On Wolf's regularity criterion of suitable weak solutions to the Navier-Stokes equations

TL;DR

This paper establishes a new epsilon-regularity criterion for suitable weak solutions to the 3D Navier-Stokes equations using Wolf's local pressure projection, improving previous regularity results.

Contribution

It introduces an improved epsilon-regularity criterion for suitable weak solutions to the Navier-Stokes equations based on Wolf's local pressure projection method.

Findings

Provides a new epsilon-regularity criterion involving the $L^{20/7}$ norm of velocity.

Improves upon previous regularity criteria established by Chae, Wolf, Guevara, and Phuc.

Enhances understanding of local regularity conditions for weak solutions.

Abstract

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].