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

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.