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

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{partialpsi^varepsilon}{partial r}Big)-frac{1}{r^2}frac{partial^2psi^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.