Global regular axially-symmetric solutions to the Navier-Stokes equations with small swirl

TL;DR

This paper proves the global regularity of axially symmetric solutions to the Navier-Stokes equations with small initial swirl in a bounded cylinder, using new weighted Sobolev space estimates.

Contribution

It introduces a novel estimate for the stream function in weighted Sobolev spaces to establish regularity under small swirl conditions.

Findings

Global regularity for small initial swirl

New weighted Sobolev space estimate for stream function

Axially symmetric solutions remain smooth over time

Abstract

Axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and he angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.