Regularity and long time behavior of one-dimensional first-order mean field games and the planning problem

Understand

We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density.

• We improve upon the existing literature by obtaining two regularity results.
• The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities.
• The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the initial density be bounded away from zero.