Shock Fronts and Hyperbolic Systems of Differential Equations

  • Christina Papenfuß


Rational extended thermodynamics leads to symmetric hyperbolic systems of differential equations for the wanted fields. A simple model equation with these properties in one dimension is the Burgers equation. Properties of solutions of such equations, especially the evolution and propagation of shock fronts, are studied on the example of this Burgers equation.


  1. 1.
    J.M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen 17(2), 1–53 (1939)MathSciNetzbMATHGoogle Scholar
  2. 2.
    P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, Bristol, 1973)CrossRefGoogle Scholar
  3. 3.
    W. Dreyer, S. Seelecke, Entropy and causality as criteria for the existence of shock waves in low temperature heat conduction. Contin. Mech. Thermodyn. 4, 23–36 (1992)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II (Interscience Publishers, New York, London, 1962)zbMATHGoogle Scholar
  5. 5.
    M.G. Boillat, Sur l’ existance et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques. C. R. Acad. Sc. Paris 278 A, 909–913 (1974)Google Scholar
  6. 6.
    K.O. Friedrichs, P.D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci USA 68, 1686–1688 (1971)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Lax, Shock waves and entropy, in Proceedings of a Symposium Conducted by the Mathematics Research Center, Contributions to Nonlinear Functional Analysis, ed. by Zarantonello (Academic Press, The University of Wisconsin, 1971), pp. 603–634Google Scholar
  8. 8.
    T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid. Acta Mech. 77(3), 167–183 (1983) (Review Article)Google Scholar
  9. 9.
    V. Ciancio, L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory. Phys. A 142, 309–320 (1987)CrossRefGoogle Scholar

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Authors and Affiliations

  • Christina Papenfuß
    • 1
  1. 1.FB 2Hochschule für Technik und WirtschaftBerlinGermany

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