Exponential attractors for extensible beam equations

A. Edein, A. J. Milani

Research output: Contribution to journalArticle

40 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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