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arXiv · 2009.13210

On the steady axisymmetric vortex rings for 3-D incompressible Euler flows

Daomin Cao·Weicheng Zhan
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Abstract

In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in $\mathbb{R}^3$. The construction is based on a study of solutions to the similinear elliptic problem \begin{equation*} -\frac{1}{r}\frac{\partial}{\partial r}\Big(\frac{1}{r}\frac{\partialψ^\varepsilon}{\partial r}\Big)-\frac{1}{r^2}\frac{\partial^2ψ^\varepsilon}{\partial z^2}=\frac{1}{\varepsilon^2}\left(g(ψ^\varepsilon)+\frac{f(ψ^\varepsilon)}{r^2}\right), \end{equation*} where $f$ and $g$ are two given functions of the Stokes stream function $ψ^\varepsilon$, and $\varepsilon>0$ is a small parameter.

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Daomin Cao, Weicheng Zhan. 2020-11-28. On the steady axisymmetric vortex rings for 3-D incompressible Euler flows. https://arxiv.org/abs/2009.13210

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