Lie brackets of nonsmooth vector fields and commutation of their flows

TL;DR

This paper investigates the relationship between Lie brackets and flow commutation for nonsmooth vector fields, establishing conditions under which the classical smooth case results extend to less regular settings.

Contribution

It demonstrates that the classical equivalence between vanishing Lie brackets and flow commutation does not hold universally for nonsmooth vector fields, but holds when combining Lipschitz and Sobolev regularity.

Findings

The classical result extends to Lipschitz vector fields with a.e. Lie brackets.

The equivalence fails for general a.e. differentiable vector fields.

The extension holds when one vector field is Lipschitz and the other Sobolev regular.

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).