On the vorticity direction and the regularity of 3D Navier–Stokes equations

Abstract

This short paper presents a simplified and alternative proof of the regularity of weak solutions to the 3D Navier–Stokes equations with ‘sufficiently small’ jumps in the vorticity direction. Although the main result is very similar to a previously proven one, there are some relevant differences. Specifically, we prove that the smallness condition regarding the angle spanned by the vorticity direction needs to be checked, for each point x in the domain, only over a discrete set of surrounding points. These points lie in the direction of the coordinate axes and have a fixed positive distance from x. This is achieved by using a more direct approach which does not rely on the use of singular integrals theory, but which requires estimates on higher-order derivatives of the velocity.