Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 153, pp. 1-7.

Title: Regularity for 3D Navier-Stokes equations
       in terms of two components of the vorticity

Author: Sadek Gala (Univ. of Mostaganem, Algeria)

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
 $$
 \tilde{\omega}\in L^{2/(2-r)}(0,T;L^{3/r}(\mathbb{R}^3))\quad
 \hbox{with }0<r<1,
 $$
 where $\tilde{\omega}$ form the two components of the vorticity,
 $\omega =\operatorname{curl}u$, then $u$ becomes the classical
 solution on $(0,T]$ (see [5]). We prove the regularity
 of Leray-Hopf weak solution $u$ under each of the following
 two (weaker) conditions:
 $$\displaylines{
 \tilde{\omega}\in L^{2/(2-r)}(0,T;\dot {\mathcal{M}}_{2,
 3/r}(\mathbb{R}^3))\quad \hbox{for }0<r<1,\cr
 \nabla \tilde{u}\in L^{2/(2-r)}(0,T;\dot {\mathcal{M}}_{2,
 3/r}(\mathbb{R}^3))\quad \hbox{for }0\leq r<1,
 }$$
 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 [5].
 	 
Submitted May 20, 2010. Published October 28, 2010.
Math Subject Classifications: 35Q35, 76C99.
Key Words: Navier-Stokes equations; regularity conditions;
           Morrey-Campaanto spaces.