New $\varepsilon$-regularity criteria and application to the box dimension of the singular set in the 3D Navier-Stokes equations

In this paper, by exploiting the energy hidden in the pressure, we present some new $\varepsilon$-regularity criterion below $$\|u\|_{L^{p,q}(Q(1))}+\|\Pi\|_{L^{1 }(Q(1))}<\varepsilon,~~1\leq 2/q+3/p <2, \, 1\leq p,q\leq\infty,~~\text{then}~~u\in L^{\infty}(Q(1/2)), $$ to suitable weak solutions of the 3D Navier-Stokes equations, which is an \,improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of this system from $975/758(\approx1.286)$ [28] to $2400/1903 (\approx1.261)$.