A review on attractive-repulsive hydrodynamics for consensus in collective behavior

TL;DR

This survey reviews hydrodynamics models for collective behavior, focusing on attractive-repulsive interactions, their connection to particle systems, and the use of Lagrangian schemes to analyze system properties and conjectures.

Contribution

It provides a comprehensive overview of qualitative properties of hydrodynamic models, emphasizing numerical schemes and their ability to replicate particle system behaviors.

Findings

Hydrodynamic models exhibit diverse qualitative behaviors.

Lagrangian schemes effectively preserve particle system properties.

Numerical results lead to new conjectures in collective behavior.

Abstract

This survey summarizes and illustrates the main qualitative properties of hydrodynamics models for collective behavior. These models include a velocity consensus term together with attractive-repulsive potentials leading to non-trivial flock profiles. The connection between the underlying particle systems to the swarming hydrodynamic equations is performed through kinetic theory modelling arguments. We focus on Lagrangian schemes for the hydrodynamic systems showing the different qualitative behavior of the systems and its capability of keeping properties of the original particle models. We illustrate known results concerning large time profiles and blow-up in finite time of the hydrodynamic systems to validate the numerical scheme. We finally explore unknown situations making use of the numerical scheme showcasing a number of conjectures based on the numerical results.