An anisotropic partial regularity criterion for the Navier-Stokes equations

TL;DR

This paper establishes an anisotropic partial regularity criterion for the Navier-Stokes equations, showing that smallness of a single velocity component ensures regularity at a point, advancing understanding of solution behavior.

Contribution

It introduces a novel regularity criterion based solely on one velocity component, providing a new anisotropic perspective in Navier-Stokes regularity theory.

Findings

Smallness of a single velocity component implies regularity.

The criterion is scale-invariant and localized.

It advances partial regularity understanding for Navier-Stokes solutions.

Abstract

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.