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

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{\partial\psi^\varepsilon}{\partial r}\Big)-\frac{1}{r^2}\frac{\partial^2\psi^\varepsilon}{\partial z^2}=\frac{1}{\varepsilon^2}\left(g(\psi^\varepsilon)+\frac{f(\psi^\varepsilon)}{r^2}\right), \end{equation*} where $f$ and $g$ are two given functions of the Stokes stream function $\psi^\varepsilon$, and $\varepsilon>0$ is a small parameter.