Inserted: 19 jan 2021
Last Updated: 19 jan 2021
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.