On the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system with shear thickening

Abstract

We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.