A continuous theory of traffic congestion and Wardrop equilibria

In the classical Monge–Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by each particle forming this mass. Thus, it does not allow for congestion effects, which depend instead on the proportion of mass passing through a same point or on a same path. Usually the travelling cost (or time) of a path depends on “how crowded” this path is. Starting from a simple network model, we shall define equilibria in the presence of congestion. We will then extend this theory to the continuous setting mainly following the recent papers [8, 10]. After an introduction with almost no mathematical details, we will give a survey of the main features of this theory.