Steady Ring-Shaped Vortex Sheets

Abstract: In this work, we construct traveling wave solutions to the two-phase Euler equations, featuring a vortex sheet at the interface between the two phases. The inner phase exhibits a uniform vorticity distribution and may represent a vacuum, forming what is known as a hollow vortex. These traveling waves take the form of ring-shaped vortices with a small cross-sectional radius, referred to as thin rings. Our construction is based on the implicit function theorem, which also guarantees local uniqueness of the solutions. Additionally, we derive asymptotics for the speed of the ring, generalizing the well-known Kelvin--Hicks formula to cases that include surface tension.