Communications in Mathematical Physics

, Volume 314, Issue 3, pp 641–670 | Cite as

Multi-scale Analysis of Compressible Viscous and Rotating Fluids

  • Eduard Feireisl
  • Isabelle GallagherEmail author
  • David Gerard-Varet
  • Antonín Novotný


We study a singular limit for the compressible Navier-Stokes system when the Mach and Rossby numbers are proportional to certain powers of a small parameter \({\varepsilon}\) . If the Rossby number dominates the Mach number, the limit problem is represented by the 2-D incompressible Navier-Stokes system describing the horizontal motion of vertical averages of the velocity field. If they are of the same order then the limit problem turns out to be a linear, 2-D equation with a unique radially symmetric solution. The effect of the centrifugal force is taken into account.


Mach Number Centrifugal Force Convective Term Limit Problem Singular Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Babin A., Mahalov A., Nicolaenko B.: Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48(3), 1133–1176 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50(Special Issue), 1–35, (2001). (Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000))Google Scholar
  3. 3.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical geophysics, Volume 32 of Oxford Lecture Series in Mathematics and its Applications. Oxford: The Clarendon Press/ Oxford University Press, 2006Google Scholar
  4. 4.
    D’Acona, P., Racke, R.: Evolution equations in non-flat waveguides. [math.Ap], 2010
  5. 5.
    Desjardins B., Grenier E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Desjardins B., Grenier E., Lions P.-L., Masmoudi N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)MathSciNetGoogle Scholar
  7. 7.
    Ebin, D.B.: Viscous fluids in a domain with frictionless boundary. In: Global Analysis - Analysis on Manifolds, H. Kurke, J. Mecke, H. Triebel, R. Thiele, eds. Teubner-Texte zur Mathematik. 57, Leipzig: Teubner, 1983, pp. 93–110Google Scholar
  8. 8.
    Feireisl, E., Novotný, A.: Singular limits in thermodynamics of viscous fluids. Basel: Birkhauser, 2009Google Scholar
  9. 9.
    Feireisl E., Gallagher I., Novotný A.: A singular limit for compressible rotating fluids. SIAMJ. Math. Anal. 44(1), 192–205 (2009)CrossRefGoogle Scholar
  10. 10.
    Feireisl E., Novotný A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Gallagher I., Saint-Raymond L.: Weak convergence results for inhomogeneous rotating fluid equations. J. d’Analyse Math. 99, 1–34 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Klein R.: Scale-dependent models for atmospheric flows. Ann. Rev. Fluid Mech. 42, 249–274 (2010)ADSCrossRefGoogle Scholar
  14. 14.
    Lighthill J.: On sound generated aerodynamically I. General theory. Proc. of the Royal Soc. London, A 211, 564–587 (1952)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Lighthill J.: On sound generated aerodynamically II. General theory. Proc. of the Royal Soc. London, A 222, 1–32 (1954)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Lions, P.-L.: Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford: Oxford Science Publication, 1998Google Scholar
  17. 17.
    Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. C.R. Acad. Sci. Paris Sér. I Math. 329(5), 230–240 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Masmoudi N.: Rigorous derivation of the anelastic approximation. J. Math. Pures et Appl. 88(5), 387–392 (2007)Google Scholar
  20. 20.
    Metcalfe, J. L.: Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle. Trans. Amer. Math. Soc. 356(12), 4839–4855 (electronic) (2004)Google Scholar
  21. 21.
    Smith H.F., Sogge C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Par. Diff. Eqs. 25(11–12), 2171–2183 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Eduard Feireisl
    • 1
  • Isabelle Gallagher
    • 2
    Email author
  • David Gerard-Varet
    • 2
  • Antonín Novotný
    • 3
  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  2. 2.Institut de Mathématiques UMR 7586, Université Paris DiderotParisFrance
  3. 3.IMATH Université du Sud Toulon-VarLa GardeFrance

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