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Mathematische Annalen

, Volume 369, Issue 3–4, pp 1573–1597 | Cite as

An anelastic approximation arising in astrophysics

  • Donatella Donatelli
  • Eduard Feireisl
Article

Abstract

We identify the asymptotic limit of the compressible non-isentropic Navier–Stokes system in the regime of low Mach, low Froude and high Reynolds number. The system is driven by a long range gravitational potential. We show convergence to an anelastic system for ill-prepared initial data. The proof is based on frequency localized Strichartz estimates for the acoustic equation based on the recent work of Metcalfe and Tataru.

References

  1. 1.
    Almgren, A.S., Bell, J.B., Monaka, A., Zingale, M.: Low Mach number modeling of type Ia supernovae. III. Reactions. Astrophys. J. 684, 449–470 (2008)CrossRefGoogle Scholar
  2. 2.
    Almgren, A.S., Bell, J.B., Rendleman, C.A., Zingale, M.: Low Mach number modeling of type Ia supernovae. I. Hydrodynamics. Astrophys. J. 637, 922–936 (2006)CrossRefGoogle Scholar
  3. 3.
    Almgren, A.S., Bell, J.B., Rendleman, C.A., Zingale, M.: Low Mach number modeling of type Ia supernovae. II. Energy evolution. Astrophys. J. 649, 927–938 (2006)CrossRefGoogle Scholar
  4. 4.
    De Bièvre, S., Pravica, D.W.: Spectral analysis for optical fibres and stratified fluids. I. The limiting absorption principle. J. Funct. Anal. 98(2), 404–436 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Bièvre, S., Pravica, D.W.: Spectral analysis for optical fibres and stratified fluids. II. Absence of eigenvalues. Commun. Partial Differ. Equ. 17(1–2), 69–97 (1992)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Donatelli, D., Feireisl, E., Novotný, A.: On incompressible limits for the Navier–Stokes system on unbounded domains under slip boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 13(4), 783–798 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feireisl, E., Novotný, A., Petzeltová, H.: Low Mach number limit for the Navier–Stokes system on unbounded domains under strong stratifications. Commun. Partial Differ. Equ. 35, 68–88 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feireisl, E.: Low Mach number limits of compressible rotating fluids. J. Math. Fluid Mech. 14, 61–78 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feireisl, E., Jin, B.J., Novotný, A.: Inviscid incompressible limits of strongly stratified fluids. Asymptot. Anal. 89(3–4), 307–329 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Feireisl, E., Klein, R., Novotný, A., Zatorska, E.: On singular limits arising in the scale analysis of stratified fluid flows. Math. Models Methods Appl. Sci. 26(3), 419–443 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)Google Scholar
  12. 12.
    Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford Science Publication, Oxford (1998)zbMATHGoogle Scholar
  13. 13.
    Maltese, D., Michálek, M., Mucha, P.B., Novotný, A., Pokorný, M., Zatorska, E.: Existence of weak solutions for compressible Navier–Stokes equation with entropy transport. 2016. arxiv preprint No. arXiv:1603.08965 [v1]
  14. 14.
    Masmoudi, N.: Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230–240 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Metcalfe, J., Tataru, D.: Global parametrices and dispersive estimates for variable coefficient wave equations. Math. Ann. 353(4), 1183–1237 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Oliver, M.: Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl. 215, 471–484 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Smith, H.F., Sogge, C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Commun. Partial Differ. Equ. 25(11–12), 2171–2183 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Information Engineering, Computer Science and MathematicsUniversity of L’AquilaL’AquilaItaly
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic

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