Time-Dependent Focusing Mean-Field Games: The Sub-critical Case

  • Marco Cirant
  • Daniela TononEmail author


We consider time-dependent viscous mean-field games systems in the case of local, decreasing and unbounded couplings. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimizers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimizers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker–Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short.


Variational formulation of mean field games Local decreasing coupling Non-uniqueness 

Mathematics Subject Classification

35K55 49N70 



The first author is partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games” and the INdAM-GNAMPA Project “Fenomeni di segregazione in sistemi stazionari di tipo Mean Field Games a più popolazioni”. The second author is partially supported by the ANR Project MFG ANR-16-CE40-0015-01, the PEPS-INSMI Jeunes Project “SOME OPEN PROBLEMS IN MEAN FIELD GAMES” for the years 2016 and 2017. This article benefited from the support of the FMJH Program Gaspard Monge for optimization and operations research and their interactions with data science and by a public grant as part of the Investissement d’avenir Project, Reference ANR-11-LABX-0056-LMH, LabEx LMH, PGMO project VarPDEMFG. We warmly thank the anonymous referee for his/her careful reading and valuable comments.


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Authors and Affiliations

  1. 1.Dipartimento di Matematica “Tullio Levi-Civita”Università di PadovaPaduaItaly
  2. 2.CEREMADE, UMR CNRS 7534Université Paris-Dauphine PSL Research UniversityParis Cedex 16France

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