Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner

TL;DR

This paper proves the uniqueness of weak solutions to the 2D Euler equation in a corner domain with non-constant vorticity near the boundary, even when the velocity is not Lipschitz.

Contribution

It establishes the first uniqueness result for 2D Euler solutions with nontrivial boundary vorticity and non-Lipschitz velocity in corner domains.

Findings

Uniqueness of weak solutions under specified conditions.

Applicable to domains with corner angles between 1/2π and π.

Handles non-constant vorticity supported on one side of the boundary.

Abstract

We consider the 2D incompressible Euler equation on a corner domain $\Omega$ with angle $\nu\pi$ with $\frac{1}{2}<\nu<1$. We prove that if the initial vorticity $\omega_0 \in L^{1}(\Omega)\cap L^{\infty}(\Omega)$ and if $\omega_0$ is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary.