A new proof of the uniqueness of the flow for ordinary differential equations with BV vector fields

TL;DR

This paper presents a novel proof establishing the uniqueness of flow solutions for ODEs with BV vector fields, avoiding the traditional reliance on the transport equation, thus offering a new perspective in the analysis of such differential equations.

Contribution

It introduces a new proof method for the uniqueness of flows in BV vector fields with divergence in L^ and nearly incompressible condition, bypassing the transport equation approach.

Findings

Proves uniqueness of flow solutions for BV vector fields with divergence in L^.

Provides a new proof technique not relying on the transport equation.

Enhances understanding of ODEs with BV vector fields.

Abstract

We provide in this article a new proof of the uniqueness of the flow solution to ordinary differential equations with $BV$ vector-fields that have divergence in $L^\infty$ (or in $L^1$) and that are nearly incompressible (see the text for the definition of this term). The novelty of the proof lies in the fact it does not use the associated transport equation.