@Article{CMAA-1-72,
author = {Du , Linglong and Ha , Seung-Yeal},
title = {Convergence Toward a Periodically Rotating One-Point Cluster in the Kinetic Thermodynamic Cucker-Smale Model},
journal = {Communications in Mathematical Analysis and Applications},
year = {2022},
volume = {1},
number = {1},
pages = {72--111},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2790-1939},
doi = {https://doi.org/10.4208/cmaa.2021-0002},
url = {http://global-sci.org/intro/article_detail/cmaa/20158.html}
}