Uniqueness for the Two-Dimensional Euler Equations on Domains with Corners

We prove uniqueness of the solution of the Euler equations with bounded vorticity for bounded simply connected planar domains with corners forming acute angles. Our strategy consists in mapping such domains on the unit disk via a biholomorphism. We then establish log-Lipschitz regularity for the resulting push-forward of the velocity field, which leads to uniqueness thanks to a Gronwall estimate involving the Lagrangian trajectories on the unit disk.