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

Annalaura Rebucci, Martina Zizza

TL;DR

The paper addresses the problem of when the commutativity of flows generated by two vector fields $X$ and $Y$ is equivalent to the vanishing Lie bracket $[X,Y]$ in rough, two-dimensional settings. It develops a geometric, Hamiltonian-based approach that, in 2D, shows $[X,Y]=0$ implies $ ext{flow}_t^X ext{flow}_s^Y= ext{flow}_s^Y ext{flow}_t^X$ for continuous $X,Y$ with bounded divergence, removing the need for weak Lie differentiability. The authors first treat the Hamiltonian divergence-free case, exploiting level-set structure and the identity $X eq 0$ and $X= abla^ot H$, $Y= abla^ot K$, with $[X,Y]=0$ equivalent to $X\,\cdot\,Y^\perp=\text{const}$, and then extend to nearly incompressible fields via invariant regions $\,\Omega_\kappa$ and a density-based reparameterization. They also discuss a BV extension and outline open questions for the reverse implication in the BV setting. Overall, the work advances Frobenius-type results for flows of rough vector fields in the plane and highlights geometric structures that govern commutativity in low dimensions with potential PDE applications.

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.

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

TL;DR

The paper addresses the problem of when the commutativity of flows generated by two vector fields and is equivalent to the vanishing Lie bracket in rough, two-dimensional settings. It develops a geometric, Hamiltonian-based approach that, in 2D, shows implies for continuous with bounded divergence, removing the need for weak Lie differentiability. The authors first treat the Hamiltonian divergence-free case, exploiting level-set structure and the identity and , , with equivalent to , and then extend to nearly incompressible fields via invariant regions and a density-based reparameterization. They also discuss a BV extension and outline open questions for the reverse implication in the BV setting. Overall, the work advances Frobenius-type results for flows of rough vector fields in the plane and highlights geometric structures that govern commutativity in low dimensions with potential PDE applications.

Abstract

In this paper, we prove that if are continuous, Sobolev vector fields with bounded divergence on the real plane and , 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 setting.
Paper Structure (10 sections, 22 theorems, 111 equations)

This paper contains 10 sections, 22 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Preliminaries
  4. BV functions
  5. Regular Lagrangian Flows
  6. Structure of level sets of $\text{Lip}_c$ functions
  7. Hamiltonian vector fields
  8. Analysis of $X\cdot Y^\perp=\text{const}$
  9. Nearly incompressible vector fields
  10. Some remarks on BV vector fields

Key Result

Theorem 1.2

[Theorem 1.4 in Eliodiff] Let $X\in W^{1,p}(\mathbb{R}^2,\mathbb{R}^2)$ be a continuous divergence-free vector field with bounded support and let $\Omega$ be an open ball of radius $R>0$ such that there exists $\delta>0$ and $\mathbf e\in\mathbb{S}^1$ for which $X\cdot \mathbf e>\delta$ in $\Omega$.

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 1.4: Nearly incompressible
  • Proposition 1.5
  • Definition 1.6: Steady nearly incompressible
  • Proposition 1.7
  • Theorem 1.8
  • Definition 1.9: Distributional Lie bracket
  • Proposition 1.10
  • ...and 31 more