Chapter 17 Global attractors in partial differential equations

Abstract

This chapter discusses global attractors in partial differential equations (PDE). Compact global attractors are robust objects with respect to perturbations. In general, the invariance and attractivity of the global attractors, combined with some additional hypotheses, imply interesting robustness and regularity properties. Often also, the flow restricted to the global attractor shows finite dimensional behavior. If the semigroups are defined on a finite or infinite dimensional vector space, for example, then considerable care must be taken to discuss the behavior of orbits at infinity. If each of the semigroups has a compact global attractor, the topological equivalence of the flows restricted to the global attractors can be considered. This is the strongest type of comparison of flows that can be expected in that it uses the very detailed properties of the flows. In the chapter, weaker concepts of comparison—such as estimates of the Hausdorff distance between the global attractors—are considered.