On 2D Euler equations. I: On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations (Q2737848)

A linearized two-dimensional Euler equation is studied. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. The eigenvalues of the linear Hamiltonian system are of four types: real pairs, purely imaginary pairs, quadruples zero eigenvalues. The spectral equation for each invariant subsystem is a Poincaré-type difference equation. NEWLINENEWLINENEWLINERelations to Lax pairs are given in more recent papers of the author [J. Math. Phys. 42, 3552--3553 (2001; Zbl 1005.37056) and Part II of the paper under review [Commun. Appl. Nonlinear Anal. 10, No.1, 1--43 (2003; Zbl 1146.37361)].