Electron. J. Diff. Equ., Vol. 2009(2009), No. 148, pp. 1-6.

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

Sadek Gala

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.

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

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