Critical thresholds in flocking hydrodynamics\\with nonlocal alignment

Abstract

We study the large-time behavior of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g., Cucker-Smale and Motsch-Tadmor models \cite{CS,MT}. We prove that in analogy with the agent-based models, the presence of non-local alignment enforces \emph{strong} solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional setups, proving global regularity for \emph{sub-critical} initial data. Indeed, we show that there exist \emph{critical thresholds} in the phase space of initial configuration which dictate the global regularity vs. a finite time blow-up. In particular, we explore the regularity of nonlocal alignment in the presence of vacuum.