Symmetries and global solvability of the isothermal gas dynamics equations

Archive for Rational Mechanics and Analysis

Volumn 175, Issue 3, 2005, Pages 389-430

  LeFloch, Philippe G ^a   Shelukhin, Vladimir ^b  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^b LAVRENTYEV INSTITUTE OF HYDRODYNAMICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21344434322     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-004-0344-3     Document Type: Review

References (33)

1. BALL, J.M.: A version of the fundamental theorem for Young measures. In: PDE's and Continuum Models of Phase Transitions, Lecture Notes in Physics, Vol. 344, 1989 RASCLE M., SERRE D. & SLEMROD M. (eds), Springer Verlag, pp. 207-215
2. BIANCHINI, S., BRESSAN, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math., 2004
3. CHEN, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 8, 243-276 (1988)
4. CHEN, G.-Q., LEFLOCH, P.G.: Compressible Euler equations with general pressure law. Arch. Rational Mech. Anal. 153, 221-259 (2000)
5. CHEN, G.-Q., LeFloch, P.G.: Existence theory for the isentropic Euler equations. Arch. Rational Mech. Anal. 166, 81-98 (2003)
6. DAFERMOS, C.M.: Hyperbolic conservation laws in continuum physics, Grundlehren Math. Wissenschaften Series 325, Springer Verlag, 2000
7. DAL MASO, G., LEFLOCH, P.G., MURAT, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483-548 (1995)
8. DING, X., CHEN, G.-Q., LUO, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics. Acta Math. Sci. 5, 483-540 (1985)
9. DIPERNA R.J.: Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82, 27-70 (1983)
10. DIPERNA, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1-30 (1983)
11. FRID, H., SHELUKHIN, V.V.: A quasilinear parabolic system for three-phase capillary flow in porous media. SIAM J. Math. Anal. 35, 1029-1041 (2003)
12. HUANG, F.-M., Wang, Z.: Convergence of viscosity solutions for isothermal gas dynamics. SIAM J. Math. Anal. 34, 595-610 (2003)
13. LADYŽENSKAJA, O.A., SOLONNIKOV, V.A., URAL'CEVA, N.N.: Linear and quasi-linear equations of parabolic type, A.M.S., Providence, 1968
14. LAX, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conf. Lecture 11, Philadelphia, 1973
15. LEFLOCH, P.G.: Existence of entropy solutions for the compressible Euler equations. International Series Numer. Math. Vol. 130, Birkäuser Verlag Bäsel, Switzerland, 1999, pp. 599-607
16. LEFLOCH, P.G.: Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves. Lectures in Mathematics, ETH Zürich, Birkäuser, 2002
17. LEFLOCH P.G.: Graph solutions of nonlinear hyperbolic systems. J. Hyper. Diff. Equa. 1 (2004)
18. LIONS, PL., PERTHAME, B., TADMOR, E.: Kinetic formulation for the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415-431 (1994)
19. LIONS, PL., PERTHAME, B., SOUGANIDIS, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49, 599-638 (1996)
20. MORAWETZ, C.J.: An alternative proof of DiPerna's theorem. Commun. Pure Appl. Math. 44, 1081-1090 (1991)
21. LU, Y.-G.: Convergence of the viscosity method for nonstrictly hyperbolic conservation laws. Commun. Math. Phys. 150, 59-64 (1992)
22. MURAT, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 5, 489-507 (1978)
23. -1,q est compacte pour tout q < 2. J. Math. Pures Appl. 60, 309-322 (1981)
24. NISHIDA, T.: Global solutions for the initial-boundary value problem of a quasilinear hyperbolic systems. Proc. Japan Acad. 44, 642-646 (1968)
25. OLVER, F.W.J.: Asymptotics and special functions. Academic Press, 1974
26. PERTHAME, B.: Kinetic formulation of conservation laws. Lecture Series in Math. and Appl., Oxford Univ. Press, 2002
27. PERTHAME, B., TZAVARAS, A.: Kinetic formulation for systems of two conservation laws and elastodynamics. Arch. Rational Mech. Anal. 155, 1-48 (2000)
28. OVSYANNIKOV, L.V.: Group analysis of differential equations. Academic Press, New York, 1982
29. SERRE, D.: La compacité par compensation pour les systèmes hyperboliques non-linéaires de deux équations à une dimension d'espace. J. Math. Pures Appl. 65, 423-468 (1986)
30. SHELUKHIN, V.V.: Existence theorem in the variational problem for compressible inviscid fluids. Manuscripta Math. 61, 495-509 (1988)
31. SHELUKHIN, V.V.: Compactness of bounded quasientropy solutions to the system of equations of an isothermal gas. Siberian Math. J. 44, 366-377 (2003)
32. TARTAR, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 1979, Vol. 39, Pitman, Boston, Mass.-London, pp. 136-212
33. TARTAR, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of Nonlinear Partial Differential Equations, J.M. BALL (ed.), NATO ASI Series, C. Reidel publishing Col., 1983, pp. 263-285