Uniqueness of the 2D Euler equation on rough domains

Abstract

We consider the 2D incompressible Euler equation on a bounded simply connected domain $\Omega$. We give sufficient conditions on the domain $\Omega$ so that for any initial vorticity ${\omega }_0\in L^{\infty }(\Omega )$, the weak solutions are unique. Our sufficient condition is slightly more general than the condition that $\Omega$ is a $C^{1,\alpha }$ domain for some $\alpha >0$, with its boundary belonging to $H^{3/2}({\mathbb{S}}^1)$. As a corollary, we prove uniqueness for $C^{1,\alpha }$ domains for $\alpha >1/2$ and for convex domains which are also $C^{1,\alpha }$ domains for some $\alpha >0$. Previously, uniqueness for general initial vorticity in $L^{\infty }(\Omega )$ was only known for $C^{1,1}$ domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below the $C^{1,1}$ regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional.