Skip to main content
Springer Nature Link
Log in

Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ2 where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306(8), 837–852 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Avalos, G., Triggiani, R.: The coupled PDE system arising in fluid/structure interaction. Part I: explicit semigroup generator and its spectral properties. Contemp. Math. 440, 15–54 (2007)

    MathSciNet  Google Scholar 

  3. Ball, J.M.: Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Am. Math. Soc. 63, 370–373 (1977)

    MATH  Google Scholar 

  4. Ball, J.M.: On the asymptotic behavior of generalized processes, with application to nonlinear evolution equations. J. Differ. Equ. 27, 224–265 (1978)

    Article  MATH  Google Scholar 

  5. Ball, J.M., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barbu, V.: Nonlinear Semigroups in Banach Spaces. Nordhoff, Groningen (1974)

    MATH  Google Scholar 

  7. Bociu, L., Zolesio, J.P.: Linearization of a coupled system of nonlinear elasticity and viscuous fluid. Modern Theory of PDE’s. ISAACS (2010, to appear)

  8. Barbu, V., Grujic, Z., Lasiecka, I., Tuffaha, A.: Existence of the energy-level weak solutions for a nonlinear fluid-structure intercation model. Contemp. Math. 440, 55–82 (2007)

    MathSciNet  Google Scholar 

  9. Barbu, V., Grujic, Z., Lasiecka, I., Tuffaha, A.: Smoothness of weak solutions to a nonlinear fluid-structure interaction model. Indiana Univ. Math. J. 57(2), 1173–1207 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brezis, H.: Asymptotic behavior of some evolutionary systems. Nonlinear Evol. Equ. 141–154 (1978)

  11. Chueshov, I., Lasiecka, I.: Long Tome Behavior of Second Order Evolutions with Nonlinear Damping. Memoires of American Mathematical Society, vol. 195. Springer, Berlin (2008). Nr. 912

    Google Scholar 

  12. Caputo, R., Hammer, D.: Effects of microvillus deformability on leukocyte adhesion explored using adhesive dynamics simulations. Biophysics 92, 2183–2192 (2002)

    Google Scholar 

  13. Coutand, D., Shokller, S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Ration. Mech. Anal. 176, 25–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Desajardins, B., Esteban, M.J., Grandmont, C., Le Tallec, P.: Weak solutions for a fluid elastic structure inetraction model. Rev. Mat. Complut. 14, 523–538 (2001)

    MathSciNet  Google Scholar 

  15. Du, Q., Gunzburger, M.D., Hou, L.S., Lee, J.: Analysis of a linear fluid-structure interaction problem. Discrete Contin. Dyn. Syst. 9, 633–650 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fernandez, M.A., Moubachir, M.: An exact Block-Newton algorithm for solving fluid-structure interaction problems. C.R. Acad. Sci. Paris, Ser. I 336, 681–686 (2003)

    MATH  MathSciNet  Google Scholar 

  17. Khismatullin, D., Truskey, G.: Three dimensional numerical simulation of receptor-medicated leukocyte adhesion to surfaces. Effects of cell deformability and viscoelasticity. Phys. Fluids 17, 031505 (2005)

    Article  Google Scholar 

  18. Glowinski, R., Pan, T., Hesha, T., Joseph, D., Periaux, J.: A fictious domain approach to the direct numerical simulation of incompressible viscuous flow past moving rigid bodies: Applications to particularte flow. J. Comput. Phys. 169, 363–426 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kukavica, I., Tuffaha, A., Ziane, M.: Strong solutions to nonlinear fluid structure interactions. J. Differ. Equ. 247, 1452–1478 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kukavica, I., Tuffaha, A., Ziane, M.: Strong solutions for a fluid structure interaction system. Adv. Differ. Equ. 247, 1452–1478 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lasiecka, I.: Control Theory of Coupled PDE’s. CBMS-SIAM Lecture Notes. SIAM, Philadelphia (2002)

    Google Scholar 

  22. Lasiecka, I., Seidman, T.: Strong stability of elastic control systems with dissipative saturating feedback. Syst. Control Lett. 48, 243–252 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. LaSalle, J.P.: Stability theory and invariance principles. In: Cerasir, L., Hale, J.K., LaSalle, J.P. (eds.) Dynamical Systems, vol. 1, pp. 211–222. Academic Press, New York (1976)

    Google Scholar 

  24. Littman, W.: Remarks on global uniqueness theorems for PDE’s. In: Differential Geometric Methods in Control of PDE’s. Contemporary Mathematics, vol. 268, pp. 363–371. AMS, Providence (2000)

    Google Scholar 

  25. Lions, J.L.: Controllabilite Exact and Stabilization de Systemes Distribues. Masson, Paris (1988)

    Google Scholar 

  26. Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, New York (1972)

    Google Scholar 

  27. Lyubich, Y.I., Phong, V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. LXXXXVII, 37–42 (1988)

    Google Scholar 

  28. Moubachir, M., Zolesio, J.: Moving Shape Analysis and Control: Applications to Fluid Structure Interactions. Chapman & Hall/CRC, London (2006)

    Book  MATH  Google Scholar 

  29. Sklyar, G.M.: Lack of maximal asymptotics for some linear equations in a Banach space. Dokl. Akad. Nauk 431(4), 464–467 (2010)

    MathSciNet  Google Scholar 

  30. Slemrod, M.: Stabilization of boundary control systems. J. Differ. Equ. 22, 402–415 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  31. Slemrod, M.: Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping. Proc. R. Soc. Edinb. A 113, 87–97 (1989)

    MATH  MathSciNet  Google Scholar 

  32. Temam, R.: Navier-Stokes Equations, Studies in Math. and its Applications. North Holland, Amsterdam (1977)

    Google Scholar 

  33. Walker, J.A.: Dynamical Systems and Evolution Equations. Plenum, New York (1980)

    MATH  Google Scholar 

  34. Zhang, X., Zuazua, E.: Long time behavior of a coupled heat-wave system arising in fluid structure interaction. Arch. Ration. Mech. Anal. 24, 49–120 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Irena Lasiecka & Yongjin Lu

  2. Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahram, 31261, KSA

    Irena Lasiecka

Authors
  1. Irena Lasiecka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongjin Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irena Lasiecka.

Additional information

Communicated by Jerome A. Goldstein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lasiecka, I., Lu, Y. Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction. Semigroup Forum 82, 61–82 (2011). https://doi.org/10.1007/s00233-010-9281-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-010-9281-7

Keywords

Search

Navigation