Acta Applicandae Mathematica

, Volume 88, Issue 3, pp 269–329

The Geometry of 2 × 2 Systems of Conservation Laws

Article

Abstract

We consider one typical two-parameter family of quadratic systems of 2 × 2 conservation laws, and study the geometry of the behaviour of the possible solutions of the Riemann problem near an umbilic point, following the geometric approach presented by Isaacson, Marchesin, Palmeira, Plohr, in A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. (1992). The corresponding phase portraits for the rarefaction curves, shock curves and composite curves are discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, V.: Singularities of Differentiable Maps, Vol. I, Birkhäuser, 1985.Google Scholar
  2. 2.
    Basto-Gonçalves, J. and Reis, H.: The Geometry of Quadratic 2 × 2 Systems of Conservation Laws, preprint 339, Université de Bourgogne, 2003.Google Scholar
  3. 3.
    Bruce, J. and Tari, F.: On binary differential equations, Nonlinearity 8 (1995), 255–271.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bruce, J. and Tari, F.: Generic 1-parameter families of binary differencial equations of Morse type, Discrete Contin. Dyn. Syst. 3 (1997), 79–90.MATHMathSciNetGoogle Scholar
  5. 5.
    Chen, G.-Q. and Kan, P. T.: Hyperbolic conservation laws with umbilic degeneracy, Arch. Ration. Mech. Anal. 160 (2001), 325–354.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Davydov, A.: Qualitative Theory of Control Systems, AMS Transl. Math. Monogr. 14, 1994.Google Scholar
  7. 7.
    Eschenazi, C. and Palmeira, C.: Local topology of elementary waves for systems of two conservation laws, Mat. Contemp. 15 (1998), 127–144.MATHMathSciNetGoogle Scholar
  8. 8.
    Eschenazi, C. and Palmeira, C.: The structure of composite rarefaction-shock foliations for quadratic systems of conservation laws, Mat. Contemp. 22 (2002).Google Scholar
  9. 9.
    Hirsch, M.: Differential Topology, Springer, Berlin, Heidelberg, New York, 1976.MATHGoogle Scholar
  10. 10.
    Holden, H.: On the Riemann problem for a prototype of a mixed type conservation law, Commun. Pure Appl. Math., XL (1987), 229–264.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Isaacson, E., Marchesin, D., Palmeira, C. and Plohr, B.: A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. 146 (1992), 505–552.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Isaacson, E., Marchesin, D. and Plohr, B.: Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), 837–866.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Marchesin, D. and Palmeira, C.: Topology of elementary waves for mixed-type systems of conservation laws, J. Dyn. Differ. Eq. 6 (1994), 427–446.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Palmeira, C.: Line fields defined by eigenspaces of derivatives of maps from the plane to itself, Proceedings of the IVth Conference of Differential Geometry, Santiago de Compostela (Spain), 1988.Google Scholar
  15. 15.
    Poston, T. and Stewart, I.: Catastrophe Theory and its Applications, Pitman, 1978.Google Scholar
  16. 16.
    Schaeffer, D. and Shearer, M.: The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery, Commun. Pure Appl. Math. XL (1987), 141–178.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Shearer, M., Shaeffer, D., Marchesin, D. and Paes-Leme, P.: Solution of the Riemann problem for a prototype 2 × 2 system of non-strictly hyperbolic conservation laws, Arch. Ration. Mech. Anal. 97 (1987), 299–320.CrossRefMATHGoogle Scholar
  18. 18.
    Smoller, J.: Shock Waves and Reaction-Diffusion Equations, Springer, Berlin, Heidelberg, New York, 1983.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Centro de Matemática da Universidade do PortoPortoPortugal

Personalised recommendations