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/

[1] B.J. Bayly Three-dimensional instability of elliptical flow, Phys. Rev. Lett., Volume 57 (1986) no. 17, pp. 2160-2163

[2] F.E. Browder On the spectral theory of elliptic differential operators. I, Math. Ann., Volume 142 (1960/1961), pp. 22-130

[3] S. Friedlander; M.M. Vishik Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., Volume 66 (1991) no. 17, pp. 2204-2206

[4] Y. Latushkin; M. Vishik Linear stability in an ideal incompressible fluid, Comm. Math. Phys., Volume 233 (2003) no. 3, pp. 439-461

[5] A. Lifschitz; E. Hameiri Local stability conditions in fluid dynamics, Phys. Fluids A, Volume 3 (1991) no. 11, pp. 2644-2651

[6] R.D. Nussbaum The radius of the essential spectrum, Duke Math. J., Volume 37 (1970), pp. 473-478

[7] S.A. Orszag; A.T. Patera Subcritical transition to turbulence in plane channel flows, Phys. Rev. Lett., Volume 45 (1980), pp. 989-993

[8] R.T. Pierrehumbert Universal short-wave instability of two-dimensional eddies in an inviscid fluid, Phys. Rev. Lett., Volume 57 (1986), pp. 2157-2159

[9] R.J. Sacker; G.R. Sell A spectral theory for linear differential systems, J. Differential Equations, Volume 27 (1978) no. 3, pp. 320-358

[10] R. Shvydkoy The essential spectrum of advective equations, Comm. Math. Phys., Volume 265 (2006) no. 2, pp. 507-545

[11] R. Shvidkoy; Y. Latushkin The essential spectrum of the linearized 2D Euler operator is a vertical band, Birmingham, AL, 2002 (Contemp. Math.), Volume vol. 327, Amer. Math. Soc., Providence, RI (2003), pp. 299-304

[12] R. Shvidkoy; Y. Latushkin Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics, J. Funct. Anal., Volume 257 (2009), pp. 3309-3328

[13] N. Steenrod The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, NJ, 1951

[14] M. Vishik Spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures Appl. (9), Volume 75 (1996) no. 6, pp. 531-557

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