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

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

doi:10.3934/dcds.2014.34.4419

Article Contents

2014, Volume 34, Issue 11: 4419-4458. Doi: 10.3934/dcds.2014.34.4419

This issue Previous Article Commensurable continued fractions Next Article Topological and ergodic properties of symmetric sub-shifts

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

1.
Department of Financial Engineering, Ajou University, Suwon 443-749
2.
Department of Mathematics, Imperial College London, London SW7 2AZ
3.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747
4.
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712

Received: October 31, 2013

Revised: November 30, 2013

Published: October 2014

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Abstract

We propose a coupled system for the interaction between Cucker-Smale 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 $\mathbb{T}^3$. Our model consists of the kinetic Cucker-Smale model for flocking particles and the isentropic compressible Navier-Stokes equations for fluids, and these two models are coupled through a drag force, which is responsible for the asymptotic alignment between particles and fluid. For the asymptotic flocking behavior, we explicitly construct a Lyapunov functional measuring the deviation from the asymptotic flocking states. 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.

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Mathematics Subject Classification: 35Q30, 35Q70, 76N10, 35Q83.

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