Macroscopic Limits, Self-Organization and Stability in Systems with Singular Interactions Arising from Hydrodynamics and Life Sciences

Abstract

This dissertation is centered around the analysis of non-linear partial differential equations that arise from models in physics, mathematical biology, social sciences and neuroscience. Specifically, we address a particular family of models that have been coined in the literature with the name of “collective dynamics models”. The main idea is that from basic rules stating how a system of particles interact, the population often has the ability to self-organize collectively as a unique entity and it amounts to different emergent phenomena depending on the particular context., e.g., swarming, flocking, schooling, synchronization, etc. Although these models appear in completely different settings, what make them so special from a mathematical point of view is the fact that their structural resemblance allow us to tackle them with common abstract mathematical tools. Indeed many relevant improvements and mathematical methods have emerged from this interface as we tackle the different effects that we can encounter (e.g., kinetic theory, stochastic equations, mean field limits, propagation of chaos, hydrodynamic limits, potential theory, optimal transport, etc).