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Journal of Dynamical and Control Systems

, Volume 25, Issue 2, pp 197–218 | Cite as

Global Stabilization of the Navier-Stokes Equations Around an Unstable Steady State with Mixed Boundary Kinetic Energy Controller

  • Abdou SèneEmail author
  • Timack Ngom
  • Evrad M. D. Ngom
Article
  • 65 Downloads

Abstract

The paper, benefiting from techniques developed in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), presents a mixed (Dirichlet-Neumann) boundary feedback controller for stabilizing the Navier-Stokes equations around a prescribed steady state, in a bounded domain \({\Omega }\). The Neumann part of the boundary controller is designed to be zero when the inflow vanishes, and to have the magnitude of the kinetic energy. Like in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), the present paper proves exponential decrease of the perturbation in \(L^{2}\), without blowup. In addition, it goes further than (Ngom et al., Evol Equ Control Theory. 2015;4:89–106) by proving, on the one hand, that the exponential convergence towards zero holds in \(H^{1}\), on the other hand, that the weak solution is unique when the computational domainis two-dimensional.

Keywords

Navier-Stokes system Boundary feedback stabilization Mixed boundary conditions Faedo-Galerkin method 

Mathematics Subject Classification (2010)

93D15 47N70 58E25 34H15 

Notes

Funding Information

This work is supported by the International Science Program, Uppsala, Sweden, and UMI-UMMISCO-IRD (Unité Mixte Internationale de Modélisation Mathématique et Informatique des Systèmes Complexes).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEA-MITICUniversité Virtuelle du SénégalDakar-FannSénégal
  2. 2.Laboratoire de Mathématiques et Applications (LMA)Université Assane SECK de Ziguinchor,ZiguinchorSénégal
  3. 3.CEA-MITICUniversité Gaston Berger (UGB)Saint-LouisSénégal

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