The Essential Spectrum of the Linearized 2D Euler Operator is a Vertical Band

TL;DR

This paper characterizes the essential spectrum of the linearized 2D Euler operator, showing it forms a vertical band whose width relates to the maximal Lyapunov exponent of the steady flow.

Contribution

It establishes a precise description of the essential spectrum for the linearized 2D Euler operator, linking it to the flow's Lyapunov exponent.

Findings

Essential spectrum forms a vertical strip in the complex plane.

Width of the spectrum band is determined by the maximal Lyapunov exponent.

Provides a spectral characterization of linearized 2D Euler flows.

Abstract

We prove that the essential spectrum of the operator obtained by linearization about a steady state of the Euler equations governing the motion of inviscid ideal fluid in dimension two is a vertical strip whose width is determined by the maximal Lyapunov exponent of the flow induced by the steady state.