  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.
