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

A variational approach to first order kinetic Mean Field Games with local couplings