Prodi--Serrin condition for 3D Navier--Stokes equations via one directional derivative of velocity

TL;DR

This paper establishes a regularity criterion for 3D Navier-Stokes solutions based on a single directional derivative of velocity in certain Lebesgue spaces, extending classical Prodi--Serrin conditions.

Contribution

It introduces a new regularity criterion involving one directional derivative of velocity, broadening the understanding of conditions ensuring solution smoothness.

Findings

Weak solutions become regular if one directional derivative satisfies specific integrability conditions.

The proof utilizes new local energy estimates from recent literature.

The criterion applies under the condition $rac{2}{p_{0}}+rac{3}{q_{0}}=2$ with $q_0>3/2$.

Abstract

In this paper, we consider the conditional regularity of weak solution to the 3D Navier--Stokes equations. More precisely, we prove that if one directional derivative of velocity, say $\partial_3 u,$ satisfies $\partial_3 u \in L^{p_0,1}(0,T; L^{q_0}(\mathbb{R}^3))$ with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=2$ and $\frac{3}{2}<q_0< +\infty,$ then the weak solution is regular on $(0,T].$ The proof is based on the new local energy estimates introduced by Chae-Wolf (arXiv:1911.02699) and Wang-Wu-Zhang (arXiv:2005.11906).