On the Serrin-type condition on one velocity component for the Navier-Stokes equations

TL;DR

This paper proves that a Serrin-type condition on just one velocity component ensures regularity of solutions to the 3D Navier-Stokes equations, advancing understanding of partial regularity criteria.

Contribution

It establishes a new regularity criterion based on one velocity component under Serrin-type conditions, extending previous partial regularity results.

Findings

Regularity of weak solutions under one-component Serrin condition

New local regularity criterion for suitable weak solutions

Conditions on one velocity component imply full solution regularity

Abstract

In this paper we consider the regularity problem of the Navier-Stokes equations in $ \R^{3} $. We show that the Serrin-type condition imposed on one component of the velocity $ u_3\in L^p(0,T; L^q(\R^{3} ))$ satisfying $ \frac{2}{p}+ \frac{3}{q} <1$, $ 3<q \le +\infty$ implies the regularity of the weak Leray solution $ u: \R^{3} \times (0,T) \rightarrow \R^{3} $ with the initial data belonging to $ L^2(\Bbb R^3) \cap L^3(\R^{3})$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.