Asymptotic flocking dynamics of a relativistic Cucker-Smale flock under singular communications

Abstract

We study collision avoidance and flocking dynamics for the relativistic Cucker-Smale (RCS) model with a singular communication weight. For a bounded and regular communication weight, RCS particles can exhibit collisions in finite time depending on the geometry of the initial configuration. In contrast, for a singular communication weight, when particles collide, the associated Cucker-Smale vector field becomes unbounded and the standard Cauchy-Lipschitz theory cannot be applied so that existence theory after collisions is problematic. Thus, the collision avoidance problem is directly linked to the global solvability of the singular RCS model and asymptotic flocking dynamics. In this paper, we present sufficient frameworks leading to the nonexistence of finite-time collisions and asymptotic flocking in terms of initial configuration and blow-up rate at the singular point of communication weight.