On the weak flocking of the kinetic Cucker-Smale model in a fully non-compact support setting
Seung-Yeal Ha, Xinyu Wang
TL;DR
This work advances the theory of the kinetic Cucker-Smale model by proving uniqueness and weak flocking for weak solutions in a fully non-compact phase-space setting, where previous methods relying on compact support fail. It introduces a trajectory-deviation functional to establish uniqueness and develops refined moment estimates to track spatial and velocity dispersions. By examining two decay classes—polynomial and exponential—the authors show that the velocity deviation converges to zero while the spatial dispersion remains uniformly bounded, demonstrating robust mono-cluster flocking beyond compact support. These results broaden the applicability of flocking theory to more realistic, unbounded configurations and provide quantitative decay rates tied to decay rates and moment orders, with implications for mean-field descriptions of large CS-type systems.
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.