Time-asymptotic interaction of flocking particles and an incompressible viscous fluid

We present a new coupled kinetic-fluid model for the interactions between CuckerSmale (C-S) flocking particles and incompressible fluid on the periodic spatial domain T ^d. Our coupled system consists of the kinetic C-S equation and the incompressible NavierStokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of individual C-S particles and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.