A Dynamical Systems Approach to Traveling Wave Solutions for Liquid/Vapor Phase Transition

  • Haitao Fan
  • Xiao-Biao LinEmail author
Part of the Fields Institute Communications book series (FIC, volume 64)


We study the existence of liquefaction and evaporation waves by the methods derived from dynamical systems theory. A traveling wave solution is a heteroclinic orbit with the wave speed as a parameter. We give sufficient and necessary conditions for the existence of such heteroclinic orbit. After analyzing the local unstable and stable manifolds of two equilibrium points, we show that there exists at least one orbit connecting the local unstable manifold of one equilibrium point to the local stable manifold of another equilibrium point. The method is known as the shooting method in the literature.



Research of Professor Lin was supported in part by the National Science Foundation under grant DMS-0708386.

Received 6/24/2009; Accepted 4/6/2010


  1. [1].
    D. Amadori, A. Corli, On a model of multiphase flow. SIMA J. Math. Anal. (To appear) SIAM J. Math. Anal., 40(1), 134–166.MathSciNetCrossRefGoogle Scholar
  2. [2].
    Bramson, Convergence of solutions of Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44(285) (1983)Google Scholar
  3. [3].
    A. Corli, H. Fan, The Riemann problem for reversible reactive flows with metastability. SIAM J. Appl. Math., 65, 426–457 (2005)MathSciNetCrossRefGoogle Scholar
  4. [4].
    H. Fan, One-phase Riemann problem and wave interactions in systems of conservation laws of mixed type. SIAM J. Math. Anal. 24(4), 840–865 (1993)MathSciNetCrossRefGoogle Scholar
  5. [5].
    H. Fan, Travelling waves, Riemann problems and computations of a model of the dynamics of liquid/vapour phase transitions. J. Differ. Equat. 150, 385–437 (1998)MathSciNetCrossRefGoogle Scholar
  6. [6].
    H. Fan, Convergence to travelling waves in two model systems related to the dynamics of liquid/vapour phase changes. J. Differ. Equat. 168, 102–128 (2000)MathSciNetCrossRefGoogle Scholar
  7. [7].
    H. Fan, On a model of the dynamics of liquid/vapour phase transitions. SIAM J. Appl. Math. 60, 1270–1301 (2000)MathSciNetCrossRefGoogle Scholar
  8. [8].
    H. Fan, Symmetry breaking, ring formation and other phase boundary structures in shock tube experiments on retrograde fluids. J. Fluid Mech. 513, 47–75 (2004)MathSciNetCrossRefGoogle Scholar
  9. [9].
    H. Fan, X.B. Lin, Collapsing and explosion waves in phase transitions with metastability, existence, stability and related Riemann problems. J. Dyn. Differ. Equat. 22, 163–191 (2010)MathSciNetCrossRefGoogle Scholar
  10. [10].
    J.K. Hale, Ordinary Differential Equations (Wiley-Interscience, New York, 1969; 2nd edn., Kreiger Publ., 1980)Google Scholar
  11. [11].
    P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964)Google Scholar
  12. [12].
    M. Shearer, Riemann problem for a class of conservation laws of mixed type. J. Differ. Equat. 46, 426–443 (1982)Google Scholar
  13. [13].
    M. Shearer, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type. Proc. Roy. Soc. Edinburgh. 93, 233–244 (1983)MathSciNetCrossRefGoogle Scholar
  14. [14].
    M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type. Arch.Rational Mech.Anal. 93, 45–59 (1986)MathSciNetCrossRefGoogle Scholar
  15. [15].
    M. Slemrod, Admissibility criterion for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. An. 81, 301–315 (1983)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsGeorgetown UniversityWashingtonUSA
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA

Personalised recommendations