Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 105, pp. 1-5.

Title: Regularity of generalized Naveir-Stokes equations in terms of
       direction of the velocity
       
Author: Yuwen Luo (Chongqing Univ. of Technology, China)

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 02, 2010.
Math Subject Classifications: 35D10, 35Q35, 76D03.
Key Words: Generalized Navier-Stokes equation;
           regularity; Serrin criteria.