Mean field games with congestion

  • Yves Achdou

    Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France
  • Alessio Porretta

    Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy


We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton–Jacobi equation and a forward Kolmogorov equation both posed in . Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T.

Cite this article

Yves Achdou, Alessio Porretta, Mean field games with congestion. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, pp. 443–480

DOI 10.1016/J.ANIHPC.2017.06.001