A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid

Abstract

We present a new hydrodynamic model for the interactions between collision-free Cucker-Smale flocking particles and a viscous incompressible fluid. Our proposed model consists of two hydrodynamic models. For the Cucker-Smale flocking particles, we employ the pressureless Euler system with a non-local flocking dissipation, whereas for the fluid, we use the incompressible Navier-Stokes equations. These two hydrodynamic models are coupled through a drag force, which is the main flocking mechanism between the particles and the fluid. The flocking mechanism between particles is regulated by the Cucker-Smale model, which accelerates global flocking between the particles and the fluid. We show that this model admits the global-in-time classical solutions, and exhibits time-asymptotic flocking, provided that the initial data is appropriately small. In the course of our analysis for the proposed system, we first consider the hydrodynamic Cucker-Smale equations (the pressureless Euler system with a non-local flocking dissipation), for which the global existence and the time-asymptotic behavior of the classical solutions are also investigated.