Direction of Vorticity and a Refined Regularity Criterion for the Navier-Stokes Equations with Fractional Laplacian

摘要

We give a refined regularity criterion for solutions of the three-dimensional Navier-Stokes equations with fractional dissipative term (-)/2v. The criterion is composed by the direction field of the vorticity and its magnitude simultaneously. Our result is a generalized of previous results by BeirAo da Veiga and Berselli (Differ Integral Equ 15(3):345-356, 2002), and Zhou (ANZIAM J 46(3):309-316, 2005, Monatsh Math 144(3):251-257, 2005). Moreover, our result mentioned about the relation between the solution of the Navier-Stokes equations and the Euler equations.