We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier–Stokes system in the whole three-dimensional space. Our primary goal is to establish that small initial velocities with critical Sobolev regularity \(H^{1/2}\) 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 incompressible Navier–Stokes equations (NS) that has been proved by Fujita and Kato in [11]. Assuming also that the initial velocity is in \(L^1,\) we establish that the total energy \(E_0\) of the system decays to 0 with the same rate \(t^{-3/2}\) as for the weak solutions of (NS), see [22, 24]. Our results partly rely on the use of a higher order energy functional \(E_1\) that controls the regularity \(H^1\) of the velocity. This idea seems to originate from the recent paper [18] by Li, Shou and Zhang, devoted to the inhomogeneous Vlasov-Navier–Stokes system. Here we show that \(E_1\) decays with the rate \(t^{-5/2}\) which, in particular, allows us to prove that the density of the particles has a strong limit when the time goes to infinity.
