Stability of oscillating boundary layers in rotating fluids
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, pp. 955-1002.

We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to ε. This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.

On prouve la stabilité linéaire et non-linéaire de couches limites oscillantes de type Ekman pour les fluides tournant dans le cas de données mal préparées sous une hypothèse spectrale. On s’intéresse au cas où la viscosité et le nombre de Rossby sont du même ordre ε. Cette étude généralise celle de [23] où une condition de petitesse était imposée et celle de [26] où les données bien préparées étaient traitées.

DOI: 10.24033/asens.2086
Classification: 35B25,  35B35,  35Q30
Keywords: incompressible Navier-Stokes equation, oscillatory perturbations, vanishing viscosity
@article{ASENS_2008_4_41_6_955_0,
     author = {Masmoudi, Nader and Rousset, Fr\'ed\'eric},
     title = {Stability of oscillating boundary layers in rotating fluids},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {955--1002},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {6},
     year = {2008},
     doi = {10.24033/asens.2086},
     zbl = {1159.76013},
     mrnumber = {2504110},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2086/}
}
TY  - JOUR
AU  - Masmoudi, Nader
AU  - Rousset, Frédéric
TI  - Stability of oscillating boundary layers in rotating fluids
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2008
DA  - 2008///
SP  - 955
EP  - 1002
VL  - Ser. 4, 41
IS  - 6
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2086/
UR  - https://zbmath.org/?q=an%3A1159.76013
UR  - https://www.ams.org/mathscinet-getitem?mr=2504110
UR  - https://doi.org/10.24033/asens.2086
DO  - 10.24033/asens.2086
LA  - en
ID  - ASENS_2008_4_41_6_955_0
ER  - 
%0 Journal Article
%A Masmoudi, Nader
%A Rousset, Frédéric
%T Stability of oscillating boundary layers in rotating fluids
%J Annales scientifiques de l'École Normale Supérieure
%D 2008
%P 955-1002
%V Ser. 4, 41
%N 6
%I Société mathématique de France
%U https://doi.org/10.24033/asens.2086
%R 10.24033/asens.2086
%G en
%F ASENS_2008_4_41_6_955_0
Masmoudi, Nader; Rousset, Frédéric. Stability of oscillating boundary layers in rotating fluids. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, pp. 955-1002. doi : 10.24033/asens.2086. http://www.numdam.org/articles/10.24033/asens.2086/

[1] A. Babin, A. Mahalov & B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mech. B Fluids 15 (1996), 291-300. | MR | Zbl

[2] A. Babin, A. Mahalov & B. Nicolaenko, Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal. 15 (1997), 103-150. | MR | Zbl

[3] A. J. Bourgeois & J. T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal. 25 (1994), 1023-1068. | MR | Zbl

[4] J.-Y. Chemin, B. Desjardins, I. Gallagher & E. Grenier, Mathematical geophysics, Oxford Lecture Series in Math. and its Appl. 32, The Clarendon Press Oxford University Press, 2006. | MR | Zbl

[5] T. Colin & P. Fabrie, Équations de Navier-Stokes 3-D avec force de Coriolis et viscosité verticale évanescente, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 275-280. | MR | Zbl

[6] B. Desjardins, E. Dormy & E. Grenier, Stability of mixed Ekman-Hartmann boundary layers, Nonlinearity 12 (1999), 181-199. | MR | Zbl

[7] B. Desjardins & E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 87-106. | Numdam | MR | Zbl

[8] M. Dimassi & J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Series 268, Cambridge Univ. Press, 1999. | MR | Zbl

[9] P. F. Embid & A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations 21 (1996), 619-658. | MR | Zbl

[10] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., 1964. | MR | Zbl

[11] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy 38, Springer, 1994. | MR | Zbl

[12] I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl. 77 (1998), 989-1054. | MR | Zbl

[13] F. Gallaire & F. Rousset, Spectral stability implies nonlinear stability for incompressible boundary layers, Indiana Univ. Math. J. 57 (2008), 1959-1975. | MR | Zbl

[14] H. Greenspan, The theory of rotating fluids, Cambridge Monographs on Mechanics and Applied Mathematics, 1969. | Zbl

[15] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. 76 (1997), 477-498. | MR | Zbl

[16] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), 1067-1091. | MR | Zbl

[17] E. Grenier & N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations 22 (1997), 953-975. | MR | Zbl

[18] E. Grenier & F. Rousset, Stability of one-dimensional boundary layers by using Green's functions, Comm. Pure Appl. Math. 54 (2001), 1343-1385. | MR | Zbl

[19] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier (Grenoble) 45 (1995), 973-1006. | Numdam | MR | Zbl

[20] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, 1981. | MR | Zbl

[21] D. Lilly, On the instability of Ekman boundary flow, J. Atmos. Sci. (1966), 481-494.

[22] N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary, Arch. Rational Mech. Anal. 142 (1998), 375-394. | MR | Zbl

[23] N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data, Comm. Pure Appl. Math. 53 (2000), 432-483. | MR | Zbl

[24] G. Métivier & K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175, 2005. | MR | Zbl

[25] J. Pedlosky, Geophysical fluid dynamics, Springer, 1979. | Zbl

[26] F. Rousset, Stability of large Ekman boundary layers in rotating fluids, Arch. Ration. Mech. Anal. 172 (2004), 213-245. | MR | Zbl

[27] F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations 210 (2005), 25-64. | MR | Zbl

[28] F. Rousset, Stability of large amplitude Ekman-Hartmann boundary layers in MHD: the case of ill-prepared data, Comm. Math. Phys. 259 (2005), 223-256. | MR | Zbl

[29] M. Sammartino & R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution, Comm. Math. Phys. 192 (1998), 463-491. | MR | Zbl

[30] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), 476-512. | MR | Zbl

[31] M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series 34, Princeton University Press, 1981. | MR | Zbl

Cited by Sources: