The mean-field limit of the relativistic Cucker--Smale model on complete Riemannian manifolds

다국어 초록 (Multilingual Abstract)

We study the mean-field limit of the relativistic Cucker--Smale (RCS) model on complete smooth Riemannian manifolds. The RCS model describes the collective dynamics of the relativistic Cucker--Smale particles on abstract manifolds, and it was first introduced as the generalization of the Cucker--Smale model in special relativity framework via a suitable ansatz for entropy, and using Boillat and Ruggeri's principle of subsystem \cite{B-R} from the Euler equations for a gas mixture on Riemannian manifolds. In this paper, we derive a Vlasov-type kinetic RCS model on Riemannian manifolds using manifold counterpart of particle-in-cell method and study its emergent dynamics. For the proposed kinetic model, we provide {\it a priori} velocity alignment estimate via the dissipation of total energy. We also adopt the concept of measure-valued solution to the kinetic RCS model in \cite{A-H-K-S-S, H-L} and show the global existence of a unique measure-valued solution.