Anisotropic regularity conditions for the suitable weak solutions to the 3d Navier-Stokes equations

TL;DR

This paper establishes new anisotropic regularity criteria for suitable weak solutions to the 3D Navier-Stokes equations, focusing on component-wise smallness conditions that improve previous results on local scaled norm regularity.

Contribution

It introduces novel interior regularity criteria based on anisotropic conditions involving velocity components and vorticity, enhancing existing regularity conditions for weak solutions.

Findings

Smallness of one velocity component ensures regularity.

Horizontal vorticity smallness suffices for regularity.

Vertical gradient boundedness alone guarantees regularity.

Abstract

We are concerned with the problem,originated from Seregin [18,19,20], what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3d Naiver-Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled $L^{2}$ norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.