Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Riemann problems for nonstrictly hyperbolic $ 2\times 2$ systems of conservation laws


Authors: David G. Schaeffer and Michael Shearer
Journal: Trans. Amer. Math. Soc. 304 (1987), 267-306
MSC: Primary 35L65; Secondary 35L67, 58C27
DOI: https://doi.org/10.1090/S0002-9947-1987-0906816-5
MathSciNet review: 906816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Riemann problem is solved for $ 2 \times 2$ systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flow in porous media, Exxon Production Research preprint, 1985. MR 866277 (87m:76058)
  • [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. MR 0288640 (44:5836)
  • [3] J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp and S. Yaniv, Front tracking and two dimensional Riemann problems, Adv. Appl. Math. 6 (1985), 259-290. MR 809028 (86m:76060)
  • [4] J. Glimm, E. Isaacson, D. Marchesin and O. McBryan, Front tracking for hyperbolic systems, Adv. Appl. Math. 2 (1981), 91-119. MR 612514 (82i:76097)
  • [5] M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory, Springer, New York, 1985. MR 771477 (86e:58014)
  • [6] H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math. 40 (1987), 229-264. MR 872386 (88d:35125)
  • [7] E. Isaacson, D. Marchesin, B. Plohr and B. Temple, The classification of solutions of quadratic Riemann problems. I. MRC report, Univ. of Wisconsin, 1985.
  • [8] E. Isaacson and B. Temple, The classification of solutions of quadratic Riemann problems. II. MRC report, Univ. of Wisconsin, 1985.
  • [9] -, The classification of solutions of quadratic Riemann problems. III, MRC Tech. Summary Report, 1986.
  • [10] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 0093653 (20:176)
  • [11] T.-P. Liu, The Riemann problem for general $ 2 \times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89-112. MR 0367472 (51:3714)
  • [12] O. A. Oleinik, On the uniqueness of the generalized solution of Cauchy problem for nonlinear system of equations occurring in mechanics, Uspehi Mat. Nauk. (Russian Math. Surveys) 12 (1957), 169-176. MR 0094543 (20:1057)
  • [13] D. G. Schaeffer and M. Shearer, The classification of $ 2 \times 2$ systems of nonstrictly conservation laws, with application to oil recovery; Appendix with D. Marchesin and P. J. Paes-Leme. Comm. Pure Appl. Math. 40 (1987), 141-178. MR 872382 (88a:35155)
  • [14] M. Shearer, D. G. Schaeffer, D. Marchesin and P. J. Paes-Leme, Solution of the Riemann problem for a prototype $ 2 \times 2$ system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97 (1987), 299-320. MR 865843 (88a:35156)
  • [15] Z. Tang and T. C. T. Ting, Wave curves for the Riemann problem of plane waves in simple isotropic elastic solids, Internat. J. Engrg. Sci. (to appear). MR 921358 (88m:73011)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L65, 35L67, 58C27

Retrieve articles in all journals with MSC: 35L65, 35L67, 58C27


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0906816-5
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society