Comptes Rendus
Partial Differential Equations
Continuous spectrum of the 3D Euler equation is a solid annulus
Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 897-900.

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 tR, 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.

On donne dans cette Note une description du spectre continu de l'équation d'Euler linearisée en dimension 3. Précisément, pour presque tout tR, 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.

Published online:
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
     author = {Roman Shvydkoy},
     title = {Continuous spectrum of the {3D} {Euler} equation is a solid annulus},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {897--900},
     publisher = {Elsevier},
     volume = {348},
     number = {15-16},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.009},
     language = {en},
AU  - Roman Shvydkoy
TI  - Continuous spectrum of the 3D Euler equation is a solid annulus
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 897
EP  - 900
VL  - 348
IS  - 15-16
PB  - Elsevier
DO  - 10.1016/j.crma.2010.07.009
LA  - en
ID  - CRMATH_2010__348_15-16_897_0
ER  - 
%0 Journal Article
%A Roman Shvydkoy
%T Continuous spectrum of the 3D Euler equation is a solid annulus
%J Comptes Rendus. Mathématique
%D 2010
%P 897-900
%V 348
%N 15-16
%I Elsevier
%R 10.1016/j.crma.2010.07.009
%G en
%F CRMATH_2010__348_15-16_897_0
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.

[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

Cited by Sources:

Comments - Policy