Uniqueness for the 2-D Euler equations on domains with corners

Abstract

For a large class of non smooth bounded domains, existence of a global weak solution of the 2D Euler equations, with bounded vorticity, was established by G\'erard-Varet and Lacave. In the case of sharp domains, the question of uniqueness for such weak solutions is more involved due to the bad behavior of $\Delta^{-1}$ close to the boundary. In the present work, we show uniqueness for any bounded and simply connected domain with a finite number of corners of angles smaller than $\pi/2$. Our strategy relies on a log-Lipschitz type regularity for the velocity field.