no. 2
Mean field games with congestion
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 443-480.

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 (0,T)×(RN/ZN). 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.

DOI : 10.1016/j.anihpc.2017.06.001
Mots clés : Mean field games, Congestion models, Local coupling, Existence and uniqueness, Weak solutions 
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Achdou, Yves; Porretta, Alessio. Mean field games with congestion. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 443-480. doi : 10.1016/j.anihpc.2017.06.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.06.001/

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