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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 154, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/154\hfil 2D micropolar fluid flows]
{Global well-posedness of the 2D micropolar fluid flows
 with mixed dissipation}

\author[Y. Jia, W. Wang, B.-Q. Dong \hfil EJDE-2016/154\hfilneg]
{Yan Jia, Wenjuan Wang, Bo-Qing Dong}

\address{Yan Jia \newline
 School of Mathematical Sciences,
 Anhui University, Hefei 230039, China}
\email{Yanjia@ahu.edu.cn}

\address{Wenjuan Wang \newline
 School of Mathematical Sciences,
 Anhui University, Hefei 230039, China}
\email{wangwenjuan@ahu.edu.cn}

\address{Bo-Qing Dong (corresponding author)\newline
 School of Mathematical Sciences,
 Anhui University, Hefei 230039, China}
\email{bqdong@ahu.edu.cn}

\thanks{Submitted May 3, 2016. Published June 21, 2016.}
\subjclass[2010]{35Q35, 76D05}
\keywords{Micropolar fluid flows;  mixed dissipation; global regularity}

\begin{abstract}
 This article concerns the global well-posedness of the 2D micropolar 
 fluid flows with mixed dissipation:
 $$
 -\partial_{yy}(-\Delta)^\alpha u_1,\quad 
 -\partial_{xx}(-\Delta)^\alpha u_2,\quad
 (-\Delta)^\beta w.
 $$
 We prove the existence and uniqueness of global smooth solution of 2D
 micropolar fluid flows when $\alpha+\beta \geq1/2$.
\end{abstract}

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

\section{Introduction}

 Micropolar fluid flows derived by Eringe \cite{Er} are an important
 mathematical model in some polymeric fluids and fluids
which contain certain additives in narrow films(\cite{PRU,St}). They
are non-Newtonian fluids with nonsymmetric stress tensor which are
coupled with the kinematic viscous effect, microrotational effects
as well as microrotational inertia. The two-dimensional (2D)
incompressible micropolar fluid flows is governed by
\begin{equation}\label{1.1}
\begin{gathered}
\partial_t u -(\nu +\kappa)\Delta u
 -2\kappa\nabla \times w +u\cdot\nabla u +\nabla \pi= 0, \\
\nabla\cdot u=0, \\
\partial_tw -\gamma\Delta w +4\kappa w
 -2\kappa\nabla\times u+ u\cdot \nabla w =0.
\end{gathered}
\end{equation}
Where $u(x,y,t)=(u_1(x,y,t),u_2(x,y,t))$ is the unknown velocity
vector field, $\pi(x,y,t)$ is the unknown scalar pressure field and
$w(x,y,t)$ is the unknown scalar micro-rotation angular velocity
of the rotation of the particles of the fluid.
$\nu,\kappa,\gamma\geq 0 $ are viscous coefficients.
$u_0$ and $w_0$ represent the prescribed initial data for the velocity and
micro-rotation fields. Here and in what follows,
$$
\nabla \times u
= \frac{\partial u_2}{\partial x } -\frac{\partial u_1}{\partial y},\quad
\nabla\times w
= \Big(\frac{\partial w}{\partial y}, -\frac{\partial w}{\partial x}\Big).
$$

Because of its importance in mathematics, there is much attention on the
well-posedness of the micropolar fluid flows \cite{CM,GR,Lu}. In
particular, when there is full dissipation, Lukaszewicz \cite{Lu}
examined the global well-posedness of smooth solution to the 2D
micropolar fluid flows \eqref{1.1}. A more explicit existence and
uniqueness result which is based on the decay estimates of the
linearized equations is recently investigated in Dong and Chen
\cite{DC2}. When there is partial dissipation, the issue on global
regularity becomes more difficult. Due to some new observation, Dong
and Zhang \cite{DZ} recently obtained the global existence and
uniqueness of classic solution of micropolar fluid flows
\eqref{1.1} with only velocity dissipation $ \Delta u$.
Xue\cite{Xue} also examined the well-posedness of the micropolar
fluid flows \eqref{1.1} with only velocity dissipation $ \Delta u$
under the Besov space framework. Yamaziki\cite{Ya} futher extended
the results of above the results to the magneto-micropolar
equations. Chen\cite{Chen} examined the existence and uniqueness of
smooth solution of micropolar fluid flows \eqref{1.1} with partial
dissipation $(\partial_{yy}u,\partial_{xx}w)$. Very recently, Dong,
Li and Wu \cite{DLW} proved the global well-posedness of 2D
micropolar fluid flows \eqref{1.1} with only angular viscosity
dissipation $ \Delta w$. One may also refer to some important and
interesting results of the 2D fluid dynamical models with partial
dissipation such as the 2D Boussinesq equations \cite{CW,Chae,HL,Xu}
and the 2D magnetohydrodynamic (MHD) equations \cite{CWY,CW11}.

 Motivated by the above results of the 2D fluid dyanmical models with partial
dissipation, the purpose of this study is
 investigate the global existence
and uniqueness of the smooth solution of 2D micropolar fluid flows
with mixed dissipation $(\partial_{yy}(-\Delta)^\alpha
u_1,\partial_{xx}(-\Delta)^\alpha u_2,-(-\Delta)^\beta
 w)$. More precisely, we will examine the global regularity issue of the
 following 2D micropolar fluid flows with unit viscosity
\begin{equation}\label{1.2}
\begin{gathered}
\partial_t u_1 +(u\cdot \nabla)u_1 - \partial_y w+\partial_x\pi
 = \partial_{yy}(-\Delta)^\alpha u_1, \\
\partial_t u_2 +(u\cdot \nabla)u_2 + \partial_x w+\partial_y \pi
 = \partial_{xx}(-\Delta)^\alpha u_2, \\
\partial_xu_1+\partial_yu_2=0, \\
 \partial_tw +2 w - \nabla\times u+ u\cdot \nabla w =-(-\Delta)^\beta w .
\end{gathered}
\end{equation}
 where $0<\alpha, \beta <1 $.
We are able to prove the following existence and uniqueness of smooth
global solutions for \eqref{1.2}.

\begin{theorem} \label{thm1.1}
Assume $(u_0,w_0) \in H^s(\mathbb{R}^2)$, $s>2$ and $\nabla\cdot u_0=0$. There
exists a unique global smooth solution $(u,w)$ for the 2D micropolar
fluid flows \eqref{1.2} with $\alpha+\beta\geq\frac{1}{2}$
\begin{gather*}
u\in C([0,\infty);H^s(\mathbb{R}^2)), \quad u\in
 L^2(0,T;H^{s+1+\alpha}(\mathbb{R}^2)),\\
w\in C([0,\infty);H^s(\mathbb{R}^2)), \quad w\in
 L^2(0,T;H^{s+\beta}(\mathbb{R}^2)), \quad \forall T>0.
\end{gather*}
\end{theorem}

Clearly, Theorem \ref{thm1.1} generalizes the global well-posedness results of
 Lukaszewicz \cite{Lu}, Dong and Chen \cite{DC2} where there is full dissipation.
Moreover, Theorem \ref{thm1.1} has no inclusion relation between previous
results of partial dissipative micropolar fluid flows
\cite{Chen,DZ,Xue,Ya}. Especially, the main trick based on a new
quantity $\nabla\times u-w$ in \cite{DZ,Xue,Ya} is not available
here any more.

We briefly summarize the main challenge and explain what we have
done to achieve the global regularity. In order to prove
Theorem \ref{thm1.1}, we need global {\it a priori} bounds of $(u,w)$ in
sufficiently smooth functional spaces. More precisely, if we can
prove the Beale-Kato-Majda criterion of solutions for the 2D
micropolar fluid flows \eqref{1.2}
\begin{equation}\label{uwn}
\int_0^T \|(\nabla u(t), \nabla w(t))\|_{L^\infty}\,dt <\infty,
\end{equation}
then the proof of Theorem \ref{thm1.1} is a more or less standard
procedure. The next natural step is to examine a global $H^1$-bound
for $(u, w)$ of \eqref{1.2} after the based $L^2$ energy estimates
 of $(u, w)$. However, unlike the previous results where the
$H^1$-bound can be derived from vorticity equation, the vorticity
structure here is destroyed due to the mixed
dissipation $(\partial_{yy}(-\Delta)^\alpha
u_1,\partial_{xx}(-\Delta)^\alpha u_2)$. If we directly take the
$L^2$ inner product \eqref{1.2} with $(\Delta u,\Delta w)$, then the
$H^1$-bound for $(u, w)$ of \eqref{1.2} at least requires higher
dissipation $\alpha+\beta\geq1$. To overcome the difficulty, we
first derive the optimal fractional-order derivative estimates of
$u$,
$$
 \|\Lambda^{\alpha+\beta}u(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{1+2\alpha+\beta}u(s)\|^2_{L^2} ds \le
 C(t, \|(u_0,w_0)\|_{H^s}),
$$
and then the optimal fractional-order derivative estimates of $w$
$$
 \|\Lambda^{2(\alpha+\beta)}w(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{2\alpha+3\beta}w(s)\|^2_{L^2} ds \le
 C(t, \|(u_0,w_0)\|_{H^s}).
$$
The Beale-Kato-Majda criterion of solutions then can be derived by
an iterative procedure.

\section{Beale-Kato-Majda criterion}

 To prove the existence of global
smooth solution for the 2D micropolar fluid flows \eqref{1.2}, we
will examine the Beale-Kato-Majda criterion in this section. For
simplicity, we only need to prove the case $
\alpha+\beta=1/2$, the case $ \alpha+\beta>1/2 $ can be
done in a slightly modification. To do so, we first examine the
following lemma which is mainly based on the divergence free of
velocity.

\begin{lemma} \label{lem2.1}
Suppose $u=(u_1,u_2)$ is divergence free and for any
$s\in \mathbb{R}$, we have
\begin{equation}\label{2.1}
\int_{\mathbb{R}^2}\left(-\partial_{yy}u_1 \Lambda^{2s}u_1
 -\partial_{xx}u_2 \Lambda^{2s}u_2 \right)\,dx\,dy
 \geq\frac12\|\Lambda^{1+s}u\|^2_{L^2}
\end{equation}
where $\Lambda=(-\Delta)^{1/2}$ denotes the Zygmund operator,
defined via the Fourier transform
$$
\widehat{\Lambda^\alpha f} (\xi) = |\xi|^\alpha\, \widehat{f}(\xi).
$$
\end{lemma}

\begin{proof}
 Integrating by parts and applying the
divergence free property of velocity, it is easy to check that
\begin{align*}
 &\int_{\mathbb{R}^2}\big(-\partial_{yy}u_1 \Lambda^{2s}u_1
 -\partial_{xx}u_2 \Lambda^{2s}u_2 \big)\,dx\,dy\\
 &= \int_{\mathbb{R}^2}\big( \partial_{yy}u_1\ \Delta\Lambda^{2s-2}u_1
 +\partial_{xx}u^2\ \Delta\Lambda^{2s-2}u_2 \big)\,dx\,dy\\
 &= \int_{\mathbb{R}^2}\big( \nabla\partial_{y}\Lambda^{s-1}u_1
 \nabla\partial_{y}\Lambda^{s-1}u_1
 + \nabla\partial_{x}\Lambda^{s-1}u_2
 \nabla\partial_{x}\Lambda^{s-1}u_2 \Big)\,dx\,dy\\
 &= \int_{\mathbb{R}^2}\big( |\partial_{yy}\Lambda^{s-1}u_1|^2+
 |\partial_{xy}\Lambda^{s-1}u_1|^2+
 |\partial_{xx}\Lambda^{s-1}u_2|^2
+ |\partial_{xy}\Lambda^{s-1}u_2|^2 \Big)\,dx\,dy\\
&= \int_{\mathbb{R}^2}\big( |\partial_{yy}\Lambda^{s-1}u_1|^2+
 |\partial_{yy}\Lambda^{s-1}u_2|^2+
 |\partial_{xx}\Lambda^{s-1}u_2|^2
 + |\partial_{xx}\Lambda^{s-1}u_1|^2 \Big)\,dx\,dy\\
&\geq \frac12 \int_{\mathbb{R}^2}\big( |\Delta\Lambda^{s-1}u_1|^2+
 |\Delta\Lambda^{s-1}u_2|^2 \Big)\,dx\,dy
 = \frac12\|\Lambda^{1+s}u\|^2_{L^2}.
\end{align*}
\end{proof}

To prove \emph{a priori} estimates of solutions, we recall the
following classical commutator estimate \cite{KP}.

\begin{lemma} \label{lem2.2}
Let $s>0$. Let $1<r<\infty$ and
$\frac{1}{r}=\frac{1}{p_1}+\frac{1}{q_1}=\frac{1}{p_2}+\frac{1}{q_2}$
with $q_1, p_2\in(1,\infty)$ and $p_1, q_2\in[1,\infty]$. Then,
\begin{align*}
 \|[\Lambda^s,f ]g \|_{L^r}
 \leq C\left(\|\nabla f \|_{L^{p_1}}\|\Lambda^{s-1} g \|_{L^{q_1}}
 + \|\Lambda^{s} f \|_{L^{p_2}} \| g \|_{L^{q_2}}\right),
\end{align*}
where $C$ is a constant depending on the indices $s, r, p_1, q_1,
p_2$ and $q_2$.
\end{lemma}

\begin{proposition} \label{prop2.1}
 Under the conditions of
Theorem \ref{thm1.1} and let $(u, w)$ be the corresponding solution of
\eqref{1.2}. Then $(u, w)$ obeys the following global bounds, for
any $0<t<\infty$,
\begin{gather}
\|u(t)\|^2_{L^2}+\|w(t)\|_{L^2}^2
 + \int_0^t(\|\Lambda^{1+\alpha} u(s)\|_{L^2}^2 +\|\Lambda^\beta w(s)\|_{L^2}^2) ds
 \le C, \label{2.2} \\
\|\Lambda^{\alpha+\beta}u(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{1+2\alpha+\beta}u(s)\|^2_{L^2} ds
 \le C,
 \label{2.3}\\
 \|\Lambda^{2\alpha+2\beta}w(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{2\alpha+3\beta}w(s)\|^2_{L^2} ds \le
 C, \label{2.4}
\end{gather}
where the positive constants $C$ depend on $t$ and $\|(u_0, w_0)\|_{H^s}$
only.
\end{proposition}



\begin{proof}
 Taking the $L^2$ inner product of
equations \eqref{1.2} with $(u_1,u_2,w)$, it is easy to verify after
applying Lemma \ref{lem2.1}, the H\"{o}lder inequality and the Young
inequality
\begin{align*}
&\frac{d}{dt}\big( \|u(t)\|^2_{L^2}+\|w(t)\|_{L^2}^2\big)
 + \|\Lambda^{1+\alpha}u(t)\|_{L^2}^2
 + 2\|\Lambda^\beta w(t)\|_{L^2}^2+4 \|w(t)\|^2_{L^2}\\
&\leq 2\int_{\mathbb{R}^2} \{ (\nabla \times w)\cdot u
 + (\nabla \times u) w \} \,dx\,dy\\
&\leq 4 \|\Lambda^{1-\beta}u\|_{L^2}\| \Lambda^\beta w \|_{L^2}
 \leq C \|u(t)\|_{L^2}^2+ \frac12\|\Lambda^{1+\alpha} u(t)\|^2_{L^2}
 + \|\Lambda^\beta w(t)\|^2_{L^2},
\end{align*}
where we have used the following fact due to the divergence free of
velocity $u$,
$$
\int_{\mathbb{R}^2} ( u\cdot\nabla u) \cdot u \,dx\,dy=0,\quad
\int_{\mathbb{R}^2}( u\cdot \nabla w) w \,dx\,dy=0.
$$
Integrating in time for $ 0<t<\infty $,
\[
\|u(t)\|^2_{L^2}+\|w(t)\|_{L^2}^2
 + \int_0^t(\|\Lambda^{1+\alpha} u(s)\|_{L^2}^2
+\|\Lambda^\beta w(s)\|_{L^2}^2) ds\leq
e^{ct}(\|u_0\|_{L^2}^2+\|w_0\|_{L^2}^2)
\]
which implies \eqref{2.2}.

To examine \eqref{2.3}, we take the $L^2$ inner product of
the first two equations of \eqref{1.2} with
$(\Lambda^{2\alpha+2\beta} u_1,\Lambda^{2\alpha+2\beta} u_2)$ and
use Lemma \ref{lem2.1} to obtain
\begin{equation} \label{2.5}
\begin{aligned}
& \frac{d}{dt} \|\Lambda^{\alpha+\beta} u(t)\|^2_{L^2}
 + \| \Lambda^{1+2\alpha+\beta} u(t)\|_{L^2}^2 \\
&= 2\int_{\mathbb{R}^2} (\nabla \times w)\cdot \Lambda^{2\alpha+2\beta} u \,dx\,dy
 + 2 \int_{\mathbb{R}^2} \Lambda^{\alpha+\beta}[u\cdot \nabla u] \Lambda^{\alpha+\beta}u
 \,dx\,dy \\
& = I_1+I_2
\end{aligned}
\end{equation}

For $I_1$, it is easy to check that
\begin{align*}
2\int_{\mathbb{R}^2} (\nabla \times w)\cdot \Lambda^{2\alpha+2\beta} u \,dx\,dy 
& \leq C \| \Lambda^{1+2\alpha+\beta} u(t)\|_{L^2}
 \|\Lambda^{\beta} w(t)\|_{L^2} \\
&\leq \frac14 \| \Lambda^{1+2\alpha+\beta} u(t)\|^2_{L^2}
 +C \|\Lambda^{\beta} w(t)\|^2_{L^2}
\end{align*}


For $I_2$, using Lemma \ref{lem2.2}, the H\"{o}lder inequality, Young
inequality, and interpolation inequality gives
\begin{align*}
 I_2
&= 2\int_{\mathbb{R}^2} \Lambda^{\alpha+\beta}[u\cdot \nabla u]
 \Lambda^{\alpha+\beta}u \,dx\,dy\\
&\leq C\|\nabla u\|_{L^{ \frac{2}{1-\beta}}}\|\Lambda^{\alpha+\beta}
 u\|_{L^{ \frac{2}{\beta}}}\|\Lambda^{\alpha+\beta} u\|_{L^2}\\
&\leq C\|\Lambda^{1+\beta} u\|_{L^2}\|\Lambda^{1+\alpha}
 u\|_{L^2}\|\Lambda^{\alpha+\beta} u\|_{L^2}\\
&\leq \frac12\|\Lambda^{1+\beta} u\|^2_{L^2}
 +C\|\Lambda^{1+\alpha} u\|^2_{L^2}\|\Lambda^{\alpha+\beta} u\|^2_{L^2}\\
&\leq \frac14\| u\|^2_{L^2}+ \frac14\|\Lambda^{1+2\alpha+\beta}
 u\|^2_{L^2}
 +C\|\Lambda^{1+\alpha} u\|^2_{L^2}\|\Lambda^{\alpha+\beta} u\|^2_{L^2}.
\end{align*}

Using the estimates of $I_1,I_2$ into \eqref{2.5} gives
\begin{align*}
 & \frac{d}{dt} \|\Lambda^{\alpha+\beta} u(t)\|^2_{L^2}
 + \frac12 \| \Lambda^{1+2\alpha+\beta} u(t)\|_{L^2}^2 \\
 &\leq C \|\Lambda^{\beta} w(t)\|^2_{L^2}+C \|\mathbf{u}(t)\|^2_{L^2}
 +C\|\Lambda^{1+\alpha} u\|^2_{L^2}\|\Lambda^{\alpha+\beta}
 u\|^2_{L^2}
\end{align*}
Taking the Gronwall inequality into consideration together with
\eqref{2.2} implies
\begin{align*}
& \|\Lambda^{\alpha+\beta}u(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{1+2\alpha+\beta}u(s)\|^2_{L^2} ds \\
 &\leq \exp{\Big(C\int_0^t\|\Lambda^{1+\alpha} u\|^2_{L^2}d \tau \Big)}
 \left(\|\Lambda^{\alpha+\beta}u_0\|^2_{L^2}+Ct\right)
 \leq C,\quad 0<t<\infty
\end{align*}
which is \eqref{2.3}.

For \eqref{2.4}, we take the inner product of fourth equation of
\eqref{1.2} with $ \Lambda^{4(\alpha+\beta)}w$ to obtain
\begin{equation}\label{2.6}
\begin{aligned}
 & \frac{1}{2}\frac{d}{dt} \|\Lambda^{2\alpha+2\beta}w \|_{L^2}^2
 + \|\Lambda^{2\alpha+3\beta}w\|_{L^2}^2
 +2 \|\Lambda^{2\alpha+2\beta}w\|^2_{L^2} \\
 &= 2\int_{\mathbb{R}^2} ( \nabla\times u) \Lambda^{4(\alpha+\beta)}w \,dx\,dy
 -\int_{\mathbb{R}^2} [\Lambda^{2\alpha+2\beta},u\cdot \nabla]w \Lambda^{2\alpha+2\beta}w \,dx\,dy \\
 & := J_1+J_2.
\end{aligned}
\end{equation}
Expression $J_1$ is bounded by applying the H\"{o}lder inequality and the
Young inequality,
\begin{align*}
 J_1 \leq C\|\Lambda^{1+2\alpha+\beta}u\|_{L^2} \|\Lambda^{2\alpha+3\beta}w\|_{L^2}
 \leq \frac14 \|\Lambda^{2\alpha+3\beta}w\|^2_{L^2}
 +C\|\Lambda^{1+2\alpha+\beta}u\|^2_{L^2} .
\end{align*}
As in the estimates for $I_2$, employing Lemma \ref{lem2.2}, the
Sobolev's imbedding inequality and the Young inequality, it follows
that
\begin{align*}
 J_2
& = - \int_{\mathbb{R}^2} [\Lambda^{2\alpha+2\beta},u\cdot \nabla]w \Lambda^{2\alpha+2\beta}w \,dx\,dy \\
& \leq \Big( \|\nabla u \|_{L^{\frac{2}{\beta}}}\|\Lambda^{2\alpha+2\beta}w
 \|_{L^{\frac{2}{1-\beta}}}
 +\|\Lambda^{ 2\alpha+2\beta} u \|_{L^{\frac{2}{\beta}}}
 \|\nabla w \|_{L^{\frac{2}{1-\beta}}}\Big)
 \|\Lambda^{2\alpha+2\beta}w \|_{L^2} \\
& \leq C \left( \|\Lambda^{2-\beta} u \|_{L^2}
 \|\Lambda^{1+\beta}w \|_{L^2} +\|\Lambda^{1+2\alpha+\beta} u \|_{L^2}
 \|\Lambda^{2\alpha+3\beta}w \|_{L^2} \right)\|\Lambda^{ 2\alpha+2\beta}w
 \|_{L^2} \\
& \leq C \|\Lambda^{1+2\alpha+\beta} u \|_{L^2} \|\Lambda^{2\alpha+3\beta}w \|_{L^2}
 \|\Lambda^{ 2\alpha+2\beta}w \|_{L^2} \\
& \leq \frac14\|\Lambda^{2\alpha+3\beta}w \|^2_{L^2} +C
 \|\Lambda^{1+2\alpha+\beta} u\|^2_{L^2}
 \|\Lambda^{2\alpha+ 2\beta}w \|^2_{L^2}
\end{align*}

Inserting the estimates for $J_1$ and $J_2$ into \eqref{2.6} gives
\begin{align*}
 \frac{d}{dt} \|\Lambda^{2\alpha+2\beta}w \|_{L^2}^2
 + \|\Lambda^{2\alpha+3\beta}w\|_{L^2}^2
 \leq C\|\Lambda^{1+2\alpha+\beta} u\|^2_{L^2}
 \left( \|\Lambda^{2\alpha+ 2\beta}w \|^2_{L^2}+1\right)
\end{align*}
and applying Gronwall's inequality, we have
\begin{align*}
 \|\Lambda^{2\alpha+2\beta}w(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{2\alpha+3\beta}w(s)\|^2_{L^2} ds \leq C(t,u_0,w_0)
\end{align*}
which gives \eqref{2.4}.
This completes the proof of Proposition \ref{prop2.1}.

To obtain the Beale-Kato-Majda criterion for the solutions.
 we further need to prove the following {\it a priori} bounds for higher-order
derivatives of the solutions.

\begin{proposition} \label{prop2.2}
Suppose $(u, w)$ is the corresponding solution in Theorem \ref{thm1.1}.
Then $(u, w)$ obeys the following global bounds, for any $0<t<\infty$,
\begin{gather}
 \|\Lambda^{3\alpha+3\beta}u(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{1+4\alpha+3\beta}u(s)\|^2_{L^2} ds
 \le C,
 \label{2.7}\\
 \|\Lambda^{4\alpha+4\beta}w(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{4\alpha+5\beta}w(s)\|^2_{L^2} ds \le
 C. \label{2.8}
\end{gather}
In particular, we have the Beale-Kato-Majda criterion of solutions
for the 2D micropolar fluid flows \eqref{1.2},
\begin{equation}\label{2.9}
\int_0^T \|(\nabla u(t), \nabla w(t))\|_{L^\infty}\,dt <\infty.
\end{equation}
\end{proposition}


\begin{proof}
 Similar to the estimation of
\eqref{2.2}, multiplying the both sides of the first two equations
of \eqref{1.2} with $ (\Lambda^{6\alpha+6\beta}
u_1,\Lambda^{6\alpha+6\beta} u_2)$ and applying
Lemmas \ref{lem2.1} and \ref{lem2.2}, we have
\begin{equation} \label{2.10}
\begin{aligned}
& \frac{d}{dt} \|\Lambda^{3\alpha+3\beta} u \|^2_{L^2}
 + \| \Lambda^{1+4\alpha+3\beta} u \|_{L^2}^2 \\
&= 2\int_{\mathbb{R}^2} (\nabla \times w)\cdot \Lambda^{6\alpha+6\beta} u \,dx\,dy
 + 2 \int_{\mathbb{R}^2} \Lambda^{3\alpha+3\beta}[u\cdot \nabla u]
 \Lambda^{3\alpha+3\beta}u \,dx\,dy
 \\
& \leq C \| \Lambda^{1+4\alpha+3\beta} u \|_{L^2}
 \|\Lambda^{2\alpha+3\beta} w \|_{L^2}\\
 &\quad +C \|\nabla u\|_{L^{\frac{2}{1-(2\alpha+\beta)}}}
 \|\Lambda^{3\alpha+3\beta} u\|_{L^{ \frac{2}{2\alpha+\beta }}}
 \|\Lambda^{3\alpha+3\beta} u\|_{L^2} \\
& \leq C \| \Lambda^{1+4\alpha+3\beta} u \|_{L^2}
 \|\Lambda^{2\alpha+3\beta} w \|_{L^2} \\
&\quad +C \|\Lambda^{1+2\alpha+\beta}u\|_{L^2}
 \|\Lambda^{1+ \alpha+2\beta} u\|_{L^{2}}
 \|\Lambda^{3\alpha+3\beta} u\|_{L^2} \\
& \leq C \| \Lambda^{1+4\alpha+3\beta} u \|_{L^2}
 \|\Lambda^{2\alpha+3\beta} w \|_{L^2} \\
&\quad +C \|\Lambda^{1+2\alpha+\beta}u\|_{L^2}
 (\|\Lambda^{1+4\alpha+3\beta} u\|_{L^{2}}+\| u\|_{L^{2}})
 \|\Lambda^{3\alpha+3\beta} u\|_{L^2} \\
& \leq \frac12 \| \Lambda^{1+4\alpha+3\beta} u \|^2_{L^2}
 +C \|\Lambda^{2\alpha+3\beta} w \|^2_{L^2} \\
&\quad +C \|\Lambda^{1+2\alpha+\beta}u\|^2_{L^2}
 \|\Lambda^{3\alpha+3\beta} u\|^2_{L^2}+C.
\end{aligned}
\end{equation}
Employing the Gronwall inequality
and Proposition \ref{prop2.1} gives
\[
\|\Lambda^{3\alpha+3\beta}u(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{1+4\alpha+3\beta}u(s)\|^2_{L^2} ds\leq C(t,u_0,w_0),\ \ \
 0<t<\infty
\]
which gives \eqref{2.7}.

To prove \eqref{2.8}, similarly, multiplying both sides of the
fourth equation in \eqref{1.2} by $ \Lambda^{8(\alpha+\beta)}w$
yields
\begin{equation}\label{2.11}
\begin{aligned}
 & \frac{1}{2}\frac{d}{dt} \|\Lambda^{4\alpha+4\beta}w \|_{L^2}^2
 + \|\Lambda^{4\alpha+5\beta}w\|_{L^2}^2
 +2 \|\Lambda^{4\alpha+4\beta}w\|^2_{L^2} \\
 &= 2\int_{\mathbb{R}^2} ( \nabla\times u) \Lambda^{8(\alpha+\beta)}w \,dx\,dy
 -\int_{\mathbb{R}^2} [\Lambda^{4\alpha+4\beta},u\cdot \nabla]w \Lambda^{4\alpha+4\beta}w \,dx\,dy \\
 & \leq  C\|\Lambda^{1+4\alpha+3\beta}u\|_{L^2}
 \|\Lambda^{4\alpha+5\beta}w\|_{L^2} 
 + C \Big( \|\nabla u \|_{L^{\frac{2}{\beta}}}
 \|\Lambda^{4\alpha+4\beta}w \|_{L^{\frac{2}{1-\beta}}}\\
 &\quad +\|\Lambda^{ 4\alpha+4\beta} u \|_{L^{\frac{2}{\beta}}}
\|\nabla w \|_{L^{\frac{2}{1-\beta}}}\Big)
 \|\Lambda^{4\alpha+4\beta}w \|_{L^2} \\
 & \leq  C\|\Lambda^{1+4\alpha+3\beta}u\|_{L^2}
 \|\Lambda^{4\alpha+5\beta}w\|_{L^2} 
  + C \Big( \|\Lambda^{2-\beta} u \|_{L^2}
 \|\Lambda^{ 4\alpha+5\beta}w \|_{L^2}  \\
&\quad +\|\Lambda^{1+4\alpha+3\beta} u \|_{L^2}
 \|\Lambda^{2-\beta}w \|_{L^2} \Big)\|\Lambda^{ 2\alpha+2\beta}w
 \|_{L^2} \\
 & \leq  C\|\Lambda^{1+4\alpha+3\beta}u\|_{L^2}
 \|\Lambda^{4\alpha+5\beta}w\|_{L^2} 
  + C ( \| w \|_{L^2}+
 \|\Lambda^{ 4\alpha+5\beta}w \|_{L^2}) \\
&\quad\times  \big(\| u \|_{L^2}+\|\Lambda^{1+4\alpha+3\beta} u \|_{L^2}
 \|\Lambda^{2-\beta}w \|_{L^2} \big)\|\Lambda^{ 2\alpha+2\beta}w
 \|_{L^2} \\
 & \leq  \frac12 \|\Lambda^{4\alpha+5\beta}w\|^2_{L^2} + C
 (1+\|\Lambda^{1+4\alpha+3\beta} u \|^2_{L^2} \|\Lambda^{ 4\alpha+4\beta}w
 \|^2_{L^2} +C.
\end{aligned}
\end{equation}
and then  Gronwall's inequality gives
\begin{align*}
 \|\Lambda^{4\alpha+4\beta}w(t)\|^2_{L^2}
 +\int_0^t \|\Lambda^{4\alpha+5\beta}w(s)\|^2_{L^2} ds \leq C(t,u_0,w_0)
\end{align*}
which is \eqref{2.8}.

Moreover, since
$$
1+4\alpha+3\beta,\quad\text{and}\quad  4\alpha+5\beta>2,
$$
a  priori estimates \eqref{2.7} and \eqref{2.8} actually imply
the following Beale-Kato-Majda criterion of solutions $u$ and $w$,
$$
\int_0^T \|\nabla u(t) \|_{L^\infty}\,dt <\infty,\quad
\int_0^T \| \nabla w(t) \|_{L^\infty}\,dt <\infty
$$
by the Sobolev imbedding inequality. The proof is complete.
\end{proof}

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

Under the Beale-Kato-Majda criterion in Section 2, the proof of
Theorem \ref{thm1.1} is more or less standard.

\subsection*{Existence} We now borrow the classic Friedrichs method to prove 
the existence of global smooth solutions. Firstly we consider the following
approximate equations of \eqref{1.2}
\begin{equation}\label{3.1}
\begin{gathered}
\partial_t u_{n1} + J_nP(J_nu_n\cdot \nabla J_nu_{n1}) 
 - P\partial_y w_n+\partial_x\pi_n
 = J_n\partial_{yy}(-\Delta)^\alpha u_{n1}, \\
\partial_t u_{n2} +J_nP(J_nu_n\cdot \nabla J_nu_{n2}) 
+ P\partial_x w_n+\partial_y \pi_n
 = J_n\partial_{xx}(-\Delta)^\alpha u_{n2}, \\
\partial_xu_{n1}+\partial_yu_{n}=0, \\
 \partial_tw_n +2 w_n - \nabla\times u+ J_nu_n\cdot \nabla J_n w_n
 =-J_n(-\Delta)^\beta  w_n, \\
u_n(x,y,0)=J_nu_0,\quad w_n(x,y,0)=J_nw_0, 
\end{gathered}
\end{equation}
where $J_n$ is defined as
 $J_n\varphi=\mathcal{F}^{-1}(\chi_{B(0,n)}(\xi)\mathcal{F}(\varphi)(\xi))$,
and $\chi_{B(0,n)}$ denotes the characteristic function on the
ball $B(0,n)$. $P$ denotes the standard projection onto
divergence-free vector fields. The standard Picard type theorem
ensures that, for some $T_n>0$, there exists a unique local smooth
solution $(u_n,w_n)\in C([0,T_n);L^2)$. Additionally, it is easy to
see that $(J_n u_n, J_n w_n)$ is also a local smooth solution of
\eqref{3.1}. Thus $ (u_n, w_n ) $ also satisfies
\begin{equation}\label{3.2}
\begin{gathered}
\partial_t u_{n1} + J_n ( u_n\cdot \nabla u_{n1}) - \partial_y w_n+\partial_x\pi_n
 = \partial_{yy}(-\Delta)^\alpha u_{n1}, \\
\partial_t u_{n2} +J_n ( u_n\cdot \nabla u_{n2}) + \partial_x w_n+\partial_y \pi_n
 = \partial_{xx}(-\Delta)^\alpha u_{n2}, \\
\partial_xu_{n1}+\partial_yu_{n}=0, \\
 \partial_tw_n +2 w_n - \nabla\times u+ u_n\cdot \nabla w_n =- (-\Delta)^\beta
 w_n, \\
u_n(x,y,0)=J_nu_0,\,\,\,\,w_n(x,y,0)=J_nw_0, 
\end{gathered}
\end{equation}

A basic $L^2$ energy estimate for $(u_n,w_n)$ in \eqref{3.2}
implies
\[
\|u_n(t)\|^2_{L^2}+\|w_n(t)\|_{L^2}^2+
 \int_0^t\left(\|\Lambda^{1+\alpha} u_n(\tau)\|_{L^2}^2
 + \|\Lambda^\beta w_n(\tau)\|_{L^2}^2\right)\tau \leq C(t,u_0,w_0),
\]
where $C$ is independent of $n$. Therefore, the local solution can
be extended into a global one by the standard Picard Extension
Theorem. Next we show that $(u_n, w_n)$ admits a uniform global
bound in $H^s(\mathbb{R}^2)$ with $(s>2)$.
 By a standard energy estimate involving \eqref{3.2}, it is easy to
obtain
\begin{align*}
& \frac{d}{d t}\left(\|u_n\|_{H^s}^2
 +\|w_n\|_{H^s}^2\right)
 + \| \Lambda^{1+\alpha} u_n\|_{H^s}^2 + \|\Lambda^\beta w_n\|_{H^s}^2 \\
&\leq  \int_{\mathbb{R}^2} \{(\nabla\times w_n)\cdot \Lambda^{2s}u_n
 +(\nabla\times u_n) \Lambda^{2s} w_n\} \ dx dy\\
 &\quad + \int_{\mathbb{R}^2}[\Lambda^s, u_n\cdot\nabla]u_n\cdot \Lambda^s u_n
 \,dx\,dy
 +\int_{\mathbb{R}^2}[\Lambda^s,u_n\cdot\nabla]w_n \Lambda^sw_n \,dx\,dy\\
&\leq  \frac12 \left( \| \Lambda^{1+\alpha} u_n\|_{H^s}^2 +
 \|\Lambda^\beta w_n\|_{H^s}^2\right) \\
&\quad + C \left(\|\nabla u_n\|_{L^\infty} 
+ \|\nabla w_n\|_{L^\infty} +1\right) \left(\|u_n\|_{H^s}^2
 +\|w_n\|_{H^s}^2\right).
\end{align*}
Applying the Gronwall's inequality and the Beale-Kato-Majda
criterion in Proposition \ref{prop2.2}, we have
 \begin{align*}
&\|u_n\|_{H^s}^2 +\|w_n\|_{H^s}^2
 +\int_0^t(\| \Lambda^{1+\alpha} u_n\|_{H^s}^2 + \|\Lambda^\beta w\|_{H^s}^2)ds \\
 &\leq (\|u_0\|_{H^s}^2 +\|w_0\|_{H^s}^2)
 e^{C\int_0^t \left(\|\nabla u_n\|_{L^\infty}
 + \|\nabla w_n\|_{L^\infty} +1\right) ds}\\
&\leq  C(t,\|(u_0,w_0)\|_{H^s}).
\end{align*}
Hence the above uniform $H^s$ estimates allows us to obtain the
global existence of the desired solution $(u,w)$ to \eqref{1.2} by a
standard compactness argument.


\subsection{Uniqueness}

We show that any two solutions $(u,w)$ and $(\bar{u},\bar{w})$ to
\eqref{1.2} must be the same. The difference $(U,W)$ with $U=u
-\bar{u} $ and $W=w -\bar{w} $ satisfies
\begin{equation}\label{3.3}
\begin{gathered}
\partial_t U_1 +(U\cdot \nabla)u_1 +(\bar{u}\cdot \nabla)U_1 
 - \partial_y W+\partial_x\pi
 = \partial_{yy}(-\Delta)^\alpha U_1, \\
\partial_t U_2 +(U\cdot \nabla)u_2 +(\bar{u}\cdot \nabla)U_2 + \partial_x W+\partial_y \pi
 = \partial_{xx}(-\Delta)^\alpha U _2, \\
\partial_xU_1+\partial_yU_2=0, \\
 \partial_tW +2 W - \nabla\times U+ (U\cdot \nabla)w +(\bar{u}\cdot \nabla)W 
=-(-\Delta)^\beta  W .
\end{gathered}
\end{equation}
Taking the $L^2$ inner product of $(U,W)$ with \eqref{3.3}, we have
 \begin{align*}
&\frac12\frac{d}{d t}\left(\|U\|_{L^2}^2+\|W\|_{L^2}^2\right)
 + \frac12\| \Lambda^{1+\alpha}U\|_{L^2}^2 + \|\Lambda^{\beta} W\|_{L^2}^2
 +2 \| W\|_{L^2}^2 \\
&= \int_{\mathbb{R}^2}(\nabla \times W\cdot U+\nabla \times U
 W) dx -2\int_{\mathbb{R}^2}(U\cdot\nabla u )\cdot U dx
 - \int_{\mathbb{R}^2}(U\cdot\nabla w )W dx \\
& \leq C \|\Lambda^{1-\beta }U\|_{L^2}\|\Lambda^\beta W \|_{L^2}
 +C\|\nabla u_1\|_{L^{\infty}}\|U\|^2_{L^2}+2\|U\|_{L^2}
 \|\nabla w_1\|_{L^\infty}\|W\|_{L^2} \\
 & \leq \frac12\|\Lambda^\beta W \|^2_{L^2}+\frac14\|\Lambda^{1+\alpha}U\|^2_{L^2} \\
&\quad  +C\left(\|\nabla u \|_{L^{\infty}}+\|\nabla w \|_{L^\infty}+1\right)
 (\|U\|^2_{L^2}+\|W\|^2_{L^2}) .
\end{align*}
Gronwall's inequality then implies
\begin{eqnarray}
\|U(t)\|_{L^2}^2+\|W(t)\|_{L^2}^2 &\leq C(t,u_0,w_0)\left(\|
U_0\|_{L^2}^2+ \| W_0\|_{L^2}^2\right),
\end{eqnarray}
which implies the uniqueness.
\end{proof}

\subsection*{Acknowledgments} 
The authors want to express their sincere
thanks to the editor and the referee for their invaluable comments
and suggestions which helped improve the paper greatly. 
Y. Jia was partially supported by NSFC (No.11526032) and by
the NSF of Anhui Education Bureau(KJ2015A045). 
B. Dong was partially supported by the NSFC (No.11271019), 
by the CSC Award (No. 201506505001) and by the
research plan of Anhui University (Nos. J10118511069, J10118520111).

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\end{document}