Constrained hydrodynamic flocking models in the limit of large attraction-repulsion interactions

TL;DR

This paper analyzes the collective behavior of particles under strong attraction-repulsion forces, showing they concentrate in a domain governed by energy minimization and evolve according to an external field, with internal flow described by hydrodynamics.

Contribution

It establishes a rigorous link between mean-field models and hydrodynamic descriptions in the limit of large interactions, including domain concentration and flow dynamics.

Findings

Particles concentrate in a domain minimizing interaction energy.

Domain's center of mass follows external velocity field.

Internal flow described by the lake hydrodynamic equation.

Abstract

We study the collective dynamics of a population of particles/organisms subject to self-consistent attraction-repulsion interactions and an external velocity field. The starting point of our analysis is a mean-field kinetic model and we investigate the singular limit corresponding to strong interaction forces. For well-prepared initial data, we show that the population asymptotically concentrates within a domain $\Omega(t)=\Omega_0+X(t)$ whose shape $\Omega_0$ is determined by the minimization of the interaction energy while the evolution of the domain's center of mass $X(t)$ is determined by the external force field. In addition, we show that the internal flow of organisms within this moving domain is described by a classical hydrodynamic model (the lake equation). The first part of our result relies only on the existence and uniqueness of minimizers for the interaction energy and holds for rather general interaction kernels. The second part is proved using a modulated energy method under more restrictive conditions on the nature of the interactions, and assuming that the limiting lake equation admits strong solutions.