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Admissibility Criteria and Admissible Weak Solutions of Riemann Problems for Conservation Laws of Mixed Type: A Summary

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Nonlinear Evolution Equations That Change Type

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 27))

Abstract

Consider the model system

$$ \left\{ {\begin{array}{*{20}{c}} {{v_t} + p{{(u)}_x} = 0} \\ {{u_t} - {v_x} = 0} \\ \end{array} } \right. $$
((1.1))

which describes the one-dimensional isothermal motion of a compressible elastic fluid or solid in Lagrangian coordinate system. Here v denotes the velocity, u the specific volume for a fluid or displacement gradient for a solid, and -p is the stress which is determined through a constitutive relation.

This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.

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References

  1. H. Hattori, The Riemann problem for a Van der Waals fluid with entropy rate admissibility criterion-Isothermal case, Arch. Rational Mech. Anal., 92 (1986), 247–263.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Hattori, The Riemann problem for a Van der Waals fluid with entropy rate admissibility criterion—Nonisothermal case, J. differential Equations, 65, 2 (1986), 158–174.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Hsiao and P. De Mottoni, Existence and uniqueness of Riemann problem for a nonlinear system of conservation laws of mixed type, Trans. Amer. Soc. (to appear).

    Google Scholar 

  4. L. Hsiao, Admissible weak solution for nonlinear system of conservation laws in mixed type, P.D.E., 2, 1 (1989), 40–58.

    MATH  Google Scholar 

  5. L. Hsiao, The uniqueness of admissible solutions of Riemann problem for system of conservation laws of mixed type, (to appear in Jour Diff Equn.).

    Google Scholar 

  6. L. Hsiao, Admissibility criteria and admissible weak solutions of Riemann problems for conservation laws of mixed type, IMA Preprint Series (1990).

    Google Scholar 

  7. R. Hagan and M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Arch. Rational Mech. Anal. 83 (1984), 333–361.

    Article  MathSciNet  Google Scholar 

  8. R.D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73 (1980), 125–158.

    Article  MathSciNet  MATH  Google Scholar 

  9. B.L. Keyfitz, Change of type in three-phase flow: A single analogue, J. Diff. E. (to appear).

    Google Scholar 

  10. B.L. Keyfitz, Admissibility conditions for shocks in conservation laws that change type, (to appear).

    Google Scholar 

  11. R. Pego, Phase transitions: Stability and admissibility in one dimensional nonlinear vis-coelasticity, (to appear).

    Google Scholar 

  12. M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Diff. E., 46 (1982), 426–443.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Shearer, Admissibility criteria for shock wave solution of a system of conservation laws of mixed type, Proc. Royal. Soc. Edinburgh 93 A (1983), 233–244.

    MathSciNet  Google Scholar 

  14. M. Slemrod, Admissibility for propagating phase boundaries in a Van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), 301–315.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    L. Hsiao

  2. Institute of Mathematics, Academia Sinica, Beijing, P.R. China

    L. Hsiao

Authors
  1. L. Hsiao
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, 77204, USA

    Barbara Lee Keyfitz

  2. Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695, USA

    Michael Shearer

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© 1990 Springer-Verlag New York Inc.

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Hsiao, L. (1990). Admissibility Criteria and Admissible Weak Solutions of Riemann Problems for Conservation Laws of Mixed Type: A Summary. In: Keyfitz, B.L., Shearer, M. (eds) Nonlinear Evolution Equations That Change Type. The IMA Volumes in Mathematics and Its Applications, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9049-7_7

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  • DOI: https://doi.org/10.1007/978-1-4613-9049-7_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9051-0

  • Online ISBN: 978-1-4613-9049-7

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