Uniqueness for two dimensional incompressible ideal flow on singular domains

TL;DR

This paper proves the uniqueness of solutions to the 2D incompressible Euler equations in singular domains with corners, extending understanding of flow behavior near domain singularities despite velocity blow-up.

Contribution

It establishes the uniqueness of solutions in domains with corners, a significant step beyond existence results, and clarifies flow properties near singularities.

Findings

Uniqueness of solutions in domains with corners.

Flow behaves as a vortex sheet near singularities.

Velocity blows up near corners, yet solutions are unique.

Abstract

The existence of a solution to the two dimensional incompressible Euler equations in singular domains was established in [G\'erard-Varet and Lacave, The 2D Euler equation on singular domains, submitted]. The present work is about the uniqueness of such a solution when the domain is the exterior or the interior of a simply connected set with corners, although the velocity blows up near these corners. In the exterior of a curve with two end-points, it is showed in [Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl \textbf{26} (2009), 1121-1148] that this solution has some interesting properties, as to be seen as a special vortex sheet. Therefore, we prove the uniqueness, whereas the problem of general vortex sheets is open.