Abstract: I will talk about coarsening of the pattern of interfaces in models of biological aggregation and nonlocal Cahn--Hilliard type equations. The model of biological aggregation by Topaz, Bertozzi, and Lewis describes the dynamics of individuals who are attracted to each other at a distance, but avoid overcrowding via a local dispersal mechanism. The continuum model for the population density is a gradient flow in Wasserstein metric.
The density profile develops interfaces between a near-constant-density aggregate state and the empty space. The interfaces evolve under surface-tension-like ``forces''. On long time scales the interfacial motion leads to coarsening of length scales present in the system. Similar phenomena occur in (nonlocal) Cahn-Hilliard type equations. The rate of coarsening can be investigated using the Kohn-Otto framework. We will describe a geometric viewpoint, which unites the coarsening results in a variety of interfacial models. The key technical element is proving appropriate interpolation inequalities.
Part of the talk is based on joint work with Andrea Bertozzi.