  We consider time-dependent mean-field games with congestion that are given by
a system of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation.
The congestion effects make the Hamilton-Jacobi equation singular. These models
are motivated by crowd dynamics where agents have difficulty moving in
high-density areas. Uniqueness of classical solutions for this problem is well
understood. However, existence of classical solutions, was only known in very
special cases - stationary problems with quadratic Hamiltonians and some
time-dependent explicit examples. Here, we prove short-time existence of
$C^\infty$ solutions in the case of sub-quadratic Hamiltonians.
