Stabilization of the non-homogeneous Navier-Stokes equations in a 2d channel

TL;DR

This paper develops a boundary feedback control method to stabilize the velocity and density of a non-homogeneous fluid flow in a 2D channel around Poiseuille flow, with a finite-dimensional control operator.

Contribution

It introduces a novel boundary feedback control strategy for the non-homogeneous Navier-Stokes equations in a 2D channel, ensuring local stabilization with finite-dimensional control.

Findings

Successful stabilization of velocity and density around Poiseuille flow.

Control operator has finite-dimensional range.

Applicable for initial densities with perturbations supported away from boundaries.

Abstract

In this article we study the local stabilization of the non-homogeneous Navier- Stokes equations in a 2d channel around Poiseuille flow. We design a feedback control of the velocity which acts on the inflow boundary of the domain such that both the fluid velocity and density are stabilized around Poiseuille flow provided the initial density is given by a constant added with a perturbation, such that the perturbation is supported away from the lateral boundary of the channel. Moreover the feedback control operator we construct has finite dimensional range.