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.

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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

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