# American Institute of Mathematical Sciences

December  2019, 39(12): 7163-7211. doi: 10.3934/dcds.2019300

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

 1 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA 3 School of Mathematics, Georgia Institute of Technology, 686 Cherry St NW, Atlanta, GA 30313, USA

* Corresponding author: Xukai Yan

Dedicated to Luis Caffarelli on his 70th birthday, with admiration and friendship

Received  January 2019 Revised  July 2019 Published  September 2019

Fund Project: The first named author is partially supported by NSFC grants 11871177. The second named author is partially supported by NSF grants DMS-1501004. The third named author is partially supported by AMS-Simons Travel Grant and AWM-NSF Travel Grant 1642548.

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.

Citation: Li Li, Yanyan Li, Xukai Yan. Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7163-7211. doi: 10.3934/dcds.2019300
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