Abstract
We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.
Similar content being viewed by others
References
Bell, J., Colella, P., and Trangenstein, J. (1989). Higher order Godunov methods for general systems of hyperbolic conservation laws.J. Comp. Phys. 82, 362–397.
Colella, P. (1982). Glimm's method for gas dynamics.S1AM J. Stat. Comput. 3, 76–110.
Fehribach, J., and Shearer, M. (1989). Approximately periodic solutions of the elastic string equations.Applic. Anal. 32, 1–14.
Garaizar, X., Glimm, J., and Guo, W. (1989). Elastic deformation and slug flow as applications of front tracking. SUNY Stony Brook, preprint.
Glimm, J. (1965). Solutions in the large for nonlinear hyperbolic systems of equations.Comm. Pure Appl. Math. 18, 697–715.
Golubitsky, M., and Schaeffer, D. G. (1985).Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, New York.
Guckenheimer, J., and Holmes, P. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
Isaacson, E., Marchesin, D., Plohr, B., and Temple, J. B. (1988). The Riemann problem near a hyperbolic singularity: The classification of quadratic Riemann problems I.SIAM J. Appl. Math. 48, 1–24.
Isaacson, E., Marchesin, D., and Plohr, B. (1990). Transitional waves for conservation laws.SIAM J. Math. Anal. 21, 837–866.
Keyfitz, B. L. (1989). Change of type in three phase flow—a simple analogue.J. Different. Eq. 80, 280–305.
Lax, P. D. (1957). Hyperbolic systems of conservation laws II.Comm. Pure Appl. Math. 10, 537–566.
Liu, T.-P. (1974). The Riemann problem for general 2×2 conservation laws.Trans. Am. Math. Soc. 199, 89–112.
Liu, T.-P., and Xin, Z.-P. (1990). Overcompressive shock waves. Nonlinear evolution equations that change type (B. L. Keyfitz and M. Shearer, eds.).IMA Vols. in Math. and Appl. 27, 139–145.
Marsden, J. E., and Hoffman, M. J. (1987).Basic Complex Analysis, 2nd ed., Freeman, New York.
Schaeffer, D. G., and Shearer, M. (1987a). The classification of 2×2 systems of nonstrictly hyperbolic conservation laws, with application to oil recovery.Comm. Pure Appl. Math. 40, 141–178.
Schaeffer, D. G., and Shearer, M. (1987b). Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws.Trans. A.M.S. 301, 267–306.
Schaeffer, D. G., and Golubitsky, M. (1981). Bifurcation analysis near a double eigenvalue of a model chemical reaction.Arch. Rat. Mech. Anal. 75, 315–347.
Schecter, S. (1990). Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral.Nonlinearity 3, 79–99.
Shearer, M., Schaeffer, D. G., Marchesin, D., and Paes-Leme, P. (1987). Solution of the Riemann problem for a prototype 2×2 system of non-strictly hyperbolic conservation laws.Arch. Rat. Mech. Anal. 97, 299–320.
Shearer, M. (1989). The Riemann problem for 2×2 systems of hyperbolic conservation laws with case I quadratic nonlinearities.J. Different. Eq. 80, 343–363.
Takens, F. (1971). Partially hyperbolic fixed points.Topology 10, 133–147.
Vegter, G. (1982). Bifurcations of gradient vector fields.Astérisque 98–99, 39–73.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schecter, S., Shearer, M. Undercompressive shocks for nonstrictly hyperbolic conservation laws. J Dyn Diff Equat 3, 199–271 (1991). https://doi.org/10.1007/BF01047709
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01047709