The unstable spectrum of the Navier-Stokes operator in the limit of vanishing viscosity
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 4, p. 713-724
@article{AIHPC_2008__25_4_713_0,
author = {Shvydkoy, Roman and Friedlander, Susan},
title = {The unstable spectrum of the Navier-Stokes operator in the limit of vanishing viscosity},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {25},
number = {4},
year = {2008},
pages = {713-724},
doi = {10.1016/j.anihpc.2007.05.004},
zbl = {1140.35550},
mrnumber = {2436790},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2008__25_4_713_0}
}

Shvydkoy, Roman; Friedlander, Susan. The unstable spectrum of the Navier-Stokes operator in the limit of vanishing viscosity. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 4, pp. 713-724. doi : 10.1016/j.anihpc.2007.05.004. http://www.numdam.org/item/AIHPC_2008__25_4_713_0/

[1] Chicone C., Latushkin Y., Evolution Semigroups in Dynamical Systems and Differential Equations, American Mathematical Society, Providence, RI, 1999, MR 2001e:47068. | MR 1707332 | Zbl 0970.47027

[2] Dickinson D., Gramchev T., Yoshino M., First order pseudodifferential operators on the torus: Normal forms, diophantine phenomena and global hypoellipticity, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 41 (1996) 51-64. | MR 1471014 | Zbl 0889.35128

[3] Friedlander S., Pavlović N., Shvydkoy R., Nonlinear instability for the Navier-Stokes equations, Commun. Math. Phys. 264 (2006) 335-347. | MR 2215608

[4] Friedlander S., Lipton-Lifschitz A., Localized instabilities in fluids, in: Handbook of Mathematical Fluid Dynamics, vol. II, North-Holland, Amsterdam, 2003, pp. 289-354, MR MR1984155 (2004g:76072). | MR 1984155 | Zbl pre02019785

[5] Goldstein J.A., Semigroups of Linear Operators and Applications, The Clarendon Press Oxford University Press, New York, 1985, MR 87c:47056. | MR 790497 | Zbl 0592.47034

[6] Lin Z., Nonlinear instability of ideal plane flows, Int. Math. Res. Not. 41 (2004) 2147-2178, MR MR2078852. | MR 2078852 | Zbl 1080.35085

[7] Lyashenko A.A., Friedlander S.J., A sufficient condition for instability in the limit of vanishing dissipation, J. Math. Anal. Appl. 221 (2) (1998) 544-558, MR MR1621750 (99g:76065). | MR 1621750 | Zbl 0916.35009

[8] Sattinger D.H., The mathematical problem of hydrodynamic stability, J. Math. Mech. 19 (1969/1970) 817-979, MR MR0261182 (41 #5798). | MR 261182 | Zbl 0198.30401

[9] Shvydkoy R., Cocycles and Mañe sequences with an application to ideal fluids, J. Diff. Eq. 229 (2006) 49-62. | MR 2265617 | Zbl 1104.37003

[10] Shvydkoy R., The essential spectrum of advective equations, Commun. Math. Phys. 265 (2006) 507-545. | MR 2231681 | Zbl 1121.76027

[11] Temam R., Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and Numerical Analysis, Reprint of the 1984 edition. MR MR1846644 (2002j:76001). | MR 1846644 | Zbl 0981.35001

[12] Vishik M., Spectrum of small oscillations of an ideal fluid and Lyapunov exponents, J. Math. Pures Appl. (9) 75 (6) (1996) 531-557, MR 97k:35203. | MR 1423046 | Zbl 0870.76017

[13] Vishik M., Friedlander S., Nonlinear instability in two dimensional ideal fluids: the case of a dominant eigenvalue, Comm. Math. Phys. 243 (2) (2003) 261-273, MR MR2021907 (2004k:76055). | MR 2021907 | Zbl 1043.76025

[14] Vishik M., Friedlander S., Dynamo theory methods for hydrodynamic stability, J. Math. Pures Appl. (9) 72 (2) (1993) 145-180, MR 95a:76036. | MR 1216094 | Zbl 1027.76699

[15] Wong M.W., An introduction to Pseudo-Differential Operators, World Scientific Publishing Co. Inc., Teaneck, NJ, 1991, MR MR1100930 (92b:47083). | MR 1100930 | Zbl 0753.35134

[16] Yudovich V.I., The Linearization Method in Hydrodynamical Stability Theory, Translations of Mathematical Monographs, vol. 74, American Mathematical Society, Providence, RI, 1989, Translated from the Russian by J.R. Schulenberger. MR MR1003607 (90h:76001). | MR 1003607 | Zbl 0727.76039