Oscillation waves in Riemann problems for phase transitions
Authors:
Hermano Frid and I-Shih Liu
Journal:
Quart. Appl. Math. 56 (1998), 115-135
MSC:
Primary 35L65; Secondary 65M12, 73B30, 80A22
DOI:
https://doi.org/10.1090/qam/1604813
MathSciNet review:
MR1604813
Full-text PDF Free Access
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J. L. Ericksen, Equilibrium of bars, J. Elasticity 5, Nos. 3 & 4, 191–201 (1975)
H. Frid and I-S. Liu, Oscillation waves in Riemann problems inside elliptic regions for systems of conservation laws of mixed type, Preprint (to appear)
R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 8, 223–270 (1985)
I. Suliciu, Some energetic properties of smooth solutions in rate-type viscoelasticity, Internat. J. Non-linear Mechanics 17, 525–544 (1984)
M. E. Gurtin, W. O. Williams, and I. Suliciu, On rate-type constitutive equations and the energy of viscoelastic and viscoplastic materials, Internat. J. Solids Structures 16, 607–617 (1980)
M. Mihǎilescu-Suliciu and I. Suliciu, On the method of characteristics in rate-type visoelasticity, Zeit. Angew. Mat. Mech.-ZAMM 65, 479–486 (1985)
M. Mihǎilescu-Suliciu and I. Suliciu, On the method of characteristics in rate-type viscoelasticity with non-monotone equilibrium curve, ZAMM 72, 667–674 (1992)
E. B. Pitman and Y. Ni, Visco-elastic relaxation with a Van der Waals type stress, Internat. J. Engrg. 22, 327–338 (1994)
C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Eqs. 14, 202–212 (1973)
R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158 (1980)
M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301–315 (1983)
R. Hagan and M. Slemrod, The viscosity-capillarity criterion for shocks and phase transitions, Arch. Rational Mech. Anal. 83, 333–361 (1983)
M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Differential Eqs. 46, 426–443 (1982)
M. Shearer, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type, Proc. Royal Soc. Edinburgh 93A, 233–244 (1983)
M. Shearer, Nonuniqueness of admissible solutions of the Riemann initial value problem for a system of conservation laws of mixed type, Arch. Rational Mech. Anal. 93, 45–59 (1986)
H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion: Isothermal case, Arch. Rational Mech. Anal. 92, 246–263 (1986)
H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion: Non-isothermal case, J. Differential Eqs. 65, 158–174 (1986)
B. L. Keyfitz, The Riemann problem for non-monotone stress-strain functions: a “hysteresis” approach, Lectures in Appl. Math. 23, 379–395 (1986)
R. L. Pego, Phase transitions: stability and admissibility in one-dimensional non-linear viscoelasticity, Arch. Rational Mech. Anal. 97, 353–394 (1987)
Ph. LeFloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm method, Arch. Rational Mech. Anal. 123, 153–197 (1993)
R. D. James, Coexistent phases in one dimensional static theory of elastic bars, Arch. Rational Mech. Anal. 72, 99–140 (1980)
M. E. Gurtin, On the theory of phase transitions with interfacial energy, Arch. Rational Mech. Anal. 87, 187–212 (1984)
L. Delay, R. V. Krishnan, H. Tas, and H. Warlimont, Thermoelasticity, pseudoelasticity and the memory effects associated with martensitic transformations, Journal of Materials Science 9, 1521–1555 (1974)
Y. Huo and I. Müller, Nonequilibrium thermodynamics of pseudoelasticity, Continuum Mech. Thermodyn. 5, 163–204 (1993)
L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics, ed. R. J. Knops, Pitman Press, New York, 1979, pp. 136–212
R. J. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108, 667–689 (1987)
M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Eqs. 7, 959–1000 (1982)
J. Shearer, Global Existence and Compactness in $L^{p}$ for Systems of Conservation Laws, Ph.D. Thesis, University of California at Berkeley, 1990
B. L. Rozhdestvenski and N. N. Yanenko, Systems of Quasilinear Equations and Applications to Gas Dynamics, Moscow: Nauka, 1978 (Russian)
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© Copyright 1998
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