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

\AtBeginDocument{{\noindent\small
International Conference on Applications of Mathematics to Nonlinear Sciences,\newline
\emph{Electronic Journal of Differential Equations},
Conference 24 (2017), pp. 103--113.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document} \setcounter{page}{103}
\title[\hfilneg EJDE-2017/conf/24\hfil 
2D micropolar equations with partial dissipations]
{Global regularity criteria for 2D micropolar equations
with partial dissipations}

\author[D. Regmi \hfil EJDE-2017/conf/24\hfilneg]
{Dipendra Regmi}

\address{Dipendra Regmi \newline
Department of Mathematics,
University of North Georgia,
3820 Mundy Mill Rd,
Oakwood, GA 30566, USA}
\email{dipendra.regmi@ung.edu}


\thanks{Published November 15, 2017.}
\subjclass[2010]{35Q35, 35B35, 35B65, 76D03}
\keywords{Global regularity; micro-polar equations; partial dissipation}

\begin{abstract}
 This article addresses the global regularity (in time) issue of two
 dimensional  incompressible  micropolar equations with various partial
 dissipations.  Micropolar fluids represent a class of fluids with
 nonsymmetric stress tensor  (called polar fluids) such as  fluids consisting
 of suspending particles, dumbbell molecules, etc. Whether or not its
 classical solutions of 2D micropolar  equations without velocity dissipation
 and micro-rotational viscosity  develop finite time singularities is a
 difficult problem, and remains open.  Here, we  mainly focus on two types
 of partial dissipation cases, and we prove the conditional global regularity.
\end{abstract}

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

\section{Introduction}

In this article we study the global existence and regularity of classical
solutions to the 2D incompressible micropolar equations
with various dissipation. The standard 3D incompressible
micropolar equations can be written as
\begin{equation} \label{MPE3}
\begin{gathered}
\partial_t u + (u\cdot\nabla) u +\nabla \pi
 = ( \gamma +\kappa) \Delta u +  2\kappa \nabla \times \omega,\\
\partial_t \omega + (u\cdot \nabla) \omega + 4\kappa \omega
 = \eta \Delta \omega+ \alpha \nabla \nabla\cdot \omega + 2 \kappa \nabla \times u,\\
\nabla \cdot u=0,
\end{gathered}
\end{equation}
where, for $\mathbf{x}\in \mathbb{R}^3$ and $t\ge 0$,
$u=u(\mathbf{x},t), \omega=\omega(\mathbf{x},t)$  and $\pi=\pi(\mathbf{x},t)$
denote the velocity field, the micro-rotation field and the pressure,
respectively, and $ \gamma$ denotes the kinematic viscosity, $\kappa$
the micro-rotational viscosity,   and  $\alpha$, and $\eta$  the
angular viscosities.

The 3D micropolar equations reduce to the 2D micropolar equations when
\[
 u=(u_1(x,y,t),u_2(x,y,t),0), \quad
 \omega=(0,0,\omega_3(x,y,t)), \quad \pi=\pi(x,y,t),
\]
 The 2D incompressible micropolar equations can be written as,
\begin{equation} \label{MPE2}
\begin{gathered}
\partial_t u + (u\cdot\nabla) u +\nabla \pi= ( \gamma +\kappa) \Delta u
 +  2\kappa \nabla \times \omega,\\
\partial_t \omega + (u\cdot \nabla) \omega + 4\kappa \omega = \eta \Delta \omega
 + 2 \kappa \nabla \times u,\\
\nabla \cdot u=0,
\end{gathered}
\end{equation}
where $u=(u_1, u_2)$, $\nabla\times \omega= (-\partial_y \omega,
\partial_x \omega)$ and $\nabla \times u = \partial_x u_2-\partial_y u_1$

Micropolar fluids represent a class of fluids with nonsymmetric stress
 tensor (called polar fluids) such as fluids
consisting of suspending particles, dumbbell molecules, etc (see, e.g.,
\cite{Cowin, Erdo, Er, Er2, Luka}). In the absence of micro-rotational
effects, this system reduces to well-known Navier Stokes equations.
A generalization of the 2D micropolar equations is given by
\begin{equation} \label{GMP}
\begin{gathered}
\partial_t u_1 + (u\cdot\nabla) u_1 +\partial_x \pi
 = \mu_{11}   \partial_{xx} u_1+ \mu_{12} \partial_{yy} u_1 +  2\kappa \partial_y \omega,\\
\partial_t u_2 + (u\cdot\nabla) u_2 +\partial_y \pi
 = \mu_{21}   \partial_{xx} u_2+ \mu_{22} \partial_{yy} u_2 -  2\kappa \partial_x \omega,\\
\partial_t \omega + (u\cdot \nabla) \omega + 4\kappa \omega
 =\eta_1 \partial_{xx} \omega+\eta_2 \partial_{yy} \omega + 2 \kappa \nabla \times u,\\
\nabla \cdot u=0.
\end{gathered}
\end{equation}
where we have written the velocity equation in its two components.
Clearly, if
$$
\mu_{11} = \mu_{12} = \mu_{21} =\mu_{22} = \gamma + \kappa, \quad
\eta_1=\eta_2=\eta
$$
then \eqref{GMP} reduces to the standard 2D micropolar
equations in \eqref{MPE2}.

We main focus on the global regularity problem on \eqref{GMP} with various
 dissipations.
 The global regularity to \eqref{GMP} with  $ \mu_{11}>0$,
$\mu_{12}>0$, $\mu_{21}>0$, $\mu_{22} >0$, and $\eta_1=\eta_2=0$ can be done easily.
Global regularity of the following cases have been established.
\begin{itemize}
\item [(I)]  $\mu_{11}>0$, $\mu_{12}>0$, $\mu_{21}>0$, $\mu_{22} >0$ and
$\eta_1=\eta_2=0$;
\item [(II)] $\mu_{11}= \mu_{12}= \mu_{21}= \mu_{22} =0$  and
$\eta_1=\eta_2>0$;
\item [(III)] $\mu_{11}=0$, $\mu_{12}>0$, $\mu_{21}>0$, $\mu_{22} =0$,
 and $\eta_1>0$, $\eta_2=0$;
\item [(IV)] $\mu_{11}>0$, $\mu_{12}>0$, $\mu_{21}=0$, $\mu_{22} =0$, and $\eta_1>0$,
$\eta_2=0$;
\item [(V)] $\mu_{11}=0$, $\mu_{12}>0$, $\mu_{21}=0$, $\mu_{22} >0$, and
$ \eta_1>0$, $\eta_2=0$
\end{itemize}

 For (I) and (II), the global regularity was established in \cite{DongZhang},
and \cite{DongB}, respectively. Very recently Regmi and Wu \cite{RegmiWu}
 studied the global regularity of the magneto-micropolar equations with
partial dissipation. The  global regularity results for cases (III)--(V) are
 included in \cite{RegmiWu}.

The global regularity to \eqref{GMP} for the case:
$ \mu_{11}>0$, $\mu_{12}=0$, $\mu_{21}>0$, $\mu_{22} =0$, and
$\eta_1>0$, $\eta_2=0$ is very difficult.
 In fact the dissipation is not sufficient to control $L^2$-norm when we
employ energy method.

In this article, we consider the global regularity to \eqref{GMP} for the
following two cases:
\begin{itemize}
 \item[Case 1:] $ \mu_{11}=0$, $\mu_{12}>0$, $\mu_{21}=0$, $\mu_{22} =0$,
 and $\eta_1> 0$, $\eta_2=0$.
\item[Case 2:]  $\mu_{11}=0$, $\mu_{12}=0$, $\mu_{21}>0$, $\mu_{22} =0$,  and
$\eta_1= 0$, $\eta_2>0$.
\end{itemize}

More precisely, we prove the following two  theorems.

\begin{theorem} \label{Main1}
Consider the   2D micropolar equations
\begin{equation} \label{2DMPE1}
\begin{gathered}
\partial_t u_1 + (u\cdot\nabla) u_1 +\partial_x \pi
= \mu_{12} \partial_{yy} u_1 +  2\kappa \partial_y \omega,\\
\partial_t u_2 + (u\cdot\nabla) u_2 +\partial_y \pi
=  -2\kappa \partial_x \omega,\\
\partial_t \omega + (u\cdot \nabla) \omega + 4\kappa \omega
=\eta_1 \partial_{xx} \omega + 2 \kappa \nabla \times u,\\
\nabla \cdot u=0.
\end{gathered}
\end{equation}
Assume $(u_0, \omega_0)\in H^2(\mathbb{R}^2)$, and $\nabla \cdot u_0=0$.
Then the system  has a unique global classical solution $(u,\omega)$
satisfying, for any $T>0$,
$(u,\omega) \in L^{\infty}([0,T]; H^{2}(\mathbb{R}^2))$
provided that
$$
\int_0^T \| \partial_x u\|_\infty^2 < \infty
$$
\end{theorem}

Another result in this article is summarized in the following theorem.

\begin{theorem} \label{Main2}
Consider the  2D micropolar equations
\begin{equation} \label{2DMPE2}
\begin{gathered}
\partial_t u_1 + (u\cdot\nabla) u_1 +\partial_x \pi =   2\kappa \partial_y \omega,\\
\partial_t u_2 + (u\cdot\nabla) u_2 +\partial_y\pi
 = \mu_{21} \partial_{xx} u_2 -2\kappa \partial_x \omega,\\
\partial_t \omega + (u\cdot \nabla) \omega + 4\kappa \omega
 =\eta_2 \partial_{yy} \omega + 2 \kappa \nabla \times u,\\
\nabla \cdot u=0.
\end{gathered}
\end{equation}
Assume $(u_0, \omega_0)\in H^2(\mathbb{R}^2)$, and $\nabla \cdot u_0=0$.
Then the system  has a unique global classical solution $(u,\omega)$
satisfying, for any $T>0$,
$ (u,\omega) \in L^{\infty}([0,T]; H^{2}(\mathbb{R}^2))$
provided that
$$
\int_0^T \|\partial_y u\|_\infty ^2 < \infty
$$
\end{theorem}

 The general approach to establish
the global existence and regularity results consists of two main steps.
The first step assesses the local (in time) well-posedness while the
second extends the local solution into a global one by obtaining
global (in time) \emph{a priori} bounds. For the systems of equations
concerned here, the local well-posedness follows from a standard
approach and shall be skipped here. Our main efforts are devoted to
proving the necessary global \emph{a priori} bounds. More precisely,
we  show that, for any $T>0$ and $t\le T$,
\begin{equation}\label{h2bd}
\|(u,\omega)(\cdot, t)\|_{H^2(\mathbb{R}^2)} \le C,
\end{equation}
where $C$ denotes a bound that depends on $T$ and the initial data.
\vskip .1in
The rest of this paper is divided into three sections.
The first section is about preliminaries. The last two sections
devoted to the proof of one of the theorems stated above.
To simplify the notation, we will write $\|f\|_2$ for $\|f\|_{L^2}$,
 $\int f$ for $\int_{\mathbb{R}^2} f \,dx dy$ and write
${\frac {\partial}{\partial x}f }$, $\partial_{x} f$ or $f_{x}$ as the first
 partial derivative, and ${\frac {\partial^2}{\partial x^2}f }$ or
$\partial_{xx} f$ as the second partial throughout the rest of this paper.
For the simplicity we consider all non zero parameters equal 1
(although we include some of these parameter in the proof)

\section{Preliminaries}

In this section we state some important results which will be used later.
In the proof of Theorem \ref{Main1} and \ref {Main2}, the following anisotropic
type Sobolev inequality will be frequently used. Its proof can be
found in \cite{CaoWu}.

\begin{lemma} \label{triple}
If $f, g, h, \partial_yg, \partial_xh\in L^2(\mathbb{R}^2)$, then
\begin{equation} \label{qtri}
 \iint_{\mathbb{R}^2}| f g h|  \,dx\,dy
\le C  \|f\|_2  \|g\|_{2}^{1/2} \|\partial_y g\|_2^{1/2} \|h\|_{{2}}^{1/2}
 \|\partial_x h\|_2^{1/2},
\end{equation}
where $C$ is a constant.
\end{lemma}

The following simple fact on the boundedness of Riesz transforms
will also be used.

\begin{lemma}
Let $f$ be divergence-free vector field such that
$\nabla f \in L^p$ for $ p\in (1,\infty)$. Then there exists a pure
constant $C>0$ (independent of $p$) such that
$$
\| \nabla f\|_{L^p} \leq \frac{C\, p^2}{p-1} \|\nabla \times f\|_{L^p}.
$$
\end{lemma}

\section{Proof of Theorem \ref{Main1}}

 As explained in the introduction, it suffices to establish the global
 a priori bound for the solution in $H^2$. For the sake of clarity,
we divide this process into two subsections. The first subsection
proves the global $H^1$-bound while the second proves the global $H^2$-bound.


\subsection{Global $L^2$-bound}

\begin{lemma}
Assume that  $(u_0, \omega_0)$ satisfies the condition stated in
Theorem \ref{Main1}. Let $(u, \omega)$ be the corresponding solution
of \eqref{2DMPE1}. Then, $(u,\omega)$ obeys the following global $L^2$-bound,
\begin{align*}
&\|u(t)\|_{L^2}^2  + \|\omega(t)\|_{L^2}^2
 +  \mu_{12} \int_0^t  \|\partial_y u_1(\tau)\|_{L^2}^2 \, d\tau\\
& + \eta_{1} \int_0^t  \|\partial_x \omega(\tau)\|_{L^2}^2\, d\tau
 + 8\kappa \int_0^t  \| \omega(\tau)\|_{L^2}^2\, d\tau\\
&\leq e^{\displaystyle 32 \kappa^2(\frac{1}{\mu_{12}}
 +\frac{1}{\eta_{1}})t}(\|u_0\|_2^2+\|\omega\|_2^2)  %\label{L2}
\end{align*}
for any $t\ge 0$.
\end{lemma}

Taking the $L^2$ inner product of \eqref{2DMPE1} with $(u,\omega)$,
integrating with respect to space variable, we obtain
\begin{align*}
&\frac {1}{2} \frac{d}{dt} \left( \|u(t)\|_2^2 +  \|\omega(t)\|_2^2 \right)
 + \mu_{12} \|\partial_{y} u_1\|_2^2+\eta_{1} \|\partial_{x} \omega\|_2^2
 + 4 \chi \|\omega(t)\|_2^2\\
&= \int_{\mathbb{R}^2} 2\kappa \{ (\nabla \times \omega) \cdot u
 + (\nabla \times u) \omega\} dx\\
&=\int_{\mathbb{R}^2} 2\kappa \{ \partial_y \omega u_1 -\partial_x \omega u_2
 + \partial_x u_2 \omega -\partial_y u_1 \omega\} dx\\
&=-\int_{\mathbb{R}^2} 4\kappa \{ \partial_x \omega u_2 + \partial_y u_1 \omega\} dx\\
&\leq 4 \kappa (\|\partial_x \omega\|_2 \|u\|_2+ \|\partial_y u_1\|_2 \|\omega\|_2)\\
&\leq \frac{\mu_{12}}{2} \|\partial_y u_1\|_2^2 +\frac{\eta_{1}}{2} \|\partial_x \omega\|_2^2
 + \frac{32 \kappa^2}{\mu_{12}} \|\omega\|_2^2
 + \frac{32 \kappa^2}{\eta_{1}} \|u\|_2^2,
\end{align*}
where we have used the following fact due to the divergence free condition
$$
 \int_{\mathbb{R}^2} ( u \cdot \nabla) u\cdot u \, dx
=\int_{\mathbb{R}^2} ( u \cdot \nabla)\omega \omega \,dx=0
$$
Applying Gronwall inequality for $0< t < \infty$,
\begin{align*}
&\|u(t)\|_2^2 + \|\omega(t)\|_2^2
 + \mu_{12}  \int_0^t  \|\partial_x u(\tau)\|_{L^2}^2 \, d\tau \\
&+ \eta_{1} \int_0^t  \|\partial_x \omega(\tau)\|_{L^2}^2\, d\tau
 + 8\kappa \int_0^t  \| \omega(\tau)\|_{L^2}^2\, d\tau\\
&\leq e^{\displaystyle 32 \kappa^2(\frac{1}{\mu_{12}}
 +\frac{1}{\eta_{1}}) t}(\|u_0\|_2^2+\|\omega\|_2^2)
\end{align*}

\subsection{$ H^1$-bound}

\begin{theorem} \label{DR}
Assume that  $(u_0, \omega_0)$ satisfies the condition stated in
Theorem \ref{Main1}. Then the solution $(u, \omega)$  corresponding
to  \eqref{2DMPE1} obeys, for any $0<t<\infty$,
\begin{align*}
& \|\nabla u(t)\|_2^2 + \|\nabla \omega(t)\|_2^2
 + \mu_{12} \int_0^t \|\nabla \partial_y u_1( \tau)\|_2^2 \, d\tau \\
&+\eta_{1} \int_0^t \|\nabla \partial_x \omega(\tau)\|_2^2 \, d\tau
 + 8 \kappa \int_0^t  \|\nabla \omega(t)\|_2^2 \, d\tau
 \leq C
\end{align*}
\end{theorem}

\begin{proof}
Taking the $L^2$ inner product of \eqref{2DMPE1} with $(\Delta u, \Delta \omega)$ yields
\begin{align*}
&\frac{1}{2} \frac{d}{dt} \left( \|\nabla u(t)\|_2^2 + \|\nabla \omega (t)\|_2^2 \right)
 + \mu_{12} \|\nabla \partial_y u_1\|_2^2+\eta_1 \|\nabla \partial_x \omega(t)\|_2^2
 + 4 \kappa \|\nabla \omega(t)\|_2^2\\
&=\int_{\mathbb{R}^2} (2\kappa (\nabla \times \omega) \cdot (-\Delta u)
 + 2 \kappa (\nabla \times u)(-\Delta \omega)) \, dx
 + \int_{\mathbb{R}^2} u \cdot \nabla \omega (-\Delta \omega) \, dx\\
&=2I_1+I_2,
\end{align*}
where we have use the fact that
$$
\int_{\mathbb{R}^2} (u\cdot\nabla) u \cdot \Delta u \, dx
=\int_{\mathbb{R}^2} \nabla \pi \cdot \Delta u\,dx=0.
$$
Note that
$$
\int_{\mathbb{R}^2} 2\kappa (\nabla \times \omega) \cdot (-\Delta u)
= \int_{\mathbb{R}^2}2 \kappa (\nabla \times u)(-\Delta \omega)) \, dx.
$$
To estimate $I_1$ we write in component-wise
\begin{align*}
I_1&=\int_{\mathbb{R}^2} 2\kappa (\nabla \times \omega) \cdot (-\Delta u)\, dx\\
&=\int_{\mathbb{R}^2} 2\kappa ( -\partial_y \omega \Delta u_1 +\partial_x \omega \Delta u_2) \, dx\\
&=\int_{\mathbb{R}^2} 2\kappa( \partial_y \omega \partial_{xx} u_1
+ \partial_y \omega \partial_{yy} u_1 -\partial_x \omega \partial_{xx}u_2 -\partial_x \omega \partial_{yy} u_2 ) \, dx\,,
\end{align*}
\begin{gather*}
\big| \int_{\mathbb{R}^2} 2 \kappa \partial_y \omega \partial_{xx} u_1 \,dx\big|
\leq \frac{\eta_1}{8} \|\nabla \partial_x \omega\|_2^2
 + \frac{32\kappa^2}{\eta_1} \|\nabla u\|_2^2,\\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_y \omega \partial_{yy} u_1 \, dx \big|
\leq \frac{\mu_2}{2} \|\nabla \partial_y u_1\|_2^2
 + \frac{16\kappa^2}{\mu_2} \|\nabla \omega\|_2^2, \\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_x \omega \partial_{xx}u_2 \,dx \big|
\leq \frac{\eta_1}{8}\|\nabla \partial_x \omega\|_2^2
 +\frac{32\kappa^2}{\eta_1} \|\nabla u\|_2^2, \\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_x \omega \partial_{yy} u_2\,dx\big|
\leq \frac{\gamma}{8} \| \nabla \partial_x \omega\|_2^2
 + \frac{32 \kappa^2}{\gamma} \|\nabla u\|_2^2, \\
I_2=\int_{\mathbb{R}^2} (u \cdot \nabla) \omega  (-\Delta \omega) \, dx
= \int_{\mathbb{R}^2} \nabla \omega \cdot \nabla u \cdot \nabla \omega \, dx,\\
I_2=\int \nabla \omega \cdot \nabla u \cdot \nabla \omega
 =\int \partial_x u_1 \omega_x^2 + 2\int u_1 \omega_y \omega_{xy}
 +\int (\partial_x u_2 + \partial_y u_1) \omega_x \omega_y ,\\
\int \partial_x u_1 \omega_x^2=-2 \int  u_1 \omega_{xx} \omega_x , \\
\big| \int  u_1 \omega_{xx} \omega_x \big|
\leq \frac{1}{48} \|\omega_{xx}\|_2^2
 + C \|u_1\|_2^2 \|\partial_y u_1\|_2^2 \|\nabla \omega\|_2^2, \\
\big|\int u_1 \omega_y \omega_{xy}\big|
 \leq \frac {1}{48} \|\nabla \omega_x\|_2^2
 + C \|u_1\|_2^2 \|\partial_y u_1\|_2^2 \|\nabla \omega\|_2^2, \\
\big| \int  \partial_y u_1 \omega_x \omega_y \big|
\leq \frac{1}{48} \|\nabla \omega_x\|_2^2+   C \|\partial_y u_1\|_2^2 \|\nabla \omega\|_2^2,\\
\big| \int \partial_x u_2  \omega_x \omega_y \big|
\leq  C \|\partial_x u_2\|_{\infty} \| \nabla \omega\|_2^2
\end{gather*}

Combining the estimates above, together with Gronwall's inequality, we obtain
\begin{align*}
&\|\nabla u(t)\|_2^2 + \|\nabla \omega(t)\|_2^2
 + \mu_{12} \int_0^t \|\nabla \partial_y u_1( \tau)\|_2^2 \, d\tau \\
&+\eta_{1} \int_0^t \|\nabla \partial_x \omega(\tau)\|_2^2 \, d\tau
 + 8 \kappa \int_0^t  \|\nabla \omega(t)\|_2^2 \, d\tau
\leq C
\end{align*}
for any $t\leq T$, where $C$ depends on $T$ and the initial $H^1$-norm.
This completes the proof of theorem.
\end{proof}

\subsection{Global $H^2$ bound and proof of Theorem \ref{Main1}}

To estimate the $H^2$-norm of $(u, b, \omega)$, we consider the equations
of $\Omega=\nabla\times u$, $\nabla \omega$,
\begin{gather}
\Omega_t + u\cdot\nabla \Omega
 =  - \mu_{12} \partial_{yyy}u_1+ 2\kappa \Delta \omega,\label{vorticity1} \\
\nabla \partial_t \omega + \nabla ( u \cdot \nabla \omega) + 4 \kappa \nabla \omega
 = \eta_{21} \nabla \omega _{xx} + 2\kappa\nabla \Omega \label{dom1}\\
\nabla \cdot u=0.
\end{gather}
Taking the $L^2$ inner product of \eqref{vorticity1} with $ \Delta\Omega$,
and integrating by parts, we obtain
\begin{equation}
\begin{aligned}
\frac{1}{2} \frac{d}{dt}\|\nabla \Omega\|_2^2+  \mu_2 \|\Delta \partial_y u_1\|_2^2
&=\int \nabla \Omega \cdot \nabla u\cdot \nabla \Omega \,dx\,dy
-2 \kappa \int \Delta \Omega  \Delta \omega \,dx \,dy\\
&\equiv L_1+L_2,
\end{aligned} \label{L12}
\end{equation}
Taking the $L^2$-inner product of  \eqref{dom1}
with $\Delta \omega$, and integrating by parts, we  obtain
\begin{equation}
\begin{aligned}
\frac{1}{2} \frac{d}{dt} \|\Delta \omega\|_2^2 + \eta_1\|\Delta \omega_x\|_2^2
+ 4 \kappa \| \Delta \omega\|_2^2
&=  - 2 \kappa \int \Delta \Omega \Delta \omega +\int \Delta ( u \cdot \nabla \omega) \Delta \omega  \notag \\
& \equiv L_2+L_3.
\end{aligned} \label{L34}
\end{equation}
Adding \eqref{L12} and \eqref{L34} yields
\begin{align*}
&\frac{1}{2} \frac{d}{dt}(\|\nabla \Omega\|_2^2+\|\Delta \omega\|_2^2)
 +  \mu_2 \|\Delta \partial_y u_1\|_2^2
+  \eta_1 \|\Delta \omega_x\|_2^2 +  4 \chi \| \Delta \omega\|_2^2 \\
&=L_1+2 L_2+L_3.
\end{align*}
We now estimate $L_1$ through $L_3$. We further split $L_1$ into 4 terms.
\begin{align*}
L_1 &= -\int \nabla \Omega \cdot \nabla u \cdot \nabla \Omega\;\,dx\,dy\\
&= - \int \left(\partial_x u_1 (\partial_x \Omega)^2
 +\partial_x u_2 \partial_x \Omega  \partial_y \Omega
 +\partial_y u_1 \partial_x \Omega  \partial_y \Omega+\partial_y u_2
  (\partial_y \Omega)^2 \right)\\
&= L_{11}+L_{12}+L_{13}+L_{14}.
\end{align*}
Since $\Omega=\nabla \times u$, we have
\begin{equation}{\label{ccc}}
\partial_{xx} \Omega= \Delta \partial_x u_2, \quad
\partial_{yy} \Omega= -\Delta \partial_y u_1, \quad
\partial_{xy} \Omega= \Delta \partial_x u_2.
\end{equation}
Therefore,
\[
L_{11} \leq \|\partial_x u_1\|_\infty \|\nabla \Omega\|_2^2, \quad
L_{12} \leq \|\partial_x u_2\|_\infty \|\nabla \Omega\|_2^2
\]
Invoking the divergence-free condition, we
note that
$$ \|\nabla \partial_x u_1\|_2^2 = \|\partial_{xx}u_1\|_2^2+ \|\partial_{yy} u_2\|_2^2,$$
$$ \|\nabla \partial_y u_1\|_2^2 = \|\partial_{yy}u_2\|_2^2+ \|\partial_{yy} u_1\|_2^2,$$
$$ \|\nabla \partial_x u_2\|_2^2 = \|\partial_{xx}u_1\|_2^2+ \|\partial_{xx} u_2\|_2^2.$$

By Lemma \ref{triple},
\begin{align*}
L_{13}&\leq \big|\int \partial_y u_1 \partial_x \Omega  \partial_y  \Omega\big|\\
&\leq C \|\partial_y u_1\|_2^{1/2} \|\partial_{xy} u_1\|_2^{1/2}
 \|\partial_x \Omega\|_2 \|\partial_y\Omega\|_2^{1/2}
 \|\partial_{yy} \Omega\|_2^{1/2}\\
&\leq C \|\partial_{yy} \Omega\|_2 \|\partial_{xy} u_1\|_2
 \|\partial_y \Omega\|_2 + C\|\partial_y u_1\|_2 \|\nabla \Omega\|_2^2\\
&\leq \frac{1}{48} \|\partial_{yy} \Omega\|_2^2
 + C (1+ \|\partial_yu_1\|_2^2+\|\partial_{xy}u_1\|_2^2) \|\nabla \Omega\|_2^2.
\end{align*}
From  $\Omega=\nabla \times u$,  \eqref{ccc}, and lemma \eqref{triple},
\begin{align*}
L_{14}
&\leq \big| 2\int u_2 \partial_y \Omega \partial_{yy} \Omega \big|\\
&\leq C \|\partial_{yy} \Omega\|_2 \|u_2\|_2^{1/2} \|\partial_{x} u_2\|_2^{1/2}
 \|\partial_y \Omega\|_2^{1/2} \|\partial_{yy} \Omega\|_2^{1/2}\\
&\leq C \|\partial_{yy} \Omega\|_2^{\frac{3}{2}} \|u_2\|_2^{1/2}
  \|\Omega\|_2^{1/2} \|\nabla \Omega\|_2^{1/2}\\
& \leq \frac{1}{48} \|\partial_{yy} \Omega\|_2^2 + C \|u_2\|_2^2
  \|\Omega\|_2^2 \|\nabla \Omega\|_2^2.
\end{align*}
Term $L_2$ can be easily bounded,
\[
 L_2=\int \Delta \Omega  \Delta \omega =\int \Omega_{xx} \Delta \omega
+  \int \Omega_{yy} \Delta \omega
\]
with
$$
\int \Omega_{xx} \Delta \omega =-\int \Omega_{x} \Delta \omega_x
\leq  \|\nabla \Omega\|_2 \|\Delta \omega_x\|_2, \quad
 \big|\int \Omega_{yy} \Delta \omega \big| \leq \|\Omega_{yy}\|_2 \|\Delta \omega\|_2.
$$
We now estimate the last term $L_3$.
\[
L_3=-\int \Delta ( u \cdot \nabla \omega) \Delta \omega
=-\int \Delta(u_1 \partial_1 \omega + u_2 \partial_y \omega) \Delta \omega\equiv L_{31}+L_{32}.
\]
We first split $L_{31}$ and $L_{32}$ each into two terms.
\begin{gather*}
L_{31}=-\int \partial_{xx} ( u_1 \partial_x \omega + u_2 \partial_y \omega ) \Delta \omega
=L_{311}+L_{312}.\\
L_{32}=-\int  \partial_{yy} ( u_1 \partial_x \omega + u_2 \partial_y \omega) \partial_{xx} \omega
 -\int \partial_{yy} ( u_1 \partial_x \omega + u_2 \partial_y \omega) \partial_{yy}\omega=L_{321}+L_{322}.
\end{gather*}
These terms are bounded as follows.
\begin{gather*}
\begin{aligned}
|L_{311}|
&= \big|-\int \partial_{x} ( u_1 \partial_x \omega) \Delta \omega_{x}\big|\\
&\leq \big|-\int \partial_x u_1 \partial_x \omega \Delta \omega_x \big|
 +  \big|\int u_1 \partial_{xx} \omega \Delta \omega_x \big|\\
&\leq C \|\Delta \omega_x\|_2 \|\partial_x u_1\|_2^{1/2} \|\partial_{xy} u_1\|_2^{1/2}
 \|\partial_x \omega\|_2^{1/2} \|\partial_{xx} \omega\|_2^{1/2}\\
&\quad + C \|\Delta \omega_x\|_2 \|\partial_{xx} \omega\|_2^{1/2}
 \|\partial_{xxx} \omega\|_2^{1/2} \|u_1\|_2^{1/2} \|\partial_y u_1\|_2^{1/2}\\
&\leq C  \|\Delta \omega_x\|_2 \|\Delta  \omega\|_2^{1/2} \|\Omega\|_2^{1/2}
 \| \nabla \Omega\|_2^{1/2} \|\nabla \omega\|_2^{1/2} \\
&\quad +  C \|\Delta \omega_x\|_2^{\frac 32} \|\nabla \omega_x\|_2^{1/2}
 \|u_1\|_2^{1/2} \|\Omega\|_2^{1/2}\\
&\leq \frac{1}{48} \|\Delta \omega_x\|_2^2
 + C \|\Omega\|_2^2 \|\nabla \Omega\|_2^2
 + \|\nabla \omega\|_2^2 \|\Delta \omega\|_2^2
 + C \|u_1\|_2^2 \|\nabla \omega_x\|_2^2 \|\Omega\|_2^2,
\end{aligned} \\   
\big| L_{312} \big|= \big|-\int \partial_x u_2 \partial_y \omega \Delta \omega_x
- \int u_2 \partial_{xy} \omega \Delta \omega_x \big|,\\ 
\big|-\int \partial_x u_2 \partial_y \omega \Delta \omega_x \big|
\leq \| \partial_x u\|_\infty \| \partial_ y \omega\|_2 \|\Delta \omega_x\|_2 ,\\ 
\begin{aligned} 
\big| \int u_2 \partial_{xy} \omega \Delta \omega_x \big|
&\leq C \|\Delta \omega_x\|_2 \|u_2\|_2^{1/2} \|\partial_x u_2\|_2^{1/2}
 \|\partial_{xy} \omega\|_2^{1/2} \|\partial_{xyy} \omega\|_2^{1/2} \\
&\leq \| \Delta \omega_x\|_2^2 +  C \|u_2\|_2^2  \|\Omega\|_2^2 \| \nabla \omega_x\|_2^2,
\end{aligned}\\  
L_{321}=\int  \partial_{yy} ( u_1 \partial_x \omega + u_2 \partial_y \omega) \partial_{xx} \omega
=\int \partial_{xx} ( u_1 \partial_x \omega + u_2 \partial_y \omega) \partial_{yy} \omega.
\end{gather*}
Obviously $L_{321}$ admits the same bound as that for $L_{311}$,
$$
|L_{321}| \leq \frac{1}{48} \|\Delta \omega_x\|_2^2 + C \|\Omega\|_2^2 \|\nabla \Omega\|_2^2
+ \|\nabla \omega\|_2^2 \|\Delta \omega\|_2^2
+ C \|u_1\|_2^2 \|\nabla \omega_x\|_2^2 \|\Omega\|_2^2.
$$
To estimate $L_{322}$, we write it out explicitly and integrate by parts,
\begin{align*}
L_{322}&=\int \partial_{yy} ( u_1 \partial_x \omega + u_2 \partial_y \omega) \partial_{yy}\omega\\
&=\int \partial_y( \partial_y u_1 \partial_x \omega + u_1 \partial_{xy} \omega ) \partial_{yy} \omega
 +\int \partial_y( \partial_y u_2 \partial_2 \omega + u_2 \partial_{yy} \omega ) \partial_{yy} \omega\\
&= \int [ \partial_{yy} u_1 \partial_x \omega+ 2 \partial_y u_1 \partial_{xy} \omega
 + u_1 \partial_{xyy} \omega] \partial_{yy} \omega\\
&\quad +\int [ \partial_{yy} u_2 \partial_y \omega+ 2 \partial_y u_2 \partial_{yy} \omega
 + u_2 \partial_{yyy} \omega] \partial_{yy} \omega.
\end{align*}
The terms on the right can be bounded as follows: 
\begin{align*}
&\big| \int \partial_{yy} u_1 \partial_x \omega \partial_{yy} \omega \big| \\
&\leq  C \|\partial_x \omega\|_2 \|\partial_{yy}u_1\|_2^{1/2}\|\partial_{yyy} u_1\|_1^{1/2}
 \|\partial_{yy} \omega\|_2^{1/2} \|\partial_{xyy} \omega\|_2^{1/2}\\
& \leq  C\|\partial_x \omega\|_2 \| \nabla \partial_y u_1 \|_2^{1/2} 
 \|\Delta \partial_y u_1\|_2^{1/2}\|\Delta \omega\|_2^{1/2} \|\Delta \partial_x \omega\|_2^{1/2}\\
&\leq \frac{1}{48} \|\Delta \partial_y u_1\|_2^2 + \frac{1}{48} 
 \|\Delta \omega_x\|_2^2 + C \|\omega_x\|_2^2 [ \|\nabla \partial_y u_1\|_2^2 + \|\Delta \omega\|_2^2) \\
&\leq \frac{1}{48} \|\Delta \partial_y u_1\|_2^2 +\frac{1}{48} \|\Delta \omega_x\|_2^2 
 + C \|\omega_x\|_2^2 (\| \nabla \Omega\|_2^2 + \|\Delta \omega\|_2^2)\,.
\end{align*}
\begin{align*}
&\big| \int \partial_y u_1 \partial_{xy} \omega \partial_{yy} \omega \big|\\
& \leq  C\|\partial_{yy} \omega\|_2 \|\partial_{xy} \omega\|_2^{1/2} \|\partial_{xyy}
  \omega\|_2^{1/2} \|\partial_y u_1\|_2^{1/2} \|\partial_{xy} u_1\|_2^{1/2}\\
& \leq C \|\Delta \omega\|_2 \|\nabla \omega_x\|_2^{1/2} \|\Delta \omega_x\|_2^{1/2} 
 \|\partial_y u_1\|_2^{1/2} \|\nabla \Omega\|_2^{1/2}\\
& \leq \frac{1}{48} \|\Delta \omega_x\|_2^2 +\,C\,\|\nabla \omega_x\|_2^{2}
 \|\nabla \Omega\|_2^2 + C \|\partial_y u_1\|_2 \|\Delta \omega\|_2^2\,.
\end{align*}
\begin{align*}
&\big|\int u_1 \partial_{xyy} \omega \partial_{yy} \omega \big|\\
&\leq  C \|\partial_{xyy} \omega\|  \|u_1\|_2^{1/2} \|\partial_y u_1\|_2^{1/2} 
 \|\partial_{yy} \omega\|_2^{1/2} \|\partial_{xyy} \omega\|_2^{1/2}\\
&\leq C \|\Delta \omega_x\|_2^{\frac 32} \|u_1\|_2^{1/2} 
 \|\partial_y u_1\|_2^{1/2} \|\Delta \omega\|_2^{1/2}\\
&\leq  \frac{1}{48} \|\Delta \omega_x\|_2^2 
 + C \|u_1\|_2^2 \|\partial_y u_1\|_2^2 \|\Delta \omega\|_2^2\,.
\end{align*}
\begin{align*}
&\big|\int \partial_{yy} u_2 \partial_y \omega \partial_{yy} \omega \big|\\
&\leq C \|\partial_{yy} u_2\|_2 \|\partial_y \omega\|_2^{1/2} 
 \| \omega_{yy}\|_2 \|\omega_{xyy}\|_2^{1/2}\\
&\leq \| \omega_y\|_2 \|\Delta \omega_x\|_2 
 +  C \|\nabla \partial_y u_1 \|_2^2 \|\Delta \omega\|_2^2 \\
&\leq  \frac{1}{48} \|\Delta \omega_x\|_2^2
  +  C \|\nabla \omega\|_2^2+ C \|\nabla \partial_y u_1 \|_2^2 \|\Delta \omega\|_2^2.
\end{align*}
Integration by parts yields
\begin{align*}
\int \partial_y u_2 \partial_{yy} \omega \partial_{yy} \omega 
=-\int \partial_x u_1\partial_{yy} \omega \partial_{yy}\omega
= 2\int u_1 \partial_{yy} \omega \partial_{xyy} \omega\,,\\
\int u_2 \partial_{yyy} \omega \partial_{yy} \omega 
= {\frac 12} \int u_2 \partial_y [\partial_{yy} \omega]^2
= -\int  u_1 \partial_{xyy} \omega \partial_{yy} \omega\,,
\end{align*}
which can be bounded as
\begin{align*}
\big|\int  u_1 \partial_{xyy} \omega \partial_{yy} \omega\big|
&\leq C \|\partial_{xyy} \omega\|_2 \|\partial_{yy} \omega\|_2^{1/2} 
 \|\partial_{xyy} \omega\|_2^{1/2} \|u_1\|_2^{1/2} \|\partial_y u_1\|_2^{1/2}\\
&\leq  C \|\Delta \omega_x\|_2^{\frac 32} \|\Delta \omega\|_2^{1/2} 
 \|u_1\|_2^{1/2} \|\partial_y u_1\|_2^{1/2}\\
&\leq \|\Delta \omega_x\|_2^2 + C\|u_1\|_2^2 \|\partial_y u_1\|_2^2 \|\Delta \omega\|_2^2.
\end{align*}

Collecting the estimates above and applying Gronwall's inequality,
 we obtain the desired global $H^2$-bound. This completes the proof for the
global $H^2$-bound and thus the proof of Theorem \ref{Main1}.

\section{Proof of Theorem \ref{Main2}}

We just prove the $H^1$-global bound here. The  rest of the proof 
is similar to the proof of Theorem \ref{Main1}.
The global $L^2$ bound can be proved easily.

\begin{lemma} \label{lem4.1}
Assume that  $(u_0, \omega_0)$ satisfies the condition stated in
Theorem \ref{Main1}. {  Let $(u, \omega)$ be the corresponding solution 
of \eqref{2DMPE1}}. Then, $(u,\omega)$ obeys the following global $L^2$-bound,
\begin{align*}
&\|u(t)\|_{L^2}^2  + \|\omega(t)\|_{L^2}^2 
 +  \mu_{21} \int_0^t  \|\partial_x u_2(\tau)\|_{L^2}^2 \, d\tau \\
&+ \eta_{2} \int_0^t  \|\partial_y \omega(\tau)\|_{L^2}^2\, d\tau  
+ 8\kappa \int_0^t  \| \omega(\tau)\|_{L^2}^2\, d\tau \leq C
\end{align*}
for any $t\ge 0$.
\end{lemma}

Global $H^2$ bound can be  obtained similar to  theorem \ref{DR}.

\begin{theorem} \label{thm4.2}
Assume that  $(u_0, \omega_0)$ satisfies the condition stated in
Theorem \ref{Main1}. Then,   the corresponding solution $(u, \omega)$ of 
\eqref{2DMPE1} obeys, for any $0<t<\infty$,
\begin{align*}
&\|\nabla u(t)\|_2^2 + \|\nabla \omega(t)\|_2^2
  + \mu_{21} \int_0^t \|\nabla \partial_x u_2( \tau)\|_2^2 \, d\tau \\
&+\eta_{2} \int_0^t \|\nabla \partial_y \omega(\tau)\|_2^2 \, d\tau 
+ 8 \kappa \int_0^t  \|\nabla \omega(t)\|_2^2 \, d\tau
\leq C
\end{align*}
\end{theorem}

\begin{proof}
Taking the $L^2$ inner product of \eqref{2DMPE2} with $(\Delta u, \Delta \omega)$ yields
\begin{align*}
&\frac{1}{2} \frac{d}{dt} \left( \|\nabla u(t)\|_2^2 
 + \|\nabla \omega (t)\|_2^2 \right) + \mu_{21} \|\nabla \partial_x u_2\|_2^2
 +\eta_2 \|\nabla \partial_y \omega(t)\|_2^2 + 4 \kappa \|\nabla \omega(t)\|_2^2\\
&=\int_{\mathbb{R}^2} (2\kappa (\nabla \times \omega) \cdot (-\Delta u)
 + 2 \kappa (\nabla \times u)(-\Delta \omega)) \, dx
 + \int_{\mathbb{R}^2} u \cdot \nabla \omega (-\Delta \omega) \, dx\\
&=2M_1+M_2,
\end{align*}
To estimate $M_1$ we write in component-wise
\begin{gather*}
M_1=\int_{\mathbb{R}^2} 2( \partial_y \omega \partial_{xx} u_1 
 + \partial_y \omega \partial_{yy} u_1 -\partial_x \omega \partial_{xx}u_2 -\partial_x \omega \partial_{yy} u_2 ) \, dx\,,\\
\big| \int_{\mathbb{R}^2} 2 \kappa \partial_y \omega \partial_{xx} u_1 \,dx\big| 
\leq \frac{1}{8} \|\nabla \partial_y \omega\|_2^2+  \|\nabla u\|_2^2\,,\\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_y \omega \partial_{yy} u_1 \, dx \big| 
\leq \frac{1}{8} \|\nabla \partial_y \omega\|_2^2+ \|\nabla u\|_2^2\,,\\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_x \omega \partial_{xx}u_2 \,dx \big|
\leq \frac{1}{8}\|\nabla \partial_x u_2\|_2^2 + \|\nabla \omega\|_2^2\,,\\
\big|\int_{\mathbb{R}^2} 2\kappa \partial_x \omega \partial_{yy} u_2\,dx\big|
\leq \frac{1}{8} \| \nabla \partial_y \omega\|_2^2+  \|\nabla u\|_2^2\,,\\
M_2 = \int_{\mathbb{R}^2} \nabla \omega \cdot \nabla u \cdot \nabla \omega \, dx\,,\\
M_2=\int \nabla \omega \cdot \nabla u \cdot \nabla \omega
 =\int \partial_x u_1 \omega_x^2 + 2\int u_1 \omega_y \omega_{xy} 
+\int (\partial_x u_2 + \partial_y u_1) \omega_x \omega_y \,,\\
| \int \partial_x  u_1 \omega_{x}^2 |
 \leq \| \partial_y u\|_\infty \|\omega_x\|_2^2\,,\\
\big|\int u_1 \omega_y \omega_{xy}\big| 
 \leq \|\partial_y u_1\|_{\infty} \|\nabla \omega_x\|_2^2\,,\\
\big| \int  \partial_y u_1 \omega_x \omega_y \big| 
\leq  \|\partial_y u_1\|_\infty \| \nabla \omega\|_2^2\,,\\
\begin{aligned}
\big| \int \partial_x u_2  \omega_x \omega_y \big| 
&\leq \|\partial_x u_2\|_2 \|\omega_x\|_2^{1/2} \|\omega_y\|_2^{1/2} \|\nabla \omega_y\|_2 \\
&\leq \frac{1}{48} \|\nabla \omega_y\|_2^2+ C \|\partial_x u_2\|_2^2 \|\nabla \omega\|_2^2
\end{aligned}
\end{gather*}
Combining the estimates above, together with Gronwall's inequalities, we obtain
\begin{align*}
&\|\nabla u(t)\|_2^2 + \|\nabla \omega(t)\|_2^2 
 + \mu_{12} \int_0^t \|\nabla \partial_y u_1( \tau)\|_2^2 \, d\tau \\
&+\eta_{1} \int_0^t \|\nabla \partial_x \omega(\tau)\|_2^2 \, d\tau 
 + 8 \kappa \int_0^t  \|\nabla \omega(t)\|_2^2 \, d\tau \leq C
\end{align*}
for any $t\leq T$, where $C$ depends on $T$ and the initial $H^1$-norm. 
This completes the proof of theorem.
\end{proof}


\subsection*{Acknowledgments}
The author expresses his gratitude to  the referee and editor for valuable review, 
comments  and suggestions, which improve the presentation of this article.

\begin{thebibliography}{10}

\bibitem{CaoReWu} C. Cao, D. Regmi, J. Wu;
 The 2D MHD equations with horizontal dissipation and horizontal
magnetic diffusion, \emph{J. Differential Equations}, \textbf{254} (2013), 2661-2681.

\bibitem{CaoWu} C. Cao, J. Wu;
 Global regularity for the 2D MHD equations with mixed partial dissipation 
and magnetic diffusion, \emph{Adv. in Math.}, \textbf{226} (2011), 1803-1822.

\bibitem{CaoWuY} C. Cao, J. Wu, B. Yuan;
 The 2D incompressible magnetohydrodynamics equations with only magnetic 
diffusion, \emph{SIAM J. Math. Anal.}, \textbf{46} (2014), 588-602.

\bibitem{ChengLiu} J. Cheng, Y. Liu;
 Global regularity of the 2D magnetic micropolar fluid flows with mixed 
partial viscosity, \emph{Computer and Mathematics with Applications},
 \textbf{70} (2015), 66-72.

\bibitem{Cowin} S. C. Cowin;
 Polar fluids, \emph{Phys. Fluids}, \textbf{11} (1968), 1919-1927.

\bibitem{DongB} B. Dong, J. Li, J. Wu; 
Global well-posedness and large-time decay for the 2D micropolar equations, 
\emph{J. Differential Equations}, 262 (2017), No.6, 3488-3523.

\bibitem{DongZhang} B. Dong, Z. Zhang;
 Global regularity of the 2D micropolar fluid flows with zero angular viscosity,
 \emph{J. Differential Equations}, \textbf{249} (2010), 200-213.

\bibitem{Erdo} M. E. Erdogan;
 Polar effects in the apparent viscosity of suspension, 
\emph{Rheol. Acta}, \textbf{9} (1970), 434-438.

\bibitem{Er} A. C. Eringen;
 Theory of micropolar fluids,  \emph{J. Math. Mech.}, \textbf{16}  (1966), 1-18.

\bibitem{Er2} A. C. Eringen;
 Micropolar fluids with stretch, \emph{Int. J. Engng. Eci.}, \textbf{7} (1969), 
115-127.

\bibitem{Jiu} Q. Jiu, J. Zhao;
  Global regularity of 2D generalized MHD equations with magnetic diffusion,
\emph{Z. Angew. Math. Phys.}, \textbf{66} (2015), 677-687.


\bibitem{LZ} Z. Lei, Y. Zhou;
 BKM's criterion and global weak solutions for magnetohydrodynamics
 with zero viscosity, \emph{Discrete Contin. Dyn. Syst.}, \textbf{25} (2009), 575-583.

\bibitem{LinZhang1} F. Lin, L. Xu, P. Zhang;
 Global small solutions to 2-D incompressible MHD system, 
\emph{J. Differential Equations}, \textbf{259} (2015), 5440--5485.


\bibitem{Luka} G. Lukaszewick;
 \emph{Micropolar Fluids: Theory and Applications}, Birkhauser, Boston, 1999.

\bibitem{Regmi} D. Regmi;
 {A regularity criterion for two-and-half-dimensional magnetohydrodynamic 
equations with horizontal dissipation and horizontal magnetic diffusion},
\emph{Mathematical Methods in the Applied Sciences}, \textbf{40}  (2017), Issue 5.

\bibitem{Regmi2} D. Regmi;
 {Global weak solutions for  the two-dimensional  magnetohydrodynamic 
equations with partial dissipation and diffusion}, 
\emph{Journal of Mathematical Study}, \textbf{49}  (2016) No. 2, 169-194.

\bibitem{RegmiWu} D. Regmi, J. Wu;
 {Global regularity for the 2D magneto-micropolar equations with partial dissipation},
 \emph{Journal of Mathematical Study}, \textbf{49} (2016) No. 2, 169-194.

\bibitem{Wu2} J. Wu;
 Generalized MHD equations, \emph{J. Differential Equations}, \textbf{195} (2003),
284-312.

\bibitem {Yamazaki1} K. Yamazaki;
 Global regularity of the two-dimensional magneto-micropolar fluid system 
with zero angular viscosity, \emph{Discrete Contin. Dyn. Syst.}, \textbf{35}
(2015), 2193-2207.

\bibitem{Yuan} B. Yuan;
 Regularity of weak solutions to magneto-micropolar
equations, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, \textbf{30B} (2010), 1469-1480.

\end{thebibliography}


\end{document}