[Le spectre continu de l'equation d'Euler 3D est un anneau]
On donne dans cette Note une description du spectre continu de l'équation d'Euler linearisée en dimension 3. Précisément, pour presque tout , le spectre continu de l'opérateur d'évolution est constitué d'un anneau de rayons et , 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 , the continuous spectrum of the evolution operator is given by a solid annulus with radii and , where μ and M are the smallest and largest, respectively, Lyapunov exponents of the corresponding bicharacteristic-amplitude system of ODEs.
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Roman Shvydkoy 1
@article{CRMATH_2010__348_15-16_897_0, 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}, }
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] Three-dimensional instability of elliptical flow, Phys. Rev. Lett., Volume 57 (1986) no. 17, pp. 2160-2163
[2] On the spectral theory of elliptic differential operators. I, Math. Ann., Volume 142 (1960/1961), pp. 22-130
[3] Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., Volume 66 (1991) no. 17, pp. 2204-2206
[4] Linear stability in an ideal incompressible fluid, Comm. Math. Phys., Volume 233 (2003) no. 3, pp. 439-461
[5] Local stability conditions in fluid dynamics, Phys. Fluids A, Volume 3 (1991) no. 11, pp. 2644-2651
[6] The radius of the essential spectrum, Duke Math. J., Volume 37 (1970), pp. 473-478
[7] Subcritical transition to turbulence in plane channel flows, Phys. Rev. Lett., Volume 45 (1980), pp. 989-993
[8] Universal short-wave instability of two-dimensional eddies in an inviscid fluid, Phys. Rev. Lett., Volume 57 (1986), pp. 2157-2159
[9] A spectral theory for linear differential systems, J. Differential Equations, Volume 27 (1978) no. 3, pp. 320-358
[10] The essential spectrum of advective equations, Comm. Math. Phys., Volume 265 (2006) no. 2, pp. 507-545
[11] 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] Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics, J. Funct. Anal., Volume 257 (2009), pp. 3309-3328
[13] The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, NJ, 1951
[14] 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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