Stability of oscillating boundary layers in rotating fluids

Masmoudi, Nader; Rousset, Frédéric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 955-1002.

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 $\epsilon$. Cette étude généralise celle de [23] où une condition de petitesse était imposée et celle de [26] où les données bien préparées étaient traitées.

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 $\epsilon$. 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.

[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 Fluids 15 (1996), 291-300. | MR | Zbl

[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. | 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