Ekman boundary layers in rotating fluids
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 441-466

In this paper, we investigate the problem of fast rotating fluids between two infinite plates with Dirichlet boundary conditions and “turbulent viscosity” for general ${L}^{2}$ initial data. We use dispersive effect to prove strong convergence to the solution of the bimensionnal Navier-Stokes equations modified by the Ekman pumping term.

DOI : https://doi.org/10.1051/cocv:2002037
Classification:  35Q30,  35Q35,  76U05
Keywords: Navier-Stokes equations, rotating fluids, Strichartz estimates
@article{COCV_2002__8__441_0,
author = {Chemin, Jean-Yves and Desjardins, Beno\^\i t and Gallagher, Isabelle and Grenier, Emmanuel},
title = {Ekman boundary layers in rotating fluids},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
pages = {441-466},
doi = {10.1051/cocv:2002037},
zbl = {1070.35505},
mrnumber = {1932959},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__8__441_0}
}

Chemin, Jean-Yves; Desjardins, Benoît; Gallagher, Isabelle; Grenier, Emmanuel. Ekman boundary layers in rotating fluids. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 441-466. doi : 10.1051/cocv:2002037. http://www.numdam.org/item/COCV_2002__8__441_0/

[1] A. Babin, A. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48 (1999) 1133-1176. | Zbl 0932.35160

[2] A. Babin, A. Mahalov and 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. | Zbl 0882.76096

[3] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity. Modél. Math. Anal. Numér. 34 (2000) 315-335. | Numdam | MR 1765662 | Zbl 0954.76012

[4] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids. Preprint of Orsay University. | MR 1935994

[5] B. Desjardins, E. Dormy and E. Grenier, Stability of mixed Ekman-Hartmann boundary layers. Nonlinearity 12 (1999) 181-199. | Zbl 0939.35151

[6] I. Gallagher, Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl. 77 (1998) 989-1054. | Zbl 1101.35330

[7] H.P. Greenspan, The theory of rotating fluids1980). | MR 639897 | Zbl 0443.76090

[8] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. 76 (1997) 477-498. | Zbl 0885.35090

[9] E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations 22 (1997) 953-975. | MR 1452174 | Zbl 0880.35093

[10] N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data. Comm. Pure Appl. Math. 53 (2000) 432-483. | MR 1733696 | Zbl 1047.76124

[11] Pedlovsky, Geophysical Fluid Dynamics. Springer-Verlag (1979). | Zbl 0429.76001