Exponential attractors for extensible beam equations

A. Edein, A. J. Milani

    Research output: Contribution to journalArticle

    38 Citations (Scopus)

    Abstract

    In this paper we establish a global fast dynamics for a class, of equations that include the beam equations as studied by Ball and von Karman equations for a thin plate. We introduce various energy functionals and show that they decay exponentially. Using the absorbing sets obtained through these energy functionals, we expose Hale’s theory of α-contractions and how it applies to this general framework and deduce the existence of a compact attractor, in parallel to his proof. We also establish the smoothness of this attractor when the damping is large. Finally, by proving the discrete squeezing property for these equations, the existence of a compact, finite dimensional exponentially attracting set is demonstrated. The use of energy methods throughout allow considerable simplification even when a natural Lyapunov functional is hard to exhibit. In closing, we also exhibit a simple alternative proof for Titi’s theorem on the existence of inertial manifolds for beam equations under suitable forces.

    Original languageEnglish
    Pages (from-to)393-415
    Number of pages23
    JournalNonlinearity
    Volume6
    Issue number3
    DOIs
    Publication statusPublished - 1993

    Fingerprint

    Exponential Attractors
    Beam Equation
    Damping
    Attractor
    functionals
    Absorbing Set
    Von Kármán Equations
    Inertial Manifolds
    Von Karman equation
    Squeezing
    Lyapunov Functional
    Energy Method
    Thin Plate
    Energy
    Simplification
    energy methods
    Deduce
    Contraction
    thin plates
    Smoothness

    All Science Journal Classification (ASJC) codes

    • Applied Mathematics
    • Statistical and Nonlinear Physics
    • Mathematical Physics
    • Physics and Astronomy(all)

    Cite this

    Edein, A. ; Milani, A. J. / Exponential attractors for extensible beam equations. In: Nonlinearity. 1993 ; Vol. 6, No. 3. pp. 393-415.
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    Exponential attractors for extensible beam equations. / Edein, A.; Milani, A. J.

    In: Nonlinearity, Vol. 6, No. 3, 1993, p. 393-415.

    Research output: Contribution to journalArticle

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