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

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 $\bar \mu$.