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Formation behaviour of the kinetic Cucker–Smale model with non-compact support

Part of: Partial differential equations

Published online by Cambridge University Press:  21 July 2022

Xinyu Wang  and
Xiaoping Xue
Xinyu Wang
Affiliation:
School of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China (20B912033@stu.hit.edu.cn, xiaopingxue@hit.edu.cn)
Xiaoping Xue
Affiliation:
School of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China (20B912033@stu.hit.edu.cn, xiaopingxue@hit.edu.cn)
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Abstract

In this paper, we focus on the formation behaviour of the kinetic Cucker–Smale model for initial datum without compact support for the position variable. Comparing with the case of compact support, the attractive force between particles is weak. First, we obtain the existence and uniqueness of the classical solution to the kinetic Cucker–Smale model by standard approximation method. Second, by using the characteristic flow, we overcome the difficulty brought by the weak attractive force between particles through some estimates and establish the formation behaviour, i.e., consensus of velocity, of the classical solution to the kinetic Cucker–Smale model. Finally, for the measure-valued solution to the kinetic Cucker–Smale model, the formation behaviour is also established.

Keywords

Cucker–Smale modelformation behaviournon-compact supportcharacteristic flow

MSC classification

Primary: 35B40: Asymptotic behavior of solutions 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1315 - 1346
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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