Porous media flow as the limit of a nonstrictly hyperbolic system of conservation laws

Communications in Partial Differential Equations

Volumn 21, Issue 1-2, 1996, Pages 1-21

  Rubino, Bruno ^a  

^a UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002609873     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1080/03605309608821172     Document Type: Article

References (33)

1. ARONSON, D., AND VAZQUEZ, J. The porous media equation as a finite speed approximation to a Hamilton-Jacobi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 203-230.
2. CHEN, G. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics III. Acta Math. Sci. 6 (1986), 75-120.
3. CHEN, G., DING, X., AND LUO, P. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics II. Acta Math. Sci. 5 (1985), 433-472.
4. CHEN, G., DING, X., AND LUO, P. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics I. Acta Math. Sci. 5 (1985), 415-452.
5. CHEN, G., DING, X., AND LUO, P. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Comm. Math. Phys. 121 (1989), 63-84.
6. CHUEH, K., CONLEY, C., AND SMOLLER, J. Positively invariant regions for systems of non-linear diffusion equations. Indiana Univ. Math. J. 26 (1977), 372-411.
7. DIPERNA, R. Convergence of approssimate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70.
8. DIPERNA, R. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1-30.
9. DIPERNA, R. Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292 (1985), 383-420.
10. DIPERNA, R. Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985), 223-270.
11. FRIEDMAN, A. Partial differential eaqutions of parabolic type. Prentice Hall, 1964.
12. GLIMM, J., LINDQUIST, B., AND ZHANG, Q. Front thacking, oil reservoirs, engineering scale problems and mass conservation. IMA Vol. Math. Appl. 29 (1991), 123-139.
13. J.L. LIONS. Perturbations singulieres dans les problèmes aux limites et en controle optimale, vol. 323. Springer, Berlin, 1973.
14. KAN, P.-T. On the Cauchy problem of a 2 × 2 systems of non-strictly hyperbolic conservation laws. PhD thesis, Courant Institute of Math. Sciences. N.Y. University, 1989.
15. KRUŽKOV, S. First order quasi-linear equations with several space variables. Mat. Sb. 123 (1970), 228-255.
16. LAX, P. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM, Philadelphia, 1973.
17. LIU, T.-P. Hyperbolic conservation laws with relaxations. Comm. Math. Phys. 1118 (1987), 153-175.
18. MARCATI, P. Approximate solutions to the conservation laws via convective parabolic equations. Comm. Partial Differential Equations 13 (1988), 321-344.
19. MARCATI, P. A result on the hyperbolic aspects in the theory of porous media. Boll. Un. Mat. Ital. A 3 (1989), 69-75.
20. MARCATI, P., AND MILANI, A. The one-dimensional Darcy's law as the limit of a compressible Euler flow. J. Differential Equations 13 (1990), 129-147.
21. MARCATI, P., MILANI, A., AND SECCHI, P. Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. Manuscripta Math. 60 (1988), 49-69.
22. MARCATI, P., AND NATALINI, R. Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal. 129 (1995), 129-145.
23. MARCATI, P., AND RUBINO, B. Hyperbolic to parabolic relaxation theory for quasilinear first order systems. Tech. rep., 1995. to appear.
24. MILANI, A. Long time existence and singular perturbations results for quasilinear hyperbolic equations with small parameter and dissipation term I. Nonlinear Anal. 10/11 (1986), 1237-1248.
25. MILANI, A. Long time existence and singular perturbations results for quasilinear hyperbolic equations with small parameter and dissipation term II. Nonlinear Anal. 11/12 (1987), 1371-1381.
26. MURAT, F. Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489-507.
27. RUBINO, B. Convergence of the fractional step method for a 2 × 2 non-strictly hyperbolic system of conservation laws. J. Math. Anal. Appl.. to appear.
28. RUBINO, B. On the vanishing viscosity approximation to the Cauchy problem for a 2 × 2 system of conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 627-656.
29. RUBINO, B. Approximate solutions to the Cauchy problem for a class of 2 × 2 notistrictly hyperbolic system of conservation laws. Boll. Un. Mat. It. B (7) 8 (1994), 583-614.
30. RUBINO, B. Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a 2 × 2 nonstrictly hyperbolic system of conservation laws. Quart. Appl. Math. 53 (1995), 401-421.
31. SERRE, D. La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations a une dimension d'espace. J. Math. Pures et Appl. (9) 65 (1986), 423-468.
32. TARTAR, L. Compensated compactnes and applications to partial differential equations, vol. 39 of IV. Research Notes in Math., 1979, pp. 136-210.
33. VAZQUEZ, J. Hyperbolic aspects in the theory of the porous media equation. Academic Press., Minneapolis, 1985.