Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains

We study the large time behaviour of small data solutions to the Vlasov-Navier-Stokes system set on Omega x R-3, for a smooth bounded domain Omega of R-3, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to 0 while the distribution function concentrates towards a Dirac mass in velocity centred at 0, with an exponential rate. The proof, which follows the methods introduced in Han-Kwan et al (2020 Arch. Ration. Mech. Anal. 236 1273-323), requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviours for the kinetic density, from total absorption to no absorption at all.