Short-time existence of solutions for mean-field games with congestion

TL;DR

This paper proves the short-time existence of smooth solutions for time-dependent mean-field games with congestion effects, extending previous results limited to special cases.

Contribution

It establishes the first general short-time existence result for classical solutions in mean-field games with congestion and sub-quadratic Hamiltonians.

Findings

Proves short-time existence of smooth solutions.

Extends existence results beyond special cases.

Addresses congestion effects in crowd dynamics models.

Abstract

We consider time-dependent mean-field games with congestion that are given by a system of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. The congestion effects make the Hamilton-Jacobi equation singular. These models are motivated by crowd dynamics where agents have difficulty moving in high-density areas. Uniqueness of classical solutions for this problem is well understood. However, existence of classical solutions, was only known in very special cases - stationary problems with quadratic Hamiltonians and some time-dependent explicit examples. Here, we prove short-time existence of $C^\infty$ solutions in the case of sub-quadratic Hamiltonians.