Exponential attractors for extensible beam equations

The authors establish a global fast dynamics for a class of equations that include the beam equations as studied by Ball (1973) and von Karman equations for a thin plate. They introduce various energy functionals and show that they decay exponentially. Using the absorbing sets obtained through these energy functionals, they expose Hale's theory of alpha contractions and how it applies to this general framework and deduce the existence of a compact attractor, in parallel to his proof. They 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, they also exhibit a simple alternative proof for Titi's theorem on the existence of inertial manifolds for beam equations under suitable forces.