Abstract
The solution of the initial value problem for the compressible Euler equation tends to the solution of the corresponding incompressible Euler equation with the corresponding initial data, as the Mach number (which is proportional to a parameter 1/λ) tends to zero. Under suitable conditions, we also obtain the asymptotic expansion theorem for those solution, when λ is large.
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References
R. Agemi, The incompressible limit of compressible fluid motion in a bounded domain. Proc. Japan Acad., Ser. A,57 (1981), 292–293.
S. Agmon, Lectures on Elliptic Boundary Value Problems. van Nostrand, Toronto, New York, London, 1965.
K. Asano, Asymptotic theory in the fluid dynamics I, Incompressible fluids. In preparation.
K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation. Lecture Notes in Numer. Appl. Math. 6, Kinokuniya/North-Holland, Tokyo/Amsterdam, 1983, 1–19.
A. P. Calderón and A. Zygmund, On the existence of certain singular integrals. Acta Math.,88 (1952), 85–139.
A. P. Calderón and A. Zygmund, On singular integrals. Amer. J. Math.,78 (1956), 289–309.
D. G. Ebin, Motion of slightly compressible fluids in a bounded domain I. Comm. Pure Appl. Math.,35 (1982), 451–485.
K. O. Friedrichs, Symmetric hyperbolic system of linear differential equations. Comm. Pure Appl. Math.,7 (1954), 345–392.
T. Kato, The Cauchy Problem for quasilinear symmetric systems. Arch. Rational Mech. Anal.,58 (1975), 181–205.
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic system with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math.,34 (1981), 481–524.
S. Klainerman and A. Majda, Compressible and incompressible fluids. Comm. Pure Appl. Math.,35 (1982), 629–651.
M. Matsumura, Comportement des solutions de quelques problèmes mixtes pour certains systèmes hyperboliques symétriques a coefficients constants. Publ. RIMS. Kyoto Univ., Ser. A,4 (1968), 309–359.
S. Mizohata, The Theory of Partial Differential Equations. Cambridge Univ. Press, 1973.
S. Mizohata, Systèmes hyperboliques. J. Math. Soc. Japan,11 (1959), 205–233.
S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ.,26 (1986), 323–331.
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Asano, K. On the incompressible limit of the compressible Euler equation. Japan J. Appl. Math. 4, 455–488 (1987). https://doi.org/10.1007/BF03167815
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DOI: https://doi.org/10.1007/BF03167815