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

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.