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 $\|\Pi\|_{ L^{ p,\infty}(0,T; L ^{ q,\infty}(\mathbb{ R}^{ 3}))} $ with $ { 2}/{ p}+{ 3}/{ q}=2$ $({ 3}/{ 2}<q<\infty)$ or $\|\nabla\Pi\|_{ 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.
Categories: math.AP