  We study the "stiff pressure limit" of a nonlinear drift-diffusion equation,
where the density is constrained to stay below the maximal value one. The
challenge lies in the presence of a drift and the consequent lack of
monotonicity in time. In the limit a Hele-Shaw-type free boundary problem
emerges, which describes the evolution of the congested zone where density
equals one. We discuss pointwise convergence of the densities as well as the
$BV$ regularity of the limiting free boundary.
