This talk presents recent findings on the regularity of first-order mean
field game 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 analyzing 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.