Lie brackets of nonsmooth vector fields and commutation of their flows

Abstract: It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show however that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).