Emergent behaviors of the kinetic Motsch-Tadmor model in a phase-spatially extended setting

RESUMO

The kinetic Motsch-Tadmor (in short, KMT) model is a kinetic flocking model with a normalized communication weight. In this paper, we study the emergent dynamics of the phase-spatially extended KMT model. We first establish a global well-posedness theory in the fully noncompact spatial-velocity setting. To this end, we introduce a direct Lagrangian formulation in which the unbounded part of the initial velocity distribution is separated from an interaction-generated bounded remainder. This decomposition allows us to construct global Lagrangian weak solutions without truncating the velocity distribution and to propagate finite phase-space moments. We then investigate the long-time collective behavior of the resulting solutions. When the initial velocity support is compact while the spatial support is allowed to be noncompact, a time-varying effective-region argument yields a uniform contraction mechanism for the normalized interaction and leads to exponential weak support flocking. When both the spatial and velocity supports are noncompact, support-level flocking is in general impossible. Nevertheless, the same effective-region mechanism, combined with the Lagrangian decomposition and a bootstrap argument for the interaction-generated remainder, yields exponential weak moment flocking. In particular, pairwise spatial moments remain uniformly controlled while velocity fluctuations converge exponentially to zero. These results provide a unified framework for the well-posedness and flocking dynamics of the KMT model beyond the compact-support regime.