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Global versus local admissibility criteria for dynamic phase boundaries

Published online by Cambridge University Press:  14 November 2011

Haitao Fan
Haitao Fan
Affiliation:
Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Geogia 30332, U.S.A.
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Synopsis

The nonuniqueness of solutions of Riemann problems for a system of conservation laws of mixed type which are admissible according to kinetic relation criteria or travelling wave criteria is proved. The above-mentioned admissibility criteria all consist of local restrictions on points of discontinuity of weak solutions. An example is given which has more than one solution admissible by these local admissibility criteria, but has only one solution, the one-phase solution, satisfying the vanishing viscosity criterion. The entropy rate criterion, however, prefers the two-phase solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 5 , 1993 , pp. 927 - 944
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Abeyaratne, R. and Knowles, J.. Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids. SIAM J. Appl. Math. 51 (1991), 1205.CrossRefGoogle Scholar
2Dafermos, C. M.. The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differential Equations 14 (1973), 202–212.CrossRefGoogle Scholar
3, Ding Xiaxi, , Chen Gui-Qiang and , Luo Peizhu. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Comm. Math. Phys. 121 (1989), 63–84.CrossRefGoogle Scholar
4DiPerna, R. J.. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1–30.CrossRefGoogle Scholar
5Fan, H.-T.. The uniqueness and stability of the solution of the Riemann problem for a system of conservation laws of mixed type. Trans. Amer. Math. Soc. 33 (1992), 913–938.CrossRefGoogle Scholar
6Fan, H.T. and Slemrod, M.. The Rieman problem for systems of conservation laws of mixed type. Conference Proceedings on Shock Induced Transitions and Phase Structure in General Media, Institute of Mathematics and its Applications, Minneapolis, October 1990 (to appear).Google Scholar
7Hattori, H.. The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion, Isothermal case. Arch. Rational Mech. Anal. 92 (1986), 247–263.CrossRefGoogle Scholar
8Isaacson, E., Marchesin, D. and Plohr, B.. Transitional waves for conservation laws. SIAM J. Math. Anal. 21 (1990), 837–866.CrossRefGoogle Scholar
9Lax, P. D.. Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10 (1957), 537–566.CrossRefGoogle Scholar
10Liu, T.-P.. The Riemann problem for general system of conservation laws. J. Differential Equations 18 (1975), 218–234.CrossRefGoogle Scholar
11Pence, T. J.. On the mechanical dissipation of solutions to the Riemann problem for impact involving a two-phase elastic material (preprint, 1990).Google Scholar
12Schecter, S. and Shearer, M.. Under compressive shocks for nonstrictly hyperbolic conservation laws. J. Dynamics Differential Equations 3 (1991), 199–271.CrossRefGoogle Scholar
13Shearer, M.. Nonuniqueness of admissible solutions of Riemann initial value problem for a system of conservation laws of mixed type. Arch. Rational Mech. Anal. 93 (1986), 45–59.CrossRefGoogle Scholar
14Slemrod., M.Admissibility criterion for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983), 301–315.CrossRefGoogle Scholar
15Slemrod, M.. Dynamic phase transitions in a van der Waals fluid. J. Differential Equations 52 (1984), 1–23.CrossRefGoogle Scholar
16Truskinovskii, L. M.. Equilibrium phase interfaces. Dokl. Akad. Nauk SSSR 265 (1982), 306–310.Google Scholar