  We prove existence theorems for strong solutions of time-dependent mean field
games with non-separable Hamiltonian. In a recent announcement, we showed
existence of small, strong solutions for mean field games with local coupling.
We first generalize that prior work to allow for non-separable Hamiltonians.
This proof is inspired by the work of Duchon and Robert on the existence of
small-data vortex sheets in incompressible fluid mechanics. Our next existence
result is in the case of weak coupling of the system; that is, we allow the
data to be of arbitrary size, but instead require that the (still possibly
non-separable) Hamiltonian be small in a certain sense. The proof of this
theorem relies upon an appeal to the implicit function theorem.
