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

Li Li; Yanyan Li; Xukai Yan

doi:10.3934/dcds.2019300

Article Contents

2019, Volume 39, Issue 12: 7163-7211. Doi: 10.3934/dcds.2019300

This issue Previous Article Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions Next Article Free boundaries subject to topological constraints

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
^* 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

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76D05.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	L. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.), 43 (1944), 286-288.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	N. A. Slezkin, On an exact solution of the equations of viscous flow, Uch. zap. MGU, 2 (1934), 89-90.
[11]	H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329. doi: 10.1093/qjmam/4.3.321.
[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.
[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.
[14]	C. Y. Wang, Exact solutions of the steady state Navier-Stokes equation, Annu. Rev. Fluid Mech., 23 (1991), 159-177.
[15]	V. I. Yatseyev, On a class of exact solutions of the equations of motion of a viscous fluid, 1950.