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