On the long-time behavior of global solutions of the Vlasov-Navier-Stokes equations
ABSTRACT
The incompressible Vlasov-Navier-Stokes system is a prototype model used to describe the dynamics of aerosols, i.e. suspensions of light particles in a surrounding viscous incompressible fluid. It comprises a kinetic transport equation for the particle distribution (depending on time, space, and kinetic variables) and the incompressible Navier–Stokes equations. These equations are coupled via a drag term known as the Brinkman force.
Over the last two decades, a number of studies have focused on the mathematical analysis of these equations. Notably, it has been observed that, as with the classical Navier–Stokes equations, global weak solutions with finite energy exist for any finite energy data and uniqueness holds in dimension two.
This course will review some recent results concerning the long-time behavior of these global solutions. It will be shown to differ significantly depending on whether the equations are considered in the whole space or under periodic boundary conditions.
