Electron. J. Diff. Equ., Vol. 2010(2010), No. 105, pp. 1-5.

Regularity of generalized Naveir-Stokes equations in terms of direction of the velocity

Yuwen Luo

Abstract:
In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms $(-\Delta)^{\alpha} u$. It is proved that if $\hbox{div} (u / |u|) \in L^p (0, T ; L^q (\mathbb{R}^3))$ with
$$ \frac{2 \alpha}{p} + \frac{3}{q} \leq 2 \alpha - \frac{3}{2},\quad \frac{6}{4 \alpha-3} < q \leq \infty . $$
then any smooth on GNS in $[0,T)$ remains smooth on $[0, T]$.

Submitted April 8, 2010. Published August 2, 2010.
Math Subject Classifications: 35D10, 35Q35, 76D03.
Key Words: Generalized Navier-Stokes equation; regularity; Serrin criteria.

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Yuwen Luo
School of Mathematics and Statistics
Chongqing University of Technology
Chongqing 400050, China
email: petitevin@gmail.com

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