Ebook: 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.

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This volume will be of interest to applied mathematicians, to researchers in Partial Differential Equations, and to those involved in Fluid Dynamics and Numerical Analysis examining models for viscoelastic flows, porous medium and granular flows, and flows exhibiting phase transitions. As papers in this volume indicate, physical processes whose simplest models may involve change of type occur also in other dynamic contexts, such as in the simulation of oil reservoirs, involving multiphase flow in a porous medium, and in granular flow.

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This volume will be of interest to applied mathematicians, to researchers in Partial Differential Equations, and to those involved in Fluid Dynamics and Numerical Analysis examining models for viscoelastic flows, porous medium and granular flows, and flows exhibiting phase transitions. As papers in this volume indicate, physical processes whose simplest models may involve change of type occur also in other dynamic contexts, such as in the simulation of oil reservoirs, involving multiphase flow in a porous medium, and in granular flow.
Content:
Front Matter....Pages i-xiv
Multiple Viscous Profile Riemann Solutions in Mixed Elliptic-Hyperbolic Models for Flow in Porous Media....Pages 1-17
On the Loss of Regularity of Shearing Flows of Viscoelastic Fluids....Pages 18-31
Composite Type, Change of Type, and Degeneracy in First Order Systems with Applications to Viscoelastic Flows....Pages 32-46
Numerical Simulation of Inertial Viscoelastic Flow with Change of Type....Pages 47-66
Some Qualitative Properties of 2 ? 2 Systems of Conservation Laws of Mixed Type....Pages 67-78
On the Strict Hyperbolicity of the Buckley-Leverett Equations for Three-Phase Flow....Pages 79-84
Admissibility Criteria and Admissible Weak Solutions of Riemann Problems for Conservation Laws of Mixed Type: A Summary....Pages 85-88
Shocks Near the Sonic Line: A Comparison between Steady and Unsteady Models for Change of Type....Pages 89-106
A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks....Pages 107-125
An Existence and Uniqueness Result for Two Nonstrictly Hyperbolic Systems....Pages 126-138
Overcompressive Shock Waves....Pages 139-145
Quadratic Dynamical Systems Describing Shear Flow of Non-Newtonian Fluids....Pages 146-163
Dynamic Phase Transitions: A Connection Matrix Approach....Pages 164-180
A Well-Posed Boundary Value Problem for Supercritical Flow of Viscoelastic Fluids of Maxwell Type....Pages 181-191
Loss of Hyperbolicity in Yield Vertex Plasticity Models under Nonproportional Loading....Pages 192-217
Undercompressive Shocks in Systems of Conservation Laws....Pages 218-231
Measure Valued Solutions to a Backward-Forward Heat Equation: A Conference Report....Pages 232-242
One-Dimensional Thermomechanical Phase Transitions with Non-Convex Potentials of Ginzburg-Landau Type....Pages 243-257
Admissibility of Solutions to the Riemann Problem for Systems of Mixed Type....Pages 258-284

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This volume will be of interest to applied mathematicians, to researchers in Partial Differential Equations, and to those involved in Fluid Dynamics and Numerical Analysis examining models for viscoelastic flows, porous medium and granular flows, and flows exhibiting phase transitions. As papers in this volume indicate, physical processes whose simplest models may involve change of type occur also in other dynamic contexts, such as in the simulation of oil reservoirs, involving multiphase flow in a porous medium, and in granular flow.
Content:
Front Matter....Pages i-xiv
Multiple Viscous Profile Riemann Solutions in Mixed Elliptic-Hyperbolic Models for Flow in Porous Media....Pages 1-17
On the Loss of Regularity of Shearing Flows of Viscoelastic Fluids....Pages 18-31
Composite Type, Change of Type, and Degeneracy in First Order Systems with Applications to Viscoelastic Flows....Pages 32-46
Numerical Simulation of Inertial Viscoelastic Flow with Change of Type....Pages 47-66
Some Qualitative Properties of 2 ? 2 Systems of Conservation Laws of Mixed Type....Pages 67-78
On the Strict Hyperbolicity of the Buckley-Leverett Equations for Three-Phase Flow....Pages 79-84
Admissibility Criteria and Admissible Weak Solutions of Riemann Problems for Conservation Laws of Mixed Type: A Summary....Pages 85-88
Shocks Near the Sonic Line: A Comparison between Steady and Unsteady Models for Change of Type....Pages 89-106
A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks....Pages 107-125
An Existence and Uniqueness Result for Two Nonstrictly Hyperbolic Systems....Pages 126-138
Overcompressive Shock Waves....Pages 139-145
Quadratic Dynamical Systems Describing Shear Flow of Non-Newtonian Fluids....Pages 146-163
Dynamic Phase Transitions: A Connection Matrix Approach....Pages 164-180
A Well-Posed Boundary Value Problem for Supercritical Flow of Viscoelastic Fluids of Maxwell Type....Pages 181-191
Loss of Hyperbolicity in Yield Vertex Plasticity Models under Nonproportional Loading....Pages 192-217
Undercompressive Shocks in Systems of Conservation Laws....Pages 218-231
Measure Valued Solutions to a Backward-Forward Heat Equation: A Conference Report....Pages 232-242
One-Dimensional Thermomechanical Phase Transitions with Non-Convex Potentials of Ginzburg-Landau Type....Pages 243-257
Admissibility of Solutions to the Riemann Problem for Systems of Mixed Type....Pages 258-284
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