2016/05/01 by José A. Carrillo, Carrillo, José A., Young-Pil Choi +3 · 5 citations
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Micro and Nano Robotics #Numerical Analysis (math.NA) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1605.00232
openalex publication_date 2016/05/01 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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.