Cocycles and Ma\~{n}e sequences with an application to ideal fluids

TL;DR

This paper establishes the equivalence between exponential dichotomy and Ma ext{~}ne sequences for cocycles, extending classical results to Banach bundles and analyzing the spectrum of Euler equations in fluid dynamics.

Contribution

It introduces a new equivalence between exponential dichotomy and Ma ext{~}ne sequences for cocycles, and applies this to spectral analysis of ideal fluids.

Findings

Exponential dichotomy is equivalent to the existence of Ma ext{~}ne sequences for cocycles.

Extended classical spectral results to Banach bundles.

Described the essential spectrum of the Euler equation in arbitrary dimensions.

Abstract

Exponential dichotomy of a strongly continuous cocycle $\bFi$ is proved to be equivalent to existence of a Ma\~{n}e sequence either for $\bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.