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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ x_3 $\end{document}</tex-math></inline-formula>-axis, and used it to model the dynamics of tornado. We extend Serrin's work to general cone regions minus <inline-formula><tex-math id="M2">\begin{document}$ x_3 $\end{document}</tex-math></inline-formula>-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.</p>