  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.
