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Nonexistence of global solutions for a class of viscoelastic wave equations
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 |
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.
References:
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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.
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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. |
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. 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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