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

Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France.

^* Corresponding author: Sourav.Mitra@math.univ-toulouse.fr

Received: 11 July 2018
Accepted: 12 May 2019

Abstract

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.