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 language | English |
|---|---|
| Pages (from-to) | 393-415 |
| Number of pages | 23 |
| Journal | Nonlinearity |
| Volume | 6 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1993 |
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All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
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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 journal › Article
TY - JOUR
T1 - Exponential attractors for extensible beam equations
AU - Edein, A.
AU - Milani, A. J.
PY - 1993
Y1 - 1993
N2 - 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.
AB - 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.
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U2 - 10.1088/0951-7715/6/3/007
DO - 10.1088/0951-7715/6/3/007
M3 - Article
VL - 6
SP - 393
EP - 415
JO - Nonlinearity
JF - Nonlinearity
SN - 0951-7715
IS - 3
ER -