On the weak flocking of the kinetic Cucker–Smale model in a fully noncompact support setting

Abstract We study the emergent behavior of the weak solutions to the kinetic Cucker–Smale (KCS) model in a noncompact spatial‐velocity support setting. Unlike the compact‐support situation, the noncompact support of a weak solution can cause a communication weight to have a zero lower bound, and position difference does not have a uniformly linear growth bound. These cause 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 high‐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 develop an time‐varying effective region approach to 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 weak flocking dynamics of the KCS model even for fully noncompact support settings in phase space and generalizes earlier results on strong flocking dynamics in a compact support setting.