence of detonation profiles with
finite reaction rate and finite diffusion for a qualitative model of combustion. The equation
\begin{equation}\label{V.R}
\begin{gathered}
\partial_t(u+q_0z)+\partial_xf(u)=\beta  \partial_x^2u\\
\partial_tz =-K \varphi (u) z
\end{gathered}
\end{equation}
depends on a diffusion parameter $\beta >0$ and a reaction rate parameter $K>0$. The chemical
reaction is exothermic, so that $q_0 >0$.
The condition satisfied by the smooth function $\varphi$,
$$\varphi(u)=
\begin{cases} 0 & \mbox{for } u\leq 0\\
\varphi(u)\in ]0,1]& \mbox{for } u> 0\\
1&\mbox{for } u\geq c_0>0\\
\end{cases}
$$
is an ignition temperature kinetics condition. The flux $f$ is smooth,
strictly increasing and strictly convex.
The travelling wave solutions  $(u,z)(t,x)=(U,Z)(\frac{x-\sigma t}{\beta})$,
 with asymptotic conditions
$$
(U,Z)(-\infty)=(u_L,0), \quad (U,Z)(+\infty)=(u_R,1), \quad
(U',Z')(\pm \infty)=0
$$
 satisfy the ordinary differential equation which depends on
$K_0:=\beta K$,
\begin{gather*}
U'=f(U)-\sigma (U+q_0Z)+(\sigma u_L-f(u_L))\\
Z'=\frac{1}{\sigma} K_0 \varphi (U) Z\\
\end{gather*}
The state $(u_R,1)$ is a rest point for this equation if $u_R\leq 0$ and
$$ f(u_R)-f(u_L)-\sigma  (u_R+q_0-u_L)=0 $$
which is the Rankine
Hugoniot condition for the discontinuous solution $u=u_L$ if $x<\sigma t$, $u=u_R$ if
$x>\sigma t$, of the combustion model
\begin{equation}\label{M}
\begin{gathered}
\partial_t (u+q_0 Y(u))+\partial_x f(u)=0,\\
 Y(u)=1  \mbox{ if }  u<0, \quad  Y(u)=0  \mbox { if } u>0\,.
\end{gathered}
\end{equation}
The boundary phase variables for the hyperbolic
equation (\ref{M}) are $(\sigma,u_-)$, $\sigma >0$, $u_->c_0$.
In this space the sonic
manifold $\Sigma_s$ is the curve
$\sigma=f'(u_-)$.

In \cite{Ma} we find the following result. First, a burnt state $(\underline u,\underline z)=(\underline u,0)$, $\underline
u>c_0$, being fixed, the Rankine-Hugoniot equation can be solved as
$u_R=u_R(\sigma,u_L,q_0)\leq 0$ for every
$(\sigma,u_L)$ close to the sonic phase boundary $(\underline \sigma,\underline u)=(f'(\underline u),\underline u)$, if
$$
q_0>\hat q(\underline\sigma,\underline u)
:=\underline u-\frac{f(\underline u)-f(0)}{\underline \sigma}\,.
$$
Let us remark that  $\hat q$ is Hugoniot dipole invariant, that is $\hat q( \sigma,v(u,\sigma))=\hat q(
\sigma,u)$ for every Hugoniot dipole $((\sigma,v(\sigma,u)),(\sigma,u))$ near $(\underline \sigma,\underline u)$. Moreover
the level curves of $\hat q$, $\hat q(\sigma,u)=Cte$, are graphs of strictly convex
smooth functions of $u$ with minimum speed $\sigma$ at the intersection with the sonic curve.

Second, $K_0$ being fixed, there exists a critical function
$q_0^{cr}(\sigma,u_L,K_0)$ defined in the indeterminate (supersonic) domain $f'(u_L)\leq \sigma$,
which satisfies
$q_0^{cr}(\sigma,u_L,K_0)\geq \hat q( \sigma,u)$, and such that:
\begin{itemize}
\item If $q_0=q_0^{cr}(\sigma,u_L,K_0)$, it exists a unique connection
$(U,Z)(\xi)$ between $(u_L,0)$ and $(u_R(\sigma,u_L,q_0),1)$,
\item If $q_0>q_0^{cr}(\sigma,u_L,K_0)$, it exists a unique connection
 between $(v(\sigma,u_L),0)$ and $(u_R(\sigma,u_L,q_0),1)$,
\item If $q_0^{cr}(\sigma,u_L,K_0)> \hat q(\sigma, u)$ and $q_0^{cr}(\sigma,u_L,K_0)>q_0> \hat q(
\sigma,u)$, no connection between $(u_L,0)$ or $(v(\sigma,u_L),0)$ and
$(u_R(\sigma,u_L,q_0),1)$ exists.
\end{itemize}

 For our purpose, we can read this result in the following way.
An exothermic reaction parameter $q_0 >0$ is fixed. For $K_0>0$, near a sonic
phase boundary
$(\underline \sigma,\underline u)$ which satisfies
$$
q_0^{cr}(f'(\underline u),\underline u,K_0)= q_0
>\hat q(f'(\underline u),\underline u)
$$
we define $\tilde q_0^{cr}$ as the extension by Hugoniot dipole
invariance $\tilde q_0^{cr}(\sigma,v(\sigma,u))=q_0^{cr}(\sigma,u)$.
Simple phase waves are obtained as the limit,
$\beta\to 0$, of viscosity-reacting travelling waves:
\begin{itemize}
\item 
 For $q_0^{cr}(\sigma ,u_L,K_0) = q_0$ in the domain
$\sigma \geq f'(u_L)$, 
\item
 For  $\tilde q_0^{cr}(\sigma,u_L,K_0) <
q_0$ in the domain $\sigma\leq f'(u_L)$.
\end{itemize}


We expect the function $q_0^{cr}$ to be smooth
enough, so the geometry of the phase boundaries space is again
 mixed sonic-kinetic.

\begin{figure}[th]
\begin{center}
\begin{picture}(160,160)(85,-65)
\setlength{\unitlength}{1.4pt}
\put(145,0){
\put(-80,-35){\vector(1,0){160}}
\put(-75,-50){\vector(0,1){120}}
\put(-66,-42){\makebox{${c_0}$}}
\put(-66,-36){\makebox{${|}$}}
\put(70,-40){\makebox{$u_L$}}
\put(-82,67){\makebox{$\sigma$}
}

\put(5,45){
%q-chapeau
\qbezier(-65,20)(-25,-20)(30,20)
\put(28,20){\makebox{$ _{\{\hat q(\sigma,u_L)=q_0\}}$}}
%s
\qbezier(-80,-70)(-29,5)(20,20)
\put(8,22){\makebox{$ _{\Sigma_s}$}}
%k
\qbezier(-65,-10)(-55,-23)(-48,-28)
\put(-147,-8){\makebox{$ _{\mathcal{P}\mathcal{B}_{ad}\supset
\{q_0^{cr}(u_L,\sigma ,K_0) = q_0\}=:\Sigma_k}$}}
%d
\qbezier(-48,-28)(-25,-25)(-10,-15)
\put(-10,-15){\makebox{$ _{\Sigma_d:=\{q_0^{cr}(v(\sigma,u_L),\sigma
,K_0) = q_0\}\not\subset\mathcal{P}\mathcal{B}_{ad}}$}}
}
}
\put(98,35){\makebox{$ _{\rm no connection}$}}
\put(60,17){\makebox{$ _{\rm no connection}$}}
\put(100,-0){\makebox{$ _{\rm connections}$}}
\put(98,12){\makebox{$ _{(\underline \sigma,\underline u_-)}$}}
\end{picture}
\end{center}
\caption{ Viscosity-reaction admissible phase
boundaries of Majda's model }
\end{figure}

\section{The analytical entropy condition}\label{entropy}

 We assume here that the system
(\ref{lc}) admits an entropy $\eta$, convex in each domain $\Omega^\pm$. If $q$ denotes the
entropy-flux, ($Dq(u)=D\eta(u)Df(u)$), the admissible weak solutions of (\ref{lc}) with
values in only one of the domains $\Omega^\pm$ satisfy the analytical entropy
condition $\partial_t\eta (u)+\partial_xq(u)\leq 0$, (which is equivalent to the geometrical
compressive one \cite{L}).

The only phase boundaries $(\sigma,u^-)$ which satisfy the geometric
entropy condition are the determinate (subsonic) ones.
For $(u^-,\sigma ,H(\sigma ,u^-))$, a simple phase  wave,
 the analytical entropy
condition $\partial_t\eta (u)+\partial_xq(u)\leq 0$ is equivalent to
$$\mathcal{E}(\sigma ,u^-) := (\sigma
\eta -q)(H(\sigma ,u^-))-(\sigma \eta -q)(u^-)\geq 0\,.$$
Since
$$\partial_{r_s}(\sigma \eta -q)(u^-)= (\sigma D\eta (u^-)-Dq(u^-))r_s(u^-)=
D\eta (u^-)(\sigma -\lambda_s(u^-))r_s(u^-)
$$
vanishes on $\Sigma_s$, as well as $\partial_{r_s}H$, we obtain
$(\partial_{r_s}\mathcal{E})(\lambda_s(u^-) ,u^-)=0$.
Otherwise the derivation of the jump equation, we get that
$$
(Df(H(\sigma,u^-))-\sigma I)\partial_{r_s}H(\sigma,u^-)=(Df(u^-)
-\sigma I)r_s(u^-)=(\lambda_s(u^-)-\sigma)r_s(u^-)
$$
implies, using (\ref{vnl}),
\begin{align*}
(Df(H(\lambda_s(u^-),u^-))-\lambda_s(u^-) I)(\partial_{r_s}^2H)
(\lambda_s(u^-),u^-)
&=(\partial_{r_s}\lambda_s)(u^-)r_s(u^-)\\
&=r_s(u^-)\,.
\end{align*}
 From what follows that for  $u^+=H(\lambda_s(u^-),u^-)$,
\begin{align*}
&\partial_{r_s}^2((\sigma \eta -q)\circ H)(\lambda_s(u^-),u^-)\\
&= D\eta (u^+)(\lambda_s(u^-)I-Df(u^+))(\partial_{r_s}^2H)
(\lambda_s(u^-),u^-)\\
&= -D\eta (u^+)r_s(u^-)\,.
\end{align*}
Since $(\partial_{r_s}^2(\sigma \eta -q))(\lambda_s(u^-),u^-)
=-D\eta (u^-)r_s(u^-)$,
we obtain
$$
(\partial_{r_s}^2\mathcal{E})(\lambda_s(u^-) ,u^-)
=-(D\eta (u^+))-D\eta (u^-))r_s(u^-)\,.
$$
Therefore, the analytical entropy condition
\begin{equation}\label{E}
\mathcal{E}(\underline \sigma ,\underline u^-)>0\,, \quad
(D\eta (\underline u^+)-D\eta (\underline u^-))r_s(\underline u^-)<0
\end{equation}
implies that near $(\underline \sigma ,\underline u^-)$, along the integral curves of $\partial_{r_s}$, the
entropy dissipation rate $\mathcal{E}(\sigma ,u^-)$ is minimum at the sonic point, where it is
positive.   As a consequence, every phase boundary $(\sigma,u^-)$ close to $(\underline \sigma ,\underline
u^-)$ satisfies the analytical entropy condition. Moreover, for an Hugoniot dipole
$((\sigma,u^-),(\sigma,v(\sigma ,u^-)))$ where
$(\sigma,u^-)$ is determinate (subsonic), we have
$$
\mathcal{E}(\sigma ,u^-) -\mathcal{E}(\sigma ,v(\sigma ,u^-))
= (\sigma \eta -q)(u^-)-(\sigma \eta -q)(v(\sigma ,u^-))\geq 0\,,
$$
because $\sigma$ is the speed of the admissible small amplitude $s$-shock from $u^-$ to $v(\sigma
,u^-)$. So, for an Hugoniot dipole, the entropy dissipation rate $\mathcal{E}$ is larger at
the determinate phase boundary than at the indeterminate one; the excess is that of the
small amplitude shock wave with the same speed, called above micro-detonation.  \\


 For lack of analytical entropy selection, and also lack of structure
profiles for complex physical systems, but indications on some relevant models, we
can use the properties of the entropy dissipation function $\mathcal{E}$ to define an
admissible set of indeterminate (supersonic) phase boundaries near a sonic one $(\underline \sigma ,\underline u^-)$. This
is an idea of Abeyaratne and Knowles \cite{A-K}, in the elasticity setting. We consider an extra function $\phi$
which is constant along the integral curves of
$\partial_{r_s}$, negative on $\Sigma_s$, and satisfies
$$
\phi (\underline \sigma ,\underline u^-)
=-\mathcal{E}(\underline \sigma ,\underline u^-)\,.
$$
As $\partial_{r_s}(\mathcal{E}+\phi )$ vanishes on
$\Sigma_s$, if $(\underline \sigma ,\underline u^-)$ is not a  critical point of $\mathcal{E}+\phi $, the set
$$K_{\phi}:= \{(\sigma ,u^-): \mathcal{E}(\sigma ,u^-) =-\phi (\sigma ,u^-)\}$$
is an Hugoniot-stationary manifold, which has to be rejected, (see the remark in the previous
section).
So we assume that
$(\underline \sigma ,\underline u^-)$ is a
 critical point of $\mathcal{E}+\phi $, that is
$D(\mathcal{E}+\phi )(\underline \sigma ,\underline u^-)=0$,
and we use the Morse theory \cite{Mi}, in the only physically
reasonable following case.
The differential of $\mathcal{E}+\phi $ vanishes not only at $(\underline
\sigma ,\underline u^-)$ but all
along an hypersurface ${\mathcal{C}}_s$ of $\Sigma_s$ which contains $(\underline \sigma ,\underline u^-)$,
$$
D(\mathcal{E}+\phi )(\sigma , u^-)=0,\quad \forall
(\sigma,u^-)\in {\mathcal{C}}_s\,.
$$
Also   the second differential at point $(\underline \sigma ,\underline u^-)$ of the restriction of $\mathcal{E}+\phi
$ to a plane transversal to ${\mathcal{C}}_s$ has signature $(1,1)$. Since every vector field
which is tangent to $\Sigma_s$, is orthogonal to $r_s$ at every point of ${\mathcal{C}}_s$, relatively
to the second differential of $\mathcal{E}+\phi $, the set $K_{\phi}$ then is the union of two
smooth hypersurfaces transversal along ${\mathcal{C}}_s$, not Hugoniot-stationary, every one
being transversal to $\Sigma_s$. 

\begin{figure}[th]
\begin{center}
\begin{picture}(170,110)(-90,-30)
\setlength{\unitlength}{1.2pt}
\put(40,8){\makebox(0,0)[l]{{ $\Sigma_s$}}}
\put(-70,58){\makebox(0,0)[l]{{ $K_\phi$}}}
\put(10,62){\makebox(0,0)[l]{{ $K_\phi$}}}
\put(-30,-15){\makebox(0,0)[l]{{ $K_\phi$}}}
\put(22,-17){\makebox(0,0)[l]{{ $K_\phi$}}}
\put(-7,30){\makebox(0,0)[l]{{ ${\mathcal{C}}_s$}}}
\put(0,0){\qbezier(-80,20)(-10,70)(40,40)}
\put(0,0){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(120,20){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(15,-45){\qbezier(-80,20)(-10,70)(40,40)}
\put(120,-20){\qbezier(-80,22)(-85,0)(-90,-20)}
\put(40,-10){\qbezier(-80,0)(-70,-10)(-65,-20)}
\put(0,50){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(60,30){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(100,55){\qbezier(-80,20)(-70,0)(-65,-25)}
\put(0,0){\qbezier(-26,-30)(-20,0)(34,30)}
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\put(-8,41){\qbezier(-70,28)(-30,21)(-12,8)}
\end{picture}
\end{center}
\caption{}
\end{figure}

 The indeterminate (supersonic) part of one of
these hypersurfaces can be selected as admissible phase boundaries subset $\Sigma_k$. This
set defines a (micro)-detonating hypersurface
$\Sigma_d$ to be rejected, and  a open domain $\mathcal{D}$ (with borderline $\Sigma_k\cup\Sigma_d$)
also to be rejected. Then only the sonic phase boundaries which do not belong to $\mathcal{D}$ may be
admissible, in a mixed sonic-kinetic setting.

\section{A non-classical Riemann problem near a sonic detonation wave.}
\label{C-J}

\subsection{ Chapman-Jouguet detonations}
We consider the one-dimensional combustion model, involving an infinite reaction
rate
between ideal polytropic gases with the same $\gamma$-law, $\gamma > 1$, and constant heat of
complete exothermic
reaction
 $Q>0$. This model is governed by the Euler equations of
gas
dynamics. In Eulerian coordinates
$V= (\rho ,\rho u,\rho e)$,  the conservative form is
 \begin{gather*}
 \partial _t \rho +  \partial _x (\rho u)  =  0\\
  \partial _t (\rho u) +\partial _x (\rho u^2+p)  =  0\\
\partial _t (\rho e) +\partial _x ((\rho e+p)u)  =  0
 \end{gather*}
A discontinuity curve $x=\chi (t)$ moving to the right, separates the burnt gas,
which is
on its left, from the unburnt
gas which is on its right. We denote with the index $-$ the functions and
variables in the
burnt gas, by the index $+$
those in the unburnt gas.

In the following, notation is taken from the book by \cite{G-R}. The specific internal energy is $\epsilon =
e-\frac {u^2}{2}$ ,   the
specific volume is $\tau =1/\rho$;
the equations of state, with constant energies of formation $Q_\pm$, are
$$\epsilon_-(\tau_-,p_-)= \frac {\tau_-p_-}{\gamma -1}+Q_-\,, \quad
\epsilon_+(\tau_+,p_+)= \frac {\tau_+p_+}{\gamma -1}+Q_+\,,\quad
Q:= Q_+-Q_->0\,.
$$
The eigenvalue which occurs for detonations is
$$\lambda_3= u_-+\sqrt{\gamma p_-\tau_-}\,.
$$
We select a  Chapman-Jouguet background  detonation wave
$(\underline V^-,\underline \sigma,\underline V^+)$. The
states $\underline V^\pm$ are constant and
they are related on each side of the discontinuity
$x=\underline \sigma t$ by the jump conditions,
where
$[x]:= x^+-x^-$,
\begin{equation}\label{RHE}
\begin{gathered}
\sigma [\rho ]   =  [\rho u] \\
\sigma [\rho u]  =  [\rho u^2+p] \\
\sigma [\rho e]  =  [(\rho e+p)u]
\end{gathered}
\end{equation}
Moreover, the wave is 3-sonic on the left hand side; it satisfies the
Chapman-Jouguet condition
$$ \sigma - u_- =  \sqrt{\gamma p_-\tau_-}$$
with the detonation property
$$ \sigma - u_+ > \sqrt{\gamma p_-\tau_-}\,,  $$
from which follows
$[p]<0<[ \tau ]$.
Using the dependent variables
$$M := \frac{u_--\sigma }{\tau_-}\,,\quad U^\pm = (u_\pm,\tau_\pm,p_\pm)\,,
$$
where $M$ is negative, the equation (\ref{RHE}) for the simple phase waves is equivalent to
\begin{equation}\label{RHM}
\tilde J(U^-,M,U^+) :=
 \begin{pmatrix}
 [u]- M  [\tau ]  \\[0pt]
 [p] + M^2[\tau ] \\[0pt]
 [\epsilon]+\frac{1}{2}[\tau](p_-+p_+)
 \end{pmatrix} =0\,,
 \end{equation}
from which $M = \frac {u_+-\sigma }{\tau_+}$ follows.
The third row also reads
$$
(\frac{\gamma +1}{\gamma -1} \tau_-- \tau_+)p_--
(\frac{\gamma +1}{\gamma -1}\tau_+- \tau_-)p_+ -2Q=0\,.
$$
We have
\begin{gather*}
D_{U^+}\tilde J (U^-,M,U^+)=  \begin{pmatrix}
1& -M
& 0\\
0 & M^2 & 1 \\
0 &
 (-p_- - \frac {\gamma +1}{\gamma -1} p_+) & - (\frac{\gamma +1}{\gamma -1}
\tau_+- \tau_-)
 \end{pmatrix}, \\
\det  D_{U^+}\tilde J(U^-,M,U^+)= \frac{2\tau_+}{\gamma-1}
(\gamma \frac{p_+}{\tau_+}-M^2) \neq 0\,,
\end{gather*}
near the Chapman-Jouguet background detonation $(\underline U^-,\underline M,\underline U^+)$. So, detonations are
semi-characteristic phase transition waves and the jump equation (\ref{RHM}) is locally
equivalent to $U^+=\tilde H(M,U^-)$ for a smooth Hugoniot function $\tilde H$. One eigenvalue of
$$
D_{U^-}\tilde J (U^-,M,U^+)=  \begin {pmatrix}
-1& M
& 0\\
0 & -M^2 & -1 \\
0 & (\frac {\gamma +1}{\gamma -1}p_- +  p_+)
 & (\frac{\gamma +1}{\gamma -1} \tau_-- \tau_+)
 \end{pmatrix},
 $$
is $-1$, the others are the
solutions $\lambda $ of the equation
$$
\lambda ^2 + (M^2 + [\tau] - \frac{2\tau_-}{\gamma -1} ) \lambda +
\frac{2\tau_-}{\gamma -1}
(\gamma \frac{p_-}{\tau _-}-M^2)=0\,.
$$
So, at a Chapman-Jouguet detonation, 0 is a simple eigenvalue of $D_{U^-}\tilde H$ and
the (sonic) set $\{det (D_{U^-}\tilde H(M,U^-))=0\}$ is the graph $\tilde \Sigma_s=\{(M,U^-): M=-
\sqrt{\gamma \frac{p_-}{\tau _-}}\}$.
For $(M,U^-)\in \tilde\Sigma_s$, the kernel of $D_{U^-}\tilde H(M,U^-)$ is $\mathbb{R} ^t(M,1, - M^2)$. Then
in the $(M,U^-)$ variables, a canonical Hugoniot's stationary vector field is
 $$
 \tilde X:= M  \partial _{u_-} +  \partial _{\tau_-} - M^2 \partial_{p_-}\,.
$$


\subsection{Hugoniot calculus, reduced phase boundary variables}

The solution $U^+=\tilde H(M,U^-)$ reads
$$(u_+,\tau_+,p_+)=(u_-+M(\tau_+-\tau_-), H(M,\tau_-,p_-))$$
and the set
$\tilde \Sigma_s$ is defined by a function independent of $u_-$.
Then a phase transition may be identified by
the \textit{reduced variables} $R := (M,\tau_-, p_-)$. In this space, the \textit{Chapman-Jouguet or sonic
set} is  the surface
$$\Sigma_s := \big\{  (M,\tau_-, p_-):
-M = \sqrt {\gamma\frac{ p_-}{\tau_-}}\big\}
$$
the \textit{indeterminate} (supersonic) combustion boundaries $R := (M,\tau_-, p_-)$ are defined
by $-M >\sqrt {\gamma \frac{p_-}{\tau_-}}$ and the
\textit{determinate} (subsonic) ones by
$-M < \sqrt {\gamma\frac{  p_-}{\tau_-}}$.
Let
$$\beta _\pm := \sqrt{\gamma \frac{p_\pm}{\tau _\pm}}\,,\quad
q:=\sqrt{2Q\frac{\gamma-1}{\gamma+1}}\,.
$$
At a sonic point
$R_s\in \Sigma_s$, the thermodynamical part
$$  X:=  \partial _{\tau_-} - M^2  \partial _{p_-}
$$
of the Hugoniot's stationary vector field writes
$X_s:= \partial _{\tau_-} -\beta^2_-\partial  _{p_-}$
and is transversal to $\Sigma_s$.
We also denote by $ J(R,\tau_+,p_+)$ the thermodynamic part of $\tilde J$,
$(\tau_+(R),p_+(R))=H(R)$ the local solution of $
J(R,\tau_+,p_+)=0$.
Using differential calculus, we obtain
\begin{equation}\label{dR}
\begin{gathered}
 \partial _M \tau_+ (R)= \frac {M [\tau] ((\gamma +1)\tau_+ -(\gamma
-1)\tau_-)}{\tau_+(\beta_+^2-M^2)}\,,\\
 \partial _M p_+(R) = \frac {- M [\tau] ((\gamma +1)p_+ +(\gamma
-1)p_-)}{\tau_+(\beta_+^2-M^2)}\,,\\
 \partial _{\tau_-} \tau_+ (R)= \frac {\tau_+ \beta^2_+-\tau_-M^2}{\tau_+(\beta_+^2-M^2)}
 \,,\quad
 \partial _{\tau_-}p_+(R) = \frac {M^2[p]}{\tau_+(\beta_+^2-M^2)}\,,\\
\partial _{p_-} \tau_+ (R)= \frac {-\gamma [\tau ]}{\tau_+(\beta_+^2-M^2)}
 \,,\quad
 \partial _{p_-}p_+(R)
 = \frac {\tau_- \beta^2_--\tau_+M^2}{\tau_+(\beta_+^2-M^2)}\,,\\
X \tau_+ (R)= \frac {\tau_-(\beta^2_--M^2)}{\tau_+(\beta_+^2-M^2)}
 \,,\quad
Xp_+(R) = \frac
{\tau_-M^2(M^2-\beta^2_-)}{\tau_+(\beta_+^2-M^2)}\,.
\end{gathered}
\end{equation}

\subsection{Entropy analysis}

The entropy inequality required for physical solutions of (\ref{RHE})
reads $\partial _t (\rho s)+  \partial _x (\rho su) \geq 0$,
where the entropy $s$ is
$$
s_\pm  = s_{0,\pm} + \frac {r_0}{\gamma -1}\
\log  (\frac {\tau _\pm ^{ \gamma } p_\pm}{ \gamma -1} )\,.$$
For the simple detonation waves, this inequality is equivalent to
$$[\rho s (u-\sigma )] = M [s] \geq 0\,,\quad {\rm or}\quad [s] \leq 0\,,$$
that is
$$ \mathcal{E}(R) := s_+(\tau_+(R), p_+(R))-s_-(\tau_-,p_-)\leq 0\,.$$
Everywhere we have
$$
Xs_-(R) = \frac{r}{\gamma-1} \frac{1}{p_-} (\beta^2_--M^2)\,,
$$
and at a sonic point $R_s$,
\begin{equation}\label{X}
X\tau_+(R_s)=0\,,\quad  Xp_+(R_s)=0\,,\quad
Xs_-(R_s)=0\,.
\end{equation}
Then $X \mathcal{E}(R_s)=0$.
Moreover we have
\begin{equation}\label{dE1}
\begin{gathered}
\partial_M \mathcal{E}(R) = r_0\frac {M[\tau ]^2}{\tau_+p_+}\,,\quad
\partial_{\tau_-} \mathcal{E}(R) =\frac{r_0}{\gamma-1}
\frac {[\tau ](M^2\tau_--\beta^2_+\tau_+)}{\tau_-\tau_+p_+}\,,\\
 X\mathcal{E}(R) = \frac{r_0}{\gamma -1} \frac{[\tau]}{\tau_+ p_-p_+}
 (\beta^2_--M^2)(\tau_+ M^2-p_-)\,,
\end{gathered}
\end{equation}
and on $\Sigma_s $,
\begin{equation}\label{det}
M^2 [\tau ]^2 =q^2\,,\quad 2\tau_- -\gamma [\tau] =\gamma
\frac{p_-+p_+}{M^2} > 0\,,
\quad X\beta^2_- = -\frac{\gamma+1}{\tau_-}\beta^2_- \,.
\end{equation}
As a consequence, at a sonic point $R_s \in \Sigma_s $, $$
X^2 \mathcal{E}(R_s)=-r_0\frac{\gamma+1}{\gamma-1} \frac{[\tau]}{\tau_-\tau_+p_-p_+}\
([\tau]+\frac{\gamma-1}{\gamma}\tau_-) \beta^4_- <0\,.
$$
Thus,
along the integral curves of $X$, the entropy dissipation rate $\mathcal{E}$ is locally maximum
at the Chapman-Jouguet detonation. From (\ref{det}) we also get at a
sonic point
$$
\frac {\tau _+ ^\gamma p_+}{ \tau _- ^\gamma p_-}=
(1+\frac{q}{\sqrt{\gamma p_-\tau_-}})^\gamma (1-\frac{\gamma q}{\sqrt{\gamma p_-\tau_-}})\,.
$$
Then the entropy inequality is satisfied near the background detonation
if we assume that its sonic speed $\underline \sigma$ satisfies
\begin{equation}\label{s0}
(1+\frac{q}{\underline \sigma-\underline u_-})^\gamma\
(1-\frac{\gamma q}{\underline \sigma-\underline u_-})<e^{-[s_0]\frac{\gamma -1}{r_0}}\,.
\end{equation}

\subsection{Entropic selection of admissible weak detonations}

Only partial results are known about the existence of structure profiles for reacting
compressible Navier-Stokes equations. In our knowledge, the most advanced analysis is that of
Gardner \cite{G}, in Lagrangian coordinates, with an ignition temperature
assumption, (see also Wagner \cite{Wa}). A rewriting of his existence theorem
\cite[Theorem 2.1, p. 438]{Wa}, where assumptions concern the
burnt states (in place of the unburnt ones, which is not correct), leads to the existence of a critical
liberated energy $q^{cr}$ in the way of \cite{Ma}. The function $q^{cr}$ defines a borderline of an
existence domain for detonation structure profiles. Then, in the reduced phase boundary variables
$(\sigma,(\tau,T))$, admissible indeterminate phase transitions have to belong to a
``kinetic'' surface.
The geometric properties of this surface at the intersection with the sonic surface should require some
estimates for the asymptotic unburnt critical point $C(z_*)$ of the weak connection of Gardner. In lack
of detailed structure profiles, but with the indication of  Majda's model in section
\ref{Majda} and Gardner's result,  we present here an example using the entropy to select the
indeterminate (supersonic) combustion boundaries near a sonic one
$\underline R = (\underline M,\underline\tau_-,\underline p_-)$, following section \ref{entropy}.\\

 With $(M,\tau_-)$ as parameters of the sonic surface $\Sigma_s$ near $\underline R = (\underline
M,\underline\tau_-,\underline p_-)$, we select the simplest curve ${\mathcal{C}}_s\subset \Sigma_s$  in regards to
calculus and geometry to illustrate our previous theory.
\begin{gather*}
\Sigma_s=\{R_-:  p_-=p_s(M,\tau_-):=\frac{1}{\gamma} M^2\tau_-\}\,,\\
{\mathcal{C}}_s=\{R_-:  R_-=R_s (\tau_-):= (\underline M\,,  \tau_-\,,
p_s(\underline M,\tau_-))\}\,.
\end{gather*}
We consider an extra
constitutive function $\phi (R)$, of thermodynamical nature, which
 is constant along the integral curves of the field $X$ and defined by
 a smooth positive  function $\phi_s$ on the sonic surface
$$
X\phi =0\,, \quad \phi(M,\tau_-,\frac{\tau_-M^2}{\gamma})
=\phi_s(M,\tau_-)\,, \quad  \phi (\underline R)
=\phi_s(\underline M,\underline \tau_-)=- \mathcal{E}(\underline R)\,.
$$
We assume that the function
$\mathcal{ E}+\phi$ is critical all along the curve ${\mathcal{C}}_s$
of the sonic surface. Using (\ref{dE1}), and
$\partial_{\tau_-} \mathcal{E}(R_s) =
\frac{\gamma r_0}{\gamma-1}\frac {[\tau ]^2M^2}{\tau_-\tau_+p_+}$,
we get the zero and first order conditions to be satisfied by $\phi_s$
along ${\mathcal{C}}_s$:
\begin{gather*}\label{d0phi}
\phi_s(\underline M,\tau_-)= -\mathcal{E}(R_s(\tau_-))\,.\\
\label{Zphi}
0=Z_s(\mathcal{E}+\phi) (R_s(\tau_-)) \equiv   r_0  \gamma
\frac{\gamma+1}{\gamma-1}  \frac{q^2}
{\underline M(\underline M\tau_-+q)(\underline M\tau_--\gamma q)}
+\partial_M\phi_s(\underline M,\tau_-))\,,
\end{gather*}
where
$$
Z_s:= \partial _{M}+\frac{2\tau_-M}{\gamma} \partial _{p_-}\,.
$$
(The vector field $Z_s$ is tangent to $\Sigma_s$, transversal to
${\mathcal{C}}_s$ and
$(Z_s\phi)(M,\tau_-,\frac{\tau_-M^2}{\gamma})=\partial _M \phi_s(M,\tau_-)$).\\

Also we assume that, at point $\underline R$, the second
differential of the restriction of $\mathcal{E}+\phi $ to the plane generated by the components
$x:=(0,1,-M^2)$ and $z:=(1,0,\frac{2\tau_-M}{\gamma})$ of $X$  and $Z_s$ has signature
$(1,1)$. As along the curve ${\mathcal{C}}_s$, we have
\begin{gather*}
D^2(\mathcal{E}+\phi)(R_s(\tau_-))(x,x)= X^2(\mathcal{E}+\phi)(R_s(\tau_-))
=X^2\mathcal{E}(R_s(\tau_-))< 0\,,\\
D^2(\mathcal{E}+\phi)(R_s(\tau_-))(x,z)=Z_sX(\mathcal{E}+\phi)(R_s(\tau_-))
=0\,,\\
\begin{aligned}
D^2(\mathcal{E}+\phi)(R_s(\tau_-))(z,z)
&=Z_s^2(\mathcal{E}+\phi)(R_s(\tau_-))\\
&=Z_s^2\mathcal{E}(R_s(\tau_-))+
\partial _M^2 \phi_s(\underline  M,\tau_-)\,,
\end{aligned}\\
\begin{aligned}
 Z_s^2\mathcal{E}(R_s(\tau_-))
 &=-r_0 \gamma  \frac{\gamma+1}{\gamma-1} \frac{q^2}{\underline
 M^2(\underline M\tau_-+q)(\underline M\tau_--\gamma q)}\\
& \quad \times(1+\frac{M\tau_-}{(\underline M\tau_-+q)}
+ \frac{M\tau_-}{(\underline M\tau_--\gamma q)})\,,
\end{aligned}
\end{gather*}
the second order condition
$X^2\mathcal{E}(\underline R) (Z_s^2\mathcal{E}(\underline R)+\partial  _M^2
\phi_s(\underline M,\underline\tau_-)) <0$, needs to be
satisfied by $\phi_s$; that is,
\begin{equation}\label{d2phi}
\begin{aligned}
\partial _M^2 \phi_s(\underline M,\underline \tau_-))
>& r_0 \gamma  \frac{\gamma+1}{\gamma-1}
 \frac{q^2}{\underline M^2(\underline M\underline \tau_-+q)
 (\underline M\underline \tau_--\gamma q)}\\
 &\times(1+\frac{\underline M \underline \tau_-}{(\underline M\underline \tau_-+q)}
+ \frac{\underline M\underline \tau_-}{(\underline M\underline \tau_--\gamma q)})\,.
\end{aligned}
\end{equation}
Under conditions (\ref{d0phi}), (\ref{Zphi}),
(\ref{d2phi}) the set $ \{ R: (\mathcal{E}+\phi) (R)=0\} $ is the union of
two smooth surfaces
$\Sigma ^\pm$, transversal along ${\mathcal{C}}_s$, which are
not Hugoniot stationary (relatively to the field $X$).

By $Z_s^2(\mathcal{E}+\phi) >0$ on ${\mathcal{C}}_s$, the zeroes of
$(\mathcal{E}+\phi)$ leave
the sonic surface in the $Z_s$ direction, and they leave by pair,
subsonic and supersonic, along the integral curves of
$X$, because $\mathcal{E}+\phi$ is maximum at the sonic points.
Therefore the $\Sigma ^\pm$ may be parametrized by
$(M,\tau_-)$ near $\underline R$ as
$$ \Sigma ^\pm=\{(M,\tau_-,p_-): p_-= p^\pm(M,\tau_-)\}
$$
for smooth  functions $p^\pm$ which satisfy
$p^\pm(\underline M,\tau_-) = p_s(\underline M,\tau_-)$ for every
$\tau_-$ and also
\begin{equation}\label{dMpk}
\begin{aligned}
\partial_Mp^\pm (\underline M,\tau_-)
&= \partial_Mp_s (\underline M,\tau_-)\\
&\pm \frac{\gamma +1}{\gamma}\frac{\underline
M^2}{(X^2\mathcal{E})(R_s(\tau_-))} \sqrt{-(X^2\mathcal{E}\,.\,Z_s^2(\mathcal{E}+\phi))(R_s(\tau_-))}\,.
\end{aligned}
\end{equation}
Hence, the $\Sigma^\pm$  intersect  the sonic surface $\Sigma_s$ transversally,
and they are located on either side of the plane $\{M=\underline M\}$.
We choose as kinetic one indeterminate (supersonic) part $\Sigma _k$ of $\Sigma^-\cup \Sigma^+$,
for example
$\Sigma^+\cap\{M\leq \underline M\}$, and we denote by $p_k$ the
restriction of
$p^+$ to the set  $M\leq \underline M$. Then the chosen \textit{kinetic manifold}
is
$$
\Sigma _k = \{R:   p_-= p_k(M,\tau_-) \}\,,
$$
where the smooth function $p_k$ is defined for $M\leq \underline M$
satisfying:
$p_k(\underline M,\tau_-) = p_s(\underline M,\tau_-)$,
$p_k(M,\tau_-)\leq p_s(M,\tau_-)$, and the (+) part of (\ref{dMpk}).


\subsection{Entropic excision of non-admissible strong detonations}

The counterpart of the admissibility of the weak detonation defined by $\Sigma_k$ is to
reject some strong detonations as non-admissible. This set is defined from the (micro)-detonating set
$\Sigma_d$ associated to $\Sigma_k$.

To describe this (micro)-detonating set, which is subsonic, we use the (micro)-detonating strength
defined in section \ref{microdet}. In the combustion setting,
the speed of the 3-shocks in the burnt gas is
$$
s_3(\epsilon_3 ,U^-) = u_-+\tau_-\sqrt {\beta^2_-+\frac{(\gamma +1)
\epsilon_3}{2\tau_-}}\,,\quad \mbox{where}\quad
\epsilon_3 = p_+-p_-\leq 0 \,.
$$
If we denote by $ \eta := (\tau, p)$ the thermodynamic variable, and
$$
m(\epsilon_3,\eta^-):= \frac{u_--s_3(\epsilon_3; U^-)}{\tau_-}
= -\sqrt {\beta ^2_-+\frac{(\gamma+1)\epsilon_3}{2\tau_-}}\,,
$$
recalling that $M= \frac{u_--\sigma }{\tau_-}$, we see that $s_3(\epsilon_3; U^-)=\sigma$ is
equivalent to $m(\epsilon_3,\eta^-)=M$. Let
$$
\epsilon_3^d(M, \eta^-):=  \frac{2\tau_-}{\gamma+1}(M^2-\beta^2_-)
$$
be the solution $ \epsilon_3 $ of the equation $m(\epsilon_3,\eta^-)=M$.
The (micro)-detonating set $\Sigma_d$ associated to $\Sigma_k$ is \
$\Sigma_d:= \{R:   (M,\ell _3(\epsilon_3 ^d(M, \eta^-), \eta^-)) \in \Sigma_k\}$,\
$\ell _3(\epsilon_3, \eta^-)$ being the thermodynamic part of the Lax function for the 3-shocks,
$$
\ell _3(\epsilon_3, \eta^-):= \begin{pmatrix}
\frac{\tau_-(2\gamma p_-+(\gamma-1)\epsilon_3)}{2\gamma p_-
+(\gamma+1)\epsilon_3}\\
p_-+\epsilon_3
\end{pmatrix}.
$$
Thus,
$$\Sigma_d= \big\{R:
  p_-+\epsilon_3 ^d(M, \tau_-,p_-)- p_k(M,\frac{1}{\gamma+1}((\gamma-1)\tau_-+\frac{2\gamma
p_-}{M^2})=0\big\}
$$
since at point $(\underline M,\underline \eta^-)$ the derivative
$\partial_{p_-}$ of the vanishing function which
defines $\Sigma_d$ is $-1$, $\Sigma_d$ is an hypersurface parametrized
by $(M,\tau_-)$.
The \textit{micro-detonating manifold} reads
$$
\Sigma_d=\big\{(M,\tau_-,p_-): p_-=p_d(M,\tau_-)\big\}\,,
$$
where the smooth function $p_d$ is defined for $M\leq \underline M$
and satisfies
\begin{gather*}
p_d(\underline M,\tau_-) = p_s(\underline M,\tau_-), \quad
p_d(M,\tau_-)\geq p_s(M,\tau_-)\,,\\
\partial_Mp_d (\underline M,\tau_-)= 2\partial_Mp_s (\underline M,\tau_-)
-\partial_Mp_k (\underline M,\tau_-)\,.
\end{gather*}
The detonations which belong to the set
$$\mathcal{D} := \big\{(M,\ell _3(\epsilon_3, \eta^-))\ :\ 
 \epsilon_3 ^d(M, \eta^-)<\epsilon_3\leq 0\,, (M,\eta)\in
\Sigma_d\big\}
$$
have to be rejected. The sonic and subsonic parts of $\mathcal{D}$ are the sets of the non-admissible
Chapman-Jouguet or strong detonations induced by the entropic selection $\Sigma_k$ for the weak
detonations.

\subsection{The phase boundary part in a non-classical combustion
Riemann problem}

Since $p_k(M,\tau_-)\leq p_s(M,\tau_-)\leq p_d(M,\tau_-)$,
a simple combustion wave $(U^b,\sigma)$ with speed $\sigma $ and state
$U^b=(u_b,\tau_b, p_b)$ on the left, close to a Chapman-Jouguet
detonation $((\underline u_-,\underline \tau_-, \underline
p_-),\underline u_-+\sqrt{\gamma  \underline \tau_-\underline p_-})$,
satisfies the properties as shown on Table 1.
\begin{table}[th]
\begin{tabular}{|l|l|l|}
\hline
Sonic & $p_b= p_s(M,\tau_b)=\frac{\tau_bM^2}{\gamma}$& \\
\hline
(strictly)supersonic & $p_b<p_s(M,\tau_b)$& \\
\hline
(strictly)subsonic  & $p_b> p_s(M,\tau_b)$& \\
\hline
Kinetic & $M\leq \underline M$ and
$p_b=p_k(M,\tau_-)$
& \parbox[t]{3cm}{then it is indeterminate, admissible, and supersonic}\\
 \hline
(micro)-detonating & $M\leq \underline M$ and
$p_b = p_d(M,\tau_-)$
& \parbox[t]{3cm}{then it is determinate, non admissible, and subsonic}\\
\hline
\parbox[t]{26mm}{Determinate \\ and\\ admissible}
&\parbox[t]{44mm}{$M\leq \underline M$ and  $p_b> p_d(M,\tau_-)$\\
or\\
 $M\geq \underline M$ and $p_b\geq p_s(M,\tau_-)$}
& \parbox[t]{3cm}{then it is strictly subsonic,\\
or subsonic}\\
\hline
\end{tabular}
\caption{Properties of a simple combustion wave}
\end{table}

We denote by $\Sigma_s^a:= \{R\in \Sigma_s: M\geq \underline M\}$ the
set of admissible sonic phase boundaries. By
$\Sigma_k\cup \Sigma_s^a$ we denote the graph of the function
$$
\tilde p_k(M,\tau_-)=\begin{cases}
p_k(M,\tau_-)& \mbox{if }M\leq \underline M\\
p_s(M,\tau_-)& \mbox{if }M\geq \underline M'\,.
\end{cases}
$$
By $\Sigma_d\cup \Sigma_s^a$ we denote the graph of the function
$$
\tilde p_d(M,\tau_-)=\begin{cases}
p_d(M,\tau_-)&\mbox{if }M\leq \underline M\,,\\
p_s(M,\tau_-)&\mbox{if }M\geq \underline M\,.
\end{cases}
$$
Coming back to the $U$-variable, $U=(u,\tau,p)=(u,\eta)$, we denote by
\begin{gather*}
\tilde \Sigma_s^a=\{(M,U^-):(M,\eta^-)\in \Sigma_s^a\}\,, \quad
\tilde \Sigma_d=\{(M,U^-): (M,\eta^-)\in \Sigma_d\}\,,\\
 \tilde \Sigma_k=\{(M,U^-): (M,\eta^-)\in\Sigma_k\}
\end{gather*}
 the unfolded admissible sonic, (micro)-detonating, and kinetic manifolds.
 We paste $\tilde \Sigma_d$
and $\tilde \Sigma_k$ with the map
$$
\tilde \Sigma_d\ni (M,U^-)\longmapsto (M, S_3(\epsilon^d_3(M,\eta^-),U^-))
 \in \tilde \Sigma_k\,,
$$
where $S_3(\epsilon_3,U^-)$ is, for $\epsilon_3\leq 0$,
 the admissible shock half-curve of the Lax curve
$$
\epsilon_3\mapsto L_3(\epsilon_3,U^-):= (\nu_3(\epsilon_3,U^-),\ell_3(\epsilon_3,\eta^-)):=
(\nu_3(\epsilon_3,U^-),\mu_3(\epsilon_3,\eta^-),p_-+\epsilon_3)\,,
$$
\begin{align*}
&\nu_3(\epsilon_3,U^-)\\
&:= \begin{cases}
u_-+ \epsilon_3\sqrt{\frac{2\tau_-}{2\gamma p_-+(\gamma+1)\epsilon_3}}
& \mbox{if } \epsilon_3\leq 0,  \mbox{ (shock)}\\
u_-+\frac{2\sqrt\gamma}{\gamma -1}(p_-\tau_-^\gamma)^{\frac{1}{2\gamma}}
((p_-+\epsilon_3)^{\frac{\gamma-1}{2\gamma}}
-p_-^{\frac{\gamma-1}{2\gamma}}) &\mbox{if } \epsilon_3\geq 0,
\mbox{ (rarefaction)}\,,
\end{cases}
\end{align*}
$$
\mu_3(\epsilon_3,\eta^-):= \begin{cases}
\tau_- \frac{2\gamma p_-+(\gamma-1)\epsilon_3}{2\gamma p_-+(\gamma+1)
\epsilon_3}& \mbox{if }\epsilon_3\leq 0,  \mbox{( shock)}\\
\tau_-(\frac{p_-}{p_-+\epsilon_3})^{\frac{1}{\gamma}}
& \mbox{if } \epsilon_3\geq 0,  \mbox{ (rarefaction).}
\end{cases}
$$
Using the arguments in section \ref{microdet}, the Hugoniot function
$\tilde H$ satisfies the pasting property
$$
\tilde H(M,U^-)=\tilde H(M, S_3(\epsilon^d_3(M,\eta^-),U^-))
\quad \forall\, (M,U^-)\in \tilde \Sigma_d\,.
$$
$(M,\eta^-)$ being fixed in the phase boundary space, the transversal
curve $\epsilon_3\mapsto (M,\ell_3(\epsilon_3,\eta^-))$ reaches
$\Sigma_k\cup \Sigma_s^a$
at a defined point $\epsilon_3= \tilde\epsilon_3(M,\eta^-)$ which is the solution of
\begin{equation}\label{eps3k}
 p_-+ \epsilon_3-\tilde p_k(M,\mu_3(\epsilon_3,\eta^-))=0\,,
\end{equation}
and the function $\tilde\epsilon_3$ is Lipschitz-continuous. Defining the
\textit{discontinuous} function
\begin{equation}\label{eps3}
\epsilon_3^{sk}(M,\eta^-)= \begin{cases}
\tilde\epsilon_3(M,\eta^-) &\mbox{if }\quad p_-\leq \tilde p_d(M,\tau_-)\\
0 & \mbox{if }\quad p_-> \tilde p_d(M,\tau_-)\,,
\end{cases}
\end{equation}
by the pasting property, we get the result that the composition
$$(M,U^-)\longmapsto \tilde H(M, L_3(\epsilon_3^{sk}(M,\eta^-),U^-))
$$
is \textit{Lipschitz continuous} near $(\underline M,\underline U^-)$.
The succession $\tilde H(M,L_3(\epsilon_3^{sk}(M,\eta^-),U^-))$ of a small amplitude 3-wave,
and of a large amplitude phase transition which is kinetic or 3-sonic on the left when
$\epsilon_3^{sk}(M,\eta^-)\neq 0$,  is the \textit{phase boundary part} in our
mixed sonic-kinetic Riemann problem. In this fan, the admissible large amplitude detonations belong to the set
$$
\mathcal{P}\mathcal{B}_{ad}^{sk}:= \tilde \Sigma_k\cup
\tilde\Sigma_s^a\cup \{(M,U^b): p_b>\tilde p_d(M,\tau_b)\}\,.
$$

\paragraph{Remark} The availability of a small 3-shock or non attached 3-rarefaction wave in a
detonation process is pointed out at the end of section 90 of \cite{C-F}.

\subsection{Two well-posed Riemann problem near a sonic detonation}

We focus here on the properties required for the well-posedness of the Riemann problem near
a  sonic detonation, using or not supersonic phase boundaries
described in the previous section.

In the unburnt gas, there is no wave in a Riemann problem near a Chapman-Jouguet
detonation. In  the burnt gas, we use the
jump of $p$ from left to right as a parameter for the $C^2-$Lax's curves $\epsilon_i\mapsto
L_i(\epsilon_i,U)$ for genuinely nonlinear
waves. $L_3$ is defined above, $L_1$ writes
$$L_1(\epsilon_1,U):= (\nu_1(\epsilon_1,U),\ell_1(\epsilon_1,\eta)):=
(\nu_1(\epsilon_1,U),\mu_1(\epsilon_1,\eta),p+\epsilon_1)\,,$$
$$
\nu_1(\epsilon_1,U):= \begin{cases}
u- \epsilon_1\sqrt{\frac{2\tau}{2\gamma p+(\gamma+1)\epsilon_1}}&
 \mbox{if }\epsilon_1\geq 0, \mbox{ (shock}\,,\\
u-\frac{2\sqrt\gamma}{\gamma-1}(p\tau^\gamma)^{\frac{1}{2\gamma}}(
(p+\epsilon_1)^{\frac{\gamma-1}{2\gamma}}
-p^{\frac{\gamma-1}{2\gamma}})&\mbox{if }\epsilon_1\leq 0,
\mbox{ (rarefaction)}
\end{cases}
$$
$$
\mu_1(\epsilon_1,\eta):= \begin{cases}
\tau \frac{2\gamma p+(\gamma-1)\epsilon_1}{2\gamma p
+(\gamma+1)\epsilon_1}& \mbox{if }\epsilon_1\geq 0,  \mbox{ (shock)}\,,\\
\tau(\frac{p}{p+\epsilon_1})^{\frac{1}{\gamma}} &\mbox{if }
 \epsilon_1\leq 0,  \mbox{ (rarefaction).}
\end{cases}
$$
The $2-$contact discontinuities are parametrized by the jump $\epsilon_2$
of $\tau$ as
$$L_2(\epsilon_2,U):= (u,\tau+\epsilon_2,p)\,.
$$
For a given burnt datum $U^-$ close to $\underline U^-$, an unburnt
datum $U^+$ close to $\underline U^+$, the general form
of the Riemann problem is
\begin{equation}\label{RPC}
U^+ = \tilde H(M, L_3(\epsilon_3^{\#}(M,\eta^{\#}),U^{\#}))\,,\quad U^{\#}:= (u_{\#},\eta^{\#})
=L_2(\epsilon_2,L_1(\epsilon_1,U^-))\,.
\end{equation}
Choosing the function $\epsilon_3^{\#}(M,\eta)$, we describe two well
posed Riemann problems, under a unique stability condition.

\subsubsection*{Mixed sonic-kinetic Riemann problem, non-classical theory}
The mixed sonic-kinetic Riemann problem selects the supersonic
detonations of
$\Sigma_s^a\cup\Sigma_k$. The function $\epsilon_3^{\#}(M,\eta)$ is here $\epsilon_3^{sk}(M,\eta)$
function defined in (\ref{eps3}).
The composition
$$(\epsilon_1,\epsilon_2,M,U^-)\mapsto \tilde H(M, L_3(\epsilon_3^{sk}(M,
\eta^{\#}),U^{\#}))\,,
$$
where $U^{\#}=(u_{\#},\eta^{\#})=L_2(\epsilon_2,L_1(\epsilon_1,U^-))$
is Lipschitz-continuous, and as for theorem \ref{skrp}, well posedness
is a consequence of the
invertibility of the differential
\begin{align*}
D_s&:= D_{(\epsilon_1,\epsilon_2,M)}(\tilde H(M,L_2
(\epsilon_2,L_1(\epsilon_1,U^-)))(0,0,\underline M,\underline U^-)\\
&=\begin{pmatrix}
2/M&*&*\\
0&\partial_{\tau_-}\tau_+ & \partial_{M}\tau_+ \\
0 & \partial_{\tau_-}p_+ & \partial_{M}p_+
\end{pmatrix}(\underline M,\underline U^-)\,,
\end{align*}
where $\partial\tau_+$, $\partial p_+$ are the functions in (\ref{dR}).
At the Chapman-Jouguet detonation point 
$(\underline U^-,\underline M,\underline U^+)$, we have
$$
\det D_s = - 2(\gamma+1) [\tau] [p] \frac{\gamma p_++M^2\tau_+}
{\beta^2_+-M^2} \neq 0\,.
$$
Clarke's inverse function theorem can be applied near
$(\underline U^-,\underline M,\underline U^+)$
and the solution
$(\epsilon_1,\epsilon_2,M)(U^-,U^+)$
is Lipschitz-continuous. Moreover, the mapping
$(U^-,U^+)\mapsto \epsilon_3^{sk}(M,\eta^{\#})$ is
Lipschitz-continuous function on the open set where
$(M,U^{\#})(U^-,U^+)\notin \tilde\Sigma_d$.
Therefore, we have proved the following theorem.

\begin{theorem}[sonic-kinetic combustion Riemann problem] \label{prop:skCrp}
\quad \\ Let
$(\underline U^-,\underline M,\underline U^+)$ be a Chapman-Jouguet detonation satisfying the entropy
condition (\ref{s0}). Let $\phi_s$ a smooth positive function satisfying (\ref{d0phi}),
(\ref{Zphi}),  (\ref{d2phi}), which defines by entropic selection the admissible detonations
$$\mathcal{P}\mathcal{B}_{ad}^{sk}= \tilde \Sigma_k\cup \tilde\Sigma_s^a\cup \{(M,U^b): p_b>\tilde p_d(M,\tau_b)\}\,.$$
There exists a neighbourhood
$\Omega^-\times\Omega^+$ of $(\underline U^-,\underline U^+)$, a neighbourhood
$\omega_1\times\omega_2\times\omega$ of $(0,0,\underline M)$, such that for every
$(U^-,  U^+)\in
\Omega^-\times\Omega^+$, it exists a unique solution
$(\epsilon_1,\epsilon_2,M)\in \omega_1\times\omega_2\times\omega$
of the Riemann problem
$$
U^+ = \tilde H(M, L_3(\epsilon_3^{sk}(M,\eta^{\#}),U^{\#}))\,,\quad U^{\#}:= (u_{\#},\eta^{\#})
=L_2(\epsilon_2,L_1(\epsilon_1,U^-))\,.
$$
where $\epsilon_3^{sk}(M, \eta^{\#})$ is defined by (\ref{eps3}),
in such a way that
$(M, L^-_3(\epsilon_3^{sk}(M, U^\#))\in \mathcal{P}\mathcal{B}_{ad}^{sk}$.
The function $(U^-,U^+)\mapsto  (\epsilon_1,\epsilon_2,M)$ is
Lipschitz-continuous and the variation of the wave solution
is estimated by
$$|\epsilon_1|+|\epsilon_2|+|\epsilon_3^{sk}(M,\eta^{\#})|+
|M-\underline M |=O(1) (|U^--\underline U^-| +|U^+-\underline U^+|)\,.
$$
\end{theorem}

\begin{figure}[t]
\begin{center}
\begin{picture}(120,100)(20,-20)
\setlength{\unitlength}{1.4pt}
%LEFT:
%axes:
\put(-12,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\thicklines\line(1,1){45}}
\put(60,45){\makebox(0,0)[tr]
{\hbox{$ _{x=\sigma  t}$}}}
\put(35,45){\makebox(0,0)[b]
{\hbox{ $U^{\#}$}}}
\put(0,0){\line(-3,1){50}}
\put(0,0){\line(-5,2){48}}
\put(0,0){\line(-5,3){47}}
\put(-30,25){\makebox(0,0)[br]
{\hbox{$\epsilon _1^-$}}}
\put(0,0){\line(1,2){28}}
\put(24,50){\makebox(0,0)[br]
{\hbox{$\epsilon_2^-$}}}
\put(-40,-6){\makebox{$U^-$}}
\put(30,-6){\makebox{$U^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{subsonic case}}}
}
%RIGHT:
\put(115,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\line(2,3){38}}
\put(0,0){\line(4,5){44}}
\put(0,0){\thicklines\line(1,1){45}}
\put(60,45){\makebox(0,0)[tr]
{\hbox{$ _{x=\sigma  t}$}}}
\put(0,0){\line(-3,1){60}}
\put(-30,18){\makebox(0,0)[br]
{\hbox{$\epsilon _1^-$}}}
\put(0,0){\line(2,5){25}}
\put(20,55){\makebox(0,0)[br]
{\hbox{$\epsilon_2^-$}}}
\put(25,45){\makebox(0,0)[b]
{\hbox{$U^{\#}$}}}
\put(50,55){\makebox(0,0)[b]
{\hbox{$ _{\epsilon_3(M,\eta^-)}$}}}
\put(-40,-6){\makebox{$U^-$}}
\put(30,-6){\makebox{$U^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{sonic case, attached rarefaction}}}
}
\end{picture}
\end{center}
\caption{}
\end{figure}


\begin{figure}[t]
\begin{center}
\begin{picture}(120,100)(20,-20)
\setlength{\unitlength}{1.4pt}
%LEFT:
%axes:
\put(-12,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\line(3,5){35}}
\put(0,0){\line(3,4){40}}
\put(0,0){\thicklines\line(3,2){50}}
\put(60,30){\makebox(0,0)[tr]
{\hbox{$ _{x=\sigma  t}$}}}
\put(22,45){\makebox(0,0)[b]
{\hbox{ $U^{\#}$}}}
\put(40,35){\makebox(0,0)[b]
{\hbox{ $U^b$}}}
\put(0,0){\line(-3,1){50}}
\put(0,0){\line(-5,2){48}}
\put(0,0){\line(-5,3){47}}
\put(-30,25){\makebox(0,0)[br]
{\hbox{$\epsilon _1^-$}}}
\put(0,0){\line(1,3){20}}
\put(15,45){\makebox(0,0)[br]
{\hbox{$\epsilon_2^-$}}}
\put(50,55){\makebox(0,0)[b]
{\hbox{$ _{\epsilon_3(M,\eta^-)}$}}}
\put(-40,-6){\makebox{$U^-$}}
\put(30,-6){\makebox{$U^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{supersonic case, non attached rarefaction}}}
}
%RIGHT:
\put(115,0){
\put(0,0){\line(1,0){50}}
\put(0,0){\line(-1,0){50}}
\put(0,0){\line(2,3){38}}
\put(0,0){\thicklines\line(1,1){45}}
\put(60,45){\makebox(0,0)[tr]
{\hbox{$ _{x=\sigma  t}$}}}
\put(0,0){\line(-3,1){60}}
\put(-30,18){\makebox(0,0)[br]
{\hbox{$\epsilon _1^-$}}}
\put(0,0){\line(1,3){19}}
\put(17,49){\makebox(0,0)[br]
{\hbox{$\epsilon_2^-$}}}
\put(25,45){\makebox(0,0)[b]
{\hbox{$U^{\#}$}}}
\put(40,45){\makebox(0,0)[b]
{\hbox{$U^b$}}}
\put(50,55){\makebox(0,0)[b]
{\hbox{$ _{\epsilon_3(M,\eta^-)}$}}}
\put(-40,-6){\makebox{$U^-$}}
\put(30,-6){\makebox{$U^+$}}
\put(0,-10){\makebox(0,0)[t]
{\hbox{supersonic case, shock}}}
}
\end{picture}
\end{center}
\caption{}
\end{figure}


\paragraph{Remark}
We find again these pictures as the numerical diagrams of figures
4 or 5 in Wood \cite{Wo}.


\subsubsection*{Sonic Riemann problem, Chapman-Jouguet theory} The fully sonic Riemann problem
selects the supersonic combustion boundaries of
$\Sigma_s$. This is the usual configuration accepted by the physicists. Its definition uses as $
\epsilon_3^{\#}$ the continuous function
$$
\epsilon_3^s(M,\eta^-):= \begin{cases}
(\frac{\tau_-M^2}{\gamma})^{\frac{\gamma}{\gamma+1}}p_-^{\frac{1}
{\gamma+1}}-p_- &
\mbox{if }p_-\leq p_s(M,\tau_-)\equiv \frac{\tau_-M^2}{\gamma}\\
0 & \mbox{if } p_-\geq p_s(M,\tau_-)\,,
\end{cases}
$$
where $
(\frac{\tau_-M^2}{\gamma})^{\frac{\gamma}{\gamma+1}}p_-^{\frac{1}{\gamma+1}}-p_-
:=\hat\epsilon_3(M,\eta^-)$
 is the  solution of the equation
 $$
 p_-+ \epsilon_3-p_s(M,\tau_-(\frac{p_-}{p_-+\epsilon_3})^{1/\gamma})=0\,.
$$
The composition
$(\epsilon_1,\epsilon_2,M,U^-)\mapsto \tilde H(M,
L_3(\epsilon_3^s(M,\eta^{\#}),U^{\#}))$, with
$$
U^{\#}=(u_{\#},\eta^{\#})=L_2(\epsilon_2,L_1(\epsilon_1,U^-))
$$
is $C^1$. Its well posedness comes from
\begin{align*}
&\det \{D_{(\epsilon_1,\epsilon_2,M)}(\tilde H(M,
L_3(\hat \epsilon_3(M,\eta^{\#}),L_2(\epsilon_2,L_1(\epsilon_1,U^-)))
(0,0,\underline M,\underline U^-)\}\\
&=\det D_s \neq 0\,;
\end{align*}
so that the classical inverse function theorem applies near the
sonic detonation $(\underline U^-,\underline M,\underline U^+)$
and the solution $(\epsilon_1,\epsilon_2,M)(U^-,U^+)$
is $C^1$. Moreover $(U^-,U^+)\mapsto \epsilon_3^s(M,\eta^{\#})$ is
Lipschitz-continuous.
In a wave solution, the phase transition discontinuity is strictly subsonic (strong detonation), or
sonic with or without a rarefaction attached
on its left. Only the two first pictures above may appear.

\begin{proposition}[sonic combustion Riemann problem] \quad\\
Let  $(\underline U^-,\underline M,\underline U^+)$ be a Chapman-Jouguet detonation.
There exists a
neighbourhood
$\Omega^-\times\Omega^+$ of $(\underline U^-,\underline U^+)$, a neighbourhood
$\omega_1\times\omega_2\times\omega$ of $(0,0,\underline M)$, such that for every
$(U^-,  U^+)\in
\Omega^-\times\Omega^+$, it exists a unique solution
$(\epsilon_1,\epsilon_2,M)\in \omega_1\times\omega_2\times\omega$
of the Riemann problem
$$
U^+ = \tilde H(M, L_3(\epsilon_3^{s}(M,\eta^{\#}),U^{\#}))\,,\quad U^{\#}:= (u_{\#},\eta^{\#})
=L_2(\epsilon_2,L_1(\epsilon_1,U^-))\,.
$$
The detonation part of the solution is a sonic or a strong detonation.
The function
$(U^-,U^+)\mapsto  (\epsilon_1,\epsilon_2,M)$ is
$C^1$  and the variation of the wave solution
is estimated by
$$
|\epsilon_1|+|\epsilon_2|+|\epsilon_3^{s}(M,\eta^{\#})|+
|M-\underline M |=O(1) (|U^--\underline U^-| +|U^+-\underline U^+|)\,.
$$
\end{proposition}


\begin{thebibliography}{00}

\bibitem{A-K} R. Abeyaratne and J. K. Knowles, {\sl Kinetic relations and the
propagation of phase boundaries in solids},  Arch. Rational Mech. Anal.
{\bf 114} (1991), 119--154.

\bibitem{C} F. H. Clarke, {\sl On the inverse function theorem}, Pacific J.
Math. {\bf 64} (1) (1976), 97--102.

\bibitem{C-F} R. Courant and K.O. Friedrichs, {\sl Supersonic flow and shock waves},
Interscience, New-York, 1948.

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\end{thebibliography}

\noindent\textsc{Monique Sabl\'e-Tougeron}\\
I.R.M.A.R., Universit\'e de Rennes 1\\
Campus de Beaulieu, F-35042 Rennes Cedex, France\\
email: sable@maths.univ-rennes1.fr

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