The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of   regular equilibria
  In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase. 
Olivier Glass, Daniel Han-Kwan and Ayman Moussa
math.AP