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A variational approach to first order kinetic Mean Field Games with local couplings

Griffin-Pickering, Megan; Mészáros, Alpár R.

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Megan Griffin-Pickering


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 regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this article 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-regularizing) 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 characterizing solutions through two convex optimization 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 minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions.


Griffin-Pickering, M., & Mészáros, A. R. (2022). A variational approach to first order kinetic Mean Field Games with local couplings. Communications in Partial Differential Equations, 47(10), 1945-2022.

Journal Article Type Article
Acceptance Date Jul 9, 2022
Online Publication Date Aug 12, 2022
Publication Date 2022
Deposit Date Jul 11, 2022
Publicly Available Date Oct 12, 2022
Journal Communications in Partial Differential Equations
Print ISSN 0360-5302
Electronic ISSN 1532-4133
Publisher Taylor and Francis Group
Peer Reviewed Peer Reviewed
Volume 47
Issue 10
Pages 1945-2022


Published Journal Article (5.7 Mb)

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Copyright Statement
&copy; 2022 The Author(s). Published with license by Taylor &amp; Francis Group, LLC<br /> This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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