  We consider the incompressible Euler equations in a (possibly multiply
connected) bounded domain of R^2, for flows with bounded vorticity, for which
Yudovich proved, in 1963, global existence and uniqueness of the solution. We
prove that if the boundary of the domain is C^infty (respectively Gevrey of
order M > 1) then the trajectories of the fluid particles are C^infty (resp.
Gevrey of order M + 2). Our results also cover the case of "slightly unbounded"
vorticities for which Yudovich extended his analysis in 1995. Moreover if in
addition the initial vorticity is Holder continuous on a part of the domain
then this Holder regularity propagates smoothly along the flow lines. Finally
we observe that if the vorticity is constant in a neighborhood of the boundary,
the smoothness of the boundary is not necessary for these results to hold.
