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), 345-356

MSC (2010):
Primary 35L45, 35L65

Published electronically:
February 29, 2012

MathSciNet review:
2953107

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Abstract | References | Similar Articles | Additional Information

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.

**1.**Alberto Bressan,*Hyperbolic systems of conservation laws*, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR**1816648****2.**Constantine M. Dafermos,*Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method*, Arch. Rational Mech. Anal.**52**(1973), 1–9. MR**0340837****3.**Constantine M. Dafermos,*The entropy rate admissibility criterion for solutions of hyperbolic conservation laws*, J. Differential Equations**14**(1973), 202–212. MR**0328368****4.**Constantine M. Dafermos,*Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws*, Arch. Rational Mech. Anal.**53**(1973/74), 203–217. MR**0348289****5.**C. M. Dafermos and R. J. DiPerna,*The Riemann problem for certain classes of hyperbolic systems of conservation laws*, J. Differential Equations**20**(1976), no. 1, 90–114. MR**0404871****6.**Hai Tao Fan,*A limiting “viscosity” approach to the Riemann problem for materials exhibiting a change of phase. II*, Arch. Rational Mech. Anal.**116**(1992), no. 4, 317–337. MR**1132765**, 10.1007/BF00375671**7.**Hai Tao Fan,*One-phase Riemann problem and wave interactions in systems of conservation laws of mixed type*, SIAM J. Math. Anal.**24**(1993), no. 4, 840–865. MR**1226854**, 10.1137/0524053**8.**Hai Tao Fan,*A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids*, J. Differential Equations**103**(1993), no. 1, 179–204. MR**1218743**, 10.1006/jdeq.1993.1046**9.**Haitao Fan and Marshall Slemrod,*Dynamic flows with liquid/vapor phase transitions*, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 373–420. MR**1942467**, 10.1016/S1874-5792(02)80011-8**10.**Barbara L. Keyfitz and Herbert 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), no. 3, 444–476. MR**0466993****11.**Barbara L. Keyfitz and Herbert C. Kranzer,*A system of nonstrictly hyperbolic conservation laws arising in elasticity theory*, Arch. Rational Mech. Anal.**72**(1979/80), no. 3, 219–241. MR**549642**, 10.1007/BF00281590**12.**Barbara L. Keyfitz and Herbert C. Kranzer,*The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy*, J. Differential Equations**47**(1983), no. 1, 35–65. MR**684449**, 10.1016/0022-0396(83)90027-X**13.**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653****14.**Tai Ping Liu,*The Riemann problem for general 2×2 conservation laws*, Trans. Amer. Math. Soc.**199**(1974), 89–112. MR**0367472**, 10.1090/S0002-9947-1974-0367472-1**15.**Tai Ping Liu,*Existence and uniqueness theorems for Riemann problems*, Trans. Amer. Math. Soc.**212**(1975), 375–382. MR**0380135**, 10.1090/S0002-9947-1975-0380135-2**16.**Hiroki Ohwa,*On shock curves in 2×2 hyperbolic systems of conservation laws*, Adv. Math. Sci. Appl.**19**(2009), no. 1, 227–244. MR**2553478****17.**Hiroki Ohwa,*The shock curve approach to the Riemann problem for 2×2 hyperbolic systems of conservation laws*, J. Hyperbolic Differ. Equ.**7**(2010), no. 2, 339–364. MR**2659740**, 10.1142/S0219891610002128**18.**H. Ohwa,*On the uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of conservation laws*, Commun. Math. Sci.**9**(2011), 161-185.**19.**M. Slemrod,*A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase*, Arch. Rational Mech. Anal.**105**(1989), no. 4, 327–365. MR**973246**, 10.1007/BF00281495**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), 201–210. MR**0247283****21.**V. A. Tupčiev,*The method of introducing a viscosity in the study of a problem of decay of a discontinuity*, Dokl. Akad. Nauk SSSR**211**(1973), 55–58 (Russian). MR**0330801**

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Additional Information

**Hiroki Ohwa**

Affiliation:
Graduate School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

Email:
ohwa-hiroki@suou.waseda.jp

DOI:
http://dx.doi.org/10.1090/S0033-569X-2012-01254-5

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.