On weighted estimates for the stream function of axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder

TL;DR

This paper develops weighted Sobolev space estimates for the stream function in axially symmetric solutions of the Navier-Stokes equations within a bounded cylinder, aiding in the analysis of solution existence.

Contribution

It introduces higher-order weighted Sobolev estimates using Kondratiev's technique for axially symmetric solutions to Navier-Stokes equations in bounded cylinders.

Findings

Establishment of weighted Sobolev estimates for the stream function.

Application of Kondratiev's technique to derive these estimates.

Foundation for proving existence of axially symmetric solutions.

Abstract

Higher-order estimates in weighted Sobolev spaces for solutions to a singular elliptic equation for the stream function in an axially symmetric cylinder are provided. These estimates are essential for investigating the existence of axially symmetric solutions to incompressible Navier-Stokes equations in axially symmetric cylinders. To derive the estimates the technique of Kondratiev is incorporated. The weight has a form of a power function of the distance to the axis of symmetry.