A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component

TL;DR

This paper establishes a new regularity criterion for the 3D Navier-Stokes equations that depends solely on one velocity component, providing insights into the conditions under which solutions remain smooth.

Contribution

It introduces a Serrin-type regularity criterion based on a single velocity component, advancing understanding of partial regularity conditions for Navier-Stokes solutions.

Findings

Proves regularity criteria involving only one velocity component.

Provides conditions ensuring solution smoothness in 3D Navier-Stokes.

Enhances understanding of component-wise regularity in fluid dynamics.

Abstract

We study the Cauchy problem for the 3D Navier-Stokes equations, and prove some scalaring-invariant regularity criteria involving only one velocity component.