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

International audience; We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on $\T^3 \times \R^3$. 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 us to get global bounds on moments.