Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids

Hyeong-Ohk Bae,Moon-Jin Kang,Seung-Yeal Ha,Young-Pil Choi

doi:10.3934/dcds.2014.34.4419

Abstract

We propose a coupled system for the interaction between Cucker-Smaleflocking particles and viscous compressible fluids, and present aglobal existence theory and time-asymptotic behavior for theproposed model in the spatial periodic domain $\mathbb{T}^3$. Our modelconsists of the kinetic Cucker-Smale model for flocking particlesand the isentropic compressible Navier-Stokes equations for fluids,and these two models are coupled through a drag force, which is responsible for the asymptotic alignmentbetween particles and fluid. For the asymptotic flocking behavior,we explicitly construct a Lyapunov functional measuring thedeviation from the asymptotic flocking states. For a large viscosityand small initial data, we show that the velocities of Cucker-Smaleparticles and fluids are asymptotically aligned to the commonvelocity.

Highlights

Collective behavior is a common natural phenomenon, such as fish swimming, birds flocking and agents synchronizing
It is expected that the developed technique as well as the physical discoveries could be potentially useful for designing automatic controllers, e.g. deep brain stimulators, to remedy some synchronization-induced mental disorders including Parkinson disease and epilepsy
In this report, we investigate the multi-cluster flocking behavior of the hierarchical Cucker-Smale model parameterized by a constant $\beta$ measuring the strength of the interaction between agents

Summary

Introduction

Collective behavior is a common natural phenomenon, such as fish swimming, birds flocking and agents synchronizing. Hyeong-Ohk Bae Department of Financial Engineering,Ajou University, Suwon 443-749, Korea Abstract: We propose a coupled system for the interaction between CuckerSmale flocking particles and viscous compressible fluids, and present a global existence theory and time-asymptotic behavior for the proposed model in the spatial periodic domain T^3 . For a large viscosity and small initial data, we show that the velocities of Cucker-Smale particles and fluids are asymptotically aligned to the common velocity.

Results

Conclusion