# 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, 2019, 39 (12) : 7163-7211. doi: 10.3934/dcds.2019300
##### References:
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##### References:
 [1] M. A. Goldshtik, A paradoxical solution of the Navier-Stokes equations, Prikl. Mat. Mekh., 24 (1960), 610-621. Transl., J. Appl. Math. Mech., 24 (1960), 913-929. doi: 10.1016/0021-8928(60)90070-8.  Google Scholar [2] L. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.   Google Scholar [3] L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅰ. One singularity, Arch. Ration. Mech. Anal., 227 (2018), 1091-1163.  doi: 10.1007/s00205-017-1181-5.  Google Scholar [4] L. Li, Y. Y. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅱ. Classification of axisymmetric no-swirl solutions, Journal of Differential Equations, 264 (2018), 6082-6108.  doi: 10.1016/j.jde.2018.01.028.  Google Scholar [5] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, vol. 6., New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.  Google Scholar [6] A. F. Pillow and R. Paull, Conically similar viscous flows. Part 1. Basic conservation principles and characterization of axial causes in swirl-free flow, Journal of Fluid Mechanics, 155 (1985), 327-341.  doi: 10.1017/S0022112085001835.  Google Scholar [7] A. F. Pillow and R. Paull, Conically similar viscous flows. Part 2. One-parameter swirl-free flows, Journal of Fluid Mechanics, 155 (1985), 343-358.  doi: 10.1017/S0022112085001847.  Google Scholar [8] A. F. Pillow and R. Paull, Conically similar viscous flows. Part 3. Characterization of axial causes in swirling flow and the one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum, Journal of Fluid Mechanics, 155 (1985), 359-379.  doi: 10.1017/S0022112085001859.  Google Scholar [9] J. Serrin, The swirling vortex, Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci., 271 (1972), 325-360.  doi: 10.1098/rsta.1972.0013.  Google Scholar [10] N. A. Slezkin, On an exact solution of the equations of viscous flow, Uch. zap. MGU, 2 (1934), 89-90.   Google Scholar [11] H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329.  doi: 10.1093/qjmam/4.3.321.  Google Scholar [12] V. Šverák, On Landau's solutions of the Navier-Stokes equations, Problems in Mathematical Analysis, No. 61. J. Math. Sci., 179 (2011), 208–228. arXiv: math/0604550. doi: 10.1007/s10958-011-0590-5.  Google Scholar [13] G. Tian and Z. P. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.  doi: 10.12775/TMNA.1998.008.  Google Scholar [14] C. Y. Wang, Exact solutions of the steady state Navier-Stokes equation, Annu. Rev. Fluid Mech., 23 (1991), 159-177.   Google Scholar [15] V. I. Yatseyev, On a class of exact solutions of the equations of motion of a viscous fluid, 1950. Google Scholar
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