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

Annalaura Rebucci; Martina Zizza

doi:10.1016/j.jfa.2025.110919

paper

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

arXiv:2406.02340 · doi:10.1016/j.jfa.2025.110919

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 $BV$ setting.