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.
Highlights
Settings of the problem We are interested in stabilizing the density dependent Navier–Stokes equations around some stationary state (where is a stationary solution) in a two dimensional channel Ω
For that we will use an appropriate boundary control uc acting on the velocity in the inflow part of the boundary ∂Ω
The plan of this section is as follows (i) In Section 2.1, we study the stabilization of the homogeneous linear system (1.13)4–(1.13)8, using a finite dimensional boundary control
Summary
For that we will use an appropriate boundary control uc acting on the velocity in the inflow part of the boundary ∂Ω. Where ucχΓc is a control function for the velocity v with χΓc denoting the characteristics function of a set Γc which is compactly supported on Γ. The triplet (ρs, vs, ps) is the Poiseuille profile defined as follows ρs is a positive constant, vs(x1, x2) =. In view of the stationary profile (1.3), it is natural to control the inflow part of the boundary, i.e. we will consider the control function uc supported on. We consider the control zone of the form (1.5) to simplify the notations. Our analysis allows to consider any subset {0} × (A, B) (0 < A < B < 1) of Γin as the control zone. To state our results precisely, we introduce some appropriate functional spaces
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
