  We investigate under which assumptions the flow associated to autonomous
planar vector fields inherits the Sobolev or BV regularity of the vector field.
We consider nearly incompressible and divergence-free vector fields, taking
advantage in both cases of the underlying Hamiltonian structure. Finally we
provide an example of an autonomous planar Sobolev divergence-free vector
field, such that the corresponding regular Lagrangian flow has no bounded
variation.
