A Nonlocal-to-Local Approach to Aggregation-Diffusion Equations

Over the past decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models, and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn-Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models, while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. Additionally, we discuss recent analytical results based on the gradient-flow structure of the model, along with open problems and future research directions.