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Weak convergence results for inhomogeneous rotating fluid equations

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Abstract

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vectorB(x); this is a generalization of the usual rotating fluid model (whereB is constant). In the case n whichB has non-degenerate critical points, we prove the weak convergence of Leray-type solutions towards a vector field which satisfies a heat equation as the rotation rate tends to infinity. The method of proof uses weak compactness arguments, which also enable us to recover the usual 2D Navier-Stokes limit in the case whenB is constant.

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References

  1. J.-P. Aubin,Un théorème de compacité, Notes aux Comptes Rendus de l'Académie des Sciences de Paris309 (1963), 5042–5044.

    MathSciNet  Google Scholar 

  2. A. Babin, A. Mahalov and B. Nicolaenko,Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mech.,15 (1996), 291–300.

    MATH  MathSciNet  Google Scholar 

  3. A. Babin, A. Mahalov and B. Nicolaenko,Resonances and regularity for Boussinesq equations, Russian J. Math. Phys.4 (1996), 417–428.

    MATH  MathSciNet  Google Scholar 

  4. 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.

    MATH  MathSciNet  Google Scholar 

  5. J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,Anisotropy and dispersion in rotating fluids, inNonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Studies in Mathematics and its Applications,31, North-Holland Amsterdam, 2002, pp. 171–191.

    Chapter  Google Scholar 

  6. J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,Ekman boundary layers in rotating fluids, ESAIM Contrôle Optimal et Calcul des Variations, Special Tribute issue to Jacques-Louis Lions,8 (2002), 441–466.

    MATH  MathSciNet  Google Scholar 

  7. P. Embid and A. Majda,Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations21 (1996), 619–658.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Gallagher,Applications of Schochet's methods to parabolic equations. J. Math. Pures Appl.77 (1998), 989–1054.

    MATH  MathSciNet  Google Scholar 

  9. A. E. Gill,Atmosphere-Ocean Dynamics, International Geophysics Series, Vol. 30, Academic Press, New York, 1982.

    Google Scholar 

  10. F. Golse and L. Saint-Raymond,The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl.78 (1999), 791–817.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. P. Greenspan,The Theory of Rotating Fluids, Cambridge University Press, 1969.

  12. E. Grenier,Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl.76 (1997), 477–498.

    MATH  MathSciNet  Google Scholar 

  13. E. Grenier and N. Masmoudi,Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations22 (1997), 953–975.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Leray,Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math.63 (1933), 193–248.

    Article  MathSciNet  Google Scholar 

  15. P.-L. Lions and N. Masmoudi,Incompressible limit for a viscous compressible fluid J. Math. Pures Appl.77 (1998), 585–627.

    MATH  MathSciNet  Google Scholar 

  16. P.-L. Lions and N. Masmoudi,Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Sér. I Math.329 (1999), 387–392.

    MATH  MathSciNet  Google Scholar 

  17. N. Masmoudi,Ekman layers of rotating fluids: the case of general initial data, Comm. Pure Appl. Math.53 (2000), 432–483.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Pedlosky,Geophysical fluid dynamics, Springer, Berlin, 1979.

    MATH  Google Scholar 

  19. S. Schochet,Fast singular limits of hyperbolic PDEs, J. Differential Equations114 (1994), 476–512.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Temam and M. Ziane,Some mathematical problems in geophysical fluid dynamics, inHandbook of Mathematical Fluid Dynamics, Vol. III, North-Holland, Amsterdam, 2004, pp. 535–657.

    Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013, Paris, France

    Isabelle Gallagher

  2. Laboratoire J.-L. Lions UMR 7598, Université Paris VI, 175, rue du Chevaleret, 75013, Paris, France

    Laure Saint-Raymond

Authors
  1. Isabelle Gallagher
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  2. Laure Saint-Raymond
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Gallagher, I., Saint-Raymond, L. Weak convergence results for inhomogeneous rotating fluid equations. J. Anal. Math. 99, 1–34 (2006). https://doi.org/10.1007/BF02789441

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  • DOI: https://doi.org/10.1007/BF02789441

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