Archive for Rational Mechanics and Analysis

, Volume 182, Issue 1, pp 1–24 | Cite as

Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

  • Dan Marchesin
  • Alexei A. Mailybaev


We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S i, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S i, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S i, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S i, j shocks and classical waves are described. As a physical application, all types of S i, j shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.


Shock Wave Porous Medium Unstable Manifold Rarefaction Wave Riemann Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brooks R.H., Corey A.T.: Properties of porous media affecting fluid flow. J. Irrig. Drain. E-ASCE 6, 61–88 (1966)Google Scholar
  2. 2.
    Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, 1983Google Scholar
  3. 3.
    Hurley J.M., Plohr B.J.: Some effects of viscous terms on Riemann problem solutions. Mat. Contemp. 8, 203–224 (1995)MathSciNetGoogle Scholar
  4. 4.
    Isaacson E.L., Marchesin D., Plohr B.J.: Transitional waves for conservation laws. SIAM J. Math. Anal. 21, 837–866 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kulikovskij A.G.: Surfaces of discontinuity separating two perfect media of different properties. Recombination waves in magnetohydrodynamics. Prikl. Mat. Mekh. 32, 1125–1131 (1968), in RussianGoogle Scholar
  6. 6.
    Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu.: Mathematical aspects of numerical solution of hyperbolic systems. Chapman & Hall/CRC, Boca Raton, 2001Google Scholar
  7. 7.
    Liu T.-P., Zumbrun K.: On nonlinear stability of general undercompressive viscous shock waves. Comm. Math. Phys. 174, 319–345 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mailybaev A.A., Marchesin D.: Dual-family viscous shock waves in systems of conservation laws: a surprising example. In: Proc. Conf. on Analysis, Modeling and Computation of PDE and Multiphase Flow. Stony Brook NY, 2004Google Scholar
  9. 9.
    Majda A., Pego R.L.: Stable viscosity matrices for systems of conservation laws. J. Differential Equations 56, 229–262 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Plohr B., Marchesin D.: Wave structure in WAG recovery. SPE J. 6, 209–219 (2001)Google Scholar
  11. 11.
    Schecter S., Marchesin D., Plohr B.J.: Structurally stable Riemann solutions. J. Differential Equations 126, 303–354 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Schecter S., Plohr B.J., Marchesin D.: Classification of codimension-one Riemann solutions. J. Dynam. Differential Equations 13, 523–588 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shearer M., Schaeffer D.G., Marchesin D., Paes-Leme P.L.: Solution of the Riemann problem for a prototype 2× 2 system of nonstrictly hyperbolic conservation laws. Arch. Ration Mech. Anal. 97, 299–320 (1987)CrossRefGoogle Scholar
  14. 14.
    Sotomayor J.: Generic bifurcations of dynamical systems. Dynamical Systems (Ed. Peixoto, M.M.) (Proc. Sympos., Univ. Bahia, Salvador, 1971). Academic Press, New York, pp 561–582, 1973Google Scholar
  15. 15.
    Stone H.L.: Probability model for estimating three-phase relative permeability. J. Petrol. Technol. 249, 214–218 (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Instituto Nacional de Matemática Pura e Aplicada – IMPARio de JaneiroBrazil
  2. 2.Institute of MechanicsMoscow State Lomonosov UniversityMoscowRussia

Personalised recommendations