Large time behavior of the Vlasov-Navier-Stokes system on the torus

Abstract

We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on ${\mathbb{T}}^3\times {\mathbb{R}}^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing to get global bounds on moments.