# 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 & Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126
##### References:

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##### References:
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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