Luigi Ambrosio
Transport Equation and Cauchy Problem for BV Vector Fields

ABSTRACT. I will present some recent work and open problems on the transport equation and the generic uniqueness of solutions of the ODE in presence of a non-smooth vector field. The lectures will include the classical theory of DiPerna-Lions as well as recent developments concerning BV vector fields. Some applications to systems of concervation laws and compactness of Lagrangian flows will be presented. The main mathematical tools involved will be the fine theory of BV functions and the theory of Young measures.