Uniqueness for Two-Dimensional Incompressible Ideal Flow on Singular Domains

Christophe Lacave (1)

1 IMJ - Institut de Mathématiques de Jussieu

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.

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Lacave. Uniqueness for Two-Dimensional Incompressible Ideal Flow on Singular Domains. SIAM Journal on Mathematical Analysis, 2015, 47 (2), pp.50. ⟨10.1137/140972238⟩. ⟨hal-00620055⟩