# American Institute of Mathematical Sciences

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. Li, 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. 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. Li, Y. 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.

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##### 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. Li, 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. 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. Li, Y. 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.
Graph of $\alpha$ by numerical computation
The graph of $\bar{K}(x_0)$
The existence graph in $(\nu^2, P)$ plane
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