Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains

TL;DR

This paper investigates the long-term behavior of small solutions to the Vlasov-Navier-Stokes system in bounded domains, showing fluid velocity decay and kinetic concentration with exponential rate, considering boundary effects and initial data variations.

Contribution

It provides a detailed analysis of the asymptotic behavior of solutions, including boundary effects and initial data influence, extending previous methods to bounded domains.

Findings

Fluid velocity homogenizes to zero

Distribution function concentrates at zero velocity

Exponential rate of convergence

Abstract

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $\Omega \times \mathbb{R}^3$, for a smooth bounded domain $\Omega$ of $\mathbb{R}^3$, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to $0$ while the distribution function concentrates towards a Dirac mass in velocity centered at $0$, with an exponential rate. The proof, which follows the methods introduced in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.