Aggregation-Diffusion Phenomena: From Microscopic Models to Free Boundary Problems

This chapter reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a result of the competition between attractive and repulsive phenomena—especially (but not exclusively) in the context of attractive chemotaxis phenomena. At microscopic scales, particles (or other agents) are represented by spheres of radius \(\delta >0\) and we discuss both soft-sphere models (with a pressure term penalizing the overlap of the particles) and hard-sphere models (in which overlap is prohibited). The first case leads to so-called blob models which have received some attention recently as a tool to approximate nonlinear diffusion by particle systems. The hard-sphere model is similar to a classical model for congested crowd motion. We will review well-posedness results for these models and discuss their relationship to classical continuum description of aggregation-diffusion phenomena in the limit \(\delta \to 0\) : the classical nonlinear drift diffusion equation and its incompressible counterpart. In the second part of the chapter, we discuss recent results on the emergence and evolution of sharp interfaces when a large population of particles is considered at appropriate space and timescales: At some intermediate timescale, phase separation occurs and a sharp interface appears which evolves according to a Stefan free boundary problem (and the density function eventually relaxes to a characteristic function—metastable steady state for the original problem). At a larger timescale the attractive forces lead to surface tension phenomena and the evolution of the sharp interface can be described by a Hele-Shaw free boundary problem with surface tension. At that same timescale, we will also discuss the emergence of contact angle conditions for problems set in bounded domains.