Existence of strong travelling wave profiles to $2\times 2$ systems of viscous conservation laws
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- by Tong Yang, Mei Zhang and Changjiang Zhu PDF
- Proc. Amer. Math. Soc. 135 (2007), 1843-1849 Request permission
Abstract:
In this paper, we prove the existence of strong travelling wave profiles for a class of $2\times 2$ 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.References
- 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 340837, DOI 10.1007/BF00249087
- 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 404871, DOI 10.1016/0022-0396(76)90098-X
- 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, DOI 10.1007/BF00281590
- 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, DOI 10.1016/0022-0396(83)90027-X
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- Tai Ping Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89–112. MR 367472, DOI 10.1090/S0002-9947-1974-0367472-1
- David G. Schaeffer and Michael Shearer, Riemann problems for nonstrictly hyperbolic $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), no. 1, 267–306. MR 906816, DOI 10.1090/S0002-9947-1987-0906816-5
- 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, DOI 10.1512/iumj.1989.38.38048
- 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 247283
- 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 236527, DOI 10.1007/BF00247508
Additional Information
- Tong Yang
- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
- MR Author ID: 303932
- 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
- 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
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2286095