A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient

TL;DR

This paper introduces an improved local regularity criterion for suitable weak solutions of the 3D Navier--Stokes equations, refining previous criteria and providing quantitative relations using a direct method.

Contribution

The paper presents a new, refined regularity criterion for Navier--Stokes solutions, enhancing previous criteria with a direct approach and quantitative relations.

Findings

Improved regularity criterion for weak solutions.

Quantitative relation in Seregin's criterion.

Refinement of the Caffarelli--Kohn--Nirenberg criterion.

Abstract

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. There have been several attempts to refine the Caffarelli--Kohn--Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin's criterion (2007).