Since the velocities of solutions to the 2D Euler equations with bounded vorticities are typically not close to Lipschitz near boundary singularities, the question of uniqueness of such solutions on possibly singular domains is a challenging one and has only been answered for some special non-smooth domains. In this talk we will present a general sufficient condition on the geometry of the domain that guarantees global uniqueness for all weak solutions 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 is showing that constancy of the vorticity near the boundary is preserved for all time because the Lagrangian trajectories of general solutions on these domains cannot reach the boundary in finite time. We also show that our condition is essentially sharp in this sense, demonstrating that trajectories on domains that come arbitrarily close to satisfying it can reach the boundary in finite time, and find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary when it is satisfied.