| Abstract: |
| In this talk 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, i.e. at converging to a common state. From the PDE viewpoint, several issues are intrinsic in this framework, mainly caused by the lack of regularizing effects induced by increasing monotonicity of the coupling. Non-existence, non-uniqueness of solutions, non-smoothness, and concentration are likely to arise. Even more
than in the competitive case, the assumptions on the Hamiltonian, the growth of the coupling and
the dimension of the state space affect the qualitative behavior of the system.
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. These results are obtained in collaboration with Marco Cirant. |
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