An anelastic approximation arising in astrophysics
Article
First Online:
Received:
Revised:
- 191 Downloads
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.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.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.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.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)MathSciNetCrossRefMATHGoogle Scholar
- 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)MathSciNetMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 8.Feireisl, E.: Low Mach number limits of compressible rotating fluids. J. Math. Fluid Mech. 14, 61–78 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 9.Feireisl, E., Jin, B.J., Novotný, A.: Inviscid incompressible limits of strongly stratified fluids. Asymptot. Anal. 89(3–4), 307–329 (2014)MathSciNetMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 11.Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)Google Scholar
- 12.Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford Science Publication, Oxford (1998)MATHGoogle Scholar
- 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.Masmoudi, N.: Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230–240 (2007)MathSciNetCrossRefMATHGoogle Scholar
- 15.Metcalfe, J., Tataru, D.: Global parametrices and dispersive estimates for variable coefficient wave equations. Math. Ann. 353(4), 1183–1237 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 16.Oliver, M.: Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl. 215, 471–484 (1997)MathSciNetCrossRefMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
Copyright information
© Springer-Verlag Berlin Heidelberg 2016