Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games
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We study the local in time existence of a regular solution of a nonlinear parabolic backward–forward system arising from the theory of mean-field games (briefly MFG). The proof is based on a contraction argument in a suitable space that takes account of the peculiar structure of the system, which involves also a coupling at the final horizon. We apply the result to obtain existence to very general MFG models, including also congestion problems.
KeywordsParabolic equations Backward–forward system Mean-field games Hamilton–Jacobi Fokker–Planck Congestion problems
Mathematics Subject Classification35K40 35K61 49N90
Marco Cirant and Paola Mannucci are members of GNAMPA-INdAM, and were partially supported by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs” and by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”.
- 3.Ambrose DM (2018) Existence theory for non-separable mean field games in Sobolev spaces. preprint arXiv:1807.02223
- 4.Bardi M, Cirant M (2019) Uniqueness of solutions in MFG with several populations and Neumann conditions. In: Cardaliaguet P, Porretta A, Salvarani F (eds) PDE models for multi-agent phenomena. Springer INdAM series, pp 1–20Google Scholar
- 6.Cardaliaguet P (2013) Notes on mean field games. https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf. Accessed 10 Apr 2019
- 12.Gomes D, Nurbekyan L, Prazeres M (2016) Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion. In: 2016 IEEE 55th conference on decision and control, CDC 2016, pp 4534–4539Google Scholar
- 17.Graber PJ (2015) Weak solutions for mean field games with congestion. preprint arXiv: 1503.04733
- 20.Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN (1968) Linear and quasilinear equations of parabolic type. Translations of Math Mon. Providence, p 23Google Scholar
- 21.Lions P-L (2007–2011) College de france course on mean-field games. https://www.college-de-france.fr/site/pierre-louis-lions/_course.htm