On the weak flocking of the kinetic Cucker-Smale model in a fully non-compact support setting

TL;DR

This paper investigates the flocking behavior of the kinetic Cucker-Smale model in non-compact support settings, establishing conditions for emergent flocking despite challenges posed by unbounded support.

Contribution

It extends previous flocking results to non-compact support scenarios by deriving new estimates and proving the robustness of flocking behavior under broader conditions.

Findings

Velocity deviation tends to zero asymptotically.

Spatial deviation remains uniformly bounded over time.

Flocking persists even with unbounded phase space support.

Abstract

We study the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short, KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can cause a communication weight to have zero lower bounds, and position difference does not have a uniformly linear growth bound. These cause the previous approach based on the nonlinear functional approach for spatial and velocity diameters to break down. To overcome these difficulties, we derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution using the estimate on the deviation of particle trajectories. For the estimate of emergent dynamics, we consider two classes of distribution functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation from an average velocity tends to zero asymptotically, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This illustrates the robustness of the mono-cluster flocking dynamics of the KCS model even for fully non-compact support settings in phase space and generalizes earlier results on flocking dynamics in a compact support setting.