Convergence Toward a Periodically Rotating One-Point Cluster in the Kinetic Thermodynamic Cucker-Smale Model
Year: 2022
Author: Linglong Du, Seung-Yeal Ha
Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 1 : pp. 72–111
Abstract
We study a relaxation dynamics of the kinetic thermodynamic Cucker-Smale (TCS) model under the effect of potential force field. For this, we present a sufficient framework on the emergence of a periodically rotating one-point cluster to the TCS model with a harmonic potential force. In the presence of external potential force, large-time behavior of the kinetic TCS model is completely different from that of the kinetic TCS model without an external force. For the relaxation toward the periodic motion in thermo-mechanical observables, we use dynamical systems theory such as the Lyapunov functional approach, method of characteristics and continuity argument. First, we derive the exponential decay estimate for the Lyapunov functionals of thermo-mechanical observables for the kinetic density $f$ with a compact support. Second, we show that the supports of $f$ in $(x,v,θ)$ shrink to a periodically rotating one-point cluster exponentially fast. Our sufficient framework is formulated in terms of initial configuration, coupling strengths and communication weight functions.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmaa.2021-0002
Communications in Mathematical Analysis and Applications, Vol. 1 (2022), Iss. 1 : pp. 72–111
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 40
Keywords: One-point cluster harmonic potential field thermodynamic Cucker-Smale ensemble kinetic model.