Critical thresholds in flocking hydrodynamics with non-local alignment.

We study the large-time behaviour 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 (2007 IEEE Trans. Autom. Control 52, 852-862. (doi:10.1109/TAC.2007.895842)) and Motsch & Tadmor (2011 J. Stat. Phys. 144, 923-947. (doi:10.1007/s10955-011-0285-9)) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces 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 set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.