New $\epsilon$-regularity criteria of suitable weak solutions of the 3D Navier-Stokes equations at one scale

Abstract

In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\epsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any $p,q\in [1,\infty ]$ satisfying $1\le 2/q+3/p<2$, there exists an absolute positive constant $\epsilon$ such that $u\in L^{\infty }(Q(1/2))$ if $$\Vert u{\Vert }_{L^{p,q}(Q(1))}+\Vert \Pi {\Vert }_{L^1(Q(1))}<\epsilon .$$ This is an improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application of these $\epsilon$-regularity criteria, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of the Navier-Stokes system from $975/758(\approx 1.286)$ [28] to $2400/1903(\approx 1.261)$.

We revised the title and the introduction

References in corpus (2)

• Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
• On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier-Stokes equations

Cited by in corpus (1)

• Improved bounds for box dimensions of potential singular points to the Navier--Stokes equations