Calculus of Variations and Geometric Measure Theory

C. Rigoni - E. Stepanov - D. Trevisan

Lie brackets of nonsmooth vector fields and commutation of their flows

created by stepanov on 14 Nov 2020
modified on 11 Jul 2023


Published Paper

Inserted: 14 nov 2020
Last Updated: 11 jul 2023

Journal: J. London Math. Soc.
Year: 2020


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).

Keywords: Frobenius theorem, Lie bracket, commuting flows, regular Lagrangian flow