Euler Equations on General Planar Domains

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. While similar existing results require domains that are \(C^{1,1}\) except at finitely many convex corners, our condition involves much less domain smoothness, being only slightly more restrictive than the exclusion of corners with angles greater than \(\pi \). In particular, it 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.