Mean field games with congestion

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) x (R-N /Z(N)). 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 +infinity 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. (C) 2017 Elsevier Masson SAS. All rights reserved.