text:
\section{Introduction}

The Navier-Stokes equations for van der Waals fluids with viscosity and capillarity effects included,
allow one to model the dynamics of liquid-vapor flows. The associated set of first-order conservation laws
is of hyperbolic or hyperbolic-elliptic type, and admits propagating discontinuities (shock waves).
The singular limit corresponding to vanishing viscosity and capillarity coefficients allows one to select physically
admissible, discontinuous solutions to the first-order conservation laws. In particular, this yields an approach to
select solutions to the so-called Riemann problem, when the initial data consists of two constant states
separated by a single jump.  Our purpose in the present paper is to derive uniform (viscosity-capillarity independent)
bounds and to justify this singular limit in the context of nonlinear elasticity and phase transition dynamics.

Attention will be concentrated on the following system of two conservation laws:
\be
\label{elastic}
\aligned
& {\del v \over \del t} - {\del \over \del x} \Big( \sigma(w) - \eps \, {\del v \over \del x} + \delta {\del^2 w \over \del x^2} \Big) = 0,
\\
& {\del w \over \del t} - {\del v \over \del x} = 0,
\endaligned
\ee
where $v=v(t,x)$ and $w=w(t,x)$ represent the velocity and deformation gradient
of the fluid or solid material under consideration.
We consider also the associated first-order system
\be
\label{elastic-first}
\aligned
& {\del v \over \del t} - {\del \over \del x}( \sigma(w) ) = 0,
\\
& {\del w \over \del t} - {\del v \over \del x} = 0,
\endaligned
\ee
and impose Riemann initial data
\be
\label{boundary}
(v,w)(0,x) = \begin{cases}
v_l, w_l,    & x < 0,
\\
v_r, w_r,    & x > 0,
\end{cases}
\ee
where $v_l, v_r, w_l, w_r \in \RR$ are given constants. Solutions of \eqref{elastic-first}-\eqref{boundary}
are known to be self-similar, that is, to depend only upon the variable $y:=x/t$.

When viscosity and capillarity terms are taken into account,
the corresponding set of differential equations reads
\be
\label{elastic2}
\aligned
& -y \, v' - \sigma(w)' = \eps \, v'' - \delta w''',
\\
& - y \, w' - v' = 0,
\endaligned
\ee
supplemented with the boundary conditions
\be
\label{boundary1}
\lim_{y \to -\infty} (v,w)(y) = (v_l, w_l),
\qquad
\lim_{y \to +\infty} (v,w)(y) = (v_r, w_r).
\ee
Our main objective in this paper is to establish :
\begin{enumerate}
\item the existence of a smooth, self-similar solution to \eqref{elastic2}-\eqref{boundary1} having uniformly bounded total variation,
\be
\label{BVbound}
TV(v_\eps, w_\eps) \lesssim |v_r - v_l| + |w_r - w_l|,
\ee
with the implied constant being {\sl independent} of $\eps, \delta$ within the range $\delta /\eps^2 <<1$, and
\item the strong convergence of $v_\eps, w_\eps$ toward a weak, self-similar
solution $v,w$ to the first-order conservation
laws \eqref{elastic-first}.
\end{enumerate}

An outline of our main results is as follows.

Assuming first that the system \eqref{elastic-first} is {\sl hyperbolic} we provide a rather direct proof of the
above two properties, first in the viscosity-only case (Section~2) and then for general viscosity-capillarity (Section~3).
In this regime, it is known that the limiting solutions in general contains nonclassical shocks, with
depend upon the ratio $\delta/\eps^2$; see \cite{LeFloch2}.

Next, we investigate the generalization to the hyperbolic-elliptic regime, and discover a concentration phenomena
near the axis $x=0$ separating the two wave fans. This seems to be consistent with numerical experiments
with the model under consideration, but it would be interesting to check numerically the feature discovered here analytically.
In the context of phase dynamics, it is also well-known that the limiting solutions contains subsonic
phase boundaries which again depend on the capillarity to viscosity ratio;
see \cite{Slemrod1,Slemrod2,Truskinovsky,AbeyaratneKnowles,LeFloch1,Shearer,ShearerYang}.

Observe that our results cover arbitrary large Riemann data and physical viscosity and capillarity terms.
Our results supplement the earlier, pioneering
work by Fan and Slemrod  \cite{FanSlemrod,FanSlemrod2} where an artificial, ``full'' viscosity was used.
Our technique of proof in the present paper is quite different from the one in \cite{FanSlemrod},
as we introduce a decomposition of the system of equations \eqref{elastic2} into two coupled scalar equations.

Finally, we provide various generalization to the boundary value problem
and more general classes of conservation laws.

Recall that the activity on self-similar vanishing viscosity limits started with an extensive research by
Dafermos \cite{Dafermos1}--\cite{Dafermos4} (see also \cite{DafermosDiPerna})
who advocated the use of self-similar regularizations to capture the whole structure
of wave fans within solutions of the Riemann problem. Self-similar approximations in the context of phase transition dynamics
were studied by Slemrod \cite{Slemrod1,Slemrod2}, Fan and Slemrod \cite{FanSlemrod},
who covered large data solution and artificial regularization terms.
Next, small data solutions to general systems
were treated by Tzavaras \cite{Tzavaras} (conservative systems) and LeFloch and Tzavaras
\cite{LeFlochTzavaras1,LeFlochTzavaras2} (nonconservative systems).

Self-similar diffusive-dispersive approximations for general systems were investigated in LeFloch and Rohde \cite{LeFlochRohde}. Joseph and LeFloch \cite{JLtwo}--\cite{JLfive} extended the technique
to cover boundary value problems, relaxation approximations, and general diffusion matrices.
The present paper is the continuation of \cite{JLtwo}--\cite{JLfive}.
For issues related to the discretization of the viscosity and capillarity terms, we refer
to \cite{Slemrod-scheme,HayesLeFloch2}.

\section{Elastodynamics with physical viscosity}
\label{section2}

We begin with the case that $\delta =0$ and $\sigma_w>0$. We will prove:

\begin{theorem} [Vanishing viscosity limit in elastodynamics]
Assume that the system \eqref{elastic-first} is uniformly hyperbolic, in the sense that
there exits a positive constant $c_0$ such that
$$
\inf_{w \in \RR} \sigma_w(w) \geq c_0^2 > 0.
$$
Then, given arbitrary Riemann data $v_l, w_l$ and $v_r, w_r$, the viscous Riemann problem
\eqref{elastic2}-\eqref{boundary1} with $\delta =0$
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation and converges pointwise to a limit $v,w$.
This limit is a weak solution of the Riemann problem
\eqref{elastic-first}-\eqref{boundary}.
\end{theorem}

It is worth observing that the solutions $w^\eps, v^\eps$ contain a mild singularity at
$y=0$: their derivatives up to order $C/\eps$ only are continuous at $y=0$.
Hence, this singularity vanishes in the limit $\eps \to 0$.
The singularity is due to a factor $y$ appearing in the key equation \eqref{scalar} below,
in front of the term
with the highest order derivative.

\begin{proof}

{\it Step 1. Reduction to a scalar equation for the component $w$. } To simplify the notation we suppress
the subscript $\eps$.
 We will first study the problem on a finite interval $[-L,L]$ with $L$ chosen to be sufficiently large
and we will show the existence of a solution to \eqref{elastic2} satisfying the boundary conditions
\be
\label{boundary2}
(v,w)(-L) = (v_l, w_l),
\qquad
(v,w)(L) = (v_r, w_r).
\ee

First, we observe that the system can be reduced to a single scalar equation for the unknown $w$.
Namely, using $v' = - y \, w'$ in the first equation of \eqref{elastic2} we find
\be
\label{scalar}
(y^2 + \eps - \sigma_w(w)) \, w' = - \eps \, y \, w'',
\ee
which will be studied in two regions $[-L,0]$ and $[0,L]$ with the following boundary conditions
\be
\label{boundl}
w(-L) = w_l, \qquad w(0-) = w_*,
\ee
\be
\label{boundr}
w(0+) = w_*,  \qquad w(L) = w_r,
\ee
where $w_*$ is a parameter to be determined so that the boundary condition on the variable $v$
(which has been eliminated from \eqref{scalar}) is satisfied. This condition is obtained by integrating over $[-L,L]$
 the second equation in \eqref{elastic2}:
\be
\label{boundv}
v_r - v_l = \int_{-L}^L v' \, dy = - L \, w_r + L w_l + \int_{-L}^L w \, dy.
\ee
This is a global, integral condition on the solution $w$. We will now solve the problem \eqref{scalar}--\eqref{boundv}.

Our boundary condition can be justified as follows. If we search for a piecewise smooth solution
of \eqref{elastic2}, then clearly the measure $v' = -y w'$ can not have a mass point at $y=0$, hence
$v$ is continuous. Similarly, by the first equation in \eqref{elastic2} the function $\sigma(w) - \eps v'$ must be continuous
at $y=0$. But, using again the equation $\eps v' = - y \eps w'$ we see that $\sigma(w)$ itself must be continuous.
Finally, since $\sigma$ is assumed to be strictly increasing, $w$ must be continuous.

Based on the function $w$ we can define $a(y) : = y - (\sigma_w(w(y)) - \eps)/y$, consider the integral
$\int_{\alpha^\pm}^y a(x)\, dx$ on $[-L,0)$  and on $(0,L]$ with $\alpha^- \in [-L,0)$  and $\alpha^+ \in (0,L]$, respectively.
Let us show that the minimum with respect to $y$ of this integral is attained at
$$
\rho^\pm := \pm (\sigma_w(w(\rho^\pm))-\eps )^{1/2}.
$$
For definiteness, consider the case $\alpha=\alpha^- \in [-L,0)$ with $\alpha^- < y$; then
$$
\int_{\alpha}^y a(x)\, dx \geq \int_{\alpha}^y (x - {(\sigma_w^{M}-\eps)\over x})\, dx
\geq (y^2 - \alpha^2)/2 + \log(\alpha/y)^{\sigma_w^{M} - \eps},
$$
which tends to infinity as $y \rightarrow 0$. Necessarily, the minimum is attained away from $0$.
As this integral has quadratic growth in $y$,
for $L$ large the minimum $\rho=\rho^-$ is attained in $(-L,0)$ and is given by the equation
$\rho^2 - \sigma_w(w(\rho))+\eps =0$.

This choice of $\rho^{\pm}$ gives   $\int_{\rho^-}^y a(x) \, dx \geq 0$.
By setting
$$
\varphi_-(y) := {e^{-(1/\eps)  \int_{\rho^-}^y a(x) \, dx} \over \int_{-L}^0 e^{-(1 /\eps) \int_{\rho^-}^y a(x) \, dx}}
\qquad
\varphi_+(y) := {e^{-(1/\eps) \int_{\rho^+}^y a(x) \, dx} \over \int_0^L e^{-(1 /\eps) \int_{\rho^+}^y a(x) \, dx}},
$$
the problem under consideration is equivalent to
\be
\label{equiv}
w(y) = \begin{cases}\displaystyle
w_l + (w_* - w_l) \int_{-L}^y \varphi_- \, dx,     &  y < 0,
\\\displaystyle
w_r + (w_* - w_r) \int_y^L \varphi_+ \, dx,     &  y > 0,
\end{cases}
\ee
together with the boundary condition \eqref{boundv}.

To find $w_*$, we used the continuity of the solution at $y=0$
namely, $w(0-) = w(0+) = w_*$ and  $v(0-) = v(0+) = v_*$.
Integrating the equation $v'= -yw'$ from $-L$ to $0$ and from $0$ to $L$
and using \eqref{equiv}, we get
$$
v_* -v_l = (w_* - w_l)\int_{-L}^{0} -y \varphi_-(y) dy,
$$
$$
v_r -v_* = (w_* - w_r)\int_{0}^{L} y \varphi_+(y) dy.
$$
Adding these formulas and solving for $w_*$, we get
\be
\label{wm}
w_* = {v_r - v_l + w_r \int_0^L y \, \varphi_{+} \, dy -
w_l \int_{-L}^{0} y \, \varphi_{-} \, dy
\over \int_0^L y \, \varphi_{+}^\delta \, dy -  \int_{-L}^{0}
y \,
\varphi_{-}^\delta \, dy},
\ee
which provides us with the value of $w_*$.

Observe that the denominator is bounded away from zero.
To evaluate its minimum value (in terms of the constitutive function $\sigma$ and the Riemann data)
we now need to analyze the functions $\varphi_{\pm}$.

\

\noindent{\it Step 2. Properties of the functions $\varphi_\pm$.} We will show now that the support of the functions
$\varphi_\pm$ is essentially concentrated away from the axis $y=0$.
Let $\sigma_w^m = \min_{-L\leq y \leq L} \sigma_w(w(y))$, $\sigma_w^M = \max_{-L\leq y \leq L} \sigma_w(w(y))$,
$\lambda_\eps^m = (\sigma_w^m-\eps)^{1/2}$ and $\lambda_\eps^M = (\sigma_w^M-\eps)^{1/2}$.
We claim that there exists a constant
$C>0$ such that
\be
\label{prop1}
0 < \varphi_{-}(y) \leq {C \over \eps} \begin{cases}
e^-{(y+\lambda_\eps^M)^2 \over 2\eps}, & -L \leq y < -\lambda_\eps^M,
\\
1, & -\lambda_\eps^M \leq y < -\lambda_\eps^m,
\\
e^-{(y+\lambda_\eps^M)^2 \over 2\eps}, & -\lambda_\eps^m \leq y < {-\lambda_\eps^m}/4,
\\
(-2y/\lambda_\eps^m)^{3 \lambda_\eps^m \over 4\eps}, & -{\lambda_\eps^m}/4 \leq y \leq 0,
\end{cases}
\ee
\be
\label{prop2}
0 < \varphi_{+}(y) \leq {C \over \eps} \begin{cases}
(2y/\lambda_\eps^m)^{3 \lambda_\eps^m \over 4\eps}, & 0 \leq y \leq {\lambda_\eps^m}/4,
\\
e^-{(y-\lambda_\eps^m)^2 \over 2\eps}, & {\lambda_\eps^m}/4 \leq y < {\lambda_\eps^m},
\\
1, & \lambda_\eps^m \leq y < \lambda_\eps^M,
\\
e^-{(y-\lambda_\eps^M)^2 \over 2\eps}, & \lambda_\eps^M <y <L.
\end{cases}
\ee

This is proved as follows. First, in the interval $0\leq y \leq \lambda_\eps^m/4$ we find
$$
\aligned
\int_{\rho^+}^y {(x^2 -\sigma_w(x)+\eps)\over x} dx
&\geq \int_{{\lambda_\eps^m}/2}^y {(x^2 -\sigma_w +\eps)\over x} dx
\\
&= \int_y^{{\lambda_\eps^m}/2} {(\sigma_w - \eps - x^2)\over x} dx
\\
&\geq  (3/4) (\lambda_\eps^m)^2\int_y^{{\lambda_\eps^m}/2} {dx \over x}
   =\log(\lambda_\eps^m /2 y)^{(3/4) {\lambda_{\eps}^m}^2},
\endaligned
$$
and, next, for $y>\lambda_\eps^M$
$$
\aligned
\int_{\rho^+}^y (x^2 -\sigma_w(x)+\eps) dx
&\geq \int_{\lambda_\eps^M}^y {(x^2 -\sigma_w(x)+\eps) \over x} dx
\\
&= \int_{\lambda_\eps^M}^y (x -(\sigma_w(x)+\eps)^{1/2})(1 +{(\sigma_w(x)+\eps)^{1/2} \over x}) dx
\\
&\geq \int_{\lambda_\eps^M}^y (x-\lambda_\eps^M) dx ={(y - \lambda_\eps^M)^2 \over 2}.
\endaligned
$$
The other cases are completely similar. Since, on the other hand, it is easy to check that
$$
 \int_{-L}^0 e^{-(1 /\eps) \int_{\rho^-}^y a(x) \, dx} dy \geq C \eps,
\qquad
\int_0^L e^{-(1 /\eps) \int_{\rho^+}^y a(x) \, dx} dy \geq C \eps,
$$
this completes the derivation of the properties of $\varphi_\pm$.

\

\noindent{\it Step 3. Existence result.}
Observe that $w_*$ consists of the sum of
$$
{{v_r - v_l}  \over {\int_0^L y \, \varphi_+ \, dy -  \int_{-L}^0 y \, \varphi_- \, dy}}
=  {{v_r - v_l} \over {\int_{-\lambda_\eps^M}^{-\lambda_\eps^m} -y \, \varphi_- \, dy + \int_{\lambda_\eps^m}^{\lambda_\eps^M} y \, \varphi_+ \, dy + O(\eps^n)}}
$$
and a convex combination of $w_l,w_r$.
In view of this observation, together with the estimates \eqref{prop1}
and \eqref{prop2}, we get
$$
\aligned
|w_*|
& \leq \max (w_l,w_r) + {{|v_r - v_l|} \over {2 \lambda_\eps^m} + O(\eps^n)}
\\
& \leq \max (w_l,w_r) + {{|v_r - v_l|} \over {c_0}} =: \Lambda_0
\endaligned
$$
and we arrive at the uniform bound
\be
\label{wstar}
| w_* | \leq \Lambda_0.
\ee

Replacing now $w_*$ (by its value given in \eqref{wm}) in the formula
\eqref{equiv} we can define a mapping
$w \in C^0([-L,L]) \mapsto T(w) \in C^0([-L,L])$. We claim that, for fixed $\eps$,  the function $T(w)$ is
of class $C^1$ and
$$
\aligned
& \|T(w)\|_{C^0} \leq \Lambda_0,
\\
& \|T(w)'\|_{C^0} \leq {C \over \eps} \, (|w_r - w_* | +|w_r - w_* |).
\endaligned
$$
Indeed, these estimates follow immmediatly from \eqref{equiv}, \eqref{prop1}, and \eqref{prop2}

In consequence, the operator $T$ is a compact map from the convex bounded set $\big\{ w \, / \,  \|w\|_{C^0([-L,L]} \leq \Lambda_0\big\}$ into
itself. By Schauder's fixed point theorem, $T$ has a fixed point which satisfies
\eqref{equiv} and, clearly, is of class $C^1$. Furthermore, in view of
\eqref{equiv} and \eqref{wm}, we have
\be
\label{es1}
\int_{-L}^L |w'(y)| dy \leq | w_r - w_* | +  | w_l - w_* |
\leq  | w_r - w_l |+ {2 \over c_0} \, | v_r - v_l |
\ee
and
\be
\label{es2}
\int_{-L}^L |v'(y)| dy \leq \int_{-L}^L |y w'(y)| dy
\leq (\lambda_0^M+1) \, \big( |w_r - w_l|+ {2  \over c_0} \, |v_r - v_l| \big).
\ee

Next, since all of the estimates are uniform in $L$, we can let $L \to \infty$ and we conclude with
existence of a solution $(v^\eps,w^\eps)$ on defined on the whole real line $(-\infty,\infty)$.
This function $(v^\eps,w^\eps)$ is a bounded solution of \eqref{elastic2}-\eqref{boundary1}
and has uniformly bounded total variation
$$
\int_{-\infty}^\infty \big( |w'(y)| + |v'(y)| \big) \, dy \leq (\lambda_0^M+2) \, \big( |w_r - w_l|+ {2 \over c_0} \, |v_r - v_l| \big).
$$
By Helly's compactness theorem, it admits a subsequence converging pointwise to a limit $(v,w)$,
as $\eps$ goes to zero. Clearly,
the limit is a weak solution of the problem \eqref{elastic-first}-\eqref{boundary}.
\end{proof}

\section{Elastodynamics with physical viscosity and capillarity}
\label{section3}

In this section, we consider the full system with physical viscosity and capillarity included.  We will prove:

\begin{theorem}  [Vanishing viscosity-capillarity limit in elastodynamics]
Assume that $\inf \sigma_w \geq c_0^2 >0$. Then,
given arbitrary Riemann data $v_l,v_r, w_l, w_r$, the problem
\eqref{elastic2}-\eqref{boundary1} with $\delta=\gamma \eps^2$,
$\gamma>0$,
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation
and converges to a limit $v,w$. This limit is a weak solution to the Riemann
problem \eqref{elastic-first}-\eqref{boundary}.
\end{theorem}

Observe that this is a large data result.

\begin{proof}

{\bf Step 1:  reduction to a scalar equation. }
We will first study the problem on a finite interval $[-L,L]$ with $L$ chosen to be sufficiently large
with boundary conditions
\be
\label{boundary3}
(v,w)(-L) = (v_l, w_l),
\qquad
(v,w)(L) = (v_r, w_r).
\ee

As before, the system can be reduced to a single scalar equation for the unknown $w$.
Namely, using $v' = - y \, w'$ in the first equation of \eqref{elastic2} we find
\be
\label{scalar-B}
(y^2 + \eps - \sigma_w(w)) \, w' = - \eps \, y \, w'' - \delta w''',
\ee
which will be studied in two regions $[-L,0]$ and $[0,L]$ with the following boundary conditions
\be
\label{boundl-BC}
w(-L) = w_l, \qquad w(0-) = w_*,
\ee
\be
\label{boundr-BC}
w(0+) = w_*,  \qquad w(L) = w_r,
\ee
where $w_*$ is a parameter to be determined so that the boundary condition on the variable $v$
(which has been eliminated from \eqref{scalar}) is satisfied. This condition is obtained by integrating over $[-L,L]$
 the second equation in \eqref{elastic2}:
\be
\label{boundv-BC}
v_r - v_l = \int_{-L}^L v' \, dy = - L \, w_r + L w_l + \int_{-L}^L w \, dy.
\ee
This is a global, integral condition on the solution $w$. We will now solve
the problem \eqref{scalar-B}--\eqref{boundv-BC}.

We will follow
a method originally introduced by LeFloch and Rohde \cite{LeFlochRohde} for small amplitude solutions of
systems of conservation laws.
Setting $\varphi = e^\beta H$, the differential equation under consideration becomes
$$
\delta [H'' + 2 H'\beta' + H \beta'' + H \beta'^2] + \eps y [H' +H \beta'] + (y^2 +\eps - \sigma_w ) H = 0
$$
Setting the coefficient of $H'$ equal to $0$, we get
$\beta' = {-\eps y \over 2 \delta}$, thus $\beta (y) = {-\eps y^2 \over 4 \delta}$,
and
\be
H'' = {\mu (w(y),y) \over \delta} H,
\ee
where
\be
\mu(w(y),y) = \sigma_w + y^2({\eps^2 \over 4 \delta} - 1) - {\eps \over 2}.
\ee

We take $\delta = \gamma \eps^2$ and, in this case,
$$
\mu(w,y) =  \sigma_w + y^2({1 \over 4 \gamma} - 1) - {\eps \over 2} > 0,
$$
provided $\eps,\gamma>0$ are sufficiently small.
Upon integrating this equation for $H$ and substituting, we get
\be
\phi(y) = { 1+\Phi(y) \over (4\gamma \mu)^{1/4}} e^{p(y,\rho) \over \eps},
\ee
where
$$
\aligned
& |\Phi(y)|+{\eps \gamma^{1/2} \over 2 \mu(y)^{1/2}}|\Phi'(y)| \leq \eps \gamma^{1/2} k,
\\
& k:=\int_{-L}^L \mu^{-5/4}|\mu'|^2 dx + \int_{-L}^L \mu^{-3/2}|\mu''| dx
\endaligned
$$
and
\be
\aligned
p(y,\rho)
&:= \int_\rho^y({-x \over 2\gamma} + \sqrt {\mu(w(x),x) \over \gamma}) dx
\\
&=\int_{\rho}^y ({-x\over 2\gamma} +\sqrt{{\sigma_w -\eps/2 + x^2(1/{4\gamma} -1) \over \gamma}} dx
\\
&={1\over 2\gamma}\int_{\rho}^y x(-1 + \sqrt{1+4\gamma (\sigma_w -\eps/2 - x^2)} ) dx.
\endaligned
\ee

Here, $\rho$ is the maximizer of the the integral
$$
 \max_{-L < y < L}\int_\alpha^y({-x \over 2\gamma} + \sqrt {\mu(w(x),x) \over \gamma})dx.
$$
Indeed $\rho$ satisfies
$$
\rho^2 - \sigma_w(w(\rho))+\eps/2 =0,
$$
which has pairs of solutions  $\rho_- \in [-L,0)$  and on $\rho _+ \in (0,L]$,  satisfying
$$
\rho^\pm = \pm (\sigma_w(w(\rho^\pm))-\eps/2 )^{1/2}.
$$

We consider $\varphi_-$ and $\varphi_+$  on  $[-L,0]$ and $[0, L]$, defined by
\be
\phi_-(y) = { 1+\Phi(y) \over (4\gamma \mu)^{1/4}} e^{p(y,\rho_-) \over
\eps}
\ee
\be
\phi_+(y) = { 1+\Phi(y) \over (4\gamma \mu)^{1/4}} e^{p(y,\rho_+) \over
\eps}
\ee
and setting
$$
\varphi_-(y) := {\phi_-(y) \over \int_{-L}^0 \phi_-(x) \, dx}
\qquad
\varphi_+(y) := {\phi_+(y) \over \int_0^L \phi_+(x) \, dx},
$$
the problem under consideration is equivalent to
\be
\label{equiv-E}
w(y) = \begin{cases}
w_l + (w_* - w_l) \int_{-L}^y \varphi_- \, dx,     &  y < 0,
\\
w_r + (w_r - w_*) \int_y^L \varphi_+ \, dx,     &  y > 0,
\end{cases}
\ee
together with the boundary condition \eqref{boundv}. The latter can be expressed in its original form involving $w'$
(see the second equation in \eqref{elastic2}) and yields
\be
\label{wm-C}
w_* := {v_r - v_l + w_r \int_0^L y \, \varphi_+ \, dy - w_l \int_{-L}^0 y \, \varphi_- \, dy
\over \int_0^L y \, \varphi_+ \, dy -  \int_{-L}^0 y \, \varphi_- \, dy}.
\ee
Observe that the denominator in the formula for $w_*$ is bounded away from zero.
To find the value of the minimum in terms of the constitutive function $\sigma$ and the Riemann data
we need the following properties of $\varphi_{\pm}$

\

\noindent{\bf Step 2: Properties of $\varphi_{\pm}$. } With the same $\sigma_\eps^m$ and $\sigma_\eps^M$
as before,
set $\lambda_\eps^m = (\sigma_w^m-\eps/2)^{1/2}$ and $\lambda_\eps^M = (\sigma_w^M-\eps/2)^{1/2}$.
There exist constants $C_1>0$, and $C>0$ depending only on $\gamma, \sigma_w^m, \sigma_w^M$
and such that
\be
\label{prop3}
0 < \varphi_{-}(y) \leq {C_1\over\eps} \begin{cases}
e^{C(y+\lambda_\eps^M)^{2} \over \eps y}, &  -L \leq y < -\lambda_\eps^M,
\\
1, & -\lambda_\eps^M \leq y < -\lambda_\eps^m,
\\
e^-{C(y+\lambda_\eps^m)^2 \over \eps}, & -\lambda_\eps^m \leq y < -\gamma^{1/2}\lambda_\eps^m,
\\
e^-{C|y-\lambda_\eps^m \gamma^{1/2}| \over \eps}, & -\gamma^{1/2}\lambda_\eps^m \leq y \leq 0,
\end{cases}
\ee
\be
\label{prop4}
0 < \varphi_{+}(y) \leq {C_1\over\eps} \begin{cases}
e^-{C|y-\lambda_\eps^m \gamma^{1/2}| \over \eps}, & 0 \leq y \leq \gamma^{1/2}\lambda_\eps^m,
\\
e^-{C(y-\lambda_\eps^m)^2 \over \eps}, &  \gamma^{1/2} {\lambda_\eps^m} \leq y < {\lambda_\eps^m},
\\
1, & \lambda_\eps^m \leq y < \lambda_\eps^M,
\\
e^-{C(y-\lambda_\eps^M)^2 \over \eps y}, & \lambda_\eps^M <y <L.
\end{cases}
\ee

First, we prove the estimates for $\varphi_+$. Consider the region
$y>\lambda_\eps^M$
$$
\aligned
p(y,\rho_+)
&=\int_{\rho^+}^y \Big({-x\over 2\gamma} +\sqrt{{\sigma_w -\eps/2 + x^2(1/{4\gamma} -1) \over \gamma}} \Big) \, dx
\\
&\leq \int_{{\lambda_\eps^M}}^y   \Big({-x\over 2\gamma} +\sqrt{{{\sigma_w -\eps/2 + x^2(1/4\gamma -1)} \over \gamma}}\Big) \, dx
\\
& = {1\over 2\gamma}\int_{{\lambda_\eps^M}}^y x \, \Big(-1 + \sqrt{1+4\gamma {(\sigma_w -\eps/2 - x^2)\over x^2}} \Big) \, dx
\\
&\leq {\lambda_\eps^M\over 2\gamma}\int_{{\lambda_\eps^M}}^y
      \Big(-1 + \sqrt{1+4\gamma {(\sigma_w -\eps/2 - x^2)\over x^2}} \Big) \, dx,
\endaligned
$$
and, for $y>\lambda_\eps^M$,
$$
\aligned
p(y,\rho_+)
&= \leq {\lambda_\eps^M \over 2\gamma}\int_{{\lambda_\eps^M}}^y
2\gamma {(\sigma_w -\eps/2 - x^2)\over x^2} dx
\\
&\leq {\lambda_\eps^M}\int_{{\lambda_\eps^M}}^y (\sigma_w -\eps/2)^{1/2} - x) {(\sigma_w -\eps/2)^{1/2} +x)\over x^2} dx
\\
&\leq {\lambda_\eps^M}\int_{{\lambda_\eps^M}}^y {(\sigma_w -\eps/2)^{1/2} - x) \over x}dx
\\
&\leq {\lambda_\eps^M \over y}\int_{{\lambda_\eps^M}}^y (\lambda_\eps^M - x)dx
= -{\lambda_\eps^M \over y}{(y -\lambda_\eps^M)^2 \over 2}.
\endaligned
$$

Now, consider the region
$\gamma^{1/2}\lambda_\eps^m/2 \leq y \leq \lambda_\eps^m$,
$$
\aligned
p(y,\rho_+)
&=\int_{\rho^+}^y \Big({-x\over 2\gamma} +\sqrt{{\sigma_w -\eps/2 + x^2(1/{4\gamma} -1) \over \gamma}}\Big) \,  dx
\\
&\leq \int_{{\lambda_\eps^m}}^y \Big({-x\over 2\gamma} +\sqrt{{{\sigma_w -\eps/2 + x^2(1/4\gamma -1)} \over \gamma}}\Big) \, dx
\\
&\leq \int_y^{\lambda_\eps^m} \Big ({x\over 2\gamma} -\sqrt{{{\sigma_w -\eps/2 + x^2(1/4\gamma -1)} \over \gamma}}\Big) \, dx
\\
&={1\over 2\gamma}\int_y^{{\lambda_\eps^m}} x \, \Big(1 - \sqrt{1+4\gamma {(\sigma_w -\eps/2 - x^2)\over x^2}} \Big) \, dx
\\
&\leq{\lambda_\eps^m\over 4\gamma^{1/2}}\int_y^{{\lambda_\eps^m}}
  \Big(1 - \sqrt{1+4\gamma {(\sigma_w -\eps/2 - x^2)\over \lambda_\eps^m}} \Big) \, dx,
\endaligned
$$
and, for $\gamma^{1/2}\lambda_\eps^m/2 \leq y \leq \lambda_\eps^m$,
$$
\aligned
p(y,\rho_+) &-\leq {\lambda_\eps^m\over 4\gamma^{1/2}}\int_y^{{\lambda_\eps^m}} 4\gamma {(\sigma_w -\eps/2 - x^2)\over \lambda_\eps^m}) dx
\\
&\leq -C \int_y^{{\lambda_\eps^m}} 4\gamma (\sigma_w -\eps/2)^{1/2} - x) (\sigma_w -\eps/2)^{1/2} +x) dx
\\
&\leq -C \int_y^{{\lambda_\eps^m}} (\sigma_w -\eps/2)^{1/2} - x)dx
\\
&\leq -C \int_y^{{\lambda_\eps^M}} (\lambda_\eps^m - x)dx = -C{(y -\lambda_\eps^m)^2 \over 2}.
\endaligned
$$

For  $0\leq y \leq \gamma^{1/2}\lambda_\eps^m/2$,
$$
\aligned
p(y,\rho_+)
&=\int_{\rho^+}^y \Big({-x\over 2\gamma} +\sqrt{{\sigma_w -\eps/2 + x^2(1/{4\gamma} -1) \over \gamma}} \Big) \, dx
\\
&\leq \int_{{\lambda_\eps^m}\gamma^{1/2}}^y
\Big({-x\over 2\gamma} +\sqrt{{{\sigma_w -\eps/2 + x^2(1/4\gamma -1)} \over \gamma}}\Big) \, dx
\\
&\leq \int_y^{\lambda_\eps^m \gamma^{1/2}}
\Big({x\over 2\gamma} -\sqrt{{{\sigma_w -\eps/2 + x^2(1/4\gamma -1)} \over \gamma}}\Big) \, dx
\\
&\leq \int_y^{{\lambda_\eps^m}\gamma^{1/2}} ({\gamma^{1/2}\lambda_\eps \over 2 \gamma} - {\lambda_\eps^m \over \gamma^{1/2}})dx
\\
&=\int_y^{{\lambda_\eps^m}\gamma^{1/2}} - {\lambda_\eps^m \over 2\gamma^{1/2}}dx
=-{C|y-\lambda_\eps^m \gamma^{1/2}|}.
\endaligned
$$
The other cases are identical. One can also check (for some $c>0$)
$$
 \int_{-L}^0 \phi_-(y) dy \geq c \, \eps,
\qquad
\int_0^L \phi_+(y) dy \geq c \, \eps,
$$
so that the desired properties of $\varphi_+,\varphi_-$, follow.

\

\noindent{\bf Existence arguments. }
Now $w_*$ is sum of convex combination of $w_l,w_r$ and the quantily
$$
 {{v_r - v_l}  \over {\int_0^L y \, \varphi_+ \, dy -  \int_{-L}^0 y \, \varphi_- \, dy}}
=  {{v_r - v_l} \over {\int_{-\lambda_\eps^M}^{-\lambda_\eps^m} -y \,
\varphi_- \, dy + \int_{\lambda_\eps^m}^{\lambda_\eps^M} y \, \varphi_+ \, dy + O(\eps^n)}}.
$$
From this we get
$$
|w_*| \leq \max (w_l,w_r) + {{|v_r - v_l|} \over {2 \lambda_\eps^m} + O(\eps^n)}\leq \Lambda_0.
$$
We can now replace $w_*$ (given by \eqref{wm-C}) in \eqref{equiv-E} and
arrive at a mapping
$w \in C^0([-L,L]) \mapsto T(w) \in C^0([-L,L])$. For fixed $\eps$,  $T(w)$ is of class $C^1$, and
$$
\aligned
& \|T(w)\|_{C^0} \leq \Lambda_0,
\\
& \|T(w)'\|_{C^0} \leq {C \over \eps} \, (|w_r - w_* | +|w_r - w_* |).
\endaligned
$$
These estimates follow from \eqref{equiv-E}  \eqref{prop3} and
\eqref{prop4}

Thus, $T$ is a compact map from a convex bounded set $\{ w : \|w\|_{C^0} \leq \Lambda_0\}$ into
itself. By Schauder's fixed point theorem, $T$ has a fixed point in $C^0[-L,L]$ and satisfies
\eqref{equiv-E} and so the solution is of class $C^1$. Furthermore, in
view of \eqref{equiv-E} and \eqref{wm-C},
\be
\label{es1-B}
\int_{-L}^L |w'(y)| dy \leq | w_r - w_* | +  | w_l - w_* | \leq  | w_r - w_l |+ 2{| v_r - v_l | \over c_0}
\ee
and
\be
\label{es2-B}
\int_{-L}^L |v'(y)| dy \leq \int_{-L}^L |y w'(y)| dy \leq (\lambda_0^M+1) \, \Big( |w_r - w_l|+ 2{|v_r - v_l| \over c_0}\Big).
\ee

As the estimates are uniform in $L$, we have existence of solution $(v^\eps,w^\eps)$ on $(-\infty,\infty)$
we have $(v^\eps,w^\eps)$ bounded solution of \eqref{elastic2} and \eqref{boundary1} with uniform total variation
$$
\int_{-\infty}^\infty (|w'(y)|+|v'(y)| dy \leq (\lambda_0^M+2)[ |w_r - w_l|+ 2{|v_r - v_l| \over c_0}]
$$
By compactness there exists a sequence converges in $L^1$ and pointwise
a.e. to a function $(v,w)$ as $\eps$ goes to zero and solves
\eqref{elastic-first} and \eqref{boundary} with $\eps =0$.
\end{proof}

\section{Phase dynamics with physical viscosity}
\label{section4}

We turn to the model of phase transition dynamics when the system is not strictly
hyperbolic, that is, when $\sigma_w(w)$ is only non-negative, or even is hyperbolic-elliptic
when $\sigma_w$ takes negative values.

To illustrate the key difficulty we will have to cope with, let us first consider the example
$$
a(y) = y + c/y,
$$
with $c$ a constant. When $c\geq 0$, $\rho_{+} =\delta$ and
$\int_{\rho^+}^y a(x) dx = {y^2-\rho_{+}^2 \over 2} +
\log(|y/\rho_{+}|^{c})$ and so
$$
e^{-{1\over\eps}\int_{\rho_{+}}^y a(x) dx} = e^{-(y^2-\rho_{+}^2) \over
2 \eps} (y/\rho_{+})^{-c/\eps} = e^{-(y^2-\delta^2) \over
2 \eps} (y/\delta)^{-c/\eps}.
$$
In this case
\[
\varphi_+(y) =  {e^{-(y^2-\delta^2)\over
2 \eps} (y/\delta)^{-c/\eps}\over \int_{\rho_{+}}^L e^{-(y^2-\delta^2)
\over 2 \eps} (y/\delta)^{-c/\eps} dy},
\]
which is concentrated at $y=\delta$. For $c<0$, we have,
$\rho_+ = (- c)^{1/2}$ and
$$
e^{-{1\over\eps}\int_{\rho_{+}}^y a(x) dx} = e^{-(y^2+c) \over
2 \eps} (y/(-c)^{1/2})^{-c/\eps},
$$
so that
\[
\varphi_+(y) =  {e^{-(y^2+c) \over
2 \eps} (y/(-c)^{1/2})^{-c/\eps} \over \int_{\rho_{+}}^L
e^{-(y^2+c)\over 2 \eps} (y/(-c)^{1/2})^{-c/\eps} dy},
\]
which is concentrated at $y=(-c)^{1/2}$.

In the previous section, to show that $w_*$ is bounded independent of
$\eps$ it was crucial that $\varphi^{+}$ and $\varphi^{-}$ be concentrated
away from $0$. In the above example, if $c \geq 0$, we have seen
that this is not the case. This is the difficulty is getting estimates when $\sigma_w(w(y))$
oscillates, i.e.~takes both negative and positive values.

We now prove the following result:

\begin{theorem}  [Vanishing viscosity limit in phase dynamics]
Suppose that the first order system \eqref{elastic-first} is uniformly
hyperbolic for large value of $w$ but
may admit elliptic regions in the phase space, that is for some constants $M,c_0 >0$
$$
\inf_{|w| \geq M} \sigma_w(w) \geq {c_0}^2.
$$
Suppose also that there exist constants $c_1>0$ and $\eta>0$ such that
$|\sigma_w (w) |\leq c_1 |w|^{2-\eta}$ for $|w|>1$.
Then for all Riemann data $v_l, w_l$ and $v_r,w_r$ in the hyperbolic region of the phase space, i.e.
$$
w_l, w_r \in \Hcal := \big\{w  \, / \, \sigma_w >0 \big\}
$$
the viscous Riemann problem \eqref{elastic2}-\eqref{boundary1},
with $\delta = 0$,
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation
at least away from $y=0$, and more precisely
$$
\int_\RR \big( |v'_\eps| + y \, |w_\eps'| \big) \, dy \lesssim |v_r - v_l| + | w_r - w_l|.
$$
The functions $v_\eps, w_\eps$ converge pointwise at all $y \neq 0$ to a limit $v,w$,
which is a solution of the Riemann problem
\eqref{elastic-first}-\eqref{boundary} {\rm away from the axis}
$y=x/t =0$. Furthermore, $v$ has bounded variation and so
admits left- and right-hand limits at $y=0$, while the function $w$ and its variation measure $dw/dy$ satisfy
$$
\aligned
& |w| \lesssim {1 \over |y|}, \quad y \neq 0,
\\
& \int_\RR |y| \, |{dw \over dy}| < \infty.
\endaligned
$$
\end{theorem}

Hence, the component $v$ only has globally bounded variation.
In turn, the conservation laws \eqref{elastic-first} are satisfied in the two regions $x < 0$
and $x>0$, but a singularity may arise on the axis.

\begin{proof}
{\it Step 1. A priori estimates for the components $v,w$.} To simplify the notation we suppress the subscript $\eps$.
We will first study the problem away from the axis $y=0$. We consider the function
\be
a(y) = y - (\sigma_w(w(y)) - \eps)/y,
\ee
and, given some $\delta>0$, we study the problem  in two regions $[-L,-\delta]$ and
$[\delta,L]$, with the following boundary conditions
\be
\label{boundl-B}
w(-L) = w_l, \qquad w(\delta-) = w_*,
\ee
\be
\label{boundr-B}
w(\delta) = w_*,  \qquad w(L) = w_r.
\ee
We set
$$
\varphi_{-}^\delta(y) := {e^{-(1/\eps) \int_{\rho_-}^y a^\delta(x) \,
dx} \over \int_{-L}^{-\delta} e^{-(1 /\eps) \int_{\rho_-}^y a^\delta(x)
\, dx}},
\qquad
\varphi_{+}^\delta(y) := {e^{-(1/\eps) \int_{\rho_+}^y a^\delta(x) \,
dx} \over \int_\delta^L e^{-(1 /\eps) \int_{\rho_+}^y a^\delta(x) \,
dx}},
$$
on $[-L,-\delta]$ and $[\delta,L]$
where $\rho_\pm$ are the points where $\int^y a(x) dx$ attains its global minimum.

Let us consider  $\lambda_\eps^{M+} = \sup_{1 \leq y \leq
L}{(\sigma_w(w)-\eps)^+}^{1/2}$
and let $c> \max\{1,\lambda_\eps^{M+}\}$. For $y>c$
$$
\aligned
\int_{\rho^+}^y (x^2 -\sigma_w(x)+\eps) dx
&\geq \int_{c}^y {(x^2 -\sigma_w(x)+\eps) \over x} dx
\\
&= \int_{c}^y ((x -(\sigma_w(x)+\eps)^{+})^{1/2})(1
+{((\sigma_w(x)+\eps)^{+})^{1/2} \over x}) dx
\\
&\geq \int_{c}^y (x-c) dx = {(y - c)^2 \over 2}.
\endaligned
$$
So, for $y>(1+ \sup_{1 \leq y \leq L}({(\sigma_w(w)-\eps)^+})^{1/2}$, we get
\be
\label{prop6}
\varphi(y) dy \leq {1\over\eps}e^{-{(y - c)^2 \over 2\eps}}
\ee
and
\be
\label{wavesupport}
\int_\delta^L y\varphi(y) dy \leq
(2+ \sup_{1 \leq y \leq
L}{((\sigma_w(w)-\eps)^+})^{1/2}.
\ee
We then set
\be
\label{equiv2}
w(y) = \begin{cases}
w_l + (w_* - w_l) \int_{-L}^y \varphi_{-}^\delta \, dx,     &  y <
-\delta,
\\
w_r + (w_* - w_r) \int_y^L \varphi_{+}^\delta \, dx,     &  y >\delta.
\end{cases}
\ee

Here, we have taken $w(-\delta) = w(\delta) = w_*$ and we also take
$v(-\delta) = v(\delta) = v_*$. Integrating the equation $v'=yw'$ from $-L$ to $-\delta$ and
$\delta$ to $L$  and using \eqref{equiv2}, we get
$$
v_* -v_l = (w_* - w_l)\int_{-L}^{-\delta} -y \phi_-(y) dy
$$
and
$$
v_r -v_* = (w_* - w_r)\int_{\delta}^{L} y \phi_+(y) dy.
$$
Adding these formulas, we get
\be
\label{wm2}
w_* := {v_r - v_l + w_r \int_\delta^L y \, \varphi_{+}^\delta \, dy -
w_l \int_{-L}^{-\delta} y \, \varphi_{-}^\delta \, dy
\over \int_\delta^L y \, \varphi_{+}^\delta \, dy -  \int_{-L}^{-\delta}
y \,
\varphi_{-}^\delta \, dy}
\ee
and, subtracting,
\be
\label{vm2}
v_* = v_l +{[v_r - v_l + (w_r - w_l) \int_\delta^L y \,
\varphi_{+}^\delta \, dy]
 \int_{-L}^{-\delta} -y \, \varphi_{-}^\delta \, dy
\over \int_\delta^L y \, \varphi_{+}^\delta \, dy -  \int_{-L}^{-\delta}
y \, \varphi_{-}^\delta \, dy}.
\ee

Now, $v$ can be expressed in terms of the boundary data $w_l,w_r,v_l,v_r$ and $\delta$:
\be
\label{equiv3}
v^\delta(y) = \begin{cases}
 v_l +{[v_r - v_l + (w_r - w_l) \int_\delta^L y \, \varphi_{+}^\delta \,
dy] \int_{-L}^y -x \, \varphi_{-}^\delta \, dx
\over \int_\delta^L y \, \varphi_{+}^\delta \, dy -  \int_{-L}^{-\delta}
-y \,  \varphi_{-}^\delta \, dy}, & -L \leq y<-\delta
\\
 v_r + {[v_l - v_r + (w_r - w_l) \int_-L^\delta -y \, \varphi_{-}^\delta
\, dy] \int_{y}^L x \, \varphi_{+}^\delta \, dx
\over \int_\delta^L y \, \varphi_{+}^\delta \, dy +  \int_{-L}^{-\delta}
\, -y\varphi_{-}^\delta \, dy}.     & L> y > \delta.
\end{cases}
\ee
Also, $w(y)$ can be written as
\be
\label{equiv4}
w^\delta(y) = \begin{cases} \displaystyle
w_l + (v_* - v_l){ \int_{-L}^y \varphi_{-}^\delta \, dx, \over
\int_{-L}^{-\delta} y\varphi_{-}^\delta \, dx}  &  y < -\delta,
\\  \displaystyle
w_r + (v_r - v_*){\int_y^L \varphi_{+}^\delta \, dx, \over \int_\delta^L
y\varphi_{+}^\delta \, dx}  &  y > \delta.
\end{cases}
\ee

Now clearly $v_{*}$ is bounded independent of $\eps>0$,$\delta>0$ :
$$
|v_* -v_l|\leq |v_r - v_l| + L (|w_r -w_l|)
$$
and from \eqref{equiv3} and \eqref{equiv4}, we get the following estimates
$$
\aligned
& |y| \, |w^\delta(y)| \leq (|v_r| + |v_l|) + 2L(|w_r| +|w_l|),
\\
& |v^\delta(y)|\leq 2 (|v_r| + |v_l|) + L (|w_r| +|w_l|),
\endaligned
$$
Also we have the following estimates on the derivatives of
$v^\delta,w^\delta$.
$$
\aligned
& \int_{L>|y|>\delta}|y w^\delta(y)'| dy \leq {(|v_r - v_l|) +
L(|w_r-w_l|)},
\\
& \int_{L>|y|>\delta}|v^\delta(y)'| dy\leq (|v_r-v_l|) + L (|w_r-w_l|).
\endaligned
$$
Furthermore, for all $L>|y|>\delta$
$$
\aligned
& |v^\delta(y)'| \leq {1  \over \eps \delta} \Big( (|v_r-v_l|) + L (|w_r-w_l|)y \Big),
\\
& |w^\delta(y)'| \leq {1 \over \eps \delta} \Big( (|v_r-v_l|) + L (|w_r-w_l|)\Big).
\endaligned
$$

The existence of solution for fixed $L$ follows from these estimates
as we will see. In order to pass $L\rightarrow \infty$, we need estimates
independent of $L$. For this
we use the growth condition on $\sigma_w$. From the expression
\eqref{vm2} for $v_{*}$, and \eqref{wavesupport}, we get

$$
|v_* -v_l|\leq |v_r - v_l| + (2+ \sup_{1 \leq y \leq
L}{(\sigma_w(w)-\eps)^+}^{1/2}) |w_r -w_l|.
$$

Using this in \eqref{equiv3} and \eqref{equiv4}, we get
$$
\aligned
&\sup_{1\leq y\leq L} |w^\delta(y)| \leq (|v_r| + |v_l|) + ((3+ \sup_{1
\leq y \leq L}{(\sigma_w(w)-\eps)^+}^{1/2})(|w_r| +|w_l|),
\\
& |v^\delta(y)|\leq 2 (|v_r| + |v_l|) + ((2+ \sup_{1
\leq y \leq L}{(\sigma_w(w)-\eps)^+}^{1/2})(|w_r| +|w_l|),
\endaligned
$$
Now using our assumption
$|\sigma_w|\leq c_1 |w|^{2-\eta}$ for $|w|>1$ in the first inequality, we
conclude that $\sup_{1\leq y\leq L}|w^\delta(y)|$ is independent of $L$. Similar
estimate holds for $[-L\leq y\leq-1]$. We get there exists a constant
$C=C(v_l,v_r,w_l,w_r)$, independent of $L$ such that
\be
\label{bound[L]}
\sup_{1\leq |y|\leq L}|w^\delta(y)| \leq C
\ee
With this constant $C$, let $\lambda^M =sup_{|w|\leq
C}{|\sigma_w(w)|}^{1/2}$.
From \eqref{prop6},
we get  $\varphi^{\delta}_{+}$ is essentially supported in
$[\delta, \lambda^M+1]$.
Similar arguments give $\varphi^{\delta}_{-}$ is essentially
supported in the interval $[-\lambda^M -1,-\delta]$.
It follows from \eqref{equiv3} and \eqref{equiv4} that
there exists a constant $C_1$
\be
\label{var}
\aligned
& \int_{|y|>\delta}|y w^\delta(y)'| dy \leq {(|v_r - v_l|) +
C_1(|w_r-w_l|)},
\\
& \int_{|y|>\delta}|v^\delta(y)'| dy\leq (|v_r-v_l|) + C_1 (|w_r-w_l|).
\endaligned
\ee

\noindent{\it Step 2. Existence proof. }
For each $\delta>0$, we can apply Schauder's fixed point theorem, as was explained earlier,
and we obtain a solution $(w^\delta,v^\delta)$ defined on the interval $[-L,-\delta] \bigcup [\delta, L]$
and satisfying uniform in $\eps>0$ total variation estimate.
The estimates \eqref{bound[L]} and \eqref{var} allows as to let $L$ tend
to infinity.
By compactness, we have a solution $(v,w)$
defined in the region $|y|>\delta$. Since $\delta>0$ is arbitrary, we obtain a well-defined
solution away from $y=0$. Indeed, $v^\delta$ has a uniform total variation on $R$ and
$v$ admits left- and right-limit at  $y=0$.
One can check easily that the limit is a weak solution of the problem \eqref{elastic-first}-\eqref{boundary}
away from the axis $y=0$ at least.
Without a control on $\sigma(w)$ near $y=0$ we can not exclude a
concentration term on the axis.
\end{proof}

\section{Boundary Riemann problem and further generalizations}
\label{section5}

In this section, we outline how the method developed in previous sections can be
generalized to the boundary-value problem and indicate several generalizations.
It is well-known that in the strictly hyperbolic case with nondegenerate function $\sigma$,
the boundary Riemann problem (in $x>0, t>0$)
\be
\label{elastic51}
v_t -\sigma(w)_x =0,
\qquad
w_t - v_x =0.
\ee
with initial and boundary conditions
\be
\label{boundary51}
w(0+,t) = w_b,
\qquad
v(x,0) = v_r, w(x,0) = w_r
\ee
is well posed. This easily follows from an analysis of the wave curves for the system corresponding to the left-moving
and right-moving characteristic families. The physical regularizations
considered in earlier sections is well-suited to handle this boundary value problem, without producing
boundary layers in the limit.

Consider the nonlinear elastodynamics with physical viscosity
\be
\label{elastic52}
-y v'- \sigma_w(w)) \, w' = \eps \, v'',
\qquad
-y \, w'-v'=0, \qquad \text{ on $[0,\infty)$, }
\ee
with boundary conditions
\be
\label{boundary52}
w(0) = w_b,
\qquad
v(\infty) = v_r, w(\infty) = w_r.
\ee
Observe that, for the "full'' viscosity approximation, we need to
prescribe the component $v$ at $y=0$ and this generates a boundary layer
at $y=0$, after passage to the limit $\epsilon \to 0$; see~\cite{JLthree,JLfour}. In
the present case, we show that no boundary layer arises.

As in Section 2,  the problem can be reduced to a scalar equation
for the unknown $w$
\be
\label{scalar51}
(y^2 + \eps - \sigma_w(w)) \, w' = - \eps \, y \, w'', \qquad \text{ on $[0,L]$,}
\ee
for sufficiently large $L$, with boundary conditions,
\be
\label{boundary53}
w(0) = w_b, w(L) = w_r.
\ee
Once we have $w$, we get  the component $v$ from the equation $yw'+v' =0$ and the boundary condition $v(L)=v_r$.

The fixed point argument in Section 2 yields a solution $w^\eps$ of
\eqref{scalar51}-\eqref{boundary53}
which is of uniformly bounded variation. Furthermore, it can be
represented by an integral formula in terms of the function $\varphi_+$
introduced in Section 2
\[
w^\eps(y) = w_r + (w_b - w_r) \int_y^L \varphi_+ \, dx,    \qquad y > 0.
\]
Using that $\int_0^L \varphi_+(x) dx =1$, we get
\be
\label{equiv51}
w^\eps(y) = w_b + (w_b - w_r) \int_0^y \varphi_+ \, dx,    \qquad y > 0.
\ee
Using \eqref{equiv51} in the equation $v'=-yw'$ and integrating from $y$
to
$L$, we get
\be
\label{equiv52}
v^\eps(y) = v_r + (w_b - w_r) \int_y^L x \varphi_+ \, dx,   \qquad   y > 0,
\ee
where we used $v(L)=v_r$. Note that
we cannot prescribe boundary condition at $v(0)$ arbitrarily, since from the above equation it follows that
\[
v(0) = v_r + (w_b - w_r) \int_0^L x \varphi_+ \, dx.
\]

As in Section 2, using the fact that $\varphi_+$ decays exponentially
we can let $L$ tend to $\infty$. Again,
using the properties \eqref{prop2} of $\varphi_+$ near $y=0$ in
\eqref{equiv51}, it follows easily that there exists $C>0$, a constant independent of $\epsilon$, such
that for all $y>0$ close to the origin
\[
|w(y) - w_b| \leq C \, y,
\]
Hence, no boundary layer arises in this approximation at $y=0$ and in the
limit as $\epsilon \rightarrow 0$, the limit function $w=w(y)$
satisfies the boundary condition $w(0+)=w_b$. We summarize our results in
the following theorem:

\begin{theorem}
Assume that $\inf \sigma_w \geq c_0^2 >0$. Then
given arbitrary Riemann boundary data, the problem
\eqref{elastic52}-\eqref{boundary52}
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation
and converges to a limit $v,w$ which is a solution of the boundary Riemann
problem
\eqref{elastic51}-\eqref{boundary51}.
\end{theorem}

\

We now discuss a general system. The basic nature of the
system \eqref{elastic51} is that $v$ appear linearly and the
characteristic speeds are equal in magnitude and opposite in sign.
Further using the second equation, the system can be reduced to uncoupled
equations for $w$. After this reduction we
solved for $w$ first and then for $v$. Our results can be
generalized for systems of first order equations for $w$ and $v$ vector
valued functions, having same structure, using the ideas of the
present work and the earlier work \cite{JLthree,JLfour} on boundary value problems.

For example
let $F : R^N \rightarrow R^N$ be a smooth function with $A(w)=D_{w}F$
has real distinct positive eigenvalues $\lambda_1(w)^2 < \lambda_2(w)^2
<...< \lambda_N(w)^2$
with a complete set of right-left normalized eigenvectors $r_j,
j=1,2,...N$, $l_j,j=1,2,...,N$, $l_j.r_k =\delta_{jk}$.

We assume that  there exists $c_0>0$ such that $\lambda_j(w) \geq c_0$ for all
$j=1, 2, ...N$ and for all $w \in B(\delta_0)$, a ball of radius $\delta_0$ around a fixed state
that we take to be $0$. We consider the system of $2N$ equations
\be
\label{general51}
\aligned
& {\del v \over \del t} - {\del \over \del x} \Big( F(w)) = 0,
\\
& {\del w \over \del t} - {\del v \over \del x} = 0,
\endaligned
\ee
where $v=v(t,x)$ and $w=w(t,x)$ are $R^N$ valued functions. The system
\eqref{elastic51} is a special case of \eqref{general51} with $N=1$ and
$\lambda(w)^2=\sigma_w(w)$

We consider boundary value problem for \eqref{general51} on $x>0, t>0$.
When physical viscosity terms are taken into account,
the corresponding set of differential equations becomes
\be
\label{general52}
\aligned
& -y \, v' - F(w)' = \eps \, v'',
\\
& - y \, w' - v' = 0,
\endaligned
\ee
supplemented with the boundary conditions
\be
\label{boundary54}
\lim_{y \to \infty} (v,w)(y) = (v_r, w_r),
\qquad
w(0+) = w_b.
\ee

We first consider \eqref{general52} on $[0,L]$ with boundary conditions
\be
\label{boundary55}
(v,w)(L) = (v_r, w_r),
\qquad
w(0+) = w_b.
\ee
As in the previous case the problem can be reduced to first solving a
system for $w(y)$ namely
\be
\label{scalar33}
(y^2 + \eps - A(w)) \, w' = - \eps \, y \, w'',
\ee
on $[0,L]$, with boundary conditions,
\be
\label{boundary7}
w(0) = w_b, w(L) = w_r.
\ee
Existence of uniformly BV solutions of this problem easily
follows from the work of Joseph and LeFloch \cite{JLthree,JLfour}. We just
outline
the main steps omitting the details.

We decompose $u'$ in the basis of eigenvectors of $A(u)$,
\be
\label{expan51}
u'(y)=\sum_{j=1}^{N}a_j r_j(u), a_j= \la l_j,  u' \ra
\ee
A straight forward calculation lead to the following system of nonliear
equations for $a_j, j=1,2, ..., N$
\be
\label{coupled51}
\eps y a_j' + (y^2 +\eps -\lambda_j^2(w))a_j = D_1(a,a)
\ee
where
\be
\label{quadra51}
D_1(a,a) = -\eps y \sum_{k,i} a_i a_k (D_u r_i.r_k)
\ee

Observe that the linearized form of the equation \eqref{coupled51} has the
form
\[
\eps \, y \, a_j'(y^2 + \eps - \lambda_j^2(w)) \, a_j' = 0,
\]
and the corresponding wave measure $\varphi_{j+}$ has exactly same
properties
\eqref{prop2} given in section 2.
These wave measures naturally appear when we invert the linear part of
the equation \eqref{coupled51}. Because of the quadratic righthand side
this leads to interaction terms with different families of wave measures
which is controlled by $\sum_{j=1}^N \varphi_{j+}$. Fixed point arguments
give a uniform BV
solution $w^\eps(y)$ for the system \eqref{scalar33} and
\eqref{boundary7}.
The details are similar to Joseph and LeFloch \cite{JLthree} and, therefore, are omitted. Then
the BV estimate for $v$ follows from the second equation in
\eqref{general52} and
\[
v(y)=v_r + \int_y^L x w'(x) dx.
\]

As the estmate \eqref{prop2} shows that
$\varphi_{j+}$ is essentially supported in $[\lambda_j^m,\lambda_j^M]$
we can let $L$ go to infinity. Here $\lambda_j^m$ and $\lambda_j^M$
denotes the minimum and maximum of $\lambda_j(w)$ on the ball
$B(\delta_0)$. We have the following result.

\begin{theorem}
Assume that $\inf \lambda_j(w) \geq c_0 >0$. Then there exists
$\delta_1>0,\delta_2>0$ such that, for $w_b, w_r \in B(\delta_1)$
and arbitrary $v_r$
the boundary Riemann problem \eqref{general52}-\eqref{boundary54}
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation
with $w^\eps(y) \in B(\delta_2)$
and converges to a limit $v,w$ which is a solution of the equation
\eqref{general51} with boundary conditions
$w(0+)=w_b, w(x,0)=w_r, v(x,0)=v_r$.
\end{theorem}

\begin{remark} This analysis is easily extended to the case with physical
viscosity and capillarity as well namely
\be
\label{general53}
\aligned
& -y \, v' - F(w)' = \eps \, v'' - \gamma \eps^2 w''', \gamma >0,
\\
& - y \, w' - v' = 0,
\endaligned
\ee
supplemented with the boundary conditions
\be
\label{boundary56}
\lim_{y \to \infty} (v,w)(y) = (v_r, w_r),
\qquad
w(0+) = w_b.
\ee
\end{remark}

As in the previous case, the problem can be reduced to first solving a
system for $w(y)$, namely
\be
\label{scalar53}
(y^2 + \eps - A(w)) \, w' = - \eps \, y \, w''-\gamma \eps^2 w''',
\ee
on $[0,L]$, for sufficiently large $L$, with boundary conditions
\be
\label{boundary57}w(0) = w_b, w(L) = w_r.
\ee
We decompose $u'$ in the basis of eigenvectors of $A(u)$,
\be
\label{expan}
u'(y)=\sum_{j=1}^{N}a_j r_j(u), a_j = \la l_j, u'\ra
\ee
 and then the nonlinear
equations for $a_j, j=1,2, ..., N$ take the form
\be
\label{coupled52}
\gamma \eps^2 a_j'' +\eps y a_j' + (y^2 +\eps -\lambda_j^2(w))a_j =
D_1(a,a)
+ D_2(a,a) +D_3(a,a,a),
\ee
where
\[
D_1(a,a) = -\eps y \sum_{k,i} a_i a_k (D_u r_i.r_k)
\]
\[
D_2(a,a) = -\gamma \eps^2 (\sum_{k,i} a_i a_k' (D_u r_i.r_k)+
          \sum_{k,i} (a_i a_k)' (D_u r_i.r_k
\]
\[
D_3(a,a,a) = -\gamma \eps^2  \sum_{k,i,l} a_i a_k a_l D_u(D_u r_i.r_k)r_l
\]

The linearized equation for \eqref{coupled52} is
\[
\gamma \eps^2 a_j'' +\eps y a_j' + (y^2 +\eps -\lambda_j^2(w))a_j = 0,
\]
and hence the wave measures $\varphi_{j+}$ in this case has the same
qualitative
properties as \eqref{prop4}.

Existence of uniformly BV solutions of the problem \eqref{scalar53} and
\eqref{boundary57} can be easily deduced from \cite{LeFlochRohde}. Then, the BV estimate for $v$ follows from
the second equation of \eqref{general53}. The details are omitted. We get the following result:

\begin{theorem}
Assume that $\inf \lambda_j(w) \geq c_0 >0$. Then there exists
$\delta_1$
and $\delta_2$ such that for $w_b, w_r \in B(\delta_1)$ and for arbitrary
$v_r$,
the problem \eqref{general53}-\eqref{boundary56}
admits a solution $v^\eps, w^\eps$, which has
uniformly bounded total variation
with $w^\eps(y) \in B(\delta_2)$
and converges to a limit $v,w$ which is a solution of the equation
 \eqref{general51} with boundary conditions
$w(0+)=w_b,w(x,0)=w_r,v(x,0)=v_r$.
\end{theorem}

Now let us consider the phase transition case with physical viscosity
and capillarity. Following
previous sections , the system can be reduced to a
single scalar equation for the unknown $w$,
\be
\label{scalar-B1}
(y^2 + \eps - \sigma_w(w)) \, w' = - \eps \, y \, w'' - \gamma \eps^2
w''', \gamma>0
\ee
which can be studied in two regions $[-L,-\delta_0]$ and $[-\delta_0,L]$
away from $y=0$, with the boundary conditions
\be
\label{boundl-BC1}
w(-L) = w_l, \qquad w(-\delta_0) = w_*,
\ee
\be
\label{boundr-BC1}
w(\delta_0) = w_*,  \qquad w(L) = w_r,
\ee
where $w_*$ is given by \eqref{wm2} with $\delta$ replaced by $\delta_0$ chosen as follows.
Following Sections 3 and 4, we can construct a BV solution $w$ for
\eqref{scalar-B1}, in $[-L,-\delta_0]$ and $[\delta_0, L]$
with the boundary conditions \eqref{boundl-BC1} \eqref{boundr-BC1}, provided
$$
\mu(w,y) =  \sigma_w + y^2({1 \over 4 \gamma} - 1) - {\eps \over 2} > 0.
$$
This is the case if we impose the restriction $|y|>\delta_0$ where
\be
\label{delta}
\delta_0> ({4 c \gamma \over 1-4\gamma})^{1/2}.
\ee
Here, $c>0$ is a constant such that $\sigma_w(w)\geq -c$ for all $w$.

We get the following result :

\begin{theorem}  [Vanishing viscosity-capillarity limit in phase dynamics]
Suppose that the first-order system \eqref{elastic-first} is uniformly
hyperbolic for large value of $w$ but may admit elliptic regions in the phase space,
that is for some constants $M,c_0 >0$
$$
\inf_{|w| \geq M} \sigma_w(w) \geq {c_0}^2.
$$
Suppose also that there exist positive constants $c_1, \eta$ such that
$|\sigma_w|\leq c_1 |w|^{2-\eta}$ for $|w|>1$.
Let $c>0$ be such that $\sigma_w(w)>-c$ for all $w$ and suppose $\delta_0$ is chosen as in \eqref{delta}.
Then, for all Riemann data $v_l, w_l$ and $v_r,w_r$ in the hyperbolic region of the phase space, i.e.
$$
w_l, w_r \in \Hcal := \big\{w  \, / \, \sigma_w >0 \big\}
$$
the viscous-capillarity Riemann problem
\eqref{elastic2}-\eqref{boundary1}, with $\delta=\gamma \eps^2$, $\gamma>0$
admits a solution $v^\eps, w^\eps$ which has uniformly bounded total variation
in $|y|>\delta_0$, and more precisely
$$
\int_{|y|>\delta_0} \big( |v'_\eps| +  |w_\eps'| \big) \, dy \lesssim
|v_r - v_l| + | w_r - w_l|.
$$
The functions $v_\eps, w_\eps$ converge pointwise at all $|y|>\delta_0$ to
a limit $v,w$, which is a solution of the Riemann problem
\eqref{elastic-first}-\eqref{boundary} for  $|y|=|x/t| >\delta_0$.
\end{theorem}

For instance, the example $\sigma(w)=u(u^2-1)$ satisfies all the assumptions of the theorem.
We can not exclude that the vanishing viscosity-capillarity approximations to the Riemann problem
could contain a highly oscillating (i.e.~weakly but not strongly converging) pattern
near the axis; its thickness would be $2\delta_0$, at most. The existence of such stationary waves
is not surprising as it has been observed numerically; they were never pointed out analytically until now.

\section{Effect of general viscosity in fluid dynamics}

Following our earlier work \cite{JLfive} we now return to the general diffusion approximation for strictly
hyperbolic systems with general diffusion matrix $B(u)$ of the form
\be
\label{system}
-y u' + A(u) u' =\eps (B(u)u')',
\ee
and we consider an associated generalized eigenvalue problem.

Let $\mu_j$, $\hatl_j$, $\hatr_j$, $j=1,2,\ldots,N$ be
the eigenvalues and left and right-eigenvectors given by
\be
\label{eigen}
\aligned
& \bigl(- y + A(v) \bigr) \, \hatr_j(v,y) = \mu_j (v,y) \,  B(v) \, \hatr_j(v,y),
\\
& \hatl_j(v,y) \cdot \bigl( - y + A(v) \bigr) = \mu_j(v,y) \, \hatl_j(v,y)
\, B(v).
\endaligned
\ee
Let us impose the normalization
$$
\hatl_i(v,y) \, B(v) \, \hatr_j(v,y) = 0  \, \text{ if } i \neq j;
\quad
\hatl_j(v,y) \, B(v) \, \hatr_j(v,y) = 1.
$$

In the special case $B(u) = I$ we have simply
\be
\label{special}
\mu_j(v,y)         = - y + \lam_j(v),
\quad \hatr_j(v,y) = r_j(v),
\quad \hatl_j(v,y) = l_j(v).
\ee
So, we expect the eigenvalues and eigenvectors in \eqref{eigen} to be close to
to that of \eqref{special}, at least in the case that
$B(v) = I + \eta \, T(v,\eta)$ where $T(v, \eta) = \bigl( t_{ij} (v, \eta) \bigr)$
with $t_{ij}(v, \eta) = O(1)$ and $\eta<<1$.

\begin{lemma} If $|B(v) - I| = O(\eta)$ is sufficiently small,
then
\be
\label{approx}
\aligned
& \mu_j(v,y) = - y + \lambda_j(u) + O(\eta),
  \quad \hatl_j(v,y) = l_j(v) + O(\eta),
  \\
& \hatr_j(v,y) = r_j(v) + O(\eta),
 \quad
 \hatl_i(v,y) \,B(v) \, \del_y \hatr_j(v,y) = O(\eta).
\endaligned
\ee
\end{lemma}

\begin{proof}
When $B$ has the form $B(v) = I + \eta \, T(v)$ where the matrix $T(v)$
does not depend on $\eta$, the desired estimates follow from classical results.
In the slightly more general case when $T= T(v, \eta)$ we may argue as follows.
First, note that the eigenvalues under consideration are given by
\be
\label{det}
\det \Big( \bigl(- y \, I + A(v) \bigr) - \mu \, B(v) \Big) = 0.
\ee
Clearly, \eqref{det} is a polynomial equation in $\mu$ whose coefficients
are polynomials in the $t_{ij}$'s. The leading coefficient $det \Bigl( I + \eta \, \bigl( t_{ij} \bigr) \Big)$
is different from zero. Since the roots of \eqref{det} are distinct for
$t_{ij} = 0$, then for $\eta$ sufficiently small the roots of \eqref{det}
(namely the $\mu_j$'s) are distinct and depend smoothly upon the $t_{ij}$'s with, in addition,
\be
\label{expan61}
\mu_j(v,y) = - y + \lam_j(v) + \eta \sum_i \lam_{ij}(v,y) \, t_{ij}(v,\eta) + \ldots,
\ee
where the coefficients $\lam_{ij}$ are smooth functions in $y$.

Since the corresponding left- and right-eigenvectors (which need not
be normalized at this stage) $\hatl_j$ and $\hatr_j$ are polynomials in
$a_{ij}$ and $\mu_j$, it follows that they are also smooth in the $a_{ij}$'s.
Substituting them in \eqref{special} we get
\be
\label{expan62}
\aligned
&\hatl_j(v,y) = l_j(v) + \eta \sum_i l_{ij}(v,y) \, t_{ij}(v,\eta) +
\ldots,\\
&\hatr_j(v,y) = r_j(v) + \eta \sum_i r_{ij}(v,y) \, t_{ij}(v,\eta) +
\ldots,
\endaligned
\ee
where $l_{ij}$ and $r_{ij}$ depend smoothly upon $y$. From \eqref{expan62}, it follows
that for $\eta$ sufficiently small the expansion holds for the normalized
vectors $\hatl_j(v,y)$ and $\hatr_j(v,y)$. Finally, the required estimates \eqref{approx} follow
from \eqref{expan61} and \eqref{expan62}. This completes the proof of \eqref{approx}.
\end{proof}

By the same technique as above we can also prove:

\begin{theorem}
Under the conditions that $|D_u B(u)| < \eta$ and the characteristic fields associated with $A(u)$
are genuinely nonlinear, the generalized Lax shock inequalities
\[
\hat\lambda_j(u(y+),y)\leq y \leq \hat\lambda_j(u(y-),y)
\]
are equivalent to the standard Lax shock inequalities,
\[
\lambda_j(u(y+))\leq y \leq \lambda_j(u(y-)),
\]
where  $\hatlam_j = \la \hatr_j, A \, \hatr_j \ra$.
\end{theorem}

\begin{proof} We assume that $\nabla \lambda_j \cdot r_j >0$.
The key observation is that $\nabla \hatlam_j \cdot \hatr_j$ is also positive
and therefore the same part of the Hugoniot curve is selected by
the standard and by the generalized Lax shock inequalities. This is so because
$\nabla \hatlam_j \cdot \hatr_j = \nabla \lam_j \cdot r_j + O(\eta)$ as we now check.

When $|D_u B(u)| < \eta$ we see that $D_v T(v,\eta)| \leq C$ and that the expansion \eqref{expan62} is valid
for the $v$-derivatives as well. This completes the proof.
\end{proof}

We end this section with a further discussion of systems of two equations, and provide some explicit calculations
which lead to sufficient conditions on the diffusion matrix $B$ allowing us to apply the techniques introduced
earlier in this paper.
Note first that \eqref{det} can be written as
\be
\label{detg}
\det \Big( \bigl( - y \, I + \diag\bigl( \lam_1(v), \lam_2(v) \bigr) - \mu
\, L(v) \, B(v) \, R(v) \Big) = 0,
\ee
where $L(u)$ and $R(u)$ are matrices of left- and right-eigenvectors
associated with $A(u)$.
Namely, $L(v):= \bigl( l_1(v), \ldots, l_N(v) \bigr)$
and $R(v) := \bigl( r_1(v), \ldots, r_N(v) \bigr)$.
Setting $L(v) \, B (v) \, R(v)  = \bigl(b_{ij}(v) \bigr)$,
\eqref{detg} becomes
$$
\bigl( \det B(v) \bigr) \, \mu^2 -  \Big( b_{11}(v) \, (\lam_2 - y)
+ b_{22}(v) \, (\lam_1 - y) \Big) \, \mu
+ (\lam_1 - y) \, (\lam_2 - y) = 0.
$$
Solving this quadratic equation in $\mu$ and using the notation
$$
\beta(v) := {b_{12}(v) \, b_{21}(v) \over b_{11}(v) \, b_{22}(v)},
\quad
a_i(v,y) := {\lam_i(v) - y \over b_{ii}(v)},
$$
we arrive at
$$
\aligned
\mu_1(v,y)
& = {\bigl( a_2(v,y) + a_1(v,y) \bigr) - \bigl( a_2(v,y) - a_1(v,y) \bigr) \,
\Big( 1 + 4 \, \beta(v) \, {a_2(v,y) \, a_1(v,y) \over a_2(v,y) - a_1(v,y)  } \Big)^{1/2}
\over 2 \, (1 - \beta(v)},
\\
\mu_2(v,y)
& = {\bigl( a_2(v,y) + a_1(v,y) \bigr) + \bigl( a_2(v,y) - a_1(v,y) \bigr) \,
\Big( 1 + 4 \, \beta(v) \, {a_2(v,y) \, a_1(v,y) \over a_2(v,y) - a_1(v,y)
}\Big)^{1/2}
\over 2 \, (1 - \beta(v)}.
\endaligned
$$
It is easy to see that $\mu_1(v)$ and $\mu_2(v)$ are real if
$0 < \beta(v) < 1$, which is guaranteed if
\be
\label{ineq}
\aligned
& b_{11}(v) > 0, \quad b_{22}(v) > 0,
\\
& b_{12} (v) \, b_{21} (v) > 0.
\endaligned
\ee
The first condition above is also a necessary condition for viscous shocks
to be strictly stable in the sense of Majda and Pego \cite{MajdaPego}.

Assuming \eqref{ineq}, an easy calculation shows that the eigenvalues
$\mu_i(v)$ are separated:
$$
\mu_2(v) - \mu_1(v) \geq C > 0.
$$
It is easy to compute the corresponding right- and left-eigenvectors, namely
$$
\aligned
& \hatr_i(v,y) = \Big( - b_{12}(v) \, \mu_i(v,y), \lam_1(v) -y - b_{11}(v)
\, \mu_i(v,y) \Big)^t,
\\
& \hatl_i(v,y) = \Big( - b_{21}(v) \, \mu_i(v,y), \lam_1(v) -y - b_{11}(v)
\, \mu_i(v,y) \Big).
\endaligned
$$
Finally, it is easily seen that the estimates \eqref{approx} hold true when $|B(u) - I| = O(\eta)$.

\section*{Acknowledgments}

KTJ and PGL were supported by a grant (Number 2601-2) from the Indo-French Centre
for the Promotion of Advanced Research, IFCPAR (Centre Franco-Indien pour la Promotion de
la Recherche Avanc\'ee, CEFIPRA).
PGL was also supported by Centre National de la Recherche Scientifique (CNRS) and
by the A.N.R. grant 06-2-134423 entitled ``Mathematical Methods in General Relativity'' (MATH-GR).

 \newcommand{\textsc}{\textsc}

redpajama_set_name: RedPajamaArXiv


A couple of weeks ago, I taught our Pre-K through 2nd grade homeschool children at our monthly gathering. I first read the book, The 3 Trees, which is a beautiful story and I highly recommend it during the Lenten or Advent season.
For the craft idea, I went to Lacy's blog at Catholic Icing for ideas and inspiration and as usual I was not disappointed with the wealth of resources she shares. So using the ideas in her post here and the link she had to here, I came up with my own version of the Stations of the Cross bags for the children to make.
I could not find purple (the liturgical color during Lent) bags so I bought small white bags and tied a purple ribbon around the bags as pictured above. On the outside of the bag the children cut out the image of Jesus as the Divine Mercy and glued it to the bag. It's difficult to see from the picture above, but on the bottom of every Divine Mercy image there is the phrase, "Jesus I trust in You," which were the words that Jesus gave Sister Maria Faustina along with how the image was to look. Divine Mercy Sunday is celebrated the first Sunday after Easter Sunday. On the back of the bags the children glued information about the Divine Mercy image that I compiled from information on a couple of websites.
Station 3: Jesus falls the first time. Symbolize this with a band-aid with a "1"
Station 4: Jesus meets his mother. For this, each child had a holy card of Mary holding the rosary that I created by finding the image on the computer, copying and pasting into a word document, printed, then laminated.
Station 6: Veronica wipes Jesus' face. Just take a scrap of white fabric, but I didn't draw Jesus' face on it (there is only so much time in a day! :) ) and said the children could do that at home.
Station 7: Jesus falls the second time. A band-aid with a "2".
Station 8: Jesus meets the women of Jerusalem. A tissue to dry their tears.
Station 9: Jesus falls the third time. Draw a "3" on a band-aid.
Station 11: Jesus is nailed to the cross. 3 nails.
Station 12: Jesus dies on the cross. Symbolize this with another holy card of Jesus on the cross. I found a crucifix image on the computer that I liked, copied and pasted into a word document, printed, then laminated.
Station 13: Jesus is taken down from the cross. A picture of Michelangelo's Pieta that I copied and pasted into a word document, printed, then laminated.
Station 14: Jesus is laid in the tomb. We symbolized this with a rock- for sealing the tomb.
After the children assembled their stations of the cross bags, we had a snack of pretzels and I read them the story, Walter the Baker by Eric Carle, another cute and highly recommended book. At the end of the story, it gives a little history about pretzels.
It says, "The word pretzel comes from the Latin word bracchium, meaning "arm". The pretzel was originally a simple bread eaten during Lent. Its shape is based on an ancient position for prayer in which the arms were folded across the chest and the hands were placed on opposite shoulders."
At the beginning of Lent, we made some pretzels following this simple recipe from Holy Heroes Lenten Adventure.
You can read Lacy's Pretzels for Lent post for further ideas and links. (Can you tell I think Lacy's blog is awesome?!!? :) ) By the way, I link up to other blogs, like Lacy's, to share so many fantastic ideas with my visitors and to give credit where credit is due, but I also link up so I can refer to these links in the future from my own blog here.
This is a great craft. I love it! It is really a great way for the the kids to be able to actually put their hands on the symbols of The Stations, rather than just looking at pictures. I am going to have to try this craft this week. I am also going to head over to Dollar Tree to see what I can find. It looks like they have a great supply. Thank you so much for sharing and have a great day!
What a great idea! I love them...such a great learning tool! Thanks for sharing.!
Thanks for posting your Stations of the Cross bags...and with all the links! I love the Nippert Artworks clipart, but had lost the link!
This is a fantastic activity. Really brings the message home for children (and even for me). Not sure I can gather everything up this week since it is already late Tuesday but I would love to do this with my own children. We were talking about Palm Sunday at breakfast(on Palm Sunday)and explaining the meaning to the kids. Super idea.
I always enjoy looking at all of Lacy's great ideas and links and I was glad to find this one. It was a bit time consuming to gather the materials and copy, laminate and cut the ones that used images, but the kids really did enjoy it. Hope you find the stickers at your Dollar Tree! They are great at a great price! I hope you are having a blessed Holy Week!
Thanks for stopping in and commenting! I know I learned a lot from preparing the craft for the children :) Hope you are having a blessed week!
Glad you enjoyed this post and useful links! Thanks for stopping in and hope you are having a blessed Holy Week!
Thanks Anna-Marie for stopping by and commenting! I hope you get a chance to make these with your children someday!


McCutcheon played for the Los Angeles Rams from 1972 to '80 and was selected to the Pro Bowl five times (1973-'77). He spent the final two years of his career with the Denver Broncos, Seattle Seahawks and Buffalo Bills. McCutcheon rushed for over 1,000 yards four time (1973, '74, '76 & '77) and finished his career with 6,578 yards. He was a member of the Rams N.F.C. Championship team in 1980.


text:
Sewell Barn Theatre
A local theatre group in Norwich. The auditorium features raked seating on three sides of an open acting space. This unusual staging helps to draw the audience deeply into the performance. Please note: these reviews come from a wide variety of audience members, including active members of the Sewell Barn community. Where we have received reviews from committee members this is stated against their reviews.
Three Russian Encounters
What happens to fictional characters when the stories they are in come to an end? Might those from one play mix with those from another, resuming their lives through a different author? Themes of illusion and theatrical conceit abound in Three Russian Encounters, providing a novel combination of three one-act plays ‒ by Russia's most famous playwright, Anton Chekhov, and the much-acclaimed modern Irish dramatist, Brian Friel.
In The Yalta Game, adapted by Friel from Chekhov's story, The Lady with the Lapdog, a couple meet, begin an affair and, as they play the game of imagining the lives of those around them, they invent their own story.
Afterplay brings together two of the characters from Chekhov's Uncle Vanya and Three Sisters. Twenty years on, they share their stories, their 'fictions' and seek the 'fortitude' to keep going.
Swansong finds an ageing actor alone on the stage of a country theatre. His cries wake the Prompter with whom he reflects on past triumphs and his doubtful future.
Photography: Sean Owen
"This is theatre at its most powerful. The performances are finely drawn and very involving, and the atmosphere, and the lighting, I could go on! I found it haunting, moving and beautifully staged. And the final piece gave me goosebumps. All in all, a haunting dream of a production which lingers in the mind long afterwards, and with good reason. Bravo!" Barnaby M
"These three short plays are triumphs of writing and are showcased by some seriously strong acting talent. It is an inspired selection of pieces by director Tony Fullwood, and a gem of an evening." James Goffin [click here to read full review]
"That was utterly compelling…. I am moved to silence having been so deeply affected… every performance was totally captivating… please convey to all involved, from two very appreciative audience members… just wow! ... Often folk feel they have to be 'nice', commend the effort….but that was truly something of another order." Charlotte C
"Thank you for a wonderful evening of theatre yesterday. Great performances and excellent choice of play. Probably most thought provoking art in Norwich in November. Excellent work." James F
"Went to see Three Russian Encounters at the Sewell Barn. Brilliant evening. Well done everyone!" Etta G
"I guarantee that it will send you home satisfied yet thoughtful, inspired by three well written pieces. "Julian Swainson, Norwich Eye [click here to read full review]
"Bravo!!" Cherry C
"Yet another great performance at the Barn!" Shelagh S
"A fascinating evening at the theatre, with an artfully chosen trilogy of plays which chimed off each other in the most interesting of ways. Beautifully handled, of course, with very strong performers." Jonathan
"Excellent show tonight - congratulations to all involved!" Trevor B
"I thought the production had an admirable clarity and a deceptive simplicity. I like Afterplay very much (I'm currently ploughing through essays on Three Sisters!). The actors were all thinking and listening to each other and there was a lot of good, but not fussy or cluttered, detailed work onstage. So well done all!" Tony (UEA Drama lecturer)
"We really enjoyed it! Well done to all involved." Tasha P
"We really enjoyed the plays the other evening. To the extent that Mike is now rediscovering a rather old copy of Chekhov from our bookshelf! Such a good production, and the actors were impressive. Thank you." Helen
"A most enjoyable Sat matinee of Three Encounters! Well cast with some beautiful introspective moments and good projection!" Robin S
"Absolutely brilliant performances by Steven Scase, Poppy Hiscox, Jen Alexander, Dawn Brindle, David White and Kevin Oelrichs. Only two performances left so I urge you to go and show them your support." Laura L
"My friend and I, as part of our weekend in Norwich, saw [Three Russian Encounters] at the Sewell Barn on Saturday 4th December, and were delighted with the quality of the acting." Roy T
"[We] attended the performance of 'Russian Encounters' yesterday evening and we wanted to thank you and everyone at the theatre for an immensely enjoyable evening. All three plays were extremely well performed and thought provoking. It never fails to amaze us at the quality of the performances at the theatre. I know it's a small theatre in the scheme of things but delivers Big performances every time, without fail. It just proves the calibre of actors and backroom staff. Please extend our grateful thanks to all involved. The talented cast members of all three plays delivered an exceedingly entertaining evening which we both appreciated enormously. Bravo." Linda & James
Labels: 2021, russians
Lost idealism and parental responsibility are the themes of Mike Bartlett's sparkling satirical play. Kenneth and Sandra meet, get married, have children, divorce and settle into retirement. We follow their trajectory in three acts, that take place in 1967, 1990 and 2011 – each capturing a specific moment in family life and British politics.
Written ten years ago, Bartlett's play eviscerates the still-mostly-with-us '60s generation with a painfully funny savagery. The pertinence of his ferociously sharp, and sometimes simply ferocious, observations about the baby boomers and their disaffected Generation X children has not changed in the intervening decade.
Photography: Barry Parsons
"Fabulous energy, excellent performances, powerful subject, thought provoking AND funny - great show, thank you!" Harriet W
"I enjoyed the play, interesting and relevant... I thought that the Gen X woman [Emma Kirkham as Rose] was exceptionally good... Phil's sets also always amaze me with their inventiveness." Chris B
"All four of us thought 'Love, Love, Love' was absolutely brilliant last night – it filled the conversation all the way back to North Walsham and beyond! PS: Also meant to say how brilliant the Staging was – the extra applause was well deserved. We are wondering whether have interchangeable wall panels for our lounge so we can change it each week!" Roy & Eva T
"If you get a chance to see this play I unhesitatingly recommend that you do. Well written, brilliantly acted and staged and proof that the Sewell Barn Theatre company are back with a winner." Julian Swainson, Norwich Eye [click here for full review]
"Congratulations Clare Williamson and cast/crew of LOVE, LOVE, LOVE (Sewell Barn Theatre). Provocative, funny, infuriating...expert portrayal of so much that needs addressing in society... left me wanting to see what these characters would be doing in 2021! Looking forward to seeing it again next week - get a ticket if you can!" Jen A
"Brilliant play, so well done... really good, thoroughly enjoyed it." Vicky F
"...plenty of funny dialogue and entertaining action, and the Sewell Barn cast make the most of it..." James Goffin, Cultivated [click here for full review]
"Just wanted to say how much Nick and I enjoyed (if that's the right word ...) last night's performance. Entertaining, witty stuff - whilst also absolutely brutal. Jonathan, I found your character both truly compelling and utterly despicable ... well done! It's a 20 minute walk home for us - and we talked about it all the way." Shelagh M [Secretary to the Sewell Barn Trustees]
"...just wanted to say how much we enjoyed the performance Thursday evening. Please extend our congratulations to all involved." Linda O
"A very entertaining evening with some uncomfortable truths! Well worth a watch." Barnaby M
"[Message to Emma, who played Rose]: I just wanted to say that I thought your performance last night was outstanding!!! You crossed the age barriers effortlessly and the passion in the final act was amazing!!! Well done you!!! My friend and I enjoyed the show immensely and we thought your brother was excellent too!! Well done to the entire cast, it is a great show!!" Wendy A
"We loved the production of this show even if we did go home somewhat emotionally bruised. Well done everyone, very well acted, brilliantly staged - great sets as always. A pleasure to be back at the Barn." Shelagh & Alan S
Amanda & Elyot are divorced, and each has married again (to Victor and Sibyl respectively). They find themselves, on their respective honeymoons, in adjacent rooms at the same hotel. The reasons for their original attraction and for their parting rapidly become very clear, and equally clearly haven't changed.
We are tremendously excited to present Coward's classic, brittle comedy as our first full-length show since February 2020.
Photography: Sean Owen of Reflective Arts
"Congratulations to Cassie, cast and crew for a wonderful, joyous and laughter-filled evening! Inspired casting and the actors' palpable chemistry brought Coward's sparkling wit to life, whilst the set and costumes reflected the period perfectly. All together a marvellous tonic for theatre deprived audiences. Thank you so much." Jan F (review written after seeing a dress rehearsal)
"What a great show you put on last night! The acting was first class, set was brilliant and costumes were perfect ... Piano too of course was first rate! You must have been so pleased. It was remarkable and left a big impression. An amazing tour de force in such a brief and unpredictable time!" Shelagh S
"It's a marvellous party to watch. You should try getting on the guest list." James Goffin, Norwich Evening News (click here for full review)
"Well acted and incredibly funny. Well done to cast, crew and all volunteers at Sewell Barn." Kat J
"It was indeed a marvellous party ... if you haven't booked tickets already, you should! It's just so marvellous seeing live theatre again! Well done all!" Emma S
"Great performance tonight, excellent pace and lots of laughs. Congratulations to everyone!" Tim S
"Fantastic evening of laughs with Private Lives last night! Thank you cast (and crew - great scene changing) for a slick and excellent version of Coward's play." Frances M
"What a lovely evening actually AT THE THEATRE! First time since March 2020. Wonderfully fun evening watching Private Lives. On until Saturday so grab your tickets you lovely Norwich lot!" Emma K
"Thoroughly enjoyable. So well done. I just loved Elyot!!" Vicky F
"A highly enjoyable evening of wit and charm! Fabulous performances and much hilarity! Just what we need at the moment!" Barnaby M
"Saw Coward's Private Lives at the Sewell Barn last night and I think we can safely say what a sparkling comedy it was, and we enjoyed it immensely. I am always surprised how well Coward survives into the 21st century, although the casual violence does jar, I must admit.
The cast were all excellent. Charming, sophisticated and funny. Harriet Waterhouse was naturally lovely and feisty, and I thought at one point was going to subject Nick Meir to fifty lashes of her commodious and luscious tresses! Nick Meir was very believable as the rather cold and arrogant Eliot and thankfully avoided any mannered Cowardisms. Verity Thomson coped very well with the rather straitlaced, buttoned-up Sybyl and I was very impressed with Lee Johnson, a fine slightly preposterous comic performance as Victor. It was a shame we had to wait a couple of acts for Christina Fernandes'.s very funny, very French Louise, but she was worth waiting for.
Excellent direction from Cassie Tillett, pacey and frothy, and altogether a lovely enjoyable evening of fine entertainment. Well done everyone." David W
Put Your Shorts On!
During the summer of 2020, the Artistic Directors of the Sewell Barn Theatre were contacted by members of a Norfolk creative writing course, and asked to provide a team of performers to bring their scripts to life on Zoom calls. This was such a success that we decided to run three more sessions, calling for the support of local writers and performers. We selected ten of those pieces, intending to present them in late 2020. Another lockdown intervened… and finally, in June 2021, we are able to perform them LIVE on the Sewell Barn stage between 9-12 June, under the title Put Your Shorts On!
To ensure the safety of our audience, the capacity of the theatre (normally 100) has been reduced to 25; each half, consisting of five short items, will be performed by a team of four actors plus director and stage manager, using separate dressing rooms; and all other necessary measures will be taken to ensure the safety of audiences and cast.
The pieces in Put Your Shorts On! are written for one, two or three performers. Their style varies from the domestic to the surreal, and from comedy to sadness; the settings include parks, cafes, supermarkets and living rooms; the subject matter covers relationships, dogs, shopping and estrangement; and the characters' ages range from 20s to 70s. The authors are local, many of whom are seeing their work performed for the very first time, and we are hugely grateful that all have agreed to waive royalties, to help make live performances possible at the Sewell Barn in such difficult times.
We limited the audiences to 25 per show, keeping it cautious for COVID safety, and were delighted that - including a few additional tickets that we were able to release due to the sizes of some parties - we entertained 125 people in the course of the run.
"Really enjoyed this last night. Well done everyone, and so lovely to be back in the Sewell Barn." Ruth H
"We did enjoy it and good to see some new acting talent as well as the excellent stalwarts. Also liked your music interludes tying the production together. We liked the way you did the photo projection, nice and simple but effective." Shelagh S
"Congratulations to the Sewell Barn and their determination to bring live theatre back to Norwich. At their current performance of Put Your Shorts On, great attention was paid to Covid regulations and carried out with care and attention to detail, but the welcome was so friendly and warm. Congratulations also to the actors and backstage support, for giving us an entertainment, very much based on our local literary talent and so professionally delivered. A thoroughly satisfying evening for all who attended. May it be a signal for the return of amateur drama to Norwich." June G
"A big 'Thank You' to everyone for the brilliant performances last night. It is so good to see you all again, and to be part of a 'real' audience." Roy & Eva T
"Very many thanks for providing a safe and stimulating theatre experience with the short plays. As a playwright myself I am greatly encouraged that a local theatre is prepared to give a platform to new writing. Please pass on our thanks to all involved." Tony V
We were especially pleased to welcome most of the authors to the show during the run, and to receive some lovely feedback:
"I just want to thank everyone concerned with Put Your Shorts On yesterday for such a wonderful theatrical experience. We received an incredibly warm welcome from everyone on our arrival, you're such a friendly group of people! The show itself was absolutely stunning, the acting was incredible. Each piece was so powerful and by the end of the afternoon I had used all my emotions, I was drained. As the writer of Don't Forget The Freddos Dad, I'd like to thank Cassie and Kevin for bringing my words to life. It could not have been performed any better. I have to say that it was one of the proudest moments of my life!" Tim Strugnell: Don't forget the Freddos, Grandad
"Terrific performances from everyone tonight. Huge congratulations to the entire Sewell Barn team for making this happen. This is always a fantastic venue to visit and we enjoyed a splendidly entertaining evening under your roof." Chris Sadler: I cried the first time I saw you
"It was such a thrill to see it performed. The actors played the parts so well, with the emotion and strength that I had intended in the dialogue. It was a play about domestic abuse, the kind that is played out so much behind closed doors without anyone realising. I think the actors really brought the message home. It was quite overwhelming to see it come to life." Julia Morris: Out of the blue
Stella Jay
Stella was, as many have said, the 'face of the Sewell Barn' right from our earliest days. Robert Little, Chair of Trustees and previous Artistic Director, shares his memories.
"I have just heard the news that Stella Jay has passed away. Our thoughts go out her family during these difficult times.
It seems that Stella was part of the foundations of the Sewell Barn. So how did it all start? Well, her son Peter was a student at the Blyth Jex School (as it was then). The headmistress at that time, Valerie Glauert, was mustering local support to help fundraise for a community Theatre. The Sewell Barn was born.
Stella, whose son was interested in drama, went along to offer her support. Peter would perform in the early days of the company; Stella came to help and loved being involved with all those friendly people starting a new, exciting chapter in the Norwich theatre scene. Quite simply, she never really left. She was neither an actor nor a stage manager; but she was involved in more productions than anyone else. Although never officially a member of the acting company, she certainly belonged to it, by default. Stella was to head the Front of House Team for over 25 years before handing the title to her friend, Jane Page. She retired twice, although she kept coming back. She would restock the bar from the cash-and-carry, sort out the cleaning rotas, and sometimes take the rubbish home to her own bin.
She was a great persuader, managing to bring on board people to fulfil various roles within the Barn family. She served on the management committee for over twenty years, and then went on to become a Trustee. In her front of house capacity she would include most of her family - husband, children and grandchildren - although I suspect that was the best place to be able to see and speak to her. She also, with her husband Wilf, managed the distribution of our newsletter for many years. Her service to the company and the Sewell Barn in general was never a task or a problem: she loved to be part of it all. She was always willing to help where needed.
A friend of mine once said of Stella. "She is the face of the Sewell Barn, the first face to greet you when you arrive to see a show; always with a big smile, you know that you are in place that cares."
Stella will be greatly missed by so many, but she will always be with us in spirit."
The photograph above was taken by Andrew Evans as part of a set of images of all Trustees and Committee in 2004. We brought Stella on stage – not in reality, but virtually – in my July 2019 production of Noel Coward's Waiting in the Wings. One character, Martha, was spoken of but never appeared. I needed, however, an image of 'Martha' as part of the display board of all the residents in their days on stage, and Stella's family were happy to oblige with the lovely photograph below. We were delighted when they brought Stella to see her friends, and to see her own image as part of our front-of-house display, between the matinee and the evening performance of the show. She was a kind, gentle, hardworking and loyal member of our theatre family. RIP, dear Stella.
Cassie Tillett
Since publishing the above tribute in our newsletter, we have received many lovely comments.
"I've been attending Sewell Barn for 35 years... she was always there.. lovely lady. RIP" Ruth M
"I was saddened to read of the death of Stella Jay. She was one of my generation of Sewell Barn people! Though she did a lot more than I did; she was always there! We shared the highs, and occasional lows, but most of all the fun and laughs of many shows together. I echo Robert - RIP Stella and thank you." Frances M
"So sorry to see Stella Jay has passed away. She truly was the Face of Sewell Barn." Cherry C
"Really sad to hear Stella passed away. My love and thoughts go with her." James B
"A lovely tribute to Stella, a character the like of which we won't see again!" June G
... and many, many tributes on the posting on our Facebook page can be seen by clicking here (you will be able to see it whether or not you have a Facebook account).
at November 16, 2020 2 comments:
The Female of the Species
Acclaimed author Margot Mason has writer's block and is struggling to meet a deadline. She is interrupted by the arrival of an unexpected guest, followed by several more uninvited visitors.
This wickedly satirical comedy, verging on farce, was inspired by a true-life incident involving Germaine Greer and a student. It is full of drama, bad language, bad behaviour and some rather deliciously ridiculous people.
Photography: Andrew Evans
Full of characters that are as colourful as the language and the set, this fabulous comedy is a great way to celebrate our 300th production at the Sewell Barn Theatre (during our 40th season).
"The Female of the Species - absolutely superb! Firstly, I'd like the chaise lounge please... Beautiful, bright bold set, fantastic strong cast, a serious yet hilarious play - a must see! Thoroughly enjoyed it from start to finish." Gwen H-C
"'The Female of the Species' is a brilliantly incisive piece of comedy theatre, penetratingly exposing the bulls**t of self-righteous glorification." Vincent G
"Brilliant production! Well done all!!" Robin W
"I enjoyed the play, well written and acted, great set too." Chris B
"We travelled 230 miles to see 'The Female of the Species' at 'Sewell Barn Theatre' last weekend. We were certainly not disappointed! We were immediately impressed and also entertained by the colourful, interesting and clever set. The cast were all excellent with some extremely witty and convincing performances and we thoroughly enjoyed the evening. Well done to everyone who took part. If you haven't seen it ..... get your tickets.... you won't be disappointed!" Jan H
"Another brilliant show with excellent performances all round! Funny too! Well done." Shelagh S
"An entertaining 90 minutes of action... Michelle Moore steals the show with a fine performance as the writer's unloved offspring - now a tired mother-of-three and at the end of her tether. Her tantrum at the insanity of parenting is brilliant, falling to the floor something like a defeated toddler herself. The design team of Pete Sessions, Jo Parker-Sessions, and Phil Williamson have had fun with the set and props, with great fake book covers and well-executed gun shot effects that build up a visually delightful production." James Goffin, EDP [click here for full review]
"I'm just home from the Sewell Barn theatre where I've been extremely well entertained by the fabulous cast of The Female of the Species! Having had nothing to do with this production, apart from reading it a year ago when Mel Sessions (the director) first suggested it, it's been a joy to discover what a great show it's turned out to be. Clever, serious, witty and with a bit of silliness thrown in, it makes you think and laugh in equal measures - the best sort of drama in my opinion. It's been beautifully directed, the cast are fantastic and the set showcases interior design at its showbizzy best - I want that chaise longue now!" Clare Williamson [Joint Artistic Director, Sewell Barn]
Labels: 2020, Female of the Species
"I want to save the children, not destroy them..."
Based on the greatest of Victorian ghost stories – Henry James' The Turn of the Screw – this is a play about a haunted mansion and a governess's attempts to save the two orphaned children who inhabit it. Are the ghosts real or a figment of the imagination? Of those two possibilities, which is the most terrifying?
The show plays 9-11 and 15-18 January at 7.30pm, with a matinee on the final Saturday at 2.30pm. Book your tickets at St Georges Music Shop on 01603 626414, or online at www.ticketsource.co.uk/sewellbarn.
"I saw the performance last night and was properly spooked! I know the original Henry James story but had not seen this version before. I thought it was a splendid play.The whole production was superb - set great as always and the quality of acting sustained and professional by all. I loved how the tension was built up so quickly and then increased! My congratulations to everyone." Jill C
"I went to see The Innocents last night. There are already many great reviews for this fabulous production, the cast were superb, but Megan Artherton, that was a brilliant performance. If a Victorian ghost story is your cup of tea, then go and see this." David R
"I thought The Innocents was absolutely brilliant - please pass on my congratulations to director, cast and crew! The build-up of menace and suspense was superb, and I had the feeling that everything came together in that performance - the timing seemed to be spot-on, and I take my hat off to all four members of the speaking cast because the delivery was seemingly faultless. Wow, what a triumph!" Sue N
"Last night I had the pleasure of one last trip to the Sewell Barn Theatre (with some of my favourite people) to see their current production: The Innocents. I am so glad I did. It was a truly excellent evening of theatre. I don't want to give too much away as it really is worth going to see for yourself. The whole play was presented with such understanding of how to create an unsettling atmosphere that really draws you in and makes for an engrossing, unnerving and riveting production that you can't take your eyes off. A sparse but detailed set with some subtle changes in the second half to reflect the unravelling of Miss Giddens mental state. Fantastic use of lighting and sound that toys with the senses in the already atmospheric space that is the Barn. And then the actors themselves. The children were wonderfully creepy and the two ghosts genuinely menacing, ominous and not over played in the slightest. Gill Tichborne as the house keeper was a marvel but the final word must go to Megan Artherton as Miss Giddens. A masterful performance delivered with such humanity which had me transfixed throughout. I loved this show and suggest as many people go to see it as possible. I will certainly miss being part of this great Company in the future." Matt S
"Just been to see "The Innocents" @sewellbarn theatre. The perfect Victorian ghost story with a wonderfully engaging and often chilling cast. They don't hold back! More performances to come. See their website for details. I'll tell you more on Monday @BBCNorfolk. Sleep well." Stephen Bumfrey - BBC Radio Norfolk [via Twitter]
"Just been to see the chilling production of The Innocents at Sewell Barn. Megan Artherton - you expertly took me on a journey of emotion with your portrayal of this story, so natural was your performance that I felt every emotion with you. You lead the play with assurance and authority and so subtly and effectively built the suspense for the audience. I wanted to applaud after each of your scenes but couldn't because it would have ruined the atmosphere you had created!! I cannot begin to tell you how proud I am of you. There were so many complexities and layers that you brought to the character that even the most seasoned actresses would have struggled to have achieved. I was particularly impressed with the vulnerability you showed whilst also maintaining the characters natural reserve and restraint, something that is not easy to pull off. Congratulations and enjoy the rest of your run." Laura W
"This retelling of a classic Henry James tale is a menacing and suspenseful production, with more chills than the winter weather... The small cast is led by Megan Artherton as governess Miss Giddens. She puts in an exceptional performance, with her initial over-cheeriness giving way to mental and physical decline. Along with her the audience can never quite be sure whether the 'ghosts' are real, or part of her imagination; Artherton's tension and nervous exhaustion are palpable... This dark and provocative tale with an outstanding performance at its heart makes for a spooky winter treat." [Click here for full review.] James Goffin, EDP
"Congratulations to Richard Hand on a wonderful production of The Innocents at the Sewell Barn. He produces an atmosphere of approaching menace from the moment the audience arrives. Three excellent performances from the child, the maiden and the crone (in the pagan connotation of the word). An absolutely outstanding performance from Megan Artherton as the governess. We witnessed her mental deterioration palpably and painfully. Where does this actor come from? Superb! One little negative from me personally. I didn't like the Addams family portrayal of Quint. Otherwise, superb! Well, well done, everyone." David W
"Fantastic evening at @sewellbarn watching 'The Innocents' starring one of our very talented students, Freya! This brilliant production is not for the faint hearted!" Triple Threat Stage School
Labels: 2020, Innocents
A Month of Sundays
Beauty Queen
Blood and ice
Brimstone and Treacle
Confusions
cowardy custard
Faustus
Female of the Species
Handbagged
History Boys
Innocents
Kissing Sid James
Lord Arthur
The Burial at Thebes
The Go Between
The Thrill of Love
There Came a Gypsy Riding
These Shining Lives
Waiting in the Wings

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