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

Abstract

We study the local in time existence of a regular solution of a nonlinear parabolic backward-forward system arising from the theory of Mean-Field Games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

“…Proof. This is a classical maximal L p regularity statement for uniformly elliptic equations with continuous coefficients, that can be deduced from results contained in [16]; see [7] for additional details. One can also rely on abstract results on maximal regularity for parabolic equations in [20].…”

On the Existence and Uniqueness of Solutions to Time-Dependent Fractional MFG

“…By standard (local) parabolic regularity theory (see [37, Theorem IV.9.1] or [16]), since the right hand side of the equation in (41) is in L p (Q τ ), (41) admits a unique solution w ∈ W 2,1 p (Q τ ) satisfying the following estimate…”

“…As for the short-time regime, the proofs proposed in [3,4] cannot be adapted to our setting, being designed for the Laplacian and established through L 2 -type estimates. Here, we follow an approach presented in [16,17] to deal with the existence problem in the local case. The idea will be to exploit decay properties of the semigroup associated to the fractional Laplacian in suitable Bessel potentials spaces.…”

On the existence of oscillating solutions in non-monotone Mean-Field Games

“…Heuristically, this condition imposes aversion to crowd within each population, and this effect to be dominant with respect to effects due to interactions between different populations. Another uniqueness regime was discussed in [28], and has been recently revived in [2,3,13]: it occurs under the "smallness" of some data. A typical example of this case is that the time horizon T be small enough.…”