Critical regularity criteria for Navier-Stokes equations in terms of one directional derivative of the velocity

TL;DR

This paper establishes new regularity criteria for the 3D Navier-Stokes equations based on one directional derivative of the velocity, extending known results and including axisymmetric solutions.

Contribution

It introduces novel inequalities and a priori estimates that expand the conditions under which solutions remain regular, especially for axisymmetric flows.

Findings

Regularity criteria in terms of one directional derivative of velocity.

Extension of the range of q for axisymmetric solutions.

Solutions are regular if the directional derivative satisfies certain integrability conditions.

Abstract

In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{u} \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q\leq 6$. Moreover, we extend the range of $q$ while the solution is axisymmetric, i.e. the axisymmetric solution $\mathbf{m}{u}$ is regular in $(0,T]$, if $ \partial_3 u^3 \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q< \infty$.