|Date||Nov 26, 2021|
In this thesis, we investigate various relativistic Cucker-Smale (Say, RCS) models. First, we present flocking dynamics and uniform(-in-time) stability of the 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. Second, we provide a sufficient framework for the uniform-in-time nonrelativistic limits for the RCS model and the relativistic kinetic Cucker-Smale (RKCS) equation. Here, we highlight that most of the previous works on the nonrelativistic limit are only done in a finite-time interval. Therefore, an interesting question is whether one can derive nonrelativistic limit which is valid on the whole time interval in RCS and RKCS models. Third, we give a relativistic counterpart of the Cucker-Smale(CS) model on Riemannian manifolds and study its collective behavior. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using the LaSalle invariance principle. As concrete examples, we further analyze the RCS models on the d-dimensional unit sphere and the d-dimensional hyperbolic space.