# Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions

## TL;DR

This paper classifies all smooth, axisymmetric, no-swirl solutions to the stationary Navier-Stokes equations with a specific homogeneity, parameterizing them in a four-dimensional space and analyzing their smoothness properties.

## Contribution

It provides a complete classification of (-1)-homogeneous axisymmetric no-swirl solutions on the sphere minus poles, forming a foundation for studying solutions with non-zero swirl.

## Key findings

- Classified solutions form a four-dimensional parameter space.
- Established smoothness properties of the solution surface.
- Set the stage for analyzing solutions with non-zero swirl.

## 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 and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^2\setminus\{S,N\}$, emanating from the four dimensional solution surface.