  We are concerned with the construction of global-in-time strong solutions for
the incompressible Vlasov-Navier-Stokes systemin the whole three-dimensional
space. One of our goals is to establish that small initial velocities with
critical Sobolev regularity and sufficiently well localized initial kinetic
distribution functions give rise to global and unique solutions. This
constitutes an extension of the celebrated result for the
incompressibleNavier-Stokes equations (NS) that has been established in 1964 by
Fujita and Kato. If in addition the initial velocity is integrable, we
establish that the total energy of the system decays to 0 with the optimal rate
t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the
use of a higher order energy functional that controls the regularity $H^1$ of
the velocityand seems to have been first introduced by Li, Shou and Zhang in
the contextof nonhomogeneous Vlasov-Navier-Stokes system. In the small data
case, we show that this energy functional decays with the rate t^{-5/2}.
