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 $\|Π\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))}$ with ${2}/{p}+{3}/{q}=2$ $({3}/{2}<q<\infty)$ or $\|\nablaΠ\|_{L^{p,\infty}(0,T; L ^{q,\infty}(\mathbb{R}^{3}))}$ 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.

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