A panoramic view of exponential attractors

TL;DR

This paper establishes necessary and sufficient conditions for the existence of T-discrete exponential attractors in metric spaces, linking covering conditions to attractor properties and comparing various construction methods.

Contribution

It provides a unified framework for understanding exponential attractors, generalizes existing methods, and clarifies their relation to classical exponential attractors.

Findings

Conditions for existence of T-discrete exponential attractors are characterized.

All known construction methods satisfy the covering condition.

The relation between T-discrete and classical exponential attractors is clarified.

Abstract

We state necessary and sufficient conditions for the existence of $T$-discrete exponential attractors for semigroups in complete metric spaces. These conditions are formulated in terms of a covering condition for iterates of the absorbing set under the time evolution of the semigroup and imply the existence and finite-dimensionality of the global attractor. We then review, generalize and compare existing construction methods for exponential attractors and show that they all imply the covering condition. Furthermore, we relate the results and concept of $T$-discrete exponential attractors to the classical notion of exponential attractors.