Boundary layers in weak solutions of hyperbolic conservation laws

Archive for Rational Mechanics and Analysis

Volumn 147, Issue 1, 1999, Pages 47-88

  Joseph, K T ^a   Lefloch, P G ^b  

^a TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033446249     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002050050145     Document Type: Article

References (49)

1. AMADORI D., Initial-boundary value problems for nonlinear systems of conservation laws, preprint, SISSA Trieste, Italy, 1995.
2. AMADORI D. & COLOMBO M., Continuous dependence for 2 × 2 systems of conservation laws with boundary, preprint, SISSA Trieste, Italy, 1995.
3. BALL J., A version of the fundamental theorem for Young measures, in Proceedings of Conf. on Partial Differential Equations and Continuum Models of Phase Transitions, Nice, 1988, ed. D. SERRE, Springer-Verlag.
4. BARDOS C. W., LEROUX A.-Y., & NEDELEC J.-C., First order quasilinear equations with boundary conditions, Comm. Part. Diff. Eqns. 4 (1979), 1018-1034.
5. BENABDALLAH A., Le p-système dans un intervalle, C.R. Acad. Sc., Paris, Série I, 303 (1986), 123-126.
6. BENABDALLAH A. & SERRE D., Problèmes aux limites pour des systèmes hyperboliques nonlinéaires de deux équations à une dimension d'espace, C.R. Acad. Sc. Paris, Série I, 305 (1987), 677-680.
7. BENABDALLAH A. & SERRE D., Problèmes aux limites pour des systèmes hyperboliques nonlinéaires de deux équations à une dimension d'espace, Preprint 69, Univ. Paris-Nord, Villetaneuse, France, 1988 (unpublished report).
8. BOURDEL F., DELORME P., & MAZET P. A., Convexity in hyperbolic problems, Application to a discontinuous Galerkin method, Proc. Inter. Conf. on Hyperbolic problems, Aachen (Germany), March 1988, Notes on Numer. Fluid Mech., Vol. 24, Vieweg (1989).
9. CHEN G.-Q. & LEFLOCH P. G., Entropy flux splittings for hyperbolic conservation laws. General framework, Comm. Pure Appl. Math. 48 (1995), 691-729.
10. COURANT R. & FRIEDRICHS K. O., Supersonic Flows and Shock Waves, Interscience (1948).
11. DAFERMOS C. M., Hyperbolic systems of conservation laws, Systems of Nonlinear Partial Differential Equations, J. M. BALL, editor, NATO Adv. Sci. Series C, 111, Reidel (1983), 25-70.
12. DIPERNA R. J., Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70.
13. DIPERNA R. J., Measure-valued solutions to conservations laws, Arch. Rational Mech. Anal. 88 (1985), 223-270.
14. DUBOIS F. & LEFLOCH P. G., Boundary condition for a system of conservation laws, C.R. Acad. Sc. Paris, Série I 304 (1987), 75-78.
15. DUBOIS F. & LEFLOCH P. G., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Diff. Eqns. 71 (1988), 93-122.
16. DUBOIS F. & LEFLOCH P. G., Boundary conditions for nonlinear hyperbolic systems of conservation laws, Proc. Inter. Conf. on Hyperbolic problems, Aachen (Germany), March 1988, Notes on Numer. Fluid Mech., Vol. 24, Vieweg (1989), 96-106.
17. DUBROCA B. & GALLICE G., Résultats d'existence et d'unicité du problème mixte pour des systèmes hyperboliques de lois de conservation monodimensionels, Comm. Part. Diff. Eqns. 15 (1990), 59-80.
18. GISCLON M., Comparaison de deux perturbations singulières pour l'équations de Burgers avec conditions aux limites, C.R. Acad. Sc., Paris, Série I, 316 (1993), 1011-1014.
19. GISCLON M., Etude des conditions aux limites pour un système strictement hyperbolique via l'approximation parabolique, J. Math. Pures Appl. 75 (1996), 485-508.
20. GISCLON M. & SERRE D., Etude des conditions aux limites pour un système strictement hyperbolique via l'approximation parabolique, C.R. Acad. Sc., Paris, Série I, 319 (1994), 377-382.
21. GLIMM J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715.
22. GOODMAN J., Initial boundary value problems for hyperbolic systems of conservation laws, Ph.D. Thesis, Stanford University, California, 1982.
23. HARTMAN P., Ordinary Differential Equations, Wiley, (1964).
24. JOSEPH K. T., Burgers equation in the quarter plane: a formula for the weak limit, Comm. Pure Appl. Math. 41 (1988), 133-149.
25. JOSEPH K. T., Boundary layers in approximate solutions, Trans. Amer. Math. Soc. 314 (1989), 709-726.
26. JOSEPH K. T. & LEFLOCH P. G., Boundary layers in weak solutions to hyperbolic conservation laws, Preprint #1402, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, May 1996.
27. JOSEPH K. T. & VEERAPPA GOWDA G. D., Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J. 62 (1991), 401-416.
28. KREISS H. O., Initial-boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298.
29. LAX P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954), 159-193.
30. LAX P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566.
31. LAX P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conf. Series in Appl. Math. 11, SIAM, Philadelphia (1973).
32. LEFLOCH P. G., Explicit formula for scalar conservation laws with boundary condition, Math. Meth. Appl. Sc. 10 (1988), 265-287.
33. LEFLOCH P. G., An introduction to nonclassical shocks of systems of conservation laws, Proc. International School on Theory and Numerics for Conservation Laws, Freiburg, Germany, 20-24 Oct. 97, D. Kröner, M. OHLBERGER & C. ROHDE ed., Lecture Notes in Computational Science and Engineering, Springer Verlag, 1999, pp. 28-72.
34. LEFLOCH P.G. & NEDELEC J.-C., Explicit formula for weighted scalar nonlinear conservation laws, Trans. Amer. Math. Soc. 308 (1988), 667-683.
35. LEROUX A.-Y., Approximation de quelques problèmes hyperboliques nonlinéaires, Thèse d'état, University of Rennes, France (1979).
36. LI T.-T. & Yu W.-C., Boundary value problem for quasilinear hyperbolic systems, Duke Univ. Math. Series (1985).
37. LIU T.P., Large time behavior of initial and initial-boundary-value problems of general systems of hyperbolic conservation laws, Comm. Math. Phys. 55 (1977), 163-177.
38. LIU T.-P., Initial-boundary value problems for gas dynamics, Arch. Rational Mech. Anal. 64 (1977), 137-168.
39. LIU T.-P., The free piston problem for gas dynamics, J. Diff. Eqns. 30 (1978), 175-191.
40. LIU T.-P., Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc. 30 (1981).
41. MAJDA A. & PEGO R., Stable viscosity matrices for systems of conservation laws, J. Diff. Eqns. 56, (1985) 229-262.
42. SABLÉ-TOUGERON M., Méthode de Glimm et problème mixte, Ann. Inst. Henri Poincaré 10 (1993), 423-443.
43. SABLÉ-TOUGERON M., Les N-ondes de Lax pour le problème mixte, to appear.
44. SERRE D., Personal communication, Paris, 1987.
45. SMOLLER J., Shock waves and reaction diffusion equations, Springer-Verlag, 1983.
46. SCHWARTZ J. T., Nonlinear functional analysis, Courant Inst. Lect. Notes, 1964.
47. SZEPESSY A., Measure-valued solutions to scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal. 107, (1989), 181-193.
48. TEMPLE B., Systems of conservation laws with coinciding shock and rarefaction curves, in Nonlinear Partial Differential Equations, J. SMOLLER, ed., Contemp. Math. 17 (1982), 143-151.
49. XIN Z. P., Personal communication, June 1994.