\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \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 }00}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