% Submitted to Jesus Ildefonso Diaz on July 30, 2017. cauchy02@naver.com
%

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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 103, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/103\hfil 3D incompressible MHD equations
 with Hall term]
{Regularity criteria for weak solutions to 3D incompressible
MHD equations with Hall term}

\author[J.-M. Kim \hfil EJDE-2018/103\hfilneg]
{Jae-Myoung Kim}

\address{Jae-Myoung Kim \newline
Department of Mathematical Sciences,
Seoul National University,
Seoul,  Korea}
\email{cauchy02@naver.com}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted July 30, 2017. Published May 7, 2018.}
\subjclass[2010]{35B65, 35Q35, 76W05}
\keywords{Magnetohydrodynamics equation; weak solution; \hfill\break\indent
regularity condition}

\begin{abstract}
 We study the regularity conditions for a weak solution to the
 incompressible 3D magnetohydrodynamic equations with Hall
 term in the whole space $\mathbb{R}^3$. In particular, we show the regularity
 criteria in view of gradient vectors in various spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We consider the incompressible 3D magneto hydro dynamic (MHD) equations
 with Hall term
\begin{gather}
\partial_tu -\Delta u+u\cdot\nabla u+\nabla\pi
=b\cdot\nabla b,\label{HMHD-1}\\
\partial_tb-\Delta b+u\cdot\nabla b-b\cdot\nabla u+\nabla
\times ((\nabla \times b)\times b)=0,\label{HMHD-2}\\
\operatorname{div} u=\operatorname{div} b=0,\label{HMHD-3}
\end{gather}
Here $u:Q_T:=\mathbb{R}^3\times [0,T)\to\mathbb{R}^3$ is the flow velocity
vector, $b:Q_T\to\mathbb{R}^3$ is the magnetic vector,
$\pi=p+ \frac{|b|^2}{2}:Q_T\to\mathbb{R}$ is the
total pressure. We consider the initial value problem of
\eqref{HMHD-1}--\eqref{HMHD-3}, which requires initial conditions
\begin{equation}\label{HMHD-4}
u(x,0)=u_0(x) \quad \text{and} \quad b(x,0)=b_0(x) \quad x\in\mathbb{R}^3
\end{equation}
The initial conditions satisfy the compatibility condition, i.e.
\[
\operatorname{div} u_0(x)=0, \quad \text{and} \quad
\operatorname{div} b_0(x)=0.
\]

\begin{definition} \rm
A weak solution pair $(u,b)$ of the incompressible 3D MHD equations
with the Hall term \eqref{HMHD-1}--\eqref{HMHD-4} is regular in
$Q_T$ provided that
$\|u\|_{L^{\infty}(Q_T)}+\|b\|_{L^{\infty}(Q_T)}<\infty$.
\end{definition}

For a long time, the effects of Hall current on fluids has been a
subject of great interest to researchers. A current induced in a
direction normal to the electric and magnetic fields is commonly
called Hall current \cite{Sato}. In particular, the effects of Hall
current are very important if the strong magnetic field is applied

The mathematical derivations of the incompressible 3D MHD equations
with the Hall term could be given in \cite{ADFL} from either
two-fluids or kinetic models. It is well-known that the global
existence of weak solutions, local existence and uniqueness of
smooth solutions to the system \eqref{HMHD-1}--\eqref{HMHD-4} were
established in \cite{CJ2,CJ14}. Recently, various results for this
equation were proved in view of partial regularity, temporary decay
and regularity or blow-up conditions
(see \cite{CJ2,CJ14,CS13,CJ215,CJ216,FLN14,FFNZ15,RY15,Weng16,Zhang15} and
references therein.)


We list only some results relevant to our concerns. In view of the
regularity conditions in Lorentz space, He and Wang \cite{HW07}
proved  that a weak solution pair $(u, b)$ becomes regular in
the presence of a certain type of the integral conditions, typically
referred to as Serrin's condition, namely,
\[
u \in L^{q,\infty}(0,\, T; L^{p,\infty}({\mathbb{R}}^3))\quad
\text{with } 3/p+2/q \leq 1, \; 3<p\leq \infty,
\]
or
\[
\nabla u \in L^{q,\infty}(0,\, T; L^{p,\infty}({\mathbb{R}}^3))
\quad \text{with} \ 3/p+2/q \leq 2,\; \frac{3}{2}<p\leq \infty,
\]
(also see \cite{BGR14,BPR14,LRM16}). Also, Wang proved in
\cite{Wang07} that a weak solution pair $(u, b)$ become regular if
$u$ satisfies
\[
u \in L^2(0,T ;BMO( \mathbb{R}^3).
\]
On the other hand, recently, Zhang  \cite{Zhang15} 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 $0<\beta<1$.
 Our study is motivated by these viewpoints, we
obtain the regularity conditions for a weak solution to the
incompressible 3D MHD equations with the Hall term
\eqref{HMHD-1}--\eqref{HMHD-4} in a whole space. Our proof of main
results is based on a priori estimate for the gradient of the
velocity field.


Our main results reads as follows.

\begin{theorem}\label{thm1}
Suppose that $(u,b)$ is a weak solution of \eqref{HMHD-1}--\eqref{HMHD-4}
with initial condition $u_0, b_0\in H^2(\mathbb{R}^3)$.
If $(u, b)$ satisfies one of the following cases:
\begin{equation} \label{eA}
\int_0^T\Big(\|\nabla u\|^{q}_{L^{p,\infty}}+\|\nabla
b\|^{m}_{L^{l,\infty}}\Big)dt<\infty
\end{equation}
with the relations $ \frac{3}{p}+\frac{2}{q}=2$,
$\frac{3}{2}< p\leq \infty$ and $\frac{3}{l}+\frac{2}{m}=1$,
$3< l\leq \infty$.
or
\begin{equation} \label{eB}
\int_0^T\Big(\|\nabla u\|^{q}_{L^{p,\infty}}+\|\nabla b\|_{\dot
B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}\Big)dt<\infty
\end{equation}
with the relations $ \frac{3}{p}+\frac{2}{q}=2$,
$\frac{3}{2}< p\leq \infty$ and $0<\beta<1$,
then $(u,b)$ is regular in $Q_T$.
\end{theorem}

\begin{theorem}\label{thm2}
Suppose that $(u,b)$ is a weak solution of \eqref{HMHD-1}--\eqref{HMHD-4}
with initial condition $u_0, b_0\in H^2(\mathbb{R}^3)$. If $(u, b)$ satisfies one
of the following two conditions:
\begin{equation} \label{ea}
 \int_0^T \|\nabla u\|_{BMO(\mathbb{R}^3)}+\|\nabla b\|^2_{BMO(\mathbb{R}^3)} \ dt < \infty,
\end{equation}
or
\begin{equation} \label{eb}
 \int_0^T \|\nabla u\|^2_{BMO^{-1}(\mathbb{R}^3)}+\|\nabla^2 b\|^2_{BMO^{-1}(\mathbb{R}^3)} \ dt < \infty.
\end{equation}
then $(u,b)$ is regular in $Q_T$.
\end{theorem}

Theorem \ref{thm1} extends the result by He and Wang \cite{HW07}
with respect to the gradient of the velocity field. Moreover, using
the estimate in \cite[Lemma A.5]{WZ12}, we obtain
$BMO^{-1}(\mathbb{R}^3)$-regularity condition.


This article is organized as follows:
In Section 2 we recall the notion of weak solutions and review some known results.
In Section 3, we present the proofs of the Theorem \ref{thm1} and \ref{thm2}.

\section{Preliminaries}

 In this section we introduce the notation and definitions to be
used in this paper. We also recall the well-known results
for our analysis. For $1 \leq q \leq \infty $,  $W^{k,q}( \mathbb{R}^3 )$
indicates the usual Sobolev space with standard norm
$\|\cdot\|_{k,q}$, i.e.
\[
W^{k,q}(\mathbb{R}^3) = \{ u \in L^{q}( \mathbb{R}^3 ): D^{ \alpha }u \in L^{q}(
\mathbb{R}^3 ), 0 \leq | \alpha | \leq k \}.
\]
When $q=2$, we denote $W^{k,q}(\mathbb{R}^3)$ by $H^{k}$.
All generic constants will be denoted by C, which may vary from line
to line.

\subsection{BMO and Lorentz spaces}

The John-Nirenberg space or  the Bounded Mean
Oscillation  space  (in short BMO space) \cite{JN} consists of
 all functions $f$ which are integrable on every ball
$B_R(x) \subset \mathbb{R}^3$ and satisfy:
$$
\|f\|^2_{BMO}= \sup_{x\in \mathbb{R}^3} \sup_{R>0}
\frac{1}{B(x,R)}\int_{B(x,R)} | f (y)- f_{B_R} (y)|dy<\infty.
$$
Here, $f_{B_R}$ is the average of $f$ over all ball $B_R(x)$ in
$\mathbb{R}^3$. It will be convenient to define BMO in terms of its dual
space, $\mathcal{H}^1$. On the other hand, following \cite{KT01} let
$w$ be the solution to the heat equation $w_t-\Delta w=0$ with
initial data $v$. Then
$$
\|v\|^2_{BMO}= \sup_{x\in \mathbb{R}^3}
\sup_{R>0}\frac{1}{B(x,R)}\int_{B(x,R)}\int_{0}^{R^2} |  w|^2\,dt\,dy.
$$
and define the  $BMO^{-1}$-norm by
$$
\|v\|^2_{BMO^{-1}}= \sup_{x\in \mathbb{R}^3}
\sup_{R>0}\frac{1}{B(x,R)}\int_{B(x,R)}\int_{0}^{R^2} | \nabla w|^2\,dt\,dy.
$$
We note that if $u$ is a tempered distribution. Then $u
\in BMO^{-1}$ if and only if there exist $f^i \in BMO$ with $u=\sum
\partial_i f^i$ in \cite[Theorem 1]{KT01}.


 Let $m(\varphi,t) $ be the Lebesgue measure of the set
$\{x\in \mathbb{R}^3:|\varphi(x)|> t\}$, i.e.
$$
m(\varphi,t):=m\{x\in \mathbb{R}^3:|\varphi(x)|> t\}.
$$
We denote by the Lorentz space $L^{p,q}(\mathbb{R}^3)$ with $1\leq p$,
$q\leq \infty $ with the norm \cite{Tr}
\begin{equation}\label{poiseuille}
\|\varphi\|_{L^{p,q}(\mathbb{R}^3)}
=\begin{cases}
\Big(\int_0^{\infty}t^q(m(\varphi,t))^{q/p} \frac{dt}{t}
\Big)^{1/q}<\infty, & \text{for } 1\leq q ,\\
\sup_{t\geq 0}\{t(m(\varphi,t))^{1/p}\} ,
&\text{for }  q=\infty\,.
\end{cases}
\end{equation}
Followed in \cite{Tr}, Lorentz space $L^{p,q}(\mathbb{R}^3)$ may be defined
by real interpolation methods
\begin{equation}\label{interpolation-lorentz}
L^{p,q}(\mathbb{R}^3) =(L^{p_{1}}(\mathbb{R}^3),\,L^{p_{2}}(\mathbb{R}^3))_{\alpha,q},
\end{equation}
with
$
\frac{1}{p}=\frac{1-\alpha}{p_{1}}+\frac{\alpha}{p_{2}}$,
$1\leq p_{1}<p<p_{2}\leq \infty$.
From the interpolation method above, we note that
\begin{equation}
L^{\frac{2p}{p-1},2}(\mathbb{R}^3)
=\Big(L^2(\mathbb{R}^3), L^{6}(\mathbb{R}^3)\Big)_{\frac{3}{2p},2}.
\end{equation}
We also need the H\"{o}lder inequality in Lorentz spaces (see \cite{ON})
for our proof.

\begin{lemma}\label{oneil}
Assume $1\leq  p_1$, $p_2\leq \infty$, $1\leq  q_1$,
$q_2\leq \infty$ and $u\in L^{p_1,q_1}(\mathbb{R}^3)$,  $v\in L^{p_2,q_2}(\mathbb{R}^3)$.
 Then $uv\in L^{p_3,q_3}(\mathbb{R}^3)$ with
$ \frac1{p_3}=\frac1{p_1}+\frac1{p_2}$ and
$\frac1{q_3}\leq \frac1{q_1}+\frac1{q_2} $, and
\begin{equation}\label{2.3}
\|uv\|_{L^{p_3,q_3}(\mathbb{R}^3)}\leq C
\|u\|_{L^{p_1,q_1}(\mathbb{R}^3)}\|v\|_{L^{p_2,q_2}(\mathbb{R}^3)}\,.
\end{equation}
\end{lemma}

\subsection{Besov space}

Following \cite{Tr}, let $\mathcal{B}=\{\xi\in\mathbb{R}^d,\
|\xi|\le\frac{4}{3}\}$ and
$\mathcal{C}=\{\xi\in\mathbb{R}^d: 3/4\le|\xi|\le8/3\}$.
Choose two nonnegative smooth radial function
$\chi, \varphi$ supported, respectively, in $\mathcal{B}$ and
$\mathcal{C}$ such that
\begin{gather*}
\chi(\xi)+\sum_{j\ge0}\varphi(2^{-j}\xi)=1, \quad \xi\in\mathbb{R}^d,\\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad \xi\in\mathbb{R}^d\setminus\{0\}.
\end{gather*}
We denote $\varphi_j=\varphi(2^{-j}\xi)$,
$h=\mathcal{F}^{-1}\varphi$ and $\tilde{h}=\mathcal{F}^{-1}\chi$,
where $\mathcal{F}^{-1}$ stands for the inverse Fourier transform.
Then the dyadic blocks $\Delta_j$ and $S_j$ can be defined as
follows
\begin{gather*}
\Delta_jf=\varphi(2^{-j}D)f=2^{jd}\int_{\mathbb{R}^d}h(2^jy)f(x-y)dy,\\
S_jf=\sum_{k\le j-1}\Delta_{k}f=\chi(2^{-j}D)f
=2^{jd}\int_{\mathbb{R}^d}\tilde{h}(2^jy)f(x-y)dy.
\end{gather*}
Formally, $\Delta_j=S_j-S_{j-1}$ is a frequency projection to
annulus $\{C_{1}2^j\le|\xi|\le C_22^j\}$, and $S_j$ is a frequency
projection to the ball $\{|\xi|\le C2^j\}$. One can easily verify
that with our choice of $\varphi$,
$$
\Delta_j\Delta_{k}f=0\text{ if } |j-k|\ge2\quad \text{and}\quad
\Delta_j(S_{k-1}f\Delta_{k}f)=0\text{ if }|j-k|\ge5.
$$
With the introduction of $\Delta_j$ and $S_j$, let us recall the
definition of the  Besov space.

Let $s\in \mathbb{R}$, $p,q\in[1,\infty]$, the homogeneous space
is defined as
$$
\dot{B}_{p,q}^{s}=\{f\in \mathcal{S}':
\|f\|_{\dot{B}_{p,q}^{s}}<\infty\},
$$
where
\[
\|f\|_{\dot{B}_{p,q}^s}
=\begin{cases}
\big(\sum_{j\in
\mathbb{Z}}2^{sjq}\|\Delta_jf\|_{L^p}^{q}\big)^{1/q},
&\text{for }  1\le q<\infty,\\
\sup_{j\in\mathbb{Z}}2^{sj}\|\Delta_jf\|_{L^p}, &\text{for }
q=\infty,
\end{cases}
\]
In particular, when $p=q=2$, the Besov space and Sobolev space are
equivalence; that is
$$
\dot{H}^s\approx\dot{B}_{2,2}^s,\quad  H^s\approx B_{2,2}^s.
$$
Now we recall first the definition of weak solutions.

\begin{definition}\label{weak-solution}\rm
Let $u_0, b_0 \in L^2(\mathbb{R}^3)$ with the divergence free conditions.
 We say that $(u,b)$ is a weak solution of Hall-MHD equations
\eqref{HMHD-1}--\eqref{HMHD-3} with initial condition $u_0, b_0\in
L^2(\mathbb{R}^3)$,  if $u$ and $b$ satisfy the following:
\begin{itemize}
\item[(i)] $u \in L^{\infty}([0, T); L^2(\mathbb{R}^3) )\cap L^2([0, T); H^{1}(\mathbb{R}^3))$,
and  $b \in L^{\infty}([0, T); L^2(\mathbb{R}^3)) \cap L^2([0, T);H^{1}(\mathbb{R}^3))$.

\item[(ii)] $(u, b)$ satisfies \eqref{HMHD-1}--\eqref{HMHD-2} in the sense
of distribution; that
is
\begin{gather*}
\int^{T}_{0}{\int_{\mathbb{R}^3}{\Big(\frac{\partial \phi}{\partial
t}+\Delta\phi+(u \cdot \nabla)\phi\Big)u}}\,dx\,dt
+ \int_{\mathbb{R}^3}{u_{0}\phi(x, 0)}\,dx
= \int^{T}_{0}{\int_{\mathbb{R}^3}(b\cdot\nabla)\phi\, b}\,dx\,dt
\\
\begin{aligned}
&\int^{T}_{0}{\int_{\mathbb{R}^3}{\Big(\frac{\partial \phi}{\partial
t}+\Delta\phi+(u \cdot \nabla)\phi\Big)b}}\,dx\,dt
+\int_{\mathbb{R}^3}{b_{0}\phi(x, 0)}\,dx  \\
&= \int^{T}_{0}{\int_{\mathbb{R}^3}{(b\cdot\nabla)\phi\, u}}\,dx\,dt
+ \int^{T}_{0}\int_{\mathbb{R}^3}(\nabla \times b)\times b\cdot (\nabla
\times \phi)\,dx\,dt,
\end{aligned}
\end{gather*}
for all $\phi \in C^{\infty}_{0}({\mathbb{R}^3 \times [0, T)})$
with $\operatorname{div}\phi = 0$, and
\begin{equation*}
{\int_{\mathbb{R}^3}{u\cdot\nabla\psi}dx}=0, \quad
{\int_{\mathbb{R}^3}{b\cdot\nabla\psi}dx}=0,
\end{equation*}
for every $\psi \in C^{\infty}_{0}({\mathbb{R}^3})$.\qed
\end{itemize}
\end{definition}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1}]
 ($L^2$-estimate or energy estimate): By the standard energy estimate, we obtain
\begin{equation}
{\frac12\frac{d}{dt}\int}(|u|^2+|b|^2)\,dx+\int(|\nabla u|^2+|\nabla
b|^2)\,dx=0.\label{4.3}
\end{equation}
$\bullet$ ($H^1$-estimate): Testing $-\Delta u$ and
$-\Delta b$ to the fluid equation and by the magnetic equation of
\eqref{HMHD-1} and \eqref{HMHD-2}, respectively, using the
integrating by parts, integrating on domain, we have
\begin{equation} \label{eq-1}
\begin{aligned} 
&\frac{1}{2}\frac{d}{dt}(\|\nabla u(\tau)\|^2_{L^2(\mathbb{R}^3)}
 +\|\nabla b(\tau)\|^2_{L^2(\mathbb{R}^3)})
 +\int_{\mathbb{R}^3}(|\Delta u|^2+|\Delta b|^2)dx \\
&\leq -\int_{\mathbb{R}^3}\nabla [(u\cdot \nabla)u]: 
 \nabla u dx +\int_{\mathbb{R}^3}\nabla [(b\cdot \nabla)b]: \nabla u\, dx\\
&\quad-\int_{\mathbb{R}^3}\nabla [(u\cdot \nabla)b]\cdot \nabla b\,dx
 +\int_{\mathbb{R}^3}\nabla [(b\cdot \nabla)u]:\nabla b dx\\
&\quad +\int \nabla((\nabla \times b)\times b)\nabla \nabla \times b\,dx\\
&:=\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3+\mathcal{I}_4+\mathcal{I}_5.
\end{aligned}
\end{equation}
We estimate separately the terms in the right hand side of
\eqref{eq-1}. The first term $\mathcal{I}_1$ is computed as follows:
\begin{equation}\label{est-i1-1000}
|\mathcal{I}_1| \leq \|\nabla u\|^3_{L^3},
\end{equation}
where the divergence free condition of $u$ is used.

On the other hand, we observe that
\[
\mathcal{I}_2+\mathcal{I}_4\leq \int_{\mathbb{R}^3}|\nabla u||\nabla b|^2.
\]
since
\begin{align*}
&\int_{\mathbb{R}^3} (b\cdot \nabla)\nabla b\cdot \nabla u\,dx
+\int_{\mathbb{R}^3}(b\cdot \nabla)\nabla u\cdot \nabla b\, dx \\
&=\sum_{j=1}^3\int_{\mathbb{R}^3}{b}_{j}\Big(\frac{\partial \nabla
b}{\partial x_j}\nabla u\,dx+\frac{\partial \nabla u}{\partial x_j}
\nabla b\Big)dx \\
&=-\sum_{j=1}^3\int_{\mathbb{R}^3}{b}_j\Big(\frac{\partial
(\nabla b \nabla u)}{\partial x_j}\Big)dx
=0,
\end{align*}
where we use the product rule and $\operatorname{div}b=0$.

Note that
\begin{gather}
\|\nabla f\|_{L^4}^2\leq C\|\nabla f\|_{\dot
B_{\infty,\infty}^{-\beta}}
\|f\|_{\dot H^{1+\beta}}\quad \text{with } 0<\beta<1,\label{1.32}\\
\|f\|_{\dot H^{1+\beta}} \leq C\|\nabla f\|_{L^2}^{1-\beta}\|\Delta
f\|_{L^2}^\beta\quad \text{with }
 0<\beta<1,\label{1.33}
\end{gather}
(e.g. see \cite{MO02, Meyer06} and \cite[Theorem 2.42]{BCD11}).

First of all, using the interpolation \eqref{interpolation-lorentz},
Lemma \ref{oneil}, H\"{o}lder and Young's inequalities, we estimate
$\mathcal{I}_3$ as follows:
\begin{equation}
\begin{aligned}\label{b-3}
|\mathcal{I}_3|\leq \int_{\mathbb{R}^3}|\nabla b|^2|\nabla u|dx &\leq
\|\nabla
u\|_{L^{p,\infty}}\||\nabla b|^2\|_{L^{\frac{p}{p-1},1}}\\
&= \|\nabla
u\|_{L^{p,\infty}}\|\nabla b\|^2_{L^{\frac{2p}{p-1},1}}\\
&\leq C\|\nabla
u\|_{L^{p,\infty}}\|\nabla b\|^{2\theta}_{L^2}\|\nabla^2 b\|^{2(1-\theta)}_{L^2}\\
&\leq C\|\nabla
u\|^{\frac{2p}{2p-3}}_{L^{p,\infty}}\|\nabla b\|^2_{L^2}+\frac{1}{16}\|\nabla^2 b\|^2_{L^2},\\
\end{aligned}
\end{equation}
where $\theta=1-\frac{3}{2p}$. Similarly, from \eqref{est-i1-1000},
we have
\[
|\mathcal{I}_1|\leq \|\nabla u\|^3_{L^3} \leq C\|\nabla
u\|^{\frac{2p}{2p-3}}_{L^{p,\infty}}\|\nabla
u\|^2_{L^2}+\frac{1}{16}\|\nabla^2 u\|^2_{L^2}.
\]
\smallskip

\noindent\textbf{Case 1:}
 Again, using the interpolation \eqref{interpolation-lorentz},
Lemma \ref{oneil}, H\"{o}lder and Young's inequalities, we bound
$\mathcal{I}_5$ as follows.
\begin{align*}
|\mathcal{I}_5| 
& \leq C\|\nabla b\|_{L^{l,\infty}}\|\nabla b\|_{L^{{\frac{2l}{l-2}},2}}
 \|\Delta b\|_{L^{2,2}}\\
&\leq C\|\nabla b\|_{L^{l,\infty}}\|\nabla b\|^{\frac{l-3}{l}}_{L^2}
 \|\Delta b\|^{\frac{l+3}{l}}_{L^2}\\
&\leq C\|\nabla b\|^{\frac{2l}{l-3}}_{L^{l,\infty}}
 \|\nabla b\|_{L^2}^2+\frac{1}{16}\|\Delta b\|_{L^2}^2.
\end{align*}
Summing  the terms $\mathcal{I}_1$--$\mathcal{I}_5$, inequality
\eqref{eq-1} becomes
\begin{equation} \label{eq-2}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|\nabla u\|^2_{L^2}+\|\nabla b\|^2_{L^2})
+\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla^2 u|^2+|\nabla^2 b|^2)\,dx \\
&\leq C\Big(\|\nabla u\|^{\frac{2p}{2p-3}}_{L^{p,\infty}}+\|\nabla
b\|^{\frac{2l}{l-3}}_{L^{l,\infty}}\Big)(\|\nabla
u\|^2_{L^2}+\|\nabla b\|^2_{L^2}).
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Case 2:}
Using \eqref{1.32} and \eqref{1.33}, we bound $\mathcal{I}_5$ as
follows:
\begin{align*}
\mathcal{I}_5 
&= -\sum_i\int(\nabla \times
 b\times\partial_ib)\partial_i\nabla \times 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*}
Summing the terms $\mathcal{I}_1$--$\mathcal{I}_5$, the
\eqref{eq-1} becomes
\begin{equation} \label{eq-2-1}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|\nabla u\|^2_{L^2}+\|\nabla b\|^2_{L^2})
+\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla^2 u|^2+|\nabla^2 b|^2)\,dx \\
&\leq C\Big(\|\nabla u\|^{\frac{2p}{2p-3}}_{L^{p,\infty}}+\|\nabla
b\|_{\dot B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}\Big)(\|\nabla
u\|^2_{L^2}+\|\nabla b\|^2_{L^2}).
\end{aligned}
\end{equation}

For Cases 1 and 2 with the given conditions, we
apply the Grownwall's inequality to estimates \eqref{eq-2} and
\eqref{eq-2-1}, respectively, to find
\begin{align*}
&\sup_{0<\tau\leq T}(\|\nabla u(\tau)\|^2_{L^2}+\|\nabla
b(\tau)\|^2_{L^2}) +\int_0^T\int_{\mathbb{R}^3}(|\nabla^2 u|^2+|\nabla^2
b|^2) \ dx\,dt \\
& \leq C(\|\nabla u(0)\|^2_{L^2}+\|\nabla b(0)\|^2_{L^2}).
\end{align*}

\noindent $\bullet$ ($H^2$-estimate) Applying the operator $\Delta$ to
\eqref{HMHD-1}--\eqref{HMHD-2}, then multiplying it by $\Delta u$
and $\Delta b$, respectively, and integrating on domain, we obtain
\begin{align*}
&\frac{1}{2} \frac{d}{dt}(\|\Delta u\|^2+\|\Delta b\|^2)dx +
(\|\nabla \Delta u|_{L^2}^2+\|\nabla \Delta b|_{L^2}^2) \\
&=-\int_{\mathbb{R}^3}\Delta (u \cdot \nabla u) \cdot \Delta
u\,dx+\int_{\mathbb{R}^3}\Delta (b \cdot \nabla b) \cdot \Delta u\,dx\\
&\quad -\int_{\mathbb{R}^3}\Delta (u \cdot \nabla b) \cdot \Delta b\,dx \\
&\quad +\int_{\mathbb{R}^3}\Delta (b \cdot \nabla u) \cdot \Delta b\,dx
 -\int_{\mathbb{R}^3}\Delta((\nabla \times b) \times b) \cdot \Delta
\nabla \times b\,dx \\
&:=\mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{J}_4+\mathcal{J}_5
\end{align*}
By the commutator estimate in \cite[Theorem 2.1 or Corollary
2.1]{FMRR14} or \cite{KP88}, we note that
\begin{gather*}
\big|\int_{\mathbb{R}^3}\Delta [(u\cdot \nabla)u],\Delta u\,dx\big|
\leq C\|\nabla u\|_{H^2}\| u\|^2_{H^2}, \\
\big|\int_{\mathbb{R}^3}\Delta [(u\cdot \nabla)b],\Delta b\,dx\big|
\leq C\|\nabla u\|_{H^2}\| b\|^2_{H^2}, \\
\big|\int_{\mathbb{R}^3}\Delta [(b\cdot \nabla)u],\Delta b\,dx\big|
\leq C\|\nabla u\|_{H^2}\| b\|^2_{H^2}.
\end{gather*}
Also,  integrating by parts we obtain the estimate for the
remaining convection term follows as:
\[
\big|\int_{\mathbb{R}^3}\Delta [(b\cdot \nabla)b],\Delta u\,dx\big|
\leq C|\langle\Delta [b\otimes b],\Delta \nabla u\,dx|
\leq C\|\nabla u\|_{H^2}\| b\|^2_{H^2}.
\]
Thus
\begin{equation}\label{thm1-H2}
\begin{aligned}
|\mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{J}_4| 
&\leq C\|\nabla u\|_{H^2}(\| u\|^2_{H^2}+\| b\|^2_{H^2}) \\
&\leq C (\|u\|^4_{H^2}+\| b\|^4_{H^2})+\frac{1}{128}\|\nabla u\|^2_{H^2}\\
&\leq C (\| u\|^2_{H^2}+\| b\|^2_{H^2})(\| u\|^2_{H^2}+\|
b\|^2_{H^2})+\frac{1}{128}\|\nabla u\|^2_{H^2}
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Case 1:}
For the term $\mathcal{J}_5$, using the chain rule, we note that 
\begin{equation}\label{j5}
\mathcal{J}_5=\int_{\mathbb{R}^3}(\nabla \times b \times \Delta b + 2
\partial_i(\nabla \times b)\times \partial_ib)\nabla \Delta b\,dx
\end{equation} 
And thus, we have
\[
|\mathcal{J}_5|\leq \|\nabla b\|_{L^{l},\infty}
\|\Delta b\|_{L^{\frac{2l}{l-2},2}} \|\nabla \Delta b\|_{L^2} \leq C
\|\nabla b\|^{\frac{2l}{l-3}}_{L^{l},\infty}
\|\Delta b\|^2_{L^2} +\frac{1}{128}\|\nabla \Delta b\|^2_{L^2}
\]
Summing  the estimate of terms $\mathcal{J}_1$--$\mathcal{J}_5$
with the energy estimate and $H^1$-estimates, we obtain
\begin{equation}\label{h2-case1}
\begin{aligned}
&\frac{1}{2} \frac{d}{dt}(\|u\|_{H^2}^2+\|b\|_{H^2}^2) +
\frac{1}{2}(\|u|_{H^3}^2+\|b|_{H^3}^2) \\
&\leq  C \Big( \|\nabla
u\|^{\frac{2q}{2q-3}}_{L^{q,\infty}}+\|\nabla b\|^{\frac{2l}{l-3}}_{L^{l},\infty}+\| u\|^2_{H^2}
+\| b\|^2_{H^2}\Big)(\| u\|^2_{H^2}+\| b\|^2_{H^2})
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Case 2:}
Using \eqref{1.32} and \eqref{1.33}, we bound $\mathcal{J}_5$ as
follows:
\begin{align*}
|\mathcal{J}_5|  
&\leq C\|\Delta b\|_{\dot B_{\infty,\infty}^{-\beta}}
 \|\nabla b\|_{\dot H^{1+\beta}}\|\nabla \Delta b\|_{L^2}\\
& \leq C\|\nabla b\|_{\dot B_{\infty,\infty}^{-\beta}}\|\nabla^2
b\|_{L^2}^{1-\beta}
 \|\nabla \Delta b\|_{L^2}^{1+\beta}\\
&\leq C\|\nabla b\|_{\dot
B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}
 \| b\|_{H^2}^2+\frac{1}{128}\|\nabla \Delta b\|_{L^2}^2.
\end{align*}
As in case 1, summing  the estimate of terms
$\mathcal{J}_1$--$\mathcal{J}_5$ with the energy estimate and
$H^1$-estimates, we obtain
\begin{equation}\label{h2-case2}
\begin{aligned}
&\frac{1}{2} \frac{d}{dt}(\|u\|_{H^2}^2+\|b\|_{H^2}^2) +
\frac{1}{2}(\|u|_{H^3}^2+\|b|_{H^3}^2) \\
&\leq  C \Big( \|\nabla u\|^{\frac{2p}{2p-3}}_{L^{p,\infty}}+\|\nabla
b\|_{\dot B_{\infty,\infty}^{-\beta}}^\frac{2}{1-\beta}+\|
u\|^2_{H^2}+\| b\|^2_{H^2}\Big)(\| u\|^2_{H^2}+\| b\|^2_{H^2})
\end{aligned}
\end{equation}

Under the assumption, we apply Grown's inequality to the estimates
\eqref{h2-case1} and \eqref{h2-case2}, respectively, we finally
obtain
\[
\sup_{0\leq \tau\leq T}(\|u(\tau)\|_{H^2}^2+\|b(\tau)\|_{H^2}^2) +
\|u\|_{H^3}^2+\|b\|_{H^3}^2\leq (\|u_0\|_{H^2}^2+\|b_0\|_{H^2}^2)
\]
Th proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2}]
 ($H^1$-estimate): Testing $-\Delta u$ and
$-\Delta b$ to the fluid equation and the magnetic equation of
\eqref{HMHD-1} and \eqref{HMHD-2}, respectively, using the
integrating by parts, integrating on domain, we have
\begin{align}
\begin{aligned} \label{thm2-eq-1}
&\frac{1}{2}\frac{d}{dt}(\|\nabla u(\tau)\|^2_{L^2(\mathbb{R}^3)}+\|\nabla
 b(\tau)\|^2_{L^2(\mathbb{R}^3)}) +\int_{\mathbb{R}^3}(|\Delta u|^2+|\Delta b|^2)dx\\
&\leq -\int_{\mathbb{R}^3}\nabla
 [(u\cdot \nabla)u]: \nabla u dx+\int_{\mathbb{R}^3}\nabla [(b\cdot
 \nabla)b]: \nabla u dx\\
&\quad-\int_{\mathbb{R}^3}\nabla [(u\cdot \nabla)b]: \nabla b\,dx
 +\int_{\mathbb{R}^3}\nabla [(b\cdot \nabla)u]:\nabla b dx\\
&\quad +\int \nabla((\nabla \times b)\times b): \nabla \nabla \times b\,dx\\
&:=\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3+\mathcal{I}_4+\mathcal{I}_5.
\end{aligned}
\end{align}
\smallskip

\noindent\textbf{Case 1:} By the H\"{o}lder, Young inequalities and the 
space duality $BMO$-$H^1$, we have
\begin{align*}
\int_{\mathbb{R}^3}|\nabla b|^2|\nabla u|dx
&\leq  \|\nabla u\|_{BMO}\||\nabla b|^2\|_{\mathcal{H}^1}
\leq  \|\nabla u\|_{BMO}\|\nabla b\|_{L^2}\|\nabla b\|_{L^2}\\
&=  \|\nabla u\|_{BMO}\|\nabla b\|^2_{L^2}.
\end{align*}
Similarly, we obtain
\[
\int_{\mathbb{R}^3}|\nabla u|^3dx \leq C\|\nabla u\|_{BMO}\|\nabla
u\|^2_{L^2}.
\]
Again, by the vector identity, the H\"{o}lder and Young
inequalities, we have
\begin{equation} \label{S-5-14}
\begin{aligned}
\mathcal{I}_5&= \int_{\mathbb{R}^3} \nabla[
(\nabla\times b)\times b]\cdot \nabla(\nabla \times b)dx\\
&=\int_{\mathbb{R}^3} \Big((\nabla\times b)\times \nabla b
 -\nabla(\nabla\times b)\times b\Big)\cdot \nabla(\nabla \times b)dx\\
&\leq C\|\nabla b\|_{BMO}\||\nabla b||\nabla^2 b|\|_{\mathcal{H}^1}\\
&\leq  C\|\nabla b\|^2_{BMO}\|\nabla
b\|^2_{L^2}+\frac{1}{8}\|\nabla^2 b\|^2_{L^2}.
\end{aligned}
\end{equation}
Summing the estimates above, the \eqref{thm2-eq-1} becomes
\begin{equation}
\begin{aligned} \label{eq-2000}
&\frac{d}{dt}(\|\nabla u\|^2_{L^2}+\|\nabla b\|^2_{L^2})
+\int (|\nabla^2 u|^2+|\nabla^2 b|^2)\, dx\\
&\leq C(\|\nabla u\|_{BMO}+\|\nabla b\|^2_{BMO})(\|\nabla
u\|^2_{L^2}+\|\nabla b\|^2_{L^2}).
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Case 2:}
 Following \cite[Lemma A.5]{WZ12}, we note that
\[
\|u\|^2_{L^4} = \|uu\|_{L^2} \leq C\|\nabla
u\|_{L^2}\|u\|_{BMO^{-1}}.
\]
Using this estimate, we have
\begin{align*}
\int_{\mathbb{R}^3}|\nabla b|^2|\nabla u|dx
&\leq  \|\nabla
u\|_{L^2}\|\nabla b\|^2_{L^4}\leq  C\|\nabla u\|_{L^2}\|\nabla^2
b\|_{L^2}\|\nabla b\|_{BMO^{-1}}\\
&\leq  C\|\nabla u\|^2_{L^2}\|\nabla
b\|^2_{BMO^{-1}}+\frac{1}{8}\|\nabla^2 b\|^2_{L^2}.
\end{align*}
Similarly, we obtain
\[
\int_{\mathbb{R}^3}|\nabla u|^3dx
\leq C\|\nabla u\|^2_{L^2}\|\nabla u\|^2_{BMO^{-1}}+\frac{1}{8}\|\nabla^2 u\|^2_{L^2}.
\]
By the vector identity, the H\"{o}lder and Young inequalities, 
we have
\begin{equation} \label{S-5-14b}
\begin{aligned}
\mathcal{I}_5
&= \int_{\mathbb{R}^3} \nabla[
(\nabla\times b)\times b]\cdot \nabla(\nabla \times b)dx\\
&=\int_{\mathbb{R}^3} \Big((\nabla\times b)\times \nabla b
 -\nabla(\nabla\times b)\times b\Big)\cdot \nabla(\nabla \times b)dx\\
&\leq C\|(\nabla\times b)\times \nabla b\|_{L^2}\|\nabla^2 b\|_{L^2}\\
&\leq  C\|\nabla^2 b\|^2_{{\rm BMO}^{-1}}\|\nabla
b\|^2_{L^2}+\frac{1}{258}\|\nabla^2 b\|^2_{L^2}\\
\end{aligned}
\end{equation}
Using the estimates above,  \eqref{thm2-eq-1} becomes
\begin{equation} \label{eq-3000}
\begin{aligned}
&\frac{d}{dt}(\|\nabla u\|^2_{L^2}+\|\nabla b\|^2_{L^2})
+\int_{}(|\nabla^2 u|^2+|\nabla^2 b|^2)\,dx \\
&\leq C(\|\nabla u\|^2_{BMO^{-1}}+\|\nabla b\|^2_{BMO^{-1}})(\|\nabla
u\|^2_{L^2}+\|\nabla b\|^2_{L^2}).
\end{aligned}
\end{equation}


\noindent $\bullet$ ($H^2$-estimate)  Taking $\Delta$ to
\eqref{HMHD-1}--\eqref{HMHD-2}, then multiplying it by $\Delta u$
and $\Delta b$, respectively, and integrating on domain, we derive
\begin{equation}\label{thm2-eq-11}
\begin{aligned}
&\frac{1}{2} \frac{d}{dt}(\|\nabla^2 u\|^2+\|\nabla^2 b\|^2)dx
 + (\|\nabla \Delta u|_{L^2}^2+\|\nabla \Delta b|_{L^2}^2) \\
&=-\int_{\mathbb{R}^3}\Delta (u \cdot \nabla u) \cdot \Delta u\,dx
+\int_{\mathbb{R}^3}\Delta (b \cdot \nabla b) \cdot \Delta u\,dx
-\int_{\mathbb{R}^3}\Delta (u \cdot \nabla b) \cdot \Delta b\,dx \\
&\quad +\int_{\mathbb{R}^3}\Delta (b \cdot \nabla u) \cdot \Delta b\,dx
-\int_{\mathbb{R}^3}\Delta((\nabla \times b) \times b) : \Delta \nabla
\times b\,dx \\
&:=\mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{J}_4+\mathcal{J}_5
\end{aligned}
\end{equation}
As in the proof of Theorem \ref{thm1}, namely \eqref{thm1-H2}, we note
that
\[
|\mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{J}_4| \leq C (\|
u\|^2_{H^2}+\| b\|^2_{H^2})(\| u\|^2_{H^2}+\|
b\|^2_{H^2})+\frac{1}{128}\|\nabla u\|^2_{H^2}.
\]
\smallskip

\noindent\textbf{Case 1.}
From \eqref{j5} with the space duality $BMO$-$\mathcal{H}^1$, we
have
\[
|\mathcal{J}_5|\leq \|\nabla b\|_{{\rm BMO}}\|\nabla^2 b\|_{L^2}
\|\nabla \Delta b\|_{L^2} \leq C \|\nabla b\|^2_{{\rm BMO}}\|\nabla^2 b\|^2_{L^2} +\frac{1}{128}\|\nabla \Delta b\|^2_{L^2}.
\]
Summing the estimates $\mathcal{J}_1$--$\mathcal{J}_5$ with the
energy estimate and $H^1$-estimates, the \eqref{thm2-eq-11} becomes
\begin{equation}\label{thm2-h2-case1}
\begin{aligned}
&\frac{d}{dt}(\| u\|_{H^2}^2+\|b\|_{H^2}^2) + (\|u|_{H^3}^2+\|b|_{H^3}^2)\\
&\leq  C \Big( \|\nabla u\|_{BMO}+\|\nabla b\|^2_{{\rm BMO}}
 +\|u\|_{H^2}^2+\|b\|_{H^2}^2\Big)(\| u\|_{H^2}^2+\|b\|_{H^2}^2).
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Case 2.}
From \eqref{thm1-H2}, we note that
\[
|\mathcal{J}_1+\mathcal{J}_2+\mathcal{J}_3+\mathcal{J}_4| \leq C (\|
u\|^2_{H^2}+\| b\|^2_{H^2})(\| u\|^2_{H^2}+\|
b\|^2_{H^2})+\frac{1}{128}\|\nabla u\|^2_{H^2}.
\]
Following \cite[Lemma A.5]{WZ12}, we note that
\[
\|uu\|_{L^2} \leq C\|\nabla u\|_{L^2}\|u\|_{BMO^{-1}}.
\]
As in the previous proof, namely, from  \eqref{j5} with the space
duality $BMO$-$\mathcal{H}^1$, we have
\[
|\mathcal{J}_5|\leq C \|\nabla^2 b\|_{{\rm BMO}^{-1}}\|\nabla^2 b\|_{L^2} +\frac{1}{128}\|\nabla \Delta b\|^2_{L^2}.
\]
Summing $\mathcal{J}_1$--$\mathcal{J}_5$ with the energy estimate
and $H^1$-estimate, the \eqref{thm2-eq-11} becomes
\begin{equation}\label{thm2-h2-case2}
\begin{aligned}
&\frac{d}{dt}(\| u\|_{H^2}^2+\|b\|_{H^2}^2) +(\|u|_{H^3}^2+\|b|_{H^3}^2) \\
&\leq C(\|\nabla u\|^2_{BMO^{-1}}+\|\nabla^2 b\|^2_{BMO^{-1}}+\|
u\|_{H^2}^2+\|b\|_{H^2}^2)(\| u\|_{H^2}^2+\|b\|_{H^2}^2).
\end{aligned}
\end{equation}
Under the assumption, we apply Gronwall's inequality to the estimates
\eqref{thm2-h2-case1} and \eqref{thm2-h2-case2}, respectively, we
finally obtain
\[
\sup_{0\leq \tau\leq T}(\|u(\tau)\|_{H^2}^2+\|b(\tau)\|_{H^2}^2) +
\|u\|_{H^3}^2+\|b\|_{H^3}^2\leq (\|u_0\|_{H^2}^2+\|b_0\|_{H^2}^2).
\]
The proof is complete.
\end{proof}


\subsection*{Acknowledgments}
J.-M. Kim was supported by grants NRF-2015R1A5A1009350 and
NRF-2016R1D1A1B03930422.


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\end{document}