Electron. J. Diff. Equ., Vol. 2011 (2011), No. 06, pp. 1-7.

A regularity criterion for the Navier-Stokes equations in terms of the horizontal derivatives of the two velocity components

Wenying Chen, Sadek Gala

Abstract:
In this article, we consider the regularity for weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$. It is proved that if the horizontal derivatives of the two velocity components
$$
 \nabla _h\widetilde{u}\in  L^{2/(2-r)}(0,T;\dot{\mathcal{M}}_{2,3/r}
 (\mathbb{R}^3)),\quad  \hbox{for }0<r<1,
 $$
then the weak solution is actually strong, where $\dot{\mathcal{M}} _{2,3/r}$ is the critical Morrey-Campanato space and $\widetilde{u} =(u_1,u_2,0)$, $\nabla_h\widetilde{u}=(\partial _1u_1,\partial _2u_2,0)$.

Submitted October 18, 2010. Published January 12, 2011.
Math Subject Classifications: 35Q30, 76F65.
Key Words: Navier-Stokes equations; Leray-Hopf weak solutions; regularity criterion.

Show me the PDF file (218 KB), TEX file, and other files for this article.

Wenying Chen
College of Mathematics and Computer Science
Chongqing Three Gorges University
Wanzhou 404000, Chongqing, China
email: wenyingchenmath@gmail.com
  Sadek Gala
Department of Mathematics
University of Mostaganem
Box 227, Mostaganem 27000, Algeria
email: sadek.gala@gmail.com

Return to the EJDE web page