The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements

Chiun-Chuan Chen; Seung-Yeal Ha; Xiongtao Zhang

doi:10.3934/cpaa.2018028

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2018, Volume 17, Issue 2: 505-538. Doi: 10.3934/cpaa.2018028

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The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements

1.
Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, No. 1 Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
3.
Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, 430074, China

* Corresponding author: Xiongtao Zhang
* Corresponding author: Xiongtao Zhang

Received: December 31, 2016

Revised: August 31, 2017

Published: June 2018

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Abstract

We present a coupled kinetic-macroscopic equation describing the dynamic behaviors of Cucker-Smale(in short C-S) ensemble undergoing velocity jumps and chemotactic movements. The proposed coupled model consists of a kinetic C-S equation supplemented with a turning operator for the kinetic density of C-S particles, and a reaction-diffusion equation for the chemotactic density. We study a global existence of strong solutions for the proposed model, when initial data is sufficiently regular, compactly supported in velocity and has finite mass and energy. The turning operator can screw up the velocity alignment, and result in a dispersed state. However, under suitable structural assumptions on the turning kernel and ansatz for the reaction term, the effects of the turning operator can vanish asymptotically due to the diffusion of chemical substances. In this situation, velocity alignment can emerge algebraically slow. We also present parabolic and hyperbolic Keller-Segel models with alignment dissipation in two scaling limits.

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Mathematics Subject Classification: Primary:92D25, 74A25, 76N10.

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