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

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.