Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 148, pp. 1-6.

Title:A remark on the regularity for the 3D Navier-Stokes equations
      in terms of the two components of the velocity

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

Abstract:
 In this note, we study the regularity of Leray-Hopf weak solutions
 to the Navier-Stokes equation, with the condition
 $$
 \nabla (u_{1},u_{2},0)
 \in L^{\frac{2}{1-r}}(0,T; \dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3) ,
 $$
 where $\dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ is the
 Morrey-Campanato space for $0<r<1$. Since
 $$
 L^{1/3}(\mathbb{R}^3)\subset \dot{X}_r(\mathbb{R}^3)
 \subset \dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3),
 $$
 the above regularity condition allows us to improve the results
 obtained by Fan and Gao [6].
 
Submitted November 5, 2009. Published November 25, 2009.
Math Subject Classifications: 35Q30, 35K15, 76D05.
Key Words: Navier-Stokes equations; regularity criterion;
           Morrey-Campanato spaces.