title: Critical thresholds in flocking hydrodynamics with non-local alignment.
abstract: 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.
authors.affiliation: Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Department of Mathematics, University of Maryland, College Park, MD 20742, USA Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA tadmor@cscamm.umd.edu.
authors.forename: Eitan
authors.initials: E
authors.lastname: Tadmor
authors.affiliation: Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
authors.forename: Changhui
authors.initials: C
authors.lastname: Tan
keywords: alignment; critical thresholds; flocking; hydrodynamics; regularity