Mathematics

We consider bounded autonomous vector fields $b:\mathbb{R}^2 \to \mathbb{R}^2$ and the associated regular Lagrangian flows, which are a suitable extension of the classical flows to non smooth vector fields.Under the additional assumptions that the vector field is divergence-free and with bounded variation, we will prove that the associated flow is approximately differentiable.We finally investigate under which assumptions the flow inherits the Sobolev or BV regularity of the corresponding vector field. The whole analysis relies on the Hamiltonian structure enjoyed by planar divergence-free vector fields. Part of this work is obtained in collaboration with Paolo Bonicatto.