Undercompressive shocks for nonstrictly hyperbolic conservation laws

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.

[1]  Dan Marchesin,et al.  Transitional waves for conservation laws , 1990 .

[2]  David G. Schaeffer,et al.  The classification of 2 × 2 systems of non‐strictly hyperbolic conservation laws, with application to oil recovery , 1987 .

[3]  M. Golubitsky,et al.  Bifurcation analysis near a double eigenvalue of a model chemical reaction , 1981 .

[4]  L. Ahlfors.  Complex Analysis , 1979 .

[5]  Phillip Colella,et al.  Glimm's Method for Gas Dynamics , 1982 .

[6]  M. Shearer,et al.  Approximately periodic solutions of the elastic string equations , 1989 .

[7]  F. Takens.  Partially hyperbolic fixed points , 1971 .

[8]  D. G. Schaeffer,et al.  Solution of the riemann problem for a prototype 2×2 system of non-strictly hyperbolic conservation laws , 1987 .

[9]  J. K. Hunter,et al.  Transverse diffraction of nonlinear waves and singular rays , 1988 .

[10]  M. Shearer.  The Riemann problem for 2 × 2 systems of hyperbolic conservation laws with case I quadratic nonlinearities , 1989 .

[11]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[12]  Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral , 1990 .

[13]  Z. Xin,et al.  Overcompressive shock waves , 1990 .

[14]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[15]  B. Keyfitz.  Change of type in three-phase flow: A simple analogue , 1989 .

[16]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[17]  Tai-Ping Liu.  The Riemann problem for general 2×2 conservation laws , 1974 .

[18]  Blake Temple,et al.  The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems 1 , 1988 .

[19]  Phillip Colella,et al.  Higher order Godunov methods for general systems of hyperbolic conservation laws , 1989 .

[20]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[21]  D. Schaeffer,et al.  Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws , 1987 .

[22]  B. Keyfitz,et al.  Nonstrictly hperbolic conservation laws , 1987 .

[23]  Jerrold E. Marsden,et al.  Basic Complex Analysis , 1973 .