Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 06, pp. 1-7.

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

Authors: Wenying Chen (Chongqing Three Gorges Univ., China)
         Sadek Gala   (Univ. of Mostaganem, Algeria)

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.