Weak flocking for phase-spatially extended kinetic Cucker-Smale equation in a confining force field

Abstract: We study the quantitative fast and slow weak flocking of the phase-spatially extended kinetic Cucker-Smale model in confining potential fields. The confining potential is allowed to be nonconvex. In contrast to the phase-spatially confined case, the communication weight may have no positive lower bounds, while spatial and velocity diameters may remain infinite. Moreover, an indefinite Hessian of confining potential prevents the direct use of convexity-based coercive estimates. To overcome these difficulties, we identify three structural assumptions that are sufficient for the noncompact hypocoercive method: quadratic confinement, a globally Lipschitz force, and a strict virial inequality. The admissible class of confining potentials includes genuinely nonconvex radially symmetric ones and localized oscillatory perturbations of the harmonic potential. Weak flocking analysis combines three key ingredients: a microscopic mechanical-energy estimate, a time-varying effective region, and a macroscopic Lyapunov functional. For this, we first establish global existence for initial data with finite second moments. In the exponential mechanical-energy class, we further show the uniqueness and finite-time Osgood-type stability in 1-Wasserstein distance. For polynomially decaying initial mechanical-energy tails, we derive the optimal algebraic decay exponent of the fluctuation energy throughout the admissible communication-decay regime. The optimality is verified by a symmetric countably infinite particle solution for a fixed nonconvex potential. For exponentially decaying tails, a two-stage localization argument yields exponential weak flocking on an optimal exponential time scale. These results show that well-posedness and weak flocking persist even for fully noncompact data under genuinely nonconvex confining forces.