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

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 $\mathrm{\parallel }\mathrm{\Pi }{\mathrm{\parallel }}_{L^{p,\mathrm{\infty }}(0,T;L^{q,\mathrm{\infty }}({\mathbb{R}}^3))}$ with $2\mathrm{/}p+3\mathrm{/}q=2$ $(3\mathrm{/}2<q<\mathrm{\infty })$ or $|\nabla\Pi|_{L^{p,\infty}(0,T; L ^{q,\infty}($\mathbb{R}$^{3}))} $with$ {2}/{p}+{3}/{q}=3$ $(1<q<\mathrm{\infty })$ 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 $\mathrm{\nabla }u$ obtained here generalize known ones to allow the time direction to belong to Lorentz spaces.