Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit

Hyunjin Ahn; Seung-Yeal Ha; Jeongho Kim

doi:10.3934/cpaa.2021156

Article Contents

2021, Volume 20, Issue 12: 4209-4237. Doi: 10.3934/cpaa.2021156

This issue Previous Article Cesaro summation by spheres of lattice sums and Madelung constants Next Article Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction

Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit

1.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
3.
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

^* Corresponding author
^* Corresponding author

Received: March 2021

Revised: August 2021

Early access: September 2021

Published: December 2021

The work of S.-Y. Ha was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881), and the work of J. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066)

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Abstract

We present a uniform(-in-time) stability of the relativistic Cucker-Smale (RCS) model in a suitable framework and study its application to a uniform mean-field limit which lifts earlier classical results for the CS model in a relativistic setting. For this, we first provide a sufficient framework for an exponential flocking for the RCS model in terms of the diameters of state observables, coupling strength and communication weight function, and then we use the obtained exponential flocking estimate to derive a uniform $ \ell_{q,p} $-stability of the RCS model under appropriate conditions on initial data and system parameters. As an application of the derived uniform $ \ell_{q,p} $-stability estimate, we show that a uniform mean-field limit of the RCS model can be made for some admissible class of solutions uniformly in time. This justifies a formal derivation of the kinetic RCS equation [18] in a rigorous setting.

Keywords:

Flocking,
kinetic model,
relativistic Cucker-Smale model,
uniform $ \ell_{q, p} $-stability uniform mean-field limit.

Mathematics Subject Classification: 82C10, 82C22, 74A25, 92D25.

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References

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