Finite-dimensional global attractors in Banach spaces

TL;DR

This paper establishes bounds on the complexity of negatively invariant sets in Banach spaces and demonstrates that global attractors for broad classes of parabolic PDEs are finite-dimensional, advancing understanding of their structure.

Contribution

It introduces bounds on the box-counting dimension of invariant sets in Banach spaces and proves finite-dimensionality of attractors for many parabolic PDEs.

Findings

Bounds on the upper box-counting dimension of invariant sets

Global attractors of broad class of parabolic PDEs are finite-dimensional

Applicable to semilinear equations in Banach spaces

Abstract

We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional.