$$\varepsilon $$-Regularity criteria for the 3D Navier–Stokes equations in Lorentz spaces

Abstract

In this paper, we are concerned with the regularity of suitable weak solutions to the 3D Navier–Stokes equations in Lorentz spaces. We obtain $$\varepsilon $$ -regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity or deformation tensor in Lorentz spaces, which generalizes the corresponding results derived by Gustafson et al. (Commun Math Phys 273:161–176, 2007) in Lebesgue spaces. As applications, this allows us to extend recent results involving Leray’s blow up rate in time, and to show that the number of singular points of weak solutions belonging to $$ L^{p,\infty }(-1,0;L^{q,l}({\mathbb {R}}^{3})) $$ is finite, where the pair (p, q) satisfies $$ {2}/{p}+{3}/{q}=1$$ with $$3<q<\infty $$ and $$q\le l <\infty $$ .