A note on non-coercive first order Mean Field Games with analytic data

TL;DR

This paper investigates first order Mean Field Games with non-coercive operators, establishing the existence of weak solutions and describing population evolution via a flow derived from optimal control, addressing cases with forbidden directions.

Contribution

It introduces a framework for analyzing non-coercive first order Mean Field Games and proves the existence of weak solutions under regularity assumptions.

Findings

Existence of weak solutions for non-coercive MFGs

Population distribution evolves as a push-forward of initial data

Framework handles cases with forbidden directions

Abstract

We study first order evolutive Mean Field Games whose operators are non-coercive. This situation occurs, for instance, when some directions are `forbidden' to the generic player at some points. Under some regularity assumptions, we establish existence of a weak solution of the system. Mainly, we shall describe the evolution of the population's distribution as the push-forward of the initial distribution through a flow, suitably defined in terms of the underlying optimal control problem.