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Genuinely Nonlinear Systems of Two Conservation Laws

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Book cover Hyperbolic Conservation Laws in Continuum Physics

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 325))

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Abstract

The theory of solutions of genuinely nonlinear, strictly hyperbolic systems of two conservation laws will be developed in this chapter at a level of precision comparable to that for genuinely nonlinear scalar conservation laws, expounded in Chapter X1. This will be achieved by exploiting the presence of coordinate systems of Riemann invariants and the induced rich family of entropy-entropy flux pairs. The principal tools in the investigation will be generalized characteristics and entropy estimates.

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  1. Glimm, J. and P.D. Lax: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs AMS, No. 101 (1970).

    Google Scholar 

  2. Lax, P.D.: Shock waves and entropy. Contributions to Functional Analysis pp. 603–634, ed. E.A. Zarantonello. New York: Academic Press, 1971.

    Google Scholar 

  3. DiPerna, R.J.: Singularities of solutions of nonlinear hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 60 (1975), 75–100.

    MathSciNet  MATH  Google Scholar 

  4. Dafermos, C.M.: Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107 (1989), 127–155.

    MathSciNet  MATH  Google Scholar 

  5. Dafermos, C.M. and X. Geng: Generalized characteristics in hyperbolic systems of conservation laws with special coupling. Proc. Royal Soc. Edinburgh 116A (1990), 245–278.

    Article  MathSciNet  MATH  Google Scholar 

  6. Dafermos, C.M. and X. Geng: Generalized characteristics, uniqueness and regularity of solutions in a hyperbolic system of conservation laws. Ann. Inst. Henri Poincaré 8 (1991), 231–269.

    MathSciNet  MATH  Google Scholar 

  7. Trivisa, K.: A priori estimates in hyperbolic systems of conservation laws via generalized characteristics. Comm. PDE 22 (1997), 235–267.

    Article  MathSciNet  MATH  Google Scholar 

  8. Temple, B.L: Decay with a rate for noncompactly supported solutions of conservation laws. Trans. AMS 298 (1986), 43–82.

    Article  MathSciNet  MATH  Google Scholar 

  9. Courant, R. and K.O. Friedrichs: Supersonic Flow and Shock Waves. New York: Wiley-Interscience, 1948.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. DiPerna, R.J.: Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws. Indiana U. Math. J. 24 (1975), 1047–1071.

    MathSciNet  MATH  Google Scholar 

  12. DiPerna, R.J.: Decay of solutions of hyperbolic systems of conservation laws with a convex extension. Arch. Rational Mech. Anal. 64 (1977), 1–46.

    MathSciNet  MATH  Google Scholar 

  13. Liu, T.-P.: Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math. 30 (1977), 585–610.

    MATH  Google Scholar 

  14. Liu, T.-P.: Linear and nonlinear large-time behavior of solutions of hyperbolic conservation laws. Comm. Pure Appl. Math. 30 (1977), 767–796.

    MATH  Google Scholar 

  15. Liu, T.-P.: Pointwise convergence to N-waves for solutions of hyperbolic conservation laws. Bull. Inst. Math. Acad. Sinica 15 (1987), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  16. Zumbrun, K.: N-waves in elasticity. Comm. Pure Appl. Math. 46 (1993), 75–95.

    MathSciNet  MATH  Google Scholar 

  17. Zumbrun, K.: Decay rates for nonconvex systems of conservation laws. Comm. Pure Appl. Math. 46 (1993), 353–386.

    MathSciNet  MATH  Google Scholar 

  18. Dafermos, C.M.: Large time behavior of solutions of hyperbolic systems of conservation laws with periodic initial data. J. Diff. Eqs. 121 (1995), 183–202.

    Article  MathSciNet  MATH  Google Scholar 

  19. Serre, D.: ntegrability of a class of systems of conservation laws. Forum Math. 4 (1992), 607–623.

    Article  MathSciNet  MATH  Google Scholar 

  20. Serre, D.: Systèmes de Lois de Conservation, Vols. I-II. Paris: Diderot, 1996. English translation: Systems of Conservation Laws, Vols. 1–2. Cambridge: Cambridge University Press, 1999.

    Google Scholar 

  21. Dafermos, C.M. and X. Geng: Generalized characteristics in hyperbolic systems of conservation laws with special coupling. Proc. Royal Soc. Edinburgh 116A (1990), 245–278.

    Article  MathSciNet  MATH  Google Scholar 

  22. Dafermos, C.M. and X. Geng: Generalized characteristics, uniqueness and regularity of solutions in a hyperbolic system of conservation laws. Ann. Inst. Henri Poincaré 8 (1991), 231–269.

    MathSciNet  MATH  Google Scholar 

  23. Heibig, A.: Existence and uniqueness of solutions for some hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 126 (1994), 79–101.

    MathSciNet  MATH  Google Scholar 

  24. Ostrov, D.N.: Asymptotic behavior of two interreacting chemicals in a chromatography reactor. SIAM J. Math. Anal. 27 (1996), 1559–1596.

    MathSciNet  MATH  Google Scholar 

  25. LeVeque, R.J. and B. Temple: Ionvergence of Godunov’s method for a class of 2 x 2 systems of conservation laws. Trans. AMS 288 (1985), 115–123.

    MathSciNet  MATH  Google Scholar 

  26. LeVeque, R.J. and B. Temple: Ionvergence of Godunov’s method for a class of 2 x 2 systems of conservation laws. Trans. AMS 288 (1985), 115–123.

    MathSciNet  MATH  Google Scholar 

  27. Serre, D.: Solutions à variation bornée pour certains systèmes hyperboliques de lois de conservation. J. Diff. Eqs. 67 (1987), 137–168.

    Article  Google Scholar 

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    Constantine M. Dafermos

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  1. Constantine M. Dafermos
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© 2000 Springer-Verlag Berlin Heidelberg

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Dafermos, C.M. (2000). Genuinely Nonlinear Systems of Two Conservation Laws. In: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol 325. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22019-1_12

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  • DOI: https://doi.org/10.1007/978-3-662-22019-1_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-22021-4

  • Online ISBN: 978-3-662-22019-1

  • eBook Packages: Springer Book Archive

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