Strong solutions for time-dependent mean field games with non-separable Hamiltonians

TL;DR

This paper establishes the existence of strong solutions for time-dependent mean field games with complex, non-separable Hamiltonians, extending previous results to broader classes of systems.

Contribution

It generalizes prior work by proving existence theorems for non-separable Hamiltonians and introduces new methods inspired by fluid mechanics for these systems.

Findings

Existence of small strong solutions for non-separable Hamiltonians.

Existence of solutions with weak coupling and small Hamiltonian data.

Use of implicit function theorem in proof techniques.

Abstract

We prove existence theorems for strong solutions of time-dependent mean field games with non-separable Hamiltonian. In a recent announcement, we showed existence of small, strong solutions for mean field games with local coupling. We first generalize that prior work to allow for non-separable Hamiltonians. This proof is inspired by the work of Duchon and Robert on the existence of small-data vortex sheets in incompressible fluid mechanics. Our next existence result is in the case of weak coupling of the system; that is, we allow the data to be of arbitrary size, but instead require that the (still possibly non-separable) Hamiltonian be small in a certain sense. The proof of this theorem relies upon an appeal to the implicit function theorem.