Remarks on stationary and uniformly-rotating vortex sheets: Rigidity results

TL;DR

This paper proves that the only finite-length, smooth vortex sheet solutions with positive vorticity, either stationary or uniformly rotating with negative angular velocity, are trivial, consisting of concentric circles with constant vorticity.

Contribution

It establishes a rigidity result for vortex sheets, showing that under certain conditions, solutions must be simple and symmetric, extending understanding of vortex sheet configurations.

Findings

Only trivial solutions are possible under specified conditions.

Solutions are concentric circles with constant vorticity.

The proof uses desingularization and calculus of variations.

Abstract

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $\Omega$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.