On the mean field limit for Cucker-Smale models

Roberto Natalini; Thierry Paul

doi:10.3934/dcdsb.2021164

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2022, Volume 27, Issue 5: 2873-2889. Doi: 10.3934/dcdsb.2021164

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On the mean field limit for Cucker-Smale models

Roberto Natalini^1, , and
Thierry Paul^2,

1.
Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini, 19, 00185, Rome, Italy
2.
CNRS & LJLL Sorbonne Université, 4, place Jussieu, 75005 Paris, France

^* Corresponding author: Roberto Natalini
^* Corresponding author: Roberto Natalini

Received: February 28, 2021

Early access: June 2021

Published: May 2022

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Abstract

In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [13]. Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [9] in the Hamiltonian setting, we show the limit providing explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.

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Mathematics Subject Classification: Primary: 92D50, 35Q83; Secondary: 35Q92, 82C05.

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