Exponential attractors for extensible beam equations

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. W e 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 0-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.

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

“…Moreover, as not difficult to see using that the identity map on the attractor A is continuous as the map of H to H 2 (due to compactness of A in H 2 ), that the vector field W is continuous on A. Slightly more delicate arguments based, e.g., on the "almost equivalence" of the H and H 2 norms on the attractor, see Proposition 3.26 below, show that W is also Hölder continuous and the Hölder exponent can be chosen arbitrarily close to one, see [13,58] and references therein.…”

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

“…In order to overcome these drawbacks, the concept of an exponential attractor has been suggested in [8]. By definition, an exponential attractor M is a compact semiinvariant set of the phase space which is finite-dimensional (in the sense of (0.2)) and attracts exponentially the images of the bounded subsets of Φ, i.e., there exist a positive constant α and a monotonic function Q such that (0.…”

“…We now recall that the original construction of exponential attractors from [8] was based on the so-called squeezing property and was highly nonconstructive (indeed, Zorn's lemma had to be used). The lower semicontinuity property was also obtained, but only up to "time-shifts", which is factually equivalent to the following:…”

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…These drawbacks led Foias et al to introduce a new object called exponential attractor. This is a compact and positively invariant set with finite fractal dimension which attracts all the trajectories starting from bounded sets at a uniform exponential rate (see [10]). Clearly, an exponential attractor contains the global attractor.…”

“…The classical construction of exponential attractors makes use of orthogonal projectors with finite rank and are valid in Hilbert spaces only (see, e.g., [1,10]). Rather recently, this construction has been extended to Banach spaces (see [11], cf.…”