Existence of strong travelling wave profiles to systems of viscous conservation laws

Authors:
Tong Yang, Mei Zhang and Changjiang Zhu

Journal:
Proc. Amer. Math. Soc. **135** (2007), 1843-1849

MSC (2000):
Primary 35L65; Secondary 74J30, 35L45

DOI:
https://doi.org/10.1090/S0002-9939-07-08747-3

Published electronically:
January 5, 2007

MathSciNet review:
2286095

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove the existence of strong travelling wave profiles for a class of viscous conservation laws when the corresponding invisid systems are hyperbolic. Besides some technical assumptions, the only main assumption is the hyperbolicity. Therefore, the existence theory can be applied to systems which are not strictly hyperbolic. Moreover, the characteristic fields can be neither genuinely nonlinear nor linearly degenerate.

**1.**Dafermos C.M., 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 (49:5587)****2.**Dafermos C.M., DiPerna R.J., The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations, 20(1976), 90-114. MR**0404871 (53:8671)****3.**Keyfitz B.L., Kranzer H.C., A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72(1980), 219-241. MR**0549642 (80k:35050)****4.**Keyfitz B.L., Kranzer H.C., The Riemann Problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations, 47(1983),35-65. MR**0684449 (84a:35162)****5.**Lax P.D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10(1957), 537-556. MR**0093653 (20:176)****6.**Liu T.P., The Riemann problem for general conservation laws, Trans. Amer. Math. Soc., 199(1974), 89-112. MR**0367472 (51:3714)****7.**Schaeffer D.G., Shearer M., Riemann problems for nonstrictly hyperbolic systems of conservation laws, Trans. Amer. Math. Soc., 304(1987), 267-305. MR**0906816 (88m:35101)****8.**Slemrod M., Tzavaras A.E., A limiting viscosity approach for the Riemann problem in isentropic gas dynamics, Indiana Univ. Math. J., 38(1989), 1047-1073. MR**1029688 (90m:35119)****9.**Smoller J.A., On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Mich. Math. J., 16(1969), 201-210. MR**0247283 (40:552)****10.**Smoller J.A., Johnson J.L., Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal., 32(1969), 168-189. MR**0236527 (38:4822)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35L65,
74J30,
35L45

Retrieve articles in all journals with MSC (2000): 35L65, 74J30, 35L45

Additional Information

**Tong Yang**

Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Email:
matyang@cityu.edu.hk

**Mei Zhang**

Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Email:
meizhang3@student.cityu.edu.hk

**Changjiang Zhu**

Affiliation:
Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 430079, People’s Republic of China

Email:
cjzhu@mail.ccnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-07-08747-3

Keywords:
Viscous conservation laws,
travelling wave profile,
{\it a priori} estimate.

Received by editor(s):
September 14, 2005

Received by editor(s) in revised form:
February 13, 2006

Published electronically:
January 5, 2007

Communicated by:
Walter Craig

Article copyright:
© Copyright 2007
American Mathematical Society