\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 153, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/153\hfil Regularity for 3D Navier-Stokes equations]
{Regularity for 3D Navier-Stokes equations
in terms of two components of the vorticity}

\author[S. Gala\hfil EJDE-2010/153\hfilneg]
{Sadek Gala} 

\address{Sadek Gala \newline
Department of Mathematics, University of Mostaganem\\
Box 227, Mostaganem 27000, Algeria}
\email{sadek.gala@gmail.com}

\thanks{Submitted May 20, 2010. Published October 28, 2010.}
\subjclass[2000]{35Q35, 76C99}
\keywords{Navier-Stokes equations; regularity conditions;
 Morrey-Campaanto spaces}

\begin{abstract}
 We establish regularity conditions for the 3D
 Navier-Stokes equation via two components of the vorticity vector.
 It is known that if a Leray-Hopf weak solution $u$ satisfies
 \[
 \tilde{\omega}\in L^{2/(2-r)}(0,T;L^{3/r}(\mathbb{R}^3))\quad
 \text{with }0<r<1,
 \]
 where $\tilde{\omega}$ form the two components of the vorticity,
 $\omega =\operatorname{curl}u$, then $u$ becomes the classical
 solution on $(0,T]$ (see \cite{DC}). We prove the regularity
 of Leray-Hopf weak solution $u$ under each of the following
 two (weaker) conditions:
 \begin{gather*}
 \tilde{\omega}\in L^{2/(2-r)}(0,T;\dot {\mathcal{M}}_{2,
 3/r}(\mathbb{R}^3))\quad \text{for }0<r<1,\\
 \nabla \tilde{u}\in L^{2/(2-r)}(0,T;\dot {\mathcal{M}}_{2,
 3/r}(\mathbb{R}^3))\quad \text{for }0\leq r<1,
 \end{gather*}
 where $\dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ is the
 Morrey-Campanato space.
 Since $L^{3/r}(\mathbb{R}^3)$ is a proper subspace of
 $\dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$, our regularity
 criterion improves the results in  Chae-Choe \cite{DC}.
\end{abstract}

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

\section{Introduction}

We consider the Navier-Stokes equations, in $\mathbb{R}^3$,
\begin{equation}
\begin{gathered}
\partial _{t}u+(u\cdot\nabla) u-\Delta u+\nabla p =0,
\quad (x,t)\in \mathbb{R}^3\times (0,T),   \\
\operatorname{div}u =0,\quad (x,t)\in \mathbb{R}^3\times (0,T), \\
u(x,0) =u_{0}(x),\quad x\in \mathbb{R}^3,
\end{gathered} \label{eqNS}
\end{equation}
where $u=u(x,t)$ is the velocity field, $p=p(x,t)$ is the
scalar  pressure and $u_{0}(x)$ with $\operatorname{div}u_{0}=0$
in the sense of distribution is the initial velocity field. For
simplicity, we assume that the external force has a scalar
potential and is included in the pressure gradient.

In their well known articles, Leray \cite{Ler} and Hopf \cite{Hop}
independently constructed a weak solution $u$ of \eqref{eqNS} for
arbitrary $u_{0}\in L^2(\mathbb{R}^3)$ with div $u_{0}=0$. The
solution is called the Leray-Hopf weak solution. Regularity of
such Leray-Hopf weak solutions is one of the most significant open
problems in mathematical fluid mechanics. Here we mean by the weak
solution a function $u\in L^{\infty }(0,T;L^2(\mathbb{R}^3))\cap
L^2(0,T;\dot{H}^1(\mathbb{R}^3))$ satisfying \eqref{eqNS} in
sense of distributions.

A weak solution of the Navier-Stokes equation that belongs to
$L^{\infty }(0,T;H^1(\mathbb{R}^3))\cap
L^2(0,T;H^2(\mathbb{R}^3))$ is called a strong solution.
Introducing the class $L^{\alpha }(0,T;L^{q}(\mathbb{R} ^3))$, it
is shown that if a Leray-Hopf weak solution $u$ belongs
to $L^{\alpha }((0,T);L^{q}(\mathbb{R}^3))$ with the exponents
$\alpha $ and $q$ satisfying $\frac{2}{\alpha }+\frac{3}{q}\leq
1$, $2\leq \alpha <\infty $, $3<q\leq \infty $, then
$u(x,t)\in C^{\infty }(\mathbb{R}^3\times (0,T))$
 \cite{Ser, ohy,pro, fab, Soh-W, tak, G}, while the limit case
$\alpha =\infty$, $q=3$ was covered much later Escauriaza, Seregin
and Sverak in \cite{sve}. See also \cite{koz0} for recent
improvements of the criteria, using the negative order
Triebel-Lizorkin spaces.

On the other hand, Beir\~{a}o da Veiga \cite{B} obtained a sufficient
condition for regularity using the vorticity rather than velocity. His
result says that if the vorticity $\omega =$curl $u$ of a weak solution $u$
belongs to the space $L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))$ with $\frac{2
}{\alpha }+\frac{3}{q}\leq 2$ and $1\leq \alpha <\infty $, then $u$ becomes
the strong solution on $(0,T]$. Later, Chae-Choe \cite{DC} obtained an
improved regularity criterion of \cite{B} imposing condition on only two
components of the vorticity, namely if
\begin{equation}
\tilde{\omega}=(\omega _1,\omega _2,0)\in L^{\alpha }
(0,T;L^{q}(\mathbb{R}^3))\quad
\mbox{with } \frac{2}{\alpha }+\frac{3}{q}\leq 2,\;
1\leq\alpha <\infty,  \label{eq0}
\end{equation}
then the weak solution becomes smooth.

The purpose of this article is to prove the result of \cite{DC}
in the other cases, proving that if
 $\tilde{\omega}\in L^{2/(2-r)}(0,T;\dot{\mathcal{M}}_{2,3/r}
(\mathbb{R}^3))$ with $0<r<1$, then the weak solution becomes smooth.
Here $\dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ is the Morrey-Campanato
space, which is
strictly bigger than $L^{3/r}(\mathbb{R}^3)$ (see the
next section for the related embedding relations). We remark that
in the limiting case $r=0$, Kozono-Yatsu \cite{koz} previously
weakened the condition
$\tilde{\omega}\in L^1(0,T;L^{\infty }(\mathbb{R}^3))$ of
\cite{DC} into $\tilde{\omega}\in L^1(0,T;
\dot{B}_{\infty ,\infty }^0(\mathbb{R}^3))$, where
$\dot{B}_{\infty ,\infty }^0$ is the Besov space.

\section{Preliminaries and the main theorems}

Now, we recall the definition and some properties of the space that
we are going to use. These spaces play an important role in studying
the regularity of solutions to partial differential equations
(see e.g. \cite{Kat,Tay}).

\begin{definition} \label{def} \rm
For $1<p\leq q\leq +\infty $, the Morrey-Campanato space
$\dot {\mathcal{M}}_{p,q}(\mathbb{R}^3)$ is defined as
\begin{align*}
&\dot {\mathcal{M}}_{p,q}(\mathbb{R}^3)\\
&=\big\{ f\in L_{\rm loc}^p(\mathbb{R}^3):\|
f\| _{\dot {\mathcal{M}}_{p,q}}
=\sup_{x\in \mathbb{R}^3} \sup_{R>0}R^{3/q-3/p}\|
f(y)1_{B(x,R)}(y)\| _{L^p(dy)}<\infty \big\}
\end{align*}
\end{definition}

It is easy to check that
\begin{gather*}
\| f(\lambda \cdot )\| _{\dot {\mathcal{M}}_{p,q}}
=\frac{1}{\lambda ^{3/q}}\| f\| _{\dot{\mathcal{M}}_{p,q}},
\quad \lambda >0, \\
\dot{\mathcal{M}}_{p,\infty }(\mathbb{R}^3)
=L^{\infty }(\mathbb{R}^3)\quad \text{for }1\leq p\leq \infty .
\end{gather*}
Additionally, for $2\leq p\leq 3/r$ and $0\leq r<3/2$ we
have the following embedding relations:
\[
L^{3/r}(\mathbb{R}^3)\hookrightarrow L^{3/r,\infty }
(\mathbb{R}^3)\hookrightarrow \dot{\mathcal{M}}_{p,3/r}
(\mathbb{R}^3),
\]
where $L^{p,\infty }$ denotes the weak $L^p-$space.
 The second relation
\[
L^{3/r,\infty }(\mathbb{R}^3)\hookrightarrow
\dot{\mathcal{M}}_{p,\frac{3}{r}}(\mathbb{R}^3)
\]
is shown as follows.
\begin{align*}
\| f\| _{\dot {\mathcal{M}}_{p,3/r}}
&\leq \sup_{E}| E| ^{\frac{r}{3}-\frac{1}{p}}\Big(
\int_{E}| f(y)| ^pdy\Big)^{1/p}\quad
 (f\in L^{3/r,\infty }(\mathbb{R}^3))\\
&=\Big(\sup_{E} | E| ^{\frac{pr}{3}-1}\int_{E}| f(y)| ^p dy\Big)^{1/p}
 \\
&\cong \Big(\sup_{R>0} R| \{ x\in \mathbb{R}
^3:| f(y)| ^p>R\} | ^{pr/3}\Big)^{1/p} \\
&=\sup_{R>0} R| \{ x\in \mathbb{R}^p:|
f(y)| >R\} | ^{r/3} \\
&\cong \| f\| _{L^{3/r,\infty }}.
\end{align*}

\begin{remark} \label{rmk2.1} \rm
For the case $q=3/2$ in \eqref{eq0}, we can show that
there exists an absolute constant $\delta $ such that if the weak
solution $u$ of \eqref{eqNS} on $(0,T)$ with energy inequality
satisfies
\[
\sup_{0<t<T} \| \tilde{\omega}(t)\| _{L^{3/2,\infty }}\leq \delta ,
\]
then $u$ is actually regular (see \cite[p. 60]{koz}, \cite{Ber1}).
As another type of criterion, Neustupa-Novotny-Penel \cite{NNP}
considered suitable weak solution $u=(u_1,u_2,u_{3})$ introduced by
Caffarelli-Kohn-Nirenberg \cite{CKN} and showed regularity of $u$
under the hypothesis that
$u_{3}\in L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))$ with
 $ \frac{2}{\alpha }+\frac{3}{q}=\frac{1}{2}$ (see  \cite{CT}).
The corresponding result for $u$ to the above case was obtained
by Berselli \cite{Ber} who proved regularity under the assumption
that $\sup_{0<t<T}\| \widetilde{u}(t)\| _{L^{3,\infty }}$
is sufficiently small, where $\widetilde{u}=(u_1,u_2,0)$.
\end{remark}

 We need the following lemma which is essentially due to Lemari\'{e}
-Rieusset \cite{Lem1}.

\begin{lemma}\label{lem 2}
For $0\leq r<3/2$, the space
$\dot {Z}_{r}( \mathbb{R}^3)$ is defined as the space of
$f(x)\in L_{\rm loc}^2(\mathbb{R}^3)$ such that
\[
\| f\| _{\dot {Z}_{r}}=\sup_{\|g\| _{\dot {B}_{2,1}^{r}}\leq 1}
\|fg\| _{L^2}<\infty .
\]
Then $f\in \dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$
if and only if $f\in \dot {Z}_{r}(\mathbb{R}^3)$ with equivalence
of norms.
\end{lemma}

Since $L^{3/r}(\mathbb{R}^3)\mathbb{\varsubsetneq }
\dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$, the
above regularity criterion is an improvement on Chae-Choe' s result and
hence our regularity criterion covers the recent result given by
 Chae-Choe \cite{DC}.
Our result on \eqref{eqNS} reads as follows.

\begin{theorem}\label{th1}
Let $u_{0}\in L^2(\mathbb{R}^3)$ with $\nabla.u_{0}=0$ and
$\omega _{0}=$curl $u_{0}\in L^2(\mathbb{R}^3) $. If
the first two components of the vorticity
$\tilde{\omega }=\omega _1e_1+\omega _2e_2$ of the Leray-Hopf
weak solution $u$, satisfies
$ \tilde{\omega }\in L^{2/(2-r)}(0,T,\dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3))$ with $0<r<1$, then $u$ becomes the
classical solution on $(0,T]$.
\end{theorem}

\begin{remark} \label{rmk2.2} \rm
As an immediate consequence of the above theorem, we find that if the
classical solution of the Navier-Stokes equations blow-up at
time $T$, then
\[
\| \tilde{\omega }\| _{L^{2/(2-r)}(0,T, \dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3))
}=\infty ,
\]
where $\tilde{\omega }$ is any two component vector of $\omega $.
\end{remark}

Our second theorem concerns the regularity criterion in terms
of gradients of the components of velocity.

\begin{theorem}\label{th2}
Let $\tilde{u}=u_1e_1+u_2e_2$ be the first two components
of a Leray-Hopf weak solution of the Navier-Stokes equation corresponding to
$u_{0}\in H^1(\mathbb{R}^3)$ with $\mathrm{div}\,u_{0}=0$.
Suppose that
$\nabla \tilde{u}\in L^{2/(2-r)}(0,T,\dot {\mathcal{M}}_{2,3/r}(\mathbb{R}^3))$ with $
0\leq r<1$, then $u$ becomes the classical solution on $(0,T]$.
\end{theorem}

\section{Proof of Theorem \ref{th1}}

Now we are in a position to prove our main result.

\begin{proof}
Taking the curl on \eqref{eqNS}, we obtain
\begin{equation}
\partial _{t}\omega -\Delta \omega +(u\cdot \nabla )\omega
-(\omega \cdot \nabla )u=0.  \label{eq1}
\end{equation}
Multiplying \eqref{eq1} by $\omega $ in $L^2(\mathbb{R}^3)$ and
integrating by parts, we obtain
\begin{equation}
\frac{1}{2}\frac{d}{dt}\| \omega (t,\cdot )\|
_{L^2}^2+\| \nabla \omega (t,\cdot )\|
_{L^2}^2=\langle \omega \cdot \nabla u,\omega \rangle .
\label{eq9}
\end{equation}
Here we have used the identity
\[
\langle u\cdot \nabla \omega ,\omega \rangle =0.
\]
Using the Biot-Savart law, $u$ is written in terms of $\omega $:
\[
u(x,t)=-\frac{1}{4\pi }\int_{\mathbb{R}^3}\frac{(x-y)
\times \omega (x,t)}{| x-y| ^3}dy.
\]
Substituting this into the right hand side of \eqref{eq9}, we obtain
\[
\langle \omega \cdot \nabla u,\omega \rangle =\frac{3}{4\pi }
\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{y}{|
y| }\cdot \omega (x,t)\big\{ \frac{y}{|
y| ^{4}}\times \omega (x+y,t)\cdot \omega (
x,t)\big\} \,dy\,dx.
\]
We decompose $\omega $ for the vorticities in $\{ \cdot\} $ as
follows
\[
\omega =\tilde{\omega}+\omega ',\quad
\tilde{\omega}=\omega _1e_1+\omega _2e_2,\quad
\omega '=\omega _{3}e_{3}.
\]
Since $\omega '=(0,0,\omega _{3})$, there holds
\[
\frac{3}{4\pi }\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{
y}{| y| }\cdot \omega (x,t)\big\{ \frac{y}{
| y| ^{4}}\times \omega '(x+y,t)
\cdot \omega '(x,t)\big\} \,dy\,dx=0
\]
for all $0<t<T$. Then, it follows
\begin{equation}
\begin{aligned}
\langle \omega \cdot \nabla u,\omega \rangle
&= \frac{3}{4\pi }
\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{y}{|
y| }\cdot \omega (x,t)\{ \frac{y}{|
y| ^{4}}\times \tilde{\omega}(x+y,t)\cdot \omega
'(x,t)\} \,dy\,dx   \\
&\quad +\frac{3}{4\pi }\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\frac{y}{| y| }\cdot \omega (x,t)\{
\frac{y}{| y| ^{4}}\times \tilde{\omega}(
x+y,t)\cdot \tilde{\omega}(x,t)\} \,dy\,dx   \\
&\quad +\frac{3}{4\pi }\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\frac{y}{| y| }\cdot \omega (x,t)\{
\frac{y}{| y| ^{4}}\times \omega '(x+y,t)\cdot \tilde{\omega}(x,t)\}
\,dy\,dx,
\end{aligned} \label{eq7}
\end{equation}
where all the integrations with respect to $y$ are in the sense
of principal value. Using the following interpolation inequality
\cite{MO}:
\[
\| w\| _{\dot {B}_{2,1}^{r}}\leq
C\| w\| _{L^2}^{1-r}\| \nabla w\| _{L^2}^{r},\quad r\in (0,1)
\]
it is easy to see that by Lemma \ref{lem 2}
\begin{align*}
&| \langle \omega \cdot \nabla u,\omega \rangle |\\
&\leq C\int_{\mathbb{R}^3}| \omega (x,t)| | P(\tilde{\omega})|
| \omega '(x,t)| dx \\
&\quad +C\int_{\mathbb{R}^3}| \omega (x,t)
| | P(\tilde{\omega})| |
\tilde{\omega}(x,t)| dx \\
&\quad +C\int_{\mathbb{R}^3}| \omega (x,t)
| | P(\omega ')|
| \tilde{\omega}(x,t)| dx \\
&\leq C\int_{\mathbb{R}^3}| \omega |
^2| P(\tilde{\omega})| dx+C\int_{
\mathbb{R}^3}| \omega | | P(\omega
')| | \tilde{\omega}| dx \\
&\leq C\| \omega \| _{L^2}\| \omega
\cdot P(\tilde{\omega})\| _{L^2}+C\|
\tilde{\omega} \cdot P(\omega ')\|
_{L^2}\| \omega
\| _{L^2} \\
&\leq C\| \omega \| _{L^2}\| \omega
\| _{\dot {B}_{2,1}^{r}}\|
P(\tilde{\omega}) \| _{\dot
{\mathcal{M}}_{2,3/r}}+C\| \omega \|
_{L^2}\| P(\omega ') \|
_{\dot {B}_{2,1}^{r}}\| \tilde{\omega}\|
_{\dot {\mathcal{M}}_{2,3/r}} \\
&\leq C\| \tilde{\omega}\| _{\dot {\mathcal{M}}
_{2,3/r}}\| \omega \| _{L^2}\| \omega
\| _{\dot {B}_{2,1}^{r}}+C\| \omega
\| _{L^2}\| \tilde{\omega}\| _{\dot
{\mathcal{M}} _{2,3/r}}\| \omega '\|
_{\dot {B}_{2,1}^{r}} \\
&\leq C\| \tilde{\omega}\| _{\dot {\mathcal{M}}
_{2,3/r}}\| \omega \| _{L^2}^{2-r}\|
\nabla \omega \| _{L^2}^{r}
\end{align*}
where $P(.)$ denotes the singular integral operator defined by the
integrals with respect to $y$ in \eqref{eq7}.

By Young' s inequality, we find
\begin{equation}
| \langle \omega \cdot \nabla u,\omega \rangle |
\leq C\| \tilde{\omega}\| _{\dot {\mathcal{M}}_{2,3/r}}^{2/(2-r)}
\| \omega \|_{L^2}^2+\frac{1}{2}\| \nabla \omega \| _{L^2}^2.
\label{eq 3.2}
\end{equation}
Substituting \eqref{eq 3.2} in \eqref{eq9}, we have
\begin{equation}
\frac{d}{dt}\| \omega (\cdot ,t)\|
_{L^2}^2+\| \nabla \omega (\cdot ,t)\|
_{L^2}^2\leq C\| \tilde{ \omega}\| _{\dot
{\mathcal{M}}_{2,3/r}}^{\frac{2 }{2-r}}\| \omega
\| _{L^2}^2.  \label{eq10}
\end{equation}
By Gronwall' s inequality we have that
\begin{equation}
\| \omega (\cdot ,t)\| _{L^2}\leq \|
\omega (0,\cdot )\| _{L^2}\exp \Big(C\int_{0}^{T}\|
\tilde{ \omega}(\cdot ,\tau )\| _{\dot
{\mathcal{M}}_{2,\frac{3}{ r}}}^{2/(2-r)}d\tau \Big).  \label{eq3.99}
\end{equation}
This implies
\[
\omega \in L^{\infty }([ 0,T);L^2(\mathbb{R}^3)) \cap
L^2([ 0,T);H^1(\mathbb{R}^3))
\]
provided that $\tilde{\omega}$ satisfies the condition
$\tilde{\omega}\in L^{\frac{2}{2-r}}(0,T;\dot {\mathcal{M}}_{2,3/r}
(\mathbb{R}^3))$. This proves Theorem \ref{th1}.
\end{proof}

\section{Proof of Theorem \ref{th2}}

Now we are in a position to prove our second result.

\begin{proof}
We set $\tilde{u}=(u_1,u_2,0)$.\ Then, taking the first two
components of the vorticity equation \eqref{eq1}, we obtain
\[
\partial _{t}\tilde{\omega}-\Delta \tilde{\omega}+(u\cdot \nabla
)\tilde{\omega}-(\omega \cdot \nabla )\tilde{u}=0.
\]
Multiplying \eqref{eq1} by $\tilde{\omega}$ in $L^2(\mathbb{R}
^3)$ and integrating by parts, we obtain
\begin{equation}
\frac{1}{2}\frac{d}{dt}\| \tilde{\omega}(t,\cdot
)\| _{L^2}^2+\| \nabla \tilde{\omega}(t,\cdot
)\| _{L^2}^2=\langle \omega \cdot \nabla
\tilde{u},\tilde{\omega} \rangle .  \label{eq 12}
\end{equation}
We first consider the case $0<r<1$. Using the H\"{o}lder inequality
and Lemma \ref{lem 2}, we estimate
\begin{equation}
\begin{aligned}
| \langle \omega \cdot \nabla
\tilde{u},\tilde{\omega} \rangle |
&= |\langle (\tilde{\omega}+\omega ')\cdot \nabla
 \tilde{u},\tilde{\omega}\rangle | \leq \|
 \tilde{\omega}\| _{L^2}\| \tilde{\omega}
 \cdot \nabla \tilde{u}\| _{L^2}   \\
&\leq C\| \nabla \tilde{u}\| _{\dot {\mathcal{M}
 }_{2,3/r}}\| \tilde{\omega}\| _{L^2}\|
 \tilde{\omega}\| _{\dot {B}_{2,1}^{r}}   \\
&\leq C\| \nabla \tilde{u}\| _{\dot {\mathcal{M}
 }_{2,3/r}}\| \tilde{\omega}\| _{L^2}\|
 \tilde{\omega}\| _{L^2}^{1-r}\| \nabla
 \tilde{\omega}\| _{L^2}^{r}   \\
&= C(\| \nabla \tilde{u}\| _{\dot {
 \mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \tilde{\omega}
 \| _{L^2}^2)^{\frac{2-r}{2}}\| \nabla \tilde{
 \omega}\| _{L^2}^{r}   \\
&\leq C\| \nabla \tilde{u}\| _{\dot {\mathcal{M}
 }_{2,3/r}}^{2/(2-r)}\| \tilde{\omega}\|_{L^2}^2
 +\frac{C}{2}\| \nabla \tilde{\omega}\|_{L^2}^2,
\end{aligned} \label{eq 13}
\end{equation}
where we used
\[
\langle \omega '\cdot \nabla \tilde{u},\tilde{\omega}
\rangle =0.
\]
Estimates \eqref{eq 13} combined with \eqref{eq 12}, yield
\[
\frac{1}{2}\frac{d}{dt}\| \tilde{\omega}(t,\cdot
)\| _{L^2}^2+\| \nabla \tilde{\omega}(t,\cdot
)\| _{L^2}^2\leq C\| \nabla \tilde{u}\|
_{\dot { \mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \tilde{\omega}
\| _{L^2}^2.
\]
By Gronwall' s inequality we have
\[
\| \tilde{\omega}(t,\cdot )\| _{L^2}\leq
\| \tilde{\omega}(0,\cdot )\| _{L^2}\exp \Big(
C\int_{0}^{T}\| \nabla \tilde{u}(\cdot ,\tau )\|
_{\overset{ \cdot }{\mathcal{M}}_{2,3/r}}^{2/(2-r)}d\tau \Big).
\]

Next we consider the case $r=0$. In this case we estimate
\begin{equation}
| \langle \omega \cdot \nabla
\tilde{u},\tilde{\omega} \rangle | \leq \|
\tilde{\omega}\| _{L^2}\| \tilde{\omega}\cdot
\nabla \tilde{u}\| _{L^2}\leq \|
\tilde{\omega}\| _{L^2}^2\| \nabla
\tilde{u}\| _{L^{\infty }}.  \label{eq14}
\end{equation}
This  estimate  combined with \eqref{eq 12}, yield
\[
\frac{1}{2}\frac{d}{dt}\| \tilde{\omega}(t,\cdot
)\| _{L^2}^2+\| \nabla \tilde{\omega}(t,\cdot
)\| _{L^2}^2\leq C\| \nabla \tilde{u}\|
_{L^{\infty }}\| \tilde{\omega}\| _{L^2}^2.
\]
By Gronwall' s inequality we have
\[
\| \tilde{\omega}(t,\cdot )\| _{L^2}\leq
\| \tilde{\omega}(0,\cdot )\| _{L^2}\exp \Big(
C\int_{0}^{T}\| \nabla \tilde{u}\| _{L^{\infty
}}d\tau \Big).
\]
This completes the proof of Theorem \ref{th2}.
\end{proof}

\subsection*{Acknowledgements}
The author would like to express his gratitude to Professor
Dongho Chae for many helpful discussions, constant encouragement
and valuable suggestions;
also to the anonymous referee for his or her useful comments
 and suggestions.

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\end{thebibliography}

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