Uniqueness of the 2D Euler equation on rough domains

TL;DR

This paper establishes new sufficient conditions on the domain's regularity to ensure the uniqueness of weak solutions to the 2D Euler equations with bounded initial vorticity, extending previous results to less regular domains.

Contribution

It introduces a more general domain regularity condition for uniqueness and develops a novel change of variable and energy functional to handle lower boundary regularity.

Findings

Uniqueness holds for domains with boundary in $H^{3/2}$ under certain conditions.

Proves uniqueness for $C^{1,eta}$ domains with $eta > 1/2$ and convex $C^{1,eta}$ domains.

Overcomes the barrier of non-log-Lipschitz velocity near less regular boundaries.

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 all 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.