December  2021, 14(12): 4231-4258. doi: 10.3934/dcdss.2021126

Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

2. 

School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

3. 

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

* Corresponding author: Li Li

Received  September 2021 Revised  October 2021 Published  December 2021 Early access  October 2021

Fund Project: The first author is partially supported by NSFC Grant No.12071439 and ZJNSF Grant No.LY19A010016. The second author is partially supported by NSFC Grant No. 11871177

In this paper, we study axisymmetric homogeneous solutions of the Navier-Stokes equations in cone regions. In [James Serrin. The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325-360, 1972.], Serrin studied the boundary value problem in half-space minus $ x_3 $-axis, and used it to model the dynamics of tornado. We extend Serrin's work to general cone regions minus $ x_3 $-axis. All axisymmetric homogeneous solutions of the boundary value problem have three possible patterns, which can be classified by two parameters. Some existence results are obtained as well.

Citation: Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126
References:
[1]

M. A. Gol'dshtik, A paradoxical solution of the Navier-Stokes equations, 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. LiY. 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. LiY. 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, J. Differential Equations, 264 (2018), 6082-6108.  doi: 10.1016/j.jde.2018.01.028.

[5]

L. LiY. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities, Discrete Contin. Dyn. Syst., 39 (2019), 7163-7211.  doi: 10.3934/dcds.2019300.

[6]

Y. Li and X. Yan, Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations, J. Differential Equations, 297 (2021), 226-245.  doi: 10.1016/j.jde.2021.06.033.

[7]

R. Paull and A. F. Pillow, Conically similar viscous flows. I. Basic conservation principles and characterization of axial causes in swirl-free flow, J. Fluid Mech., 155 (1985), 327-341.  doi: 10.1017/S0022112085001835.

[8]

R. Paull and A. F. Pillow, Conically similar viscous flows. II. One-parameter swirl-free flows, J. Fluid Mech., 155 (1985), 343-358.  doi: 10.1017/S0022112085001847.

[9]

R. Paull and A. F. Pillow, Conically similar viscous flows. III. Characterization of axial causes in swirling flow and the one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum, J. Fluid Mech., 155 (1985), 359-379.  doi: 10.1017/S0022112085001859.

[10]

J. Serrin, The swirling vortex, Trans. Roy. Soy. London Ser. A, 271 (1972), 325-360.  doi: 10.1098/rsta.1972.0013.

[11]

N. Slezkin, On an exact solution of the equations of viscous flow, Uch. Zap. MGU, 2 (1934), 89-90. 

[12]

H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329.  doi: 10.1093/qjmam/4.3.321.

[13]

V. Šverák, On Landau's solutions of the Navier-Stokes equations. Problems in mathematical analysis, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.

[14]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.  doi: 10.12775/TMNA.1998.008.

[15]

C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, in Annual Review of Fluid Mechanics, Vol. 23, Annual Reviews, Palo Alto, CA, 1991,159–177. doi: 10.1146/annurev.fl.23.010191.001111.

[16]

V. I. Yatseyev, On a class of exact solutions of the equations of motion of a viscous fluid, NACA Tech. Memo., 1953 (1953), 7pp.

show all references

References:
[1]

M. A. Gol'dshtik, A paradoxical solution of the Navier-Stokes equations, 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. LiY. 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. LiY. 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, J. Differential Equations, 264 (2018), 6082-6108.  doi: 10.1016/j.jde.2018.01.028.

[5]

L. LiY. Li and X. Yan, Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities, Discrete Contin. Dyn. Syst., 39 (2019), 7163-7211.  doi: 10.3934/dcds.2019300.

[6]

Y. Li and X. Yan, Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations, J. Differential Equations, 297 (2021), 226-245.  doi: 10.1016/j.jde.2021.06.033.

[7]

R. Paull and A. F. Pillow, Conically similar viscous flows. I. Basic conservation principles and characterization of axial causes in swirl-free flow, J. Fluid Mech., 155 (1985), 327-341.  doi: 10.1017/S0022112085001835.

[8]

R. Paull and A. F. Pillow, Conically similar viscous flows. II. One-parameter swirl-free flows, J. Fluid Mech., 155 (1985), 343-358.  doi: 10.1017/S0022112085001847.

[9]

R. Paull and A. F. Pillow, Conically similar viscous flows. III. Characterization of axial causes in swirling flow and the one-parameter flow generated by a uniform half-line source of kinematic swirl angular momentum, J. Fluid Mech., 155 (1985), 359-379.  doi: 10.1017/S0022112085001859.

[10]

J. Serrin, The swirling vortex, Trans. Roy. Soy. London Ser. A, 271 (1972), 325-360.  doi: 10.1098/rsta.1972.0013.

[11]

N. Slezkin, On an exact solution of the equations of viscous flow, Uch. Zap. MGU, 2 (1934), 89-90. 

[12]

H. B. Squire, The round laminar jet, Quart. J. Mech. Appl. Math., 4 (1951), 321-329.  doi: 10.1093/qjmam/4.3.321.

[13]

V. Šverák, On Landau's solutions of the Navier-Stokes equations. Problems in mathematical analysis, J. Math. Sci. (N.Y.), 179 (2011), 208-228.  doi: 10.1007/s10958-011-0590-5.

[14]

G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.  doi: 10.12775/TMNA.1998.008.

[15]

C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, in Annual Review of Fluid Mechanics, Vol. 23, Annual Reviews, Palo Alto, CA, 1991,159–177. doi: 10.1146/annurev.fl.23.010191.001111.

[16]

V. I. Yatseyev, On a class of exact solutions of the equations of motion of a viscous fluid, NACA Tech. Memo., 1953 (1953), 7pp.

Figure 1.  Graph of $ \alpha $ by numerical computation
Figure 2.  The graph of $ \bar{K}(x_0) $
Figure 3.  The existence graph in $ (\nu^2, P) $ plane
[1]

Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219

[2]

Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325

[3]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113

[4]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355

[5]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081

[6]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[7]

Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121

[8]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151

[9]

Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199

[10]

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

[11]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427

[12]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148

[13]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845

[14]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[15]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[16]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080

[17]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277

[18]

Kehan Shi, Ying Wen. Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022089

[19]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141

[20]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (136)
  • HTML views (119)
  • Cited by (0)

Other articles
by authors

[Back to Top]