|Date||Nov 21, 2022|
n this thesis, we investigate Cucker-Smale type (in short, CS-type) models, mainly focusing on a case of a singular kernel. The CS-type model introduces an activation function to the CS model to describe various group phenomena, and the theory of relativity can be reflected as an example.
To motivate the CS-type model, we first introduce the relativistic Cucker-Smale (in short, RCS) model with a singular kernel. More precisely, we study collision avoidance and flocking dynamics for the 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 the 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.
We then propose the CS-type model, which is a general nonlinear consensus model incorporating the RCS model. Depending on the regularity and singularity of communication weight at the origin and far-field, we provide diverse clustering patterns for collective behaviors on the real line. The singularity of the kernel induces collision avoidance or sticking property, depending on the integrability of the kernel near the origin. We study the regularity of sticking solutions of the proposed model on the real line. On the other hand, we provide a sufficient framework beyond collision avoidance property to guarantee a strict lower bound between agents in the Euclidean space. We then introduce a kinetic analog of the proposed model and study its well-posedness. We also show the structural stability in both particle and kinetic levels.