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

Abstract

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 ∞ solutions in the case of sub-quadratic Hamiltonians.

Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

“…For second-order congestion problems and a small enough T , the existence of a solution to (I.3) was established in [15] and [16] (resp. strong and weak solutions).…”

Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games

“…A general result of existence of weak solutions for arbitrary time horizon T is discussed in [1]. So far, smoothness of solutions has been verified in the short-time regimes only in [10]. All the mentioned works do rely on the MFG structure of (4.47)-(4.48).…”

Mean field games with congestion

“…Let us compare the above assumptions with previous settings used for congestion models in mean field games. In [18], congestion models are presented starting from the Lagrangian function…”

“…m and that the following matrix be positive semidefinite: In the present work, we will show that this hypothesis yields both the existence and the uniqueness of weak solutions. Except for situations in which special tricks may be applied (stationary problems and quadratic Hamiltonian, see [16]), the existence of classical solutions of suitable generalizations of (1.1) seems difficult to obtain, because generally neither upper bounds on m nor strict positivity of m are known unless one restricts the growth conditions for the nonlinearities and assumes the time horizon T to be small, see [18], [19] (see also [15] for the stationary case). Therefore, in order to get at a sufficiently general result, we aim at proving the existence and uniqueness of suitably defined weak solutions.…”