Existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy
Author:
Hiroki Ohwa
Journal:
Quart. Appl. Math. 70 (2012), 345356
MSC (2010):
Primary 35L45, 35L65
Published electronically:
February 29, 2012
MathSciNet review:
2953107
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Abstract: This paper contains a proof of the existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy. The method used in this paper is based upon the vanishing viscosity approach. This approach enables us to establish the existence of solutions to the Riemann problem for those systems which do not satisfy the genuine nonlinearity condition and the shock admissibility condition.
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 C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations 20 (1976), 90114. MR 0404871 (53:8671)
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 H.T. Fan, A limiting ``viscosity'' approach to the Riemann problem for materials exhibiting changes of phase (II), Arch. Rational Mech. Anal. 116 (1992), 317337. MR 1132765 (93a:35102)
 7.
 H.T. Fan, Onephase Riemann problem and wave interactions in systems of conservation laws of mixed type, SIAM J. Math. Anal. 24 (1993), 840865. MR 1226854 (94f:35082)
 8.
 H.T. Fan, A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids, J. Differential Equations 103 (1993), 179204. MR 1218743 (94g:35140)
 9.
 H.T. Fan and M. Slemrod, Dynamic flows with liquid/vapor phase transitions, Handbook of mathematical fluid dynamics, Vol. I, 373420, NorthHolland, Amsterdam, 2002. MR 1942467 (2003j:76085)
 10.
 B. L. Keyfitz and H. C. Kranzer, Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws, J. Differential Equations 27 (1978), 444476. MR 0466993 (57:6866)
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 B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1980), 219241. MR 549642 (80k:35050)
 12.
 B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations 47 (1983), 3565. MR 684449 (84a:35162)
 13.
 P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537566. MR 0093653 (20:176)
 14.
 T. P. Liu, The Riemann problem for general conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89112. MR 0367472 (51:3714)
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 T. P. Liu, Existence and uniqueness theorems for Riemann problems, Trans. Amer. Math. Soc. 212 (1975), 375382. MR 0380135 (52:1036)
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 H. Ohwa, On shock curves in hyperbolic systems of conservation laws, Adv. Math. Sci. Appl. 19 (2009), 227244. MR 2553478 (2010i:35236)
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 H. Ohwa, The shock curve approach to the Riemann problem for hyperbolic systems of conservation laws, J. Hyperbolic Differ. Equ. 7 (2010), 339364. MR 2659740
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 H. Ohwa, On the uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of conservation laws, Commun. Math. Sci. 9 (2011), 161185.
 19.
 M. Slemrod, A limiting ``viscosity'' approach to the Riemann problem for materials exhibiting change of phase, Arch. Rational Mech. Anal. 105 (1989), 327365. MR 973246 (89m:35186)
 20.
 J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201210. MR 0247283 (40:552)
 21.
 V. A. Tupciev, On the method of introducing viscosity in the study of problems involving the decay of discontinuity, Dokl. Akad. Nauk. SSSR 211 (1973), 5558. MR 0330801 (48:9138)
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Additional Information
Hiroki Ohwa
Affiliation:
Graduate School of Education, Waseda University, 161 NishiWaseda, Shinjukuku, Tokyo, 1698050, Japan
Email:
ohwahiroki@suou.waseda.jp
DOI:
http://dx.doi.org/10.1090/S0033569X2012012545
PII:
S 0033569X(2012)012545
Keywords:
Conservation laws,
nonlinear wave equation,
the Riemann problem,
the vanishing viscosity approach
Received by editor(s):
October 4, 2010
Published electronically:
February 29, 2012
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
