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

We give a refined regularity criterion for solutions of the three-dimensional Navier–Stokes equations with fractional dissipative term ( - Δ ) α / 2 v . The criterion is composed by the direction field of the vorticity and its magnitude simultaneously. Our result is a generalized of previous results by Beirão 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.