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 ℝ³ × ℝ³. 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 [19]: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, and (2) a bootstrap argument allows us to obtain such a control. This second 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.