In this talk, I will present recent results concerning the singular limit of the solutions of the porous medium equations with a drift. This problem appears in models of tumor growth, population dynamics in cell biology, congestion and crowd control. In these situations, a conserved quantity is being transported by an external drift field, but the amount of the quantity present in a given volume is constrained, for instance by the maximum packing density of cells. In modeling, this constraint is often relaxed, the quantity is allowed to be compressible, and its accumulation is prevented by introducing a degenerate diffusion that kicks in when the density becomes too high.
We show that, in the incompressible limit, the solutions of this problem converge to the solution of a constrained transport equation with no diffusion, under the assumption that the drift field is “compressive”. The limit solution reaches the constraint in a so-called congested set, which can be characterized by a certain Hele-Shaw type problem. This convergence result justifies the above relaxation and establishes the relationship of the diffuse interface and sharp interface models. This is joint work with Inwon Kim and Brent Woodhouse.