Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T09:01:56.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluids with anisotropic viscosity

Published online by Cambridge University Press:  15 April 2002

Jean-Yves Chemin ,
Benoît Desjardins ,
Isabelle Gallagher  and
Emmanuel Grenier
Jean-Yves Chemin
Affiliation:
Laboratoire d'Analyse Numérique, CNRS UMR 7598, Université Paris 6, place Jussieu, 75005 Paris, France.
Benoît Desjardins
Affiliation:
CEA, BP 12, 91680 Bruyères-le Châtel, France.
Isabelle Gallagher
Affiliation:
CNRS UMR 8628 , Université Paris Sud, Bâtiment 425, 91405 Orsay Cedex, France.
Emmanuel Grenier
Affiliation:
UMPA (CNRS UMR 5669), E.N.S. Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France.
Get access

Abstract

Motivated by rotating fluids, we study incompressible fluidswith anisotropic viscosity.We use anisotropic spaces that enable us to prove existencetheoremsfor less regular initial data than usual. In the case of rotatingfluids, in the whole space, we prove Strichartz-type anisotropic, dispersive estimateswhich allow us to prove global wellposedness for fast enoughrotation.

Keywords

Navier-Stokes equationsRotating fluidsStrichartz estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 315 - 335
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales de l'École Normale Supérieure 14 (1981) 209-246. CrossRef
J.-Y. Chemin, Fluides parfaits incompressibles. Astérisque 230 (1995).
Chemin, J.-Y., À propos d'un problème de pénalisation de type antisymétrique. J. Math. Pures Appl. 76 (1997) 739-755. CrossRef
Chemin, J.-Y. and Lerner, N., Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes. J. Differential Equations 121 (1992) 314-328. CrossRef
J. -Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids, preprint of Université d'Orsay (1999).
B. Desjardins and E. Grenier, On the homogeneous model of wind driven ocean circulation. SIAM J. Appl. Math. (to appear).
B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations. Adv. in Differential Equations 3 (1998), No. 5, 715-752.
Desjardins, B. and Grenier, E., Low Mach number limit of compressible flows in the whole space. Proceedings of the Royal Society of London A 455 (1999) 2271-2279. CrossRef
Fujita, H. and Kato, T., On the Navier-Stokes initial value problem I. Archiv for Rational Mechanic Analysis 16 (1964) 269-315. CrossRef
Gallagher, I., The Tridimensional Navier-Stokes Equations with Almost Bidimensional Data: Stability, Uniqueness and Life Span. International Mathematics Research Notices 18 (1997) 919-935. CrossRef
H.P. Greenspan, The theory of rotating fluids. Cambridge monographs on mechanics and applied mathematics (1969).
E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differential Equations 22, No. 5-6, (1997) 953-975.
D. Iftimie, La résolution des équations de Navier-Stokes dans des domaines minces et la limite quasigéostrophique. Thèse de l'Université Paris 6 (1997).
Iftimie, D., The resolution of the Navier-Stokes equations in anisotropic spaces. Revista Matematica Ibero-Americana 15 (1999) 1-36.
Leray, J., Essai sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1933) 193-248. CrossRef
J. Pedlosky, Geophysical fluid dynamics, Springer (1979).
Rauch, J. and Reed, M., Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension. Duke Mathematical Journal 49 (1982) 397-475. CrossRef
Sablé-Tougeron, M., Régularité microlocale pour des problèmes aux limites non linéaires. Annales de l'Institut Fourier 36 (1986) 39-82. CrossRef