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), 18431849
MSC (2000):
Primary 35L65; Secondary 74J30, 35L45
Published electronically:
January 5, 2007
MathSciNet review:
2286095
Fulltext PDF Free Access
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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.
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
(49 #5587)
 2.
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
(53 #8671)
 3.
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
(80k:35050), http://dx.doi.org/10.1007/BF00281590
 4.
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
(84a:35162), http://dx.doi.org/10.1016/00220396(83)90027X
 5.
P.
D. Lax, Hyperbolic systems of conservation laws. II, Comm.
Pure Appl. Math. 10 (1957), 537–566. MR 0093653
(20 #176)
 6.
Tai
Ping Liu, The Riemann problem for general
2×2 conservation laws, Trans. Amer. Math.
Soc. 199 (1974),
89–112. MR
0367472 (51 #3714), http://dx.doi.org/10.1090/S00029947197403674721
 7.
David
G. Schaeffer and Michael
Shearer, Riemann problems for nonstrictly
hyperbolic 2×2 systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), no. 1, 267–306. MR 906816
(88m:35101), http://dx.doi.org/10.1090/S00029947198709068165
 8.
Marshall
Slemrod and Athanassios
E. Tzavaras, A limiting viscosity approach for the Riemann problem
in isentropic gas dynamics, Indiana Univ. Math. J. 38
(1989), no. 4, 1047–1074. MR 1029688
(90m:35119), http://dx.doi.org/10.1512/iumj.1989.38.38048
 9.
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
(40 #552)
 10.
J.
A. Smoller and J.
L. Johnson, Global solutions for an extended class of hyperbolic
systems of conservation laws, Arch. Rational Mech. Anal.
32 (1969), 169–189. MR 0236527
(38 #4822)
 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), 19. 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), 90114. 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), 219241. 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),3565. MR 0684449 (84a:35162)
 5.
 Lax P.D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10(1957), 537556. MR 0093653 (20:176)
 6.
 Liu T.P., The Riemann problem for general conservation laws, Trans. Amer. Math. Soc., 199(1974), 89112. 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), 267305. 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), 10471073. 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), 201210. 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), 168189. MR 0236527 (38:4822)
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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:
http://dx.doi.org/10.1090/S0002993907087473
PII:
S 00029939(07)087473
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
