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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 137, pp. 1--7.\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-2016/137\hfil Logarithmically improved regularity]
{Logarithmically improved regularity criteria for supercritical
quasi-geostrophic equations in Orlicz-Morrey spaces}

\author[S. Gala, M. A. Ragusa \hfil EJDE-2016/137\hfilneg]
{Sadek Gala, Maria Alessandra Ragusa}

\address{Sadek Gala \newline
Department of Mathematics, University of Mostaganem,
Box 227, Mostaganem 27000, Algeria. \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Catania,
Viale Andrea Doria, 6 95125 Catania, Italy}
\email{sadek.gala@gmail.com}

\address{Maria Alessandra Ragusa \newline
Dipartimento di Matematica e Informatica,
Universit\`a di Catania,
Viale Andrea Doria, 6 95125 Catania, Italy}
\email{maragusa@dmi.unict.it}

\thanks{Submitted September 28, 2015. Published June 8, 2016.}
\subjclass[2010]{35Q35,  76D03}
\keywords{Quasi-geostrophic equations; logarithmical regularity
criterion; \hfill\break\indent Orlicz-Morrey space}

\begin{abstract}
 This article provides a regularity criterion for the surface quasi-geostrophic
 equation with supercritical dissipation. This criterion is in terms of the
 norm of the solution in a Orlicz-Morrey space.
 The result shows that, if a weak solutions $\theta $ satisfies
 \begin{equation*}
 \int_0^T\frac{\| \nabla \theta (\cdot,s)\|
 _{\mathcal{M}_{L^2\log^P L} ^{2/r}} ^{\frac{\alpha }{\alpha -r}}}
 {1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,s)\| _{L^{2/r}})}ds<\infty ,
 \end{equation*}
 for some $0<r<\alpha $ and $0<\alpha <1$, then $\theta $ is regular at
 $t=T$. In view of the embedding
 $L^{2/r}\subset {\mathcal{M}_p}^{2/r}
 \subset  \mathcal{M}_{L^2\log^PL}^{2/r}$
 with $2<p<2/r$ and $P>1$, our result extends the results
 due to Xiang \cite{X} and Jia-Dong \cite{JD}.
\end{abstract}

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

\section{Introduction}

The surface quasi-geostrophic equation is
\begin{equation}
\begin{gathered}
\partial _{t}\theta +u\cdot\nabla \theta +\Lambda ^{\alpha }\theta =0,
\\
\theta (x,0) =\theta _0(x),
\end{gathered} \label{eq1.1}
\end{equation}
where $\theta =\theta (x,t)$ is a scalar real-valued function of
$(x,t)\in\mathbb{R}^2\times \mathbb{R}^{+}$ and $u=(u^1,u_{2})$ is the associated
incompressible velocity field of the fluid with $\nabla \cdot u=0$, and
determined from $\theta $ by
\begin{equation*}
u:=(-\partial _{2}\Lambda ^{-1}\theta ,\partial ^1\Lambda ^{-1}\theta )
=\mathcal{R}^{\bot }\theta =(-\mathcal{R}_{2}\theta ,\mathcal{R}^1\theta ),
\end{equation*}
where $\Lambda =(-\Delta ) ^{1/2}$ is the Zygmund
operator and $\mathcal{R}_i$, $i=1,2$ are the Riesz transforms.

The surface quasi-geostrophic equation with subcritical $(1<\alpha \leq 2)$
or critical dissipation $(\alpha =1)$ have been shown to possess global
classical solutions whenever the initial data is sufficiently smooth.
However, the global regularity problem remains open for the supercritical
case $(0<\alpha <1)$. Various regularity (or blow-up) criteria have been
produced to shed light on this difficult global regularity problem (see e.g.
\cite{CV,Chae, CC, CDW, CW, CW1, Ju, KNV, Wu, Y} and the references
therein). The difficulties in understanding this problem are similar to
those in solving the three-dimensional Navier-Stokes equations.

For the study of the regularity criterion to \eqref{eq1.1} for the critical and
supercritical case, Constantin, Majda and Tabak \cite{CMT} obtained the
following conditions
\begin{equation}
\limsup_{t\nearrow T} \| \theta (t)\|_{H^m}<\infty \quad
\text{if and only if}\quad
\int_0^T\|\nabla ^{\bot }\theta (\cdot,t)\| _{L^{\infty }}dt<\infty ,
\label{eq01}
\end{equation}
with $m\geq 3$ and $\nabla ^{\bot }=(-\partial _{x_{2}},\partial _{x^1})$.
Later on, Chae \cite{Chae} (see also \cite{CL}) generalizes \eqref{eq01} to
obtain the regularity criterion of the supercritical quasi-geostrophic
equation \eqref{eq1.1} under the assumption
\begin{equation*}
\int_0^T\| \nabla ^{\bot }\theta (\cdot,t)\|
_{L^p}^rdt<\infty \quad \text{with  }\frac{2}{p}+\frac{\alpha }{r}\leq
\alpha \text{  and  }\frac{2}{\alpha }<p<\infty .
\end{equation*}

Recently, Xiang \cite{X} improved the Chae's result and obtained another
logarithmically improved regularity criterion in terms of the Lebesgue space
subject to the assumption
\begin{equation}
\int_0^T\frac{\| \nabla \theta (\cdot,t)\|_{L^p}^r}
{1+\ln (e+\| \nabla \theta (\cdot,t)\| _{L^{\infty}})}dt<\infty \quad
\text{with  }\frac{2}{p}+\frac{\alpha }{r}\leq \alpha
\text{ and  }\frac{2}{\alpha }<p<\infty .  \label{eq1.6}
\end{equation}

Very recently, Jia and Dong \cite{JD} improves the above regularity
criterion \eqref{eq1.6} from Lebesgue space framework to Morrey-Campanato
space framework. More precisely, they show the regularity of weak solution
when the temperature function $\theta $ satisfies the  growth
condition
\begin{equation}
\int_0^T\frac{\| \nabla \theta (\cdot,t)\| _{{
\mathcal{M}}_q^p}^r}{1+\ln (e+\| \nabla \theta
(\cdot,t)\| _{L^p})}dt<\infty \quad
\text{with  }\frac{2}{p}+\frac{
\alpha }{r}=\alpha \text{ and  }\frac{2}{\alpha }<p<\infty .
\label{eq1.7}
\end{equation}

The regularity criterion presented in this article states that, if a weak
solution of \eqref{eq1.1} satisfies
\begin{equation*}
\int_0^T\frac{\| \nabla \theta (\cdot,s)\| _{\mathcal{M}_{L^2\log ^PL}^{2/r}}
^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,s)\| _{L^{2/r}})}ds
<\infty \quad\text{with  }0<r<\alpha ,
\end{equation*}
for some $0<r<\alpha $ and $0<\alpha <1$, then $\theta $ is actually regular
in $H^2$ on $[0,T]$, where
 $\mathcal{M}_{L^2\log^PL}^{2/r}$ denotes the Orlicz-Morrey space.
Since the embedding relation
 $L^{2/r}\subset \mathcal{M}_{L^2\log^PL}^{2/r}$ with $P>1$ holds,
our regularity criterion can be understood
as an extension of the regularity results of Xiang \cite{X} and Jia-Dong
\cite{JD}. Main tools used in this paper are a weighted norm inequality for
the Riesz potential and the Gagliardo-Nirenberg inequality.

\section{Orlicz-Morrey spaces and statement of the main result}

Before stating our result, let us recall some definitions and properties of
the spaces that we are going to use (see e.g. \cite{GST, gala,  GST1, GSTR,
SSTG} and references therein).

\begin{definition}\label{def1} \rm
For $P\in {\mathbb{R}}$ and $1<w<v<\infty $, the Orlicz-Morrey
space ${\mathcal{M}}_{L^u\log ^P L}^v$ is defined by
\begin{equation}
\| f\| _{\mathcal{M}_{L^u\log ^P L}}^v
:=\sup \big\{r^{2/v}\| f\| _{B(x,r),L^u\log ^P L}:
x\in {\mathbb{R}}^2,\,r>0\big\} ,  \label{eq6}
\end{equation}
where $\| f\| _{B(x,r),L^u\log ^P L}$ denotes the
$t^{w}\log ^P (3+t)$ average given by
\begin{align*}
&\| f\| _{B(x,R),L^u\log ^P L} \\
& :=\inf \Big\{ \lambda >0:\frac{1}{|B(x,R)|}\int_{B(x,R)}\Big(\frac{
|f(x)|}{\lambda }\Big) ^{w}\log \Big(3+\frac{|f(x)|}{\lambda }\Big)
^P \,dx\leq 1\Big\} .
\end{align*}
\end{definition}

Our main result now reads as follows.

\begin{theorem}\label{thm0}
Let $\theta $ be a Leray-Hopf weak solutions of \eqref{eq1.1}
with $0<\alpha <1$, namely
\begin{equation*}
\theta \in L^{\infty }(0,T;L^2(\mathbb{R}^2))\cap L^2(0,T;\dot {H
}^{\alpha }(\mathbb{R}^2)).
\end{equation*}
and satisfies the condition
\begin{equation}
\int_0^T\frac{\| \nabla \theta (\cdot,s)\|
_{\mathcal{M}_{L^2\log ^P L}^{2/r}} ^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,s)\| _{L^{2/r}})}ds
<\infty \quad\text{with  }0<r<\alpha .  \label{eq11}
\end{equation}
Then, the solution $\theta (x,t)$ is regular on $(0,T]$.
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
This criterion is in terms of the norm of the solution in a Orlicz-Morrey
space. It is clear that Theorem \ref{thm0} gives a logarithmic improvement of
Xiang's regularity criteria \eqref{eq1.6} (see also \ref{eq1.7}). As a
consequence, this result extends several previous works.
\end{remark}

Meanwhile, the definition of classical Morrey-Campanato spaces is as follows
(see e.g. \cite{Lem}):

\begin{definition} \label{def2.3} \rm
For $1<p\leq q\leq +\infty $, the Morrey-Campanato space
is defined by
\begin{equation}
{\mathcal{M}}_q^p=\Big\{ f\in L_{\mathrm{loc}}^p(\mathbb{R}
^2) : \| f\| _{{\mathcal{M}}
_q^p}=\sup_{x\in {\mathbb{R}}^2}\sup_{R>0}| B|^{1/q-1/p}
\| f\| _{L^p(B(x,R))}<\infty \Big\} ,
\label{eq4}
\end{equation}
where $B(x,R)$ denotes the closed ball in $\mathbb{R}^2$ with center $x$
and radius $R$.
\end{definition}

In view of \eqref{eq6} and \eqref{eq4}, the definition \eqref{eq6} covers
\eqref{eq4} as a special case when $P=0$. Here and below we write
$(\log a)^P =:\log ^P a$.

Recall the following crucial result established in \cite{GST, GSTR, GST1}
(see also \cite{SST, SST1, SST2, SST3}).

\begin{theorem}\label{thm1}
Let $0<\alpha <1$ and fractional integral operator $I_{\alpha }$
be defined by
\begin{equation}
I_{\alpha }f(x)=\int_{{\mathbb{R}}^2}\frac{f(y)}{|x-y|^{2-\alpha }}\,dy.
\label{eq26}
\end{equation}
If $P>1$, then
\begin{equation}
\| g\cdot I_{\alpha }f\| _{L^2}\leq C\,\| g\| _{{\mathcal{M}}_{
L^2\log ^P L}^{3/\alpha }}\| f\| _{L^2}.  \label{eq27}
\end{equation}
\end{theorem}

Additionally, we have the following embeddings:
for $P>0$ and $0<u<\tilde{u} <v$,
\begin{equation}
L^v\hookrightarrow L^{v,\infty }\hookrightarrow {\mathcal{M}}_{\tilde{u}
}^v\hookrightarrow {\mathcal{M}}_{L^u\log ^P L
}^v\hookrightarrow {\mathcal{M}}_{u}^v  \label{eq7}
\end{equation}
in the sense of continuous embedding and the inclusion is proper, where
$L^{p,\infty }$ denotes the usual Lorentz (weak-$L^p$) space. For more
details see \cite{GST, GSTR, GST1}.
We shall use as well the following useful Sobolev inequality.

\begin{lemma}\label{lem12}
Suppose that $s>1$ and $p\in \lbrack 2,+\infty ]$. Then, there
is a constant $C\geq 0$ such that
\begin{equation*}
\| f\| _{L^p({\mathbb{R}}^2{\mathbb{)}}}\leq
C\| f\| _{H^s({\mathbb{R}}^2{\mathbb{)}}}.
\end{equation*}
In particular,
\begin{equation*}
\| f\| _{L^{2/r}({\mathbb{R}}^2{\mathbb{)}}}\leq
C\| f\| _{H^2({\mathbb{R}}^2{\mathbb{)}}}\quad \text{with }0\leq r\leq 1.
\end{equation*}
\end{lemma}

The above lemma can be proved using the well-known boundedness property of the
Riesz potential operator (see, e.g., Stein \cite{St}).
In the proof of the main result, we employ the following Gagliardo-Nirenberg
inequality having fractional derivatives contained in \cite{GT}.

\begin{lemma} \label{lem0}
Let $1<p$, $p_0,p^1\leq \infty $, $s,\gamma \in {\mathbb{R}}_{+} $,
$0\leq \beta \leq 1$. Then, there exists a constant $C$ such that
\begin{equation*}
\| f\| _{\dot {H}_p^s}\leq C\|f\| _{L_{p_0}}^{1-\beta }\| f\| _{\dot {H}
_{p^1}^{\gamma }}^{\beta },
\end{equation*}
where
\begin{equation*}
\frac{1}{p}-\frac{s}{2}=\frac{1-\beta }{p_0}+\beta \big(\frac{1}{p^1}-
\frac{\gamma }{2}\big) ,\quad s\leq \beta \gamma .
\end{equation*}
In particular,
\begin{equation}
\| f\| _{\dot {H}^s}\leq C\| f\|_{L^2}^{1-\frac{s}{\gamma }}\| f\| _{\dot {H}
^{\gamma }}^{s/\gamma}.  \label{eq89}
\end{equation}
\end{lemma}

Now we are in a position to prove our regularity criterion.

\begin{proof}[Proof of Theorem \ref{thm0}]
It suffices to prove that \eqref{eq11} ensures the a priori estimate
\begin{equation*}
\int_0^T\| \nabla ^{\bot }\theta (\cdot,t)\|_{L^{\infty }}dt<\infty ,
\end{equation*}
hence guaranteeing the desired regularity until $T$ by \eqref{eq01}.

For this, applying $\Lambda ^2$ to  \eqref{eq1.1} and taking the
 $L^2$ inner product of the resulting equation with $\Lambda ^2\theta $
and integrating by parts, we obtain
\begin{align*}
&\frac{d}{dt}\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+2\| \Lambda ^{2+\frac{\alpha }{2}}\theta
(\cdot,t)\| _{L^2}^2 \\
&= -2\int_{\mathbb{R}^2}(u\cdot\nabla \theta ) \Lambda ^{4}\theta\,dx \\
&= -2\int_{\mathbb{R}^2}\Lambda ^2(u\cdot\nabla \theta ) \Lambda ^2\theta
\,dx \\
&= -2\int_{\mathbb{R}^2}(\Lambda ^2u\cdot\nabla \theta ) \Lambda ^2\theta
\,dx-2\int_{\mathbb{R}^2}(\Lambda u\cdot\nabla \Lambda \theta ) \Lambda
^2\theta\,dx-2\int_{\mathbb{R}^2}(u\cdot\nabla \Lambda ^2\theta ) \Lambda
^2\theta\,dx \\
&= -2\int_{\mathbb{R}^2}(\Lambda ^2u\cdot\nabla \theta ) \Lambda ^2\theta
\,dx-2\int_{\mathbb{R}^2}(\Lambda u\cdot\nabla \Lambda \theta ) \Lambda
^2\theta\,dx,
\end{align*}
where we have used the following cancelation property
\begin{equation*}
\int_{\mathbb{R}^2}(u\cdot\nabla \Lambda ^2\theta ) \Lambda ^2\theta\,dx
=\frac{1}{2}\int_{\mathbb{R}^2}u\cdot\nabla | \Lambda ^2\theta| ^2dx
=-\frac{1}{2}\int_{\mathbb{R}^2}(\nabla \cdot u).| \Lambda ^2\theta | ^2dx=0.
\end{equation*}
Notice that $\Lambda ^s$ and $\nabla $  commute. Hence, by
H\"{o}lder inequality, first we estimate $J$.
By using the Schwarz inequality, the
fact that $\dot{B}_{2.2}^r=\dot {H}^r$ and the interpolation
inequality, we have
\begin{align*}
&\frac{d}{dt}\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+2\| \Lambda ^{2+\frac{\alpha }{2}}\theta
(\cdot,t)\| _{L^2}^2   \\
&= -2\int_{\mathbb{R}^2}(\Lambda ^2u\cdot\nabla \theta )
I_{r}(-\Delta )^{\frac{r}{2}}\Lambda ^2\theta\,dx
-2\int_{\mathbb{R}^2}(\Lambda u\cdot\nabla \Lambda \theta )
 I_{r}(-\Delta )^{\frac{r}{2}}\Lambda ^2\theta\,dx
\\
&\leq 2\| \mathcal{R}^{\bot }\Lambda ^2\theta (\cdot,t)\|
_{L^2}\| (\nabla \theta  I_{r}(-\Delta )^{\frac{r}{2}}\Lambda
^2\theta )(\cdot,t)\| _{L^2}   \\
&\quad +2\| \nabla \Lambda \theta (\cdot,t)\| _{L^2}\| (
\mathcal{R}^{\bot }\Lambda \theta  I_{r}(-\Delta )^{\frac{r}{2}}\Lambda
^2\theta )(\cdot,t)\| _{L^2}.
\end{align*}
If we invoke Theorem \ref{thm1} and \eqref{eq89}, then we have by Young inequality
and the boundedness of $\mathcal{R}^{\bot }$ in the space $L^2$ and
${\mathcal{M}}_{L^2\log ^P L}^{2/r}$
\begin{align*}
&\frac{d}{dt}\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+2\| \Lambda ^{2+\frac{\alpha }{2}}\theta
(\cdot,t)\| _{L^2}^2 \\
&\leq C\| \Lambda ^2\theta (\cdot,t)\| _{L^2}
\| \Lambda ^2\theta (\cdot,t)\| _{\dot {H}^r}
\| \nabla \theta (\cdot,t)\| _{\mathcal{M}_{L^2\log^PL}} ^{2/r} \\
&\quad +C\| \Lambda ^2\theta (\cdot,t)\| _{L^2}\|
\Lambda ^2\theta (\cdot,t)\| _{\dot {H}^r}\| \mathcal{
R}^{\bot }\Lambda \theta (\cdot,t)\| \\
&\leq C\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^{2-\frac{2r}{
\alpha }}\| \Lambda ^2\theta (\cdot,t)\| _{\dot {H}^{
\frac{\alpha }{2}}}^{\frac{2r}{\alpha }}\| \nabla \theta
(\cdot,t)\| _{\mathcal{M}_{L^2\log^PL}}^{2/r} \\
&= \Big(C\| \Lambda ^2\theta (\cdot,t)
\|_{L^2}^2 \| \nabla \theta (\cdot,t)
\| _{\mathcal{M}_{L^2\log ^P L}^{2/r}} ^{\frac{\alpha }{\alpha -r}}\Big)
^{1-\frac{r}{\alpha }}
\Big(\| \Lambda ^2\theta (\cdot,t)\|
_{\dot {H}^{\frac{\alpha }{2}}}^2\Big) ^{r/\alpha} \\
&\leq \frac{1}{2}\| \Lambda ^2\theta (\cdot,t)\|
_{\dot{H}^{\alpha/2}}^2 +C\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2
\| \nabla \theta (\cdot,t)\| _{\mathcal{M}_{L^2\log ^P L}^{2/r}}
^{\frac{\alpha }{\alpha -r}}.
\end{align*}
Consequently, by absorbing the diffusion term into the left hand side, we
obtain
\begin{align*}
&\frac{d}{dt}\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+\| \Lambda ^{\frac{\alpha }{2}+2}\theta
(\cdot,t)\| _{L^2}^2 \\
&\leq C\| \nabla \theta (\cdot,t)\| _{{\mathcal{M}}_{\mathrm{
L^2\log ^P L}}^{2/r}}^{\frac{\alpha }{\alpha -r}}\|
\Lambda ^2\theta (\cdot,t)\| _{L^2}^2 \\
&\leq C\frac{\| \nabla \theta (\cdot,t)\| _{{\mathcal{M}}_{
L^2\log ^P L}^{2/r}}^{\frac{\alpha }{\alpha -r}}}{1+\ln
(e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})}
[1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{\frac{2}{r}
}})]\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2 \\
&\leq C\frac{\| \nabla \theta (\cdot,t)\| _{{\mathcal{M}}_{
L^2\log ^P L}^{2/r}}^{\frac{\alpha }{\alpha -r}}}{1+\ln
(e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})}
[1+\ln (e+\| \Lambda ^2\theta (\cdot,t)\|
_{L^2})]\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2,
\end{align*}
where we have used the  Sobolev embedding (see Lemma \ref{lem12})
\begin{equation*}
\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}}
\leq C\| \Lambda ^2\theta (\cdot,t)\| _{L^2}\quad \text{for } 0<r<1.
\end{equation*}
It follows that
\[
\frac{d}{dt}\ln (e+\| \Lambda ^2\theta (\cdot,t)\|_{L^2}^2) \\
\leq C\frac{\| \nabla \theta (\cdot,t)\| _{{
\mathcal{M}}_{L^2\log ^P L}^{2/r}}^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})}
[1+\ln (e+\| \Lambda ^2\theta (\cdot,t)\|_{L^2}^2)]
\]
and thus by Gronwall's inequality,
\begin{align*}
&\ln (e+\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2)\\
&\leq \ln (e+\| \Lambda ^2\theta _0(\cdot,t)\| _{L^2}^2)
\exp \Big(C\int_0^T
\frac{\| \nabla \theta (\cdot,t)\| _{\mathcal{M}_{L^2\log^P L} ^{2/r}}
^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})} dt \Big) .
\end{align*}
This gives the uniform boundedness of
$\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2$ in the time interval $[0,T]$.
Recall that 
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+\| \Lambda ^{\frac{\alpha }{2}+2}\theta
(\cdot,t)\| _{L^2}^2   \\
&\leq C\frac{\| \nabla \theta (\cdot,t)\|
 _{\mathcal{M}_{L^2\log ^P L}^{2/r}}^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})}
[1+\ln (e+\| \Lambda ^2\theta (\cdot,t)\|
_{L^2})]\| \Lambda ^2\theta (\cdot,t)\| _{L^2}^2.
\end{aligned} \label{eq29}
\end{equation}
Integrating \eqref{eq29} over $[0,T]$, we have
\begin{align*}
&\| \Lambda ^2\theta (\cdot,t)\|
_{L^2}^2+\int_0^T\| \Lambda ^{\frac{\alpha }{2}+2}\theta (\cdot,t)\| _{L^2}^2dt \\
&\leq C\int_0^T \frac{\| \nabla \theta (\cdot,t)
\| _{\mathcal{M}_{L^2\log^PL}^{2/r}}^{\frac{\alpha }{\alpha -r}}}
{1+\ln (e+\| \nabla ^{\bot }\theta (\cdot,t)\| _{L^{2/r}})}dt\\
&\quad\sup_{0\leq t\leq T} \big\{ [1+\ln (e+\|
\Lambda ^2\theta (\cdot,t)\| _{L^2})]\| \Lambda ^2\theta
(\cdot,t)\| _{L^2}^2\big\} +\| \Lambda ^2\theta_0\| _{L^2}^2,
\end{align*}
which implies
\begin{equation*}
\int_0^T\| \Lambda ^{\frac{\alpha }{2}+2}\theta
(\cdot,t)\| _{L^2}^2dt<\infty .
\end{equation*}
On the other hand, by the Gagliardo-Nirenberg inequality in $\mathbb{R}^2$,
it follows that
\begin{align*}
\| \nabla ^{\bot }\theta \| _{L^{\infty }}
&\leq C\| \theta \| _{L^2}^{\frac{\alpha }{\alpha +4}
}\| \nabla ^{\bot }\theta \| _{\dot {H}^{1+\frac{
\alpha }{2}}}^{\frac{4}{\alpha +4}} \\
&\leq C\| \theta \| _{L^2}^{\frac{\alpha }{\alpha +4}
}\| \Lambda ^{\frac{\alpha }{2}+2}\theta \| _{L^2}^{\frac{4}{\alpha +4}}.
\end{align*}
Noting that $\| \theta \| _{L^2}\leq \| \theta_0\| _{L^2}$, implies
\begin{align*}
\int_0^T\| \nabla ^{\bot }\theta (\cdot,t)\|_{L^{\infty }}dt
&\leq C\int_0^T\| \theta
(\cdot,t)\| _{L^2}^{\frac{\alpha }{\alpha +4}}\| \Lambda ^{
\frac{\alpha }{2}+2}\theta (\cdot,t)\| _{L^2}^{\frac{4}{\alpha +4}}dt
\\
&\leq C\| \theta _0\| _{L^2}^{\frac{\alpha }{\alpha +4}
}\int_0^T\| \Lambda ^{\frac{\alpha }{2}+2}\theta
(\cdot,t)\| _{L^2}^{\frac{4}{\alpha +4}}dt \\
&\leq C\| \theta _0\| _{L^2}^{\frac{\alpha }{\alpha +4}
}T^{\frac{\alpha +2}{\alpha +4}}(\int_0^T\| \Lambda
^{\frac{\alpha }{2}+2}\theta (\cdot,t)\| _{L^2}^2dt) ^{\frac{2
}{\alpha +4}}
<\infty .
\end{align*}
By the blow-up criterion \eqref{eq11} of smooth solutions to \eqref{eq1.1},
we complete the proof.
\end{proof}

\subsection*{Acknowledgments}

This work was done, while the first author was visiting University of
Catania. He thanks the Department of Mathematics at the University of
Catania for its support and hospitality. The authors want to express their
sincere thanks to the editor and the referees for their invaluable comments
and suggestions for improving this paper.

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