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 $\hat{I}\text{c◯}$, 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.

31 pages, 4 figures