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Convergence of approximate solutions to conservation laws

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  1. 1.

    Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403 (1977).

  2. 2.

    Bahkvarov, N., In the existence of regular solutions in the large for quasilinear hyperbolic systems, Zhur. Vychisl. Mat. i. Mathemat. Fig. 10, 969–980 (1970).

  3. 3.

    DiPerna, R. J., Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 26, 1–28 (1973).

  4. 4.

    DiPerna, R. J., Existence in the large for nonlinear hyperbolic conservation laws, Arch. Rational Mech. Anal. 52, 244–257 (1973).

  5. 5.

    Federer, H., Geometric Measure Theory, Springer, N.Y., 1969.

  6. 6.

    Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18, 697–715 (1965).

  7. 7.

    Greenberg, J. M., The Cauchy problem for the quasilinear wave equation, unpublished.

  8. 8.

    Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159–193 (1954).

  9. 9.

    Lax, P. D., Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537–566 (1957).

  10. 10.

    Lax, P. D., Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press, 603–634 (1971).

  11. 11.

    Lax, P. D., & B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13, 217–237 (1960).

  12. 12.

    Liu, T.-P., Solutions in the large for the equations of non-isentropic gas dynamics, Indiana Univ. Math. J. 26, 147–177 (1977).

  13. 13.

    Majda, A., & S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32, 797–838 (1979).

  14. 14.

    Murat, F., Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 5, 489–507 (1978).

  15. 15.

    Murat, F., Compacité par compensation: Condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa 8, 69–102 (1981).

  16. 16.

    Murat, F., L'injection du cone positif de H−1 dans W−1,q est compacte pour tout q < 2. Preprint.

  17. 17.

    Nishida, T., Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44, 642–646 (1968).

  18. 18.

    Nishida, T., & J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 26, 183–200 (1973).

  19. 19.

    Tartar, L., Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. 4, ed. R. J. Knops, Pitman Press, (1979).

  20. 20.

    Vol'pert, A. I., The spaces BV and quasilinear equations, Math. USSR Sb. 2, 257–267 (1967).

  21. 21.

    Liu, T.-P., Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 213, 375–382 (1975).

  22. 22.

    Liu, T.-P., Uniqueness of weak solutions of the Cauchy problem for general 2×2 conservation laws, J. Differential Eq. 20, 369–388 (1976).

  23. 23.

    Friedrichs, K. O., & P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA 68, 1686–1688 (1971).

  24. 24.

    Ball, J. M., On the calculus of variations and sequentially weakly continuous maps, Proc. Dundee Conference on Ordinary and Partial Differential Equations (1976), Springer Lecture Notes in Mathematics, Vol. 564, 13–25.

  25. 25.

    Ball, J. M., J. C. Currie & P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal., 41, 135–175 (1981).

  26. 26.

    Tartar, L., Une nouvelle méthode de résolution d'équations aux dérivées partielles nonlinéaires, Lecture Notes in Math., Vol. 665, Springer-Verlag (1977), 228–241.

  27. 27.

    Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lefschetz Center for Dynamical Systems Lecture Notes # 81-77, Brown University (1981).

  28. 29.

    Dacorogna, B., Minimal hypersurface problems in parametric form with nonconvex integrands, to appear in Indiana Univ. Math. Journal.

  29. 30.

    Dacorogna, B., A generic result for nonconvex problems in the calculus of variations, to appear in J. Funct. Anal.

  30. 31.

    Greenberg, J., R. MacCamy & V. Mizel, On the existence, uniqueness and stability of solutions to the equation 70-001 J. Math. Mech. 17, 707–728 (1968).

  31. 32.

    Chueh, K. N., C. C. Conley & J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26, 372–411 (1977).

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This research was supported by the U.S. National Science Foundation under Grant No. MCS 77-16049, by a Fellowship of the Sloan Foundations, by a Graduate School Fellowship from the University of Wisconsin, and by U.S. National Science Foundation Grant No. MCS-7900813 while the author was a visiting member of the Courant Institute of Mathematical Sciences.

Communicated by C. Dafermos

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DiPerna, R.J. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82, 27–70 (1983).

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