Flocking dynamics and optimal classical limit of the relativistic Cucker–Smale model with infinite particles

We study the flocking behavior and classical limit of the infinite particles counterpart of the Relativistic Cucker-Smale (in short, RCS) model introduced in Ha et al., Arch. Ration. Mech. Anal. 235, 1661–1706 (2020). First, we derive the infinite RCS model from the kinetic RCS model with non-compact support through the particle-in-cell method, and then we show that the diameter and radius of the velocity variable in the RCS model are non-increasing. Next, we construct a system of dissipative differential inequalities via the finite truncation of the proposed infinite RCS model. This overcomes the difficulty brought by the inaccessibility of infinite norms, which enables us to analyze the solution’s flocking behaviors. Finally, we establish the model’s classical limit and derive the optimal convergence rate in terms of the speed of light, which is the same as a finite-time classical limit.