Continuous spectrum of the 3D Euler equation is a solid annulus

TL;DR

This paper characterizes the continuous spectrum of the linearized 3D Euler equations' evolution operator as a solid annulus determined by Lyapunov exponents, for almost all times.

Contribution

It provides a precise geometric description of the continuous spectrum of the linearized 3D Euler equations in terms of Lyapunov exponents and solid annuli.

Findings

Continuous spectrum forms a solid annulus with radii based on Lyapunov exponents.

The description holds for all but countably many times.

Spectrum is explicitly linked to the bicharacteristic-amplitude system.

Abstract

In this note we give a description of the continuous spectrum of the linearized Euler equations in three dimensions. Namely, for all but countably many times $t\in \R$, the continuous spectrum of the evolution operator $G_t$ is given by a solid annulus with radii $e^{t\mu}$ and $e^{t M}$, where $\mu$ and $M$ are the smallest and largest, respectively, Lyapunov exponents of the corresponding bicharacteristic-amplitude system of ODEs.