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

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 particle-in-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.