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

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    * Corresponding author 

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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  • 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.

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


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