## Current Progress in Hyperbolic Systems: Riemann Problems and Computations: Proceedings of the AMS-IMS-SIAM Joint Summer Conference Held July 16-22, 1988 with Support from the National Science Foundation and the Office of Naval ResearchThe study of Riemann problems has undergone a strong, steady growth in the last decade. The general direction of the research has headed toward understanding the wave structure of the solutions of more physically realistic systems. These systems fail either or both of the two main restrictions of the classical theory--that the system be strictly hyperbolic or genuinely nonlinear. The systems that have been studied tend to fall into the following broad classes: real gas dynamics (including combustion), visco-elastic materials, phase transitions, and multiphase flow in porous media. In addition to their usefulness in large-scale calculations, computational schemes have vastly improved the handling of discontinuity behavior. This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Current Progress in Hyperbolic Systems: Riemann Problems and Computations, held at Bowdoin College in July 1988. The papers presented here provide a complete picture of recent research by some of the leaders in this field. Graduate students and beginning researchers will find this book a useful introduction to current work in this area. |

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### Contents

1 | |

9 | |

Capillary energy and the entropy condition for the BuckleyLeverett equation | 21 |

Nonlinear elastoplastic waves | 27 |

An example of a Riemann problem of second kind | 55 |

Density profiles for diverging detonations | 63 |

Anomalous waves in shock wave fluid interface collisions | 77 |

Timedependent shear flow of a nonNewtonian fluid | 91 |

A criterion for certain wave structures in systems that change type | 203 |

A note on the stability of eigenvalue degeneracy in nonlinear conservation laws of multiphase flow | 215 |

Analogies between Riemann problem for 1D fluid dynamics and 2D steady supersonic flow | 225 |

Instability and illposedness in granular flow | 241 |

Wellposedness of the Riemann problem consistency of the Godunovs scheme | 251 |

The Riemann problem for a system of conservation laws modeling phase transitions | 267 |

Detonation waves and deflagration waves in the one dimensional ZND model for high Mach number combustion | 277 |

The Riemann solution to a system of conservation laws with application to a nonzero sum game | 287 |

The Riemann problem for combustion | 111 |

Transitional shock waves | 125 |

Threephase flow with gravity | 147 |

A system of conservation laws with a parabolic degeneracy | 161 |

Nonlinear surface waves | 185 |

Asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in several dimensions | 299 |

the existence and basic structure of the selfsimilar solutions | 305 |

Dynamic instability of the liquid crystal director | 325 |

On the Riemann problem for a prototype of a mixed type conservation law II | 331 |

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1989 American Mathematical admissible analysis angle asymptotic behavior bifurcation boundary Buckley-Leverett equation capillary energy Cauchy problem characteristic speeds compressive computed conservation laws constant constitutive equations Contemporary Mathematics Volume corresponding defined deflagration deformation denote density differential equations discontinuity Editors eigenvalues elastic elliptic region entropy condition existence figure fluid dynamics flux function genuine non-linearity Godunov's scheme Hugoniot locus hyperbolic conservation laws incident shock initial data interface intersection Lemma linear Marchesin material Math Mathematics Subject Classification Mathematics Volume 100 matrix Mech monotone Non-Newtonian Fluid nonlinear numerical obtain parameters phase planar plastic proof quadratic model rarefaction wave relative permeabilities Riemann problem satisfies saturation shear Shearer shock polar stable manifold steady stress system of conservation Theorem theory three-phase flow transitional shock waves travelling wave umbilic point unburned unique variables velocity viscoelastic viscosity viscous profiles wave curve wave patterns zero