A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication

Seung-Yeal Ha; Jeongho Kim; Peter Pickl; Xiongtao Zhang

doi:10.3934/krm.2019039

Article Contents

2019, Volume 12, Issue 5: 1045-1067. Doi: 10.3934/krm.2019039

This issue Previous Article Sedimentation of particles in Stokes flow Next Article The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
2.
Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea
3.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
4.
Mathematisches Institut, Ludwig-Maximilians-Universiät, Theresienstr. 39, Munich 80333, Germany
5.
Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, China

^* Corresponding author
^* Corresponding author

Received: July 31, 2018

Revised: January 31, 2019

Published: July 2019

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Abstract

We present a probabilistic approach for derivation of the kinetic Cucker-Smale (C-S) equation from the particle C-S model with singular communication. For the system we are considering, it is impossible to validate effective description for certain special initial data, thus such a probabilistic approach is the best one can hope for. More precisely, we consider a system in which kinetic trajectories are deviated from a microscopic model and use a suitable probability measure to quantify the randomness in the initial data. We show that the set of "bad initial data" does in fact have small measure and that this small probability decays to zero algebraically, as $ N $ tends to infinity. For this, we introduce an appropriate cut-off in the communication weight. We also provide a relation between the order of the singularity and the order of the cut-off such that the machinery for deriving classical mean-field limits introduced in [3] can be applied to our setting.

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Mathematics Subject Classification: Primary: 70F10, 35Q70.

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