On the existence of strong travelling wave profiles to $2 \times 2$ systems of viscous conservation laws
Author:
Hiroki Ohwa
Journal:
Quart. Appl. Math. 71 (2013), 283-288
MSC (2000):
Primary 35L65; Secondary 35L45
DOI:
https://doi.org/10.1090/S0033-569X-2012-01301-0
Published electronically:
October 18, 2012
MathSciNet review:
3087423
Full-text PDF Free Access
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Abstract: We consider strong travelling wave profiles for a class of $2\times 2$ viscous conservation laws. Our main assumption is that the product of nondiagonal elements within the Fŕechet derivative (Jacobian) of the flux is nonnegative. By using the regularization method improved by the author, we prove the existence of strong travelling wave profiles for those systems.
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 https://doi.org/10.1007/BF00249087
- Constantine M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202–212. MR 328368, DOI https://doi.org/10.1016/0022-0396%2873%2990043-0
- 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 348289, DOI https://doi.org/10.1007/BF00251384
- 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 https://doi.org/10.1016/0022-0396%2876%2990098-X
- H. Ohwa, Existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, Quart. Appl. Math. 70 (2012), 345–356.
- H. Ohwa, Existence of solutions to the Riemann problem for $2\times 2$ conservation laws, to appear in Appl. Anal.
- Tong Yang, Mei Zhang, and Changjiang Zhu, Existence of strong travelling wave profiles to $2\times 2$ systems of viscous conservation laws, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1843–1849. MR 2286095, DOI https://doi.org/10.1090/S0002-9939-07-08747-3
References
- C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method, Arch. Rational Mech. Anal. 52 (1973), 1–9. MR 0340837 (49:5587)
- C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202–212. MR 0328368 (48:6710)
- C. M. Dafermos, Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 53 (1974), 203–217. MR 0348289 (50:787)
- C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations 20 (1976), 90–114. MR 0404871 (53:8671)
- H. Ohwa, Existence of solutions to the Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, Quart. Appl. Math. 70 (2012), 345–356.
- H. Ohwa, Existence of solutions to the Riemann problem for $2\times 2$ conservation laws, to appear in Appl. Anal.
- T. Yang, M. Zhang and C. Zhu, Existence of strong travelling wave profiles to $2\times 2$ systems of viscous conservation laws, Proc. Amer. Math. Soc. 135 (2007), 1843–1849. MR 2286095 (2007j:35133)
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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
Keywords:
Viscous conservation,
strong travelling wave profiles,
existence,
vanishing viscosity approach
Received by editor(s):
June 11, 2011
Published electronically:
October 18, 2012
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
