Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games

We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games with a variational structure. We prove L infinity regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent's trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L infinity results have already been found by P.-L. Lions in his course on Mean Field Games, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) phi, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics.