Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields

TL;DR

This paper proves that continuous Sobolev vector fields with bounded divergence in the plane commute if their Lie bracket vanishes, improving previous results by removing the need for weak differentiability assumptions.

Contribution

It establishes the commutativity of flows for Sobolev vector fields with bounded divergence, extending prior work by weakening regularity conditions.

Findings

Flows commute when Lie bracket is zero for continuous Sobolev vector fields

Improves previous results by removing weak differentiability requirement

Discusses potential extensions to the BV setting

Abstract

In this paper, we prove that if $X,Y$ are continuous, Sobolev vector fields with bounded divergence on the real plane and $[X,Y]=0$, then their flows commute. In particular, we improve the previous result of Colombo-Tione (2021), where the authors require the additional assumption of the weak Lie differentiability on one of the two flows. We also discuss possible extensions to the $\text{BV}$ setting.