Nonclassical shocks and kinetic relations: Strictly hyperbolic systems

SIAM Journal on Mathematical Analysis

Volumn 31, Issue 5, 2000, Pages 941-991

  Hayes, Brian T ^a   Lefloch, Philippe G ^b  

^a DUKE UNIVERSITY   (United States)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Conservation laws; Hyperbolic entropy; Kinetic relation; Nonclassical shock; Shock wave

Indexed keywords

EID: 0034345304     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141097319826     Document Type: Article

References (68)

1. R. ABEYARATNE AND J.K. KNOWLES, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal., 114 (1991), pp. 119-154.
2. R. ABEYARATNE AND J.K. KNOWLES, Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math., 51 (1991), pp. 1205-1221.
3. A. AZEVEDO, D. MARCHESIN, B.J. PLOHR, AND K. ZUMBRUN, Non-uniqueness of solutions of Riemann problems caused by 2-cycles of shock waves, in Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications, J. Glimm, M.J. Graham, J.W. Grove, and B.J. Plohr, eds., World Scientific Editions, River Edge, N.J., 1996, pp. 43-51.
4. M. BRIO AND J. HUNTER, Rotationally invariant hyperbolic waves, Comm. Pure Appl. Math., 43 (1990), pp. 1037-1053.
5. M. BRIO AND C.C. WU, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., 75 (1988), pp. 400-422.
6. G.Q. CHEN, Hyperbolic systems of conservation laws with a symmetry, Comm. Partial Differential Equations, 16 (1991), pp. 1461-1467.
7. K.N. CHUEY, C.C. CONLEY, AND J.A. SMOLLER, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), pp. 372-411.
8. R.H. COHEN AND R.M. KULSRUD, Non-linear evolution of quasi-parallel propagating hydromagnetic waves, Phys. Fluid, 17 (1974), pp. 2215-2225.
9. C. CONLEY AND J.A. SMOLLER, On the structure of the magnetohydrodynamics shock waves, Comm. Pure Appl. Math., 28 (1974), pp. 367-375.
10. C.M. DAFERMOS, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations, 14 (1973), pp. 202-212.
11. C.M. DAFERMOS, Hyperbolic systems of conservation laws, in Proceedings Systems of Nonlinear Partial Differential Equations, J.M. Ball, ed., NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 111, D. Reidel, Dordrecht, The Netherlands, 1983, pp. 25-70.
12. C.M. DAFERMOS, Admissible fans in nonlinear hyperbolic systems, Arch. Rational Mech. Anal., 106 (1989), pp. 243-260.
13. R.J. DIPERNA, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), pp. 27-70.
14. L.C. EVANS AND R.F. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Ann Arbor, London, 1992.
15. H.T. FAN AND M. SLEMROD, The Riemann problem for systems of conservation laws of mixed type, in Shock Induced Transitions and Phase Structures in General Media, R. Fosdick, E. Dunn, and H. Slemrod, eds., IMA Vol. Math. Appl. 52, Springer-Verlag, 1993, pp. 61-91.
16. H. FREISTÜHLER, Anomale Schocks, strukturell labile Lösungen und die Geometrie der Rankine-Hugoniot-Bedingungen, Ph.D. thesis, Ruhr-Univ. Bochum, Germany, 1987.
17. H. FREISTÜHLER, Dynamical stability and vanishing viscosity: A case study of a non-strictly hyperbolic system of conservation laws, Comm. Pure Appl. Math., 45 (1992), pp. 561-582.
18. H. FREISTÜHLER, Stability of nonclassical shock waves, in Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications, J. Glimm, M.J. Graham, J.W. Grove, and B.J. Plohr, eds., World Scientific Editions, River Edge, N.J., 1996, pp. 120-129.
19. H. FREISTÜHLER AND T.P. LIU, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws, Comm. Math. Phys., 153 (1993), pp. 147-158.
20. J. GLIMM, Nonlinear waves: Overview and problems, in Multidimensional Hyperbolic Problems and Computations, J. Glimm and A. Majda, eds., IMA Vol. Math. Appl. 29, Springer-Verlag, New York, 1991.
21. B.T. HAYES AND P.G. LEFLOCH, Measure-solutions to a strictly hyperbolic system of conservation laws, Nonlinearity, 9 (1996), pp. 1547-1563.
22. B.T. HAYES AND P.G. LEFLOCH, Nonclassical shocks and kinetic relations: Scalar conservation laws, Arch. Rational Mech. Anal., 139 (1997), pp. 1-56.
23. B.T. HAYES AND P.G. LEFLOCH, Nonclassical shocks and kinetic relations: Finite difference schemes, SIAM J. Numer. Anal., 35 (1998), pp. 2169-2194.
24. B.T. HAYES AND P.G. LEFLOCH, in preparation.
25. T.Y. HOU, P. ROSAKIS, AND P.G. LEFLOCH, A level-set approach to the computation of twinning and phase transition dynamics, J. Comput. Phys., 150 (1999), pp. 302-331.
26. L. HSIAO, Uniqueness of admissible solutions of Riemann problems of systems of conservation laws of mixed type, J. Differential Equations, 86 (1990), pp. 197-233.
27. E. ISAACSON, D. MARCHESIN, C.F. PALMEIRA, AND B.J. PLOHR, A global formalism for nonlinear waves in conservation laws, Comm. Math. Phys., 146 (1992), pp. 505-552.
28. E. L. ISAACSON, D. MARCHESIN, AND B. J. PLOHR, Transitional waves for conservation laws, SIAM J. Math. Anal., 21 (1990), pp. 837-866.
29. E. ISAACSON AND B. TEMPLE, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math., 52 (1992), pp. 1260-1278.
30. D. JACOBS, W.R. MCKINNEY, AND M. SHEARER, Traveling wave solutions of the modified Korteweg-deVries Burgers equation, J. Differential Equations, 116 (1995), pp. 448-467.
31. B. KEYFITZ, A geometric theory of conservation laws which change type, Z. Angew. Math. Mech., 75 (1995), pp. 571-581.
32. B. KEYFITZ AND H. KRANZER, A system of nonstrictly hyperbolic conservation arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1980), pp. 219-241.
33. A. KULIKOVSKY, Dokl. Acad. Nauk. USSR, 275 (1984), pp. 1349-1352 (in Russian).
34. P.D. LAX, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957), pp. 537-566.
35. P.D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, PA, 1973.
36. P.D. LAX, The zero dispersion limit, a deterministic analogue of turbulence, Comm. Pure Appl. Math., 44 (1991), pp. 1047-1056.
37. P.D. LAX AND C.D. LEVERMORE, The small dispersion limit of the Korteweg-deVries equation, Comm. Pure Appl. Math., 36 (1983) I, pp. 253-290, II, pp. 571-593, III, pp. 809-829.
38. P.D. LAX AND C.D. LEVERMORE, The small dispersion limit of the Korteweg-deVries equation, Comm. Pure Appl. Math., 36 (1983) I, pp. 253-290, II, pp. 571-593, III, pp. 809-829.
39. P.D. LAX AND C.D. LEVERMORE, The small dispersion limit of the Korteweg-deVries equation, Comm. Pure Appl. Math., 36 (1983) I, pp. 253-290, II, pp. 571-593, III, pp. 809-829.
40. P.G. LEFLOCH, Propagating phase boundaries: Formulation of the problem and existence via the Glimm scheme, Arch. Rational Mech. Anal., 123 (1993), pp. 153-197.
41. P.G. LEFLOCH AND R. NATALINI, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), pp. 213-230.
42. P.G. LEFLOCH AND A.E. TZAVARAS, Nonconservative hyperbolic systems: Existence theory for the Riemann problem, in preparation.
43. T.P. LIU, The Riemann problem for general 2×2 conservation laws, Trans. Amer. Math. Soc., 199 (1974), pp. 89-112.
44. T.P. LIU, Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc. 30, 1981.
45. T.P. LIU, Nonlinear stability and instability of overcompressive shock waves, IMA Vol. Math. Appl. 52, Springer-Verlag, New York, 1993, pp. 159-167.
46. T.P. LIU AND J.H. WANG, On a nonstrictly hyperbolic system of conservation laws, J. Differential Equations, 57 (1985), pp. 1-14.
47. T.P. LIU AND Z. XIN, Stability of viscous shock waves associated with a nonstrictly hyperbolic system, Comm. Pure Appl. Math., 45 (1992), pp. 361-388.
48. T.P. LIU AND K. ZUMBRUN, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys., 168 (1995), pp. 163-186.
49. T.P. LIU AND K. ZUMBRUN, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys., 174 (1995), pp. 319-345.
50. R. MENIKOFF AND B.J. PLOHR, The Riemann problem for fluid flow of real materials, Rev. Modern Phys., 61 (1989), pp. 75-130.
51. F. MURAT, A survey on compensated compactness, in Contributions to Modern Calculus of Variation, L. Cesari, ed., Pitman Res. Notes Math. Ser. 148, Longman, Harlow, UK, 1987, pp. 145-183.
52. O. OLEINIK, Discontinuous solutions of nonlinear differential equations, Uspekhio Mat. Nauk. (N.S.), 12 (1957), pp. 3-73; Amer. Math. Transl. Ser. 2, 26, pp. 95-172 (in English).
53. O. OLEINIK, Discontinuous solutions of nonlinear differential equations, Uspekhio Mat. Nauk. (N.S.), 12 (1957), pp. 3-73; Amer. Math. Transl. Ser. 2, 26, pp. 95-172 (in English).
54. S. SCHECTER, D. MARCHESIN, AND B.J. PLOHR, Structurally stable Riemann solutions, J. Differential Equations, 126 (1996), pp. 303-354.
55. S. SCHECTER AND M. SHEARER, Undercompressive shocks for nonstrictly hyperbolic conservation laws, Dynamics Differential Equations, 3 (1991), pp. 199-271.
56. M.E. SCHONBEK, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), pp. 959-1000.
57. D. SERRE AND J.W. SHEARER, Convergence with physical viscosity for nonlinear elasticity, unpublished notes.
58. p for the quasilinear wave equation, Ph.D. thesis, University of California, Berkeley, CA, 1991.
59. M. SHEARER, The Riemann problem for a class of conservation laws of mixed type, J. Differential Equations, 46 (1982), pp. 426-443.
60. M. SHEARER AND Y. YANG, The Riemann problem for the p-system of conservation laws of mixed type with a cubic nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), pp. 675-699.
61. M. SHEARER, D.G. SCHAEFFER, D. MARCHESIN, AND P. PAES-LEME, Solution of the Riemann problem for a prototype 2 × 2 system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal., 97 (1987), pp. 299-320.
62. M. SLEMROD, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal, 81 (1983), pp. 301-315.
63. L. TARTAR, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, J.M. Ball, ed., NATO Adv. Sci. Inst. Ser. C. Math. Phys., 1983, pp. 263-285.
64. L. TRUSKINOVSKY, Dynamics of non-equilibrium phase boundaries in a heat conducting nonlinear elastic medium, J. Appl. Math. Mech., 51 (1987), pp. 777-784.
65. L. TRUSKINOVSKY, Kinks versus shocks, in Shock Induced Transitions and Phase Structures in General Media, R. Fosdick, E. Dunn, and H. Slemrod, eds., IMA Vol. Math. Appl. 52, Springer-Verlag, New York, 1993, pp. 185-229.
66. C.C. WU, Formation, structure, and stability of MHD intermediate shocks, J. Geophys. Res., 95 (1990), pp. 8149-8175.
67. G.G. WU, New theory of MHD shock waves, in Viscous Profiles and Numerical Methods for Shock Waves, M. Shearer, ed., SIAM, Philadelphia, PA, 1991, pp. 209-236.
68. G.G. WU AND C.F. KENNEL, The small amplitude magnetohydrodynamic Riemann problem, Phys. Fluids B, 5 (1993), pp. 2877-2886.