Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity

TL;DR

This paper classifies and analyzes homogeneous axisymmetric solutions to the stationary Navier-Stokes equations on the sphere with isolated singularities, revealing the structure of solution spaces and their pressure profiles.

Contribution

It provides a complete classification of (-1)-homogeneous axisymmetric solutions with singularities on the sphere and describes the existence of solution curves with nonzero swirl.

Findings

Existence of a curve of solutions with nonzero swirl emanating from interior points.

No such solution curve exists on certain boundary parts.

Asymptotic expansions near the singularity are established.

Abstract

We classify all $(-1)-$homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of $(-1)-$homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of solutions for any point on the other part of the boundary. We also establish asymptotic expansions for every (-1)-homogeneous axisymmetric solutions in a neighborhood of the singular point on the unit sphere.