The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. (English) Zbl 0612.76082

The author considers the Euler equations for nonisentropic compressible inviscid fluids in a bounded domain. The description of the fluid is completed by the equation of state \(\rho =f(P/\lambda^ 2,S)\) where \(\rho\) is the density, P the pressure, S the entropy and the parameter \(\lambda\) is essentially the inverse of the Mach number. First the author proves the local (in time) existence of a classical solution for any fixed \(\lambda\). Afterwards he shows that the solutions converge, as \(\lambda \to +\infty\), to the corresponding solution of the equations for incompressible inviscid fluids with variable density.
Reviewer: P.Secchi


76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
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[1] Agemi, R.: The initial boundary value problem for inviscid barotropic fluid motion. Hokkaido Math. J.10, 156-182 (1981) · Zbl 0472.76065
[2] Barker, J.: Interactions of first and slow moves in problems with two time scales, Thesis, California Institute of Technology 1982
[3] Bourguignon, J.P., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal.15, 341-363 (1975) · Zbl 0279.58005
[4] Browning, G., Kreiss, H.-O.: Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math.42, 704-708 (1982) · Zbl 0506.35006
[5] Ebin, D.: The initial boundary value problem for subsonic fluid motion, Commun. Pure Appl. Math.32, 1-19 (1979) · Zbl 0394.76053
[6] Ebin, D.: Motion of slightly compressible fluids in a bounded domain. I. Commun. Pure Appl. Math.35, 451-485 (1982) · Zbl 0487.76015
[7] Folland, G.: Introduction to partial differential equations. Princeton, NJ: Princeton University Press 1976 · Zbl 0325.35001
[8] Hormander, L.: Linear partial differential operators. Berlin, Heidelberg, New York: Springer 1963
[9] Ikawa, M.: Mixed problem for a hyperbolic system of first order. Publ. Res. Inst. Math. Sci.7, 427-454 (1971/2) · Zbl 0231.35051
[10] Klainerman, S., Majda, A.: Singular limits of quasilinear systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.34, 481-524 (1981) · Zbl 0476.76068
[11] Lax, P.D.: Hyperbolic systems of conservation lows and the mathematical theory of shock waves. SIAM, Philadelphia 1972
[12] Lax, P.D., Phillips, R.: Local boundary conditions for dissipative symmetric linear differential equations. Commun. Pure Appl. Math.13, 427-55 (1960) · Zbl 0094.07502
[13] Majda, A.: Compressible fluid flow and systems of conservation laws in several space dimensions. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0537.76001
[14] Rauch, J., Massey, F.: Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Am. Math. Soc.189, 303-318 (1974) · Zbl 0282.35014
[15] Rauch, J., Nishida, T.: In preparation
[16] Schochet, S.: Initial-boundary-value-problems for quasilinear symmetric hyperbolic systems, existence of solutions to the compressible Euler equations, and their incompressible limit, thesis, Courant Institute 1984
[17] Whitham, G.: Linear and nonlinear waves. New York: Wiley 1974 · Zbl 0373.76001
[18] Ebin, D.: The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math.105, 141-200 (1977) · Zbl 0373.76007
[19] Friedman, A.: Partial differential equations. Huntington, NY: Kreiger 1976 · Zbl 0323.60057
[20] Majda, A.: The existence of multi-dimensional shock fronts. Memiors AMS # 201, 1983 · Zbl 0517.76068
[21] Beirao Da Veiga, H.: Un theoreme d’existence dans la dynamique des fluids compressible. C.R. Acad. Sci. Paris289B, 297-299 (1979)
[22] Beirao Da Veiga, H.: On the barotropic motion of compressible perfect fluids. Ann. Sc. Norm. Sup. Pisa8, 317-351 (1981) · Zbl 0477.76059
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