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Convergence of approximate solutions to conservation laws

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References

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

    MATH  Google Scholar 

  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).

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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 Greenberg, J. M., The Cauchy problem for the quasilinear wave equation, unpublished.

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

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

    MathSciNet  Google Scholar 

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

    ADS  MATH  MathSciNet  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

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

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

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  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).

    ADS  MATH  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

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

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

  30. 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).

    MathSciNet  Google Scholar 

  31. 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).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison

    R. J. DiPerna

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  1. R. J. DiPerna
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Communicated by C. Dafermos

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.

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DiPerna, R.J. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82, 27–70 (1983). https://doi.org/10.1007/BF00251724

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