Global regular axially symmetric solutions to the Navier-Stokes equations

TL;DR

This paper proves the existence of global regular axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder, primarily by establishing the Holder continuity of the swirl to extend local solutions globally.

Contribution

It introduces a novel proof of Holder continuity of the swirl, enabling the extension of local solutions to global regular solutions for the Navier-Stokes equations in cylindrical domains.

Findings

Existence of global regular solutions in bounded cylinders

Holder continuity of the swirl established

Global a priori estimates derived

Abstract

The existence of global regular axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder is proved. The main step in the proof is a proof of the Holder continuity of the swirl. This gives a possibility to prove global a priori estimate for regular solutions. Then a local solution can be extended step by step in time.