# American Institute of Mathematical Sciences

November  2014, 34(11): 4419-4458. doi: 10.3934/dcds.2014.34.4419

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

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

Received  November 2013 Revised  December 2013 Published  May 2014

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.
Citation: Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419
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