Uniqueness for two dimensional incompressible ideal flow on singular domains

Abstract

The existence of a solution to the two dimensional incompressible Euler equations in singular domains was established in [Gérard-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.

References in corpus (1)

• The 2D Euler equation on singular domains

Cited by in corpus (3)

• The Two Dimensional Euler Equations on Singular Exterior Domains
• The Euler Equations in planar nonsmooth convex domains
• Topography influence on the Lake equations in bounded domains