Convergence to equilibrium for the relaxation approximations of conservation laws

Communications on Pure and Applied Mathematics

Volumn 49, Issue 8, 1996, Pages 795-823

  Natalini, Roberto ^a  

^a CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030490070     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3     Document Type: Article

References (32)

1. Aregba-Driollet, D., and Natalini, R., Convergence of relaxation schemes for conservation laws, Appl. Anal., in press.
2. Brezis, H., Analyse Fonctionelle, Masson, Paris, 1983.
3. Caflisch, R., and Liu, T. P., Nonlinear stability of shock waves for the Broadwell model, Comm. Math. Phys. 114, 1988, pp. 103-130.
4. Caflisch, R., and Papanicolaou, G. C., The fluid-dynamical limit of nonlinear model of Boltzmann equations, Comm. Pure Appl. Math. 32, 1979, pp. 589-616.
5. Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
6. Cercignani, C., Illner, R., and Pulvirenti, M., The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
7. Chen, G.-Q., and Liu, T. P., Zero relaxation and dissipation limits for hyperbolic conservation laws, Comm. Pure Appl. Math. 46, 1993, pp. 755-781.
8. Chen, G.-Q., Levermore, C., and Liu, T. P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47, 1994, pp. 787-830.
9. Chueh, K. N., Conley, C. C., and Smoller, J., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26, 1977, pp. 372-411.
10. DiPerna, R., Convergence of approximate solutions of conservation laws, Arch. Rational Mech. Anal. 82, 1983, pp. 27-70.
11. Fife, P., Mathematical Aspects of Reacting and Diffusion Systems, Lecture Notes in Biomathematics, Springer-Verlag, New York, 1979.
12. Giga, Y., and Miyakawa, T., A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J. 50, 1983, pp. 505-515.
13. Godlewski, E., and Raviart, P.-A., Hyperbolic Systems of Conservation Laws, Math. & Appl., 3/4, Ellipses, Paris, 1991.
14. Hanouzet, B., and Natalini, R., Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, in press.
15. Jin, S., and Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 1995, pp. 235-277.
16. Kružkov, S. N., First order quasilinear equations in several independent variables, Mat. Sb. (N.S.) 81(123), 1970, pp. 228-255.
17. Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, SIAM, Philadelphia, 1973.
18. Lions, P. L., Perthame, B., and Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7, 1994, pp. 169-191.
19. Liu, T. P., Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108, 1987, pp. 153-175.
20. Marcati, P. A., Relaxation and singular convergence for quasilinear hyperbolic systems, pp. 140-148 in: Proceedings of the First Italian-Spanish Meeting on Nonlinear Analysis, L. Boccardo, M. A. Herrero, and A. Tesei, eds., Quaderno IAC, Rome, 1989.
21. Perthame, B., and Tadmor, E., A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136, 1991, pp. 501-517.
22. Platkowski, T., and Illner, R., Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Rev. 30, 1988, pp. 213-255.
23. Protter, M., and Weinberger, H., Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.
24. Rhee, H. K., Aris, R., and Amundsen, N. R., First-Order Partial Differential Equations (vol. 1), Theory and Application of Single Equations (vol. 2), Prentice-Hall International Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, Englewood Cliffs, N.J., 1986.
25. Schochet. S., Hyperbolic-hyperbolic singular limits, Comm. Partial Differential Equations 12, 1987, pp. 589-632.
26. Smith, H. L., Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev. 30, 1988, pp. 87-113.
27. Tartar, L., Compensated compactness and applications to partial differential equations, pp. 136-212 in: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Research Notes in Mathematics 39, Pitman, Boston-London, 1979.
28. Van Harten, A., and Van Hassel, R. R., A quasilinear, singular perturbation problem of hyperbolic type, SIAM J. Math. Anal. 16, 1985, pp. 1258-1267.
29. Vincenti, W. G., and Kruger, C. H., Introduction to Physical Gas Dynamics, Wiley, New York, 1965.
30. Whitham, J., Linear and Nonlinear Waves, Wiley, New York, 1974.
31. Xin, Z., The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks, Comm. Pure Appl. Math. 44, 1991, pp. 679-713.
32. Yong, W.-A., Singular Perturbations of First-Order Hyperbolic Systems, Thesis, University of Heidelberg, 1992.