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\markboth{\hfil Nonlinear stability of centered rarefaction waves
\hfil EJDE--2002/57}
{EJDE--2002/57\hfil Wei-Cheng Wang \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 57, pp. 1--20. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Nonlinear stability of centered rarefaction waves
  of the Jin-Xin relaxation model for $2 \times 2$ conservation laws
 %
\thanks{ {\em Mathematics Subject Classifications:} 65M12, 35L65.
\hfil\break\indent
{\em Key words:} Jin-Xin relaxation model, conservation laws, centered rarefaction wave.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 15, 2002. Published June 18, 2002. \hfil\break\indent
Supported by the National Science Committee:
 Project number NSC88-2119-M-007-003.} }
\date{}
%
\author{Wei-Cheng Wang}
\maketitle

\begin{abstract}
  We study the asymptotic equivalence of the Jin-Xin relaxation model
  and its formal limit for genuinely nonlinear $2\times 2$ conservation
  laws.  The initial data is allowed to have jump discontinuities
  corresponding to centered rarefaction waves, which includes
  Riemann data connected by rarefaction curves.
  We show that, as long as the initial data is a small perturbation
  of a constant state, the solution for the relaxation system exists
  globally in time and converges, in the zero relaxation limit,
  to the solution of the corresponding conservation law uniformly 
  except for an initial layer.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newcommand{\bnorm}{{-\!\!\!\!\|}}
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\section{Introduction}

In this paper, we study the asymptotic behavior of the Jin-Xin
model for a semilinear hyperbolic system with relaxation:
\begin{equation}\label{jx}
\begin{gathered}
     \mathbf{u}_t + \mathbf{v}_x = \mathbf{0}\\
     \mathbf{v}_t + \alpha^2 \mathbf{u}_x = {1\over\epsilon }( \mathbf{f}(\mathbf{u}) - \mathbf{v} )
\end{gathered}
\end{equation}
where $\mathbf{u},\mathbf{v},\mathbf{f} \in \mathbb{R}^2$, $x \in
\mathbb{R}$, $t >0$. In the formal limit as $\epsilon  \to 0$, we
expect the second equation of (\ref{jx}) to be well approximated
by the local equilibrium
\begin{equation}\label{Mx}
     \mathbf{v} = \mathbf{f}(\mathbf{u})
\end{equation}
and the relaxation system reduces to
\begin{equation}\label{clw}
     \mathbf{u}_t + \mathbf{f}(\mathbf{u})_x = \mathbf{0}.
\end{equation}
The relaxation limit for $2 \times 2$ nonlinear hyperbolic systems
was first studied by Liu in \cite{Liu}, where the stability
criterion and asymptotic limit were established for some basic
wave patterns such as diffusion waves, expansion waves and
traveling waves. In Chen, Levermore and Liu \cite{CLL}, further
mathematical theories were developed including the entropic
extension and asymptotic expansions.

Distinguished by the special structure of its nonlinear terms,
(\ref{jx}) was proposed by Jin and Xin in \cite{JX} with an
interesting numerical origin. It is used as an approximation for
general conservation laws and the authors developed a new class of
numerical methods for (\ref{clw}) called relaxation schemes based
on discretizing (\ref{jx}). Due to the simplicity of the
relaxation scheme and its outstanding performance, the hyperbolic
system (\ref{jx}) has stimulated much research activities on
rigorous justification of the asymptotic equivalence between
(\ref{jx}) and (\ref{clw}).

When $\mathbf{u}$ and $\mathbf{f}$ are scalars, 
(\ref{jx}) admits compact invariant
regions. With this a priori $L^\infty$ bound, Natalini \cite{Na1},
established the asymptotic equivalence between  the Cauchy
problems of (\ref{jx}) and (\ref{clw}) within the class of $BV$
data. Teng \cite{Teng} gave an optimal $L^1$ error estimate
between (\ref{jx}) and (\ref{clw}) using the matching method
introduced in \cite{GX}. An optimal pointwise estimate was derived
in Tadmor and Tang \cite{TT} based the optimal $L^1$ error
estimate and the $Lip^+$ analysis. For initial boundary value
problems, the stiff well-posedness for (\ref{jx}), its asymptotic
equivalence to (\ref{clw}) and the boundary layer structure was
obtained in Wang and Xin \cite{WX}.

For $2 \times 2$ conservation laws (\ref{clw}), a priori
$L^\infty$ bound for (\ref{jx}) is not available in general.
Tzavaras \cite{Tz}, Gosse and Tzavaras \cite{GT} established the strong
dissipation estimate for $\mathbf{u}$ with growth assumption on
$\mathbf{f}$. A typical example is the elastodynamics equation
\begin{equation}\label{ela}
    \partial_t u_1 + \partial_x u_2 =0, \quad
    \partial_t u_2 + \partial_x g(u_1) =0,
\end{equation}
with growth assumption on the stress-strain function $g$. The
convergence result then follows from the $L^p$ theory of
compensated compactness.

With a slightly different approach, Serre \cite{Se} showed that if
(\ref{clw}) has a convex characteristic set whose boundary is
stable under $\mathbf{f}'(\mathbf{u})$, and if the following
sub-characteristic condition
\begin{equation}\label{subch}
       \alpha > \rho( \mathbf{f}'(\mathbf{u}) )
\end{equation}
holds, one can find an invariant region for (\ref{jx}) and hence
establish convergence results using compensated compactness.
Examples of $2 \times 2$ conservation laws satisfying the
assumption above include the Temple system and the elastodynamics
equation (\ref{ela}) with $g$ satisfying $g'>0$ and $sg''(s) > 0$
for $s \neq 0$. It should be noted that although the result in
\cite{Tz, GT, Se} cover quite a wide range of equations, the
isentropic Euler equation, for example, is not included.

In this paper, we take an alternative approach to establish the
convergence result for general $2 \times 2$ conservation laws
equipped with a convex entropy. We consider the Cauchy problem
with initial data allowed to have a jump discontinuity
corresponding to a centered rarefaction wave.  We show that, as
long as the initial data is a small perturbation of a non-vacuum
constant state, the solution of (\ref{jx}) exists globally in time
and converges, as $\epsilon  \to 0$, to the solution of
(\ref{clw}) uniformly except for an initial layer. This is done by
approximating the solution of (\ref{clw}) with a smooth
rarefaction wave followed by a nonlinear stability analysis of the
smooth rarefaction wave under discontinuous initial perturbations
for (\ref{jx}). The a priori $\sup$ norm control then follows from
the Sobolev embedding. The result here can be easily extended to
the case of two weak centered rarefaction waves of different
families. The result in this paper was announced in \cite{Wang}
for the special case of the isentropic $p$ system.

Other related works include Luo \cite{Luo}, where the author
studied the stability of rarefaction wave in the scalar,
multidimensional setting of the Jin-Xin model; Luo and Xin
\cite{LX} showed nonlinear stability of the traveling wave
solution also in the scalar, multidimensional case. Several
discrete velocity kinetic models have also been proposed as
relaxation approximations for (\ref{clw}), see \cite{Bou, Na1, KT}. 
In the 1-D case, the 2 speed case of these models are
equivalent to the Jin-Xin model. However, they are genuinely
different in the multidimensional case. The relaxation
approximation was later generalized to the Hamilton-Jacobi
equation \cite{JXHJ} and to curvature dependent front propagation
\cite{JKX}.

The rest of the paper is organized as follows: In section 2, we
construct the smooth approximate solution and list some
preliminary estimates. We then state the main theorem (Theorem
\ref{ap}). In sections 3, we review the entropic extension
property for the Jin-Xin system, which is the foundation for our
energy estimate. In sections 4, we proceed to prove the main
theorem by treating the Riemann initial data as a discontinuous
perturbation of the smooth approximation. We then proceed by a
piecewise $H^1$ estimate on the error. To this end, we first study
how the initial jump propagates and decays along the
characteristics in (\ref{jx}). We then finish the proof by
piecewise energy estimate and the Sobolev inequality.

\section{Smooth approximations}

We first rewrite (\ref{jx}) in a simpler form as
\begin{equation}\label{eBwell}
 \begin{gathered}
     \mathbf{U}_t^\epsilon  + \mathbf{A} \mathbf{U}_x^\epsilon  = {1\over\epsilon } \mathbf{N}(\mathbf{U}^\epsilon )
   \quad t \geq 0, \quad  x\in \mathbb{R} \\
   \mathbf{U}^\epsilon (x,0) = \begin{cases}  \mathbf{U}_r & \mbox{if } x> 0 \\
       \mathbf{U}_l & \mbox{if }x< 0 \end{cases}
 \end{gathered}
\end{equation}
and consider also the scaled version of (\ref{eBwell}):
\begin{equation}\label{Bwell}
\begin{gathered}
     \mathbf{U}_t + \mathbf{A} \mathbf{U}_x = \mathbf{N}(\mathbf{U})\quad  t \geq 0, \quad x\in \mathbb{R} \\
   \mathbf{U}(x,0) = \begin{cases}  \mathbf{U}_r & \mbox{if } x> 0 \\
      \mathbf{U}_l & \mbox{if } x< 0, \end{cases}
\end{gathered}
\end{equation}
where
\begin{equation}\label{F}
\mathbf{U} =  \begin{pmatrix} \mathbf{u} \\ \mathbf{v}
\end{pmatrix}, \quad
\mathbf{N} (\mathbf{U}) =  \begin{pmatrix} \mathbf{0} \\
\mathbf{f}(\mathbf{u}) - \mathbf{v}
     \end{pmatrix},\quad
\mathbf{A} =  \begin{pmatrix}\mathbf{0}        &  \mathbf{I}_2 \\
                \alpha^2 \mathbf{I}_2 &  \mathbf {0}
       \end{pmatrix}
\end{equation}
It is clear that $\mathbf{U}(x,t)$ is a solution of (\ref{Bwell})
if and only if
\begin{equation}\label{scale}
\mathbf{U}^\epsilon (x,t) \stackrel{\rm def}{=}\mathbf{U}({x\over
\epsilon },{t\over \epsilon })
\end{equation}
is a solution of (\ref{eBwell}).

We further assume that the Riemann Cauchy data $(\mathbf{U}_l,
\mathbf{U}_r)$ for (\ref{Bwell}) are in local equilibrium
\begin{equation}
\mathbf{U}_l =\begin{pmatrix}\mathbf{u}_l \\
                \mathbf{f}(\mathbf{u}_l)
                \end{pmatrix},\quad
\mathbf{U}_r =\begin{pmatrix} \mathbf{u}_r \\
                \mathbf{f}(\mathbf{u}_r)
                \end{pmatrix},
\end{equation}
and that $\mathbf{u}_l$ and $\mathbf{u}_r$ are connected by a
rarefaction curve in the phase space. Thus the solution of
(\ref{clw}) is a self similar centered rarefaction wave:
\begin{equation}\label{cr}
\mathbf{u}(x,t) =\check{\mathbf{u}}(\frac{x}{t})
\end{equation}
(For an introduction on Cauchy problems with Riemann initial data
and rarefaction curves, see \cite{Sm}.) 

We want to study the time
asymptotic/small mean free path limit of
(\ref{Bwell})/(\ref{eBwell}). We will show that if
$|\mathbf{u}_l-\mathbf{u}_r|$ is small enough and $\alpha$
satisfies (\ref{subch}), then (\ref{Bwell}) has a unique global in
time solution and this solution is asymptotically equivalent to a
self similar function $\check{\mathbf{U}}$:
\begin{equation}\label{FFcr}
 \lim_{t\to \infty}\sup_{x\in R}\;|\mathbf{U}(x,t) - \check{\mathbf{U}}( {x\over t} ) | = 0,
\end{equation}
where $\check{\mathbf{U}}$ is obtained by imposing local
equilibrium (\ref{Mx}) together with (\ref{cr}):
\begin{equation}
\label{FR}
     \check{\mathbf{U}} =
\begin{pmatrix} \check{\mathbf{u}} \\
     \mathbf{f}(\check{\mathbf{u}} )
\end{pmatrix}
\end{equation}
In view of (\ref{scale}), we conclude that (\ref{eBwell}) has a
unique solution satisfying
\begin{equation}
  \lim_{\epsilon  \to 0}
  \sup_{x\in R,\;t\geq \epsilon^\delta}\;|\mathbf{U}^\epsilon(x,t)
  - \check{\mathbf{U}}( {x\over t} ) | =0
\end{equation}
for any $\delta <1$ (one can replace $\epsilon^\delta $ by any
$g(\epsilon)$ such that $\epsilon = o(g(\epsilon))$ as $\epsilon
\to 0$).

To prove (\ref{FFcr}), we will construct
$\widetilde{\mathbf{U}}(x,t)$, a smooth approximation of
$\check{\mathbf{U}}$ and show that both $\mathbf{U}$ and
$\check{\mathbf{U}}$ are asymptotically equivalent to
$\widetilde{\mathbf{U}}$ (Lemma \ref{Gdecay} (b) and (\ref{FG})
below).

We proceed by constructing $\widetilde{\mathbf{U}}$ as follows: Let
$\widetilde w(x,t)$ be the solution of the Cauchy problem
\begin{equation}\label{w}
\begin{gathered}
       \widetilde w_t + \widetilde w\widetilde w_x =0  \\
       \widetilde w(x,0) = \frac12 \{(w_r + w_l) +(w_r - w_l)\tanh x\}
\end{gathered}
\end{equation}

\begin{lemma}[\cite{MN}] \label{sdcay}
Suppose $w_r > w_l$, then (\ref{w}) has a unique global smooth
solution satisfying
\begin{itemize}
 \item[(a)] $w_l < \widetilde w(x,t) < w_r$, \; $\widetilde w_x(x,t)>0$
            for $t\geq 0$, $x\in R$.
 \item[(b)] For any $p\in [1,\infty]$, there is a positive
            constant $C$ such that for $t\geq 0$,
\begin{equation}\label{wx}
\begin{gathered}
 \|\widetilde w_x(t)\|_{L^p}
 \leq C\min(|w_r-w_l|, |w_r-w_l|(1+t)^{-1+{1\over p}}), \\
 \|\widetilde w_{xx}(t)\|_{L^p},\|\widetilde w_{xxx}(t)\|_{L^p}
       \leq C\min(|w_r-w_l|, (1+t)^{-1} ).
\end{gathered}
\end{equation}

\item[(c)]
${\lim_{t\to \infty}\sup_{x\in R}}\;
 |\widetilde w(x,t) -\check w({x\over t}) | =0$.
\end{itemize}
\end {lemma}

Since the rarefaction wave solution written in the Riemann
invariant coordinate reduces to a rarefaction wave for the
Burgers' equation up to a nonlinear change of variables, we can
thus construct the corresponding solutions of the Euler equation
by inverting this change of variables. The corresponding solution
of (\ref{clw}), $\widetilde{\mathbf{u}}$ satisfies
\begin{lemma} \label{turho}
For each $\mathbf{u}_l$ satisfying the sub-characteristic
condition (\ref{subch}), there exists a $\delta_0>0$ such that if
$\mathbf{u}_r$ can be connected to $\mathbf{u}_l$ by a centered
rarefaction wave $\check{\mathbf{u}}$ and $| \mathbf{u}_l
-\mathbf{u}_r | < \delta_0$, then the corresponding smooth
approximation $\widetilde{\mathbf{u}}$ satisfies the following:
\begin{itemize}
\item[(a)] $\widetilde{\mathbf{u}}$  is a smooth global solution of
\begin{equation}
 \widetilde{\mathbf{u}}_t - \mathbf{f}(\widetilde{\mathbf{u}})_x  = \mathbf{0}.
\end{equation}
 \item[(b)] For any $p\in [1,\infty]$, there is a positive
            constant $C$ such that for $t\geq 0$,
\begin{equation}
\begin{gathered}
      \|\widetilde{\mathbf{u}}_x(t), \widetilde{\mathbf{u}}_t(t)\|_{L^p}
\leq C\min(|\mathbf{u}_r-\mathbf{u}_l|, |\mathbf{u}_r
-\mathbf{u}_l|^{1/p}(1+t)^{-1+{1\over p}}),     \\
      \|\widetilde{\mathbf{u}}_{xx}(t),\widetilde{\mathbf{u}}_{tx}(t),
      \widetilde{\mathbf{u}}_{xxx}(t),\widetilde{\mathbf{u}}_{txx}(t)\|_{L^p}
       \leq C\min(|  \mathbf{u}_r - \mathbf{u}_l|, (1+t)^{-1}).
\end{gathered}
\end{equation}
 \item[(c)]
 ${\lim_{t\to \infty}\sup_{x\in R}}\;
   |\widetilde{\mathbf{u}}(x,t) - \check{\mathbf{u}}({x\over t}) | =0 $.
\end{itemize}
\end {lemma}

Let us denote the zeroth order approximation by
\begin{equation}\label{F0}
\mathbf{U}^{(0)} =  \begin{pmatrix}\mathbf{u}^{(0)} \\
                \mathbf{v}^{(0)}
                \end{pmatrix}, \quad
\mathbf{u}^{(0)} =  \widetilde{\mathbf{u}}, \quad \mathbf{v}^{(0)} =
\mathbf{f}(\mathbf{u}^{(0)})
\end{equation}
and following the Chapman-Enskog expansion, we get the first order
correction
\begin{equation}\label{F1}
\mathbf{U}^{(1)} =  \begin{pmatrix} \mathbf{0} \\
                \mathbf{v}^{(1)}
                \end{pmatrix},
\mathbf{v}^{(1)} = -( \alpha^2 - \mathbf{f}'( \mathbf{u}^{(0)} )^2
) \mathbf{u}^{(0)}_x.
\end{equation}
We then construct $\widetilde{\mathbf{U}}$, the smooth approximation
of $\widetilde{\mathbf{U}}$, by
\begin{equation}\label{defG}
      \widetilde{\mathbf{U}} = \mathbf{U}^{(0)} + \mathbf{U}^{(1)}
=  \begin{pmatrix} \widetilde{\mathbf{u}} \\
                \widetilde{\mathbf{v}}
                \end{pmatrix}
=  \begin{pmatrix}  \mathbf{u}^{(0)}  \\
                \mathbf{v}^{(0)} + \mathbf{v}^{(1)}
                \end{pmatrix}
\end{equation}
Thus $\widetilde{\mathbf{U}}$ satisfies the following equation
\begin{equation}\label{Gs}
 \widetilde{\mathbf{U}}_t + \mathbf{A} \widetilde{\mathbf{U}}_x
  -\mathbf{N}(\widetilde{\mathbf{U}}) = \mathbf{U}^{(1)}_t + \mathbf{A}
  \mathbf{U}^{(1)}_x,
 \quad
 \mathbf{N}(\widetilde{\mathbf{U}}) = \begin{pmatrix}
                      \mathbf{0} \\
                      - \mathbf{v}^{(1)}
                     \end{pmatrix}
\end{equation}
and we have the corresponding estimates for $\widetilde{\mathbf{U}}$.

\begin{lemma} \label{Gdecay}
Under the assumptions in Lemma \ref{turho}, the smooth
approximation $\widetilde{\mathbf{U}}$ satisfies (\ref{Gs}) and
$$ \label{tucu} 
{\lim_{t\to \infty}\sup_{x\in R}}\;
|\widetilde{\mathbf{U}}(x,t) - \check{\mathbf{U}}({x\over t}) | =0.
$$
\end {lemma}

To show the asymptotic equivalence of $\mathbf{U}$ and
$\widetilde{\mathbf{U}}$, we let $\mathbf{\mho} = \mathbf{U} -
\widetilde{\mathbf{U}}$. The equation satisfied by $\mathbf{\mho}$
reads
\begin{equation}\label{feq}
\begin{gathered}
\mathbf{\mho}_t + \mathbf{A} \mathbf{\mho}_x = \mathbf{N}(\mathbf{U})
-\mathbf{N}(\widetilde{\mathbf{U}}) -(\mathbf{U}^{(1)}_t
+ \mathbf{A} \mathbf{U}^{(1)}_x)      \\
     \mathbf{\mho}(x,0) =  \mathbf{U}(x,0) - \widetilde{\mathbf{U}}(x,0)
  \end{gathered}
\end{equation}
where
\begin{multline} \label{Ndiff}
\mathbf{N}(\mathbf{U})-\mathbf{N}(\widetilde{\mathbf{U}})\\
= \begin{pmatrix}
  \mathbf{0} \\
   \mathbf{f}(\mathbf{u})- \mathbf{v} + \mathbf{v}^{(1)}
\end{pmatrix}
= \begin{pmatrix}
  \mathbf{0} \\
   \mathbf{f}(\mathbf{u}) - \mathbf{f}(\widetilde{\mathbf{u}}) 
 - \mbox{\boldmath $\nu$}
\end{pmatrix}
= -\begin{pmatrix}
  \mathbf{0} \\
   \mbox{\boldmath $\nu$} - \bar{\mathbf{a}} \mbox{\boldmath $\mu$} 
+ O( | \mbox{\boldmath $\mu$} |^3 )
\end{pmatrix}
\end{multline}
and $\bar{\mathbf{a}} = \mathbf{f}'(\bar{\mathbf{u}}) =
\mathbf{f}'( \widetilde{\mathbf{u}} + {\mbox{\boldmath $\mu$}\over 2} )$.

At this point, we briefly summarize the notations for readers'
convenience: Lowercase bold letters denote 2-vectors or 2 by 2
matrices, like $\mathbf{u}$, $\mathbf{f}$, $\mathbf{a}$, etc.
Uppercase letters are 4-vectors or 4 by 4 matrices, like
$\mathbf{U}$ and $\mathbf{A}$. Tilded quantities are smooth
approximations such as $\widetilde w$ and $\widetilde{\mathbf{u}}$. Greek
letters $\mbox{\boldmath $\mu$} = \mathbf{u}-\widetilde{\mathbf{u}}$,
$\mbox{\boldmath $\nu$} = \mathbf{v}-\widetilde{\mathbf{v}}$ will denote
perturbations. The barred quantities like $\bar{\mathbf{u}} =
(\mathbf{u}+\widetilde{\mathbf{u}})/2$ denote the average of the
smooth approximation and the true solution.

(\ref{feq}), like (\ref{jx}), is a semilinear hyperbolic system.
The discontinuities propagate along $x=\alpha t$ and $x=-\alpha
t$. Thus we adopt the piecewise energy estimate. We introduce the
following notations: Denote by $\Omega_k$, $k =1,2,3$ the regions
separated by $ x=\alpha t$ and $x=-\alpha t$ in the upper half
plane $t>0$, $\Omega_k^s = \Omega_k \cap \{ t=s\}$ and for any
interval $I \subset R$, $H^1(I)$ the usual Sobolev space with norm
$\| \cdot \|_1 = ( \| \cdot \|_{L^2(I)} + \| {\partial \over
\partial x}\cdot \|_{L^2(I)} )^{1/2} $.

Now we define the appropriate function space on which we will be
working:
\begin{multline*}
X(0,T) =\big\{ \mathbf{W} : R \times [0,T] \mapsto R^4 : \\
 \mathbf{W}
\in C^0([0,T),H^1(\Omega_k^s)) \cap C^1([0,T),L^2(\Omega_k^s))
\cap C^2(\overline{\Omega}_k ), \; k=1,2,3 \big\}
\end{multline*}
For $ \mathbf{W}  \in X(0,T)$, we define
\begin{equation*}
 \begin{aligned}
 \| \mathbf{W}(\cdot,t)\bnorm^2
 =&  \bint | \mathbf{W}(x,t) |^2 dx                   \\
\stackrel{\rm def}{=} 
  &  {\int_{-\infty}^{-\alpha t}} | \mathbf{W}(x,t) |^2 dx
   + {\int_{-\alpha t}^{\alpha t}}   | \mathbf{W}(x,t) |^2 dx
   + {\int_{\alpha t}^\infty}     | \mathbf{W}(x,t) |^2 dx
 \end{aligned}
\end{equation*}
\begin{gather*}
 \|\mathbf{W}(\cdot,t)\bnorm^2_1
 = \| \mathbf{W}(\cdot,t) \bnorm^2 +\| \mathbf{W}_x(\cdot,t) \bnorm^2 \\
 [\mathbf{W}]^+(t)=  \mathbf{W}(\alpha t+0,t)-\mathbf{W}(\alpha t-0,t)  \\
 [\mathbf{W}]^-(t)= \mathbf{W}(-\alpha t+0,t)-\mathbf{W}(-\alpha t-0,t) \\
 \langle \mathbf{W} \rangle ^+(t)
    = \frac12 (\mathbf{W}(\alpha t+0,t)+\mathbf{W}(\alpha t-0,t))   \\
 \langle \mathbf{W} \rangle ^-(t)
    = \frac12 (\mathbf{W}(-\alpha t+0,t)+\mathbf{W}(-\alpha t-0,t)).
 \end{gather*}
We want to show that
\begin{equation}\label{FG}
\lim_{t\to \infty}\sup_{x\in R}\;|\mathbf{U}(x,t) -
\widetilde{\mathbf{U}}(x,t) | = 0.
\end{equation}
A key observation is the following: The a priori bound on
${\sup_{t>0}} \|\mathbf{\mho} \|_{L^\infty(t)}$ implies
exponential decay in time of the jumps (see Lemma \ref{jumps}
below). Thus in view of the Sobolev inequality
\begin{equation} \label{Sob}
   \|\mathbf{\mho} \|_{L^\infty}^2(t) \leq C \left( \|\mathbf{\mho} \|\|\mathbf{\mho}_x\bnorm(t)
         + [\mathbf{\mho} ]_+^2(t) + [\mathbf{\mho} ]_-^2(t) \right),
\end{equation}
our task remains to estimate $\| \mathbf{\mho}  \bnorm_1$ (Theorem
\ref{ap} below). During this process, terms involving line
integrals of jumps across the discontinuities appear naturally.
Therefor the exponential decay in time of the jumps implies the a
priori bound on ${ \sup_{t\geq 0} }\|\mathbf{\mho} \bnorm^2_1 (t)$
(Lemma \ref{h1} and on). We close this bootstrapping argument by
the local existence theorem (Theorem \ref{hyp} below) to extend
$\mathbf{\mho} $ in $X(0,T+\Delta t)$ and conclude that $T=\infty$
(global in time existence).

\begin{theorem}[A priori estimate] \label{ap}
There exists positive constants $\epsilon_1$ and $C_1$ such that
if $\mathbf{\mho} \in X(0,T)$ is the solution of (\ref{feq}) in
$0\leq t\leq T$ for some $T>0$ and
\begin{eqnarray*}
 { \sup_{0\leq t \leq T} } \|\mathbf{\mho} (t)\bnorm_1 + |\mathbf{u}_r -\mathbf{u}_l| < \epsilon_1,
\end{eqnarray*}
then
\begin{equation}
\begin{array}{cl}
  {\sup_{0\leq t \leq T} } \|\mathbf{\mho} (t)\bnorm_1^2 + \int_0^T
 \|\mathbf{\mho}_x(\tau)\bnorm^2
 \leq  C_1(\|\mathbf{\mho} (0) \bnorm_1^2 + |\mathbf{u}_r-\mathbf{u}_l|^{1/6}).
\end{array}
\end{equation}
\end{theorem}
The proof of Theorem \ref{ap} will be given in section \ref{ape}.

\begin{theorem}[Local existence] \label{hyp}
Let $T \geq 0$ and $g \in X(0,T)$ be a solution to (\ref{feq}) for
$0 \leq t \leq T$. Consider the initial value problem to
(\ref{feq}) with the initial data
\begin{equation}\label{init}
\mathbf{\mho} (T,x) = \mathbf{W}_T(x) \stackrel{\rm def}{=}
\mathbf{W}(T,x).
\end{equation}
Then for any $M>0$, there exists a positive constant $\Delta t$
depending only on $M$ and ${\sup_{x,t} |\widetilde{\mathbf{U}}(x,t)|
}$ such that if
\begin{eqnarray*}
 \|\mathbf{W}_T\|_{C^0(-\infty,- \alpha T)} +
 \|\mathbf{W}_T\|_{C^0(- \alpha T, \alpha T)} + \|\mathbf{W}_T\|_{C^0( \alpha T,\infty)} < M
\end{eqnarray*}
Then (\ref{feq}) together with (\ref{init}) has a unique solution
$\mathbf{\mho} \in X(T,T+\Delta t)$ satisfying
\begin{eqnarray*}
 \sup_{T\leq t\leq T+\Delta t}\left(  \|\mathbf{\mho} (t)\|_{C^0(-\infty,- \alpha t)}
+\|\mathbf{\mho} (t)\|_{C^0(- \alpha t, \alpha t)} +
\|\mathbf{\mho} (t)\|_{C^0( \alpha t,\infty)} \right) < 2 M.
\end{eqnarray*}
As a consequence, $\mathbf{W}$ can be extended to $X(0,T+\Delta
t)$.
\end{theorem}
The proof of Theorem \ref{hyp} is standard, see \cite{RR} for the
existence and uniqueness in the piecewise $C^0$ function class.
The piecewise $C^2$ regularity is a direct consequence of the
special structure of the nonlinearity (being the lower order term
in (\ref{feq})) and the $C^2$ regularity of the nonlinear
functional $\mathbf{N}(\cdot)$.

From Theorem \ref{ap} and Theorem \ref{hyp}, we have the following.

\begin{corollary}\label{th}
For each $\mathbf{u}_l$ satisfying the sub-characteristic
condition (\ref{subch}), there exists $\epsilon_0$ and $C_0$ such
that if $\mathbf{u}_r$ can be connected to $\mathbf{u}_l$ by a
centered rarefaction wave and $\|\mathbf{\mho} (0) \bnorm_1 +
|\mathbf{u}_r -\mathbf{u}_l| < \epsilon_0$, then (\ref{feq}) has a
unique solution $\mathbf{\mho} \in X(0,\infty)$ satisfying
\begin{equation}
  { \sup_{t\geq 0} } \|\mathbf{\mho} (t)\bnorm^2_1 + \int_0^\infty
 \|\mathbf{\mho}_x(\tau)\bnorm^2       d\tau
\leq
 C_0(\|\mathbf{\mho} (0) \bnorm_1^2 + |\mathbf{u}_r-\mathbf{u}_l|^{1/6}).
\end{equation}
\end{corollary}
With Corollary \ref{th}, Lemma \ref{jumps} below and the Sobolev
inequality (\ref{Sob}), it is easy to see that
\begin{equation}
\int_0^\infty \|\mathbf{\mho}_x\bnorm^2(\tau) + \big|{d\over
d\tau}\|\mathbf{\mho}_x\bnorm^2\big| (\tau) < \infty
\end{equation}
and consequently (\ref{FG}) holds.

\section{Entropic Extension}

The Jin-Xin model has some nice
mathematical properties. The entropic extension below follows from
the idea outlined in \cite{CLL}. The proof can also be found in
\cite{Se}. The derived entropy $\Phi(\mathbf{u},\mathbf{v})$ and
the functions $\mathbf{u}^\pm$ will be used in our main energy
estimate so we briefly summarize the proof here for readers'
convenience.

\begin{proposition}\label{ENTROPY}
Let $(\phi(\mathbf{u}), \psi(\mathbf{u}))$ be a pair of
entropy-flux functions of (\ref{clw}) with $\phi$ convex, then we
can derive the corresponding entropy-flux pair $(\Phi,\Psi)$ for
the system (\ref{jx}) from the following wave equation with Cauchy
data
\begin{equation}\label{waveq}
\begin{gathered}
       \Phi_{\mathbf{u}} = \Psi_{\mathbf{v}}          \\
       \Psi_{\mathbf{u}} = \alpha^2 \Phi_{\mathbf{v}}          \\
       \Phi(\mathbf{u},\mathbf{f}(\mathbf{u})) = \phi(\mathbf{u})   \\
       \Phi_{\mathbf{v}}(\mathbf{u},\mathbf{f}(\mathbf{u})) = \mathbf{0}.
\end{gathered}
\end{equation}
The solution $(\Phi, \Psi)$ exists in a neighborhood of the
equilibrium $\mathbf{v} = \mathbf{f}(\mathbf{u})$ and there it
satisfies
\begin{itemize}
\item[(a)] $\Phi$ is convex if $\alpha$ is sufficiently large.
\item[(b)] $\Psi(\mathbf{u},\mathbf{f}(\mathbf{u})) = \psi(\mathbf{u})$.
\item[(c)]
$\Phi_t(\mathbf{u},\mathbf{v}) + \Psi_x(\mathbf{u},\mathbf{v})
  = -{1\over\epsilon}\Phi_{\mathbf{v}}(\mathbf{u},\mathbf{v})\cdot(\mathbf{v} - \mathbf{f}(\mathbf{u})) \leq 0
$ for any smooth solution of (\ref{jx}).
\end{itemize}
\end{proposition}

\paragraph{Proof.} From first two equation of (\ref{waveq}), $\Phi$
satisfies the wave equation
$$
\Phi_{\mathbf{u}\mathbf{u}} = \alpha^2 \Phi_{\mathbf{v}\mathbf{v}}
$$
Therefore the general solution is given by
\begin{equation}\label{defPhi0}
\Phi(\mathbf{u},\mathbf{v}) = h_+( \mathbf{v} + \alpha\mathbf{u} )
+ h_-( \mathbf{v} -\alpha \mathbf{u}),
\end{equation}
The last two equation in (\ref{waveq}) gives:
\begin{equation}\label{hpmp}
\begin{gathered}
h_+'( \mathbf{f}(\mathbf{u}) + \alpha\mathbf{u} ) - h_-'(
\mathbf{f}(\mathbf{u}) - \alpha\mathbf{u} ) = { \phi'(\mathbf{u})
\over \alpha}  \\
h_+'( \mathbf{f}(\mathbf{u}) + \alpha\mathbf{u}
) + h_-'( \mathbf{f}(\mathbf{u}) - \alpha\mathbf{u} ) =  \mathbf{0},
\end{gathered}
\end{equation}
Thus
\begin{equation}\label{hpmp2}
h_\pm' ( \mathbf{f}(\mathbf{u}) \pm \alpha\mathbf{u} )  = \pm {
\phi'(\mathbf{u}) \over 2\alpha}
\end{equation}
and
\begin{equation}\label{hpmp3}
  {d \over d\mathbf{u}} h_\pm ( \mathbf{f}(\mathbf{u}) \pm \alpha\mathbf{u} )
= \pm {\phi'(\mathbf{u}) \over 2\alpha}( \mathbf{f}'(\mathbf{u})
\pm \alpha I) = \frac12 \big( \phi'(\mathbf{u}) \pm {
\psi'(\mathbf{u}) \over \alpha} \big)
\end{equation}
Thus
\begin{equation}\label{defPhi}
  h_\pm ( \mathbf{f}(\mathbf{u}) \pm \alpha\mathbf{u} )
= \frac12 \big( \phi(\mathbf{u}) \pm { \psi(\mathbf{u}) \over
\alpha} \big)
\end{equation}
and the solution to (\ref{waveq}) is given by
\begin{equation}\label{defPhi2}
\begin{gathered}
\Phi(\mathbf{u},\mathbf{v}) = h_+( \mathbf{v} + \alpha\mathbf{u} )
+ h_-( \mathbf{v} -\alpha \mathbf{u})             \\
\Psi(\mathbf{u},\mathbf{v}) = \alpha \left( h_+( \mathbf{v} +
\alpha\mathbf{u} ) - h_-( \mathbf{v} -\alpha \mathbf{u}) \right)
\end{gathered}
\end{equation}
and part (b) of the assertion follows.

From the Inverse Function Theorem, we can define two functions
$$
\mathbf{u}^+=\mathbf{u}^+(\mathbf{v}+\alpha\mathbf{u}), \quad
\mathbf{u}^-=\mathbf{u}^-(\mathbf{v}-\alpha\mathbf{u})
$$
in a neighborhood of $(\mathbf{u}_0,\mathbf{f}(\mathbf{u}_0) )$
implicitly by
\begin{equation}\label{upm}
\mathbf{f}( \mathbf{u}^\pm) \pm \alpha \mathbf{u}^\pm = \mathbf{v}
\pm \alpha \mathbf{u}
\end{equation}
with
$\mathbf{u}^\pm(\mathbf{f}(\mathbf{u}_0) \pm \alpha \mathbf{u}^\pm)
= \mathbf{u}_0$
since
$$
{d \over d \mathbf{u}^\pm} (\mathbf{f}(\mathbf{u}^\pm) \pm \alpha
\mathbf{u}^\pm) \left |_{\mathbf{u}_0} = \mathbf{f}'(\mathbf{u}_0)
\pm \alpha \right.
$$
is nonsingular.
In view of (\ref{defPhi}-\ref{defPhi2}) and (\ref{upm}), we can
write
\begin{equation}
\Phi(\mathbf{u},\mathbf{v}) = \frac12 \Big( (\phi(\mathbf{u}^+) +
{\psi(\mathbf{u}^+) \over \alpha} )
+(\phi(\mathbf{u}^-) - {\psi(\mathbf{u}^-) \over \alpha} ) \Big)
\end{equation}
and
\begin{gather}
  {\partial \over \partial \mathbf{u}} \mathbf{u}^\pm
= \pm \alpha \left(\mathbf{f}'(\mathbf{u}^\pm) \pm \alpha
\right)^{-1} \\
  {\partial \over \partial \mathbf{v}} \mathbf{u}^\pm
= \left(\mathbf{f}'(\mathbf{u}^\pm) \pm \alpha \right)^{-1}.
\end{gather}
Therefore
\begin{align*}
\Phi_{\mathbf{u}}(\mathbf{u},\mathbf{v})
=& \frac12 \Big( { (\phi'(\mathbf{u}^+) + {\psi'(\mathbf{u}^+)
\over \alpha} )
     {\partial \over \partial \mathbf{u}} \mathbf{u}^+
     +(\phi'(\mathbf{u}^-) - {\psi'(\mathbf{u}^-) \over \alpha} )
     {\partial \over \partial \mathbf{u}} \mathbf{u}^- } \Big) \\                     \\
=& \frac12 \left( \phi'(\mathbf{u}^+) + \phi'(\mathbf{u}^-)\right),
\end{align*}
where we have used $\psi' = \phi'\mathbf{f}'$.
Similarly
\begin{gather*}
  \Phi_{\mathbf{v}}(\mathbf{u},\mathbf{v})
= \frac{1}{2\alpha} \left( \phi'(\mathbf{u}^+) -
\phi'(\mathbf{u}^-) \right)
\\
  \Phi_{\mathbf{u}\mathbf{u}}(\mathbf{u},\mathbf{v})
= \alpha^2( h_+''( \mathbf{v} + \alpha \mathbf{u} ) + h_-''(
\mathbf{v} - \alpha \mathbf{u} ) )  \\
  \Phi_{\mathbf{u}\mathbf{v}}(\mathbf{u},\mathbf{v})
= \alpha  ( h_+''( \mathbf{v} + \alpha \mathbf{u} ) - h_-''(
\mathbf{v} - \alpha \mathbf{u} ) )  \\
  \Phi_{\mathbf{v}\mathbf{v}}(\mathbf{u},\mathbf{v})
=       h_+''( \mathbf{v} + \alpha \mathbf{u} ) + h_-''(
\mathbf{v} - \alpha \mathbf{u} )
\\
h_+''(\mathbf{v} + \alpha \mathbf{u}) = \phi''(\mathbf{u}^+)(
\alpha + \mathbf{f}'(\mathbf{u}^+) )^{-1}  \\ h_-''(\mathbf{v} -
\alpha \mathbf{u}) = \phi''(\mathbf{u}^-)( \alpha -
\mathbf{f}'(\mathbf{u}^-) )^{-1}
\end{gather*}
To show the convexity of $\Phi$, we compute
\begin{multline} \label{Phipp}
   (\mathbf{x}^T, \mathbf{y}^T)
\begin{pmatrix} \Phi_{\mathbf{u}\mathbf{u}} &\Phi_{\mathbf{u}\mathbf{v}}\\
\Phi_{\mathbf{u}\mathbf{v}} &\Phi_{\mathbf{v}\mathbf{v}}
\end{pmatrix}
\begin{pmatrix} \mathbf{x}  \\ \mathbf{y} \end{pmatrix}
=  (\alpha \mathbf{x} + \mathbf{y})^T \phi''(\mathbf{u}^+)(
\alpha + \mathbf{f}'(\mathbf{u}^+) )^{-1}  (\alpha \mathbf{x} + \mathbf{y})\\
  + (\alpha \mathbf{x} - \mathbf{y})^T \phi''(\mathbf{u}^-)( \alpha
  - \mathbf{f}'(\mathbf{u}^-) )^{-1}  (\alpha \mathbf{x} - \mathbf{y})
\end{multline}
Since $\phi''  \mathbf{f}'$ is symmetric, so are $\phi'' (\alpha
\pm \mathbf{f}')^{-1}$. This makes $(\alpha \pm \mathbf{f}')^{-1}$
self adjoint operators on $R^2$ with respect to the inner product
induced by  $\phi''$:
$$
  \langle \mathbf{y}, (\alpha \pm \mathbf{f}')^{-1} \mathbf{x} \rangle
= \langle (\alpha \pm \mathbf{f}')^{-1} \mathbf{y}, \mathbf{x}
\rangle, \qquad \langle \mathbf{y}, \mathbf{x} \rangle =
\mathbf{y}^T \phi'' \mathbf{x}
$$
Therefore $\phi''(\alpha \pm \mathbf{f}')^{-1}$ are positive
definite under the sub-characteristic condition (\ref{subch}). We
conclude that $\Phi$ is convex in a neighborhood of 
$(\mathbf{u}_0, \mathbf{f}(\mathbf{u}_0))$ 
provided $\mathbf{u}_0$ satisfies
(\ref{subch}). This proves (a).

Part (c) is a direct consequence of (a) since
$$
\Phi_{\mathbf{v}}(\mathbf{u},\mathbf{v}) =
\Phi_{\mathbf{v}}(\mathbf{u},\mathbf{v})
-\Phi_{\mathbf{v}}(\mathbf{u},\mathbf{f}(\mathbf{u})) =
(\mathbf{v}-\mathbf{f}(\mathbf{u}) )^T \int_0^1
\Phi_{\mathbf{v}\mathbf{v}}(
\mathbf{u},\mathbf{f}(\mathbf{u})+\theta(\mathbf{v}-\mathbf{f}(\mathbf{u}))
) d\theta
$$
\section{A priori Estimates}\label{ape}

To proceed with the energy estimate, we
first study how the singularity propagates. (\ref{jx}) is a
semilinear hyperbolic system, the jump discontinuity in the
initial data propagate along the characteristic curves and, due to
the relaxation effect, decays exponentially in time. To show this,
we write (\ref{jx}) in diagonal form in the characteristic
variables $ \mathbf{w}^{\pm} = \mathbf{v} \pm \alpha \mathbf{u}$,
\begin{equation} \label{wpm}
\begin{gathered}
\mathbf{w}^+_t + \alpha \mathbf{w}^+_x =  \mathbf{f}(\mathbf{u}) -
\mathbf{v}                  \\ \mathbf{w}^-_t - \alpha
\mathbf{w}^-_x =  \mathbf{f}(\mathbf{u}) - \mathbf{v}
\end{gathered}
\end{equation}
We denote by $[\cdot]^{\pm}$ the jumps along the characteristic
line $ dx/dt = \pm \alpha$. Taking the jump in the `+' family on
both sides of the first equation in (\ref{wpm}), we have
\begin{equation}\label{F11}
[ \mathbf{w}^+_t +  \alpha \mathbf{w}^+_x ]^+
 =  [ \mathbf{f}( {\mathbf{w}^+ - \mathbf{w}^-_r \over 2 \alpha }) ]^+ -[ {\mathbf{w}^+ \over 2}]^+
\end{equation}
where $\mathbf{w}^-_r = \mathbf{v}_r - \alpha \mathbf{u}_r$ is the
corresponding Riemann data on $x >0$. Since $\mathbf{U}$ has
continuous first derivatives up to the boundary on either side of
the jumps, we can interchange the tangential derivative with the
jump to get an ODE along the characteristics
\begin{equation} \label{F12}
\begin{aligned}
\frac{d}{dt} [ \mathbf{w}^+]^+
=&  [ \mathbf{f}( {\mathbf{w}^+ -
\mathbf{w}^-_r \over 2 \alpha }) ]^+  -[ {\mathbf{w}^+ \over 2}]^+\\
=&  \mathbf{f}( {\mathbf{w}^+_r - \mathbf{w}^-_r \over 2 \alpha })
    -\mathbf{f}( {  -[ \mathbf{w}^+ ]^+ + \mathbf{w}^+_r - \mathbf{w}^-_r
    \over 2 \alpha })    - {[\mathbf{w}^+]^+ \over 2} .
\end{aligned}
\end{equation}
The equilibrium for (\ref{F12}) is given by $[ \mathbf{w}^+]^+ =\mathbf{0}$
with $(\mathbf{u}, \mathbf{v}) = (\mathbf{u}_r, \mathbf{v}_r)$ on
either side of the characteristic. The linearization of the right
hand side around this equilibrium state is thus
$$
\frac12 \big( {\mathbf{f}'(\mathbf{u}_r) \over \alpha} - \mathbf{I} \big)
[ \mathbf{w}^+]^+.
$$
From (\ref{subch}), we conclude that the local equilibrium
$(\mathbf{u}_r, \mathbf{v}_r)$ is a sink and $[
\mathbf{w}^+]^+(t)$ decays exponentially in $t$. The same applies
to $[ \mathbf{w}^{\pm}_x ]^+(t)$, 
$[ \mathbf{w}^- ]^-(t)$
and $[ \mathbf{w}^{\pm}_x ]^-(t)$. 
We therefore have the following
\begin{lemma}\label{jumps}
Let $\beta = | \mathbf{u}_r-\mathbf{u}_l |$ and
$$E =\sup_{0\leq t \leq T}\left( \|\mathbf{\mho} \bnorm_1(t) +
 |[\mathbf{\mho} ]^+(t)| + |[\mathbf{\mho} ]^-(t)| \right),
$$
then there exist positive constants $\epsilon_1$, $C$, $C_1$
such that if $0\leq t\leq T$, $E\leq\epsilon_1$ and $\alpha$
sufficiently large, we have
\begin{equation}\label{jmp}
   | [\mathbf{\mho} ]^{\pm} (t) | + | [\mathbf{\mho}_x]^{\pm} (t) |
   \leq C \beta e^{-C_1 t}
\end{equation}
\end{lemma}

To proceed, we define
\begin{gather*}
  \mathcal{E} = \Phi(\widetilde{\mathbf{U}}+\mathbf{\mho} ) -
  \Phi(\widetilde{\mathbf{U}}) - \Phi'(\widetilde{\mathbf{U}}) \mathbf{\mho}\\
  \mathcal{E}_{\scriptstyle \mho }
= \Phi'(\widetilde{\mathbf{U}}+\mathbf{\mho} ) 
 -\Phi'(\widetilde{\mathbf{U}})    \\
  \mathcal{E}_{\scriptstyle \widetilde U}
= \Phi'(\widetilde{\mathbf{U}}+\mathbf{\mho} ) -
  \Phi'(\widetilde{\mathbf{U}}) -\mathbf{\mho}^T
  \Phi''(\widetilde{\mathbf{U}}) \\
  \mathcal{J} = \Psi(\widetilde{\mathbf{U}}+\mathbf{\mho} ) -
  \Psi(\widetilde{\mathbf{U}}) - \Psi'(\widetilde{\mathbf{U}}) \mathbf{\mho}\\
  \mathcal{J}_{\scriptstyle \mho }
= \Psi'(\widetilde{\mathbf{U}}+\mathbf{\mho} ) 
 -\Psi'(\widetilde{\mathbf{U}})     \\
  \mathcal{J}_{\scriptstyle \widetilde U }
= \Psi'(\widetilde{\mathbf{U}}+\mathbf{\mho} ) -
  \Psi'(\widetilde{\mathbf{U}}) -\mathbf{\mho}^T\Psi''(\widetilde{\mathbf{U}})\\
\end{gather*}
 It is easy to see that, for $E  < \epsilon_2$, we have
 \begin{equation}\label{PhifG}
 \begin{gathered}
    c |\mathbf{\mho} |^2 \leq \mathcal{E} \leq C |\mathbf{\mho} |^2     \\
    \mathcal{E}_{\scriptstyle \mho} \leq C |\mathbf{\mho} |      \\
    \mathcal{E}_{\scriptstyle \widetilde U} \leq C |\mathbf{\mho} |^2
 \end{gathered}
 \end{equation}
In addition to (\ref{PhifG}), a more refined estimate for
$\mathcal{E}_{\scriptstyle \mho }$ is needed due to the structure of the
nonlinear term. Denote by
$$
\mathcal{E}_{\scriptstyle \mho } 
= ( \mathcal{E}_{\mu}, \mathcal{E}_{\nu} )
$$
In the following Lemma, we expand $\mathcal{E}_{\nu}$ in
terms of the perturbation and identify the quadratic term, which
will be of use in our energy estimate.
%%
\begin{lemma} For $\bar{\mathbf{u}} 
= \widetilde{\mathbf{u}} + \mbox{\boldmath $\mu$}/2$
and $\bar{\mathbf{a}} = \mathbf{f}'(\bar{\mathbf{u}})$,
\begin{equation}\label{Gs6}
\mathcal{E}_{\nu} =
 \left(  (\alpha^2 - \bar{\mathbf{a}}^2)^{-1} 
( \mbox{\boldmath $\nu$} - \bar{\mathbf{a}} \mbox{\boldmath $\mu$}) \right)^T 
\phi''(\bar{\mathbf{u}})
 + O( |\mathbf{v}^{(1)}||\mathbf{\mho} | 
+ |\mathbf{\mho} | 
  |\mbox{\boldmath $\nu$} -\bar{\mathbf{a}} \mbox{\boldmath $\mu$}|  )
 + O( |\mathbf{v}^{(1)}|^3 + |\mathbf{\mho} |^3 ).
\end{equation}
\end{lemma}

\paragraph{Proof.} Since
  $ \mathcal{E}_{\scriptstyle \mho }
 = \Phi'(\mathbf{U} ) - \Phi'(\widetilde{\mathbf{U}}) $
involves the difference of functions evaluated at $\mathbf{U} $
and $\widetilde{\mathbf{U}}$ respectively, we will, in the remainder
of the proof, consider $(\bar{\mathbf{u}},\bar{\mathbf{v}}) =
\left( \frac{ \widetilde{\mathbf{u}} + \mathbf{u} }{2},
                   \mathbf{f}( \frac{ \widetilde{\mathbf{u}} + \mathbf{u} }{2} ) \right)$
fixed as the base point for Taylor expansion.
We take $(\mathbf{u}_0, \mathbf{f}( \mathbf{u}_0) ) =
(\bar{\mathbf{u}},\bar{\mathbf{v}})$ and recall the two functions
$$
\mathbf{u}^+=\mathbf{u}^+(\mathbf{v}+\alpha\mathbf{u}), \quad
\mathbf{u}^-=\mathbf{u}^-(\mathbf{v}-\alpha\mathbf{u})
$$
defined in (\ref{upm}) in a neighborhood of
$(\bar{\mathbf{u}},\bar{\mathbf{v}})$. To estimate
$\Phi_{\mathbf{v}}$, we let
\begin{equation}
\mbox{\boldmath $\eta$}^\pm = \mathbf{v} - \bar{\mathbf{v}} \pm \alpha
(\mathbf{u} -\bar{\mathbf{u}})
\end{equation}
and expand $\mathbf{u}^\pm$ in Taylor series of
$\mbox{\boldmath $\eta$}^\pm$
\begin{equation}\label{upm12}
\mathbf{u}^\pm = \bar{\mathbf{u}} + \mathbf{u}_1^\pm +
\mathbf{u}_2^\pm + O( |\mbox{\boldmath $\eta$}^\pm|^3 )
\end{equation}
with $\mathbf{u}^\pm_1 = D \mathbf{u}^\pm(\bar{\mathbf{u}})
\mbox{\boldmath $\eta$}^\pm$ and $\mathbf{u}^\pm_2 = \frac12 D^2
\mathbf{u}^\pm(\bar{\mathbf{u}})
(\mbox{\boldmath $\eta$}^\pm,\mbox{\boldmath $\eta$}^\pm)$.
Substitute (\ref{upm12}) back to (\ref{upm}), expand in power
series of $\mbox{\boldmath $\eta$}^\pm$ and equate the powers of $
|\mbox{\boldmath $\eta$}^\pm| $, we get
\begin{gather} \label{upm1}
\mathbf{u}_1^\pm = (\mathbf{f}'( \bar{\mathbf{u}} ) \pm \alpha
)^{-1}  \mbox{\boldmath $\eta$}^\pm
      = (\bar{\mathbf{a}} \pm \alpha )^{-1}  \mbox{\boldmath $\eta$}^\pm \\
 \label{upm2}
\mathbf{u}_2^\pm = - (\bar{\mathbf{a}} \pm \alpha )^{-1}{
\mathbf{f}''(\bar{\mathbf{u}}) \over 2} (\mathbf{u}_1^\pm,
\mathbf{u}_1^\pm)
\end{gather}
Introduce the notation
$$
\langle \cdot \rangle_{\widetilde U}^{ U}
= \cdot |_{\scriptstyle U} + \cdot |_{\scriptstyle\widetilde U }, 
\quad
[\cdot ]_{\widetilde U}^U 
= \cdot |_{\scriptstyle U } - \cdot |_{\scriptstyle \widetilde U}
$$
we have
\begin{equation*}
 [ \mbox{\boldmath $\eta$}^\pm ]^U_{\widetilde U}  
= [\mathbf{v}-\bar{\mathbf{v}}]^U_{\widetilde U} 
 \pm \alpha [\mathbf{u} -\bar{\mathbf{u}}]^U_{\widetilde U}
  = [\mathbf{v}]^U_{\widetilde U} \pm \alpha [\mathbf{u}]^U_{\widetilde U}
           = \mbox{\boldmath $\nu$} \pm \alpha \mbox{\boldmath $\mu$}
\end{equation*}
and
\begin{equation*}
  \langle \mbox{\boldmath $\eta$}^\pm \rangle^U_{\widetilde U}
= \langle \mathbf{v}-\bar{\mathbf{v}}\rangle^U_{\widetilde U}
  \pm \alpha \langle \mathbf{u} -\bar{\mathbf{u}}\rangle^U_{\widetilde U}
= \langle \mathbf{v}-\bar{\mathbf{v}}\rangle^U_{\widetilde U}
\end{equation*}
Since
\begin{multline}
   \mathbf{v} -\bar{\mathbf{v}} |_U
= \mathbf{f}(\widetilde{\mathbf{u}}) + \mathbf{v}^{(1)} + \mbox{\boldmath $\nu$}
- \mathbf{f}(\bar{\mathbf{u}}) = \mathbf{f}(\bar{\mathbf{u}} -
{\mbox{\boldmath $\mu$} \over 2} ) + \mathbf{v}^{(1)} + \mbox{\boldmath $\nu$} 
- \mathbf{f}(\bar{\mathbf{u}}) \\
= -\bar{\mathbf{a}} {\mbox{\boldmath $\mu$} \over 2} 
 + O(|\mbox{\boldmath $\mu$}|^2 ) + \mathbf{v}^{(1)}  + \mbox{\boldmath $\nu$}
\end{multline}
and similarly
\begin{equation}
   \mathbf{v} -\bar{\mathbf{v}} \left|_{\scriptstyle \widetilde U }    \right.
= \mathbf{f}(\widetilde{\mathbf{u}}) - \mathbf{f}(\bar{\mathbf{u}}) 
= -\bar{\mathbf{a}} {\mbox{\boldmath $\mu$} \over 2} 
  + O(|\mbox{\boldmath $\mu$}|^2 ) + \mathbf{v}^{(1)}
\end{equation}
we have
\begin{gather*}
\langle \mathbf{v} - \bar{\mathbf{v}} \rangle^U_{\widetilde U} 
= \mbox{\boldmath $\nu$} - \bar{\mathbf{a}} \mbox{\boldmath $\mu$} 
  + 2 \mathbf{v}^{(1)} + O( |\mbox{\boldmath $\mu$}|^2 )\\
\begin{aligned}
  \Phi_{\mathbf{v}}
=&  \frac{1}{2\alpha} \left( \phi'(\mathbf{u}^+) -
\phi'(\mathbf{u}^-) \right) \\
=&  \frac{1}{2\alpha} \left( \phi'(
\bar{\mathbf{u}} + \mathbf{u}_1^+ + \mathbf{u}_2^+ + O(|\mbox{\boldmath $\eta$}^+|^3 ) )
 -\phi'( \bar{\mathbf{u}} + \mathbf{u}_1^- + \mathbf{u}_2^- +
 O( |\mbox{\boldmath $\eta$}^-|^3 ) )  \right)
\end{aligned}
\end{gather*}
Since
\begin{equation*}
 \phi'( \mathbf{u}^\pm )
= \phi'( \bar{\mathbf{u}}  ) +
  (\mathbf{u}_1^\pm)^T \phi''( \bar{\mathbf{u}}  ) +
  \left( (\mathbf{u}_2^\pm)^T \phi''(\bar{\mathbf{u}}) + {\phi'''(\bar{\mathbf{u}}) \over 2} (\mathbf{u}_1^\pm,\mathbf{u}_1^\pm) \right)
  + O( |\mbox{\boldmath $\eta$}^\pm|^3 )
\end{equation*}
we have the linear term for $\mathcal{E}_{\nu}$:
\begin{equation}
\left[ {1 \over 2 \alpha}  ( \mathbf{u}_1^+ -\mathbf{u}_1^- )^T
\phi''(\bar{\mathbf{u}}) \right]^U_{\widetilde U} 
=\left(  (\alpha^2 - \bar{\mathbf{a}}^2)^{-1} 
         ( \mbox{\boldmath $\nu$} -\bar{\mathbf{a}}
         \mbox{\boldmath $\mu$} ) \right)^T \phi''(\bar{\mathbf{u}})
\end{equation}

To estimate the quadratic term in $\mathcal{E}_{\nu}$,
observe that from (\ref{upm1}) and (\ref{upm2}), both
$$
  (\mathbf{u}_2^\pm)^T \phi''(\bar{\mathbf{u}})
 =  -\Big( (\bar{\mathbf{a}} \pm \alpha )^{-1}{ \mathbf{f}''(\bar{\mathbf{u}})
 \over 2}  \big( (\bar{\mathbf{a}} \pm \alpha )^{-1}  \bullet ,
 (\bar{\mathbf{a}} \pm \alpha )^{-1}  \bullet \big)
\Big)^T |_{(\mbox{\boldmath $\eta$}^\pm,\mbox{\boldmath $\eta$}^\pm)}
   \phi''(\bar{\mathbf{u}})
$$
and
$$
  {\phi'''(\bar{\mathbf{u}}) \over 2} (\mathbf{u}_1^\pm,\mathbf{u}_1^\pm)
= {\phi'''(\bar{\mathbf{u}}) \over 2}
  \left( (\bar{\mathbf{a}} \pm \alpha )^{-1}  \bullet , (\bar{\mathbf{a}} \pm \alpha )^{-1}  \bullet \right)
|_{(\mbox{\boldmath $\eta$}^\pm,\mbox{\boldmath $\eta$}^\pm)}
$$
are (vector valued) symmetric bilinear forms evaluated at
$(\mbox{\boldmath $\eta$}^+,\mbox{\boldmath $\eta$}^+)$ and
$(\mbox{\boldmath $\eta$}^-,\mbox{\boldmath $\eta$}^-)$ respectively. In addition,
both are symmetric in their arguments. For any symmetric bilinear
form $Q(\cdot,\cdot)$, we have
$$
Q(\mathbf{p},\mathbf{p}) - Q(\mathbf{q},\mathbf{q}) =
Q(\mathbf{p}+\mathbf{q},\mathbf{p}-\mathbf{q}),
$$
Thus
\begin{equation*}
 \left[ (\mathbf{u}_2^\pm)^T \phi''(\bar{\mathbf{u}})
 + {\phi'''(\bar{\mathbf{u}}) \over 2} (\mathbf{u}_1^\pm,\mathbf{u}_1^\pm)
 \right] _{\widetilde U}^U
=O(1) \left( \left| \langle \mbox{\boldmath $\eta$}^\pm \rangle_{ \widetilde U}^{ U}
\right| \cdot \left|
[\mbox{\boldmath $\eta$}^\pm]_{ \widetilde U }^{ U}
\right| \right)
\end{equation*}
and (\ref{Gs6}) follows. \hfill      $\Box$

We now proceed with the main energy estimate.
Let $\beta$ and $E$ be defined as in Lemma \ref{jumps}.

\begin{lemma}\label{l2}
There exist positive constants $\epsilon_2$ and $C$ such that for
$0\leq t\leq T$ and $E \leq\epsilon_2$,
\begin{multline} \label{l2est}
\|\mathbf{\mho} (t)\|^2 + \int_0^t\| \mbox{\boldmath $\nu$} -\bar{\mathbf{a}}
\mbox{\boldmath $\mu$} \|^2 d\tau \\
\leq \|\mathbf{\mho} (0) \|^2 +
C \int_0^t \bint \big(| \mathbf{v}^{(1)}|^4+ | \mathbf{v}^{(1)}_x|^2
+ |\mathbf{\mho} |^2|\mathbf{v}^{(1)}|
+ |\mathbf{\mho} | | \mathbf{v}^{(1)}_x|
+ | \mathbf{\mho}  |^6 \big)  dx d\tau + C \beta
\end{multline}
\end{lemma}

\paragraph{Proof.} We multiply 
both sides of (\ref{feq}) by 
$\mathcal{E}_{\scriptstyle \mho }$ from the left to get
\begin{equation} \label{Phi}
  \mathcal{E}_t + \mathcal{J}_x
= \mathcal{E}_{\scriptstyle \widetilde U}
 ( \widetilde{\mathbf{U}}_t + \mathbf{A} \widetilde{\mathbf{U}}_x ) 
 + \mathcal{E}_{\scriptstyle \mho}
 ( \mathbf{N}(\mathbf{U})-\mathbf{N}(\widetilde{\mathbf{U}} ) ) 
- \mathcal{E}_{\scriptstyle \mho }
 ( \mathbf{U}^{(1)}_t + \mathbf{A} \mathbf{U}^{(1)}_x )
\end{equation}
where we have used $\Phi' \mathbf{A} = \Psi'$.
Next we calculate each term on the right hand side of (\ref{Phi}):
\begin{gather}\label{Gs1}
  \widetilde{\mathbf{U}}_t + \mathbf{A} \widetilde{\mathbf{U}}_x
  = \mathbf{N}(\widetilde{\mathbf{U}}) + \mathbf{U}^{(1)}_t
  + \mathbf{A} \mathbf{U}^{(1)}_x
  = O(\mathbf{v}^{(1)},\mathbf{v}^{(1)}_t,\mathbf{v}^{(1)}_x),\\
\label{Gs2}
\mathbf{N}(\mathbf{U})-\mathbf{N}(\check{\mathbf{U}} ) 
= - \begin{pmatrix}
        \mathbf{0}\\
\mbox{\boldmath $\nu$} - \bar{\mathbf{a}} \mbox{\boldmath $\mu$}
 + O( |\mbox{\boldmath $\mu$}|^3)
  \end{pmatrix}
\end{gather}
Thus (\ref{l2est}) follows after integrating (\ref{Phi}) over 
$ \mathbb{R} \times (0,T) $ 
and applying the Cauchy-Schwartz inequality and
(\ref{jmp}). \hfill          $\Box$

\begin{lemma}\label{h1}
There exist positive constants $\epsilon_2$ and $C$ such that if
$E \leq\epsilon_2$ and $0\leq t\leq T$, then
\begin{equation} \label{fx}
\begin{aligned}
 \|\mathbf{\mho}_x(t)\bnorm^2 + \int_0^t\|\mathbf{\mho}_x(\tau) \bnorm^2 d\tau
\leq&   \| \mathbf{\mho}_x(0) \bnorm^2
  + C \int_0^t \| \mbox{\boldmath $\nu$} - \bar{\mathbf{a}} 
\mbox{\boldmath $\mu$} \|^2\\
 &+ C {\int_0^t} \bint \Big(
 |\mathbf{v}^{(1)}|^2 |\mathbf{\mho} |^2  + |\mathbf{\mho} |^6
 + |\mathbf{v}^{(1)}_x|^2\\
&+ |\mathbf{v}^{(1)}_{xx}|^2  + |\mathbf{\mho}_x|^2 |\mathbf{v}^{(1)}|
   \Big) dx d\tau + C \beta                     \\
\end{aligned}
\end{equation}
\end{lemma}

\paragraph{Proof.} We first derive the equation for $\mathbf{\mho}_x$ by
differentiating (\ref{feq}) with respect to $x$,
\begin{equation} \label{fxeq}
\begin{aligned}
\mathbf{\mho}_{xt} + \mathbf{A} \mathbf{\mho}_{xx}
=&-\begin{pmatrix} O( \mathbf{v}^{(1)}_{xx} ) \\
    \mbox{\boldmath $\nu$}_x - \mathbf{a}( \mathbf{u}^{(0)}_x 
   + \mbox{\boldmath $\mu$}_x )
    + \widetilde{\mathbf{a}} \mathbf{u}^{(0)}_x
    + O( \mathbf{v}^{(1)}_{xx} )\end{pmatrix}    \\
=&-\begin{pmatrix}  O( \mathbf{v}^{(1)}_{xx} ) \\
    \mbox{\boldmath $\nu$}_x - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x
     + O( |\mathbf{v}^{(1)}| |\mathbf{\mho} |
     + |\mathbf{\mho} | |\mathbf{\mho}_x| + |\mathbf{v}^{(1)}_{xx}| )
    \end{pmatrix},
\end{aligned}
\end{equation}
where $\widetilde{\mathbf{a}} = \mathbf{f}'(\widetilde{\mathbf{u}}) $ and
$\mathbf{a} = \mathbf{f}'(\mathbf{u})$. Observe that
$$
\begin{pmatrix} \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}}  \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}
$$
is a symmetrizer for $\mathbf{A}$, we therefore multiply
(\ref{fxeq}) from the left by
\begin{equation} \label{weight}
\mathbf{\mho}^T_x \begin{pmatrix}
     \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}}  \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}, \quad \widetilde\phi'' = \phi''(\widetilde{\mathbf{u}}),
\end{equation}
to get
\begin{equation} \label{5.10}
\begin{aligned}
 \widetilde{\mathcal{E}}(\mathbf{\mho}_x,&\mathbf{\mho}_x)_t
 + \widetilde{\mathcal{J}}(\mathbf{\mho}_x,\mathbf{\mho}_x)_x\\
 =&  - O(1)| \mbox{\boldmath $\nu$}_x - 
       \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x |^2
 + O(1) \Big( | \alpha^2 \mbox{\boldmath $\mu$}_x - \widetilde{\mathbf{a}}
 \mbox{\boldmath $\nu$}_x | | \mathbf{v}^{(1)}_{xx} |\\
&+| \mbox{\boldmath $\nu$}_x - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x |
 \big( |\mathbf{v}^{(1)}||\mathbf{\mho} | + |\mathbf{\mho} ||\mathbf{\mho}_x|
 + |\mathbf{v}^{(1)}_{xx}| \big) + |\mathbf{\mho}_x|^2 |\mathbf{v}^{(1)}|  \Big)
\end{aligned}
\end{equation}
where
\begin{gather*}
\widetilde{\mathcal{E}}(\mathbf{\mho}_x,\mathbf{\mho}_x)
= \mathbf{\mho}^T_x \begin{pmatrix}
     \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}} \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}  \mathbf{\mho}_x, \\
\widetilde{\mathcal{J}}(\mathbf{\mho}_x,\mathbf{\mho}_x) =
\mathbf{\mho}^T_x \begin{pmatrix}
     -\alpha^2 \widetilde\phi''\widetilde{\mathbf{a}} & \alpha^2 \widetilde\phi''         \\
      \alpha^2 \widetilde\phi''    & -\widetilde\phi'' \widetilde{\mathbf{a}}
  \end{pmatrix}\mathbf{\mho}_x.
\end{gather*}
Here we have used the fact that $\widetilde\phi'' \widetilde{\mathbf{a}}$
is a symmetric matrix and
$$
{\partial \over \partial x} \begin{pmatrix}
     \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}}           \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}
 = O(\mathbf{v}^{(1)}).
$$
The symmetrizer
$$
\begin{pmatrix}
     \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}}           \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}
$$
is positive definite, since
\begin{align*}
(\mathbf{x}^T, \mathbf{y}^T)&\begin{pmatrix}
     \alpha^2 \widetilde\phi'' & -\widetilde\phi'' \widetilde{\mathbf{a}}  \\
     -\widetilde\phi'' \widetilde{\mathbf{a}} & \widetilde\phi''
  \end{pmatrix}
\begin{pmatrix}
 \mathbf{x}  \\  \mathbf{y}  \end{pmatrix}    \\
 =& (\alpha \mathbf{x} - \mathbf{y})^T \widetilde\phi'' (\alpha \mathbf{x}
 - \mathbf{y})
 + \mathbf{x}^T ( \alpha^2 \widetilde\phi - \widetilde{\mathbf{a}}^T
 \widetilde\phi''\widetilde{\mathbf{a}} ) \mathbf{x}.
\end{align*}
We can similarly show that $ \alpha^2 \widetilde\phi'' -
\widetilde{\mathbf{a}}^T\widetilde\phi''\widetilde{\mathbf{a}} 
= \widetilde\phi''( \alpha^2 - \widetilde{\mathbf{a}}^2 ) $ 
is positive definite under the
sub-characteristic condition (\ref{subch}) using the same argument
following (\ref{Phipp}). Thus
$$
c |\mathbf{\mho}_x |^2 \leq
\widetilde{\mathcal{E}}(\mathbf{\mho}_x,\mathbf{\mho}_x) \leq C
|\mathbf{\mho}_x |^2
$$
We then integrate (\ref{5.10}) over $\mathbb{R} \times [0,t]$ to get, for
$E $ sufficiently small and any $m>0$, there exists a $C>0$ such
that
\begin{equation} \label{fx1}
\begin{aligned}
 \|\mathbf{\mho}_x(t) &\bnorm^2 + \int_0^t\| \mbox{\boldmath $\nu$}_x
 -\widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x \bnorm^2 d\tau  \\
\leq& \|\mathbf{\mho}_x(0) \bnorm^2
    + \int_0^t \bint  m  | \alpha^2 \mbox{\boldmath $\mu$}_x 
     - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x |^2 \\
&+ C \left( |\mathbf{v}^{(1)}|^2 |\mathbf{\mho} |^2 +
 |\mathbf{v}^{(1)}_{xx}|^2 + |\mathbf{\mho}_x|^2 |\mathbf{v}^{(1)}|
 \right) dx d\tau  + C \beta
\end{aligned}
\end{equation}
where we have used (\ref{jmp}).

We now proceed to estimate $ \alpha^2 \mbox{\boldmath $\mu$}_x
-\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x$.
First, let us rewrite (\ref{feq}) as
\begin{equation} \label{thla}
\begin{gathered}
\mbox{\boldmath $\mu$}_t + \mbox{\boldmath $\nu$}_x  = - \mathbf{v}^{(1)}_x \\
\mbox{\boldmath $\nu$}_t + \alpha^2 \mbox{\boldmath $\mu$}_x  = -(\mbox{\boldmath $\nu$}
-\bar{\mathbf{a}} \mbox{\boldmath $\mu$}) + O( |\mbox{\boldmath $\mu$}|^3 ) + O(
|\mathbf{v}^{(1)}_x| ).
\end{gathered}
\end{equation}
We now multiply the first equation of (\ref{thla}) by
$(-\widetilde{\mathbf{a}})$ from the left and add it to the second:
\begin{equation} \label{fx2}
 (\mbox{\boldmath $\nu$}_t - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_t ) + ( \alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x)
= -(\mbox{\boldmath $\nu$} -\bar{\mathbf{a}} \mbox{\boldmath $\mu$}) + O(
|\mbox{\boldmath $\mu$}|^3 ) + O( |\mathbf{v}^{(1)}_x| ),
\end{equation}
so
\begin{equation} \label{fx3}
\begin{aligned}
 | \alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x |^2
=& -(\mbox{\boldmath $\nu$}_t -\widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_t )^T(
\alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x )\\
  &-\bigl( \mbox{\boldmath $\nu$}-\bar{\mathbf{a}} \mbox{\boldmath $\mu$}  + O( |\mbox{\boldmath $\mu$}|^3 ) + O(\mathbf{v}^{(1)}_x) \bigr)^T
        ( \alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x ).
\end{aligned}
\end{equation}
The second term on the right hand side of (\ref{fx3}) is bounded by
\begin{multline} \label{fx4}
-\Big( \mbox{\boldmath $\nu$}-\bar{\mathbf{a}} \mbox{\boldmath $\mu$}  
+ O( |\mbox{\boldmath $\mu$}|^3 )
+ O(|\mathbf{v}^{(1)}_x|) \Big)^T(\alpha^2 \mbox{\boldmath $\mu$}_x 
-\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x ) \\
\leq   \frac14 | \alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}}
\mbox{\boldmath $\nu$}_x |^2
+ C \left( |\mbox{\boldmath $\nu$}-\bar{\mathbf{a}} \mbox{\boldmath $\mu$}|^2 
+ |\mbox{\boldmath $\mu$}|^6 + |\mathbf{v}^{(1)}_x|^2 \right)
\end{multline}
while the first term is bounded by
\begin{equation} \label{fx5}
\begin{aligned}
 \big( (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} 
        \mbox{\boldmath $\mu$})_t&
+ O(|\mathbf{v}^{(1)}||\mathbf{\mho} |)      \big)^T
\left( ( \alpha^2 \mbox{\boldmath $\mu$} 
- \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_x
+ O(|\mathbf{v}^{(1)}||\mathbf{\mho} |) \right)       \\
 \leq & (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} 
         \mbox{\boldmath $\mu$})_t^T 
    ( \alpha^2 \mbox{\boldmath $\mu$} - 
      \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_x \\
      & + C |\mathbf{v}^{(1)}| \; |\mathbf{\mho} |
\Big( | \mbox{\boldmath $\nu$}_t 
       - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_t| 
    + | \alpha^2 \mbox{\boldmath $\mu$}_x  
       -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x|
      + O(|\mathbf{v}^{(1)}||\mathbf{\mho} |) \Big)
\end{aligned}
\end{equation}
Substituting (\ref{fx2}) into the second term on the right hand
side of (\ref{fx5}), we can estimate it by
\begin{equation} \label{fx6}
\begin{aligned}
 C |&\mathbf{v}^{(1)}|\;|\mathbf{\mho} | \left( |
\mbox{\boldmath $\nu$}_t - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_t| 
+ | \alpha^2 \mbox{\boldmath $\mu$}_x 
   -\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x|
      + |\mathbf{v}^{(1)}||\mathbf{\mho} | \right)    \\
 \leq & C |\mathbf{v}^{(1)}|\;|\mathbf{\mho} | \left( 2 |
\alpha^2 \mbox{\boldmath $\mu$}_x 
-\widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x| +
|\mbox{\boldmath $\nu$}-\bar{\mathbf{a}}\mbox{\boldmath $\mu$}|
           + |\mbox{\boldmath $\mu$}|^3 + |\mathbf{v}^{(1)}_x|
+ |\mathbf{v}^{(1)}||\mathbf{\mho} | \right) \\
\leq & {1\over 4} | \alpha^2 \mbox{\boldmath $\mu$}_x -\widetilde{\mathbf{a}}
\mbox{\boldmath $\nu$}_x |^2 + C \left( |\mathbf{v}^{(1)}|^2|\mathbf{\mho}
|^2 + |\mbox{\boldmath $\nu$}-\bar{\mathbf{a}} \mbox{\boldmath $\mu$}|^2 +
|\mbox{\boldmath $\mu$}|^6 + |\mathbf{v}^{(1)}_x|^2  \right)
\end{aligned}
\end{equation}
while
\begin{equation*} %\label{fx7}
\begin{aligned}
 (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$})_t^T (
\alpha^2 \mbox{\boldmath $\mu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_x
=&\{ (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$})^T ( \alpha^2
\mbox{\boldmath $\mu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_x \}_t
 -\{ (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$})^T
 ( \alpha^2 \mbox{\boldmath $\mu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_t \}_x\\
 &+   (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$})_x^T ( \alpha^2
 \mbox{\boldmath $\mu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_t.
\end{aligned}
\end{equation*}
The last term on the right hand side of the above equation
can be treated similarly as (\ref{fx6}):
\begin{equation} \label{fx8}
     (\mbox{\boldmath $\nu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$})_x^T ( \alpha^2 \mbox{\boldmath $\mu$} - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$})_t
\leq C ( | \mbox{\boldmath $\nu$}_x - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x |^2
+ |\mathbf{v}^{(1)}|^2 |\mathbf{\mho} |^2 )  .
\end{equation}
After integrating (\ref{fx3}) over $\mathbb{R} \times (0,t)$ and applying
(\ref{fx3}-\ref{fx8}) and (\ref{jmp}), we have
\begin{equation} \label{fx9}
\begin{aligned}
\int_0^t \| \alpha^2 \mbox{\boldmath $\mu$}_x - \widetilde{\mathbf{a}}
\mbox{\boldmath $\nu$}_x \bnorm^2 d\tau     \leq &
C\Big\{  {\int_0^t} \| \mbox{\boldmath $\nu$} - \bar{\mathbf{a}}
\mbox{\boldmath $\mu$} \|^2 d\tau
 + \int_0^t \| \mbox{\boldmath $\nu$}_x - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x
 \bnorm^2d\tau \\
&  + \int_0^t \bint \left(  |\mathbf{v}^{(1)}|^2|\mathbf{\mho} |^2
+ |\mathbf{\mho} |^6 + |\mathbf{v}^{(1)}_x|^2  \right)  dx d\tau\Big\}
             + C \beta
\end{aligned}
\end{equation}
Since
\begin{equation} \label{fx10}
    c_1 ( | \mbox{\boldmath $\nu$}_x |^2 + | \mbox{\boldmath $\mu$}_x |^2 )
\leq | \alpha^2 \mbox{\boldmath $\mu$}_x 
       - \widetilde{\mathbf{a}} \mbox{\boldmath $\nu$}_x |^2 
  + | \mbox{\boldmath $\nu$}_x 
      - \widetilde{\mathbf{a}} \mbox{\boldmath $\mu$}_x |^2
\leq
    c_2 ( | \mbox{\boldmath $\nu$}_x |^2 + | \mbox{\boldmath $\mu$}_x |^2 ),
\end{equation}
we conclude with (\ref{fx}) from (\ref{fx1}) and (\ref{fx8}).
\hfill          $\Box$

Combining Lemma \ref{l2} and Lemma \ref{h1} with suitably chosen
$m$, $\beta$ and $E$, for $0\leq t\leq T$, we have
\begin{equation}\label{almost}
\begin{aligned}
\|\mathbf{\mho} (t)\bnorm_1^2 + \int_0^t \|\mathbf{\mho}_x\bnorm^2   d\tau
\leq & \|\mathbf{\mho} (0) \bnorm_1^2
+ C_1 \int_0^t \bint \Big(  |\mathbf{v}^{(1)}|^4
   + |\mathbf{v}^{(1)}_x|^2
   + |\mathbf{\mho} |    |\mathbf{v}^{(1)}_x|\\
 &  + |\mathbf{\mho} |^6
   + |\mathbf{v}^{(1)}_{xx}|^2
   + |\mathbf{\mho}_x |^2 | \mathbf{v}^{(1)} |
   + |\mathbf{\mho} |^2 | \mathbf{v}^{(1)} |
\Big)   dx d\tau + C \beta
\end{aligned}
\end{equation}
Finally, we can estimate the right hand side of (\ref{almost}).

\begin{lemma}
There exist a positive constant $C$ such that for $E  < \epsilon_2$
and $0\leq t\leq T$, the right hand side of (\ref{almost})is bounded
by
\begin{equation*}
 C\Big( \|\mathbf{\mho} (0) \bnorm_1^2 + \beta^{1/6}
 + (E +\beta) \int_0^t\| \mathbf{\mho}_x \bnorm^2 d\tau \Big).
\end{equation*}
\end{lemma}

\paragraph{Proof.} We estimate each term by virtue of Lemma \ref{turho}
and the Sobolev inequality (\ref{Sob}) as follows:
% 1
 $$
\int_0^t\!\!\bint |\mathbf{v}^{(1)}|^4  dx d\tau  \leq C \int_0^t
\| \mathbf{v}^{(1)} \|_{L^\infty}^2 \| \mathbf{v}^{(1)} \|^2 d
\tau \leq C \beta
$$
% 2
$$
\int_0^t\!\!\bint | \mathbf{v}^{(1)}_x |^2 + |
\mathbf{v}^{(1)}_{xx} |^2 dx d \tau \leq C \beta^{1/4}.
$$
% 3
 \begin{align*}
\int_0^t\!\!\bint |\mathbf{\mho} |\:|\mathbf{v}^{(1)}_x| dx d\tau
\leq & C\int_0^t ( \|\mathbf{\mho} \|^{1/2}
\|\mathbf{\mho}_x\bnorm^{1/2} +\beta e^{-C \tau}) \|
\mathbf{v}^{(1)}_x \|_{L^1}  d \tau    \\
\leq & C\int_0^t (E^2\|\mathbf{\mho}_x\bnorm^2
 + \| \mathbf{v}^{(1)}_x \|_{L^1}^{4/3} ) d \tau  + C \beta
 \\
\leq & C\int_0^t (E ^2\|\mathbf{\mho}_x\bnorm^2
         + \beta^{1/6}(1+\tau)^{-7 \over 6} ) d \tau  + C \beta
 \\
\leq & C E \int_0^t  \|\mathbf{\mho}_x\bnorm^2 d \tau  + C \beta^{1/6}
\end{align*}
% 4
$$
\int_0^t\!\!\bint | \mathbf{\mho}  |^6  dx d\tau  \leq
   \int_0^t  \| \mathbf{\mho}  \|^4 \| \mathbf{\mho}_x\bnorm^2  d \tau
   + C \beta
\leq   E  \int_0^t  \| \mathbf{\mho}_x\bnorm^2  d \tau + C \beta
$$
% 5
$$
\int_0^t\!\!\bint | \mathbf{\mho}_x |^2 |\mathbf{v}^{(1)}| dx d\tau
 \leq   \beta \int_0^t  \| \mathbf{\mho}_x\bnorm^2  d \tau
$$
 \begin{align*}
\int_0^t\!\!\bint | \mathbf{\mho} |^2|\mathbf{v}^{(1)}| dx d\tau
\leq & C\int_0^t \|\mathbf{\mho} \|^{3/2}
\|\mathbf{\mho}_x\bnorm^{1/2}
                  \| \mathbf{v}^{(1)} \|_{L^\infty}   d \tau  + C \beta \\
 \leq &
C\int_0^t ( E  \| \mathbf{\mho}_x\bnorm^2 + \| \mathbf{v}^{(1)}
\|^{4/3}_{L^\infty} ) d \tau   + C \beta   \\
 \leq & C E \int_0^t  \| \mathbf{\mho}_x\bnorm^2 d \tau  + C
\beta^{1/6}
\end{align*}
Which completes the proof.

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\end{thebibliography}

\noindent\textsc{Wei-Cheng Wang}\\
Department of Mathematics\\
National Tsing Hua University\\
HsinChu, Taiwan \\
e-mail: wangwc@duet.am.nthu.edu.tw

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