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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 85, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/85\hfil Wave map and related systems]
{Regularity criteria for the wave map and related systems}

\author[J. Fan, Y. Zhou \hfil EJDE-2016/85\hfilneg]
{Jishan Fan, Yong Zhou}

\address{Jishan Fan \newline
Department of Applied Mathematics,
Nanjing Forestry University, Nanjing 210037,  China}
\email{fanjishan@njfu.com.cn}

\address{Yong Zhou (corresponding author) \newline
School of Mathematics,
Shanghai University of Finance and Economics, \newline
Shanghai 200433,  China. \newline
Department of Mathematics,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia}
\email{yzhou@sufe.edu.cn}

\thanks{Submitted February 18, 2016. Published March 29, 2016.}
\subjclass[2010]{35K55, 35Q35, 70S15}
\keywords{Regularity criterion; wave map; liquid crystals; Hall-MHD}

\begin{abstract}
 We obtain some regularity criteria for the wave map,
 a liquid crystals model, and the Hall-MHD with ion-slip effect. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

First, we consider the $n$D wave maps 
$d:\mathbb{R}^{1+n}\to\mathbb{S}^m\subset\mathbb{R}^{1+m}$ which 
obey the nonlinear wave equation
\begin{equation}
\partial_t^2d-\Delta d=d(|\nabla d|^2-|\partial_td|^2)\label{1.1}
\end{equation}
with the initial conditions
\begin{equation}
(d,\partial_td)(\cdot,0)=(d_0,d_1),\; d_0\in\mathbb{S}^m,\quad
 d_0\cdot d_1=0.\label{1.2}
\end{equation}

Wave maps have wide applications in physics from the harmonic gauge in 
general relativity to the nonlinear $\sigma$-models in particle physics.

Local well-posedness of \eqref{1.1} \eqref{1.2} has been proved 
by Tao \cite{1}. Shatah \cite{2} showed that solutions to the Cauchy problem 
for wave maps may blow up in finite time. However, some smallness assumption
 on the initial data or integrability condition on the solution itself 
are sufficient to guarantee the regularity. Fan and Ozawa \cite{3} obtained
the regularity criterion
\begin{equation}
\nabla d,\partial_t d\in L^1(0,T;{\dot{B}^0_{\infty,\infty}}(\mathbb{R}^n))\label{1.3}
\end{equation}
when $n=2$.

The first aim of this article is to prove a following  regularity criterion 
when $n\geq3$.

\begin{theorem} \label{thm1.1} 
Let $n\geq3$ and $(\nabla d_0,d_1)\in H^{1+s}(\mathbb{R}^n)$ with 
$s>\frac{n}{2}$, $|d_0|=1$, $d_0\cdot d_1=0$ and $d$ be a smooth solution 
of \eqref{1.1}, \eqref{1.2}. If \eqref{1.3} and
 $\partial_td\in L^\infty(0,T;L^n(\mathbb{R}^n))$ hold true with
 $0<T<\infty$, then the solution $d$ can be extended beyond $T>0$.
\end{theorem}

Next, we consider the  liquid crystals model \cite{4,5,6,7}:
\begin{gather}
\partial_tu+u\cdot\nabla u+\nabla\pi-\Delta u
 =-\nabla\cdot(\nabla d\odot\nabla d),\label{1.4}\\
\partial_td+u\cdot\nabla d-\Delta d=d|\nabla d|^2,\quad |d|=1,\label{1.5}\\
\operatorname{div} u=0,\label{1.6}\\
(u,d)(\cdot,0)=(u_0,d_0)\quad \text{in }\mathbb{R}^n,\; |d_0|=1.\label{1.7}
\end{gather}
Here $u$ is the velocity, $\pi$ is the pressure, $d$ is the direction vector, 
and $(\nabla d\odot\nabla d)_{i,j}:=\sum_k\partial_id_k\partial_jd_k$,
and hence 
$$
\nabla\cdot(\nabla d\odot\nabla d)
=\sum_k\Delta d_k\nabla d_k+\frac12\nabla|\nabla d|^2.
$$
If $u=0$, then \eqref{1.5} is the harmonic heat flow.

Fan-Gao-Guo \cite{8} proved the blow-up criterion
\begin{equation}
u,\nabla d\in L^2(0,T;{\dot{B}^0_{\infty,\infty}})\label{1.8}
\end{equation}
when $n=3$. 
One can find other related results in \cite{FAHNZ-AA, ZF} and references therein.
We will prove the following theorem.

\begin{theorem} \label{thm1.2} 
Let $n\geq3$ and $s>\frac{n}{2}$ be an integer. Let $u_0$ and $d_0$ 
satisfy $u_0,\nabla d_0\in H^s, \operatorname{div} u_0=0$, and $|d_0|=1$ in $\mathbb{R}^n$. 
Let $(u,d)$ be a local strong solution to the problem \eqref{1.4}-\eqref{1.7}. 
If $\nabla u$ and $\nabla^2d$ satisfy
\begin{equation}
\nabla u,\nabla^2d\in L^\frac{2}{2-\alpha}(0,T;\dot B_{\infty,\infty}^{-\alpha}
(\mathbb{R}^n))\label{1.9}
\end{equation}
with $0<\alpha<1$ and $0<T<\infty$, then the solution $(u,d)$ can be extended 
beyond $T>0$.
\end{theorem}

Also we consider the  incompressible MHD with the Hall or ion-slip system
\begin{gather}
\partial_tu+u\cdot\nabla u+\nabla\Big(\pi+\frac12|b|^2\Big)
 -\Delta u=b\cdot\nabla b,\label{1.10}\\
\partial_tb+u\cdot\nabla b-b\cdot\nabla u+h\operatorname{rot}(\operatorname{rot} b\times b)
 -\gamma\operatorname{rot}[(\operatorname{rot} b\times b)\times b]=\Delta b,\label{1.11}\\
\operatorname{div} u=\operatorname{div} b=0,\label{1.12}\\
(u,b)(\cdot,0)=(u_0,b_0)\quad \text{in }\mathbb{R}^3.\label{1.13}
\end{gather}
Here $b$ is the magnetic field. $h$ is the Hall effect coefficient, 
and $\gamma\geq0$ the ion-slip effect coefficient, respectively.

Applications of the Hall-MHD system cover a very wide range of physical subjects, 
such as,  magnetic reconnection in space plasmas, star formation, neutron stars, 
and geo-dynamos.

Very recently, Zhang \cite{9} obtained the  regularity criterion
\begin{equation}
u\in L^\frac{2}{1-\alpha}(0,T;\dot B_{\infty,\infty}^{-\alpha}),\quad
\nabla b\in L^\frac{2}{1-\beta}(0,T;\dot B_{\infty,\infty}^{-\beta})\label{1.14}
\end{equation}
with $-1<\alpha<1$ and $0<\beta<1$ when $h=1$ and $\gamma=0$.

Local well-posedness of strong solutions to  \eqref{1.10}-\eqref{1.13} has been 
proved by Fan, Jia, Nakamura and Zhou \cite{10}, they also obtained the regularity 
criterion
\begin{equation}
u\in L^\frac{2q}{q-3}(0,T;L^q),\quad b\in L^\infty(0,T;L^\infty),\quad
\nabla b\in L^\frac{2p}{p-3}(0,T;L^p)\label{1.15}
\end{equation}
with $3<p,q\leq\infty$. For standard Hall-MHD system we refer to
\cite{CWW,FAHZ-AMC, FAHZ-NONRWA,FAHNZ,HAHZ,W,WZ} and references therein.

By the method in \cite{9}, we will refine \eqref{1.15} as follows.

\begin{theorem} \label{thm1.3} 
Let $u_0,b_0\in H^2$ with $\operatorname{div} u_0=\operatorname{div} b_0=0$ in $\mathbb{R}^3$. Let $(u,b)$
 be a local strong solution to the problem \eqref{1.10}-\eqref{1.13}. 
If $u$ and $b$ satisfy \eqref{1.14} and $b\in L^\infty(0,T;L^\infty)$ 
with $0<T<\infty$, then the solution $(u,b)$ can be extended beyond $T>0$.
\end{theorem}

In the following proofs, we  use the logarithmic Sobolev inequality \cite{11}:
\begin{gather}
\|\nabla d\|_{L^\infty}\leq C(1+\|\nabla d\|_{\dot{B}^0_{\infty,\infty}}
 \log(e+\|\nabla d\|_{H^{1+s}})),\label{1.16}\\
\|\partial_td\|_{L^\infty}\leq C(1+\|\partial_td\|_{\dot{B}^0_{\infty,\infty}}
 \log(e+\|\partial_td\|_{H^{1+s}}))\label{1.17}
\end{gather}
for $s>\frac{n}{2}-1$, and the bilinear product and commutator estimates 
due to Kato-Ponce \cite{12}:
\begin{gather}
\|\Lambda^s(fg)\|_{L^p}\leq C(\|\Lambda^sf\|_{L^{p_1}}
 \|g\|_{L^{q_1}}+\|f\|_{L^{p_2}}\|\Lambda^sg\|_{L^{q_2}}),\label{1.18}\\
\|\Lambda^s(fg)-f\Lambda^sg\|_{L^p}\leq C(\|\nabla f\|_{L^{p_1}}
\|\Lambda^{s-1}g\|_{L^{q_1}}+\|\Lambda^sf\|_{L^{p_2}}\|g\|_{L^{q_2}}),\label{1.19}
\end{gather}
with $s>0$, $\Lambda:=(-\Delta)^\frac12$ and 
$\frac1p=\frac{1}{p_1}+\frac{1}{q_1}=\frac{1}{p_2}+\frac{1}{q_2}$.

We  also use the  Gagliardo-Nirenberg inequalities
\begin{gather}
\|\nabla d\|_{L^{2p}}^2\leq C\|d\|_{L^\infty}\|\nabla^2d\|_{L^p},\label{1.20}\\
\|\nabla^2d\|_{L^p}\leq C\|\nabla d\|_{L^\infty}^{1-\theta}
 \|\Lambda^{2+s}d\|_{L^2}^\theta,\label{1.21}\\
\|\Lambda^{1+s}d\|_{L^\frac{2p}{p-2}}\leq C\|\nabla d\|_{L^\infty}^\theta
\|\Lambda^{2+s}d\|_{L^2}^{1-\theta}\label{1.22}
\end{gather}
with $p:=2s+2$ and $\theta:=1/(1+s)$.

We  also use the  improved Gagliardo-Nirenberg inequalities \cite{13,14,15}:
\begin{gather}
\|\nabla u\|_{L^{q_1}}\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}
 }^{1-\theta_1}\|u\|_{\dot H^{s+\alpha}}^{\theta_1},\label{1.23}\\
\|\Lambda^su\|_{L^\frac{2q_1}{q_1-2}}
\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^{\theta_1}
\|u\|_{\dot H^{s+\alpha}}^{1-\theta_1},\label{1.24}
\end{gather}
with $q_1:=\frac{2(s-1+2\alpha)}{\alpha}$ and $\theta_1:=2/q_1$, and
\begin{gather}
\|\nabla d\|_{L^{q_2}}
\leq C\|\nabla d\|_{\dot B_{\infty,\infty}^{-\alpha}}^{1-\theta_2}
\|\nabla d\|_{\dot H^{s+\alpha}}^{\theta_2},\label{1.25}\\
\|\Lambda^s\nabla d\|_{L^\frac{2q_2}{q_2-2}}
\leq C\|\nabla d\|_{\dot B_{\infty,\infty}^{-\alpha}}^{\theta_2}
\|\nabla d\|_{\dot H^{s+\alpha}}^{1-\theta_2},\label{1.26}
\end{gather}
with $q_2:=\frac{2(s+2\alpha)}{\alpha}$ and $\theta_2:=\frac{2}{q_2}$, 
\begin{equation}
\|\nabla d\|_{\dot B_{\infty,\infty}^{-\alpha}}
\leq C\|d\|_{L^\infty}^\frac{1}{2-\alpha}
\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{1-\alpha}{2-\alpha},\label{1.27}
\end{equation}
and
\begin{gather}
\|D^ku\|_{L^{p_k}}
\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^{1-\tilde\theta_k}
 \|u\|_{\dot H^{s+\alpha}}^{1-\tilde\theta_k}, \label{1.28}\\
\|D^{s+2-k}d\|_{L^\frac{2p_k}{p_k-2}}
\leq C\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^{\tilde\theta_k}
\|\nabla d\|_{\dot H^{s+\alpha}}^{1-\tilde\theta_k}, \label{1.29}
\end{gather}
with $p_k:=\frac{2}{\tilde\theta_k}$ and 
$\tilde\theta_k:=\frac{k+\alpha-1}{s+2\alpha-1}$, and
\begin{gather}
\|\nabla u\|_{L^3}^3\leq C\|u\|_{\dot B_{\infty,\infty}^{-\alpha}}\|u\|_{\dot H^\frac{3+\alpha}{2}}^2\ \ \mathrm{with}\ \ -1<\alpha<1,\label{1.30}\\
\|u\|_{\dot H^\frac{3+\alpha}{2}}\leq C\|\nabla u\|_{L^2}^\frac{1-\alpha}{2}\|\Delta u\|_{L^2}^\frac{1+\alpha}{2}\ \ \mathrm{with}\ \ -1<\alpha<1,\label{1.31}
\end{gather}
and
\begin{gather}
\|\nabla b\|_{L^4}^2\leq C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}
\|b\|_{\dot H^{1+\beta}}\quad \text{with } 0<\beta<1,\label{1.32}\\
\|b\|_{\dot H^{1+\beta}}
\leq C\|\nabla b\|_{L^2}^{1-\beta}\|\Delta b\|_{L^2}^\beta\quad \text{with }
 0<\beta<1.\label{1.33}
\end{gather}

\section{Proof of Theorem \ref{thm1.1}}

Testing \eqref{1.1} by $\partial_td$ and using $|d|=1$ and $d\cdot\partial_td=0$,
 we easily get the conservation of the energy:
\begin{equation}
\frac{d}{dt}\int(|\partial_td|^2+|\nabla d|^2) dx=0.\label{2.1}
\end{equation}

Applying the operator $\Lambda^{1+s}$ to equation \eqref{1.1},
 testing by $\Lambda^{1+s}\partial_td$, using 
\eqref{1.18}, \eqref{1.16}, \eqref{1.17}, \eqref{1.20}, \eqref{1.21} 
and \eqref{1.22}, we reach
\begin{align*}
&{\frac12\frac{d}{dt}\int}(|\Lambda^{1+s}\partial_td|^2+|\Lambda^{2+s}d|^2)dx\\
&= \int\Lambda^{1+s}(d|\nabla d|^2-d|\partial_td|^2)\Lambda^{1+s}\partial_t d dx\\
&\leq (\|\Lambda^{1+s}(d|\nabla d|^2)\|_{L^2}
 +\|\Lambda^{1+s}(d|\partial_td|^2)\|_{L^2})\|\Lambda^{1+s}\partial_td\|_{L^2}\\
&\leq C(\|d\|_{L^\infty}\|\Lambda^{1+s}(|\nabla d|^2)\|_{L^2}
 +\|\nabla d\|_{L^{2p}}^2\|\Lambda^{1+s}d\|_{L^\frac{2p}{p-2}})\|\Lambda^{1+s}
 \partial_td\|_{L^2}\\
&\quad +C(\|d\|_{L^\infty}\|\Lambda^{1+s}(|\partial_td|^2)\|_{L^2}
 +\|\partial_td\|_{L^{2n}}^2\|\Lambda^{1+s}d\|_{L^\frac{2n}{n-2}}) 
 \|\Lambda^{1+s}\partial_td\|_{L^2}\\
&\leq C(\|\nabla d\|_{L^\infty}\|\Lambda^{2+s}d\|_{L^2}
 +\|\nabla^2d\|_{L^p}\|\Lambda^{1+s}d\|_{L^\frac{2p}{p-2}})\|\Lambda^{1+s}
 \partial_td\|_{L^2}\\
&\quad +C(\|\partial_td\|_{L^\infty}\|\Lambda^{1+s}\partial_td\|_{L^2}
 +\|\partial_td\|_{L^n} \|\partial_td\|_{L^\infty}\|\Lambda^{2+s}d\|_{L^2})
 \|\Lambda^{1+s}\partial_td\|_{L^2}\\
&\leq C\|\nabla d\|_{L^\infty}\|\Lambda^{2+s}d\|_{L^2}
 \|\Lambda^{1+s}\partial_td\|_{L^2}\\
&\quad +C\|\partial_td\|_{L^\infty}\|\Lambda^{1+s}\partial_td\|_{L^2}^2
 +C\|\partial_td\|_{L^\infty} \|\Lambda^{2+s}d\|_{L^2}\|\Lambda^{1+s}
 \partial_td\|_{L^2}\\
&\leq C(\|\nabla d\|_{L^\infty}+\|\partial_td\|_{L^\infty})
 (\|\Lambda^{2+s}d\|_{L^2}^2  +\|\Lambda^{1+s}\partial_td\|_{L^2}^2)\\
&\leq C(1+\|\nabla d\|_{\dot{B}^0_{\infty,\infty}}+\|\partial_td\|_{\dot{B}^0_{\infty,\infty}})\log(e+y^2)y^2,
\end{align*}
with $y^2:=\|\Lambda^{1+s}\partial_td\|_{L^2}^2+\|\Lambda^{2+s}d\|_{L^2}^2$, 
which gives 
$$
\sup_{0\leq t\leq T}(\|\Lambda^{1+s}\partial_td\|_{L^2}^2
+\|\Lambda^{2+s}d\|_{L^2}^2)\leq C.
$$
This completes the proof.


\section{Proof of Theorem \ref{thm1.2}}

 Since it is easy to prove that there are $T_0>0$ and a unique strong 
solution $(u,\pi,d)$ to the problem \eqref{1.4}-\eqref{1.7} in $[0,T_0]$, 
we only need to prove a priori estimates.
Testing \eqref{1.4} by $u$ and using \eqref{1.6}, we see that
\begin{equation}
{\frac12\frac{d}{dt}\int}|u|^2 dx+\int|\nabla u|^2 dx=-\int(u\cdot\nabla)d\cdot\Delta d dx.\label{3.1}
\end{equation}

Testing \eqref{1.5} by $-\Delta d$, using $d\Delta d=-|\nabla d|^2$ and $|d|=1$, 
we find that
\begin{equation}
\begin{aligned}
{\frac12\frac{d}{dt}\int}|\nabla d|^2 dx+\int|\Delta d|^2 dx
&= \int(u\cdot\nabla)d\cdot\Delta d dx+\int(d\Delta d)^2 dx\\
&\leq \int(u\cdot\nabla)d\cdot\Delta d dx+\int|\Delta d|^2 dx.
\end{aligned} \label{3.2}
\end{equation}
Summing up \eqref{3.1} and \eqref{3.2}, we have
\begin{equation}
\int(|u|^2+|\nabla d|^2)dx\leq\int(|u_0|^2+|\nabla d_0|^2)dx.\label{3.3}
\end{equation}

Applying $D^s$ to \eqref{1.4}, testing by $D^su$, using \eqref{1.6}, 
\eqref{1.18}, \eqref{1.19}, \eqref{1.23}, \eqref{1.24}, \eqref{1.25}, 
\eqref{1.26} and \eqref{1.27}, we obtain
\begin{equation}
\begin{aligned}
&{\frac12\frac{d}{dt}\int}|D^su|^2 dx+\int|D^{1+s}u|^2 dx\\
&= -\int(D^s(u\cdot\nabla u)-u\nabla D^su)D^s u dx
 +\int D^s(\nabla d\odot\nabla d):\nabla D^su dx\\
&\leq C\|\nabla u\|_{L^{q_1}}\|D^su\|_{L^\frac{2q_1}{q_1-2}}\|D^su\|_{L^2}
 +C\|\nabla d\|_{L^{q_2}}\|D^s\nabla d\|_{L^\frac{2q_2}{q_2-2}}\|\nabla D^su\|_{L^2}\\
&\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}\|D^{s+\alpha}u\|_{L^2}\|D^su\|_{L^2}
 +C\|\nabla d\|_{\dot B_{\infty,\infty}^{-\alpha}}\|D^{s+\alpha}\nabla d\|_{L^2}\|D\Lambda^su\|_{L^2}\\
&\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}\|D^su\|_{L^2}^{2-\alpha}\|D^{1+s}u\|_{L^2}^\alpha\\
&\quad +C\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{1-\alpha}{2-\alpha}
 \|D^{s+1}d\|_{L^2}^{1-\alpha}\|D^{s+2}d\|_{L^2}^\alpha\|D^{1+s}u\|_{L^2}\\
&\leq \frac18\|D^{1+s}u\|_{L^2}^2+\frac18\|D^{s+2}d\|_{L^2}^2
 +C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}\|D^su\|_{L^2}^2\\
&\quad+C\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}\|D^{s+1}d\|_{L^2}^2.
\end{aligned} \label{3.4}
\end{equation}
Applying $D^{s+1}$ to \eqref{1.5}, testing by $D^{s+1}d$ and using \eqref{1.6},
 we obtain
\begin{equation}
\begin{aligned}
&{\frac12\frac{d}{dt}\int}|D^{s+1}d|^2 dx+\int|D^{s+2}d|^2 dx\\
&=\int D^{s+1}(d|\nabla d|^2)D^{s+1}d dx\\
&\quad -\int(D^{s+1}(u\cdot\nabla d)-u\nabla D^{s+1}d)D^{s+1}d \,dx
=:I_1+I_2.
\end{aligned}\label{3.5}
\end{equation}

Using \eqref{1.18}, $|d|=1$, \eqref{1.25}, \eqref{1.26}, and \eqref{1.27}, 
we bound $I_1$ as follows.
\begin{align*}
I_1&\leq \|D^{s+1}(d|\nabla d|^2)\|_{L^\frac{2q_2}{q_2+2}}
 \|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}}\\
&\leq C(\|d\|_{L^\infty}\|D^{s+1}(|\nabla d|^2)\|_{L^\frac{2q_2}{q_2+2}}
 +\|\nabla d\|_{L^{q_2}}^2\|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}})
 \|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}}\\
&\leq C(\|\nabla d\|_{L^{q_2}}\|D^{s+2}d\|_{L^2}
 +\|\nabla d\|_{L^{q_2}}^2\|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}})
 \|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}}\\
&\leq C\|\nabla d\|_{L^{q_2}}^2\|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}}^2
 +\frac{1}{16}\|D^{s+2}d\|_{L^2}^2\\
&\leq C\|\nabla d\|_{\dot B_{\infty,\infty}^{-\alpha}}^2\|\nabla d\|_{\dot H^{s+\alpha}}^2
 +\frac{1}{16}\|D^{s+2}d\|_{L^2}^2\\
&\leq \frac18\|D^{s+2}d\|_{L^2}^2+C\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}
 \|D^{s+1}d\|_{L^2}^2.
\end{align*} %3.6
Using the Leibniz rule, we write $I_2$ as follows.
\begin{equation}
\begin{aligned}
I_2
&= -\int(C_1DuD^{s+1}d+\sum_{k=2}^sC_kD^kuD^{s+2-k}d
 +C_{s+1}D^{s+1}u\cdot\nabla d)D^{s+1}d dx\\
&=:I_2^1+\sum_{k=2}^sI_2^k+I_2^{s+1}.
\end{aligned} \label{3.7}
\end{equation}
By the same calculations as that of $I_1$, we have
\begin{equation}
\begin{aligned}
I_2^{s+1}
&\leq C\|\nabla d\|_{L^{q_2}}\|D^{s+1}d\|_{L^\frac{2q_2}{q_2-2}}\|D^{s+1}u\|_{L^2}\\
&\leq \frac{1}{16}\|D^{s+1}u\|_{L^2}^2
 +C\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}\|D^{s+1}d\|_{L^2}^2.
\end{aligned} \label{3.8}
\end{equation}

Using \eqref{1.23} and \eqref{1.24}, we bound $I_2^1$ as follows.
\begin{equation}
\begin{aligned}
I_2^1
&\leq C\|\nabla u\|_{L^{q_1}}\|D^{s+1}d\|_{L^\frac{2q_1}{q_1-2}}\|D^{s+1}d\|_{L^2}\\
&\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^{1-\theta_1}\|u\|_{\dot H^{s+\alpha}}^{\theta_1}
 \cdot\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^{\theta_1}\|\nabla d\|_{\dot H^{s+\alpha}}^{1-\theta_1}
 \|D^{s+1}d\|_{L^2}\\
&\leq C(\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}+\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}})(\|u\|_{\dot H^{s+\alpha}}
+\|\nabla d\|_{\dot H^{s+\alpha}})\|D^{s+1}d\|_{L^2}\\
&\leq \frac{1}{16}\|D^{s+1}u\|_{L^2}^2+\frac{1}{16}\|D^{s+2}d\|_{L^2}^2\\
&\quad +C(\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}
+\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha})(\|D^su\|_{L^2}^2+\|D^{s+1}d\|_{L^2}^2).
\end{aligned}\label{3.9}
\end{equation}
Using \eqref{1.28} and \eqref{1.29}, we bound $\sum_{k=2}^sI_2^k$ as follows.
\begin{equation}
\begin{aligned}
\sum_{k=2}^sI_2^k
&\leq C\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^{1-\tilde\theta_k}
 \|u\|_{\dot H^{s+\alpha}}^{\tilde\theta_k}\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^{\tilde\theta_k}
 \|\nabla d\|_{\dot H^{s+\alpha}}^{1-\tilde\theta_k}\|D^{s+1}d\|_{L^2}\\
&\leq C(\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}+\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}})(\|u\|_{\dot H^{s+\alpha}}
 +\|\nabla d\|_{\dot H^{s+\alpha}})\|D^{s+1}d\|_{L^2}\\
&\leq C(\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}+\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}})(\|u\|_{\dot H^{s+\alpha}}
 +\|\nabla d\|_{\dot H^{s+\alpha}})\\
&\quad\times (\|D^su\|_{L^2}+\|D^{s+1}d\|_{L^2})\\
&\leq \frac{1}{16}\|D^{s+1}u\|_{L^2}^2+\frac{1}{16}\|D^{s+2}d\|_{L^2}^2\\
&\quad +C(\|\nabla u\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha}
 +\|\nabla^2d\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{2-\alpha})(\|D^su\|_{L^2}^2
 +\|D^{s+1}d\|_{L^2}^2).
\end{aligned}\label{3.10}
\end{equation}

Inserting the above estimates into \eqref{3.5} and combining with \eqref{3.4} 
and using the Gronwall inequality, we arrive at 
$$
\|D^su\|_{L^\infty(0,T;L^2)}+\|D^{s+1}d\|_{L^\infty(0,T;L^2)}\leq C.
$$
This completes the proof.

\section{Proof of Theorem \ref{thm1.3}}

We only need to show a priori estimates. For simplicity, we will 
take $h=\gamma=1$.
First, testing \eqref{1.10} by $u$ and using \eqref{1.12}, we see that
\begin{equation}
{\frac12\frac{d}{dt}\int}|u|^2 dx+\int|\nabla u|^2 dx=\int(b\cdot\nabla)b\cdot u dx.\label{4.1}
\end{equation}
Testing \eqref{1.11} by $b$ and using \eqref{1.12}, we find that
\begin{equation}
{\frac12\frac{d}{dt}\int}|b|^2 dx+\int|\nabla b|^2 dx+\int|b\times\operatorname{rot} b|^2 dx
=\int(b\cdot\nabla)u\cdot b dx.\label{4.2}
\end{equation}
Summing up \eqref{4.1} and \eqref{4.2}, we obtain
\begin{equation}
{\frac12\frac{d}{dt}\int}(|u|^2+|b|^2) dx+\int(|\nabla u|^2+|\nabla b|^2+|b\times\operatorname{rot} b|^2) dx=0.\label{4.3}
\end{equation}

Testing \eqref{1.10} by $-\Delta u$, using \eqref{1.12}, \eqref{1.30} 
and \eqref{1.31}, we infer that
\begin{equation}
\begin{aligned}
&{\frac12\frac{d}{dt}\int}|\nabla u|^2 dx+\int|\Delta u|^2 dx\\
&= \int(u.\nabla)u\cdot\Delta u dx-\int(b\cdot\nabla)b\cdot\Delta u dx\\
&= -\sum_{i,j}\int\partial_ju_i\partial_iu\partial_ju dx
 -\int(b\cdot\nabla)b\cdot\Delta u dx\\
&\leq C\|\nabla u\|_{L^3}^3+\|b\|_{L^\infty}\|\nabla b\|_{L^2}\|\Delta u\|_{L^2}\\
&\leq C\|u\|_{\dot B_{\infty,\infty}^{-\alpha}}\|u\|_{\dot H^\frac{3+\alpha}{2}}^2
 +C\|\nabla b\|_{L^2}\|\Delta u\|_{L^2}\\
&\leq C\|u\|_{\dot B_{\infty,\infty}^{-\alpha}}\|\nabla u\|_{L^2}^{1-\alpha}\|\Delta u\|_{L^2}^{1+\alpha}
 +C\|\nabla b\|_{L^2}\|\Delta u\|_{L^2}\\
&\leq \frac18\|\Delta u\|_{L^2}^2
+C\|u\|_{\dot B_{\infty,\infty}^{-\alpha}}^\frac{2}{1-\alpha}\|\nabla u\|_{L^2}^2+C\|\nabla b\|_{L^2}^2.
\end{aligned}\label{4.4}
\end{equation}

Testing \eqref{1.11} by $-\Delta b$ and using \eqref{1.12}, we deduce that
\begin{equation}
\begin{aligned}
&{\frac12\frac{d}{dt}\int}|\nabla b|^2 dx+\int|\Delta b|^2 dx\\
&=\int(u\cdot\nabla)b\cdot\Delta b dx-\int(b\cdot\nabla)u\cdot\Delta b dx\\
&\quad +\int(\operatorname{rot} b\times b)\operatorname{rot}\Delta b dx-\int[(\operatorname{rot} b\times b)\times b]\operatorname{rot}\Delta b dx\\
&=:\ell_1+\ell_2+\ell_3+\ell_4.
\end{aligned}\label{4.5}
\end{equation}

We bound $\ell_1$ and $\ell_2$ as follows.
\begin{align*}
\ell_1&= \sum_{i,j}\int u_i\partial_ib\partial_j^2b dx=-\sum_{i,j}\int\partial_ju_i\partial_ib\partial_jb dx\leq C\|\nabla u\|_{L^2}\|\nabla b\|_{L^4}^2\\
&\leq C\|\nabla u\|_{L^2}\|b\|_{L^\infty}\|\Delta b\|_{L^2}\leq C\|\nabla u\|_{L^2}\|\Delta b\|_{L^2}\leq\frac{1}{16}\|\Delta b\|_{L^2}^2+C\|\nabla u\|_{L^2}^2.\\
\ell_2&\leq \|b\|_{L^\infty}\|\nabla u\|_{L^2}\|\Delta b\|_{L^2}\leq C\|\nabla u\|_{L^2}\|\Delta b\|_{L^2}\leq\frac{1}{16}\|\Delta b\|_{L^2}^2+C\|\nabla u\|_{L^2}^2.
\end{align*}

Using \eqref{1.32} and \eqref{1.33}, we bound $\ell_3$ and $\ell_4$ as follows.
\begin{align*}
\ell_3
&= -\sum_i\int(\operatorname{rot} b\times\partial_ib)\partial_i\operatorname{rot} b dx
 \leq C\|\nabla b\|_{L^4}^2\|\Delta b\|_{L^2}\\
&\leq C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}
 \|b\|_{\dot H^{1+\beta}}\|\Delta b\|_{L^2}
 \leq C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}\|\nabla b\|_{L^2}^{1-\beta}
 \|\Delta b\|_{L^2}^{1+\beta}\\
&\leq \frac{1}{16}\|\Delta b\|_{L^2}^2
 +C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}
 \|\nabla b\|_{L^2}^2.
\end{align*}
\begin{align*}
\ell_4
&= \sum_i\int\partial_i[(\operatorname{rot} b\times b)\times b]\partial_i\operatorname{rot} b dx
 \leq\sum_i\int[(\operatorname{rot} b\times\partial_ib)\times b]\partial_i\operatorname{rot} b dx\\
&\quad +\sum_i\int[(\operatorname{rot} b\times b)\times\partial_ib]\partial_i\operatorname{rot} b dx
 \leq C\|b\|_{L^\infty}\|\nabla b\|_{L^4}^2\|\Delta b\|_{L^2}\\
&\leq \frac{1}{16}\|\Delta b\|_{L^2}^2
 +C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}
 \|\nabla b\|_{L^2}^2.
\end{align*}
Inserting the above estimates into \eqref{4.5}, and combining this 
with \eqref{4.4}, and using the Gronwall inequality, we conclude that
\begin{equation}
\|\nabla u\|_{L^\infty(0,T;L^2)}+\|\nabla b\|_{L^\infty(0,T;L^2)}\leq C.\label{4.6}
\end{equation}
This completes the proof by \eqref{1.15}.


\subsection*{Acknowledgment}
This work is partially supported by NSFC (No. 11171154). 
The authors would like to thank the referees for their careful reading 
and helpful suggestions.

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