  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.
