Comptes Rendus
no. 15-16
Partial Differential Equations
Continuous spectrum of the 3D Euler equation is a solid annulus
[Le spectre continu de l'equation d'Euler 3D est un anneau]

Présenté par : Pierre-Louis Lions

Roman Shvydkoy1
1 Department of Mathematics, Statistics and Computer Science, 851 S. Morgan St., M/C 249, University of Illinois at Chicago, Chicago, IL 60607, United States
Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 897-900.

On donne dans cette Note une description du spectre continu de l'équation d'Euler linearisée en dimension 3. Précisément, pour presque tout t∈R, le spectre continu de l'opérateur d'évolution Gt est constitué d'un anneau de rayons etμ et etM, où μ et M sont, respectivement, le plus petit et le plus grand exposant de Lyapunov du système d'EDO bicaractéristique-amplitude associé.

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∈R, the continuous spectrum of the evolution operator Gt is given by a solid annulus with radii etμ and etM, where μ and M are the smallest and largest, respectively, Lyapunov exponents of the corresponding bicharacteristic-amplitude system of ODEs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.009

Roman Shvydkoy 1

1 Department of Mathematics, Statistics and Computer Science, 851 S. Morgan St., M/C 249, University of Illinois at Chicago, Chicago, IL 60607, United States 
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Roman Shvydkoy. Continuous spectrum of the 3D Euler equation is a solid annulus. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 897-900. doi : 10.1016/j.crma.2010.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.009/

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  • Roman Shvydkoy On the Rayleigh–Taylor Instability in Presence of a Background Shear, Journal of Mathematical Fluid Mechanics, Volume 20 (2018) no. 3, p. 1195 | DOI:10.1007/s00021-018-0362-9
  • Elizabeth Thoren Linear Instability Criteria for Ideal Fluid Flows Subject to Two Subclasses of Perturbations, Journal of Mathematical Fluid Mechanics, Volume 14 (2012) no. 3, p. 541 | DOI:10.1007/s00021-011-0081-y

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