The derivation of swarming models: Mean-field limit and Wasserstein distances

By José Antonio Carrillo, Young-Pil Choi, M. Hauray. 2013

261 citations recorded by the source corpus.

These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.

Subfields: Global and Planetary Change, Modeling and Simulation, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/3229eea3dfb6d8838da303f9ace8d09e25cf5b85

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Global and Planetary Change

1. Pierre-Emmanuel Jabin
2. Zhenfu Wang

In Modeling and Simulation

1. S. Lisini
2. Pierre-Emmanuel Jabin
3. Giuseppe Savar'e
4. Massimo Fornasier
5. Zhenfu Wang

In Ocean Engineering

1. Pierre-Emmanuel Jabin
2. Zhenfu Wang

Citations in the corpus

• Mean-field and graph limits for collective dynamics models with time-varying weights (2020, unknown): “φ(x) := ∇W (‖x‖) (see for instance [9]).”
• Mean Field Limit for Stochastic Particle Systems (2017, background): “One can see [17, 18, 40] and the references therein for a more detailed discussion of flocking or swarming models in the literature.”
• A review of the mean field limits for Vlasov equations (2014, methodology): “The W∞ distance was first used in [122] and has started to be used extensively for mean field limit (see [37] or [87] for example).”
• Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, background): “The Lipschitz case, K ∈ W 1 , ∞ loc can still be handled through Gronwall like inequalities, see for instance [6, 11].”
• Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “1) retains simple additive interactions, contrary to the more complex structure found for example in [40, 41]; but it still includes a large range of first order models, such as swarming, opinion dynamics or aggregation equations, see for instance [2, 5, 10, 13] or [32].”
• Mean-field optimal control as Gamma-limit of finite agent controls (2018, background): “As it is very hard to be exhaustive in accounting all the developments of this very fast growing field, we refer to [18, 19, 21, 22, 61] for recent surveys.”