In this talk, we present new findings on the regularity of first-order mean field games systems with a local coupling. We focus on systems where the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function for small densities. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, when f(m) = mθ with θ > 0, we prove that the speed of propagation is finite. In this case, we establish that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.
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