Remarks on Regularity Criteria for Axially Symmetric Weak Solutions to the Navier-Stokes Equations, II

TL;DR

This paper investigates conditions under which axially symmetric weak solutions to the Navier-Stokes equations remain regular, focusing on weighted Serrin conditions for velocity components.

Contribution

It establishes new regularity criteria involving the radial and angular velocity components under axial symmetry assumptions.

Findings

Radial velocity's positive part satisfying weighted Serrin condition ensures regularity.

Additional conditions on angular velocity component contribute to regularity.

Provides criteria that extend previous regularity results for axially symmetric solutions.

Abstract

We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin condition and in addition angular component satisfies some condition, then the solution is regular.