References
J. M. Burgers. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1 (1948), 171–199.
S. N. Chow, J. K. Hale, & J. Mallet-Paret. Applications of generic bifurcation II. Arch. Rational Mech. Anal. 62 (1976), 209–236.
M. C. Crandall & P. H. Rabinowitz. Bifurcation from simple eigenvalues. J. Functional Anal. 8 (1971), 321–340.
R. L. Foy. Steady state solutions of hyperbolic systems of conservation laws with viscosity terms. Comm. Pure Appl. Math. 17 (1964), 177–188.
M. Golubitsky & D. G. Schaeffer. A theory for imperfect bifurcation via singularity theory. Comm. Pure Appl. Math. 32 (1979), 21–98.
M. Golubitsky & D. G. Schaeffer. Singularities and Groups in Bifurcation Theory, Volume 1. Springer-Verlag, 1985.
J. Guckenheimer & P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.
E. Isaacson, D. Marchesin, B. Plohr & B. Temple. Classification of the solutions of quadratic Riemann problems. I. In preparation.
J. P. Keener & H. B. Keller. Perturbed bifurcation theory. Arch. Rational Mech. Anal. 50 (1973), 159–175.
P. D. Lax. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math. 10 (1957), 537–566.
T.-P. Liu. The Riemann problem for general 2×2 conservation laws. Trans. Amer. Math. Soc. 199 (1974), 89–112.
T.-P. Liu. The Riemann problem for general systems of conservation laws. J. Differential Equations 18 (1975), 218–234.
J. B. McLeod & D. H. Sattinger. Loss of stability and bifurcation at a double eigenvalue. J. Functional Anal. 14 (1973), 62–84.
D. H. Sattinger. Branching in the Presence of Symmetry. CBMS-NSF Conference Notes, 40 SIAM, Philadelphia, 1983.
D. G. Schaeffer. Unpublished notes.
D. G. Schaeffer & M. Shearer. The classification of 2×2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Math., to appear.
M. Shearer. Secondary bifurcation near a double eigenvalue. SIAM J. Math. Anal. 11 (1980), 365–389.
M. Shearer. Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type. Proc. Royal Soc. Edinburgh 93 A (1983), 233–244.
M. Shearer. Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type. Arch. Rational Mech. Anal. 93 (1986), 45–59.
M. Slemrod. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983), 301–315.
B. Wendroff. The Riemann problem for materials with nonconvex equations of state. I: Isentropic flow. J. Math. Anal. Appl. 38 (1972), 454–466.
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Shearer, M., Schaeffer, D.G., Marchesin, D. et al. Solution of the riemann problem for a prototype 2×2 system of non-strictly hyperbolic conservation laws. Arch. Rational Mech. Anal. 97, 299–320 (1987). https://doi.org/10.1007/BF00280409
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DOI: https://doi.org/10.1007/BF00280409