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

  • * Corresponding author: Li Li

    * 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
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  • 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:

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  • 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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