Uniqueness for two dimensional incompressible ideal flow on singular domains

Details Uniqueness for two dimensional incompressible ideal flow on singular domains Uniqueness for two dimensional incompressible ideal flow on singular domains

2011-09-06

arxiv.org/abs/1109.1153v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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