THE MEAN-FIELD LIMIT OF THE RELATIVISTIC CUCKER-SMALE MODEL ON COMPLETE RIEMANNIAN MANIFOLDS

초록

. 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 [9] 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 particlein-cell method and study its emergent dynamics. For the proposed kinetic model, we provide 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 [4,32] and show the global existence of a unique measure-valued solution.

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