Global regularity criterion for the Navier-Stokes equations based on the direction of vorticity

We study the regularity criterion for the Navier-Stokes equations and show that the (beta (1),beta (2),beta (3))-Holder continuity assumption in (x(1),x(2),x(3)) on the direction of the vorticity ensures the regularity of the solution. This may be viewed as an extension of many previous results, since some of the beta (i) can be arbitrarily small.