On the weak flocking of the spatially extended generalized kinetic Justh–Krishnaprasad equation

Source abstract

Abstract We study the emergent weak flocking behaviors of the generalized kinetic Justh–Krishnaprasad (GKJK) model in a spatially extended setting. For this, we provide a sufficient framework to ensure the emergence of weak flocking behaviors. Weak flocking behaviors refer to the situation in which the second moment for heading angle fluctuation around the heading angle average tends to zero asymptotically, whereas the second moment for spatial fluctuations around the center of mass remains bounded uniformly in time. On an unbounded spatial domain, the nonlocal alignment kernel may lose a uniform positive lower bound, so standard energy‐type argument breaks down as it is. To overcome this difficulty, we impose a suitable tail– decay ansatz on the one‐particle distribution function and introduce the method of an effective time‐varying domain whose complement carries vanishing mass asymptotically. Our results show that polynomial spatial decay of the initial distribution yields at least polynomial convergence to weak flocking, while exponential decay implies exponential convergence. In particular, we also demonstrate that even if the heading angle diameter remains constant, weak flocking behavior can still emerge in the GKJK model. This is the new phenomenon that was not observed in the literature.