Acta Mathematica Vietnamica

, Volume 42, Issue 2, pp 357–367 | Cite as

On the Stationary Solutions to 2D g-Navier-Stokes Equations

  • Dao Trong QuyetEmail author
  • Nguyen Viet Tuan


We consider g-Navier-Stokes equations in a two-dimensional smooth bounded domain Ω. First, we study the existence and exponential stability of a stationary solution under some certain conditions. Second, we prove that any unstable steady state can be stabilized by proportional controller with support in an open subset \(\omega \subset {\Omega }\) such that Ω∖ω is sufficiently “small.”


g-Navier-Stokes equations Stationary solutions Stabilizable Feedback controller 

Mathematics Subject Classification (2010)

35B35 35Q35 35D35 



This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.10.

The authors would like to thank Cung The Anh for suggestions and stimulating discussions on the subject of the paper.


  1. 1.
    Anh, C.T., Quyet, D.T.: Long-time behavior for 2D non-autonomous g-Navier-Stokes equations. Ann. Pol. Math. 103, 277–302 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anh, C.T., Quyet, D.T., Tinh, D.T.: Existence and finite time approximation of strong solutions of the 2D g-Navier-Stokes equations. Acta Math. Vietnam. 28, 413–428 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anita, S.: Internal stabilizability of diffusion equation. Nonlinear Studies 8, 193–202 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barbu, V., Lefter, C.: Internal stabilizability of the Navier-Stokes equations. Systems Control Lett. 48, 161–167 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barbu, V., Sritharan, S.S.: Feedback stabilization of the magneto-hydrodynamic system. Semigroups of Operators: Theory and Applications (Rio de Janeiro, 2001), pp 45–53. Optimization Software, New York (2002)zbMATHGoogle Scholar
  6. 6.
    Bae, H., Roh, J.: Existence of solutions of the g-Navier-Stokes equations. Taiwanese J. Math. 8, 85–102 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Jiang, J., Hou, Y.: The global attractor of g-Navier-Stokes equations with linear dampness on \(\mathbb {R}^{2}\). Appl. Math. Comp. 215, 1068–1076 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jiang, J., Hou, Y.: Pullback attractor of 2D non-autonomous g-Navier-Stokes equations on some bounded domains. App. Math. Mech. -Engl. Ed. 31, 697–708 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jiang, J., Hou, Y., Wang, X.: Pullback attractor of 2D nonautonomous g-Navier-Stokes equations with linear dampness. Appl. Math. Mech. Engl. Ed. 32, 151–166 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jiang, J., Wang, X.: Global attractor of 2D autonomous g-Navier-Stokes equations. Appl. Math. Mech. (English Ed.) 34, 385–394 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kwak, M., Kwean, H., Roh, J.: The dimension of attractor of the 2D g-Navier-Stokes equations. J. Math. Anal. Appl. 315, 436–461 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kwean, H.: The H 1-compact global attractor of two-dimensional g-Navier-Stokes equations. Far East J. Dyn. Syst. 18, 1–20 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kwean, H., Roh, J.: The global attractor of the 2D g-Navier-Stokes equations on some unbounded domains. Commun. Korean Math. Soc. 20, 731–749 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Quyet, D.T.: Asymptotic behavior of strong solutions to 2D g-Navier-Stokes equations. Commun. Korean Math. Soc. 29, 505–518 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Robinson, J.C.: Introduction to infinite-dimensional dynamical systems. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Roh, J.: Dynamics of the g-Navier-Stokes equations. J. Differ. Equ. 211, 452–484 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Roh, J.: Derivation of the g-Navier-Stokes equations. J. Chungcheon Math. Soc. 19, 213–218 (2006)Google Scholar
  18. 18.
    Wang, G.: Stabilization of the Boussinesq equation via internal feedback controls. Nonlinear Anal. 52, 485–506 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wu, D., Tao, J.: The exponential attractors for the g-Navier-Stokes equations. J. Funct. Spaces Appl. Art. ID, 503454, 12 p (2012)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Faculty of Information TechnologyLe Quy Don Technical UniversityHanoiVietnam
  2. 2.Department of MathematicsSao Do UniversityHai DuongVietnam

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