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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

Li Li 1, , Yanyan Li 2, and  Xukai Yan 3,,

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.

Keywords: Homogeneous solutions, stationary Navier-Stokes equations, isolated singularities, existence, uniqueness.
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76D05.
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 and Continuous Dynamical Systems, 2019, 39 (12) : 7163-7211. doi: 10.3934/dcds.2019300
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.

show all references

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.

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