Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a two-dimensional bounded domain, for flows with bounded vorticity, for which Yudovich proved global existence and uniqueness of the solution. We prove in particular that if the boundary of the domain is $C^\infty$ (respectively Gevrey of order $M$) then the trajectories of the fluid particles are $C^\infty$ (resp. Gevrey of order $M+2$).