Euler Equations on General Planar Domains

TL;DR

This paper establishes a geometric condition on planar domains that ensures the uniqueness and boundary behavior of weak solutions to the Euler equations, extending understanding to domains with singularities and providing sharp bounds on particle trajectories.

Contribution

It introduces a nearly optimal geometric condition for uniqueness of Euler solutions on singular domains, showing boundary vorticity constancy preservation and trajectory bounds.

Findings

Uniqueness holds for domains satisfying the geometric condition.

Boundary vorticity remains constant over time under the condition.

Sharp bounds on particle trajectories approaching the boundary are derived.

Abstract

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $\pi$ and, in particular, is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.