A Limiting “Viscosity” Approach to the Riemann Problem for Materials Exhibiting Change of Phase

  • M. Slemrod


The one-dimensional isothermal motion of a compressible elastic fluid or solid can be described in Lagrangian coordinates by the coupled system
$${u_{1}} + p{\left( w \right)_{x}} = 0,$$
$${w_{1}} - {u_{x}} = 0.$$


Local Minimum Local Maximum Extremal Point Riemann Problem Convergent Subsequence 
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  1. 1.
    R. D. James, The propagation of phase boundaries in elastic bars, Archive for Rational Mechanics and Analysis.Google Scholar
  2. 2.
    M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Differential Equations46 (1982), 426–443.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Archive for Rational Mechanics and Analysis93 (1986), 45–59.MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Archive for Rational Mechanics and Analysis81 (1983), 301–315.MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    R. Hagan & M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Archive for Rational Mechanics and Analysis83 (1984), 333–361.MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    M. Slemrod, Dynamics of first order phase transition, in Phase Transformations and Material Instabilities in Solids, ed. M. Gurtin, Academic Press: New York (1984), 163–203.Google Scholar
  7. 7.
    C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Archive for Rational Mechanics and Analysis52 (1973), 1–9.MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    C. M. Dafermos & R. J. Di Perna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations20 (1976), 90–114.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    C. S. Morawetz, On a weak solution for a transonic flow, Comm. Pure and Applied Math.38 (1985), 797–818.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    J. Mawhin, Topological degree methods in nonlinear boundary value problems, Conference Board of Mathematical Sciences Regional Conference Series in Mathematics, No. 40, American Mathematical Society (1979).Google Scholar
  11. 11.
    I. P. Natanson, Theory of functions of a real variable, Vol. 1, F. Ungar Publishing Co., New York (1955).Google Scholar
  12. 12.
    M. Shearer, Dynamic phase transitions in a van der Waals gas, to appear Quarterly of Applied Math.Google Scholar
  13. 13.
    A. S. Kalasnikov, Construction of generalized solutions of quasilinear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Akad. Nauk. SSR127 (1959), 27–30 (Russian).MathSciNetMATHGoogle Scholar
  14. 14.
    V. A. Tupciev, The asymptotic behavior of the solution of the Cauchy problem for the equation \({\varepsilon ^{2}}t{u_{{xx}}} = {u_{t}}\left[ {\varphi \left( u \right)} \right]x\) that degenerates for ξ = 0 into the problem of the decay of an arbitrary discontinuity for the case of a rarefraction wave. Z. Vycisl. Mat. Fiz.12 (1972), 770–775; English translation in USSR Comput. Math. and Phys.12.MathSciNetMATHGoogle Scholar
  15. 15.
    V. A. Tupciev, On the method of introducing viscosity in the study of problems involving decay of a discontinuity, Dokl. Akad. Nauk. SSR211 (1973), 55–58; English translation in Soviet Math. Dokl.14.MathSciNetGoogle Scholar
  16. 16.
    C. M. Dafermos, Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws, Arch. for Rational Mechanics and Analysis53 (1974), 203–217.MathSciNetADSMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • M. Slemrod
    • 1
  1. 1.Center for the Mathematical SciencesUniversity of WisconsinMadisonUSA

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