Continuous alignment of vorticity direction prevents the blow-up of the Navier–Stokes flow under the no-slip boundary condition

Abstract

This paper is concerned with a regularity criterion based on vorticity direction for Navier–Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the first and the last authors (2014). The result of this paper is its natural but non-trivial extension based on L∞ theory of the Stokes and the Navier–Stokes equations recently developed by K. Abe and the first author.