The IMA Volumes in Mathematics

and Its Applications

Volume 27

Series Editors A vner Friedman Willard Miller, Jr.

Institute for Mathematics and its Applications

IMA

The Institute for Mathematics and its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The IMA seeks to encourage the development and study of fresh mathemat­ical concepts and questions of concern to the other sciences by bringing together mathematicians and scientists from diverse fields in an atmosphere that will stim­ulate discussion and collaboration.

The IMA Volumes are intended to involve the broader scientific community in this process.

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Statistical and Continuum Approaches to Phase Transition Mathematical Models for the Economics of

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Barbara Lee Keyfitz Michael Shearer Editors

Nonlinear Evolution Equations That Change Type

With 96 Figures

Springer-Verlag New York Berlin Heidelberg

London Paris Tokyo Hong Kong

Barbara Lee Keyfitz Department of Mathematics University of Houston Houston, Texas 77204 USA

Series Editors A vner Friedman Willard Miller, Jr.

Michael Shearer Department of Mathematics North Carolina State University Raleigh, North Carolina 27695 USA

Institute for Mathematics and Its Applications University of Minnesota Minneapolis, MN 55455 USA

Mathematical Subject Classification Codes: Primary: 35M05, 76AIO, 35L65, 35L67: Secondary: 35K65, 65N99, 73E05, 76A05, 76H05, 76505, 82A25.

Library of Congress Cataloging-in-Publication Data Nonlinear evolution equations that change type / [edited by] Barbara

Lee Keyfitz, Michael Shearer. p. cm. - (The IMA volumes in mathematics and its

applications ; v. Tl) "Based on the proceedings of a workshop which was an integral part

of the 1988-89 IMA program on nonlinear waves''--Foreword. 1. Evolution equations, Nonlinear. 1. Keyfitz, Barbara Lee.

11. Shearer, Michael. 111. Series. QA377.N664 1990 515'.353-dc20 90-9970

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© 1990 Springer-Verlag New York Inc. Softcover reprint ofthe hardcover I st edition 1990

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Volume 27: Nonlinear Evolution Equations that Change Type Editors: Barbara Lee Keyfitz and Michael Shearer

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FOREWORD

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 prob­lems 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 Coordinat­ing 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 Non­linear Waves, the issue of change of type in nonlinear partial differential equations appeared frequently. Discussion began with the January 1989 workshop on Two­Phase 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. As the 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.

There is also considerable recent mathematical work on simple model problems involving systems of conservation laws in space and time that change type. Some of this work addresses the theoretical issues, in particular the loss of linearized well­posedness of initial value problems; but there are interesting numerical problems also. Much of the mathematical work was not previously known to applied math­

ematicians or fluid dynamicists looking at models for specific flows. In addition, recent work on both steady and unsteady models of viscoelasticity has indicated

the importance of composite systems in the study of steady visco-elastic flows, and has exhibited change of type in these steady models; unsteady change of type (change of type in the evolution equations) has even been conjectured to describe

some instabilities in viscoelastic flows. The general theme of the March 1989 \Vork­shop on Evolution Equations that Change Type was the relationship between the analytical and numerical issues posed by equations that change type, and the ap­

plications modelled by these equations.

The papers in these proceedings by Coleman and by Cook, Schleiniger and Weinacht discuss the current status of modelling of viscoelastic fluids, including change of type for both steady and unsteady flows, while the poper of Crochet and Delvaux details how numerical computations can be performed on steady viscoelas­tic flows that change type. This includes adapting the concept of an upwind scheme from transonic flow calculations. Renardy's paper, and that of Malkus, Nohel and Plohr, obtain analytical results which help to compare different models of viscoelas­tic fluids. An explanation of how multiphase flow in porous media leads to con­servation laws that change type can be found in Lars Holden's paper. There are dynamic models for phase transitions which exhibit change of type, and the propa­gation of phase boundaries in equations arising this way is analysed by Mischaikow and by Sprekels. Models of granular flow give rise to linearly ill-posed equations; the paper of Schaeffer and Shearer contains an analytical treatment of change of

type in yield-vertex models of plasticity.

Mathematical background may be found in papers of Keyfitz and of ·Warnecke, which include comparison of classical steady transonic with unsteady models. Math­ematical properties of model equations which exhibit change of type, and constrnc-

xi

tion of solutions, are discussed by Holden, Holden and Risebro, by Hsiao and by Azevedo and Marchesin. Theoretical issues of well-posedness, weak formulations, and admissibility of shock waves arise naturally if one tries to relate the linear ill-posedness of the Cauchy problem to nonlinear considerations, or to formulate correct boundary conditions for equations of mixed type. An approach to this anal­ysis through examples which are not strictly hyperbolic is given in the papers of LeFloch, of Liu and Xin, and of Shearer and Schecter. The example considered by Kranzer and Keyfitz is strictly hyperbolic, but is evidently related to nonstrictly hy­perbolic problems. Well-posedness for a nonlinear model which is linearly ill-posed is described by Slemrod.

One of the conclusions which emerged from the workshop was that at least some dynamic instabilities in viscoelastic flows can be explained by a simpler mechanism than change of type, namely a bifurcation of attractors. However, change of type of the transonic kind, in steady flows, remains of interest in viscoelasticity.

Among promising mathematical approaches which were displayed at the work­shop, Riemann problems played a prominent role in many of the talks, with new phenomena, loss of uniqueness of solutions, and constructive solutions being dis­cussed in detail. One classic result on equations of mixed type, Friedrichs' 1958 theory of symmetric positive systems, emerged as a potential tool to discuss well­posedness of boundary-value problems.

New uses for the qualitative theory of planar dynamical systems appear in the work of Liu and Xin, of Malkus, Nohel, and Plohr, of Azevedo and Marchesin, of Shearer and Schecter and of Keyfitz; higher-dimensional vectorfields appear in the papers of Kranzer and Keyfitz and of Mischaikow.

As organizers of the workshop and editors of the proceedings, we extend a special word of thanks to Dan Joseph, whose papers on loss of hyperbolicity in viscoelastic models provided an important link between specialists in viscoelasticity and participants working in other areas related to equations that change type. In addition to introducing the participants to each other and organizing a lab tour, Dan presented a summary of Fraenkel's work on change of type in steady flow.

We are also pleased to thank Avner Friedman and \Villard Miller, Jr. and the IMA staff for their smooth organization of the details of the workshop and the visits of the participants. Finally, we thank all the participants in this vol­ume, who submitted their papers so promptly, and we thank the editorial staff of Patricia V. Brick, Stephan Skogerboe, Kaye Smith and Marise Ann Widmer who completed the manuscript preparation.

Barbara Lee Keyfitz

Michael Shearer

xii

CONTENTS

Foreword ....................................................... ix

Preface ......................................................... xi Multiple viscous profile Riemann solutions in mixed elliptic-hyperbolic models for flow in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1

A. V. Azevedo and D. Marche8in On the loss of regularity of shearing flows of viscoelastic fluids ................................................ 18

Bernard D. Coleman

Composite type, change of type, and degeneracy in first order systems with applications to viscoelastic flows ................................................ 32

L. Pamela Cook, G. Schleiniger and R.I. Weinacht

N uinerical simulation of inertial viscoelastic flow with change of type ............................................. 47

M.I. Crochet and V. Delvaux

Some qualitative properties of 2 x 2 systems of conservation laws of mixed type ................................. 67

H. Holden, L. Holden and N.H. Ri8ebro

On the strict hyperbolicity of the Buckley-Leverett equations for three-phase flow ................................... 79

Lar8 Holden

Admissibility criteria and admissible weak solutions of Riemann problems for conservation laws of mixed type: a summary . . . . . .. 85

L. Hsiao

Shocks near the sonic line: a comparison between steady and unsteady models for change of type .................. 89

Barbara Lee K eyfitz

A strictly hyperbolic system of conservation laws admitting singular shocks .................................. 107

Herbert C. Kranzer and Barbara Lee Keyfitz

An existence and uniqueness result for two nonstrictly hyperbolic systems ................................... 126

Philippe Le Floch

Overcompressive shock waves .................................... 139 Tai-Ping Liu and Zhouping Xin

Quadratic dynamical systems describing shear flow of non-Newtonian fluids .............................. 146

D.S. Malku8, I.A. Nohel and B.I. Plohr

xiii

Dynamic phase transitions: a connection matrix approach ................................................ 164

Konstantin Mischaikow

A well-posed boundary value problem for supercritical flow of viscoelaStic fluids of Maxwell type ........................................... 181

Michael Renardy

Loss of hyperbolicity in yield vet'tex plasticity models under nonproportionalloading .................................. 192

David G. Schaeffer and Michael Shearer

Undercompressive shocks in systems of conservation laws ............................................... 218

Michael Shearer and Stephen Schecter

Measure valued solutions to a backward-forward heat equation: a conference report ............................... 232

M. Slemrod

One-dimensional thermomechanical phase transitions with non-convex potentials of Ginzburg-Landau type ............ 243

Jurgen Sprekels

Admissibility of solutions to the Riemann problem for systems of mixed type -transonic small disturbance theory- ............................. 258

Gerald Warnecke

xiv