Electron. J. Diff. Equ., Vol. 2010(2010), No. 153, pp. 1-7.

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

Sadek Gala

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:
 \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.

Show me the PDF file (219 KB), TEX file, and other files for this article.

Sadek Gala
Department of Mathematics, University of Mostaganem
Box 227, Mostaganem 27000, Algeria
email: sadek.gala@gmail.com

Return to the EJDE web page