  We study first order evolutive Mean Field Games where the Hamiltonian is
non-coercive. This situation occurs, for instance, when some directions are
"forbidden" to the generic player at some points. We establish the existence of
a weak solution of the system via a vanishing viscosity method and, mainly, we
prove that the evolution of the population's density is the push-forward of the
initial density through the flow characterized almost everywhere by the optimal
trajectories of the control problem underlying the Hamilton-Jacobi equation. As
preliminary steps, we need that the optimal trajectories for the control
problem are unique (at least for a.e. starting points) and that the optimal
controls can be expressed in terms of the horizontal gradient of the value
function.
