Nonlinear Evolution Equations That Change Type

This IMA Volume in Mathematics and its Applications NONLINEAR EVOLUTION EQUATIONS THAT CHANGE TYPE is based on the proceedings of a workshop which was an integral part of the 1988-89 IMA program on NONLINEAR WAVES. The workshop focussed on prob­ lems of ill-posedness and change of type which arise in modeling flows in porous materials, viscoelastic fluids and solids and phase changes. We thank the Coordinat­ ing Committee: James Glimm, Daniel Joseph, Barbara Lee Keyfitz, Andrew Majda, Alan Newell, Peter Olver, David Sattinger and David Schaeffer for planning and implementing an exciting and stimulating year-long program. We especially thank the workshop organizers, Barbara Lee Keyfitz and Michael Shearer, for their efforts in bringing together many of the major figures in those research fields in which theories for nonlinear evolution equations that change type are being developed. A vner Friedman Willard Miller, J r. ix PREFACE During the winter and spring quarters of the 1988/89 IMA Program on Non­ linear Waves, the issue of change of type in nonlinear partial differential equations appeared frequently. Discussion began with the January 1989 workshop on Two­ Phase Waves in Fluidized Beds, Sedimentation and Granular Flow; some of the papers in the proceedings of that workshop present strategies designed to avoid the appearance of change of type in models for multiphase fluid flow.

Multiple viscous profile Riemann solutions in mixed elliptic-hyperbolic models for flow in porous media.- On the loss of regularity of shearing flows of viscoelastic fluids.- Composite type, change of type, and degeneracy in first order systems with applications to viscoelastic flows.- Numerical simulation of inertial viscoelastic flow with change of type.- Some qualitative properties of 2 × 2 systems of conservation laws of mixed type.- On the strict hyperbolicity of the Buckley-Leverett equations for three-phase flow.- Admissibility criteria and admissible weak solutions of Riemann problems for conservation laws of mixed type: a summary.- Shocks near the sonic line: a comparison between steady and unsteady models for change of type.- A strictly hyperbolic system of conservation laws admitting singular shocks.- An existence and uniqueness result for two nonstrictly hyperbolic systems.- Overcompressive shock waves.- Quadratic dynamical systems describing shear flow of non-Newtonian fluids.- Dynamic phase transitions: a connection matrix approach.- A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell type.- Loss of hyperbolicity in yield vertex plasticity models under nonproportional loading.- Undercompressive shocks in systems of conservation laws.- Measure valued solutions to a backward-forward heat equation: a conference report.- One-dimensional thermomechanical phase transitions with non-convex potentials of Ginzburg-Landau type.- Admissibility of solutions to the Riemann problem for systems of mixed type-transonic small disturbance theory-.