New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier-Stokes equations

Abstract

In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $\Vert \Pi {\Vert }_{L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))}$ with $2/p+3/q=2$ $(3/2<q<\infty )$ or $\Vert \nabla \Pi {\Vert }_{L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))}$ with $2/p+3/q=3$ $(1<q<\infty )$ is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of $\nabla u$ obtained here generalize known ones to allow the time direction to belong to Lorentz spaces.

10 pages