Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations

TL;DR

This paper establishes a regularity criterion for Navier-Stokes solutions based on a single velocity component satisfying a scaling invariant condition, advancing understanding of partial regularity in fluid dynamics.

Contribution

It introduces a new local regularity criterion for suitable weak solutions based on one velocity component, extending the Serrin condition.

Findings

Regularity of Leray weak solutions under one component condition

Extension of Serrin criterion to a single velocity component

New local regularity criterion for Navier-Stokes equations

Abstract

In this paper, we prove that the Leray weak solution $u : \mathbb{R}^3\times (0, T)\rightarrow\mathbb{R}^3 $ of the Navier-Stokes equations is regular in $\mathbb{R}^3\times (0,T)$ under the scaling invariant Serrin condition imposed on one component of the velocity $u_3\in L^{q,1}(0, T;L^p(\mathbb{R}^3))$ with \[ \frac{2}{q}+\frac{3}{p}\leq 1,\quad 3<p<+\infty. \] This result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.