Stability of oscillating boundary layers in rotating fluids

Nader Masmoudi; Frédéric Rousset

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 6, page 955-1002
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Masmoudi, Nader, and Rousset, Frédéric. "Stability of oscillating boundary layers in rotating fluids." Annales scientifiques de l'École Normale Supérieure 41.6 (2008): 955-1002. <http://eudml.org/doc/272108>.

@article{Masmoudi2008,
abstract = {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 $\varepsilon $. 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.},
author = {Masmoudi, Nader, Rousset, Frédéric},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {incompressible Navier-Stokes equation; oscillatory perturbations; vanishing viscosity},
language = {eng},
number = {6},
pages = {955-1002},
publisher = {Société mathématique de France},
title = {Stability of oscillating boundary layers in rotating fluids},
url = {http://eudml.org/doc/272108},
volume = {41},
year = {2008},
}

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
PB - Société mathématique de France
VL - 41
IS - 6
SP - 955
EP - 1002
AB - 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 $\varepsilon $. 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.
LA - eng
KW - incompressible Navier-Stokes equation; oscillatory perturbations; vanishing viscosity
UR - http://eudml.org/doc/272108
ER -

References

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  1. [1] A. Babin, A. Mahalov & B. Nicolaenko, Global splitting, integrability and regularity of 3 D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mech. B Fluids15 (1996), 291–300. Zbl0882.76096MR1400515
  2. [2] A. Babin, A. Mahalov & B. Nicolaenko, Regularity and integrability of 3 D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal.15 (1997), 103–150. Zbl0890.35109MR1480996
  3. [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. Zbl0811.35097MR1278890
  4. [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. Zbl1205.86001MR2228849
  5. [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. Zbl0882.76098MR1438399
  6. [6] B. Desjardins, E. Dormy & E. Grenier, Stability of mixed Ekman-Hartmann boundary layers, Nonlinearity12 (1999), 181–199. Zbl0939.35151MR1677778
  7. [7] B. Desjardins & E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal. Non Linéaire20 (2003), 87–106. Zbl1114.76024MR1958163
  8. [8] M. Dimassi & J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Series 268, Cambridge Univ. Press, 1999. Zbl0926.35002MR1735654
  9. [9] P. F. Embid & A. J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations21 (1996), 619–658. Zbl0849.35106MR1387463
  10. [10] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., 1964. Zbl0144.34903MR181836
  11. [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. Zbl0949.35004MR1284205
  12. [12] I. Gallagher, Applications of Schochet’s methods to parabolic equations, J. Math. Pures Appl.77 (1998), 989–1054. Zbl1101.35330MR1661025
  13. [13] F. Gallaire & F. Rousset, Spectral stability implies nonlinear stability for incompressible boundary layers, Indiana Univ. Math. J.57 (2008), 1959–1975. Zbl1158.35073MR2440888
  14. [14] H. Greenspan, The theory of rotating fluids, Cambridge Monographs on Mechanics and Applied Mathematics, 1969. Zbl0443.76090
  15. [15] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl.76 (1997), 477–498. Zbl0885.35090MR1465607
  16. [16] E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math.53 (2000), 1067–1091. Zbl1048.35081MR1761409
  17. [17] E. Grenier & N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations22 (1997), 953–975. Zbl0880.35093MR1452174
  18. [18] E. Grenier & F. Rousset, Stability of one-dimensional boundary layers by using Green’s functions, Comm. Pure Appl. Math.54 (2001), 1343–1385. Zbl1026.35015MR1846801
  19. [19] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier (Grenoble) 45 (1995), 973–1006. Zbl0831.34023MR1359836
  20. [20] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, 1981. Zbl0456.35001MR610244
  21. [21] D. Lilly, On the instability of Ekman boundary flow, J. Atmos. Sci. (1966), 481–494. 
  22. [22] N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary, Arch. Rational Mech. Anal.142 (1998), 375–394. Zbl0915.76017MR1645962
  23. [23] N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data, Comm. Pure Appl. Math.53 (2000), 432–483. Zbl1047.76124MR1733696
  24. [24] G. Métivier & K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175, 2005. Zbl1074.35066MR2130346
  25. [25] J. Pedlosky, Geophysical fluid dynamics, Springer, 1979. Zbl0713.76005
  26. [26] F. Rousset, Stability of large Ekman boundary layers in rotating fluids, Arch. Ration. Mech. Anal.172 (2004), 213–245. Zbl1117.76026MR2058164
  27. [27] F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations210 (2005), 25–64. Zbl1060.35015MR2114123
  28. [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. Zbl1075.76033MR2169974
  29. [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. Zbl0913.35103MR1617538
  30. [30] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations114 (1994), 476–512. Zbl0838.35071MR1303036
  31. [31] M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series 34, Princeton University Press, 1981. Zbl0453.47026MR618463

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