Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on $\R^3 \times \R^3$. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at $0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.