Weak solutions for mean field games with congestion

Original · EN

We study the short-time existence and uniqueness of solutions to a coupled system of partial differential equations arising in mean field game theory. It has the generic form { arrayc -∂ₜ u - Δu + H(t,x,m,∇ u) = f(t,x,m) ∂ₜ m - Δm - div (m∇ₚ H(t,x,m,∇ u)) = 0 array. plus initial-final and boundary conditions. The novelty of the problem is that the Hamiltonian H(t,x,m,p) may take such forms as m⁻α|p|ʳ for some α≥ 0 and r > 1. Our main result is the existence of weak solutions for small times T so long as r is not too large, and uniqueness under additional constraints. The main ingredient in the proof is an a priori estimate on solutions to the Fokker-Planck equation. We also briefly consider existence and uniqueness of solutions to an optimal control problem related to mean field games.