Remarks on Interior Regularity Criterion for an Axially Symmetric Suitable Weak Solution to the Navier Stokes Equations

TL;DR

This paper establishes that for axially symmetric suitable weak solutions to the Navier-Stokes equations, higher regularity of the radial component ensures the regularity of all velocity components, advancing understanding of solution regularity criteria.

Contribution

It introduces a new interior regularity criterion based on the radial component's weighted Serrin-Prodi condition for axially symmetric solutions.

Findings

Radial component regularity implies full velocity regularity.

Weighted Serrin-Prodi condition suffices for regularity.

Enhances criteria for solution regularity in Navier-Stokes equations.

Abstract

We show that if v is an axially symmetric suitable weak solution to the Navier Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg) such that the radial component of v has a higher regularity (i.e. satisfies weighted Serrin-Prodi type condition), then all components of v are regular.