On the commutativity of flows of rough vector fields

TL;DR

This paper characterizes when flows of Sobolev vector fields with bounded divergence commute, extending Frobenius' classical result to a non-smooth setting using Lie brackets and flow regularity.

Contribution

It provides a new characterization of flow commutativity for Sobolev vector fields, generalizing classical Frobenius theorem to less regular contexts.

Findings

Flow commutativity is characterized by Lie brackets and flow regularity.

Extension of Frobenius' theorem to Sobolev vector fields with bounded divergence.

Provides conditions under which non-smooth vector fields' flows commute.

Abstract

In the class of Sobolev vector fields in $\mathbb{R}^n$ of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the vector fields whose flow commute in terms of the Lie bracket and of a regularity condition on the flows themselves. This extends a classical result of Frobenius in the smooth setting.