In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an “unstable disk theorem” through a simple energy–Casimir argument [Holm et al., Phys. Rep. 123, 1–116 (1985)]. The eigenvalues of the linear Hamiltonian system are of four types: real pairs purely imaginary pairs quadruples and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincaré-type difference equation, i.e., it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constant-coefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory.
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February 2000
Research Article|
February 01 2000
On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations
Yanguang (Charles) Li
Yanguang (Charles) Li
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
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J. Math. Phys. 41, 728–758 (2000)
Article history
Received:
July 10 1998
Accepted:
September 30 1999
Citation
Yanguang (Charles) Li; On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations. J. Math. Phys. 1 February 2000; 41 (2): 728–758. https://doi.org/10.1063/1.533176
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