Global regularity of two-dimensional flocking hydrodynamics

Abstract

We study the systems of Euler equations that arise from agent-based dynamics driven by velocity alignment. It is known that smooth solutions to such systems must flock, namely the large-time behavior of the velocity field approaches a limiting “flocking” velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration that characterizes the global regularity and hence the flocking behavior of such two-dimensional systems. Specifically, we prove for that a large class of sub-critical initial conditions such that the initial divergence is “not too negative” and the initial spectral gap is “not too large”, global regularity persists for all time.