Issues of Chaos and Recurrence in Infinite Dimensions

TL;DR

This paper discusses the presence of chaos and recurrence in infinite-dimensional Sobolev spaces, highlighting differences between bounded and unbounded domains and emphasizing the complexity of local Sobolev spaces in fluid dynamics.

Contribution

It clarifies the conditions under which chaos and recurrence occur in Sobolev spaces over different spatial domains, providing insights into their mathematical structure and challenges.

Findings

Sobolev spaces over bounded domains host chaos and recurrence.

Sobolev spaces over unbounded domains lack chaos and recurrence.

Local Sobolev spaces can host chaos but are difficult to analyze.

Abstract

Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. for fluid problems, but are very challenging to study.