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).