On existence of Borel flow for ordinary differential equation with a non-smooth vector field

TL;DR

This paper investigates the existence of Borel measurable flows for bounded, non-smooth vector fields, establishing an equivalence with measure-valued solutions to the associated continuity equation.

Contribution

It extends the classical connection between ODEs and PDEs to non-smooth vector fields by proving the existence of measurable flows under minimal regularity assumptions.

Findings

Existence of Borel measurable flow is equivalent to measure-valued solutions.

The results generalize classical characteristics method to non-smooth vector fields.

Provides a framework for analyzing ODEs with non-smooth vector fields.

Abstract

For smooth vector fields the classical method of characteristics provides a link between the ordinary differential equation and the corresponding continuity equation (or transport equation). We study an analog of this connection for merely bounded Borel vector fields. In particular we show that, given a non-negative Borel measure $\bar \mu$ on $\mathbb{R}^d$, existence of $\bar \mu$-measurable flow of a bounded Borel vector field is equivalent to existence of a measure-valued solution to the corresponding continuity equation with the initial data $\bar \mu$.