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.

We consider extended stationary mean field games, that is mean-field games which depend on the velocity field of the players. We establish existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where existence of classical solutions can be proved.

In this paper, we prove the existence of classical solutions for second order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density. In the case of polynomial couplings, we recur to an iterative method. An integral method builds upon the properties of the logarithmic function in the setting of logarithmic nonlinearities. This work extends substantially previous results by allowing for more general classes of Hamiltonians and mean-field assumptions.

In this paper, we consider mean-field games where the interaction of each player with the mean-field takes into account not only the states of the players but also their collective behavior, To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.