\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 148, 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-2009/148\hfil A remark on regularity] {A remark on the regularity for the 3D Navier-Stokes equations in terms of the two components of the velocity} \author[S. Gala\hfil EJDE-2009/148\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 November 5, 2009. Published November 25, 2009.} \subjclass[2000]{35Q30, 35K15, 76D05} \keywords{Navier-Stokes equations; regularity criterion; \hfill\break\indent Morrey-Campanato spaces} \begin{abstract} In this note, we study the regularity of Leray-Hopf weak solutions to the Navier-Stokes equation, with the condition $\nabla (u_{1},u_{2},0) \in L^{\frac{2}{1-r}}(0,T; \dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3) ,$ where $\dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ is the Morrey-Campanato space for $03$. Later, Chae-Choe \cite{DC} obtained an improved regularity criterion of \cite{B} imposing condition on only two components of the velocity, namely if \begin{gather*} \nabla \tilde u \in L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))\quad \text{with }\frac{2}{\alpha }+\frac{3}{q}\leq 2,\; 1\leq\alpha <\infty,\\ \tilde u =(u_{1},u_{2},0) \end{gather*} then the weak solution becomes smooth. See also \cite{ZP1, ZP2} for recent improvements of these criteria, via one velocity component. Recently, Fan and Gao \cite{FG} improved the regularity criterion in \cite{DC}, under the condition $\nabla \tilde u \in L^{\frac{2}{2-r}}(0,T; \dot{X}_r(\mathbb{R}^3)\mathbb{)}\quad \text{for some }00. \end{gather*} Also we have the imbedding \[ L^{1/3}(\mathbb{R}^3) \hookrightarrow \dot{X}_r(\mathbb{R}^3) \quad \text{for }0\leq r<\frac{3}{2}\,.$ Now we recall the definition of the Morrey-Campanato spaces. \begin{definition} \label{def2.2} \rm For $10} R^{3/q-3/p}\|f\| _{L^{p}(B(x,R))} <\infty \} \label{eq1.13} \end{definition} It is easy to check the equality $\| f(\lambda \cdot)\| _{\dot{\mathcal{M}} _{p,q}}=\frac{1}{\lambda ^{3/q}} \| f\| _{\dot{\mathcal{M}}_{p,q}}, \quad \lambda >0.$ For$20} R|\{ x\in \mathbb{R}^3:| f(y)| ^{p}>R\} | ^{pr/3}\Big)^{1/p} \\ &=\sup_{R>0} R| \{ x\in \mathbb{R}^{p}:|f(y)| >R\} | ^{r/3} \\ &\cong \| f\| _{L^{3/r,\infty }}. \end{align*} For 0k}2^{j(r-1)}2^{j}\| \Delta _{j}f\| _{L^{2}} \\ &\leq (\sum_{j\leq k}2^{2jr})^{1/2}( \sum_{j\leq k}\| \Delta _{j}f\| _{L^{2}}^{2}) ^{1/2}+(\sum_{j>k}2^{2j(r-1)})^{\frac{1}{2} }(\sum_{j>k}2^{2j}\| \Delta _{j}f\| _{L^{2}}^{2})^{1/2} \\ &\leq C\Big(2^{rk}\| f\| _{L^{2}}+2^{k(r-1)}\| f\| _{\dot{H}^{1}}\Big)\\ &= C(2^{rk}A^{-r}+2^{k(r-1)}A^{1-r})\| f\| _{L^{2}}^{1-r}\| f\| _{\dot{H}^{1}}^{r}, \end{align*} whereA=\| f\| _{\dot{H}^{1}}/ \|f\| _{L^{2}}$. Choose$k$such that$2^{rk}A^{-r}\leq 1$; that is,$k\leq [ \log A^{r}] . Then \begin{align*} \| f\| _{\dot{B}_{2,1}^{r}} &\leq C( 1+2^{k(r-1)}A^{1-r})\| f\| _{L^{2}}^{1-r}\| f\| _{\dot{H}^{1}}^{r} \\ &\leq C\| f\| _{L^{2}}^{1-r}\| \nabla f\|_{L^{2}}^{r}, \end{align*} and so the proof is complete. \end{proof} SinceL^{1/3}(\mathbb{R}^3)\subset \dot{X}_r(\mathbb{R}^3)\mathbb{\subset } \dot{\mathcal{M}}_{2,\frac{3}{r}}(\mathbb{R}^3)$, the above regularity criterion alloy us to improve the results obtained by Fan and Gao \cite{FG}. Our main result on \eqref{eqNS} reads as follows. \begin{theorem}\label{th1} Let$\tilde{u}=u_{1}e_{1}+u_{2}e_{2}$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$\mathop{\rm div}u_0=0$. Suppose that$\nabla \tilde{u}\in L^{\frac{2}{1-r}}(0,T,\dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3))$with$0