Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Abstract

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity Ω , 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.