\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 121, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/121\hfil
 Regularity for the Navier-Stokes equations]
{Two component regularity for the Navier-Stokes equations}

\author[J. Fan, H. Gao\hfil EJDE-2009/121\hfilneg]
{Jishan Fan, Hongjun Gao}  % in alphabetical order

\address{Jishan Fan \newline
School of Mathematical Science\\
Nanjing Normal University, Nanjing, 210097, China\newline
Department of Applied Mathematics\\
Nanjing Forestry University, Nanjing 210037, China}
\email{fanjishan@njfu.com.cn}

\address{Hongjun Gao \newline
School of Mathematical Science\\
Nanjing Normal University, Nanjing, 210097, China}
\email{gaohj@njnu.edu.cn}

\thanks{Submitted August 28, 2009. Published September 29, 2009.}
\thanks{Supported by grants 10871097 from the NSFC,
20080441062 from the China \hfill\break\indent
Postdoctoral Research Foundation, and
0802020C from Jiangsu Planned Projects for \hfill\break\indent
Postdoctoral Research
Foundation.}
\subjclass[2000]{35Q30, 35K15, 76D03}
\keywords{Navier-Stokes equations; regularity criterion;
 two component; \hfill\break\indent multiplier spaces; Besov spaces}

\begin{abstract}
 We consider the regularity of weak solutions to the Navier-Stokes
 equations in $\mathbb{R}^3$. Let $u:=(u_1,u_2,u_3)$ be a weak
 solution and $\widetilde{u}:=(u_1,u_2,0)$. We prove that $u$ is
 strong solution if $\nabla\widetilde{u}$ satisfy Serrin's type criterion.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this article we study the regularity of the weak solutions
of the Navier-Stokes equations:
\begin{gather}
u_t+u\cdot\nabla u+\nabla p-\Delta u=0,\label{1.1}\\
\mathop{\rm div}u=0\quad\text{in }(0,\infty)\times\mathbb{R}^3,\label{1.2}\\
u|_{t=0}=u_0,\quad \mathop{\rm div}u_0=0\quad\text{in }
 \mathbb{R}^3\label{1.3}
\end{gather}
where $u:=(u_1,u_2,u_3)$ represents the velocity and $p$ represents
the pressure.

The existence of global weak solutions for any initial data with
finite energy is known since the work of Leray \cite{l1}. The
smoothness of Leray's weak solution is not known. While the
existence of a regular solution is still an open problem, there are
many interesting sufficient conditions which guarantee that a given
weak solution is smooth. A well-known condition states that if
$u\in L^r(0,T;L^s(\mathbb{R}^3))$ with $\frac{2}{r}+\frac{3}{s}=1$ and
$s\in[3,\infty]$, then the solution $u$ is actually regular
\cite{e1,f1,g1,o1,s1,s2,s3}.
 A similar condition $\omega=\mathop{\rm curl} u\in
L^r(0,T;L^s(\mathbb{R}^3))$ with $\frac{2}{r}+\frac{3}{s}=2$ where
$\frac{3}{2}\leq s\leq\infty$ also implies the regularity as shown
by Bei\~{a}o da Veiga \cite{b2}. Chae and Choe \cite{c1} proved that
if $\widetilde{\omega}=(\omega_1,\omega_2,0)\in
L^r(0,T;L^s(\mathbb{R}^3))$ with $\frac{2}{r}+\frac{3}{s}=2$ and
$\frac{3}{2}\leq s<\infty$, then the solution is regular. Kozono and
Yatsu \cite{k1} showed that if $\widetilde{\omega}\in L^1(0,T;BMO)$,
then the solution remains smooth. Zhang and Chen \cite{z1} proved
that $u$ is regular if $\widetilde{\omega}\in
L^1(0,T;\dot{B}^0_{\infty,\infty})$. Bae and Choe \cite{b1} proved
that $u$ is strong if $\widetilde{u}\in L^r(0,T;L^s(\mathbb{R}^3))$
with $\frac{2}{r}+\frac{3}{s}=1$ with $s>3$. In \cite{c1}, the
authors also proved that $u$ is strong if $\nabla\widetilde{u}\in
L^r(0,T;L^s)$ with $\frac{2}{r}+\frac{3}{s}\leq 1,2\leq r\leq\infty$
and $3\leq s\leq\infty$, which is not optimal from the scaling
argument. Here we would like to improve the regularity criterion on
$\nabla\widetilde{u}$ in such a way that it undergoes the correct
scaling.

There have been many efforts  to show that analogous conditions on
only one component of the velocity or the gradient of velocity imply
the regularity of solutions but all the results are partial, see
\cite{k2}, cite{z2, z3} and the references there in.

We say that a function belongs to the multiplier spaces
$M(\dot{H}^r,L^2)$ if it maps, by point-wise multiplication,
$\dot{H}^r$ in $L^2$:
\begin{equation}
\dot{X}_r:=M(\dot{H}^r,L^2):=\{f\in\mathcal{S}',\|f\phi\|_{L^2}\leq
C\|\phi\|_{\dot{H}^r}\}.\label{1.4}
\end{equation}
Similarly we can define $\dot{Y}_{1+r}:=M(\dot{H}^r,\dot{H}^{-1})$
and $\dot{Y}_2^{(0)}$ denotes the closure of the Schwartz class
$\mathcal{S}$ in $\dot{Y}_2$. We denote
$\Lambda:=(-\Delta)^{\frac{1}{2}}$, then $\dot{Y}_2=\Lambda^2BMO$
\cite{m1}, $\dot{X}_r$ and $\dot{Y}_{1+r}$ have been characterized
in \cite{m1,m2}.

Now we are in a position to state the main result in this paper.

\begin{theorem}\label{th1.1}
Let $u_0\in H^1$. Assume that one of the following four
conditions holds:
\begin{gather}
\nabla\widetilde{u}\in L^\frac{2}{2-r}(0,T;\dot{X}_r)\quad
\text{ for  some }r\in[0,1),\label{1.5}\\
 \nabla\widetilde{u}\in L^\frac{2}{1-r}(0,T;\dot{Y}_{1+r})\quad
\text{for some } r\in[0,1),\label{1.6}\\
 \nabla\widetilde{u}\in C([0,T];\dot{Y}_2^{(0)}),\label{1.7}\\
 \nabla\widetilde{u}\in L^1(0,T;\dot{B}^0_{\infty,\infty}).\label{1.8}
\end{gather}
Then
\begin{equation}
u\in L^\infty(0,T;H^1)\cap L^2(0,T;H^2).\label{1.9}
\end{equation}
\end{theorem}

Here and thereafter, $\dot{B}^s_{p,q}$ stands for the homogeneous
Besov space, see below for the definition.

\begin{remark}\label{re1.1} \rm
Since $L^\infty\subsetneq BMO\subsetneq\dot{B}^0_{\infty,\infty},
L^\frac{3}{r}\subset L^{\frac{3}{r},\infty}\subset\dot{X}_r$ and
$L^\frac{3}{1+r}\subset
L^{\frac{3}{1+r},\infty}\subset\dot{Y}_{1+r}$, our results improve
that given in \cite{c1}.
\end{remark}

\section{Preliminaries}

We introduce the Littlewood-Paley decomposition. Let
$\mathcal{S}(\mathbb{R}^3)$ be the Schwartz class of rapidly
decreasing function. Given $f\in\mathcal{S}(\mathbb{R}^3)$, its
Fourier transform $\mathscr{F}f=\hat f$ is defined by $$\hat f(\xi)
=\int_{\mathbb{R}^3}e^{-ix\xi}f(x)dx,$$ and its inverse Fourier
transform $\mathscr{F}^{-1}f=f^\vee$ is defined by
$$
f^\vee(x)=(2\pi)^{-3}\int_{\mathbb{R}^3}e^{ix\xi}f(\xi)d\xi.
$$
Let us choose a nonnegative radial function
$\phi\in\mathcal{S}(\mathbb{R}^3)$ such that
$$
0\leq\hat\phi(\xi)\leq 1,\quad
\hat\phi(\xi)=\begin{cases}
1,&\text{if }|\xi|\leq 1,\\
0,&\text{if }|\xi|\geq 2,
\end{cases}
$$
and let
$$
\psi(x)=\phi(x)-2^{-3}\phi\big(\frac{x}{2}\big),\quad
\phi_j(x)=2^{3j}\phi(2^jx), \quad
\psi_j(x)=2^{3j}\psi(2^jx),\quad j\in\mathbb{Z}.
$$
For $j\in\mathbb{Z}$, the Littlewood-Paley projection operators
$S_j$ and $\Delta_j$ are, respectively, defined by
\begin{gather}
S_jf=\phi_j*f,\label{2.1}\\
\Delta_jf=\psi_j*f.\label{2.2}
\end{gather}

Observe that $\Delta_j=S_j-S_{j-1}$. Also, if $f$ is an $L^2$
function, then $S_jf\to 0$ in $L^2$ as $j\to -\infty$
and $S_jf\to  f$ in $L^2$ as $j\to +\infty$ (this is
an easy consequence of Parseval's theorem). By telescoping the
series, we thus have the Littlewood-Paley decomposition
\begin{equation}
f=\sum_{j=-\infty}^{+\infty}\Delta_jf,\label{2.3}
\end{equation}
for all $f\in L^2$, where the summation is in the $L^2$ sense.
Notice that
$$
\Delta_jf=\sum_{l=j-2}^{j+2}\Delta_l(\Delta_jf)
=\sum_{l=j-2}^{j+2}\psi_l*\psi_j*f,
$$
then from the Young
inequality, it follows that
\begin{equation}
\|\Delta_jf\|_{L^q}\leq
C2^{3j(\frac{1}{p}-\frac{1}{q})}\|\Delta_jf\|_{L^p},\label{2.4}
\end{equation}
where $1\leq p\leq q\leq\infty$, $C$ is a constant independent of
$f,j$.

Let $s\in\mathbb{R},p,q\in[1,\infty]$, the homogeneous Besov space
$\dot{B}^s_{p,q}$ is defined by the full-dyadic decomposition such
as
$$
\dot{B}^s_{p,q}=\{f\in\mathcal{Z}'(\mathbb{R}^3):
\|f\|_{\dot{B}^s_{p,q}}<\infty\},
$$
where
$$
\|f\|_{\dot{B}^s_{p,q}}
=\Big(\sum_{j=-\infty}^{+\infty}2^{jsq}\|\Delta_jf\|^q_{L^p}
\Big)^{1/q}
$$
and $\mathcal{Z}'(\mathbb{R}^3)$ denotes the dual space of
$\mathcal{Z}(\mathbb{R}^3)=\{f\in\mathcal{S}(\mathbb{R}^3):D^\alpha\hat
f(0)=0;\forall\alpha\in\mathbb{N}^3\}$. We refer to \cite{t1} for
more detailed properties.

\section{Proof of Theorem \ref{th1.1}}

We set
$$
|\nabla u|^2=\sum_{i,k}|\partial_ku_i|^2,\quad
|\nabla^2 u|^2=\sum_{i,j,k}|\partial_k\partial_ju_i|^2.
$$
Differentiating both sides of equation (\ref{1.1}) with respect to
$x_k$, taking the scalar product with $\partial_ku$, adding over $k$
and, finally, integrating by parts over $\mathbb{R}^n$, we show that
\begin{align*}
\frac{1}{2}\frac{d}{dt}\int|\nabla u|^2dx+\int|\nabla^2u|^2dx
&=-\int\nabla[(u\cdot\nabla)u]\cdot\nabla u dx\\
&=\sum_{i,j,k}\int\partial_ku_i\cdot\partial_iu_j\cdot\partial_ku_j
dx.
\end{align*}

Following \cite{b1}, we consider separately the three cases
$i\neq 3$; $i=3$ and $j\neq3$; $i=j=3$. We only need to deal
with the case
$i=j=3$. Since $\partial_3u_3=-\partial_1u_1-\partial_2u_2$, it
readily follows that
\begin{align*}
\int\partial_ku_i\cdot\partial_iu_j\cdot\partial_ku_j dx
&=-\int\partial_ku_3\cdot(\partial_1u_1+\partial_2u_2)\cdot\partial_k
u_3dx\\
&\leq 2\int|\nabla\widetilde u|\cdot|\nabla u|^2dx.
\end{align*}
And thus we get
\begin{equation}
\frac{1}{2}\frac{d}{dt}\int|\nabla u|^2dx+\int|\nabla^2u|^2dx\leq
2\int|\nabla\widetilde u|\cdot|\nabla u|^2dx=:I.\label{3.1}
\end{equation}
Now we assume that (\ref{1.5}) holds. Then
\begin{align*}
I&\leq 2\|\nabla u\|_{L^2}\cdot\|\ |\nabla\widetilde u|\cdot|\nabla
u|\ \|_{L^2}\\
&\leq 2\|\nabla u\|_{L^2}\cdot\|\nabla\widetilde
u\|_{\dot{X}_r}\|\nabla u\|_{\dot{H}^r}\\
&\leq C\|\nabla u\|_{L^2}\cdot\|\nabla\widetilde
u\|_{\dot{X}_r}\|\nabla u\|_{L^2}^{1-r}\|\nabla^2u\|^r_{L^2}
\end{align*}
by the interpolation inequality
\begin{equation}
\|w\|_{\dot{H}^r}\leq C\|w\|_{L^2}^{1-r}\|\nabla
w\|_{L^2}^r,\label{3.2}
\end{equation}
whence
$$
I\leq\epsilon\|\nabla^2u\|_{L^2}^2+C\|\nabla\widetilde
u\|^{\frac{2}{2-r}}_{\dot{X}_r}\|\nabla u\|^2_{L^2},
$$
for any $\epsilon>0$ by  Young's inequality.
Inserting the above estimates into (\ref{3.1}) and taking $\epsilon$
small enough and the Gronwall's inequality yield (\ref{1.9}).

Next we assume that (\ref{1.6}) holds. Then
\begin{align*}
I&\leq 2\|\nabla u\|_{\dot{H}^1}\|\ |\nabla\widetilde u|\cdot|\nabla
u|\ \|_{\dot{H}^{-1}}\\
&\leq C\|\nabla u\|_{\dot{H}^1}\|\nabla\widetilde
u\|_{\dot{Y}_{1+r}}\|\nabla u\|_{\dot{H}^r}\\
&\leq C\|\nabla^2 u\|_{L^2}\|\nabla\widetilde u\|_{\dot{Y}_{1+r}}
\|\nabla u\|_{L^2}^{1-r}\|\nabla^2 u\|_{L^2}^r\ \ \ ({\rm by}\ \
(\ref{3.2}))\\
&\leq \epsilon\|\nabla u\|_{L^2}^2+C\|\nabla\widetilde
u\|_{\dot{Y}_{1+r}}^\frac{2}{1-r}\|\nabla u\|_{L^2}^2
\end{align*}
for any $\epsilon>0$ by Young's inequality.
Inserting the above estimates into (\ref{3.1}) and taking $\epsilon$
small enough and the Gronwall's inequality give (\ref{1.9}).

Now we assume that (\ref{1.7}) holds. For any $\epsilon>0$,
then there exist $\alpha$ and $\beta$ such that
$\nabla\widetilde u=\alpha+\beta,\|\alpha\|_{L^\infty(0,T;Y_2)}
\leq\epsilon$ and
$\beta\in L^\infty((0,T)\times\mathbb{R}^3)$,
\begin{align*}
I&\leq 2\int|\alpha|\cdot|\nabla
u|^2dx+2\|\beta\|_{L^\infty}\|\nabla u\|^2_{L^2}\\
&\leq 2\|\nabla u\|_{\dot{H}^1}\||\ \alpha|\cdot\nabla
u\|_{\dot{H}^{-1}}+C\|\nabla u\|_{L^2}^2\\
&\leq 2\|\nabla u\|_{\dot{H}^1}\|\alpha\|_{\dot{Y}_2}\|\nabla
u\|_{\dot{H}^1}+C\|\nabla u\|_{L^2}^2\\
&\leq 2\epsilon\|\nabla^2u\|_{L^2}^2+C\|\nabla u\|_{L^2}^2.
\end{align*}
Inserting the above estimates into (\ref{3.1}) and taking $\epsilon$
small enough and then the Gronwall's inequality show
(\ref{1.9}).

Finally we assume that (\ref{1.8}) holds. Then using the
Littlewood-Paley decomposition (\ref{2.3}), we decompose
$\nabla\widetilde u$ as follows:
$$
\nabla\widetilde u=\sum_{\ell
=-\infty}^{+\infty}\Delta_\ell(\nabla\widetilde u)
=\sum_{\ell<-N}\Delta_\ell(\nabla\widetilde u)
 +\sum_{\ell=-N}^N\Delta_\ell(\nabla\widetilde u)
 +\sum_{\ell>N}\Delta_\ell(\nabla\widetilde u).
$$
Here $N$ is a positive integer to be chosen later. Substituting this
into $I$, we have
\begin{equation}
\begin{aligned}
I&\leq 2\sum_{\ell<-N}\int|\Delta_\ell(\nabla\widetilde u)|\
|\nabla u|^2 dx+2\sum_{\ell=-N}^N\int|\Delta_\ell(\nabla\widetilde
u)|\ |\nabla u|^2dx\\
&\quad +2\sum_{\ell>N}\int|\Delta_\ell(\nabla\widetilde u)|\
|\nabla u|^2dx\\
&=:I_1+I_2+I_3.
\end{aligned}\label{3.3}
\end{equation}
For $I_1$, from the H\"{o}lder inequality and (\ref{2.4}), it
follows that
\begin{align*}
I_1&\leq 2\sum_{\ell<-N}\|\Delta_\ell(\nabla\widetilde
u)\|_{L^\infty}\|\nabla u\|_{L^2}^2\\
&\leq 2\|\nabla u\|_{L^2}^2\sum_{\ell<-N}2^{\frac{3}{2}\ell}
\|\Delta_\ell(\nabla\widetilde u)\|_{L^2}\\
&\leq C2^{-\frac{3}{2}N}\|\nabla u\|_{L^2}^2\|\nabla\widetilde
u\|_{L^2}\leq C2^{-\frac{3}{2}N}\|\nabla u\|_{L^2}^3.
\end{align*}
For $I_2$, from the H\"{o}lder inequality, it follows that
\begin{align*}
I_2&\leq 2\|\nabla
u\|_{L^2}^2\sum_{\ell=-N}^N\|\Delta_\ell(\nabla\widetilde
u)\|_{L^\infty}\\
&\leq C\|\nabla u\|_{L^2}^2\cdot N\|\nabla\widetilde
u\|_{\dot{B}^0_{\infty,\infty}}\\
&= C N\|\nabla u\|_{L^2}^2\|\nabla\widetilde
u\|_{\dot{B}^0_{\infty,\infty}}.
\end{align*}
For $I_3$, from the H\"{o}lder inequality and (\ref{2.4}), it
follows that
\begin{align*}
I_3&\leq 2\|\nabla
u\|_{L^3}^2\sum_{\ell>N}\|\Delta_\ell(\nabla\widetilde
u)\|_{L^3}\\
&\leq C\|\nabla
u\|_{L^3}^2\sum_{\ell>N}2^\frac{\ell}{2}\|\Delta_\ell(\nabla\widetilde
u)\|_{L^2}\\
&\leq C\|\nabla
u\|_{L^3}^2\Big(\sum_{\ell>N}2^{-\ell}\Big)^{1/2}\Big(\sum_{\ell>N}
2^{2\ell}\|\Delta_\ell(\nabla\widetilde
u)\|_{L^2}^2\Big)^{1/2}\\
&\leq C\|\nabla u\|_{L^3}^22^{-\frac{N}{2}}\|\nabla^2\widetilde
u\|_{L^2}\\
&\leq C2^{-\frac{N}{2}}\|\nabla u\|_{L^3}^2\|\nabla^2u\|_{L^2}\\
&\leq C2^{-\frac{N}{2}}\|\nabla u\|_{L^2}\|\nabla^2u\|_{L^2}^2
\end{align*}
by the Gagliardo-Nirenberg inequality
$$
\|\nabla u\|_{L^3}^2\leq
C\|\nabla u\|_{L^2}\|\nabla^2u\|_{L^2}.
$$
Inserting the above estimates into (\ref{3.3}), we find that
$$
I\leq C2^{-\frac{3}{2}N}\|\nabla u\|_{L^2}^3+C N\|\nabla\widetilde
u\|_{\dot{B}^0_{\infty,\infty}}\|\nabla
u\|_{L^2}^2+C2^{-\frac{N}{2}}\|\nabla
u\|_{L^2}\|\nabla^2u\|_{L^2}^2.
$$
Now we choose $N$ so that
$C2^{-\frac{N}{2}}\|\nabla u\|_{L^2}\leq\frac{1}{2}$; i.e.,
$$
N\geq2+\frac{2\log^+(2C\|\nabla u\|_{L^2})}{\log2}.
$$
Then
$$
I\leq C+C\|\nabla u\|_{L^2}^2+C\|\nabla
u\|_{L^2}^2\log(e+\|\nabla
u\|_{L^2})+\frac{1}{2}\|\nabla^2u\|_{L^2}^2.
$$
Insetting the above estimates into (\ref{3.1}) and the Gronwall's
inequality give (\ref{1.9}).
This completes the proof.


\subsection*{Acknowledgements}
The authors are indebted to Professor H. O. Bae who kindly
sent us the preprint \cite{b1}.


\begin{thebibliography}{99}

\bibitem{b1} H. O. Bae and H. J. Choe;
\emph{A regularity criterion for the
Navier-Stokes equations}, (Preprint 2005).

\bibitem{b2} H. Beir\~{a}o da Veiga;
\emph{A new regularity class for the
Navier-Stokes equations in $\mathbb{R}^n$}, Chinese Ann. Math. 16
(1995), 407-412.

\bibitem{c1} D. Chae and H. J. Choe;
\emph{Regularity of solutions to the
Navier-Stokes equations}, Electronic J. Differential Equations, Vol.
1999 (1999), No. 5, pp. 1-7.

\bibitem{e1} L. Escauriaza, G. A. Seregin and V.\v{S}ver\'{a}k;
\emph{$L^{3,\infty}$-solutions of the Navier-Stokes equations and backward
uniqueness}, Russ. Math. Surv. 58(2) (2003), 211-248.

\bibitem{f1} E. Fabes, B. Jones and N. M. Riviere;
\emph{The initial value
problem for the Navier-Stokes equations with data in $L^p$}, Arch.
Rat. Mech. Anal. 45 (1972), 222-248.

\bibitem{g1} Y. Giga;
\emph{Solutions for semilinear parabolic equations in
$L^p$ and regularity of weak solutions of the Navier-Stokes
equations}, J. Diff. Eqns. 62 (1986), 186-212.

\bibitem{k1} H. Kozono and N. Yatsu;
\emph{Extension criterion via
two-Components of vorticity on strong solutions to the 3-D
Navier-Stokes equations}, Math. Z., 246 (2004), 55-68.

\bibitem{k2} I. Kukavica and M. Ziane;
\emph{One component regularity for
the Navier-Stokes equations}, Nonlinearity 19 (2006), 453-469.

\bibitem{l1} J. Leray;
\emph{Sur le mouvement d'un liquide visqueux
emplissant l'espace}, Acta Math. 63 (1934), 193-248.

\bibitem{m1} V. G. Maz'ya and T. O. Shaposhnikova;
\emph{Theory of Multipliers
in Spaces of Differentiable Functions}. Monographs and Studies in
Mathematics 23, Pitman, Boston, MA, 1985.

\bibitem{m2} V. G. Maz'ya;
\emph{On the theory of the $n$-dimensional
Schr\"{o}dinger operator}, Izv. Akad. Nauk SSSR (Ser. Mat.) 28
(1964), 1145-1172.

\bibitem{o1} T. Ohyama;
\emph{Interior regularity of weak solutions to the
Navier-Stokes equations}, Proc. Japan Acad. 36 (1960), 273-277.

\bibitem{s1} J. Serrin;
\emph{On the interior regularity of weak solutions
of the Navier-Stokes equations}, Arch. Rat. Mech. Anal. 9 (1962),
197-195.

\bibitem{s2} H. Sohr and W. Von Wahl;
\emph{On the regularity of the pressure
of weak solutions of Navier-Stokes equations}, Archiv Math. 46
(1986), 428-439.

\bibitem{s3} M. Struwe;
\emph{On partial regularity results for the
Navier-Stokes equations}, Comm. Pure Appl. Math. 41 (1988), 437-458.

\bibitem{t1} H. Triebel;
\emph{Interpolation Theory, Function Spaces,
Differential Operators}, North Holland, Amsterdam-New York-Oxford,
1978.

\bibitem{z1} Z. Zhang and Q. Chen;
\emph{Regularity criterion via two
components of vorticity on weak solutions to the Navier-Stokes
equations in $\mathbb{R}^3$}, J. Differential Equations 216 (2005),
470-481.

\bibitem{z2} Y. Zhou, A new regularity result for the Navier-Stokes equations
in terms of the gradient of one velocity component, Methods Appl.
Anal. 9 (2002), 563--578.

\bibitem{z3} Y. Zhou, A new regularity criterion for weak solutions to the
Navier- Stokes equations, J. Math. Pures Appl. 84 (2005),
1496--1514.


\end{thebibliography}

\end{document}