Exponential attractors for extensible beam equations

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.