New $\varepsilon$-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 $\varepsilon$-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 $2400/1903 (\approx1.261)$.