  First order kinetic mean field games formally describe the Nash equilibria of
deterministic differential games where agents control their acceleration,
asymptotically in the limit as the number of agents tends to infinity. The
known results for the well-posedness theory of mean field games with control on
the acceleration assume either that the running and final costs are
regularizing functionals of the density variable, or the presence of noise,
i.e. a second-order system. In this article we construct global in time weak
solutions to a first order mean field games system involving kinetic transport
operators, where the costs are local (hence non-regularizing) functions of the
density variable with polynomial growth. We show the uniqueness of these
solutions on the support of the agent density. This is achieved by
characterizing solutions through two convex optimization problems in duality.
As part of our approach, we develop tools for the analysis of mean field games
on a non-compact domain by variational methods. We introduce a notion of
`reachable set', built from the initial measure, that allows us to work with
initial measures with or without compact support. In this way we are able to
obtain crucial estimates on minimizing sequences for merely bounded and
continuous initial measures. These are then carefully combined with $L^1$-type
averaging lemmas from kinetic theory to obtain pre-compactness for the
minimizing sequence. Finally, under stronger convexity and monotonicity
assumptions on the data, we prove higher order Sobolev estimates of the
solutions.
