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

Zaihong Jiang; Li Li; Wenbo Lu

doi:10.3934/dcdss.2021126

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2021, Volume 14, Issue 12: 4231-4258. Doi: 10.3934/dcdss.2021126 open access

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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
^* Corresponding author: Li Li

Received: August 31, 2021

Revised: September 30, 2021

Early access: October 2021

Published: December 2021

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

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Abstract

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.

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Mathematics Subject Classification: 35Q30, 35Q35, 76D03, 76D05.

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Figure 1. Graph of $ \alpha $ by numerical computation

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Figure 2. The graph of $ \bar{K}(x_0) $

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Figure 3. The existence graph in $ (\nu^2, P) $ plane

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