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

TL;DR

This paper establishes new regularity criteria for the 3D Navier-Stokes equations based on pressure and velocity gradient in Lorentz spaces, extending previous conditions and affirmatively answering a question by Suzuki.

Contribution

It introduces novel regularity criteria involving pressure and gradient of velocity in Lorentz spaces, broadening the scope of known conditions for solution regularity.

Findings

Regularity is guaranteed if pressure or its gradient in Lorentz spaces is sufficiently small.

The criteria generalize previous conditions by allowing the time variable to be in Lorentz spaces.

Answers a question posed by Suzuki regarding regularity conditions in Lorentz spaces.

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 $\|\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.