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Article Contents

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

• * Corresponding author: Li Li
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.

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

 Citation:

• 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

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