The porous medium equation describes diffusion whose strength depends on the density. As the diffusion exponent becomes large, this mechanism approaches a hard density constraint, connecting the equation to models of congestion and incompressible motion. Understanding this limit raises a quantitative question: how quickly do solutions approach their limiting behavior? In this talk, we study this question for stationary states in the presence of a confining potential.

We establish a sharp convergence rate in Wasserstein distances under mild assumptions on the potential. The argument uses the geometry of the stationary states to distinguish the movement of their supports from the redistribution of mass within them. We will explain how these two effects can be controlled through transport estimates, leading to convergence even in a distance that controls the largest displacement of mass. We will also discuss what the sharp stationary rate tells us about possible convergence rates for the time-dependent equation.

This is a joint work with Filippo Santambrogio.