The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria

TL;DR

This paper investigates the existence and stability of stationary solutions in a 2D Vlasov-Navier-Stokes system with boundary conditions, demonstrating their asymptotic stability near small Poiseuille flows.

Contribution

It establishes the existence and asymptotic stability of regular equilibria in a 2D Vlasov-Navier-Stokes system with boundary conditions, using geometric control conditions.

Findings

Existence of stationary states near small Poiseuille flows.

Asymptotic stability of these states under perturbations.

Use of geometric control conditions to prevent kinetic concentration.

Abstract

In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase.