Approximate Controllability of Second-Grade Fluids


This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids is approximately controllable by a finite-dimensional control force.

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  1. 1.

    Agrachev AA, Kuksin S, Sarychev AV, Shirikyan A. On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations. Ann Inst H Poincaré Probab Stat 2007;43(4):399–415.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Agrachev AA, Sarychev AV. Navier-Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 2005;7(1):108–52.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Agrachev AA, Sarychev AV. Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing. Commun Math Phys 2006;265(3):673–97.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bernard J-M. Solutions globales variationnelles et classiques des fluides de grade deux. C R Acad Sci Paris 1998;327(Série I):953–8.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bresch D, Lemoine J. On the existence of solutions for non-stationary second grade fluids. Navier-Stokes equations and related non-linear problems, Palanga, 1997. Utrecht: VSP; 1998. p. 1530.

  6. 6.

    Cioranescu D, Girault V. Weak and classical solutions of a family of second grade fluids. Int J Non-Linear Mech 1997;32:317–35.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cioranescu D, Ouazar EH. Existence and uniqueness for fluids of second grade. Nonlinear partial differential equations and their applications. Collège de France seminar, Vol VI (Paris, 1982/1983). Boston: Pitman; 1984. p. 178–97.

  8. 8.

    Dunn JE, Fosdick RL. Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade. Arch Ration Mech Anal 1974; 56:191–252.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Foias C, Holm D, Titi ES. The Navier-Stokes-alpha model of fluid turbulence, Special Issue in Honor of V. E. Zakharov on the Occasion of His 60th Birthday. Phys D 2001;152:505–19.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Foias C, Holm D, Titi ES. The three-dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory. J Dyn Differ Equ 2002;14:1–35.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Galdi GP, Rajagopal KR. Slow motion of a body in a fluid of second grade. Int J Eng Sci 1997;35:33–54.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Galdi GP, Sequeira A. Further existence results for classical solutions of the equations of second grade fluids. Arch Ration Mech Anal 1994;128:297–312.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Galdi GP, Grobbelaarvandalsen M, Sauer N. Existence and uniqueness of classical solutions of the equations of motion for 2nd-grade fluids. Arch Ration Mech Anal 1993;124:221–37.

    Article  Google Scholar 

  14. 14.

    Galdi GP, Grobbelaarvandalsen M, Sauer N. Existence and uniqueness of solutions of the equations of motion for a fluid of second grade with non-homogeneous boundary conditions. Int J Non-Linear Mech 1995;30:701–9.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Girault V, Saadouni M. On a time-dependent grade-two fluid model in two dimensions. Comput Math Appl 2007;53:347–60.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Girault V, Scott LR. Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition. J Math Pures Appl 1999;78:981–1011.

    MathSciNet  Article  Google Scholar 

  17. 17.

    Jurdjevic V. Geometric control theory. Cambridge: Cambridge University Press; 1997.

    Google Scholar 

  18. 18.

    Le Roux C. Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions. Arch Rational Mech Anal 1999;148:309–56.

    MathSciNet  Article  Google Scholar 

  19. 19.

    Moise I, Rosa R, Wang X. Attractors for non-compact semigroups via energy equations. Nonlinearity 1998;11(5):1369–93.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Nersisyan H. Controllability of 3D incompressible Euler equations by a finite-dimensional external force. ESAIM Control Optim Calc Var 2010;16 (3):677–94.

    MathSciNet  Article  Google Scholar 

  21. 21.

    Nersisyan H. Controllability of the 3D compressible Euler system. Commun Partial Differ Equ 2011;36(9):1544–64.

    MathSciNet  Article  Google Scholar 

  22. 22.

    Nersesyan V. Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force. Nonlinearity 2015;28(3): 825–48.

    MathSciNet  Article  Google Scholar 

  23. 23.

    Paicu M, Raugel G. Dynamics of second grade fluids: the Lagrangian approach. Recent trends in dynamical systems springer proceedings in mathematics & statistics; 2013. p. 517–53.

  24. 24.

    Paicu M, Raugel G, Rekalo A. Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations. J Differ Equ 2012;252(6):3695–751.

    MathSciNet  Article  Google Scholar 

  25. 25.

    Phan D, Rodrigues SS. Approximate controllability for 3D Navier-Stokes equations under Lions boundary conditions. J Dyn Control Syst 2019;25(3): 351–76.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Rivlin RS, Ericksen JL. Stress-deformation relations for isotropic materials. J Ration Mech Anal 1955;4(4):323–425.

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Rodrigues SS. Navier-Stokes equation on the rectangle: controllability by means of low modes forcing. J Dyn Control Syst 2006;12(4):517–62.

    MathSciNet  Article  Google Scholar 

  28. 28.

    Shirikyan A. Approximate controllability of three-dimensional Navier-Stokes equations. Commun Math Phys 2006;266(1):123–51.

    MathSciNet  Article  Google Scholar 

  29. 29.

    Shirikyan A. Contrôlabilité exacte en projections pour les équations de Navier-Stokes tridimensionnelles [Exact controllability in projections for three-dimensional Navier-Stokes equations]. Ann Inst H Poincaré Anal Non Linéaire 2007; 24(4):521–37.

    MathSciNet  Article  Google Scholar 

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This work was partially done during the visit of the first author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The first author is thankful to the VIASM for the support and for the very kind hospitality of the institute and of all the staff.


The first author received financial support from VIASM.

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Correspondence to Van-Sang Ngo.

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Ngo, V., Raugel, G. Approximate Controllability of Second-Grade Fluids. J Dyn Control Syst (2020).

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  • Second grade fluid equations
  • Approximate controllability
  • Agrachev-Sarychev method

Mathematics Subject Classification (2010)

  • 35Q35
  • 93B05
  • 93C20