  We prove well-posedness of a class of kinetic-type Mean Field Games, which
typically arise when agents control their acceleration. Such systems include
independent variables representing the spatial position as well as velocity. We
consider non-separable Hamiltonians without any structural conditions, which
depend locally on the density variable. Our analysis is based on two main
ingredients: an energy method for the forward-backward system in Sobolev
spaces, on the one hand and on a suitable vector field method to control
derivatives with respect to the velocity variable, on the other hand. The
careful combination of these two techniques reveals interesting phenomena
applicable for Mean Field Games involving general classes of drift-diffusion
operators and nonlinearities. While many prior existence theories for general
Mean Field Games systems take the final datum function to be smoothing, we can
allow this function to be non-smoothing, i.e. also depending locally on the
final measure. Our well-posedness results hold under an appropriate smallness
condition, assumed jointly on the data.
