On 2D Euler Equations: Part I. On the Energy-Casimir Stabilities and The Spectra for Linearized 2D Euler Equations
Li, Yanguang Charles
arXiv, 0010240 / Harvested from arXiv
Text on arXiv
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. The eigenvalues of the linear Hamiltonian system are of four types: real pairs ($c,-c$), purely imaginary pairs ($id,-id$), quadruples ($\pm c\pm id$), and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincar\'{e}-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.
Publié le : 2000-10-25
Classification:  Mathematics - Analysis of PDEs,  Mathematical Physics,  Mathematics - Dynamical Systems,  47.27 47.52 47.20 02.30.J 05.45 
@article{0010240,
     author = {Li, Yanguang Charles},
     title = {On 2D Euler Equations: Part I. On the Energy-Casimir Stabilities and The
  Spectra for Linearized 2D Euler Equations},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0010240}
}

Li, Yanguang Charles. On 2D Euler Equations: Part I. On the Energy-Casimir Stabilities and The
  Spectra for Linearized 2D Euler Equations. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0010240/