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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 63, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/63\hfil Global smooth solutions]
{Global smooth solutions of the spin polarized transport equation}

\author[B. Guo, X. Pu\hfil EJDE-2008/63\hfilneg]
{Boling Guo, Xueke Pu}  % in alphabetical order

\address{Boling Guo \newline
Institute of Applied Physics and Computational Mathematics \\
P. O. Box 8009, Beijing, 100088, China}
\email{gbl@iapcm.ac.cn}

\address{Xueke Pu \newline
Graduate School of China Academy of Engineering Physics \\
P. O. Box 2101, Beijing,  100088, China}
\email{xuekepu@tom.com}

\thanks{Submitted July 4, 2007. Published April 27, 2008.}
\subjclass[2000]{35K15, 35K20}
\keywords{Spin-polarized transport equations; global smooth solutions;
\hfill\break\indent Landau-Lifshitz equations}

\begin{abstract}
 In this paper, we prove the existence of global smooth solutions
 of the spin-polarized transport equation using energy estimates
 method in dimension two.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 It is well known that the Landau-Lifshitz equation has a fundamental
importance in understanding the ferromagnetism in materials, see \cite{Landau}.
It reads as
\begin{equation*}
m_t=m\times \Delta m-\mu m\times (m\times \Delta m),
\end{equation*}
where $m\in \mathbb{S}^2$ is the magnetization field and $\mu>0$ is the
Gilbert damping coefficients. There is a long list of work
contributed to this equation regarding its existence, uniqueness,
self-similar solutions, blowup and partially regularities, the
interested reader can refer to
\cite{AlougesSoyeur,DingGuo,GuoHong,GH,Landau,Mo} for more details.
Mathematically, it is closely related to harmonic maps to the
sphere, see \cite{H,GuoHong}. There are also several equations
closely related to Landau-Lifshitz equation, such as
Landau-Lifshitz-Maxwell equation \cite{DingGuoLinZeng}, and the
equations in ferrimagnetic materials \cite{GuoPu} and so on. Many
authors also take the spin polarize into account, to consider the
spin polarized current-driven magnetization in materials, see
\cite{Shpiro,Zhang}. Recently, C.J. Garcia-Cervera and X.P.
Wang\cite{GarciaWang} considered the weak solutions of the following
spin-polarized transport equations in materials
\begin{equation} \label{equ1}
\begin{gathered}
\frac{\partial s}{\partial
t}=-\mathop{\rm div}J_s-D_0(x)s-D_0(x)s\times m\\
\frac{\partial m}{\partial t}=-m\times(h+s)+\alpha
m\times\frac{\partial m}{\partial t}\\
s(x,0)=s_0(x),\quad m(x,0)=m_0(x).
\end{gathered}
\end{equation}
In this equation, $(s,m)$ is the unknown, $s=(s_1,s_2,s_3):
\Omega\to \mathbb{R}^3$ denotes the spin accumulation and $m=(m_1,m_2,m_3):
\Omega\to \mathbb{S}^2$ is the magnetization field, where $\mathbb{S}^2$
is the unit sphere in $\mathbb{R}^3$. $J_s$ is the spin current
\begin{equation} \label{e1.2}
J_s=m\otimes J_e-D_0(x)[\nabla s-\beta m\otimes(\nabla s\cdot m)],
\end{equation}
where $J_e$ is the applied electronic current, $0<\beta<1$ is the
spin polarized parameter and $D_0(x)$ is the diffusion parameter
depending on the material. In the second equation of \eqref{equ1},
$h=-\nabla_m\Phi+\Delta m+h_d$ denotes the anisotropy, exchange and
self-induced energy respectively.

As can be seen, this equation is closely related to the
Landau-Lifshitz equation in the magnetization field $m$ and to
quasilinear parabolic equations in the spin accumulation $s$. It has
great importance physically and interesting mathematical structures
as well. In \cite{GarciaWang}, the authors considered the global
weak solutions of this equation and obtained interesting results by
Galerkin's approximating method. A natural question is whether or
not this system has global smooth solutions. Taking into account the
results in \cite{GuoHong}, in particular Theorem 2.6, it is not
right in general by technical reasons. However, we can show that
under some additional conditions on the initial data, this system
admits global smooth solutions. For simplicity, we consider the
periodic boundary conditions and assume that the diffusion material
$D_0(x)$ is constant, which is set to be $D_0(x)\equiv 1$ in this
paper. Furthermore, we neglect to consider the anisotropic energy
$\nabla_m\Phi$ and self-induced energy $h_d$. Under these
assumptions and simplifications, we show that there exists a global
smooth solution for this system under smallness condition
\eqref{equ78} in space dimension two. Our main result is the
following theorem.

\begin{theorem} \label{thm1}
Let $k\geq 2$, $J_e\in H^k(R^+\times\Omega)$, $0<\beta<1$ and
$s_0\in H^k(\Omega)$, $m_0\in H^k(\Omega)$. Then there exists a
positive constant $\lambda_0>0$, such that if the smallness
condition
\begin{equation}\label{equ78}
\|s_0\|^2_{L^2}+\|J_e\|^2_{L^2(R^+\times \Omega)}+\|\nabla
m_0\|^2_{L^2}<\lambda_0
\end{equation}
holds, then there exists a global solution $(s,m)$ of \eqref{equ1}
in $H^k(\Omega)$ satisfying
\[
\partial_t^j\partial_x^{\alpha}\in L^{\infty}([0,T];L^2{\Omega});\quad
\partial_t^k\partial_x^{\beta}\in L^{2}([0,T];L^2{\Omega}),
\]
with $2j+|\alpha|\leq k$ and $2k+|\beta|\leq k+1$. In particular, if
the initial data are smooth, the solution is globally smooth.
\end{theorem}

In the sequel, we  denote by $\|\cdot\|_X$ the norm of functions on
space $X$. In particular, if $X=L^2(\Omega)$, we simply denote
$\|\cdot\|$ instead of $\|\cdot\|_{L^2(\Omega)}$.

This paper is organized as follows. In the next Section, we show
that there exists a local solution in Sobolev spaces $W^{2,p}$ for
$p>2$. By standard bootstrap method, this solution is indeed
smooth. In Section 3, we give some a priori global estimates of
the solution in $H^2$. Together with the local theory, we show
that the system has a global $H^2$ solution provided that the
initial data are in $H^2$, and small in the sense of
\eqref{equ78}. Finally in Section 4, we give some \emph{a priori}
estimates in $H^k$ to deduce that the solution remains as smooth
as the initial data for all time if the initial data are small in
the sense of \eqref{equ78}. In particular, if the solution is
smooth, we get a global smooth solution for the system under
consideration.

\section{Local smooth solutions}

 In this section, we
show that there exists a local smooth solution for the system we
considered in this paper. For this purpose, we rewrite
\[
\mathop{\rm div}(m\otimes (\nabla s\cdot m))=(\Delta s\cdot
m)m+\widetilde{\mathop{\rm div}}(m\otimes (\nabla s\cdot m)),
\]
where $\widetilde{\mathop{\rm div}}(m\otimes (\nabla s\cdot m))=\nabla
m\cdot(\nabla s\cdot m)+(\nabla s\cdot \nabla m)m$. For the rest
of this paper, we set
\begin{equation*}
A=A(m)=\begin{pmatrix} 1-\beta m_1^2&-\beta m_1m_2&-\beta
m_1m_3\\-\beta m_2m_1&1-\beta m_2^2&-\beta m_2m_3\\
-\beta m_3m_1&-\beta m_3m_2&1-\beta m_3^2
\end{pmatrix}.
\end{equation*}
Since $0<\beta<1$, we know there two exists positive numbers
$\lambda,\Lambda>0$, such that
\begin{equation}\label{posA}
\lambda|\xi|^2\leq \xi A(m) \xi^T\leq \Lambda|\xi|^2,\quad \forall
\xi\in  \mathbb{R}^3,\ \xi\neq0.
\end{equation}
On the other hand, since $m\in \mathbb{S}^2$, we can write the equation
for $m$ in its equivalent form
\begin{equation}\label{equ11}
(1+\alpha^2)\frac{\partial m}{\partial t}=-m\times(\Delta
m+s)-\alpha m\times(m\times(\Delta m+s)).
\end{equation}
Using this notation, system \eqref{equ1} can be rewritten as
\begin{equation} \label{equ79}
\begin{gathered}
\frac{\partial s}{\partial t}-A(m)\Delta s+s=-\mathop{\rm div}(m\otimes
J_e)-\beta\widetilde{\mathop{\rm div}}(m\otimes(\nabla s\cdot
m))-s\times m\\
(1+\alpha^2)\frac{\partial m}{\partial t}-\alpha\Delta
m=-m\times\Delta m -m\times s+\alpha|\nabla m|^2m-\alpha
m\times(m\times s).
\end{gathered}
\end{equation}
Obviously, this system is strongly parabolic
\cite{Fredman,GuoHong}.

In the following, we use Hamilton's idea  in \cite{H} to derive
the local existence of the solution of \eqref{equ1} with smooth
initial data $(s_0,m_0)$. As in \cite{GuoHong}, we prove the following
result.

\begin{lemma} \label{lem2.1}
Let $1<p<\infty$. For every function $g_1, g_2\in
L^p(\Omega\times[\gamma,\omega])$, there exists a unique solution
$(l,k)$ of
\[
\begin{pmatrix}\partial_tl\\ (1+\alpha^2)\partial_tk
\end{pmatrix}=\begin{pmatrix}A(m)\Delta l+d(s,m)k+e(s,m,J_e)\nabla
k+f(m)l+h(m)\nabla l+ g_1
\\ \alpha\Delta k+m\times\Delta k+a(s,m)\nabla k+b(s,m)k+c(m)l+g_2
\end{pmatrix}
\]
on $\Omega\times[\gamma,\omega]$, where $a,b,c,d,e,f,s,m$ are given
smooth functions.
\end{lemma}

\begin{proof}
Denote
\begin{gather*}
H\begin{pmatrix} l\\ k
\end{pmatrix}=\begin{pmatrix}
\partial_tl-A(m)\Delta l\\
(1+\alpha^2)\partial_tk-\alpha\Delta k-m\times\Delta k
\end{pmatrix},
\\
K\begin{pmatrix}l\\ k
\end{pmatrix}=
\begin{pmatrix}d(s)k+e(s,m,J_e)\nabla k+f(m)l+h(m)\nabla l
\\a(s,m)\nabla k+b(s,m)k+c(m)l
\end{pmatrix}.
\end{gather*}
Since $H$ is a strong parabolic linear operator, the map
$(l,k)\to H(l,k)$ is an isomorphism from
$W^{2,p}(\Omega\times[\gamma,\omega])$ onto
$L^{p}(\Omega\times[\gamma,\omega])$. On the other hand,
$K:W^{2,p}(\Omega\times[\gamma,\omega])\to
L^{p}(\Omega\times[\gamma,\omega])$ is compact. By the Fredholm
theory, the indices of the map $(H-K)$ is 0 by definition of the
indices of a Fredholm operator. Thus it suffices to show that its
kernel is zero to deduce that it is an isomorphism.

Let $(l,k)\in W^{2,p}(\Omega\times [\gamma,\omega])$,
and $f|_{\Omega\times\gamma}=0$.
Since $(H-K)((l,k)^T)=0$, taking the
scalar product of this equation with $(l,k)$, we have respectively
\begin{align*}
\frac12\frac{d}{dt}\int_{\Omega}|l|^2+\lambda \int_{\Omega}|\nabla
l|^2
\leq &\int_{\Omega}|\nabla A(m)||\nabla
l||l|+C_1\int_{\Omega}|\nabla l||l|\\
& +C_2\int_{\Omega}|\nabla
k||l|+C_3\int_{\Omega}|k||l|+C_4\int_{\Omega}|l|^2\\
\leq & \frac\lambda4\int_{\Omega}|\nabla
l|^2+\frac\alpha4\int_{\Omega}|\nabla
k|^2+C_5\int_{\Omega}|l|^2+|k|^2
\end{align*}
and similarly,
\begin{align*}
(1+\alpha^2)\frac12\frac{d}{dt}\int_{\Omega}|k|^2+\alpha
\int_{\Omega}|\nabla k|^2
\leq &\int_{\Omega}k\cdot(m\times \Delta k)
 +C_6\int_{\Omega}|\nabla k||k|\\
&+C_7\int_{\Omega}|k||l|+C_8\int_{\Omega}|k|^2\\
\leq & \frac\lambda4\int_{\Omega}|\nabla
 l|^2+\frac\alpha4\int_{\Omega}|\nabla
 k|^2+C_9\int_{\Omega}|l|^2+|k|^2.
\end{align*}
In the last inequality, we used the fact that
$$
\int_{\Omega} k
\cdot (m\times \Delta k)=-\int_{\Omega}k\cdot \nabla m\times \nabla k.
$$
Adding the two inequalities together we obtain a Gronwall type
inequality
\begin{align*}
\frac{d}{dt}\int|l|^2+(1+\alpha^2)\int|k|^2+\lambda\int|\nabla
l|^2+\alpha\int|\nabla k|^2\leq C\int|l|^2+|k|^2.
\end{align*}
Hence by Gronwall inequality, we have $(l,k)=0$, which shows that
the kernel of the map $(H-K)$ is zero and finishes the proof the
lemma.
\end{proof}



\begin{lemma} \label{lem2.2}
Let $(s_0,m_0)$ be a smooth map. Then there exists a positive number
$\varepsilon>0$ and a unique local solution $(s,m)\in
W^{2,p}(\Omega\times[0,\varepsilon])$ for $p>2$ strictly.
\end{lemma}

\begin{proof}
Define a nonlinear map $\mathcal{L}:
W^{2,p}(\Omega\times[0,\omega])\to
L^p({\Omega\times[0,\omega]})$ as follows:
\begin{equation*} %2.4
\mathcal{L}\begin{pmatrix}s\\ m
\end{pmatrix}
=\begin{pmatrix}A(m)\Delta s-\mathop{\rm div}{(m\otimes
J_e)}-\beta\widetilde{\mathop{\rm div}}(m\otimes(\nabla s\cdot
m))-s-s\times m
\\\alpha\Delta m+\alpha|\nabla m|^2m-\alpha
m\times(m\times s)-m\times(\Delta m+s)
\end{pmatrix}.
\end{equation*}
Then for a smooth map $(s,m)$, the derivative of $L$ at $(s,m)$ is
given by
\begin{equation*}
D\mathcal{L}(s,m)\begin{pmatrix}l\\ k
\end{pmatrix}
=\begin{pmatrix}A(m)\Delta l+d(s,m)k+e(s,m,J_e)\nabla
k+f(m)l+h(m)\nabla l \\
\alpha\Delta k+m\times\Delta k+a(s,m)\nabla k+b(s,m)k+c(m)l
\end{pmatrix},
\end{equation*}
where $a,b,c,d,e,f$ are smooth matrix-valued functions.
Denote
\begin{align*}
\mathcal{R}\begin{pmatrix}s\\ m
\end{pmatrix}=\begin{pmatrix}\partial_ts\\
 (1+\alpha^2)\partial_tm \end{pmatrix}
-\mathcal{L}\begin{pmatrix}s\\
 m \end{pmatrix}.
\end{align*}
Let $(s_{\omega}, m_{\omega}): \Omega\times[0,\omega] \to \mathbb{R}^3$ be
smooth maps with $(s_{\omega}, m_{\omega})=(s_0,m_0)$ on
$\Omega\times 0$, we denote $(s,m)=(s_{\omega},
m_{\omega})+(s_*,m_*)$. The derivative of $\mathcal{R}$ at $(s,m)$
is given by
\begin{equation}\label{equ15}
D\mathcal{R}(s,m)\begin{pmatrix}l\\ k
\end{pmatrix}=\begin{pmatrix}\partial_tl\\ (1+\alpha^2)\partial_tk
\end{pmatrix}-D\mathcal{L}(s,m)\begin{pmatrix}l\\ k
\end{pmatrix}.
\end{equation}
Consider $(s_*,m_*)$ as a variable function, then
$(s_*,m_*)\to \mathcal{R}\begin{pmatrix}s_{\omega}+s_*\\ m_{\omega}+m_*
\end{pmatrix}$ defines a continuously differential map of
$W^{2,p}(\Omega\times[0,\omega])\to L^p(\Omega\times[0,\omega])$.
Its derivative at $[(s_*,m_*)=0]$ is given by \eqref{equ15} with
$(s,m)$ replaced by $(s_{\omega}, m_{\omega})$ and from the above
lemma we know it is an isomorphism from
$W^{2,p}(\Omega\times[0,\omega])$ onto
$L^p(\Omega\times[0,\omega])$.

Therefore by the inverse function theorem, the set of all
$\mathcal{R}\begin{pmatrix}s_{\omega}+s_*\\ m_{\omega}+m_*
\end{pmatrix}$ for $(s_*,m_*)$ in a neighborhood $\mathscr{N}\subset W^{2,p}(\Omega\times [0,\omega])$ of 0 covers a
neighborhood $\mathscr{O}$ of $\mathcal{R}(s_{\omega}, m_{\omega})$
in $L^p(\Omega\times[0,\omega])$. If we choose $\varepsilon>0$ small
enough, the function
\[
(\tilde{l},\tilde{k})=
\begin{cases}
(0,0), & 0\leq t\leq \varepsilon;\\
\mathcal{R}(s_{\omega}, m_{\omega}), &\varepsilon<t\leq
\omega,
\end{cases}
\]
will be in this neighborhood $\mathscr{O}$. So
$(s,m)|_{\Omega\times{[0,\varepsilon]}}$ solves the equation
\eqref{equ1}. On the other hand, from our choice $p>2$ and $m\in
\mathbb{S}^2$, we can show the smoothness of the solution by standard
bootstrap method. We stress that $p>2$ here plays an important role
in the bootstrap method, because for $p>2$, we have
$W^{k,p}(\Omega)\hookrightarrow W^{k-1,\infty}(\Omega)$ in space
dimension 2.
\end{proof}

\section{ A priori estimates}

 In this section, we
show some \emph{a priori} estimates for the solution. The following
Lemma can be found in \cite{GarciaWang} and we omit its proof here.

\begin{lemma}\label{lem1}
Let $J_e\in(H^1(R^+\times\Omega))^3$ and
$s(x,t)\in L^{\infty}(R^+,L^2(\Omega))$ and
$m(x,t)\in L^{\infty}(R^+,H^1(\Omega))$ be a weak solution to the problem
\eqref{equ1}. Then the solution $(s,m)$ satisfies the following
relations:
\begin{gather*}
m\in L^{\infty}(R^{+},H^1(\Omega)),\quad
\frac{\partial m}{\partial t}\in L^2(R^{+},L^2(\Omega)), \\
s\in L^{\infty}(R^{+},L^2(\Omega))\cap L^{2}(R^{+},H^1(\Omega)),\quad
\frac{\partial s}{\partial t}\in L^2(R^{+},H^{-1}(\Omega))
\end{gather*}
with the following estimates: For any $T>0$,
\begin{equation}\label{equ9}
\sup_{0\leq t\leq
T}\int_{\Omega}|s(x,t)|^2+\int^T_0\int_{\Omega}|s|^2+\int^T_0\int_{\Omega}|\nabla
s|^2\leq C\int_{\Omega}|s_0|^2+C\int^T_0\int_{\Omega}|J_e|^2,
\end{equation}
and
\begin{equation}\label{equ10}
\int^T_0\int_{\Omega}|\frac{\partial m}{\partial t}|^2+\sup_{0\leq
t\leq T}\int_{\Omega}|\nabla m(x,t)|^2\leq
C\int_{\Omega}|s_0|^2+C\int^T_0\int_{\Omega}|J_e|^2+C\int_{\Omega}|\nabla
m_0|^2,
\end{equation}
where $C$ only depends on the coefficients $(\alpha, \beta)$ of
the equation \eqref{equ1}, in particular, $C$ is independent of
$T>0$.
\end{lemma}

\begin{remark} \label{rmk1} \rm
Because of the highly nonlinear terms and that the embedding
$H^1(\Omega)\hookrightarrow L^{\infty}(\Omega)$ fails for dimension
greater than one, we cannot expect more regularity than that stated
in Lemma \ref{lem1}. One can also see from \eqref{equ10} that if the
initial data are sufficiently small
\begin{equation}
C\|s_0\|^2+C\|J_e\|^2_{L^2(R^+\times \Omega)}+C\|\nabla
m_0\|^2<\lambda_0,  \tag{\ref{equ78}}
\end{equation}
one can keep $\|\nabla m(\cdot, t)\|^2<\lambda_0$ for all time
$t>0$.
\end{remark}

Below we will give the $H^2$ \emph{a priori} estimates. We will use
the following equivalent form for $m$,
\[
(1+\alpha^2)\frac{\partial m}{\partial t}=-m\times(\Delta
m+s)-\alpha m\times(m\times(\Delta m+s)).
\]

\begin{lemma}\label{lem6}
Assume that $J_e\in H^2(R^+\times\Omega)$, $s_0\in H^2(\Omega)$ and
$m_0\in H^2(\Omega)$. Then the smooth solution $(s,m)$ satisfies
\begin{equation}
\nabla^2 m\in L^{\infty}(R^+,L^2)\quad  \text{and}\quad
\nabla^3m\in L^2(R^+,L^2),
\end{equation}
provided the smallness condition \eqref{equ78} holds.
Further more, one can get $\nabla m\in L^{2}(R^+,L^{\infty})$ by
sobolev embedding $H^2\hookrightarrow L^{\infty}$ in dimension 2.
\end{lemma}

\begin{proof}
Denote $D^{\alpha}=\partial^{\alpha_1}_{x_1}\partial^{\alpha_2}_{x_2}$,
where $\alpha$ is a multi-index, $\alpha_1,\ \alpha_2$ are
nonnegative integers and $|\alpha|=\alpha_1+\alpha_2$. For
simplicity, we will use $D^k$ to denote any kind of the differential
operator $D^{\alpha}$ with $|\alpha|=k$.

Differentiating equation \eqref{equ11} with $D^2$, taking inner
product with $D^2u$ and integrating over $\Omega$, we have
\begin{equation} \label{equ32}
\begin{aligned}
\frac{1+\alpha^2}{2}\frac{d}{dt}\|D^2m\|^2
&=-\int_{\Omega}\langle D^2(m\times(\Delta m+s)),D^2m\rangle \\
&\quad +\alpha\int_{\Omega}\langle \Delta
D^2m,D^2m\rangle +\alpha\int_{\Omega}\langle D^2(|\nabla
m|^2m),D^2m\rangle \\
&\quad-\alpha\int_{\Omega}\langle D^2((m\cdot
s)m),D^2m\rangle +\alpha\int_{\Omega}\langle D^2s,D^2m\rangle \\
&=I+II+III+IV+V .
\end{aligned}
\end{equation}

\noindent $\bullet$  Estimates of term $I=:VI+VII$.
  \begin{align*}
  VI&=|\int_{\Omega}\langle D^2(m\times\nabla m),\nabla D^2m\rangle| \\
  &=|\int_{\Omega}\langle D^2m\times\nabla m+2Dm\times D\nabla
  m+m\times\nabla D^2m,\nabla D^2m\rangle | \\
  &\leq C\|\nabla m\|_{L^4(\Omega)}\|\nabla^2
  m\|_{L^4(\Omega)}\|\nabla^3 m\|.
  \end{align*}
  Using Gagliardo-Nirenberg inequality, we have (in dimension
  $d=2$),
  \begin{gather}
  \|\nabla m\|_{L^4}\leq C\|\nabla m\|_{L^2}^{\frac34}\|\nabla
  m\|_{H^2}^{\frac14} \label{GN1}\\
  \|D^2 m\|_{L^4}\leq C\|\nabla m\|_{L^2}^{\frac14}
\|\nabla m\|_{H^2}^{\frac34}\label{GN2}.
  \end{gather}
  Then writing the $H^2-$norm explicitly and using the $\varepsilon$-Young's
inequality, we can bound the term $VI$ by
\begin{equation} \label{equ37}
  \begin{aligned}
  |VI|&\leq \left(C\|\nabla m\|_{L^2}+\varepsilon\right)\|\nabla\Delta m\|_{L^2}^2 \\
  &\quad +C\left(\|\nabla   m\|_{L^2}+\|\nabla m\|_{L^2}^2\right)
\|\Delta m\|_{L^2}^2+C\|\nabla   m\|_{L^2}^4,
  \end{aligned}
\end{equation}
  where the constant $C$ can be different from line to line.
\begin{equation} \label{equ39}
  \begin{aligned}
  |VII|&=|\int_{\Omega}\langle D(m\times s),D^3m\rangle | \\
  &\leq   \varepsilon\|D^3m\|^2+C(\varepsilon)\|D s\|^2
 +C(\varepsilon)(\|Dm\|^4+\|s\|^4) \\
  &\leq\varepsilon\|D^3m\|^2+C(\varepsilon)(\|\nabla
  m\|^2\|\Delta m\|^2+\|s\|^2\|\nabla s\|^2\\
  &\quad  +\|\nabla s\|^2+\|s\|_{L^2}^4+\|\nabla m\|_{L^2}^4).
  \end{aligned}
\end{equation}
  Since $s,\nabla m\in L^{\infty}(0,T;L^2)$ and
$\nabla s, \Delta m \in L^2(0,T;L^2)$, the last term on the right
is integrable on $[0,T]$   for any $T>0$.

\noindent$\bullet$ Estimates of term II.
  Since we consider the periodic case, no boundary terms appears, thus
  \begin{equation}\label{equ34}
  \alpha\int_{\Omega}\langle \Delta D^2m,D^2m\rangle
=-\alpha\|\nabla D^2m\|^2.
  \end{equation}

\noindent$\bullet$  Estimates of term III.
  Since we have
  \begin{equation*}
  \|D(|\nabla m|^2m)\|\leq \|\nabla
  m\|^3_{L^6(\Omega)}+2\|m\|_{L^{\infty}(\Omega)}\|\nabla
  m\|_{L^4(\Omega)}\|D\nabla m\|_{L^4(\Omega)},
  \end{equation*}
  the term III can be estimated by
\begin{equation} \label{equ35}
\begin{aligned}
  |III|&=|\int_{\Omega}<D((|\nabla m|^2m)),D^3m>| \\
  &\leq \left(C\|\nabla m\|+\varepsilon\right)\|\nabla\Delta m\|^2
 +C\left(\|\nabla
  m\|_{L^2}+\|\nabla m\|_{L^2}^2\right)\|\Delta m\|_{L^2}^2\\
&\quad +C\|\nabla  m\|_{L^2}^4,
  \end{aligned}
\end{equation}
  thanks to the Gagliardo-Nirenberg inequality (GN).

\noindent$\bullet$   Estimates of the last two terms IV and V.
\begin{equation} \label{equ40}
  \begin{aligned}
  |IV|+|V|&\leq \varepsilon\|D^3m\|^2+C(\varepsilon)\left(\|D((m\cdot
  s)m)\|_{L^2}^2+\|Ds\|_{L^2}^2\right) \\
 &\leq \varepsilon\|D^3m\|^2+C(\varepsilon)(\|\nabla s\|_{L^2}^2+\|s\|_{L^2}^4+\|\nabla
  m\|_{L^2}^4) \\
 & \leq \varepsilon\|D^3m\|^2+C(\varepsilon)
 (\|\nabla s\|_{L^2}^2+\|s\|_{L^2}^4+\|s\|_{L^2}^2\|\nabla s\|_{L^2}^2 \\
  &\quad +\|\nabla m\|_{L^2}^4+\|\nabla m\|_{L^2}^2\|\nabla^2
  m\|_{L^2}^2),
  \end{aligned}
\end{equation}
  where in the last inequality, we used the Gagliardo-Nirenberg
  inequality (GN).
Summarizing \eqref{equ32}-\eqref{equ40}, we have
\begin{equation} \label{equ41}
\begin{aligned}
&\frac{1+\alpha^2}{2}\frac{d}{dt}\|\nabla^2m\|^2_{L^2(\Omega)}
 +\alpha\|\nabla^3m\|^2_{L^2(\Omega)}  \\
&\leq (C\|\nabla m\|_{L^2}+4\varepsilon)\|\nabla^3
m\|_{L^2}^2+C(\|\nabla m\|_{L^2}+\|\nabla
m\|_{L^2}^2)\|\nabla^2m\|_{L^2}^2 \\
&\quad +C(\|\nabla m\|_{L^2}^4+\|\nabla
s\|_{L^2}^2+\|s\|_{L^2}^4+\|s\|_{L^2}^2\|\nabla s\|_{L^2}^2).
\end{aligned}
\end{equation}
By Remark \ref{rmk1}, one can keep $\|\nabla m(t)\|^2_{L^2(\Omega)}\leq
\lambda_0$ for all time $t>0$ provided \eqref{equ78} holds. Thus
there exists a $\lambda_0>0$ such that if \eqref{equ78} holds, we
have
\begin{equation*}
C\|\nabla m\|_{L^2}<\frac{\alpha}{4}.
\end{equation*}
Furthermore, set $4\varepsilon<\frac{\alpha}{4}$, we then have
\begin{equation*}
C\|\nabla m\|_{L^2}+4\varepsilon<\frac{\alpha}{2}.
\end{equation*}
This together with \eqref{equ41} insures us to deduce the
Gronwall-type inequality for $\|\nabla^2m\|^2_{L^2(\Omega)}$:
\begin{equation} \label{equ42}
\begin{aligned}
&\frac{d}{dt}\|\nabla^2m\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|
\nabla^3m\|^2_{L^2(\Omega)}\\
&\leq C(\|\nabla m\|_{L^2}+\|\nabla
 m\|_{L^2}^2)\|\nabla^2m\|_{L^2}^2\\
&\quad +C(\|\nabla m\|_{L^2}^4+\|\nabla
s\|_{L^2}^2+\|s\|_{L^2}^4+\|s\|_{L^2}^2\|\nabla s\|_{L^2}^2).
\end{aligned}
\end{equation}
Since both the coefficient of $\|\nabla^2m\|^2_{L^2(\Omega)}$ and
the second term on the right hand side are integrable by Lemma
\ref{lem1}, we deduce by Gronwall inequality that for any $T>0$,
\begin{equation}\label{equ43}
\sup_{0\leq t\leq T}\|\nabla^2m\|^2_{L^2(\Omega)}\leq C,\quad
\forall T>0.
\end{equation}
Then integrating \eqref{equ42}, we have
\begin{equation}\label{equ44}
\|\nabla^3m\|^2_{L^2(0,T;L^2(\Omega))}\leq C,\quad \forall T>0,
\end{equation}
and by Sobolev embedding $H^2(\Omega)\hookrightarrow
L^{\infty}(\Omega)$, we get $\nabla m\in
L^2(0,T;L^{\infty}(\Omega))$.
\end{proof}

\begin{remark}\label{rmk2} \rm
It can be easily seen from this Lemma and the expression
\eqref{equ11}, that $\partial_t m\in
L^{\infty}(0,R^+;L^2(\Omega))$ for any positive times $T>0$.
Furthermore, differentiating equation \eqref{equ11} with respect
to spatial variable $x$, we have $\partial_t\nabla m\in
L^2(0,R^+;L^2(\Omega))$. This observation is rather simple, thus
we omit the details here.
\end{remark}

Below, we will focus ourselves on  the estimates for the
$s$ variable.

\begin{lemma}\label{lem4}
Under the conditions of lemma \ref{lem6}, we have
$$
\frac{\partial s}{\partial t}\in L^2(R^+,L^2(\Omega)),\quad
 s\in L^{\infty}(R^+,H^1(\Omega))
$$
with the estimation: for any $T>0$,
\begin{equation} \label{equ28}
\begin{aligned}
&\int^T_0\|\frac{\partial s}{\partial
t}\|^2_{L^2(\Omega)}+\sup_{0\leq t \leq T}\|s\|^2_{H^1(\Omega)}\\
&\leq Ce^{C\|\nabla
m\|^2_{L^2(0,T;L^{\infty})}}\Big(\|s_0\|^2_{H^1(\Omega)}
+\sup_{0\leq t\leq T}\|\nabla m\|^2_{L^2(\Omega)}\|\nabla
m\|^2_{L^2(0,T;L^{\infty}(\Omega))}\\
&\quad +\|\nabla J_e\|^2_{L^2([0,T]\times\Omega)}\Big).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Taking the inner product of $s$-equation with $\frac{\partial
s}{\partial t}$, and then integrating over $\Omega$, we get
\begin{equation} \label{equ18}
\int_{\Omega}|\frac{\partial s}{\partial
t}|^2+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|s|^2
=-\int_{\Omega}\langle \mathop{\rm div}J_s,\frac{\partial
s}{\partial t}\rangle
-\int_{\Omega}\langle s\times m,\frac{\partial s}{\partial t}\rangle
=:I+II.
\end{equation}
Since the second term on the right can be easily controlled by
\begin{equation}\label{equ19}
|II|\leq\varepsilon\int_{\Omega}|\frac{\partial s}{\partial
t}|^2+C(\varepsilon)\int_{\Omega}|s|^2,
\end{equation}
we need only to handle the first term $I$ on the right hand side
carefully. Using the notations introduced in Section 2, we can
rewrite $I$ as follows
\begin{equation}\label{equ21}
I=-\int_{\Omega}\langle \mathop{\rm div}(m\otimes J_e),
\frac{\partial s}{\partial t}\rangle
+\int_{\Omega}\langle A(m)\Delta s,\frac{\partial s}{\partial
t}\rangle -\beta\int_{\Omega}
\langle \widetilde{\mathop{\rm div}}(m\otimes (\nabla
s\cdot m)),\frac{\partial s}{\partial t}\rangle .
\end{equation}
The first and the last term on the right side of \eqref{equ21} can
be estimated by
\begin{equation}\label{equ22}
|\int_{\Omega}\langle \mathop{\rm div}(m\otimes J_e),
\frac{\partial s}{\partial t}\rangle |
\leq \varepsilon\int_{\Omega}|\frac{\partial s}{\partial
t}|^2+C(\varepsilon)\Big(|J_e|^2_{L^{\infty}(\Omega)}\int_{\Omega}|\nabla
m|^2+\int_{\Omega}|\nabla J_e|^2\Big),
\end{equation}
and
\begin{equation}\label{equ23}
|\int_{\Omega}\langle \widetilde{\mathop{\rm div}}(m\otimes (\nabla s\cdot
m)),\frac{\partial s}{\partial t}\rangle|
\leq \varepsilon\int_{\Omega}|\frac{\partial s}{\partial
t}|^2+C(\varepsilon)|\nabla m|^2_{L^{\infty}}\int_{\Omega}|\nabla
s|^2
\end{equation}
respectively, where $\nabla m\in L^{\infty}(0,T;L^2(\Omega))$ and
$J_e\in H^2(R^+\times\Omega)$.

In the following, we focus  on estimates of the second term
of \eqref{equ21}. For this purpose, write
\begin{equation*}
A\Delta s=A\nabla\cdot\nabla s=\nabla\cdot(A\nabla s)-(\nabla\cdot
A)\cdot\nabla s.
\end{equation*}
Then the second term on the right of \eqref{equ21} can be rewritten
as
\begin{align*}
\int_{\Omega}\langle A\Delta s,\frac{\partial s}{\partial
t}\rangle
&=-\frac{1}{2}\frac{d}{dt}\int_{\Omega}\langle A\nabla s, \nabla
s\rangle
-\frac{1}{2}\int_{\Omega}\langle (\partial_tA)\nabla s, \nabla s\rangle
-\int_{\Omega}\langle \nabla\cdot A\cdot\nabla
s,\partial_ts\rangle \\
&=:III+IV+V
\end{align*}

\noindent$\bullet$
Since $A$ is strictly positively definite, we can write $III$ as
  \begin{equation}
  III=-\frac{1}{2}\frac{d}{dt}\int_{\Omega}\langle A^{1/2}\nabla
  S,A^{1/2}\nabla S\rangle
  \end{equation}

\noindent$\bullet$  Estimates of term IV.
 From the positivity of $A$ in \eqref{posA}, we know that for any
vector $\xi\in\mathbb{R}^3$, we have $\lambda^2|\xi|^2\leq |A\xi|^2$.
This together with the $s$-equation in \eqref{equ79}
  \begin{equation}\label{equ29}
  A\Delta s=\frac{\partial s}{\partial t}+\mathop{\rm div}(m\otimes
  J_e)+\beta\widetilde{\mathop{\rm div}}(m\otimes(\nabla s\cdot
  m))+s+s\times m
  \end{equation}
  implies
  \begin{align*}
  \lambda^2\|\Delta s\|^2_{L^2}
&\leq  \|A\Delta s\|_{L^2}^2\\
&\leq \|\frac{\partial s}{\partial  t}\|^2_{L^2}+2\|s\|^2_{L^2}
 +C\|\nabla  m\|^2_{L^2}+\|\nabla
  J_e\|^2_{L^2}+C\|\nabla m\|^2_{L^{\infty}}\|\nabla
  s\|^2_{L^2}.
  \end{align*}
  On the other hand, by lemma \ref{lem1}, lemma \ref{lem6} and \eqref{equ11}, we can choose
  $\varepsilon_1$ small enough to satisfy
  $\varepsilon_1\|m_t\|_{L^2(\Omega)}<\frac18\lambda^2$ for all $t\geq0$.
  Therefore, we can estimate $IV$ by
  \begin{align*}
  |IV|&\leq \int_{\Omega}|m_t||\nabla s|^2\leq
  \|m_t\|_{L^2}\|\nabla s\|^2_{L^4} \\
  &\leq C\|m_t\|_{L^2}\|\nabla s\|_{H^1}\|\nabla
  s\|_{L^2} \\
  &\leq \varepsilon_1\|m_t\|_{L^2}\|\Delta
  s\|^2_{L^2}+C(\varepsilon_1)\|m_t\|_{L^2}\|\nabla
  s\|^2_{L^2}\\
  &\leq \frac{1}{8}\|\frac{\partial s}{\partial
  t}\|^2_{L^2}+\left(\|s\|^2_{L^2}+C\|\nabla m\|^2_{L^2}+\|\nabla
  J_e\|^2_{L^2}\right)\\
 &\quad +C\left(\|\nabla
  m\|^2_{L^{\infty}}+\|m_t\|_{L^2}\right)\|\nabla s\|^2_{L^2}.
  \end{align*}

\noindent$\bullet$  Estimates of term V.
  \begin{equation}\label{equ25}
  |V|\leq \varepsilon\|\frac{\partial s}{\partial
  t}\|_{L^2}^2+C(\varepsilon)\|\nabla m\|_{L^{\infty}}^2\|\nabla
  s \|_{L^2}^2.
  \end{equation}

Finally, since $A$ is positively definite and can be bounded from
above and below by
\begin{equation}
\sqrt{\lambda}\|\nabla s\|_X\leq\|A^{1/2}\nabla s\|_X\leq
\sqrt{\Lambda}\|\nabla s\|_X,
\end{equation}
we can regard $\|A^{1/2}\nabla s\|_X$ and $\|\nabla s\|_X$ as the
same in the sequel. From \eqref{equ18} and the above estimates, if
we set $\varepsilon$ small enough, say $\varepsilon<\frac{1}{8}$, we
can get the Gronwall inequality:
\begin{equation} \label{equ26}
\begin{aligned}
&\frac{1}{2}\int_{\Omega}|\frac{\partial s}{\partial
t}|^2+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|s|^2+\frac{d}{dt}\int_{\Omega}|\nabla
s|^2 \\
&\leq  C\left(\|\nabla
m\|^2_{L^{\infty}}+\|m_t\|_{L^2}+\|m_t\|_{L^2}\|\nabla
m\|^2_{L^{\infty}}\right)\|\nabla
s\|_{L^2}^2 \\
&\quad +C\|s\|_{L^2}^2+C \Big(\|J_e\|^2_{L^{\infty}}\|\nabla m\|^2_{L^2}
+\|\nabla J_e\|^2_{L^2}+\|m_t\|_{L^2}\|s\|^2_{L^2}\\
&\quad +\|m_t\|_{L^2}\|\nabla
m\|^2_{L^2}\Big).
\end{aligned}
\end{equation}
By the previous lemmas, $|\nabla m|\in L^2(0,T;L^{\infty})$,
$\nabla m\in L^{\infty}(0,T;L^2)$ and
$|m_t|\in L^{\infty}_tL^2_x\cap L^2_tL^2_x$, thus both the coefficients
before $\|s\|_{L^2}^2$ and $\|\nabla s\|_{L^2}^2$ and the last
term are integrable in time. Then the Gronwall inequality
immediately implies
\begin{equation}\label{equ27}
\frac{\partial s}{\partial t}\in L^2(R^+,L^2),\quad
 s\in L^{\infty}(R^+,H^1(\Omega)),
\end{equation}
with the estimates \eqref{equ28}. This concludes the proof.
\end{proof}

\begin{lemma}\label{lem5}
Under the conditions of Lemma \ref{lem6}, we have $s\in
L^2(R^+,H^2(\Omega))$. Indeed, by Lemma \ref{lem1} and Lemma
\ref{lem4}, we get that the right hand side of \eqref{equ29} belongs
to $L^2(\Omega)$ for a.e. $0\leq t\leq T$, which implies that
$\Delta s\in L^2{(\Omega)}$ for a.e. $0\leq t\leq T$. On the other
hand
\begin{align*}
\|\widetilde{\mathop{\rm div}}(m\otimes(\nabla s\cdot
m))\|^2_{L^2(0,T;L^2)}&\leq C\int^T_0\|\nabla
m\|^2_{L^{\infty}}\|\nabla s\|^2_{L^2}dt \\
&\leq C\|\nabla m\|^2_{L^2(0,T;L^{\infty})}\sup_{0\leq t\leq
T}\|\nabla s\|^2_{L^2}.
\end{align*}
By equation \eqref{equ29}, we immediately have $\|\Delta
s\|_{L^2(\Omega)}\in L^2(0,T)$ for any $T>0$. We can also get
$s\in L^2(R^+,L^{\infty}(\Omega))$ by embedding
$H^2\hookrightarrow L^{\infty}$ in dimension 2.
\end{lemma}

However we can expect more about the regularity of the
$s$-variable. In the following lemma, we improve the regularity of
the $s$-variable.

\begin{lemma}\label{lem9}
Suppose that $J_e$ is smooth, and $(s_0,m_0)\in H^2(\Omega)\times
H^2(\Omega)$ is small in the sense of \eqref{equ78}, then for any
$T>0$, the solution of \eqref{equ1} satisfy
\begin{gather}\label{equ54}
\partial_ts\in L^{\infty}(0,T;L^2),\quad \partial_t\nabla s\in
L^2(0,T;L^2),\\
\label{equ62}
s\in L^{\infty}(0,T;H^2),\quad  s\in L^2(0,T;H^3).
\end{gather}
Also, for $s''$, we have
\begin{equation}\label{equ63}
s''\in L^2(0,T;H^{-1}(\Omega)).
\end{equation}
\end{lemma}

\begin{proof}
1. Differentiating the $s$-equation with respect to $t$, taking
inner product with $\partial_ts$ and then integrating over
$\Omega$, we obtain
\begin{equation}\label{equ49}
\frac{1}{2}\frac{d}{dt}\|\partial_ts\|^2
=\int_{\Omega}\langle \partial_tJ_s,\partial\nabla
s\rangle -\|\partial_ts\|^2-\int_{\Omega}\langle
s\times\partial_tm,\partial_ts\rangle.
\end{equation}
The last term can be estimated as
\begin{equation}\label{equ77}
|\int_{\Omega}\langle s\times\partial_tm,\partial_ts\rangle |\leq
C\|\partial_ts\|^2+C\|\partial_tm\|^2\|s\|^2_{L^{\infty}(\Omega)},
\end{equation}
where the second term on the right is integrable on $[0,T]$ since
$s\in L^2(R^+,L^{\infty}(\Omega))$ by Lemma \ref{lem5} and
$\partial_tm\in L^{\infty}(0,T;L^2(\Omega))$ by Remark \ref{rmk1}.
The first term on the right of \eqref{equ49} can be estimated as
\begin{equation} \label{equ76}
\begin{aligned}
\int_{\Omega}\langle \partial_tJ_s,\partial_t\nabla
s\rangle &=\int_{\Omega}\langle \partial_t(m\otimes J_e),\partial_t\nabla
s\rangle -\|\partial_t\nabla
s\|^2 \\
&\quad +\beta\int_{\Omega}\langle
\partial_t(m\otimes(\nabla s\cdot m)),\partial_t\nabla s\rangle  \\
&\leq-(1-\beta-\varepsilon)\|\partial_t\nabla
s\|^2+C(\varepsilon)\|\partial_t(m\otimes J_e)\|^2 \\
&\quad +C(\varepsilon)\|\partial_tm\|_{L^2}^2\|\partial_tm\|_{H^1}^2
+C(\varepsilon)\|\nabla s\|^2\|\nabla s\|_{H^1}^2,
\end{aligned}
\end{equation}
where in the last step, we used the Gagliardo-Nirenberg
inequality, and $\|\partial_t(m\otimes J_e)\|^2$ can be bounded by
\begin{equation}\label{equ61}
\|\partial_t(m\otimes J_e)\|^2\leq
\|\partial_tJ_e\|^2+C\|J_e\|^2_{L^{\infty}(\Omega)}\|\partial_tm\|^2.
\end{equation}
Combining \eqref{equ49}-\eqref{equ61}, and setting $\varepsilon$
small enough, say $\varepsilon=\frac{1-\beta}{2}$, we have
\begin{equation} \label{equ52}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\partial_ts\|^2+\delta\|\partial_t\nabla
s\|^2+\frac{1}{2}\|\partial_ts\|^2 \\
&\leq C\|\partial_ts\|^2+C\Big(\|\partial_tm\|^2\|s\|^2_{L^{\infty}
(\Omega)}+\|\partial_tJ_e\|^2+\|J_e\|^2_{L^{\infty}(\Omega)}\|
\partial_tm\|^2 \\
&\quad +(\|\partial_tm\|^2\|\partial_tm\|_{H^1}^2+\|\nabla
s\|^2\|\nabla s\|_{H^1}^2\Big),
\end{aligned}
\end{equation}
where $\delta=(1-\beta)-\varepsilon>0$ and the terms in the
parentheses $(\cdots)$ is integrable due to the above lemmas.
Applying the Gronwall lemma and integrating on $[0,T]$, we have
\eqref{equ54}.

2. Differentiating the $s$-equation in \eqref{equ1} with $\Delta$,
taking inner product with $\Delta s$ and integrating over $\Omega$,
we have
\begin{equation}\label{equ55}
\frac{1}{2}\frac{d}{dt}\|\Delta s\|^2
=\int\langle \Delta J_s,\nabla\Delta s\rangle
-\|\Delta s\|^2-\int\langle \nabla(s\times m),\nabla\Delta s\rangle .
\end{equation}
The last term of the right hand side can be controlled by
\begin{equation}\label{equ56}
|\int\langle \nabla(s\times m),\nabla\Delta s\rangle|
\leq \varepsilon\|\nabla\Delta s\|^2+C(\varepsilon)(\|\nabla
s\|^2+\|s\|^2_{L^{\infty}}\|\nabla m\|^2),
\end{equation}
where the second term on the right of which is integrable on $[0,T]$
thanks to lemma \ref{lem5}. For the first term on the right of
\eqref{equ55}, we have
\begin{equation} \label{equ57}
\begin{aligned}
&\int\langle \Delta J_s,\nabla\Delta s \rangle\\
&=\int\langle \Delta(m\otimes J_e,\nabla\Delta s)\rangle
-\|\nabla\Delta s\|^2
+\beta\int\langle \Delta(m\otimes(\nabla s\cdot m)),\nabla\Delta
s\rangle \\
&=:I+II+III
\end{aligned}
\end{equation}
For $I$, we have
\begin{equation} \label{equ58}
\begin{aligned}
|I|&\leq \varepsilon\|\nabla\Delta
s\|^2+C(\varepsilon)(\|J_e\|^2_{L^{\infty}}\|\Delta m\|^2+\|\Delta
J_e\|^2) \\
&\quad +C(\varepsilon)(\|\nabla m\|^4_{L^4}+\|\nabla
J_e\|^4_{L^4}) \\
&\leq \varepsilon\|\nabla\Delta
s\|^2+C(\varepsilon)(\|J_e\|^2_{L^{\infty}}\|\Delta m\|^2+\|\Delta
J_e\|^2) \\
&\quad +C(\varepsilon)(\|\nabla m\|^2\|\nabla
m\|_{H^1}^2+\|\nabla J_e\|^2\|\nabla J_e\|_{H^1}^2),
\end{aligned}
\end{equation}
where the last two terms are integrable on $[0,T]$. For $III$, we
have
\begin{equation} \label{equ59}
\begin{aligned}
|III|&\leq (\beta+\varepsilon)\|\nabla\Delta
s\|^2+C(\varepsilon)\Big(\|\nabla m\|^2_{L^{\infty}}\|\Delta
s\|^2+\|\nabla s\|^2\|\nabla s\|_{H^1}^2 \\
&\quad +\|\Delta m\|^2\|\Delta m\|_{H^1}^2+\|\nabla
m\|^2_{L^{\infty}}(\|\nabla s\|^2\|\nabla s\|_{H^1}^2+\|\nabla
m\|^2\|\nabla m\|_{H^1}^2)\Big).
\end{aligned}
\end{equation}
Setting $\varepsilon$ small enough, say
$\varepsilon=\frac{1-\beta}{4}$,  from
\eqref{equ55}--\eqref{equ59}, we have
\begin{equation}\label{equ60}
\frac{1}{2}\frac{d}{dt}\|\Delta s\|^2+\delta\|\nabla\Delta
s\|^2\leq C\Big(\|\nabla m\|^2_{L^{\infty}}+\|\nabla
s\|^2+\|\nabla m\|^2_{L^{\infty}}\|\nabla s\|^2\Big)\|\Delta
s\|_{L^2}^2+R,
\end{equation}
where $\delta=(1-\beta)-3\varepsilon>0$ and the remainder term $R$
is integrable on $[0,T]$. Thus the Gronwall inequality implies our
result \eqref{equ62}.

3. Prove \eqref{equ63}. Differentiating the $s$-equation with
respect to $t$, we get
\begin{equation}\label{equ64}
\frac{\partial^2s}{\partial
t^2}=-\mathop{\rm div}\partial_tJ_s-\partial_ts-\partial_t(s\times m).
\end{equation}
Then for any $v\in H^1_0(\Omega)$, we have
\begin{gather}\label{equ65}
|\int\langle \partial_ts,v\rangle |\leq \|\partial_ts\|\|v\|,\\
\label{equ66}
|\int\langle \partial_t(s\times m),v\rangle |\leq
(\|\partial_ts\|+\|s\|_{L^{\infty}(\Omega)}\|\partial_tm\|)\|v\|.
\end{gather}
For the first term on the right of \eqref{equ64}, we have
\begin{equation}\label{equ67}
-\int\langle \mathop{\rm div}\partial_tJ_s,v\rangle
=\int\langle \partial_tJ_s,\nabla v\rangle ,
\end{equation}
which can be bounded by
\begin{equation}\label{equ68}
|\int\langle \partial_tJ_s,\nabla v\rangle|\leq C\|\nabla v\|_{L^2}
\end{equation}
thanks to the above Lemmas \ref{lem4}, Remark \ref{rmk2} and the
first two parts of this lemma. Thus $s''\in L^2(0,T;H^{-1})$ for
all $T>0$. This completes the proof of the lemma.
\end{proof}

Finally, These \emph{a priori} estimates imply well-posedness of the
problem in $H^2$. We can do this by approximating the initial data
$(s_0,m_0)$ by smooth data $(s^l_0,m^l_0)\in C^{\infty}$ such that
$$
(s^l_0,m^l_0)\to (s_0,m_0)\quad \text{in }H^2\times H^2
$$
as $l\to\infty$. Thus the \emph{a priori} estimates imply a
uniform (in $l$) $H^2$ bound on the approximating solutions $(s^l,
m^l)$. Therefore we can extract a subsequence (if necessary)
$\{(s^l, m^l)\}$ which converges weakly in $H^2$ to an $H^2$
solution $(s,m)$ of the problem. The \emph{a priori} estimates also
implies that the solution is global.

For the reader's convenience, we summarize the above lemmas in the
following theorem.

\begin{theorem}\label{thm2}
Suppose that $J_e\in H^2(R^+\times\Omega)$. Let $(s_0, m_0)\in
H^2(\Omega)$ satisfy the smallness condition \eqref{equ78}. Then
there exists a global solution in $H^2$ such that for any $T>0$, we
have
\begin{gather*}
m\in L^{\infty}(0,T;H^2),\quad  m\in L^2(0,T;H^3),\\
\partial_tm\in L^{\infty}(0,T;L^2),\quad \partial_t\nabla m\in
L^2(0,T;L^2);
\end{gather*}
for $s$,
\begin{gather*}
s\in L^{\infty}(0,T;H^2),\quad  s\in L^2(0,T;H^3),\\
\partial_ts\in L^{\infty}(0,T;L^2),\quad \partial_t\nabla s\in
L^2(0,T;L^2),
\end{gather*}
and for $s''$, we have $ s''\in L^2(0,T;H^{-1}(\Omega))$.
\end{theorem}


\section{Conclusions}

 In the previous section, we obtained some \emph{a priori} estimates
of the solution to ensure that the solution is in
$L^{\infty}(0,T; H^2{\Omega})$ for
all positive times $T>0$ for given initial data in $H^2(\Omega)$. In
this section, we provide the higher order \emph{a priori} estimates
of the solution of \eqref{equ1} to show that the solution is indeed
smooth. In the sequel, we will abuse the notation of $D$ with
$\nabla$ and we get our result by induction.

\begin{lemma}\label{lem2}
Let $k\geq 3$. Suppose $J_e\in H^k(R^+\times\Omega)$, and $(s,m)$ is
a smooth solution of the equation \eqref{equ1} with initial data
$(s_0, m_0)$. Furthermore, if the smallness condition \eqref{equ78}
holds, then
\begin{gather*}
m\in L^{\infty}(0,T;H^k(\Omega)),\quad m\in L^{2}(0,T;H^{k+1}(\Omega))\\
s\in L^{\infty}(0,T;H^k(\Omega)),\quad s\in L^{2}(0,T;H^{k+1}(\Omega)).
\end{gather*}
\end{lemma}

\begin{proof}
We prove this lemma by induction. Let us first assume that for the
initial data $s_0,m_0\in H^{k-1}$, we have proved our result; i.e.,
\begin{gather*}
m\in L^{\infty}(0,T;H^{k-1}(\Omega)),\quad m\in L^{2}(0,T;H^k(\Omega))\\
s\in L^{\infty}(0,T;H^{k-1}(\Omega)),\quad s\in L^{2}(0,T;H^k(\Omega)).
\end{gather*}
We divide this proof into two parts and we omit the constant
$\alpha$, which will not influence the result.

1. In this part, we prove the regularity for $m$. Differentiating
\eqref{equ11} with $D^k$, and taking inner product with $D^km$, then
integrating over $\Omega$ leads us to
\begin{equation} \label{equ69}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|D^km\|^2+\|\nabla D^km\|^2\\
&=\langle D^k(m\times\nabla m),\nabla D^km\rangle-\langle D^k(m\times
s),D^km\rangle \\
&+\langle D^{k-1}(|\nabla m|^2m),D^{k+1}m\rangle-\langle
D^{k-1}((m\cdot s)m),D^{k+1}m\rangle+\langle D^{k}s,D^km\rangle,
\end{aligned}
\end{equation}
where $\langle\cdot,\cdot\rangle$ denotes the inner product over
$\Omega$. Below we will bound the terms one by one.
The last term can be estimated
\[
|\langle D^{k}s,D^km\rangle|\leq\|D^ks\|^2+\|D^km\|^2,
\]
where the first term on the right is integrable on $[0,T]$.
Recalling Kato's inequality, we have
\[
\|D^l(fg)-fD^lg\|_{L^p}\leq
C\|Df\|_{L^{\infty}}\|D^{l-1}g\|_{L^p}+C\|D^lf\|_{L^p}\|g\|_{L^{\infty}}
\]
for any two functions $f$ and $g$, see \cite{GuoHong}. Then the
second term can be estimated by
\begin{equation} \label{equ70}
\begin{aligned}
&|\langle D^k(m\times s),D^km\rangle|\\
&\leq \|D^km\|^2+\|D^k(m\times s)\|^2 \\
&\leq C\|Ds\|^2_{L^{\infty}}\|D^{k-1}m\|^2+C\|D^ks\|^2
 +C\left(\|s\|^2_{L^{\infty}}+1\right)\|D^km\|^2,
\end{aligned}
\end{equation}
where both the coefficients of $\|D^km\|^2$ and the the first two
terms are integrable on $[0,T]$ by the induction assumption. The
fourth term can be estimated the same.
Since
\begin{equation}
D^k(m\times\nabla m)=D^km\times\nabla m+m\times D^k\nabla
m+\sum^{k-1}_{h=1}c_hD^hm\times D^{k-h}\nabla m,
\end{equation}
the first term can be estimated by
\begin{align*}
&|\langle D^k(m\times\nabla m),\nabla D^km\rangle| \\
&\leq |\langle D^km\times\nabla m,\nabla
D^km\rangle|+\sum^{k-1}_{h=1}c_h|\langle D^hm\times D^{k-h}\nabla
m,\nabla D^km\rangle| \\
&\leq \frac{1}{4}\|\nabla D^km\|^2+C\|\nabla^km\|^2+R,
\end{align*}
where the remainder term $R$ depends on the lower order
derivatives of $m$ and is integrable on $[0,T]$.

Finally from \eqref{equ70}, we have
\begin{align*}
&\|D^{k-1}(|\nabla m|^2m)\|_{L^2}\\
&\leq \|mD^{k-1}(|\nabla m|^2)\|+C\|\nabla m\|_{L^{\infty}}\|D^{k-2}|\nabla
m|^2\|
 +C\|D^{k-1}m\|\||\nabla m|^2\|_{L^{\infty}} \\
&\leq C_1\|\nabla^km\|+C_2.
\end{align*}
Then the third term of \eqref{equ69} can be estimated by
\begin{equation}\label{equ71}
|\langle D^{k-1}(|\nabla m|^2m),D^{k+1}m\rangle|\leq
\frac{1}{4}\|\nabla D^km\|^2+C\|\nabla^km\|^2+C.
\end{equation}
Summarizing \eqref{equ69}-\eqref{equ71}, we arrive at the Gronwall
type inequality
\[
\frac{d}{dt}\|D^km\|^2+\|D^{k+1}m\|^2\leq C\|\nabla^km\|^2+C,
\]
from which we deduce that for all $T>0$,
\[
\sup_{0\leq t\leq T}\|\nabla^km(\cdot,t)\|_{L^2(\Omega)}\leq C,\quad
|\nabla^{k+1}m\|_{L^2(0,T;L^2(\Omega))}\leq C.
\]
Furthermore, we have $\|\nabla m\|_{L^{\infty}(\Omega)}\leq C$.


2. In this part, we prove the regularity for $s$. This is similar
with that in the first part. Differentiating the $s$ equation with
$D^k$, and then taking inner product with $D^ks$ over $\Omega$, we
have
\begin{equation} \label{equ73}
\frac{1}{2}\frac{d}{dt}\|D^ks\|^2=\langle D^kJ_s,\nabla
D^ks\rangle-\|D^ks\|^2-\langle D^k(s\times m),D^ks\rangle=I+II+III.
\end{equation}

\noindent$\bullet$  Estimates of I.  We expand $I$ as
  \[
  \langle D^kJ_s,\nabla D^ks\rangle+\|\nabla D^ks\|^2=\langle D^k(m\times J_e),\nabla
  D^ks\rangle+\beta\langle D^k(m\otimes(\nabla
  s\cdot m)),\nabla D^ks\rangle,
  \]
  where the terms on right hand side can be estimated by
\[
  |\langle D^k(m\times J_e),\nabla D^ks\rangle|
\leq   \varepsilon\|\nabla D^ks\|^2+C\|D^k(m\times J_e)\|^2 \\
\leq \varepsilon\|\nabla D^ks\|^2+C
\]
  and
\[
  |\beta\langle D^k(m\otimes(\nabla s\cdot m)),\nabla
  D^ks\rangle|\leq (\beta+\varepsilon)\|\nabla
  D^ks\|^2+C\|D^ks\|^2+C.
\]

\noindent$\bullet$  Estimates of III:
  \begin{equation}\label{equ74}
  |III|\leq\|D^ks\|^2+C,
  \end{equation}
where $C$ is integrable on $[0,T]$.

Letting $\varepsilon$ sufficiently small, we can deduce from
\eqref{equ73}-\eqref{equ74} the Gronwall type inequality
\begin{equation*}
\frac{d}{dt}\|D^ks\|^2+\|\nabla D^ks\|^2\leq C\|D^ks\|^2+C,
\end{equation*}
from which we can deduce
\begin{equation*}
\nabla^ks\in L^{\infty}(0,T;L^2(\Omega)),\quad
\nabla^{k+1}s\in L^{2}(0,T;L^2(\Omega)),
\end{equation*}
which concludes the proof.
\end{proof}

One can also give the regularity in time as what we did in Section
3, which is standard and we omit the details here. The local
existence theorem in Section 2 and the \emph{a priori} estimates
then imply global existence of solutions for system \eqref{equ1} in
$H^k$. In particular, the above estimates hold for any $k\geq 2$.
Thus if the initial data is smooth, we have global smooth solutions
as stated in Theorem \ref{thm1}.

\subsection*{Acknowledgements}
The authors thank the anonymous referee for his/her useful suggestions
and comments, as well as the journal editors for their hard work.

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\end{document}