Abstract

At the beginning of the 1990?s, DiPerna and Lions studied the connection between transport equations and ordinary differential equations in detail. In particular, by proving existence and uniqueness at the level of the transport equation, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend this result to BV vector fields. In recent years, several new extensions have been obtained, that give rise to applications to PDEs which include the linear Schrodinger equation and the Vlasov-Poisson equation. The aim of this talk is to give an introductory overview of these results.