Existence theory for non-separable mean field games in Sobolev spaces

The mean field games system is a coupled pair of nonlinear partial differential equations arising in game theory, as limit the number agents tends to infinity. We prove existence and uniqueness classical solutions for time-dependent with Sobolev data. Many works literature assume additive separability Hamiltonian, well further structure such convexity monotonicity resulting components. Problems practice, however, may not have this separable structure; we therefore consider non-separable problem. For our results, introduce new smallness constraints which simultaneously size time horizon, data, strength coupling system.