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

## TL;DR

This paper investigates the long-term behavior of solutions to the Vlasov-Navier-Stokes system on a three-dimensional torus, proving convergence of the distribution function to a velocity Dirac mass with exponential rate under small initial energy.

## Contribution

It establishes the exponential convergence of the distribution function to a Dirac mass in velocity for the Vlasov-Navier-Stokes system on the torus, under small initial modulated energy.

## Key findings

- Distribution function converges to a Dirac mass in velocity.
- Convergence occurs at an exponential rate.
- Global bounds on moments are achieved through bootstrap analysis.

## 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.