Dynamics of Partial Differential Equations

Volume 1 (2004)

Number 1

On spectrum of the linearized 3D Euler equation

Pages: 49 – 63

DOI: http://dx.doi.org/10.4310/DPDE.2004.v1.n1.a2

Authors

Roman Shvydkoy (Department of Mathematics, University of Texas at Austin, Texas, U.S.A.)

Misha Vishik (Department of Mathematics, University of Texas at Austin, Texas, U.S.A.)

Abstract

We investigate essential spectrum of the Euler equation linearizedabout an arbitrary smooth steady flow in dimension 3. It is provedthat for every Lyapunov-Oseledets exponent $\m$ of the associatedbicharacteristic-amplitude system, the circle of radius $e^{\mut}$ has a common point with the spectrum. If, in addition, $\mu$is attained on an aperiodic point, then the spectrum contains theentire circle.

Keywords

linearized Euler equation, essential spectrum, Lyapunov-Oseledets exponents

2010 Mathematics Subject Classification

Primary 76E09. Secondary 34D09.

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