Uniqueness of Positive Vorticity Solutions to the 2D Euler Equations on Singular Domains

TL;DR

This paper proves that positive vorticity solutions to the 2D Euler equations in complex bounded domains cannot reach the boundary in finite time, ensuring their Lagrangian nature and uniqueness under certain initial conditions.

Contribution

It establishes the boundary behavior and uniqueness of positive vorticity solutions in irregular domains with corners, extending previous results to more general geometries.

Findings

Particle trajectories do not reach the boundary in finite time.

Positive vorticity solutions are Lagrangian in these domains.

Uniqueness holds if initial vorticity is constant near the boundary.

Abstract

We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$ boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.