Emergent dynamics of spatially extended relativistic kinetic Cucker-Smale model

Abstract: We study the emergent dynamics of the relativistic kinetic Cucker-Smale (RKCS) model without assuming compactness in spatial and velocity support. In this setting, the lower bound of the kernel function in the nonlocal velocity alignment force can be zero so that the previous approach based on the energy method does not provide a quantitative flocking estimate. To overcome this difficulty, we introduce a suitable decay ansatz for the one-particle distribution function and an {\it effective domain} by identifying a time-varying region in which the total mass outside of it decays to zero asymptotically. Using these two ingredients, we show that weak flocking dynamics emerges asymptotically in the sense that the second moment for the velocity fluctuation around the velocity average tends to zero asymptotically, whereas the second moment for spatial fluctuations around the center of mass remains bounded uniformly in time. Our results demonstrate the robustness of the emergent dynamics in the RKCS model across various non-compact physically important distributions, including Gaussian, sub-Gaussian, and $D$-th moment integrable distributions.