# Mean field games with congestion

## TL;DR

This paper studies a class of coupled PDE systems modeling mean field games with congestion, proving existence and uniqueness of weak solutions without restrictions on the time horizon.

## Contribution

It introduces a framework for weak solutions in mean field games with congestion, addressing cases where the Hamiltonian vanishes or is undefined.

## Key findings

- Existence of weak solutions under general assumptions
- Uniqueness of solutions in the congestion setting
- No restriction on the time horizon T

## Abstract

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 $(0,T)\times (\mathbb{R}^N /\mathbb{Z}^N)$. 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 $+\infty$ 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 $T$.