A $\varepsilon$-regularity criterion without pressure of suitable weak solutions to the Navier-Stokes equations at one scale

Abstract

In this paper, we continue our work in [15] to derive {\epsilon}-regularity criteria at one scale without pressure for suitable weak solutions to the Navier-Stokes equations. We establish a $\varepsilon$-regularity criterion below of suitable weak solutions, for any $\delta>0$, $$\iint_{Q(1)}|u|^{\frac{5}{2}+\delta}dxdt\leq \varepsilon.$$ As an application, we extend the previous corresponding results concerning the improvement of the classical Caffarelli--Kohn--Nirenberg theorem by a logarithmic factor.