# Rotating Equilibria of Vortex Sheets

## TL;DR

This paper develops a complex analysis-based method to construct rotating equilibrium solutions of the 2D Euler equations with vortex sheets, generalizing known solutions and exploring limits as the number of sheets increases.

## Contribution

It introduces a novel approach using Riemann-Hilbert problems to find and analyze rotating vortex sheet equilibria, including configurations with multiple sheets and their limiting behavior.

## Key findings

- Constructed a family of rotating vortex sheet equilibria with multiple sheets.
- Derived explicit integral formulas for circulation densities in these configurations.
- Showed convergence of multi-sheet equilibria to a hollow vortex as the number of sheets increases.

## Abstract

We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.