Local regularity criteria in terms of one velocity component for the Navier-Stokes equations

TL;DR

This paper introduces new interior regularity criteria for 3D Navier-Stokes solutions based on the behavior of a single velocity component, linking local regularity to scaled norm conditions.

Contribution

It provides novel criteria that determine regularity near a point using only one velocity component's scaled norm, advancing understanding of partial regularity in Navier-Stokes equations.

Findings

Regularity near a point is guaranteed if scaled norms of certain velocity quantities are finite.

Smallness of the scaled norm of one velocity component ensures local regularity.

New criteria connect single component behavior to overall solution regularity.

Abstract

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier-Stokes equations in three dimensions. It is shown that the velocity is regular near a point $z$ if its scaled $L^p_tL^q_x$-norm of some quantities related to the velocity field is finite and the scaled $L^p_tL^q_x$-norm of one velocity component is sufficiently small near $z$.