On the emergence of local flocking phenomena in Cucker-Smale ensemble

Abstract

We study the Cucker-Smale model which describes the flocking phenomena. In detail, we focus on the sufficient conditions to achieve the local flocking phenomena in various scenarios and models. The Lyapunov functional approach and bootstrapping argument play key roles to prove the asymptotic stability on emergence of flocking along time evolution. In the Cucker-Smale model, the dynamics of the particles are presented by the couplings proportional to the relative velocities between each pair of particles. The sufficient condition to the global flocking was suggested by Cucker and Smale, while necessary condition or local flocking was not analytically studied. Our interests covers not only the Cucker-Smale particle model, but also the unit-speed model and the hydrodynamic model. The unit-speed model breaks the symmetry of equations and the hydrodynamic model needs the existence of solutions. The hydrodynamic equations are the macroscopic description through the mean-field limit process of the particle model. We avoid free boundary problems by describing Lagrangian variables.