A quantiative approach to the DiPerna--Lions theory for transport equations

In their celebrated theory of renormalized solutions, DiPerna and Lions (1989) establish well-posedness and stability properties for transport equations with Sobolev vector fields. In this talk, I present a new approach to well-posedness that is based on stability estimates for certain logarithmic Kantorovich--Rubinstein distances. The new approach recovers some of DiPerna's and Lions's old results. In addition, it allows for two new major applications that were very inaccessible before: 1) We extend the theory to vector fields with $L^1$ vorticities and present applications to the 2D Euler equation (joint work with G. Crippa, C. Nobili and S. Spirito). 2) We derive optimal estimates on the error of the numerical upwind scheme (joint work with A. Schlichting).