Renormalization of contact velocity fields with hori-zontal Sobolev regularity in Heisenberg groups

The classical Cauchy-Lipschitz theory ensures well-posedness of the flow equation associated with Lipschitz vector fields. A major breakthrough in extending this theory to rough velocity fields was achieved by DiPerna-Lions in the Sobolev setting, and later by Ambrosio in the BV framework. Since then, the theory has been significantly developed under various structural and regularity assumptions, both in Euclidean and metric measure settings.
In this talk, after reviewing the existing theory, we present a new well-posedness result for a class of rough velocity fields in the genuinely sub-Riemannian setting of the Heisenberg group. We describe the main ideas of our approach, and we explain why our result cannot be deduced either from existing Euclidean techniques or from available results in the metric measure framework. Based on a joint work with L. Ambrosio, G. Somma and D. Vittone.