Singular limit of the porous medium equation with a drift

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.