Uniqueness for Two-Dimensional Incompressible Ideal Flow on Singular Domains

We prove uniqueness of the weak solution of the Euler equations for compactly supported, single signed and bounded initial vorticity in simply connected planar domains with corners forming angles larger than π/2. In this type of domain, the velocity is not log-lipschitz and does not belong to W 1,p for all p, which is the standard regularity for the Yudovich’s arguments. Thanks to the explicit formula of the Biot-Savart law via a biholomorphism, we construct a Lyapunov function to prove that the vorticity never reach the boundary which is the place where the velocity is not regular. Next we adapt the proof of Yudovich although the velocity may blow up near corners. We also obtain a uniqueness result for exterior domains with large corners.