Constrained hydrodynamic flocking models in the limit of large attraction-repulsion interactions
Thierry Goudon, Antoine Mellet
TL;DR
The paper analyzes a kinetic model with nonlocal attraction-repulsion under an external velocity field and derives the hydrodynamic limit in the regime of large interaction strength. Using energy methods and a modulated-energy framework, it shows that the density concentrates on a translated energy minimizer ρ_0 supported on Ω(t)=Ω_0+X(t), with the center of mass X(t) evolving according to $X'(t)=V(t)$ and $V'(t)=\lambda(\frac{1}{m}\int ρ_0(x-X(t))u_{ext}(t,x)\,dx - V(t))$, while the internal flow is governed by the lake equation on Ω(t). The results are obtained in two steps: first, establishing convergence of ρ_ε to ρ(t,·)=ρ_0(·-X(t)) and of the flux to j=ρ𝒱 under general conditions; second, proving convergence of the flux to the lake-system solution when a strong lake solution exists, via a modulated energy argument together with coercivity estimates for the interaction energy. The analysis highlights how strong attraction-repulsion can constrain the population to a moving domain whose shape is fixed by the minimizer and whose center evolves by the external forcing, providing a rigorous link between nonlocal aggregation and boundary-limited hydrodynamics.
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 $Ω(t)=Ω_0+X(t)$ whose shape $Ω_0$ 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.