We explore a mechanism of decision-making in Mean Field Games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N -players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic Mean Field Games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.
We give a direct proof of the fact that the L p -norms of global solutions of the Boussinesq system in R 3 grow large as t → ∞ for 1 < p < 3 and decay to zero for 3 < p ≤ ∞, providing exact estimates from below and above using a suitable decomposition of the space-time space R + × R 3 . In particular, the kinetic energy blows up as u(t) 2 2 ∼ ct 1/2 for large time. This constrasts with the case of the Navier-Stokes equations. 2000 Mathematics Subject Classification. Primary 76D05; Secondary 35B40. Key words and phrases. Kato spaces, the Boussinesq equation and asymptotic behavior. 1 1 Indeed, it was claimed in [5] that u(t) 2 → ∞ for solutions of Boussinesq system. However, this article contained an erratum that was pointed out in [2].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.