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Emergence of bi-cluster flocking for the Cucker–Smale model

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

Department of Mathematical Sciences and Research, Institute of Mathematics, Seoul National University, Seoul 151-747, Korea

E-mail Address: syha@snu.ac.kr

Corresponding author

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Feimin Huang

Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190, P. R. China

Beijing Center of Mathematics and Information Sciences, Beijing 100048, P. R. China

E-mail Address: fhuang@amt.ac.cn

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Chunyin Jin

Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190, P. R. China

Beijing Center of Mathematics and Information Sciences, Beijing 100048, P. R. China

E-mail Address: jinchunyin@163.com

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, and

Dongnam Ko

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

E-mail Address: pyeong@snu.ac.kr

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https://doi.org/10.1142/S0218202516500287Cited by:72 (Source: Crossref)

Abstract

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

Communicated by J. Soler

Keywords:

AMSC: 34D05, 70E55, 92B25

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