# Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities

## TL;DR

This paper investigates the existence, uniqueness, and non-existence of axisymmetric stationary Navier-Stokes solutions with two isolated singularities on the sphere, extending previous classifications to solutions with nonzero swirl.

## Contribution

It extends prior work by analyzing solutions with nonzero swirl near the no-swirl solution surface, providing new existence and uniqueness results.

## Key findings

- Established existence of solutions with nonzero swirl near the no-swirl surface.
- Proved non-existence of certain solutions under specified conditions.
- Demonstrated uniqueness of solutions in a specific setting.

## Abstract

All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.