\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 33, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/33\hfil H\"older continuity]
{H\"older continuity for a periodic 2-component \\ $\mu$-b system}

\author[X. Wang \hfil EJDE-2014/33\hfilneg]
{Xiaohuan Wang}  % in alphabetical order

\address{Xiaohuan Wang \newline
College of Mathematics and Information Science, Henan University\\
 Kaifeng 475001, China}
\email{xiaohuanw@126.com}

\thanks{Submitted September 27, 2013. Published January 27, 2014.}
\subjclass[2000]{35G25, 35B30, 35L05}
\keywords{ H\"older; $\mu$-Hunter-Saxton system; energy
estimates; \hfill\break\indent initial value problem; $\mu$-b system}

\begin{abstract}
 In this article, we consider the Cauchy problem of a periodic
 2-component $\mu$-b system. We show that the date to solution
 for the periodic 2-component $\mu$-b system is H\"older continuous from
 bounded set of Sobolev spaces with exponent $s>5/2$ measured in
 a weaker Sobolev norm with index $r<s$ for the periodic case.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In this article, we reconsider the Cauchy problem of the following 
two-component periodic $\mu$-b system
  \begin{equation}
\begin{gathered}
\mu(u)_t-u_{txx}=bu_x(\mu(u)-u_{xx})-uu_{xxx}+\rho\rho_x,\quad
 t>0,x\in\mathbb{R},\\
 \rho_t=(\rho u)_x,\quad t>0,\; x\in\mathbb{R},\\
 u(0,x)=u_0(x),\quad \rho(0,x)=\rho_0(x), \quad  x\in\mathbb{R},\\
 u(t,x+1)=u(t,x),\quad \rho(t,x+1)=\rho(t,x), \quad  t\geq0,\; x\in\mathbb{R},
 \end{gathered} \label{1.1}
 \end{equation}
 where $b\in\mathbb{R}$, $\mu(u)=\int_\mathbb{S}udx$ and
 $\mathbb{S}=\mathbb{R}/\mathbb{Z}:=(0,1)$.

Recently, Zou \cite{Z} introduced the  system
  \begin{equation}
\begin{gathered}
\mu(u)_t-u_{txx}=2\mu(u)u_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x-\gamma_1u_{xxx},\quad
 t>0,\; x\in\mathbb{R},\\
 \rho_t=(\rho u)_x-2\gamma_2\rho_x,\quad t>0,\; x\in\mathbb{R},\\
 u(0,x)=u_0(x),\quad \rho(0,x)=\rho_0(x), \quad  x\in\mathbb{R},\\
 u(t,x+1)=u(t,x),\quad \rho(t,x+1)=\rho(t,x), \quad t\geq0,\; x\in\mathbb{R},
 \end{gathered}\label{1.2}
 \end{equation}
where $\mu(u)=\int_\mathbb{S}udx$,
$\mathbb{S}=\mathbb{R}/\mathbb{Z}$ and $\gamma_i\in\mathbb{R}$,
$i=1,2$. By integrating both sides of the first equation in the
system \eqref{1.2} over the circle $\mathbb{S}$ and using the
periodicity of $u$, one obtains
\[
\mu(u_t)=\mu(u)_t=0,
\]
which implies the following 2-component periodic
$\mu$-Hunter-Saxton system
\begin{equation}
\begin{gathered}
 -u_{txx}=2\mu(u)u_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x-\gamma_1u_{xxx},\quad
 t>0,\; x\in\mathbb{R},\\
 \rho_t=(\rho u)_x-2\gamma_2\rho_x,\quad t>0,x\in\mathbb{R},\\
 u(0,x)=u_0(x),\quad  \rho(0,x)=\rho_0(x), \quad  x\in\mathbb{R},\\
 u(t,x+1)=u(t,x),\quad \rho(t,x+1)=\rho(t,x), \quad t\geq0,\; x\in\mathbb{R}.
 \end{gathered}\label{1.3}
 \end{equation}
This system is a 2-component generalization of the generalized
Hunter-Saxton equation obtained in \cite{KLM}. Zou \cite{Z}
shows that this system is both a bi-Hamiltonian Euler equation
and a bi-variational equation. Liu-Yin \cite{LYna} established
the local well-posedness, precise blow-up scenario and global
existence result to the system \eqref{1.3}.

If $b=2$, then system \eqref{1.1} becomes the system
\eqref{1.3} with $\gamma_1=\gamma_2=0$. Therefore, system
\eqref{1.1} generalizes system \eqref{1.3} in some sense.

If $\rho\equiv0$, then system \eqref{1.1}
becomes the  system
 \begin{equation} 
\begin{gathered}
\mu(u_t)-u_{xxt}+uu_{xxx}-bu_x(\mu(u)-u_{xx})=0,\quad t>0,\; x\in\mathbb{S},\\
 u(0,x)=u_0(x),  \quad x\in\mathbb{S}.
 \end{gathered}
 \label{1.4}
\end{equation}
The above equation is called $\mu$-b equation. If
$b=2$, then equation \eqref{1.4} becomes the well-known
$\mu$-CH equation. Lenells, Misio\l{}ek and Ti\u{g}lay
\cite{LMT2010} introduced the $\mu$-CH, the $\mu$-DP as well as
$\mu$-Burgers equations, and the $\mu$-$b$ equation (see also
\cite{K}). In the case $b=3$, the $\mu$-$b$ equation reduces to
the $\mu$-DP equations. In addition, if $\mu(u)=0$, they reduce
to the HS and $\mu$-Burgers equations, respectively. It is
remarked that the $\mu$-Hunter-Saxton equation has a very
close relation with the periodic Hunter-Saxton and
Camassa-Holm equations, that is, \eqref{1.4} will reduce to
the Hunter-Saxton equation \cite{HunS,Wjp,Y2004} if $\mu(u)=0$
and $b=2$.

The local well-posedness of the $\mu$-CH and $\mu$-DP Cauchy
problems have been studied in \cite{KLM} and \cite{LMT2010}.
Recently, Fu et. al. \cite{FLQ11} described precise blow-up
scenarios for $\mu$-CH and $\mu$-DP.

When $\rho\not\equiv0$ and $\gamma_i=0$ $(i=1,2)$,
Constanin-Ivanov \cite{CI} considered the peakon solutions of
the Cauchy problem of system \eqref{1.3}. In paper \cite{Wun},
Wunsch studied the the Cauchy problem of 2-component periodic
Hunter-Saxton system, see also \cite{Km}. The local
well-posedness of system \eqref{1.1} was established in our
paper \cite{W1}.


Recently, some properties of solutions to the Camassa-Holm
equation have been studied by many authors. Himonas et al.
\cite{HMPZ} studied the persistence properties and unique
continuation of solutions of the Camassa-Holm equation, see
\cite{FL,ZC} for the similar properties of solutions to other
shallow water equation. Himonas-Kenig \cite{HK} and Himonas et
al. \cite{HKM} considered the non-uniform dependence on initial
data for the Camassa-Holm equation on the line and on the
circle, respectively. Lv et al. \cite{LPW} obtained the
non-uniform dependence on initial data for $\mu$-$b$ equation.
Lv-Wang \cite{LWjmp} considered the system \eqref{1.1} with
$\rho=\gamma-\gamma_{xx}$ and obtained the non-uniform
dependence on initial data. Just recently, Chen et al.
\cite{CLZ} and Himonas et al. \cite{HH} studied the H\"older
continuity of the solution map for shallow water equations.
Thompson \cite{T} also studied the H\"older continuity for the CH system,
which is obtained from \eqref{1.1} by replacing the operator $\mu-\partial_x^2$
with the operator $1-\partial_x^2$.

Our work has been inspired by \cite{CLZ,HH}. In this paper, we
shall study the problem \eqref{1.1}. We remark that there is
significant difference between system \eqref{1.1} and CH system
because of the two operators $1-\partial_x^2$ and
$\mu-\partial_x^2$. Moreover, the properties of $u$ and
$\gamma$ are different, see Proposition \ref{p2.1}. So the
system \eqref{1.1} will have the properties unlike the signal
equation, for example, $\mu$-b equation. And this is different
from the CH system.

This paper is organized as follows. In section 2, we will
recall some known results about the well-posedness and then
state out our main results. Section 3 is concerned with the
proof of the main results.


{\bf Notation} In this paper, the symbols
$\lesssim,\,\thickapprox$ and $\gtrsim$ are used to denote
inequality/equality up to a positive universal constant. For
example, $f(x)\lesssim g(x)$ means that $f(x)\leq cg(x)$ for
some positive universal constant $c$. In the following, we
denote by $\ast$ the spatial convolution. Given a Banach space
$Z$, we denote its norm by $\|\cdot\|_Z$. Since all space of
functions are over $\mathbb{S}$, for simplicity, we drop
$\mathbb{S}$ in our notations of function spaces if there is no
ambiguity. Let $[A,B]=AB-BA$ denotes the commutator of linear
operator $A$ and $B$. Set $\|z\|_{H^s\times
H^{s-1}}^2=\|u\|_{H^s}^2+\|\rho\|_{H^{s-1}}^2$, where
$z=(u,\rho)$.

\section{Some known results and main result}

 In this section we first recall the
known results, and then state out our main result.


As $\mu(u)_t=0$ under spatial periodicity, we can re-write
\eqref{1.1} as follows:
  \begin{equation}
\begin{gathered}
 u_t-uu_x=\partial_xA^{-1}\Big(b\mu(u)u+\frac{3-b}{2}u^2_x+\frac{1}{2}\rho^2
\Big),\quad  t>0,\;x\in\mathbb{S},\\
 \rho_t-u\rho_x=u_x\rho,\quad t>0,\; x\in\mathbb{S},\\
 u(0,x)=u_0(x),\quad \rho(0,x)=\rho_0(x), \quad  x\in\mathbb{S},
 \end{gathered}  \label{2.1}
\end{equation}
where $A=\mu-\partial_x^{2}$ is an isomorphism between
$H^s(\mathbb{S})$ and $H^{s-2}(\mathbb{S})$ with the inverse
$v=A^{-1}u$ given by
\begin{align*}
v(x)&=  (\frac{x^2}{2}-\frac{x}{2}+\frac{13}{12} )\mu(u)+ (x-{1}/{2} )
\int_0^1\int_0^yu(s){\rm d}s{\rm d}y\\
&\quad -\int_0^xu(s){\rm d}s{\rm d}y+\int_0^1\int_0^y
\int_0^su(r){\rm d}r{\rm d}s{\rm d}y.
\end{align*}
Since $A^{-1}$ and $\partial_x$ commute, the following
identities hold:
 \begin{gather}
A^{-1}\partial_xu(x) =  (x-{1}/{2} )\int_0^1u(x){\rm
d}x-\int_0^xu(y){\rm d}y+\int_0^1\int_0^xu(y){\rm d}y{\rm d}x,
   \label{2.2}\\
A^{-1}\partial_x^2u(x) = -u(x)+\int_0^1u(x){\rm d}x.
   \label{2.3}
\end{gather}
It is easy to show that $\mu(\Lambda^{-1}\partial_xu(x))=0$.

\begin{proposition}[{\cite[Theorem 2.1]{W1}}] \label{p2.1}  
 Given $z_0=(u_0,\rho_0)\in H^s\times H^{s-1}$,
$s\geq2$. Then there exists a maximal existence time
$T=T(\|z_0\|_{H^s\times H^{s-1}})>0$ and a unique solution
$z=(u,\rho)$ to system {\rm\eqref{2.1}} such that
\[
z=z(\cdot,z_0)\in C([0,T);H^s\times H^{s-1})\cap
C^1([0,T);H^{s-1}\times H^{s-2}).
\]
Moreover, the solution depends continuously on the initial
data, i.e. the mapping
\[
z_0\rightarrow z(\cdot,z_0): H^s\times H^{s-1}\rightarrow
C([0,T);H^s\times H^{s-1})\cap C^1([0,T);H^{s-1}\times H^{s-2})
\]
is continuous.
 \end{proposition}

Next, an explicit estimate for the maximal existence time $T$
is given.



\begin{proposition}\label{p2.2} 
Let $s>\frac{5}{2}$. If $z=(u,\rho)$ is a
solution of system {\rm\eqref{2.1}} with initial data $z_0$
described in Proposition {\rm\ref{p2.1}}, then the maximal
existence time $T$ satisfies
\[
T\geq T_0:=\frac{1}{2C_s\|z_0\|_{H^s\times H^{s-1}}},
\]
where $C_s$ is a constant depending only on $s$. Also, we have
\[
\|z(t)\|_{H^s\times H^{s-1}}\leq 2\|z_0\|_{H^s\times H^{s-1}},
\quad 0\leq t\leq T_0.
\]
 \end{proposition}

Now, we state our main result.

 \begin{theorem}\label{t2.1}
Assume $s>5/2$ and $3/2< r<s$.
 Then the solution map to \eqref{2.1} with \eqref{2.2} is H\"older
 continuous with exponent $\alpha=\alpha(s,r)$ as a map from
 $B(0,h)$ with $H^r(\mathbb{S})$ norm to
 $C([0,T_0],H^r(\mathbb{S}))$, where $T_0$ is defined as in Proposition
 {\rm\ref{p2.2}}. More precisely, for initial data
 $(u(0),\rho(0))$ and $(\hat u(0),\hat\rho(0))$ in a ball 
$B(0,h):=\{u\in H^s:\,\|u\|_{H^s}\leq  h\}$ of $H^s$, the solutions of 
\eqref{2.1} with \eqref{2.2}
 $(u(x,t),\rho(x,t)$ and $\hat u(x,t),\hat\rho(x,t)$ satisfy the inequality
 \begin{equation}
\begin{gathered}
\|u(t)-\hat u(t)\|_{C([0,T_0];H^r)}\leq c\|u(0)-\hat
 u(0)\|_{H^r}^\alpha, \\
\|\rho(t)-\hat \rho(t)\|_{C([0,T_0];H^r)}\leq c\|\rho(0)-\hat
 \rho(0)\|_{H^r}^\alpha,
 \end{gathered}    \label{2.4}
\end{equation}
where $\alpha$ is given by
\begin{equation}
 \alpha=\begin{cases}
1  & \text{if }  (s,r)\in \Omega_1,\\
s-r   & \text{if }  (s,r)\in\Omega_2
 \end{cases} \label{2.7}
\end{equation}
and the regions $\Omega_1$ and $\Omega_2$ are defined by
\begin{gather*}
\Omega_1= \{(s,r):\,s>5/2,\,3/2<r\leq s-1\},\\
\Omega_2= \{(s,r):\,s>5/2,\,s-1< r< s\}.
\end{gather*}
 \end{theorem}

\section{Proof of Theorem \ref{t2.1}}

In this section, we  prove Theorem
\ref{t2.1} by using energy method.  We shall prove that
\[
\|z(t)-\hat z(t)\|_{C([0,T_0];H^r\times H^{r-1})}\leq
c\|z(0)-\hat z(0)\|_{H^r\times H^{r-1}}^\alpha,
\]
 where $\|z(t)\|_{H^r\times
H^{r-1}}=\|u(t)\|_{H^r}+\|\rho(t)\|_{H^{r-1}}$.

We note that $\|u(0)-\hat  u(0)\|_{H^r}>0$ and 
$\|\rho(0)-\hat  \rho(0)\|_{H^{r-1}}>0$. Indeed, due to $r>3/2$, 
it follows from  Sobolev embedding 
$H^{\frac{1}{2}+}(\mathbb{S})\hookrightarrow  C^0(\mathbb{S})$ that
 \[
\|u(0)-\hat u(0)\|_{C^0}\lesssim\|u(0)-\hat  u(0)\|_{H^r}.
 \]
Hence $u(0)\equiv \hat u(0)$ if $\|u(0)-\hat
 u(0)\|_{H^r}=0$, and it follows from Proposition \ref{p2.1}
 that $u(x,t)=\hat u(x,t)$. Therefore, we will assume that 
$\|u(0)-\hat  u(0)\|_{H^r}>0$ and $\|\rho(0)-\hat \rho(0)\|_{H^{r-1}}>0$.
To prove Theorem \ref{t2.1}, we need the following Lemmas.

\begin{lemma}[{\cite[Lemma 1]{HH}}] \label{l3.1}  If $r+1\geq0$, then
   \[
\|[\Lambda^r\partial_x,f]v\|_{L^2}\leq c\|f\|_{H^s}\|v\|_{H^r}
    \]
provided that $s>3/2$ and $r+1\leq s$.
\end{lemma}



\begin{proof}[Proof of Theorem \ref{t2.1}]
 Let $u_0(x),\,\rho(0),\hat u_0(x),\,\hat\rho(0)\in B(0,h)$ and 
$(u(x,t),\rho(x,t))$ and $(\hat u(x,t),\hat\rho(x,t))$ be the two solutions to
\eqref{2.1} with initial data $(u_0(x),\rho(0))$ and 
$(\hat u_0(x),\hat\rho(0))$, respectively. Let
   \[
v=u-\hat u,\quad  \sigma=\rho-\hat\rho,
   \]
then $v$ and $\sigma$ satisfy that
\begin{equation}
\begin{gathered}
\begin{aligned}
 v_t-\frac{1}{2}\partial_x[v(u+\hat u)]
&=-\partial_xA^{-1}\big[b\mu(u)v+b\mu(v)\hat u\\
 &\quad +\frac{3-b}{2}\left(v_x(u+\hat u)_x\right)
 +\frac{1}{2}\sigma(\rho+\hat\rho)\big],\quad  t>0,\; x\in\mathbb{S},
\end{aligned}
\\
 \sigma_t=(v\rho+\sigma\hat  u)_x,\quad  t>0,\; x\in\mathbb{S},\\
  v(0,x)=u_0(x)-\hat u_0(x), \quad x\in\mathbb{S},\\
 \sigma(0,x)=\rho_0(x)-\hat \rho_0(x), \quad x\in\mathbb{S}.
 \end{gathered}\label{3.1}
 \end{equation}


Let $\Lambda=(1-\partial_x)^{1/2}$. Applying
$\Lambda^r$ and $\Lambda^{r-1}$ to both sides of the first and
second equation of \eqref{3.1}, then multiplying both sides by
$\Lambda^rv$ and $\Lambda^{r-1}\sigma$, respectively, and
integrating, we obtain
    \begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|v(t)\|_{H^r}^2\\
&= \frac{1}{2}\int_\mathbb{S}\Lambda^r
\partial_x[v(u+\hat u)]\cdot \Lambda^rv{\rm d}x
-\int_\mathbb{S}\Lambda^{r}\partial_xA^{-1}
\Big[b\mu(u)v+b\mu(v)\hat u\\
&\quad +\frac{3-b}{2}\big(v_x(u+\hat u)_x\big)
 +\frac{1}{2}\sigma(\rho+\hat\rho)\Big]\cdot \Lambda^rv{\rm d}x,
\end{aligned}\label{3.2}
\end{equation}
and
\begin{equation}
\frac{1}{2}\frac{d}{dt}\|\sigma(t)\|_{H^{r-1}}^2= \int_\mathbb{S}\Lambda^{r-1}
(v\rho+\sigma\hat  u)_x\cdot \Lambda^{r-1}\sigma{\rm d}x.
    \label{3.3}
\end{equation}
It follows from Lemma \ref{l3.1} that
\begin{equation}
\begin{aligned}
&\big|\frac{1}{2}\int_\mathbb{S}\Lambda^r
\partial_x[v(u+\hat u)]\cdot \Lambda^rv{\rm d}x\big| \\
&= \frac{1}{2}\big|\int_\mathbb{S}[\Lambda^r\partial_x,u+\hat
u]v\cdot \Lambda^rv{\rm d}x-\int_\mathbb{S}(u+\hat u)\Lambda^r
\partial_xv\cdot \Lambda^rv{\rm d}x\big| \\
&\lesssim \big|\int_\mathbb{S}[\Lambda^r\partial_x,u+\hat
u]v\cdot \Lambda^rv{\rm d}x\big|+\big|\int_\mathbb{S}(u+\hat
u)\Lambda^r\partial_xv\cdot \Lambda^rv{\rm d}x\big| \\
&\lesssim \big|\int_\mathbb{S}[\Lambda^r\partial_x,u+\hat
u]v\cdot \Lambda^rv{\rm
d}x\big|+\big|\int_\mathbb{S}\partial_x(u+\hat
u)\cdot (\Lambda^rv)^2{\rm d}x\big| \\
 &\lesssim \|[\Lambda^r\partial_x,u+\hat
u]v\|_{L^2}\|v(t)\|_{H^r}+\|\partial_x(u+\hat
u)\|_{L^\infty}\|v(t)\|_{H^r}^2 \\
&\lesssim (\|u+\hat u\|_{H^s}+\|\partial_x(u+\hat
u)\|_{L^\infty})\|v(t)\|_{H^r}^2 \\
&\lesssim (\|u+\hat u\|_{H^s})\|v(t)\|_{H^r}^2,
\end{aligned}  \label{3.4}
\end{equation}
where we have used the facts that
$H^{\frac{1}{2}+}\hookrightarrow L^\infty$ and $s>3/2$.
It is easy to show that
\begin{equation}
\begin{aligned}
&\big|-b\int_\mathbb{S}\Lambda^r\partial_xA^{-1}[\mu(u)v+\mu(v)\hat
u]\cdot \Lambda^rv{\rm d}x\big| \\
&\lesssim \|\partial_xA^{-1}[\mu(u)v+\mu(v)\hat u]\|_{H^r}\cdot\|v(t)\|_{H^r}.
\end{aligned}   \label{3.5}
\end{equation}
By \eqref{2.2} and \eqref{2.3}, we have
\begin{align*}
 \|\partial_xA^{-1}u\|_{H^r}
&= \big\|\big(x-\frac 12\big)\int_0^1u(x){\rm d}x
 -\int_0^xu(y){\rm d}y+\int_0^1\int_0^xu(y){\rm d}y{\rm d}x\big\|_{H^r}\\
&\lesssim \|x-\frac 12\|_{H^r}\int_0^1|u(x)|{\rm d}x
 +\|u(t)\|_{H^{r-1}} +\int_0^1\int_0^x|u(y)|{\rm d}y{\rm  d}x.
\end{align*}
Using the above inequality, we have
\begin{equation}
\begin{aligned}
&\|\partial_xA^{-1}[\mu(u)v+\mu(v)\hat u]\|_{H^r} \\
&\lesssim |\mu(u)|\Big(\|x-\frac 12\|_{H^r}\int_0^1|v(x)|{\rm d}x
 +\|v(t)\|_{H^{r-1}} +\int_0^1\int_0^x|v(y)|{\rm d}y{\rm
 d}x\Big) \\
 &\quad +|\mu(v)|\Big(\|x-\frac 12\|_{H^r}\int_0^1|\hat u(x)|{\rm d}x
 +\|\hat u(t)\|_{H^{r-1}} +\int_0^1\int_0^x|\hat u(y)|{\rm d}y{\rm
 d}x\Big) \\
 &\lesssim (\|u\|_{H^s}+\|\hat u\|_{H^s})\|v(t)\|_{H^r},
\end{aligned}    \label{3.6}
\end{equation}
where we have used the  inequality
   \[
|\mu(v)|=\big|\int_\mathbb{S}v(x,t){\rm
d}x\big|\leq\int_\mathbb{S}|v(x,t)|{\rm d}x\leq\|v(t)\|_{H^r}
   \]
provided that $r\geq0$. Substituting \eqref{3.6} into
\eqref{3.5}, we obtain
   \begin{equation}
\big|-b\int_\mathbb{S}\Lambda^r\partial_xA^{-1}\left[\mu(u)v+\mu(v)\hat
u\right]\cdot \Lambda^rv{\rm
d}x\big|\lesssim(\|u\|_{H^s}+\|w\|_{H^s})\|v(t)\|_{H^r}^2.
   \label{3.7}
\end{equation}
Similarly, integrating by parts, we have
\begin{equation}
\begin{aligned}
&\big|\frac{1}{2}\int_\mathbb{S}\Lambda^{r}\partial_xA^{-1}\left(
 \sigma(\rho+\hat\rho)\right)\cdot
\Lambda^rv{\rm d}x\big| \\
&\lesssim \|\partial_xA^{-1}\sigma(\rho+\hat\rho)\|_{H^r}\cdot\|v(t)\|_{H^r} \\
&\lesssim \|\sigma(t)\|_{L^2}(\|\rho\|_{H^1}
 +\|\hat\rho\|_{H^1})\cdot\|v(t)\|_{H^r} \\
&\lesssim (\|\rho\|_{H^{s-1}}+\|\hat\rho\|_{H^{s-1}})\cdot(\|v(t)\|_{H^r}^2
 +\|\sigma(t)\|_{H^{r-1}}^2);
\end{aligned}   \label{3.8}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\big|-\frac{3-b}{2}\int_\mathbb{S}\Lambda^r\partial_xA^{-1}v_x(u+\hat
u)_x\cdot
\Lambda^rv{\rm d}x\big| \\
&\lesssim \|\partial_xA^{-1}v_x(u+\hat u)_x\|_{H^r}\cdot\|v(t)\|_{H^r} \\
&\lesssim \|v(t)\|_{L^2}(\|u\|_{H^2}+\|\hat u\|_{H^2})\cdot\|v(t)\|_{H^r} \\
&\lesssim (\|u\|_{H^s}+\|\hat u\|_{H^s})\cdot\|v(t)\|_{H^r}^2
\end{aligned}   \label{3.9}
\end{equation}
provided that $s\geq2$. In the above inequality, we used
   \[
\big|\int_{\mathbb{S}}v_x(x,t)u_x(x,t){\rm d}x\big|
=\big|\int_{\mathbb{S}}v(x,t)u_{xx}(x,t){\rm d}x\big|
\leq\|v(t)\|_{L^2}\|u\|_{H^2}.
   \]
It follows from Lemma \ref{l3.1} that
\begin{equation}
\begin{aligned}
&\big|\int_\mathbb{S}\Lambda^r (v\rho+\sigma\hat u)_x\cdot
  \Lambda^r\sigma{\rm d}x\big| \\
&\leq \|v\rho\|_{H^r}\|v(t)\|_{H^r}+\|[\partial_x\Lambda^{r-1},\hat u]
 \sigma\|_{L^2}\|\sigma(t)\|_{H^{r-1}}+
 \|\hat u_x\|_{L^\infty}\|\sigma(t)\|_{H^{r-1}}^2 \\
&\lesssim (\|\hat  u\|_{H^s}+\|\rho\|_{H^s})(\|v(t)\|_{H^r}^2
 +\|\sigma(t)\|_{H^{r-1}}^2),
\end{aligned}  \label{3.10}
\end{equation}
where we used the fact $H^r\hookrightarrow H^s$ $(r\leq s)$
again.
\smallskip

\noindent{\bf Lipschitz continuous $\Omega_1$.} 
Substituting \eqref{3.4}-\eqref{3.9} and \eqref{3.10} into \eqref{3.2} and
\eqref{3.3}, respectively, and adding the resulting equalities,
we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\left(\|v(t)\|_{H^r}^2+\|\sigma(t)\|_{H^{r-1}}^2\right)\\
&\lesssim(\|u\|_{H^s}+\|\hat
 u\|_{H^s}+\|\rho\|_{H^{s-1}}+\|\hat\rho\|_{H^{s-1}})(\|v(t)\|_{H^r}^2
+\|\sigma(t)\|_{H^{r-1}}^2).
\end{align*}
It follows from Proposition \ref{p2.2} that
\begin{align*}
&\|u\|_{H^s}+\|\hat
 u\|_{H^s}+\|\rho\|_{H^{s-1}}+\|\hat\rho\|_{H^{s-1}}\\
&\lesssim \|u(0)\|_{H^s}+\|\hat
 u(0)\|_{H^s}+\|\rho(0)\|_{H^{s-1}}+\|\hat\rho(0)\|_{H^{s-1}}\lesssim1
\end{align*}
since $u_0,\rho_0,\,\hat u_0,\hat\rho_0\in B(0,h)$.
Consequently, we obtain
   \[
\frac{1}{2}\frac{d}{dt}\|z(t)\|_{H^r\times H^{r-1}}^2\lesssim
c\|z(t)\|_{H^r\times H^{r-1}}^2,
   \]
which implies that
\begin{equation}
\|z(t)\|_{H^r\times H^{r-1}}\leq e^{cT_0}\|z(0)\|_{H^r\times H^{r-1}}.
  \label{3.11}
\end{equation}
Or equivalently
\begin{equation}
\|u(t)-\hat u(t)\|_{H^r}+\|\rho(t)-\hat \rho(t)\|_{H^{r-1}}\leq
e^{cT_0}(\|u(0)-\hat u(0)\|_{H^r}+\|\rho(0)-\hat
\rho(0)\|_{H^{r-1}}).
 \label{3.12}
\end{equation}
In the beginning of section 3, we obtain that
$\|u(0)-\hat u(0)\|_{H^r}>0$ and $\|\rho(0)-\hat \rho(0)\|_{H^r}>0$. Indeed,
if $\|u(0)-\hat u(0)\|_{H^r}=0$ or $\|\rho(0)-\hat
\rho(0)\|_{H^r}=0$, it follows from the Sobolev embedding
Theorem and Proposition \ref{p2.1} that $u(x,t)\equiv \hat
u(x,t)$ or $\rho(x,t)\equiv \hat \rho(x,t)$, respectively. Thus
we can assume that
   \[
\|u(0)-\hat u(0)\|_{H^r}=O(\|\rho(0)-\hat \rho(0)\|_{H^{r-1}}).
   \]
By \eqref{3.11}, we have
   \[
\|u(t)-\hat u(t)\|_{H^r}\leq C(\|u(0)-\hat u(0)\|_{H^r}),
   \]
which is the desired Lipschitz continuity in $\Omega_1$.
\smallskip

\noindent\textbf{H\"older continuous in $\Omega_2$.} 
Since $s-1<r<s$, by
interpolating between $H^{s-1}$ and $H^s$ norms, we obtain
   \[
\|z(t)\|_{H^r\times H^{r-1}}\leq\|z(t)\|_{H^{s-1}
\times H^{s-2}}^{s-r}\|z(t)\|_{H^s\times H^{s-1}}^{r-s+1}.
   \]
Moreover, from the Proposition \ref{p2.2}, we have that
  \[
\|z(t)\|_{H^s\times H^{s-1}}\lesssim\|u_0\|_{H^s}+\|\hat
u_0\|_{H^s}+\|\rho_0\|_{H^{s-1}}+\|\hat
\rho_0\|_{H^{s-1}}\lesssim h,
   \]
and thus we have
 \begin{equation}
\|z(t)\|_{H^r\times H^{r-1}}\lesssim\|z(t)\|_{H^{s-1}\times H^{s-2}}^{s-r}.
   \label{3.13}
\end{equation}
We see that \eqref{3.11} is valid for $r=s-1$, $s>5/2$.
Therefore, applying \eqref{3.11} into \eqref{3.13}, we obtain
  \[
\|z(t)\|_{H^r\times H^{r-1}}\lesssim\|z(0)\|_{H^{s-1}\times H^{s-2}}^{s-r},
  \]
which is the desired H\"older continuity (similar to the
discussion in $\Omega_1$). The proof of Theorem \ref{t2.1} is
completed.
\end{proof}


\subsection*{Acknowledgments} 
This research was supported in part by PRC Grants NSFC 11301146 and 11226168.


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 \end{document}