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
MSC:
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35Q30 | Navier-Stokes equations |
Keywords:
Euler equations; nonisentropic compressible inviscid fluids; bounded domain; local (in time) existence; classical solution; incompressible inviscid fluids with variable densityReferences:
| [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 · doi:10.1016/0022-1236(74)90027-5 |
| [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 · doi:10.1137/0142049 |
| [5] | Ebin, D.: The initial boundary value problem for subsonic fluid motion, Commun. Pure Appl. Math.32, 1-19 (1979) · doi:10.1002/cpa.3160320102 |
| [6] | Ebin, D.: Motion of slightly compressible fluids in a bounded domain. I. Commun. Pure Appl. Math.35, 451-485 (1982) · doi:10.1002/cpa.3160350402 |
| [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 · doi:10.2977/prims/1195193549 |
| [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 · doi:10.1002/cpa.3160340405 |
| [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 · doi:10.1002/cpa.3160130307 |
| [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 · doi:10.2307/1971029 |
| [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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
