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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 06, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/06\hfil A regularity criterio]
{A regularity criterion for the Navier-Stokes equations in terms
of the horizontal derivatives of the two velocity components}

\author[W. Chen, S. Gala\hfil EJDE-2011/06\hfilneg]
{Wenying Chen, Sadek Gala}  % in alphabetical order

\address{Wenying Chen \newline
 College of Mathematics and Computer Science, 
 Chongqing Three Gorges University,
 Wanzhou 404000, Chongqing, China}
\email{wenyingchenmath@gmail.com}

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

\thanks{Submitted October 18, 2010. Published January 12, 2011.}
\subjclass[2000]{35Q30, 76F65}
\keywords{Navier-Stokes equations; Leray-Hopf weak solutions;
\hfill\break\indent regularity criterion}

\begin{abstract}
 In this article, we consider the regularity for weak solutions
 to the Navier-Stokes equations in $\mathbb{R}^3$.
 It is proved that if the horizontal derivatives of the two
 velocity components
 \[
 \nabla _h\widetilde{u}\in  L^{2/(2-r)}(0,T;\dot{\mathcal{M}}_{2,3/r}
 (\mathbb{R}^3)),\quad \text{for }0<r<1,
 \]
 then the weak solution is actually strong, where
 $\dot{\mathcal{M}} _{2,3/r}$ is the critical Morrey-Campanato
 space and $\widetilde{u} =(u_1,u_2,0)$,
 $\nabla_h\widetilde{u}=(\partial _1u_1,\partial _2u_2,0)$.
\end{abstract}

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


\section{Introduction}

We consider the following Cauchy problem for the incompressible
Navier-Stokes equations in $\mathbb{R}^3\times (0,T)$,
\begin{equation}
\begin{gathered}
\partial _{t}u+(u \cdot \nabla )u+\nabla p=\Delta u, \\
\nabla .u=0, \\
u(x,0)=u_{0}(x),
\end{gathered}  \label{eq1.1}
\end{equation}
where $u=(u_1,u_2,u_3)$ is the velocity field, $p(x,t)$ is a
scalar pressure, and $u_{0}(x)$ with $\nabla .u_{0}=0$ in the
sense of distribution is the initial velocity field.

Although a global weak solution of \eqref{eq1.1} was first
constructed by Leray~\cite{Le} in 1934, the fundamental problem on
uniqueness and regularity of weak solutions still remains open,
although huge contributions have been made in an effort to
understand regularities of the weak solution. It is well-known
that regularity can be persistent under certain condition, which
was introduced in the celebrated work of Serrin \cite{Se}, and can
be described as follows (see also Struwe \cite{Str}).

A \emph{weak solution} $u$ is regular if it satisfies the growth
condition
\begin{equation}
u\in L^p(0,T;L^q(\mathbb{R}^3))\equiv L_{t}^pL_{x}^q,\quad
\text{for } \frac{2}{p}+\frac{3}{q}=1,\; 3<q\leq \infty \,.
\label{eq1.2}
\end{equation}

Regularity was also extended by Beir\~{a}o da Veiga \cite{Bdv}
with \eqref{eq1.2} replaced by the velocity gradient growth
condition:
\begin{equation}
\nabla u\in L_{t}^pL_{x}^q,\quad \text{for }
\frac{2}{p}+\frac{3}{q} =2,\; \frac{3}{2}<q\leq \infty .
\label{eq1.3}
\end{equation}

We recall that the condition \eqref{eq1.2} is important from the
point of view of the relation between scaling invariance and
partial regularity of weak solutions. In fact, the conditions
\eqref{eq1.2} and \eqref{eq1.3} involve all components of the
velocity vector field $u=(u_1,u_2,u_3)$ and are known as degree
$-1$ growth condition, since
\begin{align*}
\| u(\lambda x,\lambda ^2t)\|
_{L^p(0,T;L^q(\mathbb{R}^3))}
&= \lambda ^{-(\frac{2}{p}+\frac{3 }{q})}\| u(x,t) \|
_{L^p(0,\lambda^2 T;L^q(\mathbb{R}^3))} \\
&= \lambda ^{-1}\| u\| _{L^p(0,\lambda
^2T;L^q( \mathbb{R}^3))},
\end{align*}
\begin{align*}
\| \nabla u(\lambda x,\lambda ^2t)\|
_{L^p(0,T;L^q(\mathbb{R}^3))}
&= \lambda ^{1-(
\frac{2}{p}+\frac{3 }{q})}\| \nabla u(x,t)\|
_{L^p(0,\lambda ^2T;L^q(\mathbb{R}^3))} \\
&= \lambda ^{-1}\| \nabla u\| _{L^p(0,\lambda
^2T;L^q( \mathbb{R}^3))}.
\end{align*}
The degree $-1$ growth condition is critical due to the scaling
invariance property. That is, $u$ solves \eqref{eq1.1} if and only
if $u_{\lambda }(x,t)=\lambda u(\lambda x,\lambda ^2t)$ is also
a solution of \eqref{eq1.1}.

Regularity criteria in terms of only one component of the velocity were
given in celebrated works by Zhou.
It was proved in \cite{Zhou1} (see also
\cite{M} and \cite{Zhou2})
that regularity keeps under one of the following two
conditions:
\begin{gather*}
\nabla u_3\in L_{t}^pL_{x}^q,\quad \text{for }
\frac{2}{p}+\frac{3 }{q}=\frac{3}{2},\; 2<q\leq \infty ,\\
u_3\in L_{t}^pL_{x}^q,\quad \text{for }
\frac{2}{p}+\frac{3}{q}= \frac{1}{2},\; 6<q\leq \infty .
\end{gather*}
Later on, some improvements and extensions were given by many
authors, say \cite{CaoT,DSC,Ku,ZP1,ZP2}.
Recently, Dong and Zhang \cite{BZ} proved that if the horizontal
derivatives of the two velocity components
\[
\int_{0}^{T}\| \nabla _h\widetilde{u}(.,s)\|
_{\dot {B}_{\infty ,\infty }^{0}}ds<\infty ,
\]
where $\widetilde{u}=(u_1,u_2,0)$ and
$\nabla _h \widetilde{u}=(\partial _1\widetilde{u},
\partial _2\widetilde{u},0)$, then the solution keeps
smoothness up to time $T$.

In this paper we want to prove the analogous result in the critical
Morrey-Campanato space. More precisely, we show that
the Leray-Hopf weak solution is regular on $(0,T]$ if the following
growth condition with degree $-1$ is satisfied.,
\[
\int_{0}^{T}\| \nabla _h\widetilde{u}(.,s)\|
_{\dot {\mathcal{M}}_{2,3/r}}^{2/(2-r)}ds<\infty .
\]

\section{Preliminaries and main result}

Now, we recall the definition and some properties of the space that
will be useful in the sequel. These spaces play an important role
in studying the regularity of solutions to partial differential
equations; see e.g. \cite{GL} and references therein.

\begin{definition} \label{def2.1} \rm
For $0\leq r<3/2$, the space $\dot{X}_{r}$ is defined as
the space of $f(x)\in L_{\rm loc}^2(\mathbb{R}^3)$ such that
\[
\| f\| _{\dot{X}_{r}}=\sup_{\|g\| _{\dot{H}^r}\leq 1}
\| fg\|_{L^2}<\infty .
\]
where we denote by $\dot{H}^r(\mathbb{R}^3)$ the completion of
the space $C_{0}^{\infty }(\mathbb{R}^3)$ with respect to the norm
$\| u\| _{\dot{H} ^r}=\| (-\Delta)^{r/2}u\| _{L^2}$.
\end{definition}

We have the homogeneity properties:
 for all $x_{0}\in \mathbb{R}^3$,
\begin{gather*}
\| f(.+x_{0})\| _{\dot{X}_{r}} =\|f\| _{\dot{X}_{r}}\,, \\
\| f(\lambda .)\| _{\dot{X}_{r}}=\frac{1}{\lambda ^r}\| f\|
_{\dot{X}_{r}}, \quad \lambda >0.
\end{gather*}
The following imbedding holds
\[
L^{3/r}\subset \dot{X}_{r},\quad 0\leq r<\frac{3}{2}\,.
\]
Now we recall the definition of Morrey-Campanato spaces (see e.g.
\cite{Kat}).

\begin{definition} \label{def2.2} \rm
For $1<p\leq q\leq +\infty $, the Morrey-Campanato space
is
\begin{equation}
\dot{\mathcal{M}}_{p,q}=\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\|_{L^p(B(x,R))}<\infty \big\}.
\label{eq1.13}
\end{equation}
\end{definition}

It is easy to check that
\[
\| f(\lambda .)\| _{\dot{\mathcal{M}}_{p,q}}
= \frac{1}{\lambda ^{3/q}}\|f\| _{\dot{ \mathcal{M}}_{p,q}},
\quad \lambda >0.
\]

We have the following comparison between Lorentz
and Morrey-Campanato spaces: For $p\geq 2$,
\[
L^{\frac{3}{r}}(\mathbb{R}^3)\subset \text{ }L^{3/r,\infty }
(\mathbb{R}^3)\subset \dot{\mathcal{M}}_{p,3/r}(\mathbb{R}^3).
\]
The relation
\[
L^{\frac{3}{r},\infty }(\mathbb{R}^3)\subset \dot{
\mathcal{M}}_{p,\frac{3}{r}}(\mathbb{R}^3)
\]
is shown as follows. Let $f\in L^{3/r,\infty }(\mathbb{R} ^3)$.
 Then
\begin{align*}
\| f\| _{\dot{\mathcal{M}}_{p,\frac{3}{r}}}
&\leq  \sup_{E}| E|^{\frac{r}{3}-\frac{1}{2}}
\Big(\int_{E}| f(y)| ^pdy\Big)^{1/p} \\
&= \Big(\sup_E | E| ^{\frac{pr}{3}-1}\int_{E}| f(y)| ^pdy\Big)^{1/p} \\
&\cong \Big(\sup_{R>0} R| \{ x\in \mathbb{R}
^3:| f(y)| ^p>R\} |^{\frac{pr}{3}}Big)^{1/p} \\
&= \underset{R>0}{\sup }R\big| \big\{ x\in
\mathbb{R}^p:| f(y)| >R\big\} \big| ^{r/3} \\
&\cong \| f\| _{L^{3/r,\infty }}.
\end{align*}

For $0<r<1$, we use the fact that
\[
L^2\cap \dot{H}^{1}\subset \dot{B}_{2,1}^r\subset
\dot {H}^r.
\]
Thus we can replace the space $\dot{X}_{r}$ by the pointwise
multipliers from Besov space $\dot{B}_{2,1}^r$ to $L^2$. Then
we have the following lemma given in \cite{Lem1}.

\begin{lemma}\label{lem 2}
For $0\leq r<3/2$, the space $\dot{Z}_{r}$ 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 $f\in \dot{\mathcal{M}}_{2,3/r}$ if and only if
$f\in \dot{Z}_{r}$ with equivalence of norms.
\end{lemma}

To prove our main result, we need the following lemma.

\begin{lemma} \label{lem3}
For $0<r<1$, we have
\[
\| f\| _{\dot{B}_{2,1}^r}\leq C\| f\| _{L^2}^{1-r}\| \nabla f\|
_{L^2}^r.
\]
\end{lemma}

\begin{proof}
The idea comes from \cite{MO}.
According to the definition of Besov spaces,
\begin{align*}
&\| f\| _{\dot{B}_{2,1}^r}\\
&= \sum_{j\in \mathbb{Z} }2^{jr}\| \Delta _{j}f\| _{L^2} \\
&\leq \sum_{j\leq k}2^{jr}\| \Delta _{j}f\|
_{L^2}+\sum_{j>k}2^{j(r-1)}2^{j}\| \Delta _{j}f\| _{L^2}
\\
&\leq \Big(\sum_{j\leq k}2^{2jr}\Big)^{1/2}
\Big( \sum_{j\leq k}\| \Delta _{j}f\| _{L^2}^2\Big)^{1/2}
+\Big(\sum_{j>k}2^{2j(r-1)}\Big)^{1/2}
\Big(\sum_{j>k}2^{2j}\| \Delta _{j}f\| _{L^2}^2\Big)^{1/2} \\
&\leq C\big(2^{rk}\| f\|_{L^2}+2^{k(r-1)}\|f\| _{\dot{H}^{1}}\big)\\
&= C\big(2^{rk}A^{-r}+2^{k(r-1)}A^{1-r}\big)\| f\|
_{L^2}^{1-r}\| f\| _{\dot{H}^{1}}^r,
\end{align*}
where $A=\| f\| _{\dot{H}^{1}}/ \|f\| _{L^2}$.
Choose $k$ such that $2^{rk}A^{-r}\leq 1$; that is,
$k\leq [ \log A^r] $, we thus obtain
\[
\| f\| _{\dot{B}_{2,1}^r}
\leq C\big(1+2^{k(r-1)}A^{1-r}\big)\| f\|
_{L^2}^{1-r}\|f\| _{\dot{H}^{1}}^r \\
\leq C\| f\| _{L^2}^{1-r}\| \nabla f\|_{L^2}^r.
\]
\end{proof}

Additionally, for $2<p\leq \frac{3}{r}$ and
$0\leq r<\frac{3}{2}$, we have
the following inclusion relations \cite{Lem,Lem1},
\[
\dot{\mathcal{M}}_{p,3/r}(\mathbb{R}^3)\subset
\dot{X}_{r}(\mathbb{R}^3)\subset \dot{\mathcal{M
}}_{2,3/r}(\mathbb{R}^3)=\dot{Z}_{r}( \mathbb{R}^3).
\]
The relation
\[
\dot{X}_{r}(\mathbb{R}^3)\subset \dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)
\]
is shown as follows. Let $f\in \dot{X}_{r}(\mathbb{R} ^3)$,
$0<R\leq 1$, $x_{0}\in \mathbb{R}^3$ and $\phi \in C_{0}^{\infty
}(\mathbb{R}^3)$, $\phi \equiv 1$ on $B(\frac{ x_{0}}{R},1)$. We
have
\begin{align*}
R^{r-\frac{3}{2}}\Big(\int_{| x-x_{0}| \leq
R}| f(x)| ^2dx\Big)^{1/2}
& =R^r\Big(\int_{| y-\frac{x_{0}}{R}| \leq 1}|f(Ry)| ^2dy\Big)^{1/2} \\
& \leq R^r(\int_{y\in \mathbb{R}^3}| f(Ry)\phi
(y)| ^2dy)^{1/2} \\
& \leq R^r\| f(R.)\| _{\dot{X}_{r}}\| \phi\| _{H^r} \\
& \leq \| f\| _{\dot{X}_{r}}\| \phi
\| _{H^r} \\
& \leq C\| f\| _{\dot{X}_{r}}.
\end{align*}

We recall the following definition of Leray-Hopf weak solution.

\begin{definition} \label{def2.5} \rm
Let $u_{0}\in L^2(\mathbb{R}^3)$ and $\nabla \cdot u_{0}=0$. A
measurable vector field $u(x,t)$ is called a Leary-Hopf weak
solution to the Navier-Stokes equations \eqref{eq1.1} on $(0,T)$,
if $u$ has the following properties:
\begin{itemize}
\item[(i)] $u\in L^{\infty }(0,T;L^2(\mathbb{R}^3))\cap L^2(0,T;H^{1}(
\mathbb{R}^3));$

\item[(ii)] $\partial _{t}u+(u\cdot \nabla )u+\nabla \pi =\Delta u$ in
$\mathcal{D}'((0,T)\times \mathbb{R}^3)$;

\item[(iii)] $\nabla \cdot u=0$ in $\mathcal{D}'((0,T)\times
\mathbb{R}^3)$;

\item[(iv)] $u$ satisfy the energy inequality
\begin{equation}
\Vert u(t)\Vert
_{L_{x}^2}^2+2\int_{0}^{t}\int_{\mathbb{R}^3}|\nabla
u(x,s)|^2\,dxds\leq \Vert u_{0}\Vert _{L_{x}^2}^2,\quad
\text{for} \quad 0\leq t\leq T.  \label{eq1.9}
\end{equation}
\end{itemize}
\end{definition}

By a strong solution we mean a weak solution $u$ of the
Navier-Stokes equations \eqref{eq1.1} that satisfies
\begin{equation}
u\in L^{\infty }(0,T;H^{1}(\mathbb{R}^3))\cap
L^2(0,T;H^2(\mathbb{R} ^3)).  \label{eq1.10}
\end{equation}
It is well known that strong solutions are regular and unique in
the class of weak solutions.

The following theorem is the main result of this article.

\begin{theorem} \label{th1}
Suppose $u_{0}\in H^{1}(\mathbb{R}^3)$ and $\nabla
\cdot u_{0}=0 $ in the sense of distributions. Assume that
$u(x,t)$ is a Leray-Hopf weak solution of \eqref{eq1.1} on
$(0,T)$. If
\begin{equation}
\nabla _h\widetilde{u}\in L^{2/(2-r)}(0,T;\dot{\mathcal{M}}
_{2,3/r}(\mathbb{R}^3)),\,\quad \text{for}\,\,0<r<1,
\label{eq1.11}
\end{equation}
then $u$ is a regular solution in $(0,T]$ in the sense that
$u\in C^{\infty }((0,T)\times \mathbb{R}^3).$
\end{theorem}

\section{A priori estimates}

Now we want to establish an a priori estimate for the smooth solution.

\begin{theorem}\label{th2}
Suppose $T>0$, $u_{0}\in H^{1}(\mathbb{R}^3)$ and
$\nabla \cdot u_{0}=0$ in the sense of distributions. Assume that
$u$ is a smooth solution of \eqref{eq1.1} on
$\mathbb{R}^3\times (0,T)$ and satisfies any one of of the
three degree $-1$ growth
conditions \eqref{eq1.11}. Then
\begin{equation}
\begin{aligned}
&\sup_{0\leq t<T} \Vert \nabla u(.,t)\Vert
_{L^2}^2+\int_{0}^{T}\| \Delta u(.,t)\|_{L^2}^2\,ds\\
&\leq C\| \nabla u_{0}\|
_{L^2}^2\exp\Big(\int_{0}^{t}\| \nabla
_h\widetilde{u}(.,s)\| _{
\dot{\mathcal{M}}_{2,3/r}}^{2/(2-r)}ds\Big),
\end{aligned}\label{eq2.1}
\end{equation}
for $0<t<T$, holds  for some constant $C>0$.
\end{theorem}

To prove this theorem, we need the following lemma.

\begin{lemma}[\cite{BZ}] \label{lem1}
Let $u$ be a smooth solution to the Navier-Stokes
system \eqref{eq1.1} in $\mathbb{R}^3$. Furthermore, let
$\widetilde{u}=(u_1,u_2,0)$ and $\nabla _h\widetilde{u}=(\partial
_1 \widetilde{u},\partial _2\widetilde{u},0)$. Then
\[
\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial
_{i}u_{j}\partial _{kk}u_{j}dx\big|
\leq C\int_{\mathbb{R} ^3}| \nabla _h\widetilde{u}| | \nabla u| ^2
\]
for some constant $C>0$.
\end{lemma}

The proof of this lemma is simple; see \cite[Lemma 2.2]{BZ}).

\begin{proof}[Proof of Theorem \ref{th2}]
Multiply the first equation of \eqref{eq1.1} by $\Delta u$, and
integrating on $\mathbb{R}^3$, after suitable integration by
parts, we obtain for $t\in (0,T)$,
\begin{equation}
\frac{1}{2}\frac{d}{dt}\| \nabla u\|
_{L^2}^2+\| \Delta u(t)\| _{L^2}^2\leq
2\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial
_{i}u_{j}\partial _{kk}u_{j}dx\big| .  \label{eq2.11}
\end{equation}
Due to H\"{o}lder' s inequality and Lemma \ref{lem3}, the
right-hand side \eqref{eq2.11} can be estimated as
\begin{align*}
\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial
_{i}u_{j}\partial _{kk}u_{j}dx\big|
&\leq \| \nabla_h \widetilde{u} \cdot \nabla u\| _{L^2}\|
\nabla u\| _{L^2} \\
&\leq C\| \nabla _h\widetilde{u}\| _{\dot{
\mathcal{M}}_{2,3/r}}\| \nabla u\| _{_{\dot{B
}_{2,1}^r}}\| \nabla u\| _{L^2} \\
&\leq C\| \nabla _h\widetilde{u}\| _{\dot{
\mathcal{M}}_{2,3/r}}\| \nabla u\|
_{L^2}^{2-r}\| \Delta u\| _{L^2}^r \\
&\leq C\Big(\| \nabla _h\widetilde{u}\| _{\dot{
\mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \nabla
u\| _{L^2}^2\Big)^{(2-r)/2}(\| \Delta
u\| _{L^2}^2)^{r/2}.
\end{align*}
By Young' s inequality, we obtain
\begin{equation}
\big| \sum_{i,j,k=1}^3\int_{\mathbb{R}^3}u_{i}\partial
_{i}u_{j}\partial _{kk}u_{j}dx\big|
\leq \frac{1}{2}\| \Delta u\| _{L^2}^2+C\|
\nabla _h\widetilde{u}\| _{
\dot{\mathcal{M}}_{2,3/r}}^{2/(2-r)}\| \nabla
u\| _{L^2}^2  \label{eq6}
\end{equation}
Substituting \eqref{eq6} into \eqref{eq2.11}, it follows that
\[
\frac{d}{dt}\| \nabla u(.,t)\|
_{L^2}^2+\| \Delta u(.,t)\| _{L^2}^2\leq
C\| \nabla _h\widetilde{u }\|
_{\dot{\mathcal{M}}_{2,3/r}}^{2/(2-r) }\| \nabla
u\| _{L^2}^2.
\]
Then Gronwall' s inequality yields
\begin{align*}
&\| \nabla u(t)\|
_{L^2}^2+\int_{0}^{T}\int_{\mathbb{R} ^3}\| \Delta
u(x,s)\| _{L^2}^2\,dx\,ds\\
&\leq C\| \nabla
u_{0}\| _{L^2}^2\exp \Big(\int_{0}^{t}\| \nabla
_h\widetilde{u}(.,s)\| _{\dot{\mathcal{M}}_{2,3/r}}
^{2/(2-r)}ds\Big).
\end{align*}
This completes the proof .
\end{proof}

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

After we established the key estimate in section 2, the proof of
Theorem \ref{th1} is straightforward.

It is well known \cite{FK} that there is a  unique strong
solution $\overline{u}\in C([0,T^{\ast }),H^{1}(\mathbb{R}^3))\cap
C^{1}([0,T^{\ast }),H^{1}(\mathbb{R}^3))\cap C([0,T^{\ast
}),H^3(\mathbb{R}^3))$ to \eqref{eq1.1} for some $T^{\ast }>0$,
for any given $u_{0}\in H^{1}(\mathbb{R}^3)$ with
$\nabla.u_{0}=0$. Since $u$ is a Leray-Hopf weak solution which
satisfies the energy inequality \eqref{eq1.9}, we have according
to the Serrin's uniqueness criterion \cite{Se},
\[
\bar{u}\equiv u\quad \text{on } [0,T^{\ast }).
\]
By the assumption \eqref{eq1.11} and standard continuation
argument, the local strong solution can be extended to time $T$.
So we have proved $u$ actually is a strong solution on $[0,T)$.
This completes the proof of Theorem \ref{th1}.

\begin{thebibliography}{00}

\bibitem{Bdv} H. Beir\~{a}o da Veiga;
\emph{A new regularity class for the
Navier-Stokes equations in $R^{n}$}.
Chin. Ann. Math. 16 (1995) 407-412.

\bibitem{BZ} B. Dong and Z. Zhang;
\emph{The BKM criterion for the 3-D Navier
Stokes equations via two velocity components}.
To appear in Nonlinear Anal.:
Real World Applications (2010), doi:10.1016/j.nonrwa.2009.07.013.

\bibitem{CaoT} C. Cao and E. S. Titi;
\emph{Regularity criteria for the
three-dimensional Navier-Stokes equations}.
Indiana University Mathematics Journal 57 (2008), 2643-2662.

\bibitem{DSC} B. Dong, S. Gala and Z.-M. Chen;
\emph{On the regularity criterion of the 3-D Navier-Stokes
equations in weak spaces}. To appear in Acta
Mathematica Scientia.

\bibitem{FK} H. Fujita, T. Kato;
\emph{On the nonstationary Navier Stokes initial
value problem}. Arch. Rational. Mech. Anal. 16 (1964) 269 315.

\bibitem{GL} S. Gala and P. G. Lemarie-Rieusset;
\emph{Multipliers between Sobolev spaces and fractional
differentiation}. J. Math. Anal. Appl. 322 (2006),
1030-1054.

\bibitem{Kat} T. Kato;
\emph{Strong $L^p$ solutions of the Navier-Stokes
equations in Morrey spaces}. Bol. Soc. Bras. Mat. 22 (1992), 127-155.

\bibitem{Ku} I. Kukavica and M. Ziane;
\emph{One component regularity for the
Navier-Stokes equations}. Nonlinearity 19 (2006) 453-469.

\bibitem{Lem} P. G. Lemari\'e-Rieusset;
\emph{Recent developments in the Navier-Stokes problem.}
 Research Notes in Mathematics, Chapman \& Hall, CRC,
2002.

\bibitem{Lem1} P. G. Lemari\'e-Rieusset;
\emph{The Navier-Stokes equations in the
critical Morrey-Campanato space}. Rev. Mat. Iberoam. 23 (2007), no. 3,
897--930.

\bibitem{Le} J. Leray;
\emph{Sur le mouvement d'un liquide visqueux emplissant
l'espace}. Acta. Math. 63 (1934), 183-248.

\bibitem{MO} S. Machihara and T. Ozawa;
\emph{Interpolation inequalities in Besov spaces}.
Proc. Amer. Math. Soc. 131 (2003), 1553-1556.

\bibitem{M} M. Pokorn\'y;
\emph{On the result of He concerning the smoothness of
solutions to the Navier-Stokes equations}. Electron. J. Differential
Equations ,Vol. 2003 (2003), no. 11, 1--8.

\bibitem{Se} J. Serrin;
\emph{On the interior regularity of weak solutions of the
Navier Stokes equations}. Arch. Rational Mech. Anal. 9 (1962) 187-195.

\bibitem{Str} M. Struwe;
\emph{On partial regularity results for the Navier-Stokes
equations}. Comm. Pure Appl. Math. 41 (1988) 437-458.

\bibitem{Zhou1} Y. Zhou;
\emph{A new regularity criterion for the Navier-Stokes
equations in terms of the gradient of one velocity component},
Methods Appl. Anal. 9 (2002) 563-578.

\bibitem{Zhou2} Y. Zhou;
\emph{A new regularity criterion for weak solutions to
the Navier-Stokes equations}. J. Math. Pures Appl. 84 (2005) 1496-1514.

\bibitem{ZP1} Y. Zhou and M. Pokorn\'y;
\emph{On a regularity criterion for the
Navier--Stokes equations involving gradient of one velocity component}.
J. Math. Phys. 50 (2009), 123514.

\bibitem{ZP2} Y. Zhou and M. Pokorn\'{y};
\emph{On the regularity to the solutions of the Navier-Stokes
equations via one velocity component}, Nonlinearity 23
(2010), no. 5, 1097.

\end{thebibliography}


\end{document}