Composite Wave–Plate Systems

Part of the Springer Monographs in Mathematics book series (SMM)


The aim of this chapter is to discuss long-time behavior of composite wave–plate models. These include:

• Structural acoustic model with isothermal von Karman plate.

• Structural acoustic model with thermoelastic Karman plate.

• Flow-structure interaction.

We recall that these models involve coupling between the wave equation and plate equation. The coupling occurs on the interface separating two media: acoustic field and the structure (wall). The existence and uniqueness theory associated with these models has been already developed in Chapter 6. The present chapter is devoted to long-time dynamics and global attractors associated with the models.


Global Attractor Plate System Plate Equation Airy Stress Function Composite Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 17.
    A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.MATHGoogle Scholar
  2. 28.
    V.V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963.MATHGoogle Scholar
  3. 32.
    L. Boutet de Monvel and I. Chueshov, Oscillations of von Karman plate in a potential flow of gas, Izvestiya: Mathematics, 63(2) (1999), 219–244.MATHCrossRefMathSciNetGoogle Scholar
  4. 37.
    F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discr. Cont. Dyn. Sys., 22 (2008), 557–586.MATHMathSciNetGoogle Scholar
  5. 38.
    F. Bucci, I. Chueshov and I. Lasiecka, Global attractor for a composite system of nonlinear wave and plate equations, Commun. Pure Appl. Anal., 6 (2007), 113–140.MATHMathSciNetGoogle Scholar
  6. 48.
    I. Chueshov, Asymptotic behavior of the solutions of a problem of the aeroelastic oscillations of a shell in hypersonic limit, J. Soviet Math., 52 (1990), 3545–3548.CrossRefMathSciNetGoogle Scholar
  7. 61.
    I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also
  8. 69.
    I. Chueshov and I. Lasiecka, Attractors for second order evolution equations, J. Dynam. Diff. Eqs., 16 (2004), 469–512.MATHCrossRefMathSciNetGoogle Scholar
  9. 74.
    I. Chueshov and I. Lasiecka, Long time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents. In Control Mehtods in PDE - Dynamical systems, F. Ancona et al., (Eds.), Contemporary Mathematics, vol.426, AMS, Providence, RI, 2007, 153–193.Google Scholar
  10. 75.
    I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008.Google Scholar
  11. 99.
    E.H. Dowell, Aeroelastisity of Plates and Shells, Noordhoff International, Leyden, 1975.Google Scholar
  12. 102.
    A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Appl. Math. 37, Masson, Paris, 1994.MATHGoogle Scholar
  13. 147.
    A.A. Il’ushin, The plane sections law in aerodynamics of large supersonic speeds, Prikladnaya Matem. Mech., 20(6) (1956), 733–755, in Russian.Google Scholar
  14. 162.
    A. K. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlin. Anal., (2010), in press.Google Scholar
  15. 232.
    A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Handbook of Differential Equations: Evolutionary Equations, vol. 4, C.M. Dafermos, and M. Pokorny (Eds.), Elsevier, Amsterdam, 2008, 103–200.Google Scholar
  16. 252.
    I. Ryzhkova, Stabilization of von Karman plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl., 294 (2004), 462–481.MATHCrossRefMathSciNetGoogle Scholar
  17. 254.
    I. Ryzhkova, Dynamics of a thermoelastic von Karman plate in a subsonic gas flow, Z. Angew. Math. Phys., 58 (2007), 246–261.MATHCrossRefMathSciNetGoogle Scholar
  18. 273.
    R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.MATHGoogle Scholar
  19. 275.
    H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North-Holland, Amsterdam, 1978.Google Scholar

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© Springer Science+Business Media 2010

Authors and Affiliations

  1. 1.Department of Mechanics & MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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