Volterra and Functional Differential Equations

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volterra and functional differential equations Kenneth B. Hannsgen Terry L. Herdman Harlan W. Stech Robert L. Wheeler

Volterra and Functional Differential Equations

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LECTURE NOTES IN PURE AND APPLIED MA THE MA TICS I. 2. 3. 4.

5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

N. Jacobson, Exceptional Lie Algebras L. -A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis I. Sa take, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch, W. D. Newmann, and S. S. Koh, Differentiable Manifolds and Quadratic Forms (out of print) I. Chavel, Riemannian Symmetric Spaces of Rank One (out of print) R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald, A. R. Magid, and K. C. Smith, Ring Theory: Proceedings of the Oklahoma Conference Y.-T. Siu, Techniques of Extension of Analytic Objects S. R. Caradus, W. E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces E. 0. Roxin, P.-T. Liu, and R. L. Sternberg, Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thomeier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups: An Introduction G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis: Proceedings of the Brazilian Mathematical Society Symposium L. Cesari, R. Kannan, and J. D. Schuur, N onlinear Functional Analysis and Differential Equations: Proceedings of the Michigan State University Conference J. J. Schaffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Frank/in and B. V. S. Thomas, Topology: Proceedings of the Memphis State University Conference S. K. lain, Ring Theory: Proceedings of the Ohio University Conference B. R. McDonald and R. A. Morris, Ring Theory II: Proceedings of the Second Oklahoma Conference R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems: Theory and Applications H.-C. Wang, Homogeneous Banach Algebras E. 0. Roxin, P.-T. Liu, and R. L. Sternberg, Differential Games and Control Theory Il R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Go/an, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis: Proceedings of the S.U.N.Y. Brockport Conference R. Cordon, Representation Theory of Algebras: Proceedings of the Philadelphia Conference M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda, N. C. A. da Costa, and R. Chuaqui, Mathematical Logic: Proceedings of the First Brazilian Conference

40. F. Van Oystaeyen, Ring Theory: Proceedings of the 1977 Antwerp Conference 41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization: Application to Sheaf Theory 42. M. Satyanarayana, Positively Ordered Semigroups 43. D. L. Russell, Mathematics of Finite-Dimensional Control Systems 44. P.-T. Liu and E. Roxin, Differential Games and Control Theory III: Proceedings of the Third Kingston Conference, Part A 45. A. Geramita and J. Se berry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices 46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics: Proceedings of the Third Kingston Conference, Part B 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oystaeyen, Ring Theory: Proceedings of the 1978 Antwerp Conference 52. B. Kedem, Binary Time Series 53. J. Barros-Neto and R. A. Artino, Hypoelliptic B'oundary-Value Problems 54. R. L. Sternberg, A. J. Kalinowski, and J. S. Papadakis, Nonlinear Partial Differential Equations in Engineering and Applied Science 55. B. R. McDonald, Ring Theory and Algebra Ill: Proceedings of The Third Oklahoma Conference 56. J. S. Golan, Structure Sheaves over a Noncommutative Ring 57. T. V. Narayana, J. G. Williams, and R. M. Mathsen, Combinatorics, Representation Theory and Statistical Methods in Groups: YOUNG DAY Proceedings 58. T. A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. 0. A. Nielsen, Direct Integral Theory 62. J. E. Smith, G. 0. Kenny, and R. N. Ball, Ordered Groups: Proceedings of the Boise State Conference 63. J. Cronin, Mathematics of Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. T. G. McLaughlin, Regressive Sets and the Theory of Isols 67. T. L. Herdman, S. M. Rankin, Ill, and H. W. Stech, Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra: Analytic Methods 69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Gee/, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz, C. Praeger, and R. Sullivan, Algebraic Structures and Applications: Proceedings of the First Western Australian Conference on Algebra 75. L. Bican, T. Kepka, and P. Nemec, Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry: Proceedings of the Second University of Oklahoma Conference

77. 78. 79. 80. 81.

P. F/etcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions 0. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen, T. L. Herdman, H. W. Stech and R. L. Wheeler, Volterra and Functional Differential Equations Other Volumes m Preparation

Volterra and Functional Differential Equations edited by

KENNETH TERRY

L.

B.

HANNSGEN

HERDMAN

HARLAN W. STECH RDBERT

L.

WHEELER

Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 g'j 1982 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an lnforma business

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Library of Congress Cataloging in Publication Data Main entry under title: Volterra and functional differential equations. (Lecture notes in pure and applied mathematics v. 81) Proceedings of the Conference on Volterra and Functional Differential Equations, held June 10-13, 1981, at Virginia Polytechnic Institute and State University in Blacksburg, Virginia. 1. Volterra equations--Congresses. 2. Functional differential equations--Congresses. I. Hannsgen, 11. Conference on Volterra Kenneth B., [date]. and Functional Differential Equations (1981 : Virginia Polytechnic Institute and State University) QA43l.V63 1982 515.4'5 82-13998 ISBN 0-8247-l72l-X Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRe Press Web site at http://www.crcpress.com

DOI: 10.1201/9781003420026

PREFACE

The Conference on Volterra and Functional Differential Equations was held June 10 - 13, 1981 at Virginia Polytechnic Institute and State University in Blacksburg, Virginia. Over sixty - five participants, including mathematicians from thirty - three academic institutions and research laboratories, attended lectures and informal sessions in McBryde Hall on the Virginia Tech campus. Kenneth B. Hannsgen, Terry L. Herdman and Robert L. Wheeler constituted the Organizing Committee; John A. Burns and.Harlan W. Stech served as advisors. This volume represents the contents of the twenty - four invited lectures presented at the conference. Papers based on the eleven one - hour lectures appear in Part One. Part Two contains the thirteen half - hour lectures. The conference was made possible by the National Science Foundation through grant No. MCS - 8023198. In addition, the editors gratefully acknowledge the assistance and support of many individuals and organizations. The Virginia Tech Department of Mathematics and its Head, C. Wayne Patty, generously supplied facilities and personnel both for the administration of the conference and the preparation of this proceedings; the Donaldson Brown Center for Continuing Education handled the housing arrangements; Susan Anderson, Robert Powers and Leslie Ratliff contributed in the day - to - day running of the conference; and a considerable amount of the necessary paperwork was managed with the invaluable assistance of Sharon Irvin and Debbie Wardinski. Finally, with special thanks, we acknowledge Cynthia Duncan for her skillful preparation of the final manuscript for this volume. Kenneth B. Hannsgen Terry L. Herdman Harlan W. Stech Robert L. Wheeler iii

CONTENTS

Preface

iii

Contributors

vii

Conference Participants

X

PART ONE A Survey of Some Problems and Recent Results for Parameter Estimation and Optimal Control in Delay and Distributed Parameter Systems H. Thomas Banks

3

A Model Riemann Problem for Volterra Equations James M. Greenberg, Ling Hsiao and Richard C. MacCamy

25

An Example of Boundary Layer in Delay Equations Jack K. HaZe and Luis MagaZhaes

45

Some Results on the Liapunov Stability of Functional Equations Ettore F. Infante

51

Some Synthesis Problems for Integral Equations Jacob J. Levin

61

On Volterra Equations with Locally Finite Measures and L00 - Perturbations Stig-OZof Londen

79

iv

Contents

V

A Nonlinear Conservation Law with Memory

John A. Nohel

91

Functional Equations as Control Canonical Forms for Distributed Parameter Control Systems and a State Space Theory for Certain Differential Equations of Infinite Order

125

Exponential Dichotomies in Evolutionary Equations: A Preliminary Report

167

On Weighted Measures and a Neutral Functional Differential Equation

175

David L. Russell

Robert J. Sacker and George R. Sell Olof J. Staffans

PART

mo

A Parameter Dependence Problem in Functional Differential Equations

187

Approximation of Functional Differential Equations by Differential Systems

197

Integrable Resolvent Operators for an Abstract Volterra Equation

207

Asymptotic Expansions of a Penalty Method for Computing a Regulator Problem Governed by Volterra Equations

217

Integrodifferential Equations with Almost Periodic Solutions

233

Dennis W. Brewer

Stavros N. Busenberg and Cza,tis C. Travis

Ralph W. Carr

Goong Chen and Ronald Grimmer

Constantin Corduneanu 1

Weighted L -Spaces and Resolvents of Volterra Equations

245

A Volterra Equation with Time Dependent Operator

255

Formation of Singularities for a Conservation Law with Damping Term

263

On Obtaining Ultimate Boundedness for a-Contractions

273

G. Sarrruel Jordan

Thomas R. Kiffe

Reza Malek-Madani Paul Massatt

Contents

vi Cosine Families, Product Spaces and Initial Value Problems with Nonlocal Boundary Conditions

281

Theory and Numerics of a Quasilinear Parabolic Equation in Rheology

287

On Periodic Solutions in Systems of High Order Differential Equations

311

Differentiability Properties of Pseudoparabolic Point Control Problems

323

Samuel M. Rankin, III

Michael Renardy

Sherwin J. Skar, Richard K. Miller and Anthony N. Michel Luther W. White

CONTRIBUTORS

H. THOMAS BANKS, Division of Applied Mathematics, Brown University, Providence, Rhode Island. DENNIS W. BREWER, Department of Mathematics, University of Arkansas, Fayetteville, Arkansas. STAVROS N. BUSENBERG, Department of Mathematics, Harvey Mudd College, Claremont, California. RALPH W. CARR, Department of Mathematics, St. Cloud State University, St. Cloud, Minnesota. GOONG CHEN, Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania. CONSTANTIN CORDUNEANU, Department of Mathematics, University of Texas at Arlington, Arlington, Texas. JM4ES M. GREENBERG, Division of Mathematical and Computer Sciences, National Science Foundation, Washington, D.C. RONALD GRIMMER, Department of Mathematics, Southern Carbondale, Illinois.

University,

JACK K. HALE, Division of Applied Mathematics, Brown University, Providence, Rhode Island. LING HSIAO, Division of Applied Mathematics, Brown University, Providence, Rhode Island.

vif

Contributors

viii

ETTORE F. INFANTE*, Division of Applied Mathematics, Brown University, Providence, Rhode Island. G. SAMUEL JORDAN, Department of Mathematics, University of Tennessee, Knoxville, Tennessee. THOMAS R. KIFFE, Department of Mathematics, Texas A Station, Texas.

&M University,

College

JACOB J. LEVIN, Department of Mathematics, University of Wisconsin, Madison, Wisconsin. STIG-OLOF LONDEN, Institute of Mathematics, Helsinki University of Technology, Otaniemi, Finland. RICHARD C. MacCAMY, Department of Mathematics, Carnegie - Mellon University, Pittsburgh, Pennsylvania. LUIS MAGALHAES, Division of Applied Mathematics, Brown University, Providence, Rhode Island. REZA MALEK-MADANit, Department of Mathematics, University of Wisconsin, Madison, Wisconsin. PAUL MASSATT, Department of Mathematics, University of Oklahoma, Norman, Oklahoma. ANTHONY N. MICHEL, Department of Electrical Engineering, Iowa State University, Ames, Iowa. RICHARD K. MILLER, Department of Mathematics, Iowa State University, Ames, Iowa. JOHN A. NOHEL, Mathematics Research Center, University of Wisconsin - Madison, Madison, Wisconsin. M. RANKIN, III, Department of Mathematics, West Virginia University, Morgantown, West Virginia. MICHAEL RENARDY, Mathematics Research Center, University of Wisconsin - Madison, Madison, Wisconsin. DAVID L. RUSSELL, Department of Mathematics, University of Wisconsin, Madison, Wisconsin. ROBERT J. SACKER, Department of Mathematics, University of Southern California, Los Angeles, California. GEORGE R. SELL, School of Mathematics, University of Minnesota, Minneapolis, Minnesota. *Current affiliation: Division of Mathematical and Computer Sciences, National Science Foundation, Washington, D.C. tCurrent affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia.

Contributors

ix

SHERWIN J. SKAR, Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma. OLOF J. STAFFANS, Institute of Mathematics, Helsinki University of Technology, Otaniemi, Finland. CURTIS C. TRAVIS, Oak Ridge National Laboratory, Health and Safety Research Division, Oak Ridge, Tennessee. LUTHER W. WHITE, Department of Mathematics, University of Oklahoma, Norman, Oklahoma.

CONFERENCE PARTICIPANTS

Joseph Ball Virginia Tech

Ralph W. Carr St. Cloud State University

H. Thomas Banks Brown University

Goong Chen Pennsylvania State University

John B. Bennett Arkansas State University

Eugene M. Cliff Virginia Tech

Steve Bins Clemson University

Kenneth L. Cooke Pomona College

Steven Black Clemson University

Constantin Corduneanu University of Texas at Arlington

Dennis Brewer University of Arkansas

Martin Day Virginia Tech

John A. Burns Virginia Tech

Rodney Driver University of Rhode Island

Theodore A. Burton Southern Illinois University

Robert Fennell Clemson University

Stavros Busenberg Harvey Mudd College

William Grasman Virginia Tech

Dean A. Carlson University of Delaware

David Green, Jr. General Motors Institute X

xi

Participants James M. Greenberg National Science Foundation

Robert Martin North Carolina State University

George Hagedorn Virginia Tech

Paul Massatt University of Oklahoma

Jack K. Hale Brown University

Jim Mosely West Virginia University

Kenneth B. Hannsgen Virginia Tech

Beny Neta Texas Tech University

Terry L. Herdman Virginia Tech

John A. Nohel University of Wisconsin

Herbert Hethcote University of Iowa

Mary E. Parrott University of South Florida

Ettore F. Infante Brown University

Raymond Plaut Virginia Tech

Harry Johnson Virginia Tech

Carl Prather Virginia Tech

G. Samuel Jordan University of Tennessee

T. Gilbert Proctor Clemson University

Thomas Kiffe Texas A &M University

Samuel M. Rankin, Ill West Virginia University

Werner Kohler Virginia Tech

Russell M. Reid University of Missouri

Jacob J. Levin University of Wisconsin

Michael Renardy University of Wisconsin

James Lightbourne, Ill West Virginia University

Charles Rennolet Rose - Hulman Institute of Technology

Chih-Bing Ling Virginia Tech

Dave Reynolds Carnegie - Mellon University

Stig-Olof Londen Helsinki University of Technology

David L. Russell University of Wisconsin

Michael Lyons North Carolina State University

Stephen Saperstone George Mason University

Richard C. MacCamy Carnegie - Mellon University

George R. Sell University of Minnesota

Joe Mahaffy North Carolina State

J. Kenneth Shaw Virginia Tech

Reza Malek-Madani University of Wisconsin

Henry C. Simpson University of Tennessee

Participants

xii Sherwin J. Skar Oklahoma State University

Robert L. Wheeler Virginia Tech

Olof J. Staffans Helsinki University of Technology

Luther White University of Oklahoma

Harlan W. Stech Virginia Tech

Robert White Bell Laboratories

Kenneth E. Swick Queens College

Donald F. Young Agnes Scott College

Curtis C. Travis Oak Ridge National Laboratory

PART ONE

A SURVEY OF PROBLEMS AND RECENT RESULTS FOR PARAMETER ESTIMATION AND OPTmAL CONTROL IN DELAY AND DISTRIBUTED PARAMETER SYSTEMS H. T. Banks

Lefschetz Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, Rhode Island

1.

INTRODUCTION

In this lecture we shall first present a brief account of several areas of applications which have motivated our recent efforts, both theoretical and numerical, on approximation methods for estimation and control of infinite dimensional systems.

We then shall sketch the general theoretical ideas we

have employed to establish convergence results for related iterative schemes. Finally we return to two of the applications and illustrate the use of these ideas by explaining in more detail our investigations for these problems. As we shall make clear, our efforts on many of the problems mentioned below involve joint endeavors with colleagues and students.

In addition to a well-

deserved thank you to Richard Ambrasino, James Crowley, Patti Daniel, Mary Garrett, Karl Kunisch, and Gary Rosen, we would also like to publicly acknowledge E. Armstrong (NASA Langley Research Center), R. Ewing and G. Moeckel (Mobil Research and Development Corp.), P. Kareiva (Brown University), J. P. Kernevez (Universite de Technologie de Compiegne), W. T. Kyner (University of New Mexico), and G. A. Rosenberg (V. A. Medical Center, U. N. M. School of Medicine) for numerous stimulating discussions and suggestions which have substantially affected the investigations of our group at Brown University.

3

DOI: 10.1201/9781003420026-2

Banks

4

Our discussions here focus on a general class of systems including non­ linear delay systems X(t)

= f(a,t,X(t),Xt,X(t-T 1), ... ,X(t-TV))

x(9)

0, 0 q1 (The smoothness hypothesized will guarantee certain smoothness properties for the eigenfunctions to be discussed momentarily.) We rewrite (9) as an equation z(t) = A(q)z(t) + F(q,f)

(10)

in the state space Z = X(q) = 12 (0,1) where we take as inner product = (Here the spaces do depend on the unknown parameters, a complicating possibility we mentioned earlier.) The operators in (10) are given by F = f and A(q)w = D(q 2Dw), where Dom(A(q)) = H2 n ql ql Simple integration by parts yields 0 so that A(q) is uniformly dissipative in X(q). ates a

c0 -semigroup,

q

In fact A(q) is maximal dissipative and gener-

and we are thus in a position to consider (6), (7) and

the Trotter - Kato approach to approximation schemes. To describe the spline methods we need to recall the definition of some standard cubic spline basis elements.

For any positive integer N, we let

N .· -N t. = j/N, j = -3, ... ,N+3, and let B., j = -l, ... ,N+l, be the cubic spline

J N N J N that vanishes outside (t. 2 ,t. 2), has value 4 and slope 0 at t., value 1 N JJ+ N J and slope 3N at t. 1 , and value 1 and slope -3N at t. 1 . (See [25], p. 73 JJ+ and note that our elements here differ from those of Schultz only by a

multiplicative factor of 24.)

Banks

20 For our modified basis elements the following: BN 0

-N Bo

4BN -1

BN 1

-N Bo

4BN 1

J

-N Bj'

j = 2, ... ,N-2

J

we take the restriction to [0,1] of

-N -N N 8N-1 = BN - 4BN-1 -N -N BN 4BN+l" N BN We note that these elements are in Dom(A(q)). We define our approximation subspaces XN(q) ...

c

X(q) by XN(q) =

and let PN(q) be the canonical projection of X(q) onto XN(q), i.e., =

I

j=O

q J

Finally, as usual, we take AN= AN(q) = PN(q)A(q)PN(q).

Under an assumption that Q is compact in H0

x

H0 , one can argue in this

case that solutions to the estimation problems for (5) (or (7)) do exist. We in fact assume that Q is compact in the C

1

x H

topology so that we hence-

forth assume without loss of generality (possibly by taking a subsequence) that we have a sequence {qN} of solutions to the estimation problems satisfying qN

+

q- in C x H1 with q- s Q.

We briefly indicate the steps to verify (Si) - (Siii) to ensure convergence of the semigroups generated by AN(qN) to the semigroup generated by A(q).

(As we have noted before, this is the fundamental convergence result

needed for both state and parameter convergence.)

The stability requirement

(Si) follows from

N N

N q

N N N N N N q

0 as t +

but one must have a(t)

+

00 •

For (H) condition (1.1) is still reasonable

0 as t

+

oo.

Interest in the models (V) and (H) derives from the fact that in the no memory case, a(t)

= a(O),

there can be global smooth solutions only for

A Model Riemann Problem

27

very special (and physically unrealistic) initial data.

The general situation

is that second derivatives of solutions become infinite in finite time ([1], [5] and [8]).

In contrast, (V) and (H) will have global smooth solutions,

under hypothesis (1.1), provided the initial data are small [7].

It is con-

jectured, but not quite proved, that if the data become too large, solutions

.

of (V) and (H) will also break down.

(See [10] for a related result.)

For the no memory case of (V) and (H) there has been developed a theory of generalized solutions of shock type, see [2] and [6].

A central feature

of this theory is the treatment of the Riemann problem which consists of initial data which are piecewise constant with a single jump. Many of the ideas of the no memory version of (V) and (H) are modeled by the corresponding no memory version of the problem studied in this paper; see [6].

Our object here is to continue the analogy to the equation with

memory. We list the hypotheses under which we operate.

The function f is to

satisfy f'

>

o,

f"(s) >

o,

f(O)

0.

(F)

We suppose the initial data fall into one of two categories: (C)

(x,T) _ 0,

(J)

which we term the continuous and jump situations. The conditions on a are taken from [7] and are: 0,1,2, a(t)

a 00

and a 00 > 0

+

A(t), or

We denote the cases a

a 00

CO

where A,A

E

L1 (0,oo)

0. > 0 and a CO

0 by (E) and (H) to conform with the

analogy to elasticity and heat flow in the introduction.

Greenberg et al.

28

In order that (C) make sense under condition (A 1 ), we require that the initial history satisfy L (0,oo)

€

for each x.

00

We also need a technical hypothesis on a. Volterra operator

f

L· [r;] (t)

a

t

0

Let L· denote the linear a

a(t-T)s(T)dT.

For convenience we assume a(O)

(1. 2)

1.

Then the operator I + La_ has an inverse

of the form

(I+L·) a

-1

(1. 3)

where k is the resolvent of -a. k(t) :> 0,

k(t)

:

o.

Under condition (A 2 ) a more complete analysis of k is possible by means of Laplace transforms as described in [7]. One result from [7] is this. Assuming that not only are A and A in L1 but that a certain number of their moments are finite, one can show that k has the form k(t)

koo

+ K,

K,K

€

L1 (O,oo) (1.4)

if

a

00

0

if

a

00

> 0.

A

2.

29

1 Ri emann Prob 1em

SMOOTH SOLUTIONS

We consider first solutions in which class C(l) on t u

t

+

(j

>

0

X

0.

and u 0 are continuous and u is of In this case (B) implies the local balance law

on

t

> 0.

(LB)

I f we put

f

1>(x, t)

(2 .1)

0

then our problem is equivalent to u

t

+ (I+L·)[f(u) ] a

x

-

x'

u (x, o+)

(P)

uo(x).

The corresponding no memory problem is 0,

u(x,o+)

We recall the well known results for (P 0) (see [6]). THEOREM 1. (ii) (iii)

(P 0 ) has a unique solution locally in t.

(i)

0 u 0(x 0)

If u (x) If

0, then (P 0) has a unique solution globally in t. < 0 for some x 0 , then there exists a time T < oo, and

an x 1 such that the local solution exists fort< T but ut(x,t)-+

oo

as

(x,t)-+ (x 1 ,T). We indicate now a result for the memory case which illustrates our remarks in the introduction. THEOREM 2. (ii) (iii)

(i)

0

(P) has a unique solution locally in t.

If u (x)

0, then (P) - (J) has a unique solution globally in t.

Suppose f satisfies the condition

. f" (u) llm f' (u) u-+0

y,

O 0 and satisfies (I) at points of continuity of u 0 . are admissible.

*x(s+(t),t)

We say it is admissible if all shocks

lim x(x,t) and xcs-(t),t)- lim x(x,t). x+s (t) x+s (t) x>s(t) x

0.

The essential thing to note here is the presence of the term o:u 0 . This is the product of the memory effect and, since u 0 is discontinuous, we can expect it to have an influence on the nature of the solution. We proceed to analyze this problem. selves to the situation where

and

For simplicity we restrict ourare both positive.

A Model Riemann Problem

35

We make two preliminary observations. (4.2) is u(x,t) =

First, for x < 0 a solution of

If the right side remained equal to

as we cross

x = 0, this would continue to be the solution for x
f'(ur)t (just as in the no memory case). the region Q

= ur u(x,t) = ur

Second, a solution for x > f'(ur)t is u(x,t)

= {0

$

x

$

f'(ur)t}.

have to distinguish cases:


u .

Our first clue as to how to fill in Q comes from the proposition of Section 3 which tells us that x = 0 cannot be a contact discontinuity; that is, u must be continuous across x = 0. aur,

x

>

Hence we have

0

(4.3) u(O,t)

u .

The problem (4.3) can be solved by characteristics.

Define X(t,t), U(t,t)

by dX dt dU dt

f' (U),

+

r

aU

au ,

X(t, t)

0

U(t, t)

u ,·

This yields

u (t, t)

( 4. 4)

(ur +

X(t, t) t

Then to obtain u one solves x = X(t,t) fort= T(x,t) and u(x,t) = U(t,T(x,t)). It is not too difficult to verify that the above solution has the form r

u(x,t) = u(x;u ,u ), where = X(t,OJ =

r

0 < x < x(t;u iu ),

f

t 0

r

f'(ur + (u -u )e

-en

(4.5)

)dT

(4.6)

Greenberg et al.

36 !i.

r

and u(x;u ,u ) is the solution of !1,

u.

( 4. 7)

The solution of (4.7) is determined from u

f' (I;) di; u -i; u!i. r

I I

u!i.

u

en

f' ur-1;

ax

u!i. < u r

if

( 4. 8)

dr;

-ax

u!i. > u. r

if

We need a second ingredient when u

!1,




0 shows that

X < 0

u(x;u ,0),

x

>

0

oo; that is, u tends to a steady state.

For Example (II) such steady solutions do not exist.

What is possible

are traveling waves (something not present in Example (I)). A

A

(4.17) support solutions of the form u(s), er(s), where s

The equations

= x-

A

c(cjl )t, satis-

fying the limit relations lim u(s) = cjl , A

A

lim u(s)

0.

(4.19)

41

A Model Riemann Problem These traveling waves are determined as follows: ]Jf(q/')

_O

This last possibility is, however, ex-

cluded by assumption. There are two serious drawbacks in Theorem 2.

The first is the moment

condition on the variation of r'(t) which, in general, is a very difficult hypothesis to check.

It is satisfied if

= a(t)dt with (-l)ka(k)(t)

? 0

fork= 0,1,2,3; but apart from this particular example it is hard to find classes of locally finite measures which do satisfy this assumption. second drawback is the requirement that /0Cw)/ be finite for w excludes some typical positive definite measures like

The

0 which

= t-acos t dt,

0 < a < 1. Note that in applications the moment condition frequently implies that is finite for w

0.

This is seen as follows.

measure generated by the variation of r' (t). From the moment condition it follows that v(w)

Let -v denote the finite

Thus v(w) £

=-I

R+

e-iwtdr'(t).

C1 (R) and, in particular,

that

Iiw

Re v(w)

+

A

,-2 w2Re

A

£

C1 (R).

(8)

0 and

Consequently, if ((9) does hold in most applications) 1im sup /w-w 0 /+0

/

A

,-2, w-w 0 ,-1 Re

A

> 0,

(9)

then a contradiction follows and /0Cw 0 ) / < oo. To realize why /0(w)/ < oo, w 0, and the moment condition are crucial to the proof, it suffices to at first observe that the proof of Theorem 1 basically consists of at first transforming (1) to an equation of the same type as (1) but with a finite measure, namely to

84

Londen x 1 (t)

+

J

[ 0' t

h(x(t-s))dv(s)

Z

I

l

(10)

(t),

In (10), h(x)

and then of applying previously known results [2,9]. and, by (8), vis a positive definite measure.

= g(x)

- x

The existing asymptotic re-

sults on (10) (assuming v positive definite, finite; h s C(R), z 1 (oo)

=

0)

do, however, require the imaginary part of the transform to vanish on the set where the real part vanishes.

To check this requirement one notices at first

that {wiRe

(11)

0}

where o}, Obviously v(w) = 0 for wE zl.

z2

= {w f

ol 10Cw)l

= oo}.

(12)

But for wE z2 we have v(w) = iw f 0 and

consequently we are led to the assumption that

z2

be empty.

Secondly, one observes that the existing asymptotic results on (10) do, in case 0 E {wiRe v(w)

= 0}

which now holds, require the measure to be not

only finite but also to satisfy a first moment condition. So far it has not been possible to remove the assumption I0Cw)l < oo, w f 0, and the moment condition without adding some other hypotheses.

What

we do have., however, is the following: THEOREM 3, [4,5]. xg(x)

>

0,

Let g(x) be locally Lipschitzian and assume x f 0;

and

lim inf x lxi+O

-1

g(x)

> 0,

(13)

Im Furthermore, suppose that the solution r(t) of (7) satisfies r

1

s (L 1 n NBV)(R+)

and that lim z 1 (t) t-roo Finally, let y _

0,

lim z(t) t-roo

lim lwi+O

exists and is finite.

]-l be finite.

Then lim t-roo

X

1

(t)

0 and, in

Volterra Equations

85

addition, lim {x(t)

t-+oo

+

y- 1g(x(t))}

lim

f(s)

s->-0+

(14)

y > 0

if

s real lim

lim g(x(t))

t-+oo

s->-0+

{s[0(s) l- 1 f(s)}

if

y

(15)

0.

s real

Let us make a few comments on Theorem 3. Note at first that the assumption r' s L1 (R+) and the positive definiteness of v yield that y exists and is nonnegative. lent to r(oo) for r(oo)




0.

Define, for any positive constant K, YK

= {YIY

LAC(R),

IIYII

y satisfies (16) on R,

Then sup ysYK

11

g 1 (y(t))

11

2

L (R)


0. (Z 3 := z2 u {0}.) We may now formulate 1

THEOREM 4, [5]. lim g(x(t))

=

+

A

A

Let (22) hold and assume that

z(oo),

Thus the points of

z1

is empty.

Then

z2 ,

if alone (except for {0}) on the right side of

nor

z2

(21), do not create any oscillatory behavior at infinity. In case neither surprising result: THEOREM 5.

;'(w)

z1

is empty we obtain the following somewhat

Let (22) hold and assume that is Holder continuous with exponent 1/2 on (23)

Then o(r(x(t)))

c

{0} u

z2 ,

cr(r(g(x(t))))

c

{0} u

z1 .

(24)

88

Londen

The condition (23) is needed as the proof makes use of a result by Pollard [6].

Requiring Holder continuity on a neighbornood of

z3

course, be unrealistic and (23) does indeed exclude a vicinity of

would, of

z3

from

the set where Holder continuity is to hold. How to proceed beyond (24) without making additional assumptions on is at this moment an open problem.

z2

It appears that with appropriately cho-

z1 , z2 and g one may have an oscillatory behavior at infinity. Note, however, that if the points of z2 are isolated and the technical condition

sen

(25) is satisfied, then the conclusion (24) may be strengthened by using [8, p. 252]. THEOREM 6.

In fact, we obtain Let (22) hold and suppose that

Furthermore, for each w2

z2 ,

E

Re 0Cw)

lim

-------':--'--''-'----=-

Ill (w) 12 (w-w 2 ) 2

E-i-0

z2

contains only isolated points.

let

dw

(25)

A

Then cr(r(h(x(t))))

c

{O} u

z1 .

If in addition (23) is satisfied, then lim g(x(t)) t-+co

z(oo).

Note that (25) holds in most cases arising in applications. AC KNOl4L EDGEMENT

This research was sponsored by the United States Army under Contract No. DAAG29-80-C-0041. REFERENCES

1.

G. Gripenberg, On nonlinear Volterra equations with nonintegrable kernels, SIAM J. Math. Anal. 11 (1980), 668-682.

2.

S.-0. Londen, On a Volterra integrodifferential equation with Looperturbation and noncountable zero-set of the transformed kernel, J. Integral Eqs. 1 (1979), 275-280.

3.

S.-0. Londen, On an integral equation with L00 -perturbation, J. Integral Eqs., to appear.

Volterra Equations

89

4.

S.-0. Londen, On some integral equations with locally finite measures and L00 -perturbations, MRC Technical Summary Report #2224, Mathematics Research Center, University of Wisconsin-Madison, Madison, Wisconsin, 1981.

5.

S.-0. Londen, manuscript in preparation.

6.

H. Pollard, The harmonic analysis of bounded functions, Duke Math. J. 20 (1953), 499-512.

7.

0. J. Staffans, Positive definite measures with applications to a Volterra equation, Trans. Amer. Math. Soc. 218 (1976), 219-237.

8.

0. J. Staffans, Tauberian theorems for a positive definite form, with applications to a Volterra equation, Trans. Amer. Math. Soc. 218 (1976), 239-259.

9.

0. J. Staffans, On a nonlinear integral equation with a nonintegrable perturbation, J. Integral Eqs. 1 (1979), 291-307.

A NONLINEAR CONSERVATION LAW WITH MEMORY J. A. Nohel Mathematics Research Center University of Wisconsin - Madison Madison, Wisconsin

1.

INTRODUCTION

In this paper we study the model nonlinear Volterra functional differential equation (with infinite memory)

ut+

where

+

lR

+

t

f(t,x),

-oo 0.

(2. 6)

Assumption (2.3), which requires that a(t) - a exp(-t) be a positive definite kernel on [O,oo) for some a > 0, expresses the dissipative character of viscosity.

.a(t)

Smooth, integrable, nonincreasing, convex relaxation functions, e.g., K

L

vk

k=l

>

0,

>

(2. 7)

0,

which are commonly employed in the applications of the theory of viscoelasticity, satisfy (2.3). We now consider a homogeneous, one-dimensional body (string or bar) with reference configuration [0,1) of density p

=1

(for simplicity) and

constitutive relation (2.1), which is moving under the action of an assigned body force g(t,x), -

00




0

The proof of Lemma 4.2 is given in [5]. The first estimate needed for the proof that (4.8) implies (4.9) is obtained from equation (3.5). (Recali that problem (3.5), (3.2), (3.3) is equivalent to the original problem (3.1)- (3.3), and it is assumption (3.4) concerning x which will play an important role.) Multiply (3.5) by w(u)xt and integrate over (-oo,s] x [0,1], s < T0 . After several integrations by parts in which the boundary condition is invoked, we obtain

1 1 2 f x'(u(s,x))w'(u(s,x))u (s,x)dx

2

0

X

1

- - -2

s

1

f f 0 [x' (u)w"(u) _ 00

+

+

1

f 0Q[a,w(u) 2

t;s]dx

X

x"(u)W' (u) ]u u dxdt t X

( 4 .15)

Nohel

108 1

1 2 1)J"(u)uxutdxdt -oo 0

s

-2f f s

1

-00

0

-f f

f

+

1 0

f(s,x)1jJ(u(s,x)) dx X

ft1jJ(u)xdxdt.

In contrast to the analogous calculation in [5, see (3.21)], no useful information is extracted here from integration of the term ut1jJ(u)xt over (-oo,s]

x

[0,1].

REMARK 4.3.

When obtaining the analogous estimates for the boundary-initial

value problem (3.12) (see Remark 3.3), the analogue of equation (3.5) from which the global estimates are calculated is

U t

+

X(U)

t

X

+

f Oa(t-T)\jJ(U(T,X)) XT dT

h(t,x) - a(t)1)J(u 0 ) . X

To simplify several of the estimates which follow, we make the additional assumption that

(and hence also x)

c3 (R);

E

the alternative is to employ

difference quotients and pass to limits as in [5]. respect tot (use (a*g)(t)

f

0

Differentiate (3.5) with

a(s)g(t-s)ds, differentiate, and then change

variables) to obtain (4.16)

Multiply (4.16) by 1jJ(u)xtt and integrate over (-oo,s]

x

[0,1].

After several

integrations by parts the result of this tedious calculation is

1

1

2

2 f 0x•(u(s,x))1)J'(u(s,x))uxt(s,x)dx -CI 1

+

s

1

2 I2 1

+

I 3 ) - (J 1

- f -oof 0ftt1jJ(u)xtdxdt

+

the terms Ik in (4.17) come from

+

1

f 0Q[a,1)J(u)xtt;s]dx

+ ...

+ J 7) +

1

1

2

s

f_ f 00

(4.17)

1

f 0 f t (s,x)1)J(u(s,x)) xs dx

2 [x"(u)1jJ' (u) + x' (u)1)J"(u) ]u u dxdt; xt t 0

109

A Conservation Law with Memory I

=

s

1

= 11

+ 12 + 1 3 +1 4

where 1

s

11

J J

12

f J0

2 u u dxdt, tt t X

-oo O

s

1

2 u dxdt, tt X

-00

s

1

-00

0

J J

13

2

14

f J

s

1

-00

0

1

2 12;

the terms Jk in (4.17) come from s

J

1

where s

1

J8

=

1

2

f

1

0

2

x' Cu(s,x)H' (u(s,x))u t(s,x)dx X

1

- -2

s

1

f J0 _ 00

(u)

+

x'

2

]u xt u t dxdt

which respectively give the first term on the left side and last term on the right side of (4.17),

2

s

1

2

f f 0 x"CuH"Cu)uXtutu Xdxdt, -oo

110

Nohel 1

f 0 x"(u(s,x))l/J' (u(s,x))ut(s,x)uX (s,x)uXt(s,x)dx s

1

-oo

0

-f f

s

2

x"(u)l/1' (u)uxtutdxdt -

s

1

-oo

0 X

f f

u tu [x"(u)l/J' (u)ut] dxdt, X

t

1

f f Ox'(u)l/l"(u)uXtu tt uXdxdt. _ 00

The next estimate follows from the identity a(O)l/J(u)

= a'*l/l(u) xt

xt

+ a*l/J(u)

(4.18)

xtt

which is derived by integrating a*l/l(u)xtt by parts. Multiply (4.18) by uxt and integrate over (-co,s] x [0,1]; then in each term on the right hand side use the Cauchy Schwarz inequality and (4.13), (4.14).

This leads to the

estimate a(O)j

s

l

-CO

"'(u(t,x))J, lx'(u(t,x))J, lx"(u(t,x))l, lx"' (u(t,x))J are bounded by a constant

C > 0

in order to simplify the basic estimates derived above; here we have used the additional simplifying assumption that

and hence also x

E:

3

C (R) ;

this can be avoided as in [5]. Thus (4.15) becomes K2

1 2

Ll f 0uxt(s,x)dx

+

f

1 0

X

t;s]dx

0 (5. 3)

The reader should note that (5.3) is no longer of convolution type, because of the term

X

=

(u(T,x(t,t;:)))u (T,x(t,t;:)) under the integral. X

Let x(t,t;:) be the characteristic of (5.1) through t;: and define v(t,t;:) =

a x(t,t;:); note that v(O,t;:) = 1. 3[

The function v measures the growth of the

characteristics with respect to t;:.

f 0.

According to Lemma 2.1

of [14], a singularity will develop in the solution u of (5.1) in finite time, if it can be shown that there exists a number t, 0 < t < v(T,t;:)

0.

For, in this case there exists a 0




0.

smooth solution u(t,x) of the Cauchy problem (5.6) (equivalently, of (5.1) if

w=

will develop a shock in finite time.

bound for the time at which a shock develops is 1 k(O) log

+

k(O)

S

>

0, an upper

119

A Conservation Law with Memory The following considerations provide examples of kernels a in (5.6) (equivalently, of (5.1) REMARK 5.2.

for which Theorem 5.1 can be applied.

If a(t) =Se

-at

(0

S




=0

is given by

0.

p

(This may be undefined for t

0.)

The derivatives

162

Russell

are all defined for t

q(\) =

IT

k=l

>

0 and, in fact, with

+: ),

(1

k

the partial product differential polynomials K ( IT

(qk (D)y) (t)

1 = --21Tl.

k=l

I

c

(1 + __!__ D)y) (t) ]lk

qK(\)e u w(\)d\

p

are defined forK= 1,2,3, ... and lim (qK(b)y)(t) = (q(D)y)(t)

K-+«>

0

uniformly for t in any interval [T,oo) with T > 0 - thus y satisfies q(D)y 0 in a very strong sense for positive t. Finally, there is the matter of the inhomogeneous equation (5.2), i.e., q(D)y = u. It is clearly enough to discuss the solution corresponding to the initial state {0} in E. We proceed formally, at first, in the sense that we assume that a solution y(t) of (5.2), with initial state equal to zero in E, exists and has the usual properties.

Let T > 0 and let yT(t) be the solution of (5.2), still with zero initial state, corresponding to the same u(t) for 0 t but satisfying 0,

t > T.

T

163

Canonical Forms for Control Systems Then for t

>

T we should have

indicating that the state at time T is the one which, in E, corresponds to -ll (t-T) the equivalence class of the function I qke k The Laplace transform

k=l

of Yr(t) is

T -At e Yr(t)dt

J

0

From this the desired transform

n (v, T)

can be obtained, at least for v outside C , p

p

chosen large enough so that

the zeros of q lie inside C , from the formula p

1n (v, T) -- -2rri

f

cp

Since the solution of q(D)y = u which corresponds to u forme -At /q(A), we see that we must have

o(t) has the trans-

so that n(v,T) - 1 - 2rri

JC p

1 (v-A)q(A)

T

J eA(T-t)u(t)dtdA. 0

(5.51)

From this, assuming the convergence of the integral (5.51) and the sum below, one can derive the alternate expression

Russe 11.

164

n(v,T)

(5.52)

consistent with the fact that q(D)y

u is equivalent to the infinite dimen-

sional first order system

k

1,2,3, . . . .

It is possible to show, using an argument which appears in [6], that 2

(5.52) is not, in general, convergent for uEL [O,T].

If u is measurable and

bounded on [O,T] we have, for some M> 0,

I

q

'(l

-]lk

)

T

f 0e

-11

k

(T-t)

u(t)dtl

Mk-l

and the convergence of (5.52) then follows from the uniform basis property 1-} in E. of the elements {---

y+]lk

ACKNOWLEDGEMENTS This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR 79-0018.

Part of this material appeared origi-

nally in the Proceedings of the Third IMA Conference on Control Theory, Sheffield, 1980.

REFERENCES 1.

Boas, R., Entire Functions, Academic Press, New York, 1954.

2.

Dunford, N. and J. T. Schwartz, Linear Operators, Part I: Theory, Interscience Pub., New York, 1958.

3.

Duren, P. L., Theory of HP-spaces, Academic Press, New York, 1970.

4.

Hoffman, K., Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, N. J., 1962.

5.

Ho, L. F., Thesis, Department of Mathematics, University of Wisconsin, Madison, August, 1981.

6.

Ho, L. F., and D. L. Russell, Admissible input elements for linear systems in Hilbert Space and a Carleson measure criterion, SIAM J. Cont. Opt., to appear.

7.

Lee, R. C. K., Optimal Estimation, Identification and Control, M. I. T. Press, Cambridge, 1964.

General

Canonical Forms for Control Systems

165

8.

Levinson, N., Gap and Density Theorems, Amer. Math. Soc. Colloq. Pub., Vol. 26, Providence, 1940.

9.

Paley, R. E. A. C., and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Pub., Vol. 19, Providence, 1934.

10.

Reid, R. M., Thesis, Department of Mathematics, University of Wisconsin, Madison, August, 1979.

11.

Russell, D. L., Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems, J. Math. Anal. Appl. 62 (1978), 186-225.

12.

Russell, D. L., Closed-loop eigenvalue specification for infinite dimensional systems: augmented and deficient hyperbolic cases, Technical Summary Report #2021, Mathematics Research Center, University of Wisconsin, Madison, August, 1979.

13.

Russell, D. L., Control canonical structure for a class of distributed parameter systems, Proc. Third IMA Conference on Control Theory, Sheffield, September, 1980.

14.

Russell, D. L., Uniform bases of exponentials, neutral groups, and a transform theory for Hm[a,b], Technical Summary Report #2149, Mathematics Research Center, University of Wisconsin, Madison, July, 1980.

15.

Russell, D. L., and R. M. Reid, Water waves and problems of infinite time control, Proc. IRIA International Symposium on Syst. Anal. & Optim., Rocquencourt, December, 1978.

16.

Schwartz, L., Etude des sommes d'exponentielles, Hermann, Paris, 1950.

17.

Teglas, R. G., Thesis, Department of Mathematics, University of Wisconsin, Madison, June, 1981.

EXPONENTIAL DICHOTOMIES IN EVOLUTIONARY EQUATIONS: A PRELIMINARY REPORT Robert J. Sacker Department of Mathematics University of Southern California Los Angeles, California

George R. Sell School of Mathematics University of Minnesota Minneapolis, Minnesota

1.

INTRODUCTION

In this lecture we want to present some preliminary results on the general theory of linear evolutionary equations.

While the theory we discuss here

concerns linear equations only, our investigations should be viewed as the first step in a somewhat broader program that includes the local behavior of nonlinear evolutionary equations near an invariant finite dimensional manifold. In order to motivate our theory it is helpful to review several problems which arise in functional differential equations (FDE) and in parabolic partial differential equations (PPDE).

Let us begin with some linear FDEs.

First consider

(t)

k

L

Here we have x

(I)

Ai(t)x(t-ri).

i=l E:

JRn, Ai: lR -+ Hom(JRn, JRn) is a real (n

x

n) -matrix valued

function that is periodic in t (perhaps with period that depends on i) and r.

1

2

0.

Equation (1) is a special case of the combined FDE - ordinary dif-

ferential equation (ODE) given by

167

DOI: 10.1201/9781003420026-10

168

Sacker and Se 11 k

I

:X et)

where x

lRn,

E:

e

A. (e)x(t-r.), 1

i=l

e

E:

1

F (6)

(2)

M and M is some compact manifold, A. : M ->- Hom(lRn, lRn) is

continuous and ri

1

0.

In the case where Ai(t) is periodic in tin equation

(1), then equation (1) is included in equation (2) when M is a torus with a suitable twist flow.

Another special case of equation (2), which is of

particular interest here, is x(t)

a(r,e)x(t-1)

b(r,e)x(t)

+

(3)

r

=

r(l-r),

where M =

{

8

(r, e): 0

11/2

r

1} is the unit disc in JR 2 represented in polar

coordinates. The basic problem in the study of these equations is to give a qualitative description of the solution in terms of exponential growth rates, cf. [12 - 14].

Let us now look at the nonlinear FDE

xCt)

(4)

where a is a real parameter and f satisfies certain technical conditions to insure the local existence and uniqueness of the solutions of the initial value problem associated with (4), as well as the complete continuity of the solution operator for t

r

manifold for equation (4) when a

0. =

a0.

Assume that M0 is a compact invariant We are interested in sufficient con-

ditions on the semi-flow near M0 in order that there exists a family of comM0 . pact invariant manifolds Ma that varies continuously in a with Ma 0 Such a result, if valid, would be a generalization of theorems of Sacker [10], Fenichel (1] and Pugh and Shub [9]. In the area of PPDE, the Navier - Stokes equations for fluid flow represent the type of equations that concern us.

One problem of especial in-

terest arises in the study of the Couette flow, which is the flow of a fluid between two coaxial cylinders where the outer cylinder is fixed and the inner cylinder rotates with an angular velocity w.

It is known that the

behavior of the flow changes radically as w crosses various critical values

Exponential Dichotomies

169

w1 , ... ,w 4 where 0 < w1 < w2 < w3 < w4 , cf. [2,7,17]. In the region 0 < w < w1 one observes the classical Couette flow. The Taylor's cells appear in the region w1


0

(1.5)

On a Volterra Equation u(O)

257

= u0

u' (0) = u 1

u(t)

E:

(1. 6)

V(t)

a.e.

t

::> 0.

2

THEOREM. (Local Existence) Suppose a E: LAC[O,oo), f E: L10 c[O,oo; H], f 1 E: w1loc • 2 [o oo• W'), u 0 E: V(O), u 1 E: Hand A satisfies (1.3). Then there exists ' ' 2 a T0 > 0 and a unique function u E: L2 [o,T 0 ; W) with u' E: L [O,T 0 ; H), u" E: L2 [o,T ; V' (O)] satisfying (1.5) and (1.6) on [0, T ). 0

0

V' (0) is the dual space of V(O) and all derivatives are to be taken in the sense of distributions. Since u E: L2 [o,T 0 ; H) and u' E L 2 [o,T 0 ; H), we have, by [1, Proposition A.7), that u is equal almost everywhere to a continuous function on H, so that u(O) = u 0 has a meaning. Since u" E: L2 [o,T 0 ;v• (0)), we have, by [4, p. 19), that u' is equal almost everywhere

to a continuous function in [H, V'(O)J 112 , an intermediate space of Lions; hence u'(O) = u 1 has a meaning. Without loss of generality we may assume that u 0 not, simply replace f 1 (t) with Proof.

f1 (t) - Au 0 -

f2 (t)

t

J

0

0; for i f

a(t-s)Au 0 ds.

Then f 2 satisfies the same conditions as f 1 and, if u(t) satisfies

u" (t) + Au(t) +

u(O)

f

t

0

a (t-s)Au(s) ds

0, u' (0) = u 1 , u(t)

E:

V(t), then u(t)

u(t)

+

uo satisfies (1.5) and

(1.6).

Our method of proof is an extension of the technique used by Dafermos [2].

ForT> 0, consider the set ET

{v 0

C [0, T; W] 00

E

t

T,

I v(O)

= 0,

i = 0,1,2, ... }

v(i) (t)

E:

V(t)

for

Kiffe

258

equipped with two inner products

f

T 0

{H

+

w}dt

+

T

and [v,w] 2

T

f {H + w}dt,

0

which induce norms 11·11 1 and 11·11 2 , respectively. pletion of ET under the norm 11 · 11 2 •

By FT, we denote the com-

Multiply (1.5) by a test function of the form (t-T)v'(t) and integrate

over [O,T].

-f

An integration by parts gives us

T dt-

f

0

- (v'(t), Au(t)) -

f

T

0

(t-T)(f(t)

+

T 0

(t-T){.

Our first goal is to prove that there is a function u satisfying (1.7) and

(1.6).

To this end, choose a number k > 1\ (\given by (1.3)) and define a

bilinear form B(w,z) on FT

x

ET by

T dt-

f

B(w, z)

-f

0

T 0

kf

T - k 2"G

-1

u 0v

0

(3.11)

V(O) We note that G- 1 (t) assume that V(t)




1 since a(x)

0 for t

E

>

0.

Without loss of generality, we can

[O,T] since otherwise there exists T*

> 0

such

that V(T * ) = 0 and, by Rolle's theorem, there exists a time T** such that V(T ** ) = 0. With the same argument as above, we can assume that V(t) < -o for t

>

T1 , for some

.' a'

.

V -

o>

a' k > 0. 0 and 1 u 0 is c smooth with u 0 (x) > 0.

(u)

>

0

Moreover, suppose that there exists a point !; such that u (1;) is sufficiently

0

negative and u (x)

s;

0 for all x

> !; •

Then a shock develops in finite time .

Professor Malek-Madani's research was sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

REFERENCES

1.

Dafermos, C. M., Can dissipation prevent the breaking of waves?, Transactions of the Twenty-Sixth Conference of Army Mathematicians, ARO Report 81-1, 187-198.

2.

Klainerman, S. and Majda, A., Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure and Appl. Math. 33 (1980), 241-263.

3.

Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure and

Appl. Math. 10 (1957), 537-566.

4.

Lin, S. S. and Malek-Madani, R., Solution to the Riemann problem for the equations of gas dynamics in a tube with varying cross section, Transactions of the Twenty-sixth Conference of Army Mathematicians, ARO Report 81-1, 341-354.

5.

Malek-Madani, R. and Nohel, J. A., in preparation.

6.

Nohel, J. A., A nonlinear conservation law with memory, Volterra and Functional Differential Equations, Marcel Dekker, New York, 1982 (this volume).

ON OBTAINING UL TH1ATE BOUNDEDNESS FOR

a-

CONTRACTIONS

Paul Massatt Department of Mathematics University of Oklahoma Norman, Oklahoma

The use of Lyapunov functions, maximum principles, and invariance principles for the study of the limiting behavior of evolution equations has generated interest in the study of dissipative systems.

A dissipative system is a

system where there is some bounded set which all trajectories enter into and remain in.

In the study of dissipative systems we make hypotheses as

general as possible -- hypotheses which are likely to occur in the applications; we seek to obtain as sharp as possible results on the limiting behavior of solutions.

The results are generally concerned with autonomous,

periodic, and certain types of nonautonomous flows (see [ 1], [ 2], [ 3], [ 4], and [21]). Let X be a Banach space and let T: X+ X be continuous. be bounded sets in X.

Let Br(y)

{x s X

I

Let B and C

llx - Yll : 0 such that n :> n 0 1mp that B attracts C if for all E > 0, B + BE (O) dissipates C. B is invariant i f TB = B. B is positively invariant if TB c B. B is negatively invariant i f for all X E B there exists y E B with Ty

X.

The orbit of B, y+(B), is

00

defined by y+(B)

u Tn(B).

n=O

Thew- limit set of B, w(B), is defined by

w(B) =

n Cl{ u TnB}. If B is negatively invariant, the alpha limit set m=O n=m of B, alpha(B), is defined by alpha(B) = {y s X I there exists {yj} c B,

273

DOI: 10.1201/9781003420026-21

Massatt

274

k. {x.} c B, and {k.} + with T Jy. = x. and {y.} + y}. B is stable if for J J J J + J every s > 0 there is a o > 0 such that y (B + B0 (0)) c B + BE(O). B is

uniformly asynrptoticaUy stable if it is stable mtd there is an s

>

0 such

that B attracts B + B (0). E

We say T is point dissipative if there is a bounded set which dissipates all points.

T is conrpact dissipative if there is a bounded set which dissi-

pates all compact sets.

T is local dissipative if there is a bounded set

which dissipates a neighborhood of any point.

T is local conrpact dissipative

if there is a bounded set which dissipates a neighborhood of any compact set. T is bounded dissipative, or ultimately bounded, if there is a bounded set which dissipates all bounded sets.

If T is continuous (as we always assume),

then compact dissipative, local dissipative, and local compact dissipative are all equivalent. Normally, in order to obtain results on the limiting behavior of solutions, we need some type of compactness condition on T.

Hale, LaSalle and

Slemrod [12] have shown that if T is completely continuous and point dissipative, then there exists a maximal compact invariant set (MCI) which is uniformly asymptotically stable, attracts bounded sets, and has a fixed point.

If T is conditionally conrpletely continuous, i.e., B, TB s B implies

TB is precompact where B is the collection of all bounded sets in X, then the same conclusions hold except that the MCI attracts only those sets B E B for which TB s B also. However, in applications the map T is often not conditionally completely continuous.

Examples are stable neutral functional differential equations

(SNFDE's), retarded functional differential equations (RFDE's) with infinite delay, strongly damped nonlinear wave equations, various parabolic and hyperbolic partial differential equations, etc.

In many of these problems the

Kuratowski measure of noncompactness, or a-measure, has proved useful.

a-measure is a map a: B

+

[O,oo) defined by a(B)

by a finite collection of sets of diameter d}.

inf{d

J

The

B can be covered

T is an a-contraction if

there exists a k s [0,1) such that for all BE B we have TB E Band a(TB) s ka(B).

T is a conditional a-contraction if there exists a k s [0,1) such

that for all B, TB s

B we have a(TB) s ka(B).

Hale and Lopes ([13], [10]) have shown that if T is a conditional acontraction and compact dissipative, then there exists an MCI which is uniformly asymptotically stable, attracts a neighborhood of any compact set, and has a fixed point.

One can obtain similar results under more general

assumptions on T (see [ 7], [ 11], [ 15], and [ 17]) .

Ultimate Roundedness

275

In this paper we show for certain a-contractions which allow for backward existence of solutions, and satisfy some uniform Lipschitz condition in the backward direction, that point dissipative implies the existence of a maximal compact invariant set which attracts bounded sets.

We also assume

that the equation is defined on two Banach spaces, with one compactly imbedded in the other.

The assumptions will be shown to be natural enough to include

stable neutral function differential equations of the form

(1)

where f is periodic in t and uniformly Lipschitz in xt' xt is defined on C[-r,O] and D is linear and atomic at both -r and 0.

We will also apply

the results to strongly damped nonlinear wave equations of the form

(2)

where f is uniformly Lipschitz, s JR + x A + JRn where A u(t,x)

=0

c

>

0 and A

2

0.

We also assume u(t,x):

JRm is a bounded domain with smooth boundary, and

for all x s 8A.

Equation (2) arises in the modelling of longitudinal vibrations in a homogeneous bar in which there are inertial and viscous effects ([6], [8], [9], [22], and [23]).

The term s6utt takes into account the inertia of

lateral motions in which the cross-sections are extended or contracted in their own planes.

The term A6ut indicates that the stress is proportional

not only to the strain, but also the strain rate, as in a linearized Kelvin material. In two recent papers, I have shown that for most a-contractions which arise'in the applications, point dissipative and compact dissipative are equivalent.

In this paper we obtain the much stronger result of ultimate

boundedness, and that the maximal compact invariant set attracts bounded sets.

However, we do this at the sacrifice of making more restrictive

assumptions.

The class of equations which satisfy these hypotheses is suf-

ficiently large to make the results interesting.

I expect that similar

results should be obtainable under more general assumptions. an open area for further research (see [17], [18], and [19]). we are interested in proving the following theorem.

This is still In this paper

276

Massatt

THEOREM.

Let X1 and X2 be two Banach spaces with X1 compactly imbedded into x 2 , and let T, C and U: Xi + Xi be continuous and uniformly Lipschitz in

each space.

Let C be a linear contraction in both spaces and let U: x 2 + x 1 -1

be uniformly Lipschitz.

We also assume that T

, C-

and are continuous and uniformly Lipschitz, where T is uniformly Lipschitz.

1

-1

and U: X. + X. exist

=

-1

C

+ U, and

U: X2 +x 1

Under these assumptions, if T is point dissipative

in x 2 , then T is bounded dissipative in x 1 and x 2 . Before proving this theorem, it is important to point out some of its applications.

As we mentioned before, this result will prove for equations

(1) and (2) the equivalence of point dissipative and bounded dissipative.

x2

C[-r,O] and x 1 = w1 •""[-r,O] (see [17] for Other possibilities, such as LP x JRn and w1 •P x JRn are also

In equation (1) we may let details).

=

admissible.

The hypothesis that U,U: x 2 + x 1 are uniformly Lipschitz comes from the assumption that f is uniformly Lipschitz. The hypothesis on T- 1 , -1

c '

and U is satisfied by the fact that D is atomic at -r.

(2), we may let x2 = LP

X

LP and xl = w1 ·P

iv 1 ·P

X

In equation

This equation fits into

the theory of analytic semigroups (see, for example, [14]). The equation -1 -1 -1 may be written as u - aA{l- EA) u - A(l- EA) u = (1- EA) f(u). tt !1 -1 The homogeneous part utt - aA(l - sA) ut - A(l - EA) u = 0 has a solution map which is a contraction in some equivalent norm.

Since (1 - EA)

-1

f(u)

is compact, one can ascertain from the variation of parameters formula that the solution map for the entire equation is an a-contraction.

The uniform

Lipschitz condition on f insures that U,U: x 2 + x 1 are uniformly Lipschitz. It is also easy to obtain the existence of T- 1 , C- 1 and U here. The fact that T is bounded dissipative in x 1 has

Proof of theorem.

already been proven in [17].

Also, in [18] we showed that point dissipative

and compact dissipative are equivalent in x 2 . compact invariant set K

Hence, there is a maximal

attracts a neighborhood of any compact set.

In fact, it attracts any bounded set whose orbit is bounded.

We shall assume

that T is not bounded dissipative and arrive at a contradiction. There must exist some M> 0 such that there exist sequences {xj} and {k.} + J

oo

with Tkx. +

J

t

K

+

BR(O) for 1

k

kJ..

Let R c

K

+

>

Otherwise, it is easy

to show that y (K + BR(O)) is bounded and dissipates all bounded sets. In the following, for convenience and without loss of generality, we shall assume that i f x E x 1 , then 11 xll 2 11 xll 1 . Now let k be a contraction constant for C in both spaces and let -1

a uniform Lipschitz constant forT, T

-1

, C

, U, and U: Xi +Xi, i

=

0.

BM(O)

be 1,2,

277

Ultimate Boundedness and for U,U: x 2 n

I

= cn+1 +

+

CjUTn-j.

j=O Tn+1 = Cn+1 + Un+1' Un+ 1 : X2

+

Since T = C + U, we have T2 = c 2 + CU +UT, and Tn+ 1

x1.

We shall define un+1 =

1

with {U 1yj} converging to some u 1 s x 2 .

I

j=O

CjUTn-j' n

0, so that

x1 is Lipschitz and, hence, 1 Let {y.} c {x.} be a subsequence J J 2 1 Let {yj} c {yj} be a subsequence

We notice that Un+ 1 : x 2

X2 is completely continuous.

n

+

2

with {U 2yj} converging to some u 2 . Continuing this process, and then using a diagonalization process, we can find sequences {z.} and {u } with {U z.} J n n J + un in x 2 for all n 2 1. From this, we get

Using this and the fact that U: X2

/I

x 1 is uniformly Lipschitz gives us that

n

n

j=O

j=O

11 I

un+1 z q - un+1 zp /I 1

We may also choose a sequence {un} lim //U n+1 zq -

+

I

c

x 1 with

n+1 // 1 - L (0,1r)

= {g

B: D(A) ->-

E

by

=

g"

= 0}

Bg = g(O)

281

DOI: 10.1201/9781003420026-22

Rank in

282 2 F: L (O,Tr) + JR

A: D(A) + JR =

D(A)

{(z,g)

=

Fg

L2 (0,Tf)

X

A(z,g)

by

=

JR

x

Tf

J

D(A)

I

Bg

f(x)g(x)dx

0

by

(Fg, Ag) E:

-

z}.

If one assumes that y(x,t) is a solution of equation (1) and defines z(t)

= y(O,t), z"(t)

then

+

u(t)

for

t

E:

JR.

Thus equation (1) is related to the following evolution equation on JR

x

2

L (O,Tr):

-

i2

dt

(z(t),y(·,t))

A(z(t),y(·,t)) + (u(t),O)

(z(O),y(·,O))

(2)

d

dt (z(O),y(·,O)) The second component of the mild solution of equation (2) is interpreted as the weak solution of equation (1). continuous cosine family C(t), t

E:

If the operator A generates a strongly JR,

with associated sine family S(t),

then the mild solution of equation (2) can be written as t

(z(t),y(·,t))

C(t) (z 0 ,y 0 )

+

S(t) (z 1 ,y 1)

+

J S(t

- s) (u(s) ,O)ds,

t

E:

0

The purpose of this note is to give conditions on A, F, and B such that A will be the generator of a strongly continuous cosine family.

The fact

that A generates a strongly continuous cosine family is equivalent to equa­ tion (2), and therefore equation (1), being well-posed; that is, we have existence, uniqueness and continuous dependence of solutions.

JR.

Cosine Families and Product Spaces 2.

283

STATEMENT AND OUTLINE OF PROOF OF RESULT

Let X and Y be Banach spaces, A: D(A) +X, D(A) + Y, D(A)

c

D(F)

c

X, be linear operators.

c

X, B: D(A) + Y, and F: D(F)

Let A be the linear operator A

restricted to the kernel of B, that is, D(A)

=

D(A) n ker B.

Throughout,

we make the following assumptions on A, B and F: i)

F is A- bounded on D(A), that is, there exist constants

c1 , c2

(with c2 < 1) such that IIF1/JIIy sc 1 II1/JIIx + c2 IIX1/JIIx for al11/J s D(A); ii) there exists a right inverse D s L(Y,X) of B such that AD: Y + X and FD: Y + Y are bounded linear operators. Define the operator A on Y D(A)

x

X by A(z,1/!)

{(z,1/!) s Y x D(A)I B1/!

(F1/J, A1/!) with

z}.

A one-parameter family C (t), t s R,

of bounded linear operators from

a Banach space X into itself is called a strongly continuous cosine family i f and only if

i)

ii) iii)

C(O) = I, C(t + s) + C(t - s) = 2C(t)C(s), s,t s R, C(t)x is continuous in t for each x s X.

The associated sine family S(t), t s R,

of C(t) is given by S(t)x

t

=J

0

for all x s X, and the infinitesimal generator G of C(t) is given by Gx

C(r)xdr =

lim C(Zt)x - x for all x s D(G) = {x s xl C(t)x is twice continut+O .2t 2 ously differentiable}. See the paper [1] for more detailed information on C"(O)x

=

strongly continuous cosine families.

THEOREM. If A - DF with domain D(A) generates a strongly continuous cosine family on X, then A generates a strongly continuous cosine family on Y x X. Conversely, if A generates a strongly continuous cosine family on Y x X, then A - DF has a closed linear extension which generates a strongly continuous cosine family on X.

If IIDII < 1, then A- DF itself generates a strongly

continuous cosine family on X.

Rank in

284 The basic idea of the proof is to show that

A

[

]

transforms into

A- [

FD

F

AD - DFD

A -

A generates

under a change of coordinates, and then to show that continuous cosine family if and only if by the lemma below.

-

A does.

a strongly

This technique is justified

See the paper [2] by R. Vinter in the case where

A gen-

erates a strongly continuous semigroup. LEMMA.

Let X be a Banach space, P: X + X a linear homeomorphism, and G a

linear operator-with D(G)

c

X.

Then the operator G

=

- -1

PGP

with D(G)

PD(G)

generates a strongly continuous cosine family if and only if G does.

Proof.

The proof follows from the cosine generation theorem as found

in [ 1]. Define P on Y Dz). PD(A)

x

X into Y

PAP-l

+

Thus we have +

Dz)l z

+

Dz),

Y,

E

+

{(Bw,w)l wE D(A)}

and

X by

x

[J

E

Dz)i z

D(A)} E

Y,

E

D(A)}

Dz) and P

-1

by P

-1

Cosine Families and Product Spaces 3.

285

OUTLINE OF PROOF OF THEOREM

First assume that A- DF generates a strongly continuous cosine family C(t), t s R,

on X with associated sine family S(t).

Rewrite

A as

A - DFD Since D, FD and AD are bounded linear operators, so is L.

By perturbation

A generates a

results for cosine families [1], it suffices to show that strongly continuous cosine family.

Without loss of generality assume that

JJnJJ < 1, for if this is not the case, one could rescale the norm of X.

Now

we can show that since F is A - bounded, it is also A-DF - bounded and from this deduce that

C(t) (z, 1)!)

Note that

t

A generates t

(z + F(f S(r)l)!dr), C(t)l)!), 0

J S(r)l)!dr 0

the lemma, we deduce that Suppose now that t

E

(z,l)J) s Y x X,

s D(A- DF) = D(A) for each t

and, therefore, since D(A)

the lemma,

the strongly continuous family

c

E

R

and 1)!

t s R

E

X (see [1])

D(F), C(t) is well defined onYx X.

A generates

Using

a strongly continuous cosine family.

A generates a strongly

A does, and again by

cosine

A does as well.

By Let C(t),

be the cosine family generated by A, and define the family of bounded

R,

linear operators C(t), t

E:

R,

on X by

C(t)l)! where 1r 2 denotes the projection onto the X component. One can show that c (t)' t E R, is a strongly continuous cosine family generated by a closed linear extension of A- DF. Furthermore, one can show that if llnll < 1, then this extension coincides with A- DF. REMARKS.

1 In the example of the introduction, D(A) = H2 n H0 and A generates

L cos nt(h,h )h , where h E: L (0,1r) and hn (s) n=1 n n Furthermore, if f E L2 (0,1f), then F is a bounded

the cosine family C(t)h = ns, s

E

[0,1r].

2

286

Rankin

linear operator and the operator D: R -+ D(A) defined by Dz a bounded right inverse of B.

Thus, by perturbation, If f E (H 2 n

strongly continuous cosine family.

A-

=

z ( rr - x) m is 1T

m

OF generates a

then F is no longer

bounded; however, F is A- bounded and, by Theorem 4.3 of [1], A- OF generates a strongly continuous cosine family. The details of the proof of the Theorem and an expanded version of this paper will appear elsewhere.

ACKNOHLEDGEMENTS The author would like to thank Professor John Burns for bringing this problem to his attention. Professor Rankin's research was supported by the National Science Foundation under Grant ISP-8011453-15 and by the United States Army under Contract No. DAAG29-80-C-0041.

REFERENCES 1.

C. Travis and G. Webb, Second order differential equations in Banach space, NonZinear Equations in Abstract Spaces, edited by V. Lakshmikantham, Academic Press, New York, 1978, 331-361.

2.

R. B. Vinter, On a problem of Zabczyk concerning semigroups generated by operators with nonlocal boundary conditions, Imperial College, Department of Computing and Control Report 77/8 (1977).

THEORY AND NUMERICS OF A QUASILINEAR PARABOLIC EQUATION IN RHEOLOGY M. Renardy Mathematics Research Center University of Wisconsin Madison, Wisconsin

1.

INTRODUCTION

The following problem occurs in polymer processing:

A thin filament of a

viscoelastic liquid is stretched by pulling its ends as indicated in the diagram:

We investigate a model equation for the temporal evolution of the displacement, which is derived from the "rubberlike liquid" constitutive assumption for the stress-strain law [5] and a one-dimensional approximation based on the thinness of the filament.

pu

32

. 1

.)n 3x3t [-

u ] X

+

This equation is as follows (see [9]):

()

3x

t

J

ux (t) a(t - s) [ - 2 - - u (s)

-CO

]ds

(1. 1)

X

Here u(x,t) is a real-valued function of x s [-1,1] and t s R.

A subscript

x denotes partial differentiation with respect to x and a "dot" denotes

287

DOI: 10.1201/9781003420026-23

288

Renardy Equation (1.1) is supplemented

partial differentiation with respect tot. by the nonlinear Neumann boundary condition

3n

·a 1 at[- u- l

+

f

t

X

at the ends x

-+-

u (s) ux(t) a(t- s)[ -2--]ds u (t) ux(s) X

f(t)

(1. 2)

±1.

=

Physically, u(t,x) denotes the position of a fluid particle at time t, which occupies the position x in a certain reference state. state is chosen such that the filament has uniform thickness. !owing, it will be assumed that in the limit reference state (u = x).

This reference For the fol-

the filament is in the

-"" ' The variable p denotes the density of the fluid, t +

multiplied by the square of half the length of the filament (this scaling factor arises from the normalization o-f x to the interval [-1,1]), n is a Newtonian contribution to the viscosity, and f is the force acting on the ends of the filament divided by the cross-sectional area in the reference state.

The memory kernel a is assumed to have the following properties:

(i)

a has the representation a(t) =

f e -At

for some complex-valued Borel measure variation

I I

that supp real,

of

is finite, and there

contained in {A E

on the complex plane.

. exlst

some

s arg A s


0, such E}.

Since a is

can always be chosen such that (ii)

a(t)

(iii)

a is monotone decreasing.

0 fortE [O,oo).

Note that (i) implies, in particular, that a is analytic for t > 0, continuous at t

= 0,

and that a can be estimated by a decaying exponential.

In

physical theories derived from structural models of polymers (see [5]), a is a finite sum of decaying exponentials, which is clearly a special case for our assumptions

as a finite sum of Dirac measures). n -At

examples of functions satisfying (i) are e.g., t e

, e

-At 2

Other

In the latter

cases the form (i) can be established by using complex contour integrals.

An Equation in Rheology

289

The boundary condition (1.2) agrees with the equation governing the evolution of the length of the filament when inertial forces are neglected. Prior to the results that will be sketched in this paper (cf. [7], [9]), this problem was investigated by Lodge, McLeod and Nohel in [6]. They study problems where u is a given nondecreasing function for t < .0, and the force X

f vanishes fort > 0. Theorems concerning existence of solutions, asymptotic behavior and various monotonicity properties are proved. Whereas their arguments rely on monotonicity properties, the basic tools in [7], [9] are the implicit function theorem and Liapunov functions. In Section 2 of this paper, I state the existence and uniqueness results for (1.1), (1.2) which I proved in [9]. Results concerning the boundary problem (1.2) are stated first.

Then (1.1) is transformed to a system of equations that can be classified as "quasilinear parabolic" in the sense of

Sobolevskii (see [2], [10]).

This fact can be used to establish an existence

theorem locally in time, and a global existence theorem for "small" forces f. For the details, the reader is referred to [9]. In Section 3, some numerical results are shown that are based on joint work with P. Markowich [7], [8]. We discretized (1.1), using an implicit Euler-type finite difference approximation.

For the kernel a, we chose

numbers given in [4] for a polyethylene melt at 150°C. The qualitative behavior of solutions is studied as the parameters p and n vary.

2. ANALYTICAL THEORY We begin this section by stating some theorems concerning the solution of the boundary problem (1.2). For brevity, we omit the proofs, and the reader is referred to [9]. Before we can state the first theorem, we need the following definition. DEFINITION 2.1.

Let Z be a Banach space.

Then

Z) denotes the space

of all functions R + Z whose first n derivatives are square integrable in the sense of Bochner [3]. Moreover, let

Xn0 (Z) Yn0 (Z) = {v: e

+crt

(v -

V)

ZJ e -crtv

R +

E

n

H (R, Z)}.

E

Hn (R, Z) and there exists a v oo

E

Z such that

290

Renardy

The spaces Xcr(Z) and Ycr(Z) have natural norms which make them Banach spaces. n n We quote the following two theorems from [9]. THEOREM 2.2.

Let cr

>

0 be sufficiently small.

Then the following holds:

If f E Xcr(R) has sufficiently small norm, then (1.2) has a unique solution n Acr ux(t) which satisfies ux- 1 E Yn+l(R). The function ux depends smoothly on f. THEOREM 2.3.

In addition to (a), assume that supp

real axis and lim e-crtf(t)

t+-eo

is positive real.

Let f: R

0 for some cr > 0 and f(t)

=

=

+

is contained in the

R be continuous and such that

0 for t

t 0.

equation (1.2) has a unique solution satisfying lim u (t)

t+-eo

tion exists globally in time; moreover, lim u (t) t+co

X

X

= u (eo) X

For any such f, =

1.

This solu-

exists and u (eo) > 0. X

The proof for Theorem 2.2 is based on the implicit function theorem. In the situation of Theorem 2.3, the implicit function theorem yields the existence of a solution on some interval

Estimates show that solu-

tions cannot blow up in finite time, and the result about the limiting behavior at infinity follows from a Liapunov function argument (see [9]). We now turn to the study of (1.1). we consider ux(t) function oft.

= b(t)

>

According to the theorems above,

0 as being given for x

= ±1, where b is a smooth

In [9], equation (1.1) was reformulated in such a way that

it fits into the theory of quasilinear parabolic equations. by the following substitutions: p

u

q

u

X

XX

r = u t

J

t

J

e-A(t-s)(u (s) - u (t))ds X

X

u (t)u (s) XX - X ] -----. 4 - - ds u (t) X

This was achieved

An Equation in Rheology t

J

g3 (le)

e

t

-!c(t-s)

(u

e -lc(t-s) [u

J

g4 (A)

291 -2 X

XX

(s)

u u

(s)

-2

(t)) ds

X

XX

2 (t)u (t) X

2 ux (s)

]ds.

Equation (1.1) now assumes the following form: p

r

q

r

pr

X

XX

3np

-2

rxx

-3

6np r

-

qrx

p

X

(2. 1)

-!cg1 (!c) - T r

XX [ - Ag2 (le) - -4 g1 (le) + E._ le

p

2r

X

+-

lcp3

with boundary condition p

= b(t),

rx

= b(t)

at x

= ±1.

Provided it is sat-

isfied initially, the first boundary condition follows from the second and the first equation of (2.1), and it can therefore be ignored. Before we state our theorems, we need to define some function spaces. Let Hk denote the Sobolev space of those functions on [-1,1] which have k square integrable derivatives, and tions defined on

is the space of Hk-valued func-

are s-integrable with respect to

in the Bochner

sense (see [3]). Equation (2.1) is regarded as an evolution problem in the space\ = H2 x (H 1) 2 x Let us substitute r = r- b(t)x, and introduce the abbreviation y

=

(p,q,r,g 1 ,g 2 ,g 3 ,g 4).

We split (2.1) into a

"quasilinear" and a "perturbing" part, y

A(y)y

+

f(y,t),

(2. 2)

292

Renardy

where A(y) is defined as the following linear operator A(y)y'

A incorporates the boundary conditions

= 0 at x

±1.

In [9], I proved the following theorem. THEOREM 2.4. Let 1 s < 3oo arbitrary. Let y 0 = (p 0 , q 0 , r 0 i gi,O) s X5 be given such that r 0 s H , r 0 = 0 at x = ±1, Ag.l , 0 s H ) and ,x min p 0 (x) > 0. Then, for some T > O, equation (2.2) has a unique solu-

xd -1, 1]

1

tion y s C ([O,T],Xs) such that y(O)

= y0 .

The result follows from the Sobolevskii theory (2],

Idea of Proof.

[10] after it is proved that A generates an analytic semigroup and that cer-

For the details, see [9].

tain smoothness conditions hold.

It is also possible to establish an analogue of Theorem 2.2 for the equation (1.1), i.e., to prove existence of solutions globally in time for small forces f. b (t)

ux(t)

The case f = 0 now corresponds to the boundary condition 1.

Then (2 .1) has the trivial solution p

=

1, q

= 0,

r

= 0,

0. We substitute y = (p- 1 ,q,r,g1,g2,g3,g4)' and split the right hand gi (A) side of (1. 1) into the linearization at the trivial solution and a non linear

perturbation, leading to an equation of the form y

Ay+ f(y,t).

(2.3)

In [9], it is proved that the operator A generates an analytic semigroup, and that its spectrum consists of the semisimple eigenvalue zero and a remainder in the left half-plane.

(At this point, assumptions (ii), (iii) on

the kernel a enter essentially.

In determining the spectrum, we use

f

1

A(A+a)

f 0 a(t)

-at a

dt

'

An Equation in Rheology

293

and (ii), (iii) yield that this has positive real part for Re a

0.)

Using

this fact and the implicit function theorem, I proved in [9] THEOREM 2.5.

Let a> 0 be small enough.

If bE

has sufficiently

small norm, then equation (2.3) has a solution yE n

3.

Xn0 (D(A)

n R(A))).

Yn+ 1 (N(A)) 0

(Xn+ 1 (R(A)) 0

NUMERICAL RESULTS

The results presented in this section were obtained in joint work with P. Markowich [8].

We discretized (1.1), (1.2) using a finite difference method,

which is second order in space and first order in time.

The scheme is im-

plicit, i.e., time derivatives are approximated by backward differences. In [8] we gave a convergence analysis for our scheme in the case that f is small.

The stability analysis essentially copies the analysis used for the

continuous problem with the spaces logues.

Xn'0 Yn0 replaced by

and I proceed straight to some results. kernel

their discrete ana-

I refer the reader to [8] for details about the numerical scheme, 8

L

i=l

K.e

-A. u

In these computations, we chose the

with the constants Ki,Ai listed in Table 1.

TABLE 1 i

A. (sec

-1

)

Ki (Nm

-2

sec

-1

1

lo- 3

1

X

10- 3

2

lo- 2

1.8

X

10°

3

10-l

1.89

X

10 2

4

1

9.8

X

10 3

5

10

2.67

X

10 5

6

10 2

5.86

X

10 6

7

10 3

9.48

X

10 7

8

10 4

1. 29

X

10 9

)

Renardy

294

These numbers were obtained by Laun [4] from an experimental fit for a polyethylene melt at 150°C, which he calls "Melt 1". The parameter n is physically identified as a Newtonian contribution to the viscosity.

Experimental values are not available, and theoretically

n is either a solvent viscosity (for polymer solutions), or it results from

The value of n should be com-

fractions of low molecular weight (for melts).

pared to the viscosity resulting from the memory, which, for constant shear -2 -2 L. KiA.i = 50,000 Nm sec. One would expect n to influence i=l the solution significantly only if it is at least comparable to this value.

rate, is given by

This heuristic argument was confirmed by our computations.

The numbers

given in the following are understood to be in the following units: (i)

n is given in Nm- 2sec.

(ii)

f denotes the force acting on the ends of the filament divided

by the cross-sectional area in the reference state (u in Nm- 2 . (iii)

= x).

It is measured

p denotes the density of the filament multiplied by the square

of half the length in the reference state (this latter scaling factor arises from the normalization of the variable x to the interval [-1,1]). -1

measured in kg m

pis

Time is measured in seconds.

In the accompanying figures, we have chosen f = 100,000 e the first twelve figures, we have the initial condition u(x,t The first three plots show u, u

X

and u

XX

for p

= 1, n = 1.

-t 2/25

In

= -oo) = x.

It can be seen

that uxx is negligibly small, and that u is almost linear in x.

This means

that the solution is determined by the evolution of the boundary condition, and inertial forces can be neglected.

This changes if p is increased.

Physically, this cor-:esponds to changing the length of the filament. realistic values of the density, p of a few millimeters, p

= 1,000

=1

For

would correspond to an initial length

would correspond to an initial length of

about 1m. In figures 4 - 6, we have p of u, u

X

and u

XX

are shown.

= 1,000,

n

= 1.

Three-dimensional plots

It can be seen that uxx has increased by a

factor of 1,000 compared to the previous plots. behavior remains roughly the same.

Otherwise the qualitative

295

An Equation in Rheology The next three plots (figures 7 - 9) were made for the same 100,000 (for 0 s n s 10,000, the solutions changed very little).

parison to n

p

and n

=

In com-

1, it was found here that the boundary value for ux increases

more slowly up to t

= -2, and then increases rather suddenly around t = 0.

In this region inertial forces become very important, a fact manifested in the plots by a rather pronounced spike in uxx In Figure 10 we have

p

= 1,000,

In this case the behav-

1,000,000.

n

iour becomes almost Newtonian, and there is hardly any elastic recovery (the maximal value for u(x

= l,t) is

1.404, and the value at t

= 60 is still

The dependence of u on xis again almost linear.

1.398).

Several calculations were done for p = 10,000. u and uxx for n

= 100,000.

Figures 11 and 12 show

It can be seen that uxx becomes rather large.

When one looks carefully at the plot for u, one also finds that a little "overshoot" occurs in the relaxation, whereas in the previous plots u decreased monotonically after reaching its maximal value, this is no longer true here, as the following calculations show: 6

t

u(x

t

-83.6

-1,t)

8

-22.7

10

t

t

12

-4.3

-3.8

If smaller values of n are chosen, this "overshoot" becomes even more pronounced.

.The mesh size becomes very crucial here, and computations with

adaptive meshes would be more appropriate. In the last plot (figure 13), a different initial state was chosen, namely u(x,t

= -oo) =

x

+

x

3

This corresponds physically to a filament which

is getting thinner towards the ends.

We have

p =

1,000, n = 1.

It can be

observed that u makes several oscillations during the elastic recovery. I conclude this paper by mentioning an interesting open problem. theory presented here relies essentially on the fact that n f 0.

The

On the

other hand, our computations indicate that even very big viscosities can still be considered small parameters (for comparison: n water 10 -3 , n 2).

Besides the mathematical interest, there is thus also a

very strong physical motivation for developing a theory which remains valid as n

+

0.

Renardy

296

({)

X

0

a: I

:=;)

'so 6'o.

u FIGURE 1

An Equation in Rheology

297

lo0 &o

(j)

So

>