The starting point is a model introduced by Beckmann in the ’50s, called continuous model of transportation, which takes the form of a divergence constraint optimization problem. The Beckmann’s problem is studied in relation to a proper Monge-Kantorovich one, and it is shown that, under some hypothesis, they admit the same solution.

Consequently, the model is complicated in order to take care of congestion effects, understanding if there are still connections with the optimal transport theory and if a meaningful notion of equilibrium can be found. Avoiding deep technicalities, that are only mentioned, an interesting result is presented: the solution of the congestion traffic model solves a peculiar Monge-Knatorovich one, where the cost function depends on the solution of the problem itself. Moreover, the solution also satisfies a very famous notion of non-cooperative equilibrium.
http://cvgmt.sns.it/seminar/620/