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.
Highlights
A vortex sheet is a weak solution of the 2D Euler equations: vt + v · ∇v = −∇ p, ∇ · v = 0, (1.1)
One may think of a solution to (1.1) with one incompressible fluid
The proof is inspired by our recent rigidity result in the paper [14] on stationary and rotating solutions of the 2D Euler equations both in the smooth and vortex patch settings
Summary
In the paper [15] we prove the existence of a family of vortex sheet rotating solutions with non-constant vorticity density supported on a non-radial curve, bifurcating from the circle with constant density. The proof is inspired by our recent rigidity result in the paper [14] on stationary and rotating solutions of the 2D Euler equations both in the smooth and vortex patch settings. We constructed an appropriate functional and showed, on one hand, that any stationary solution had to be a critical point, and on the other, for any curve which is not a circle there existed a vector field along which the first variation was non-zero This vector field is defined in terms of an elliptic equation in the interior of the patch.
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