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Article Dans Une Revue Journal of Dynamics and Differential Equations Année : 2019

Time-dependent focusing Mean-Field Games: the sub-critical case

Résumé

We consider time-dependent viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. 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 minimisers 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 minimisers 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.
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Dates et versions

hal-01508506 , version 1 (14-04-2017)
hal-01508506 , version 2 (30-08-2018)

Identifiants

  • HAL Id : hal-01508506 , version 2

Citer

Marco Cirant, Daniela Tonon. Time-dependent focusing Mean-Field Games: the sub-critical case. Journal of Dynamics and Differential Equations, 2019, 31 (1), pp.49-79. ⟨hal-01508506v2⟩
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