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.

Cited by in corpus (12)

• Eulerian dynamics in multi-dimensions with radial symmetry
• Anticipation breeds alignment
• Global regularity for a 1D Euler-alignment system with misalignment
• Control to flocking of the kinetic Cucker-Smale model
• Global solutions for the two dimensional Euler-Poisson system with attractive forcing
• Critical Thresholds in One Dimensional Damped Euler-Poisson Systems
• Traffic flow models with looking ahead-behind dynamics
• Global dynamics of the Euler-alignment system with weakly singular kernel
• Sharp critical thresholds in a hyperbolic system with relaxation
• Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the $L^2$ norm
• Wave turbulence and collective behavior models for wave equations with short- and long-range interactions
• Critical thresholds in a nonlocal Euler system with relaxation