Uniqueness and stability of steady vortex rings

Abstract

In this paper, we are concerned with the uniqueness and nonlinear stability of vortex rings for the 3D Euler equation. We prove the uniqueness of a special family of vortex rings with a small cross-section, which are global classical solutions to the 3D Euler equation. We also establish the nonlinear stability of these vortex rings based on a combination of the uniqueness and a general stability criteria.