Steady ring-shaped vortex sheets

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 interior 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.