Accepted Paper

Inserted: 7 dec 2021
Last Updated: 12 jul 2022

Journal: Comm. Partial Differential Equations
Year: 2022

Abstract:

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularising functionals of the density variable, or the presence of noise, i.e. a second-order system. In this manuscript we construct global in time weak solutions to a first order mean field games system involving kinetic transport operators, where the costs are local (hence non-regularising) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. This is achieved by characterising solutions through two convex optimisation problems in duality. As part of our approach, we develop tools for the analysis of mean field games on a non-compact domain by variational methods. We introduce a notion of `reachable set', built from the initial measure, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimising sequences for merely bounded and continuous initial measures with finite first velocity moment. These are then carefully combined with $L^1$-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimising sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions.