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Multiple scales and singular limits for compressible rotating fluids with general initial data. (English) Zbl 1307.35220

This article deals with a compressible, rotating fluid, occupying a three-dimensional infinite layer. The problem depends on a small parameter, \(\varepsilon\), and the characteristic numbers are expressed by means on the small parameter, as follows: Rossby number \(=\varepsilon\), Mach number \(=\varepsilon^m\), Reynolds number \(=\varepsilon^{-\alpha}\), Froude number \(=\varepsilon^n\), with \(\frac{m}{2}>n\geq 1\) and \(\alpha >0\). The goal of this work is to analyze the limit problem when \(\varepsilon \rightarrow 0.\) It is proved that the limit problem is the incompressible Euler system for the planar velocity.

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76U05 General theory of rotating fluids
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[1] Babin A., Indiana Univ. Math. J. 48 pp 1133– (1999)
[2] DOI: 10.1512/iumj.2001.50.2155 · Zbl 1013.35065
[3] DOI: 10.3934/dcds.2004.11.47 · Zbl 1138.76446
[4] Chemin J.-Y., Mathematical Geophysics, volume 32 of Oxford Lecture Series in Mathematics and its Applications (2006)
[5] DOI: 10.1088/0951-7715/11/6/011 · Zbl 0911.76014
[6] DOI: 10.1051/m2an:2005019 · Zbl 1080.35067
[7] DOI: 10.1175/1520-0477(1993)074<2179:ITCFRR>2.0.CO;2
[8] DOI: 10.2307/1971029 · Zbl 0373.76007
[9] DOI: 10.1137/100808010 · Zbl 1263.76077
[10] Feireisl E., J. Math. Fluid Mech. 14 pp 712– (2012)
[11] Feireisl , E. Novotný , A. Scale interactions in compressible rotating fluids.Anal. Mat. Pura Appl.In press . · Zbl 1302.76200
[12] DOI: 10.1007/PL00000976 · Zbl 0997.35043
[13] DOI: 10.1512/iumj.2011.60.4406 · Zbl 1248.35143
[14] Gallagher , I. ( 2005 ). Résultats récents sur la limite incompressible.Astérisque, (299):Exp. No. 926, vii, 29–57, 2005. Séminaire Bourbaki. Vol. 2003/2004 .
[15] DOI: 10.1007/s00021-009-0006-1 · Zbl 1270.35342
[16] DOI: 10.1016/j.jfa.2007.12.010 · Zbl 1145.35032
[17] Kato , T. ( 1984 ). Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Chern, S.S., ed.Seminar on PDE’s, Vol. 2. New York: Springer, pp. 85–98 .
[18] DOI: 10.1016/0022-1236(84)90024-7 · Zbl 0545.76007
[19] DOI: 10.1002/cpa.3160340405 · Zbl 0476.76068
[20] Klein R., Annu. Rev. Fluid. Mech pp 249– (2010)
[21] Lions P.-L., Mathematical Topics in Fluid Dynamics, Vol. 2. Compressible models (1998)
[22] DOI: 10.1016/S0021-7824(98)80139-6 · Zbl 0909.35101
[23] DOI: 10.1007/s002050050097 · Zbl 0915.76017
[24] DOI: 10.1016/S0294-1449(00)00123-2 · Zbl 0991.35058
[25] Masmoudi , N. ( 2006 ). Examples of singular limits in hydrodynamics. In: Dafermos, C., Feireisl, E. eds.Handbook of Differential Equations, III.Amsterdam: Elsevier .
[26] DOI: 10.1006/jmaa.1997.5647 · Zbl 0893.35107
[27] DOI: 10.1007/s002200050304 · Zbl 0913.35102
[28] DOI: 10.1007/s002200050305 · Zbl 0913.35103
[29] DOI: 10.1051/m2an:2005017 · Zbl 1094.35094
[30] Stein E.M., Singular Integrals and Differential Properties of Functions (1970) · Zbl 0207.13501
[31] Swann H.S.G., Trans. Amer. Math. Soc. 157 pp 373– (1971)
[32] Temam R., Annali Scuola Normale Pisa 25 pp 807– (1997)
[33] DOI: 10.1006/jdeq.2001.4038 · Zbl 0997.35042
[34] Vallis , G.K.Atmospheric and Oceanic Fluid Dynamics.Cambridge: Cambridge University Press . · Zbl 1374.86002
[35] Watson G.N., A Treatise on the Theory of Bessel Functions (1961)
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