Weak and Sharp Stretched-Exponential Flocking in the Kinetic Cucker-Smale Model with Noncompactly Supported Initial Data

Abstract

We study the long-time behavior of the kinetic Cucker-Smale model under initial data with unbounded support in both position and velocity. Such data invalidate classical diameter-based flocking arguments and make sustained spatial cohesion difficult to establish. We prove an estimate for spatial characteristics under a finite spatial second moment and a sufficiently high velocity moment, without any requirement on coupling strength. We also prove weak flocking, i.e. control of spatial variance and integrability of velocity fluctuations in time. Under stronger assumptions on the initial tails, we establish uniform propagation of spatial exponential moments of explicitly determined order, a substantially stronger conclusion than bounded spatial variance. These estimates lead to explicit stretched-exponential rates of velocity alignment and convergence of the centered spatial distribution toward a limiting profile. Another contribution is the construction of countably atomic solutions that show optimality of the decay exponent in the spatial-tail-limited class. Together, the results clarify how initial phase-space tails govern spatial confinement and quantitative flocking for kinetic distributions without compact support.