Fluids with anisotropic viscosity. (English) Zbl 0954.76012

Summary: Motivated by rotating fluids, we study incompressible fluids with anisotropic viscosity. We use anisotropic spaces that enable us to prove existence theorems for less regular initial data than usual. In the case of rotating fluids, in the whole space, we prove Strichartz-type anisotropic, dispersive estimates which allow us to prove global well-posedness for fast enough rotation.


76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
76U05 General theory of rotating fluids
35Q35 PDEs in connection with fluid mechanics
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[1] A. Babin, A. Mahalov and B. Nicolaenko, Global Splitting, Integrability and Regularity of 3D Euler and Navier-Stokes Equations for Uniformly Rotating Fluids. Eur. J. Mech.15 (1996) 291-300. · Zbl 0882.76096
[2] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales de l’École Normale Supérieure14 (1981) 209-246. · Zbl 0495.35024
[3] J.-Y. Chemin, Fluides parfaits incompressibles. Astérisque230 (1995). · Zbl 0829.76003
[4] J.-Y. Chemin, À propos d’un problème de pénalisation de type antisymétrique. J. Math. Pures Appl.76 (1997) 739-755. · Zbl 0896.35103
[5] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes. J. Differential Equations121 (1992) 314-328. · Zbl 0878.35089
[6] J. -Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids, preprint of Université d’Orsay (1999). · Zbl 1034.35107
[7] B. Desjardins and E. Grenier, On the homogeneous model of wind driven ocean circulation. SIAM J. Appl. Math. (to appear). · Zbl 0958.76092
[8] B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations. Adv. in Differential Equations3 (1998), No. 5, 715-752. · Zbl 0967.76096
[9] B. Desjardins and E. Grenier, Low Mach number limit of compressible flows in the whole space. Proceedings of the Royal Society of London A455 (1999) 2271-2279. Zbl0934.76080 · Zbl 0934.76080
[10] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I. Archiv for Rational Mechanic Analysis16 (1964) 269-315. · Zbl 0126.42301
[11] I. Gallagher, The Tridimensional Navier-Stokes Equations with Almost Bidimensional Data: Stability, Uniqueness and Life Span. International Mathematics Research Notices18 (1997) 919-935. · Zbl 0893.35098
[12] H.P. Greenspan, The theory of rotating fluids. Cambridge monographs on mechanics and applied mathematics (1969). Zbl0177.42401 · Zbl 0177.42401
[13] E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations22, No. 5-6, (1997) 953-975. · Zbl 0880.35093
[14] D. Iftimie, La résolution des équations de Navier-Stokes dans des domaines minces et la limite quasigéostrophique. Thèse de l’Université Paris 6 (1997).
[15] D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matematica Ibero-Americana15 (1999) 1-36. Zbl0923.35119 · Zbl 0923.35119
[16] J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.63 (1933) 193-248. Zbl60.0726.05 · JFM 60.0726.05
[17] J. Pedlosky, Geophysical fluid dynamics, Springer (1979). · Zbl 0429.76001
[18] J. Rauch and M. Reed, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension. Duke Mathematical Journal49 (1982) 397-475. Zbl0503.35055 · Zbl 0503.35055
[19] M. Sablé-Tougeron, Régularité microlocale pour des problèmes aux limites non linéaires. Annales de l’Institut Fourier36 (1986) 39-82. · Zbl 0577.35004
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