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

We present a new coupled kinetic-fluid model for the interactions between Cucker–Smale (C–S) flocking particles and incompressible fluid on the periodic spatial domain . Our coupled system consists of the kinetic C–S equation and the incompressible Navier–Stokes 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.