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On the incompressible limit of the compressible Navier-Stokes equations. (English) Zbl 0816.35105
Summary: Many interesting problems in classical physics involve the behavior of solutions of nonlinear hyperbolic systems as certain parameter and coefficients becomes infinite. Quite often, the limiting solution (when it exists) satisfies a completely different nonlinear partial differential equation. The incompressible limit of the compressible Navier-Stokes equations is one physical problem involving dissipation when such a singular limiting process is interesting.
In this article we study the time-discretized compressible Navier-Stokes equation and consider the incompressible limit as the Mach number tends to zero. For \(\gamma\)-law gas, \(1< \gamma\leq 2\), \(D\leq 4\), we show that the solutions \((\rho_ \varepsilon, \mu_ \varepsilon/ \varepsilon)\) of the compressible Navier-Stokes system converge to the solution \((1, \overline {\mathbf v})\) of the incompressible Navier-Stokes system. Furthermore we also prove that the limit also satisfies the Leray energy inequality.

MSC:
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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[1] Agemi R., Proc. Japan Acad. Ser.A 57 pp 291– (1981) · Zbl 0494.76061
[2] Arnold V.I., Mathematical Methods of Classical Mechanies (1978)
[3] Asano K., Jpn. J. Appl. Math. 4 pp 455– (1987) · Zbl 0638.35012
[4] DOI: 10.1007/BF01026608
[5] DOI: 10.1002/cpa.3160460503 · Zbl 0817.76002
[6] Bardos C., Advances in Kinetic Theory and Continuum Mechanics pp 57– (1991)
[7] Bardos C., Mathematical Model and Methods in Applied Sciences 1 pp 235– (1991) · Zbl 0758.35060
[8] DOI: 10.2307/2044999 · Zbl 0526.46037
[9] Constantin P., Chicago Lecture in Mathematics (1988)
[10] DOI: 10.2307/1971423 · Zbl 0698.45010
[11] DOI: 10.1007/BF01214424 · Zbl 0626.35059
[12] DOI: 10.2307/1971029 · Zbl 0373.76007
[13] DOI: 10.1002/cpa.3160350402 · Zbl 0478.76011
[14] DOI: 10.2307/1970699 · Zbl 0211.57401
[15] Evans L.C., CBMS Regional Conf. Ser. in Math. (1990)
[16] Fujiwara, D. and Movimato, H. 1977.An Lp- theorem of the Helmholtz decomposition of vector field,, 685–700. Tokyo: J. Fac. Sci. Univ. Sect.1-A
[17] DOI: 10.1016/0022-1236(72)90003-1 · Zbl 0229.76018
[18] DOI: 10.1007/BF00280740 · Zbl 0343.35056
[19] Kato T., with Applications to weak Solutions, Math. Z. 187 pp 471– (1984) · Zbl 0545.35073
[20] DOI: 10.1002/cpa.3160340405 · Zbl 0476.76068
[21] Klainerman S., Comm. Pure and Appl. Math., 35 pp 637– (1982) · Zbl 0478.76091
[22] Krasnoselskii, M.A. and Rutickii, J.B. 1961. ”Convex Functions and Orlicz Spaces”. Noordhoff Groningen.
[23] DOI: 10.1002/cpa.3160330310 · Zbl 0439.35043
[24] Kreiss H.O., SIAM J. Appl. Math. 42 pp 704– (1982) · Zbl 0506.35006
[25] Kreiss H.O., Pure and Appl. Math. (1989)
[26] DOI: 10.1016/0196-8858(91)90012-8 · Zbl 0728.76084
[27] Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, (1969) · Zbl 0184.52603
[28] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803
[29] Lax P.D., C.B.M.S 11 (1973)
[30] DOI: 10.1007/BF02547354 · JFM 60.0726.05
[31] Lions J.L., quelques m’ethodes de r’esolution des probl’emes aux limites non lin’eaires (1969)
[32] Lions, P.L. 1993.Limites incompressible et acoustique pour des fluides visqueux,compressibles et isentropiques, Vol. 317, 1197–1202. Paris: C.R. Acad. Sci. · Zbl 0795.76068
[33] Majda A., Appl. Math. Sci 53 (1984)
[34] DOI: 10.1002/cpa.3160390711 · Zbl 0595.76021
[35] DOI: 10.1007/BF01229377 · Zbl 0651.73004
[36] DOI: 10.1007/BF01210792 · Zbl 0612.76082
[37] DOI: 10.1080/03605308608820478 · Zbl 0651.35047
[38] Schochet S., Journal of Diff. Equa. 68 pp 400– (1987) · Zbl 0633.35047
[39] Serrin J., Non linear Problems (1963)
[40] Teman R., Navier-Stokes Equations, (1984)
[41] Temam, R. and Ukai, S. 1986.The incompressible limit and the initial layer of the compressible Euler equation, Vol. 26-2, 323–331. J. Math. Kyoto Univ. · Zbl 0618.76074
[42] Ukai S., J. Math. Kyoto Univ. 26 (2) pp 323– (1986) · Zbl 0618.76074
[43] da Veiga H.B., The Equilibrium Solutions, Comm. Math. Phys. 109 pp 229– (1987) · Zbl 0621.76074
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