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

TL;DR

This paper introduces new epsilon-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale, improving previous results and refining the upper box dimension estimate of the singular set.

Contribution

It presents novel epsilon-regularity criteria based on pressure decomposition, enhancing prior conditions and reducing the upper bound of the singular set's box dimension.

Findings

Established new epsilon-regularity criteria involving pressure decomposition.

Improved the upper box dimension estimate of the singular set from approximately 1.286 to 1.261.

Extended the class of solutions for which regularity can be guaranteed.

Abstract

In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\varepsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any $p,q\in [1,\infty]$ satisfying $1\leq 2/q+3/p <2$, there exists an absolute positive constant $\varepsilon$ such that $u\in L^{\infty}(Q(1/2))$ if $$\|u\|_{L^{p,q}(Q(1))}+\|\Pi\|_{L^{1 }(Q(1))}<\varepsilon.$$ 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 $\varepsilon$-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 $975/758(\approx1.286)$ [28] to $2400/1903 (\approx1.261)$.