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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \ell_{q,p} $\end{document}</tex-math></inline-formula>-stability of the RCS model under appropriate conditions on initial data and system parameters. As an application of the derived uniform <inline-formula><tex-math id="M2">\begin{document}$ \ell_{q,p} $\end{document}</tex-math></inline-formula>-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 [<xref ref-type="bibr" rid="b18">18</xref>] in a rigorous setting.</p>