A survey on dynamical transport distances

In this paper we review some transport models based on the continuity equation, starting with the so-called Benamou–Brenier formula, which is nothing but a fluid mechanics reformulation of the Monge–Kantorovich problem with cost $c(x,y)=|x-y|^2$. We discuss some of its applications (gradient flows, sharp functional inequalities …), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered.