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

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 T³. 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.