Abstract

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 HKMM: (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.

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