Approximation of Nonlinear Evolution Systems (Mathematics in Science and Engineering)

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Approximation of Nonlinear Evolution Systems

This is Volume 164 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, Universiiy of Southern California The complete listing of books in this series is available from the Publisher upon request.

Approximation of Nonlinear Evolution Systerns Joseph W. Jerome Department of Mathematics Northwestern University Evanston, Illinois

1983

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco SBo Paulo Sydney Tokyo Toronto

COPYRIGHT @ 1983, BY ACADEMIC PRESS,INC.

ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMIITED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kinadom Edition oublished bv ACADEMfC PRESS, INC. (LONDON) LTD.

24/28 Oval Road, London NWl

7DX

Library of Congress Cataloging in Publication Data

Jerome, W . Joseph. Approximation o f nonlinear evolution systems. (Mathematics in science and engineering) Includes bibliographical references and index. 1 . Evolution equations, Nonlinear--Numerical solutions. 2. Approximation theory. I . Title. I I . Series. QA374.Jk7 1982 515.3'53 82-8808 ISBN 0-12-384680-3 AACR2

PRINTED IN THE UNITED STATES OF AMERICA

83 84 85 86

98765 4 32 1

for my Parents

This page intentionally left blank

Preface xi Acknowledgments xiii List of Symbols and DeJinitions

Introduction I

GLOBAL ~

xv

1

A SOLUTIONS K

1. Problem Formulations and Uniqueness for Dissipative Parabolic Models 1.0 Introduction 11 12 1.1 Heat Conduction with Change of Phase: Stefan Problems 1.2 Unsaturated Fluid Infiltration in Porous Media 20 1.3 Reaction-Diffusion Systems 25 1.4 Incompressible, Viscous Fluid Dynamics at Constant Temperature: Navier-Stokes Equations and Generalizations 32 1.5 Uniqueness of Solutions 39 1.6 Bibliographical Remarks 42 References 43 vii

Contents

viii

2. Convergent Regularizations and Pointwise Stability of Implicit Schemes 2.0 Introduction 45 46 2.1 Regularization in the Stefan Problem 2.2 Semidiscrete Regularization and Maximum Principles in the Stefan Problem 52 62 2.3 Regularization in the Porous-Medium Equation 2.4 Nonnegative Semidiscrete Solutions of the Porous-Medium Equation and Maximum Principles 66 2.5 Invariant Rectangles and Maximum Principles for Reaction-Diffusion Systems in Semidiscrete Form 68 2.6 Bibliographical Remarks 73 References 14

3. Nonlinear Elliptic Equations and Inequalities 3.0 3.1 3.2 3.3 3.4

Introduction 76 General Operator Results in Banach Spaces and Ordered Spaces Applications and Examples 89 Semidiscretizations Defined by Quadrature 99 Bibliographical Remarks 109 References 111

77

4. Numerical Optirnality and the Approximate Solution of Degenerate Parabolic Equations 4.0 4.1 4.2 4.3 4.4 4.5

Introduction 113 Representations of Sobolev-Type and Upper-Bound Estimates Lower-Bound Estimates and N-Widths 126 Convergence Rates for the Continuous Galerkin Method 142 154 Convergence Rates for Semidiscrete Approximations Bibliographical Remarks 158 References 159

5. Existence Analysis via the Stability of Consistent Semidiscrete Approximations 5.0 Introduction 162 5.1 Stability in Sobolev Norms for Semidiscretizations of Degenerate 163 Parabolic Equations 5.2 Existence of Weak Solutions for the Stefan Problem and the Porous-Medium Equation and Approximation Results 177 5.3 Existence for Reaction-Diffusion Systems 190

115

Contents

ix

5.4 Existence for the Generalized Form of the Navier-Stokes Equations for Incompressible Fluids 195 5.5 Bibliographical Remarks 197 References 201

I1

LOCAL SMOOTH SOLUTIONS

6. Linear Evolution Operators Introduction 207 Semigroup Preliminaries 208 The Linear Evolution Equation and Evolution Operators 213 Perturbations of Generators and Regularity of Evolution Operators 223 6.4 The Inhomogeneous Problem and an Application to Linear Symmetric Hyperbolic Systems 230 235 6.5 Bibliographical Remarks References 235

6.0 6.1 6.2 6.3

7. Quasi-linear Equations of Evolution 7.0 7.1 7.2 7.3 7.4 7.5 7.6

Introduction 237 Perturbation of the Linear Problem and Nonlinear Preliminaries 238 The Quasi-linear Cauchy Problem in Banach Space 241 Quasi-linear Second-Order Hyperbolic Systems 247 The Vacuum Field Equations of General Relativity 257 Invariant Time Intervals for the Artificial Viscosity Method 262 Bibliographical Remarks 271 References 272

Subject Index

275


PREFACE

The evolution of complicated physical and biological systems is sufficiently subtle that it requires intricate analysis of the underlying mathematical models. This analysis must be focused so as to impart clear and original insights, yet it must be intrinsically generic, permitting examination of general questions such as existence, uniqueness, regularity, wellposedness, stability, and regularization, all related to the soundness of the models. The models themselves must possess sufficient novelty and importance to justify generality and suggest new directions for study. We believe that the models selected here accomplish this objective. The book is directed primarily toward applied mathematical analysts who are concerned with the mathematical preliminaries to numerical computations in nonlinear partial differential equations, and toward physical, biological, and engineering scientists who are interested in phase transition, diffusion in porous media, diffusion with reaction, incompressible fluid flow, and general relativity. It is aimed, secondarily, at mathematicians with interests in partial differential equations, approximation theory, numerical analysis, convex analysis, or operator theory. Readers with some basic acquaintance with linear functional analysis and the notion of a weak solution of a partial differential equation will find parts of Chapters 1, 2 , 4 , 5 , and 6 accessible. A stronger background in nonlinear functional analysis, vector measure and integration, partial differential xi

xii

Preface

equations, convex analysis, geometry, and operator theory is needed for Chapters 3 and 7, as well as some parts of Chapters 4, 5 , and 6. Part I of the book deals with the global existence in time of weak solutions of nonlinear evolution systems, and Part I1 deals with the local classical theory. The models are developed, at least in summary form, from basic principles, such as those of conservation. Where some subtleties arise, such as in the free-boundary problems associated with degenerate parabolic evolutions, distribution formulations are presented. Approximation via regularization is discussed in detail. Various levels of discrete analysis are presented, including semidiscrete maximum principles and invariant regions, an existence analysis based on semidiscretization in Parts I and 11, and numerical estimates for finite-element approximations and regularizations. Some topics particularly worth mentioning are the “entropy” determination (specifically, n-widths) of function classes and the relation of this concept to the complexity of partial differential equations; the construction of the evolution operators for linear stable systems and their use in developing a local theory for quasi-linear equations in Banach spaces; the alternative stability analysis for quasilinear equations via semidiscrete approximations and its application to invariant time interval determination in hydrodynamics; the evolution of systems that are not spatially gradient in nature but are nonetheless not far from such systems as measured by pseudo-monotone perturbations; the use of abstract integral operators, called lifting operators, in the recasting of the partial differential equations and their approximate formulations; and the use of regular Baire measures with duality to obtain convergence estimates. This book shares with explicitly numerical texts the use of such tools as discrete analysis; with explicitly applied texts the introduction of physical and biological models; and with theoretical texts the use of various techniques to formulate and prove major existence, stability, and approximation theorems. These evolution systems encompass topics of interest to the applied and theoretical scientist.

This book is based on lectures presented at Northwestern University, in which preliminary versions of six of the seven chapters were presented to small numbers of colleagues and students. Thanks are due to Stephen Fisher, Mark Pinsky, and Laurent Veron, who listened to some of these lectures and made many helpful comments. The policy of upper-level teaching seminars at Northwestern is a fruitful one, producing an atmosphere conducive to learning and research. Thanks are due also to T. Kato, J. E. Marsden, and J. R. Dorroh for their help in supplying appropriate literature and references. The author expresses his appreciation for the kind hospitality shown him by Leslie Fox during a leave at Oxford University; he also thanks Gunter Meyer, whose understanding of the quantitative aspects of the Stefan problem greatly assisted in the subsequent development of this volume. Gratitude is also due those who have helped in certain technical matters related to the book. These include Garrett Birkhoff, David Fox, Richard Graff, Klaus Hollig, C. Ionescu-Tulcea, Ken Jackson, Mitchell Luskin, Ridgway Scott, Vidar ThomCe, and Lars Wahlbin. Avner Friedman acted as a helpful conduit to the analytical aspects of the subject in his regular P. D. E. seminar and provided useful general information. Finally, the author expresses his appreciation to Miss Antoinette Trembinska for her very careful reading of the manuscript, and to Mrs. Alice Wagner for her excellent preparation of the manuscript. xiii


LIST OF' SYMBOLS AND DEFINITIONS

Term Arguments Part I Part I1 Boundary regularity for bounded sets R in Part I

Closure (of G ) Coercive Bilinear form Functional Commutator (of operators A and S) Connection coefficients Convergence Ordinary Weak Weak-*

Symbol or Definition (x, t ) (x E R c R", t E (0,T o ) )

( t , x , p ) ( t E ~ = [ O , T 0 ] , x E ~ " , p E RRn(n+2)) c Restricted cone condition (see Ref. [4.1] p. 11)or the equivalent strong Lipschitz condition (see Ref. [4.25] p. 72) assumed throughout, unless replaced by stronger specific assumptions such as star-shaped, smooth, etc. CI(G), G

Usage as in (3.2.17b) Usage as in (3.2.6) or (3.2.29)(for weaker alternate, see Remark 3.1.1) [A,S]:=AS-SA

rfj

Sequential

(continued )

xv

List of Symbols and Definitions

xvi

Symbol or Definition

Term Convolution Functions Operators Covariant derivative (in direction X) Curvature Einstein tensor Operator Ricci tensor Riemann tensor Scalar Dispersion (of B from J d in X) Distribution

* (see (6.3.4)) DX

Differentiation

Functional space Pairing Test space Domain (of mappings U) Duality Pairing Symbol for space Function/Measure spaces on Euclidean domain 0 (or 8)with range in normed linear space X Baire'

(k 2 0) Graded smooth

i =I

(.>.>

*

Real-valued. In this list, P is a generic symbol: special choices occur in the text

M(@): = { p : p is measurable on Baire subsets of 6 and {gldpl < w} Ck(O;X),C:(O;X) (see Def. 7.3.1 in the case = R" x n) ('1

Lebesgue Lipschitz Local Lebesgue (k b 0) Sobolev'

' The application is to a closed subset.

L m ( @ ; X ) : ={f:ess,supllfllx < w} Lip@; X ) Lf,,(O: X) W'*P(O;X):= { J : D ~ E L ~ ( O ; X )k) ,~~~
0) (nonintegral s, k < s < k

(k

+ I)

Uniformly local Lebesgue and Sobolev Functions Point action Set action Interior (of G) Lie bracket Matrix dot product

Wp: (see Remark 4.2.8) H k ( B ; X ) : =Wk.2(B;X) H"(6;X) := real interpolation space (Hk(C;X),

H k + ' ( 6 ;X) )*--II.* LEAB; X),W$;P(B;X) (see Definition 7.3.3)

Measure (of Q) Modular operation Monomial Multinomial coefficient N-width (of d in X) Negative Infinitesimal generator class Part of f Norms

Ikll*ln) I I U l l ~ I oX):

k ( k l , . . . , k,)

k! =

k,!...k,!

(k

=

ki)

Variation of p sup {X norm of derivatives of order C.lal

(a1 Q k taken with respect to spatial variables comprising a fixed hyperplane in span @I}

lI~IIw*.Pl~:xl

(Alternatively) 112

ll~IIW*48:Xl

with C ,

=

(

a,,

la'

...

.) a,

(see Theorem 4.2.14) (c'onrinued )


xviii


Term

Either, interpolation space norm or equivalent Bessel potential norm defined via Fourier transform if 0 = 08" eSSDSUPllUIlx max llD"~lIL-(o;x)

lIUIIL'(QX)

llull X) II. IIL,PlCo:X)l 11. W*."(Q;

Il.ll1.X)

Ilf IlLIP

Id1 6 k

IlWbP(0: X )

Il.llm.x

(see Definition 7.3.3) (see Definition 6.3.1) inf{Lipschitz constants o f f }

Numbering format Definitions Equations Lemmas, Propositions, Corollaries and Theorems Remarks Operators Bounded linear operators from X into Y Bounded linear operators on X Evolution Extension Format Inversion' Dirichlet Neumann Robin Projection Energy space Pivot space Solenoidal (generalized) Structure Diagonal Order Big 0 Little o Partition Length Sequence Positive part (off) Also called lifting or Riesz mappings.

Separate category Separate category As a category Separate category

B[X] (with norm ll.llx) U(t, s) (0 Q s Q t Q To) AzB Roman

Ell Ph P Scalar operator "multiplying" the identity matrix.


xix


Term Product Cartesian (of sets) Direct (of operators) Proximity mapping (induced by convex 4) Range (of mapping U) Real abstract space Referencing

[nl, . . . ,

Chapter m [ n ] , . . . , Resolvent Operator (U) Set Riesz Mapping Potential Schwartz class

R,, Rnnge(U)' Assumed unless indicated otherwise Refers to . . . in Reference n of the references of the current chapter Refers to . . . in Reference n of the references of Chapter m

Isomorphism T:G* + G satisfying (t,u ) = (Tf, u), 1, Y

Semi-norm

Sequence Piecewise linear (in time) Step function (in time) Set complementation

{ 0") I uNI

0

\ 1, x > 0,

Signum function -1,

Spaces Format

x 0} n g a and (8 < 0} n a,,

+ Jaant+ {s(8+)cos(vU,l,)- k ( 8 + ) V 8 +

a

nv,)c$do

(1.1.5)

1. Formulations for Parabolic Models

14

where v, is the normal to C,, L, respectively, outward for (0 > 0} and inward for (0 < O}; nv, is its projection into the plane of Q; and s(O'), k(O+), s(O-), k(O-) are understood in the limiting sense as O + , 8- + 0. Since &? and 4 are arbitrary, we conclude that the pointwise relations

a

- (a(O)

at

+ s(O)) - V - (k(O)VO) + q(O) = 0,

(1.1.6a)

in 9\C,

-

-

s(e-)cos(v,,it) - k(e-)ve- nv,I r - = s(e+)coS(v,,i,) - k(o+)ve+ xvu/r+ ( 1.1.6b) hold, and that the second term in (1.1.5) is zero for 4 E C,"(O,To).However, none of the physical motivation nor any of the preceding calculations is affected if we include, in the test class for (1.1.4), functions in C,"(Xo),that is, the test class is the union C,"(O,To)u C,"(Zo).Thus, we have as(@ ~

at

+ 4(O) = 0

(distributionally) in C,.

(1.1.6~)

Equation (1.1.1) is clearly a distributional form of (1.1.6), as can be seen by a set of calculations similar to (1.1.4) and (1.1.5). However, we wish to introduce a generalized temperature u, and define a weak solution of (l.l.l), which involves initial and boundary conditions. Thus, set u=

J:

k(5)dl = K

0

(1.1.7a)

8

via a standard Kirchhoff transformation. If the standard enthalpy is defined by Q ( t ) = a(t) 4 t h t # 0, (1.1.7b)

+

set H = Q 0 K - ' . The mathematical assumptions now follow after a brief remark concerning the physical properties of Q. There is some reason to believe (see [l] pp. 95-99) that Q can be absolutely continuous when thawed moisture is accounted for in s at temperatures lower than the nominal freezing point. A discontinuous Q results when some bulk freezing occurs, a situation expected away from soil surfaces. In the latter case, the apparent specific heat capacity dQ/dt is a distribution not represented by a function. In order to provide the maximum possible generality, we shall permit Q to be discontinuous. For simplicity, s is normalized by translation by a constant so that s(0-) = 0.

Remark 1.1.1. We assume that the function H = Q K - ' , defined on R'\{O), is a monotone increasing function, C' in R'\{O}, with a jump dis0

1.1 Stefan Problems

15

continuity of height A at zero, and a derivative satisfying

0 < A < H‘(5) 0.

(1.1.8~)

We also assume that the function f = 4 K - ’ can be written 0

f=g+h,

(1.1.8d)

where g is a locally Lipschitz continuous monotone-increasing function vanishing at zero, and h is a Lipschitz continuous function. The distribution transform of (1.1. l ) is now given by (1.1.9) In the example of permafrost thawing, the constant A represents the product of the heat of fusion B with that fraction of moisture content solidified at the nominal freezing point; at lower temperatures, s - A is defined as the negatively-signed part of B specified by the ratio of unthawed moisture = ~ R) is content to total moisture content. The specification of H ( U ) ~ , (on an apt initial condition; H(.) is closely related to enthalpy, as we have seen. Since J = H - ’ exists (as a continuous function), specifying H(u)l,=, also specifies = o. Various physical boundary conditions are possible. Perhaps the most realistic allows for a time-dependent Dirichlet condition on a portion of an, and a time-dependent Neumann condition on the remainder of We shall, however, select a homogeneous Neumann boundary condition on all of aR, and compensate by including the source/sink term f (*), which easily can be modified to be position dependent, also. In the permafrost problem, where R typically is bounded by the atmosphere, by a (partially) insulated pipeline inducing a developing thaw bulb, and by (arbitrary) underground boundaries assumed virtually insulated, we are substituting for atmospheric and pipeline fluxes, appropriate sources/sinks.

uI,

Definition 1.1.1. Suppose the initial “enthalpy” H(u)l,=, = H(uo) E L“(R) is specified. Then a function u in the class ‘X, defined by (l.l.llc), with H(u)

’

As distinct from the Stefan-Signorini problem, where the liquid is removed and the flux condition is imposed on the moving boundary.


16

a selection satisfying H ( u ( x ) )E [ @ A ]

H(u) E L"(9) n H'([O,

if u ( x ) = 0, 7-01;

[H'(Q)]*),

(1.1.10a) ( 1.1.lob)

is said to be a weak solution of the initialboundary-value problem for (1.1.9) with homogeneous Neumann condition, if f(u) E L"(Q), and if the relation

-

jnx(To) H ( u ) $ d x + jnxl0, H(uo)$dx = 0

(1.1.114

holds for all $ E Vo, where

9 = g T o = Q x (0,TO),

(1.l.llb)

and the solution class V and test class Vo are given by V

= L"((O,T,);H'(Q)) n H'([O,

To];L2(Q)) n L"(9); (1.1.11~)

g o = C([O,7-01;H1(Q))n H'([0,7-,];L2(Q)).

(1.1.1Id)

Here, Q is a bounded, strongly Lipschitz, domain in R" and, for consistency with the definition of '3, uo E H'(0) is required. Moreover, the second integral in (1.1.1la) has the initial interpretation of a functional, since it is guaranteed that H(u) E W([O, To]; [H'(Q)]*) by the regularity of (l.l.lOb), but it will be subsequently observed that H(u) is weakly continuous from [O,To] into L2(Q)(see Definition 5.2.1 to follow).

Remark 1.1.2. A classical solution of the initialboundary-value problem for (1.1.9) is a function on 9,satisfying (1.1.9) pointwise in ( u > 0} and {u < 0}, for which C = C, = C-,and for which the energy conservation principle (see (1.1.5)) -[k

2 1

= s(o+)cos(v,,1,)'

(1.1.12)

holds across C. In addition, of course, the initial condition and boundary conditions are satisfied pointwise. Here [[I denotes the discontinuity of C' across C directed from ( z : u > 0} to { z : u < O}. It is a straightforward exercise in integration by parts to verify that every classical solution, satisfying the regularity requirements of Definition 1.1.1, is a weak solution, and every sufficiently regular weak solution is a classical solution. In the special case when C is a smooth surface, defined as the zero set of a smooth function 4,

' Differentiation with respect to nv, is interpreted as the gradient dot product with RV,

1.1 Stefan Problems

17

then multiplication of (1.1.12) by JV4Jgives the usual relation, relating the front velocity and the discontinuity of k V 8 across X.

Remark 1.1.3. It is possible to lift the equation (1.l.lla) to obtain a pointwise relation on 9, via an appropriate inversion of - A. We shall introduce this linear, Riesz mapping and carry out the lifting. In fact, we find it advantageous to introduce a one-parameter family Nu of such mappings. We begin with No. The reader may view this class of mappings as inducing equivalent abstract integral equation relations. Definition 1.1.2. Given 8 E F := [H'(R)]*, we define the element w = N o t E H'(0) as the Riesz representer of C, when the inner product (1.1.13a) is employed. Here, (1.1.13b) and we have ( 8 , ~= )

jQ VNo8

*

Vv

1 +j(v)j(NoC), PI

v E H'(O).

(1.1.14)

The notation (.;) represents the duality pairing on F x H'(R). For r~ > 0,: w = No/is defined as the Riesz representer of C, with respect to the inner product (v,

w)[H1(Q)]m

=

jQ vv

'

vw

+ 0 jQ "w-

(1.1.15)

In this case, ( 8 ,V )

=

jnVNJ

*

Vv

+ jnvNJ. CJ

(1.I .16)

Remark 1.1.4. The norm induced by (1.1.13) is equivalent to the standard norm on H'(R) (see Sobolev [22] or Section 4.1). The equivalence induced by (1.1.15) is, of course, trivial. We have writen [H'(Q)], to emphasize the explicit inner product chosen. The linear mapping Nu, 0 2 0, whose restriction to L2(R) is positive definite and symmetric by (1.1.13)-(1.1.16),

' Differential symbol suppressed; convention when no ambiguity is possible. *

We have used du, when appropriate to designate the surface differential. There should be no confusion with the present usage of u.


18

induces the following (negative) norm on F : lltll[F1a

(1.1.17)

= ( e ,N u t ) " 2 *

The mappings Nu are continuous from [Flu into [H'(R)], and have closed range. These facts follow directly from the relation (1.I .I 8)

llNatl~WIW,l~= llellV%.

Moreover, (1.1.18)may also be used in conjunction with (1.1.14)and (1.1.16) to deduce that llletllF = sup{)(e,u)I:IIuII~~(~) 1} (1.1.19) isequivalent to(1.1.17)with constants dependent upon Q. The relation(l.l.19) defines the standard duality norm, of course. It follows immediately that the mappings NalL2(*) are bounded into H'(R), and hence are self-adjoint, positive definite and compact as L2 operators. Because of the equivalence of the spaces [H'(R)], and H'(R), and of [Flu and F, we shall suppress the subscripts Q in the sequel. Remark 1.1.5. An equivalent way of viewing N,:F + H'(R) is as follows: w = N o t is the unique element satisfying the (weak form of the) linear elliptic Neumann problem (1.1.20a)

aw = 0, av

on dR,

-

(1.1.20b)

n

For a > 0, w

( 1.1.20c)

= Nut satisfies

in

(-A+a)w=t,

aw = 0 , av -

(1.1.21a)

R,

on 22.

(1.1.21b)

One may deduce directly from (1.1.20)and (1.1.21) that -ANot

=

1

e - -( t ,l),

(-A

PI

+ Q)NJ = t ,

No(-Aw) = w Nu(-A

1

- -j(w),

PI

+ CJ)W = W,

(1.1.22a) (1.1.22b)

for t E F and w E Range(N,). It is easily seen that the latter is characterized as H'(R) for Q > 0, and as { w E H'(R):j(w) = 0} for a = 0. Function identically one.

1.1 Stefan Problems

Remark 1.1.6. that

19

The mappings Nu for o > 0 possess the important property

t 2 0 N u t 2 0, (1.1.23) where t 2 0 is defined by (t,u ) 2 0 for u 2 0. The property (1.1.23) follows directly from setting

+

N u t = (Nut)+ (Nut)-,

(Nut)+= sup(O,N,t), and making the choice o = -(Nu/)- in (1.1.16) to obtain (Nut)- = 0. We now conclude this section with the pointwise relation(s) referred to earlier.

Proposition 1.1.1. Let u be a weak solution satisfying Definition 1.1.1. Then (a/at)N,H(u) E L2(9),o 2 0, and for all t, 0 < t < T o ,the equations (1.1.24) aNUH(4 at

+ u + NUfu(u)= 0,

o > 0,

(1.1.25)

hold almost everywhere in R, where fu(r) = f(r) - or,

rE

[w'.

(1.1.26)

Moreover, the initial condition holds.

Proof: We give the proof explicitly only for o = 0. For 4 E Corn@), let I) = No4 in(l.l.1la). Using the commutative relation [(a/at)N, - No(d/ar)]4 = 0 (see (1.1.14)), the self-adjointness of No, and (1.1.14) directly, we obtain

which shows that the distribution derivative (a/at)N,H(u) is equal to - u Nof(u) (l/lRl)j(u). Since this function is in Lz(9), it follows that N,H(u) E H'(9) and, hence, (1.1.28) may be integrated by parts to obtain (1.1.24) for almost all t. To obtain (1.1.27),let I) E V,, I)(-, To)= 0. Then No# E V,, and this substitution into (l.l.lla), coupled with (1.1.24), yields, after an integration by parts in t,

+

1.

20

Formulations for Parabolic Models

from which (1.1.27)follows. Since Nof(u) E C([O,T o ] ;L2(Q)),the final three, hence all, terms in (1.1.24) are in C([O,T o ] L2(Q)), ; which establishes the identity (1.1.24)for all 0 < t < T o .

Remark 1.1.7. The regularity of the initial datum uo plays a decisive role in the regularity properties of the solution. For example, in Section 5.2, it is shown that (dH(u)/dt)is essentially bounded in t, as a regular Baire measure on a, if uo E W2.'(Q), and this property is exploited in Section 4.3 to obtain error estimates for the two-phase Stefan problem. In Section 2.2, it is shown that the semidiscretization defined there inherits pointwise stability, since uo eLm(S2),and this property is used in Section 5.2 to deduce that the solution is essentially bounded on its space-time domain in this case. Accordingly, it may be of interest to weaken the hypothesis on uo to uo E L2(0).In this case, it is not to be expected that uZtis an L2(9)function, but a solution, in the sense of Definition 1.1.1 exists, nonetheless, if the regularity properties defining V are accordingly adjusted. This is discussed in Section 5.2, where the major existence theorems are deduced for (1.1.10)and (1.1.11) (see especially Corollary 5.2.7 and Proposition 5.2.12). Remark 1.1.8. Analogous to the two-phase Stefan problem just described is the one-phase problem, in which diffusion processes are assumed to hold in only one of the phases; the temperature is assumed constant at the phasechange temperature in the other phase. There is an elementary change of variable, (1.1.29) U(X, t ) = O(X, t)dt,

Jd

n.

which converts the formulation corresponding to (1.1.1) into a parabolic variational inequality in this case. The reader is referred to Kinderlehrer and Stampacchia [lo] for details. Solutions of such inequalities are considerably more regular than the corresponding solutions for the two-phase problem. 1.2

UNSATURATED FLUID INFILTRATION IN POROUS MEDIAx

The density p of a gas expanding in a porous medium and the volumetric moisture content O ofa liquid of constant density infiltrating a porous medium au

tu,=-. at

* The flow is assumed such that gravity can be neglected.

1.2 Porous-Medium Equation

21

satisfy similar equations. In the case of the gas, the pressure serves as a potential; Darcy's law asserts that the velocity is given by (1.2.1) v = - ( k / p )V p . Here, k is the permeability, with units of area; p is the viscosity, with units of mass per unit length per unit time; and p is the pressure, with units of force per unit area. In the case of the liquid, there is assumed a total potential, or piezometric head, a, which is measured in units of length. The analog of (1.2.1)simply is a flux defined for horizontal flow by W =

(1.2.2)

-kV@.

Here k is the hydraulic conductivity, with units chosen so that k V @ has units of mass per unit length per unit time (the units of viscosity). Similar to the heat-balance equation (1.1.2), one sets up mass-balance equations of the forms d

-

s

dt a

bpdx

= -

he

k pv v d a = JaiB p - V p * vda, c1

(1.2.3)

and

-dtd j

Odx= - h e w * v d a = LekV@*vda.

(1.2.4)

Here, (1.2.3) and (1.2.4) express the equality of the rate of mass crossing 8 9 into 9, with the rate of increase of mass within 9l with convective effects assumed negligible. The quantity p is the dimensionless porosity of the medium, and the units of 8 are expressed in mass per unit area. As in the previous section, the choice of the open set 9 c 0 is essentially arbitrary and v is the outward normal to 9.Before the above equations can be utilized, so-called equations of state of the forms P:P

+

P(P),

e:a-,ep),

P > 0,

(1.2.5a)

a 2 0,

( 1.2.5b)

must be specified. In (1.2.3) and (1.2.4), /I,k, and p are assumed smooth and nonnegative; and, physically, p, 8 > 0. For example, a common form of (1.2.5a)is P ( P ) = PoP1"y-l'(Y > 11,

P B 0,

(1.2.6)

where y = 2 for an isothermal process, and y > 2 for an adiabatic process. We may rewrite (1.2.3) and (1.2.4), respectively, via the compact expressions (1.2.7a)


22

where the functions K and L are defined by (1.2.7b) and d L ( @ ) d x= dt a

(1.2.8a)

V K @ * vdo,

-

0

where

0 2 0,

(1.2.8b)

@ 2 0.

(1.2.8~)

Under the assumption that K is an invertit.,: function, (1.2.7)and (1.2.8)may be unified in the single statement d dt

-

jH ( u ) d x

=

La

-

(1.2.9)

Vu v d a ,

where u = K ( . ) and H = L K - ' . In the case of (1.2.6), if positive constants, we obtain (1.2.9), with 0

K(A) = cAY/(Y-'),

c = kPdY - l)/(PY),

B, k, and

p are

(1.2.10a) (1.2.lob)

H(A) = f l p o c - w / y ,

for A 2 0. Just as in the interpretation of (1.1.2), it is necessary to interpret since the free boundary, (1.2.9)as a distribution on (0, To), a { u > 01 = c, is a potential singularity set analogous to the set c of (1.1.3a). Under the assumption that Z is a sufficiently smoothly oriented hypersurface, integration by parts applied to the distribution equation

-

j

84

B x (0,To) H(u)-at

d x dt

= L B x ( 0 , To)

Vu * v$ d o dt,

4 E Cz(0,To), (1.2.11)

yields, as in the previous section, since 98 and

a~(')

--

at

AU =

o

vu IIv.Iz

in = 0,

4 are arbitrary (see (1.1.6)), {u > 01,

(1.2.12a) (1.2.12b)

1.2 Porous-Medium Equation

23

where v, is directed by the mass flux. Thus, if (1.2.13) is interpreted in a distribution sense on 9 = gTo, (1.2.12) is generalized by (1.2.13). We shall now state the precise hypotheses on H ( .). These are chosen so that we may consider the more delicate case, where H is not Lipschitz continuous. Other models are included in the format of Chapter 5. Remark 1.2.1 We shall assume that H has properties which generalize (1.2.lob). Specifically, we assume H

E

C(R'),

H(-t)

=

c l t ( l i v ) - l< 1 H'(t) \ < czt(l'y)-l,

E

(1.2.14a)

C(R'\{O}),

-H(t),

H ' ( t ) > 0,

H

H'

(1.2.14b)

t E R',

( 1.2.14~)

t # 0,

is concave on [0, a), some y > 1,

(1.2.14d) all t > 0, (1.2.14e)

where c1 and cz are positive constants. In particular, from H ( 0 ) = 0 and the concavity of H , we have that H ( t ) / t is decreasing for t > 0, so that H is subadditive (see Lorentz [14] pp. 43-44), that is, H(tl

+ t 2 ) G H ( t , ) + H(t,),

t , , t , 2 0.

(1.2.14f)

We shall consider now the initial/boundary-value problem for (1.2.13) with a homogeneous Dirichlet boundary condition. Since the model involves unsaturated filtration, specifying u = 0 on dQ amounts to the assumption that the liquid or gas diffuses from some initial configuration with compact support in Q. We shall now define weak solutions for all time; the requirement on the support of the initial datum is somewhat relaxed. Definition 1.2.1. Given 0 G uo E L"(Q), a function u k 0 in the class $?? (see (1.2.16a)) is said to be a weak solution of the initialboundary-value problem (1.2.13)with a homogeneous Dirichlet condition if, for every T o > 0,

(1.2.15)

holds for all

+ E Wo, where the solution class $?? and the test class W0 are


24

given by %? = L'((0, m);H;(R)) n H'([O, oo);L'(R)) n . . .

(1.2.16a)

L"((0, CO);HA(R)) n L"((0, CO);L"(Q)),

and

go= C([O,TO];Hh(Q))n H'([O, T0];L2(Q)).

(1.2.16b)

Here R is a bounded, strongly Lipschitz domain in R" and, for consistency with the definition of %?, uo E HA@) is required.

Remark 1.2.2. It is possible to lift (1.2.15), as in the previous section. The corresponding linear, Riesz mapping is somewhat different, since the homogeneous Neumann condition of the previous section has been replaced by a homogeneous Dirichlet condition. We shall close this section with such a description. Definition 1.2.2. Given the element t E H - '(a)= [HA(R)]*, we define the element w = Doe€HA@) as the Riesz representer oft, when the inner product (1.2.17) is employed. We, thus, have

(e, u )

=

sn

VDol * V v .

(1.2.18)

Remark 1.2.3. The linear mapping Do, which has the positivity property e B o + DofB 0, and whose restriction to L'(R) is a symmetric mapping, induces the following (negative) norm on H-'(R):

Ilq"- '(0)= (4 D0O1".

(1.2.19)

This norm is equivalent to the standard H-'(R) norm and Do is bounded as a mapping of H-'(R) onto HA@). In particular, DO(L2(R) is self-adjoint, positive-definite, and compact.

Remark 1.2.4. A more graphic way of viewing Do: H - '(Q) -, HA(R) is as follows. The quantity w = D o t is the unique element satisfying the weak form of the linear elliptic Dirichlet problem -Aw=t w=O

Note that -ADo[

=

e and Do(- A ) w

=

in

R,

(1.2.20a)

on

an.

(1.2.20b)

w, that is Do is the inverse of - A .

1.3 Reaction-Diffusion Systems

25

Proposition 1.2.1. Let u be a weak solution satisfying Definition 1.2.1. Then DoH(u)Ia,, E H1(gT0),T o > 0, and the equation

(1.2.21) holds almost everywhere in IR for each 0 < t < 00. Moreover, the initial condition

DoH(u)J1 =

= DoH(u0)

(1.2.22)

holds.

Proof: For 4 E C ; ( 9 T o )let , i,b = Do4in (1.2.15)and use the self-adjointness of Do and its commutation with a/& to conclude that

jgToDoH(u)a4 - dx dt = j at

9 T O

u 4 dx dt,

(1.2.23)

which shows that the distribution derivative (a/at)D,H(u) is equal to - u . Since H(u) E Lz(gT0),it follows that (a/axi)DoH(u)E Lz(!BTo),i = 1, . . . ,n, that is, DoH(u)E H1(gTo).In particular, (1.2.23) can be integrated by parts to obtain (1.2.19) for 0 < t < T o , since 4 is arbitrary and the left side of (1.2.21) is a member of C([O,T o ] Lz(IR)). ; Of course, To < co is arbitrary. The verification of (1.2.22) parallels that of (1.1.27).

Remark 1.2.5. The reason uo E L"(0) is required is that the convex inverse of H is of superlinear growth at infinity. Thus, to conclude that U , E LZ(Rx (0, a)), we require not only that uo E HA@), in analogy with the two-phase Stefan problem, but also require pointwise estimates on u over its space-time domain. These are deduced in Section 5.2, via the pointwise estimates of the semidiscrete schemes of Section 2.4. The existence result is presented in Corollary 5.2.8. Similar properties for (aH(u))/(at)also hold in this case as well (see Remark 1.1.7).

1.3

REACTION-DIFFUSION SYSTEMS

Reaction in conjunction with diffusion is found in chemical catalysis, biological transmission of nerve impulses, and ecological and genetic models, among many others. Usually such physical and biological systems are modeled by a simple continuity equation for the (vector) concentration of a


26

physical or biological species, whose flux (determined by Fick's law analogously to (1.2.1))and rate of generation describe the diffusion and reaction of the system. We shall briefly mention some examples.

Example 1.3.1. Chemically Reacting Systems

Chemical reactions are frequently expedited by catalysis. The catalytic agent, and possibly several promoting substances, are mixed and prepared in the form of a porous pellet. One or more chemical reactants are adsorbed at sites on the surface of the pellet, and react spontaneously and begin to diffuse within the pellet, where further reaction occurs. Though the medium is porous, it and the process are viewed as continuous. When transport effects are considered negligible, the conservation equations typically assume the forms

ac.

1 -

at

1

m

EL= V k=l

(DikVCk)

ae = V .(kVB) + pat

+ 1 pjirj,

i

j= 1

1

1 hjrj,

=

I , . . . , rn,

(1.3.1a) ( 1.3.1b)

j= 1

where ci is the concentration of the ith reactant, with units of moles per unit volume, and 8 denotes the system temperature. The first rn equations represent the mass balance for each reactant, and the last equation is an enthalpy-balance equation. It is assumed that 1 rn reactions occur, and r j is the reaction rate of the jth reaction, with units of moles per unit time per unit volume. The pji terms are the dimensionless stoichiometric integer coefficients for the ith reaction, satisfying pji = 0, i = 1, . . . ,rn; p and k are the volumetric heat capacity and thermal conductivity, and E is the total pore volume as a fraction of the total volume. Also, hj is the heat of reaction of the jth reaction, and Dik is the binary diffusion coefficient of the ith reactant in thejth, with units of area per unit time. We shall not consider in the sequel the cases when Dik # 0.

-=

Cf=

Example 1.3.2. FitzHugh-Nagumo Equations

The Hodgkin-Huxley equations model the conduction of electrical impulses along nerve fibers, or axons. Potential differences are created by ionic activation, including sodium and potassium ions. The system involves four


27

coupled equations, which we shall not display explicitly. For this, the reader is referred to the original article [8]. The first equation is a reaction-diffusion equation for the electromotive potential, and the remaining three are reaction equations for the sodium activation, sodium inactivation, and potassium activation, respectively. The FitzHugh-Nagumo model compresses the system to two linked equations in two dependent variables, which are in some sense equivalent, respectively, to the first two and the final two of the Hodgkin-Huxley variables. When the FitzHugh-Nagumo model is simulated by electric circuits, a so-called distributed line with interstage coupling resistances, to simulate diffusion, is employed in conjunction with a tunnel diode to simulate reaction. The two FitzHugh-Nagumo variables are essentially voltage and current in this simulated case. The model may be written explicitly as

-aul _ - A% - f b l ) at

(1.3.2a)

- k(u,,u2),

(1.3.2b) where the Lipschitz functions h and k have been chosen typically as

h(u,,u,) = gu1 - Y

U ~ ,

k(u1,uJ = u2,

0

> 0, Y 2 0, (1.3.2~)

and f is typically a cubic polynomial satisfying f ' 2 - c. The explicit form off depends on the number of stable equilibrium solutions which the model is expected to possess.

Example 1.3.3. Predator-Prey Models

If u1 and u2 denote population densities of prey and predator in a system including saturation and migration effects, their time evolution is governed by

(1.3.3b) Here a, b, h, and k are positive functions, typically linear, and f and g are monotone decreasing and increasing, respectively. These latter functions describe saturation. Migration is described by the diffusion terms, particularly the diffusion coefficients DUIand D,,,assumed nonnegative.


28

Remark 1.3.1. Let u = ( u l , . . . ,u,). The general reaction-diffusion model we shall consider is of the form au

-=

at

D ' - A u - f(u)

in Q,

(1.3.4)

where D' = ( I l l , .. . ,D,) is an m-tuple of nonnegative constants and f:R" + R" is C'. The additional hypotheses upon f are motivated by the natural grouping of the variables u, into those of concentration type, which are naturally nonnegative, possibly even further restricted, and those of potential type, which are unrestricted in sign. We now define these two classes of variables.

Definition 1.3.1. Let the integer io satisfy 0 6 i, 6 m. If i, > 0, we say that the variables u l , . . . , uio are of concentration type if there exists a slab

c=

n io

(1.3.5a)

c RiO,

[Ui,bi]

i= 1

a, d 0 6 b,, (b, - a,) > 0, i = 1, . . . , i, such that the vector field (fi,

is bounded on R" and satisfies certain properties on Q = c x RrnPio.

. . . ,Ao) (1.3.5b)

We describe these now. Define the faces Qai= {V

of Q for i

=

E

Q : u=~ a,},

(1.3.6a)

Qbi = {V E Q:ui = bi}

1, . . . , i,. We assume explicitly that fi(v) G 0,

v E Qai,

h(v) Z 0,

v4

fi(V)

>, 0,

(1.3.6b)

V E Qbi,

Q,

(1.3.6~)

i = 1, . . . , i,. Finally, we assume an ordering of sign regions property described by

..., U i - l , U : , u i + l , . . . , u m ) G O ,

h(U1,

u; G u;

h(U1,..., U i - l , O : I , U , + l , . . - ,

u,)20,

for each fixed { u ~ } ~ + , .

(1.3.6d)

The remaining variables are assumed to be of potential type, that is fi(u)

= gi(ui)

+ hi(u),

i

=

io

+ 1, . . . , m,

(1.3.7)

where hi is a Lipschitz continuous function on Q ; and g, is monotone increasing in the variable u i , such that [gi(ui) - gi 0 (note that Di = 0 is permitted), a boundary condition of the form

aui + wiui = 0 av

on 8 0 (v = outward normal)

-

(1.3.9)

is specified, where w i is a nonnegative, bounded function on an, satisfying wi > 0-

(1.3.10)

Moreover, in (1.3.11a)to follow, we set wi = 0 if Di = 0. Definition 1.3.2. Given uo E L"(R; Rm)with range in the set Q, defined by (1.3.5), and given, for each i, such that Di > 0, functions 0 < wi E L"(an), then a function U E % (see (1.3.11b)) is said to be a weak solution of the initialpoundary-value problem for (1.3.4) with Robin condition if the range of u is in Q; if uI,=, = u,; and if, for each i = 1,. . . ,rn,

Jn[z

-

3

ui+DiVui Vui+fi(u)ui d x + D i L n wiuiuida=O (1.3.11a)

holds for all 0

-= t c T o and all v E H'(R; Rm),where

% = L"((0, T,);H'(R,R")) n H'([O, To];L2(R;Rm))n L"(3; Rm).

(1.3.11b) Here Q is a bounded, strongly Lipschitz domain in R" and, for consistency with the definition of $9 and Q, uo E Lip@; Rm)is required. The domain R is also required to satisfy the regularity conditions specified by (2.5.7) and Definition 1.3.3 to follow. Remark 1.3.3. We shall briefly comment on the relation of the earlier models, introduced in Examples 1.3.1-1.3.3, to the general model of Definitions 1.3.1-1.3.2. In the FitzHugh-Nagumo model, it is feasible to select u1 and u2 as potential variables, with D1= 1, D 2 = 0, although one might wish

1.

30


the alternate choice of u2 as a concentration variable if the initial datum permits this choice. The assumption f ' > - c in this model means that f differs from a monotone increasing function by a linear function. The model as described, with h and k Lipschitz continuous, is, thus, included, if L" n C' initial datum is prescribed, with a Robin boundary condition for ul. In the case of the predator-prey model, the choice of u1 and u2 as concentration variables appears to be dictated in many cases of interest. The assumptions h(a1)[f(a 1) - a(b2)120,

[ - d a z )+ b(a1112 0,

k(a2)

h(bl)[f(b 1) - 4a2)l d 0, W 2 ) [

- db,)

+b(b1)Id 0,

(1.3.124 (1-3.12b)

guarantee (1.3.6b) if, in addition to the properties already stated, a and b are monotone increasing (note the sign off in (1.3.4)).The monotonicity properties off and g in (1.3.12), together with the sign properties of h and k (nonnegative), guarantee that (1.3.6d) holds. The presence and/or amount of diffusion will vary with the model. The model of the chemically reacting systems is already general, and we shall not attempt a specific analysis except to make two observations. The first involves the r j terms, which are typically nonnegative rational functions of ul, . . . , u j . The second concerns the choice of variables. Here, ul, . . . , urn are typically concentration variables and urn+ is a potential variable. This model demonstrates clearly why a partitioning of the variables is desirable and, indeed, necessary. Given a Robin boundary condition specified by o,we shall introduce the linear, Riesz lifting mapping R, in analogy with the mappings N, and Do introduced previously. We first make some preparatory remarks. Remark 1.3.4. It is known that, under certain regularity conditions on dC2, the trace operator T:H'(C2) -,L2(dQ)exists as a continuous linear mapping and coincides with the restriction mapping on Cm(n),which is dense in H'(C2) in this case. Moreover, the range of r is H"2(d12) and, in fact, r is an isomorphism between H'(R)/ker r a n d H"'(dC2). These facts are documented in Lions and Magenes (see [13]). Let 0 d o E L"(dC2) be such that Then the inner product

sari-0.

(1.3.13) (1.3.14)

induces an equivalent Hilbert space norm on H'(C2). To prove completeness, let { u j } be a Cauchy sequence. In particular, by the Cauchy-Schwarz in-

' Suppressed da.


31

equality, there is a p E R', such that n

( 1.3.15)

and, by taking the quotient space of H'(0) with the constants, we deduce the existence of u E H'(R) and {aj}c R', such that uj

+

aj -+

in

u

L~(R),

(1.3.16a)

From (1.3.16), we deduce that

lruj + aj - ru12-,0, so that

( 1.3.17)

Lno[ruj+ aildo hno r u d a . -,

Combining (1.3.15) and (1.3.17), we deduce that

It follows from (1.3.16) that uj -, u - a. in the standard H'(R) norm and, hence, in the norm induced by (1.3.14), since the standard norm dominates this norm up to a constant multiplier. A standard application of the open mapping theorem now gives the equivalence of norms on H'(R), and justifies the notation in (1.3.14). Definition 1.3.3. Under the assumption of the existence of the linear continuous trace operator T:H'(R) -, L2(dR)described above, let 0 < o E L"(sZ) be given satisfying (1.3.13). Given l' E F = [H'(R)]*, we define the element w = R,l' E H'(R) as the Riesz representer of .tin the inner product (1.3.14). We have, then,

( t ,U)

= Jn

VR,t

- Vu dx + in wr(R,tp

da.

( 1.3.18)

Remark 1.3.5. The restriction of R, to L2(R) is symmetric by (1.3.14) and (1.3.18). The mapping R, induces the following (negative) norm on F: (ll'll[F,"

=

(l'?R,t)"2.

( 1.3.19)

As in Section 1.1, we may conclude that the mappings R, are continuous and have closed range, and that [F], is topologically equivalent to F, with


32

the standard duality norm. In particular, RolL2(R)is self-adjoint, positivedefinite, and compact. The subscript o will be frequently suppressed in the sequel. Remark 1.3.6. An equivalent way of viewing R,:F + H'(R) is as follows. The element w = R u t is the unique element satisfying the (weak form of the) linear elliptic Robin problem ( 1.3.20a) -Aw=t inn, a- w ++w=o

on dR.

(1.3.20b)

R,( - A)w = W ,

(1.3.21)

av

One may deduce directly that - AR,& = &,

for L' E F and w E Range(R,). The latter is easily seen to be H'(R). Remark 1.3.7.

We may deduce the property & 3 O* RJ

20

(1.3.22)

directly from (1.3.18) by making the choice u = -(R,&)-, as in Remark 1.1.6. We now close this section with the lifting relations.

Proposition 1.3.1. Let u be a weak solution satisfying Definition 1.3.2 and let i, 1 0. Then the equation

a

-

at

+

Rwiui

+ R,,fi(u) = 0

Di~i

(1.3.23)

holds almost everywhere in R for 0 < t < T o . Proof: For 0 < t < T o ,we set vi = R,ic$i in (1.3.11a), for c$i a component of a suitable test function in Coo(@apply (1.3.18) and the self-adjointness of RUi,together with the commutation relation RUi(a/at)- (a/at)Rmi= 0, to deduce (1.3.23). 1.4

INCOMPRESSIBLE, VISCOUS FLUID DYNAMICS AT CONSTANT TEMPERATURE: NAVIER-STOKES EQUATIONS AND GENERALIZATIONS

A moving fluid transferring heat is described by a system of five equations, characterized by conservation of mass, momentum, and energy, including the continuity equation of mass balance, the three equations of motion, and

1.4

Incompressible, Viscous Fluid Dynamics

33

the energy-balance equation, augmented by an equation of state. The standard dependent variables are density, pressure, temperature, and the three velocity components of the fluid. For this analysis, temperature variations will be assumed negligible. For incompressible fluids, the equation of state in piezotropic flow, wherein density is a function of pressure, reduces to an expression of constant density p. In this case, the continuity equation aP

-

at

+v

*

(pu) = 0,

derived by a mass balance with respect to a motionless set 9 c R, reduces to the equation v.u=o (1.4.1) for the velocity u. For a so-called ideal fluid, which neglects the irreversible processes created by internal friction, the set g, in which the balance is computed, may be selected to move with the fluid to obtain the force law, in a distribution sense on 9 = R x (0,To),given by du p - = -vp, dt where p denotes pressure and duldt is computed with respect to moving coordinates. With respect to stationary Cartesian coordinates, du dt

-=

[a

au

-

at

+ (u - V)u,

so that the equations of motion assume the form p

-

]

+ (U'V)U + v p = 0.

(1.4.2)

If a stationary set 9 is now employed to set up a momentum balance of the form

aat

9

pui dx

= -

saia

(nil,n,,,ni3)vdo,

(1.4.3)

then (1.4.2)may be written according to the format, upon use of (1.4.1), (1.4.4a) where the momentum flux tensor

nik

has components

nik = p6ik

+ puiuk.

(1.4.4b)

Thus, the equations of motion for an incompressible ideal fluid are given by (1.4.4). Usual derivations of the (Navier-Stokes) equations of motion for a


34

uiscous fluid employ the format of (1.4.3)and (1.4.4),and proceed by modifying (1.4.4b) to account for internal friction. One writes (1.4.5) nik = p6ik puiuk - oik?

+

where aik is called the viscosity stress tensor and, on physical grounds, is assumed to be proportional to the rate of (shear) deformation, that is (1.4.6) which measures the amount of nonuniform rotation of the fluid. A second (isotropic) component of &, involving the divergence of u, is missing according to (1.4.1).The relationship (1.4.6)is the analog of a stress-strain relationship in elasticity, and characterizes incompressible Newtonian fluids. Altogether, then, we have the equations au 1 (1.4.7) - - q AU + (U V)U + - Vp = 0, at P when (1.4.5)and (1.4.6)are substituted into (1.4.4).Here, q > 0 is the (assumed constant) viscosity, and external forces, such as gravity, are neglected or assumed conservative. Because of the assumption that the motion is isothermal, the energybalance equation is unnecessary, and we are left with the Navier-Stokes system of (1.4.1)and (1.4.7) for the equations of motion of an incompressible fluid in a region R, with velocity u, pressure p , and constant (positive) viscosity and density, q and p, respectively. When these equations are understood variationally, or distributionally, a significant reduction due to Lerayt is possible. We shall describe this reduction in the case where the fluid adheres to its retaining boundary. Following Temam [23], we shall base the analysis on a distribution surjectivity property of the gradient mapping due to DeRham:, and a corresponding isomorphism property of this operator in Sobolev spaces. The following two results are discussed and proved in Temam ([23] Chapter 1, Sections 1 and 2). 0

Proposition 1.4.1. Let R be an open subset of R“ and let 1, . . . , n. A necessary and sufficient condition that f = (fi,

..

f

,fn)

= VP

for some p E W(R) is that

’J . Math. Pures er Appl. XI1 (1933), 1-82;

&id., XI11 (1934), 331-418.

* “Varittts Differentiables.” Paris, Hermann, 1960.

fi E 9’(R), i = (1.4.8a)

1.4


for all 4 E

35

where

v = { 4 E 9 ( R ; R"): v 4 = O } . f;. E H-'(R), i = 1, . . . ,n, then p E Ltc(R). The *

(1.4.8~)

Moreover, if conclusion p E Lz(R) holds if R is a bounded, strongly Lipschitz domain and, in this case, the mapping f + p is continuous from H-'(R; R") to L2(n)/R'. The following corollary is then immediate. Corollary 1.4.2.

Suppose that

AU + (U * V)U E Lz((0, To);.9'(R; R")), and that, for almost all t, 0 < t < T o ,the relation U, - q

(u, - q Au

+ (u .V)u, 4 ) = 0

for all

4 E -Y-

(1.4.9a) (1.4.9b)

holds distributionally. Then there exists p E Lz((O,To);B'(R)), such that

(u,

- q Au

+ (u - V)u + -1 V p , 4 ) = 0, P

for all

4 E B(R; R"). (1.4.10)

Remark 1.4.1. The reduction alluded to earlier is that of (1.4.10) to (1.4.9), provided (1.4.9a) holds. However, (1.4.9a) is weaker than the regularity expected to hold; in fact, the left-hand side of (1.4.9a) is in L'((0, To); H-'(R; R")) for n < 4, and p E Lz(9,,) in this case. To obtain a (weak) variational formulation of maximum sharpness, we have the following proposition, also discussed in Temam ([23] Chapter 2, Section 1). For completeness, and because of modifications in presentation, we present a proof.


36

Proof: have

To prove (1.4.12), we first set n

> 3. By

Holder's inequality, we

Since HA(Q) c L2ni(n-2) (Q) by Sobolev's inequality, (1.4.12) follows directly from (1.4.15). For n = 2, Holder's inequality is applied to give

and (1.4.12) is now immediate from HA(Q) c L4(Q). Now if s 2 ( 4 2 ) - 1, it follows from Sobolev's inequality that Hs(Q) c L"(R),since (lln) 2 (1/2) - (s/n) in this case. This remark applied to (1.4.12) gives (1.4.13).Prior to the derivation of (1.4.14), we observe that

b(u, V, W)

(1.4.16)

= - b(u,W, V)

on W" x W" x V .This follows from

= -

i

i,J= 1

aw.

Jnui&ujdx

=

-b(u,w,v),

-

where we have used integration by parts and V u = 0. Now suppose n 2 3 and s 2 4 2 . By (1.4.16) and Holder's inequality, we have

By Sobolev's inequality applied to the second and third terms in the product in (1.4.17), we have

from which (1.4.14) is immediate. Note that the application of Sobolev's inequality to the second term in the product in (1.4.17) used the inclusion H"-'(Q) c L"(Q).The proof is now complete. Remark 1.4.2. The result of the previous proposition may be used to define b(.;;) on the completion of W" x V x V in

HA@; R") x Hk(Q; R") x (Hs(Q;Rn)n

W))

1.4


37

for s 2 4 2 - 1. It follows from (1.4.16) that the extended form b ( . , . ; ) satisfies b(u, V, V) = 0 (1.4.18) on this completion, which we denote by V x V x V,. In fact, ifR is a bounded strongly Lipschitz domain, then (see [23] Chapter 1, Section 1) V

V,

= {V E HA(R; R"): V =

*

v

= 0},

(1.4.19)

{v E HS(R;R") n HA@; R"):V * v = O}.

In conclusion, we may sharpen (1.4.9b), by requiring that 4 belong to the extended test class V,, provided u is required to belong to L2((0, To);V).

Remark 1.4.3. We shall now consider appropriate generalizations of (1.4.9b). Let P be an orthogonal projection in L2(S2;R"), such that V = closure -Ir

in HA(R;R"),

(1.4.20a)

where V := {V

E

HA(S2;R"):Pv = 0},

(1.4.20b)

and -Ir := {v E C,"(R;[W"):Pv = O}.

( 1.4.20~)

For example, we have seen above that (1.4.20) is satisfied by the projection onto the divergence-free functions. Furthermore, let a( ., ., be a trilinear form on V x V x V,, where s 2 1, and a )

V, = {v E Hs((SZ;R")n

R"):Pv = 0},

(1.4.21)

such that the continuity relation la(u, v9 w)l d ~ll~llvllvllvllwllv, (1.4.22a) holds on V x V x V, for some constant c, as well as the dissipation relation a(v, v, v) 2 0,

for all v E V,.

(1.4.22b)

Finally, set

H = {U E L2(Q;R"):Pu = O}.

(1.4.23)

We assume, explicitly, that

urn- u (weakly in V), u,+u (strongly in H) 3 a(u,,u,,w)+a(u,u,w) forall W E V , . (1.4.24a) This relation does, in fact, hold for the trilinear form defined by (1.4.11) (see Temam [23] pp. 165 and 166).We also require an analog of (1.4.24a) on

1.

38


9,which, in certain cases, is a corollary of (1.4.24a), but is assumed explicitly

here.

-u

(weakly in L2((0,To);V)), u, + u (strongly in L2((0,To);H))

u,

* JOT" u(u,, u,,

W)

dt + JOT" a(u, u, w) dt,

(1.4.24b)

for all w E C([0, To];V,). Again, this relation holds for a as defined by (1.4.11) (see Temam [23] p. 289). As a complement of (1.4.24b), we state a further convergence result, to be used in the sequel, which follows directly from (1.4.22a). u E B(0,I ) c V,

* JOT"

w,

U(U, U, w,)

--f

(strongly in L2((0,To);V,))

w

(1.4.24~)

dt + JOT" a(u, U, w)dt

uniformly in u. We now wish to consider the weak form of the initial/boundary-value problem for au (1.4.25a) _ - v AU U(U,U, .) = 0,

+

at

Pu = 0,

(1.4.25b)

constituting the generalized Navier-Stokes system. For ease of expression in the following definition, we introduce the notation (1.4.26) if ai and b' are the rows of two n x n matrices A and B, for example, A = Vu, B = V4. Here, a b is the usual dot product; we shall frequently suppress the dot for such vectors.

-

Definition 1.4.1. Given initial datum uo E V and a trilinear form a ( . , * ; ) : V x V x V, + R' satisfying (1.4.22) and (1.4.24), we say that u E V (see (1.4.28a)) is a weak solution of the constrained initialboundary-value problem for (1.4.25), if J9[u

- vVu

*

V4

]

dxdt -

so"

a(u,u,4)dt (1.4.27)

1.5 Uniqueness of Solutions

30

holds for all Q E go.Here the solution space 't: and the test function space V0 are defined by 't: = L2((0,To);V) n H'([O, T o ] ;V 3 ,

(1.4.28a)

and 't:, = C([o, TO];V,) n H'([O, T0];L2(R)).

(1.4.28b)

Moreover, the interpretation of the third integral is that of a functional; the regularity condition (1.4.28a) guarantees that u E C([0,To];V:).

Remark 1.4.4. The case of compressible fluids requires a modification in the equations of motion to include a term of the form grad(divu). In this case, the form b( *, ., .) of (1.4.11) may be perturbed by the term (div v, div w), which is insensitive to u. Although our generalized format for the trilinear form can absorb this case, the reduction to (1.4.9b) is no longer valid and the continuity equation must be adjoined.

UNIQUENESS OF SOLUTIONS

1.5

We shall begin with a general uniqueness theorem for equations of the form aTH(u) at

+ u +Tg(u)+Th(u)=0

almost everywhere in R,

0 0, or Do, introduced in Sections 1.1 and 1.2, respectively. The choice T = R, of Section 1.3 is also acceptable with G = H'(R). By a solution of (1.51) is meant a pair [u,H(u)] satisfying (1.51) almost everywhere in R for 0 < t c T o ,where it is required that u E C([O, rO];Lz(Q))n Lz((O,To);G),

g(4, TH(4 E

H(u(x,t)) E [O,H(O+)]

C(10, 7-01 ;L2(Q 1,

if u ( x , t) = 0,

(1.5.2a) (1.5.2b)

H(u) E L2(R), (1.5.2~)


40

a

(1.5.2d)

- THb) E L2(%J,

at

[H(t) - ~ ( s ) ] / ( t- s) 2

if llhllLiP> 0.

A >0

(1.5.2e)

In (1.5.2e),H(0) is understood as the set [O,H(O+)].

Proposition 1.5.1. The solution pair of (1.5.1) is unique within the class described in Definition 1.5.1, provided TH(u)I,= is specified.

,

Proof: We recall that the operator T has the property of being pointwise nonnegative: q B0

* Tq 2 0,

(1.5.3)

4 E LZ(rZ).

Now let u, H(u) and w, H(w) be solution pairs satisfying TH(u)[,=,= TH(w)l,=, . Multiplying the relations (1.5.1) satisfied by u and w, respectively, by H(u) - H(w), and subtracting, we find, after integration over rZ, that

2zllH(4

+ (u - w, H(u) - H(W))LZ(Q)

- H(w)IIk*

+ (TCS(4 - S(W)Il H(u) - H(w)),z,*, + (T[h(u) - h(w)I, H(u) - H(W))LZ(*)= 0.

(1.5.4)

Here, we have used the relation

(&

)

T[H(u) - H(w)], H(u) - H(w)

W*)

d

- 2 zI(H(u) -

- H(w)II&.

(1.5.5)

However, the second and third terms of (15 4 ) are nonnegative by the monotonicity of g and H, and by (1.5.3). By (1.5.2e),we have, thus, +IIHCU)

G

J;

- H(w)ll;* + 1 sgnllhllLip IIu - Wll&(*)dZ

J; I(h(4 - Ww), T[H(u) -

~(W)I)LZ(*)l

dz,

(1.5.6)

after integration from z = 0 to z = t, for 0 c t c T o .By the Cauchy-Schwarz inequality in G*, the Lipschitz property of h, and the continuous injection of L2(rZ)into G, we conclude that I(h(4 - h(w), TCfW - H(W)I)LZ(*)lG llh(4 - ~ ( w ) l l G * l p ( 4- H(W)IlG* G + W l l U - WJl&

+ rI-'IIH(u) - H(w)IIk*)

(1.5.7)

holds for every q > 0, where C contains the explicit factor Ilhl(Lip.If (1.5.7)

1.5 Uniqueness of Solutions

41

is applied to (1.5.6),we obtain

tllH(4 - H(w)ll& + 3E.sgnllhll,*,

< qfor qC

J;

IIU

- W11Zqn)dz

Jot ((H(u)- H(w)(($dz

(1.5.8)

< A. By Gronwall's inequality applied to (1.5.Q we obtain 0 < IIHCU) - H ( w ) ( l Z m ( ( o , T o ) ; C *d) 0,

( 1.5.9)

so that H(u) = H(w). By the invertibility of H , we conclude that u = w. Corollary 1.5.2. The solution u of (1.1.11) is unique within the class V. In fact, if u and w are solutions of (1.1.11) in V, then u = w and H(u) = H(w). Proof: By Proposition 1.1.1, the initial condition (1.1.27) and the relation (1.1.25) hold for any (weak) solution of (1.1.11). The former is simply (1.5.1), with T = Nu, any 0 > 0, and h replaced by the Lipschitz function

h,(t) = h(t) - ot.

( 1.5.10)

The defining properties of V, together with Proposition 1.1.1, show that such weak solutions are solutions of 1.5.1, in the sense of Definition 1.5.1. Note that (1.5.2e)holds via (1.1.8a).The result now follows from Proposition 1.5.1. Corollary 1.5.3.

The solution u of (1.2.15) is unique within the class V.

Proof: We use Proposition 1.2.1 in conjunction with Proposition 1.5.1, with T = Do. Here f = 0 and (1.5.2e) is vacuous.

Remark 1.5.1. The relations (1.1.25)and (1.2.21)hold without the condition L2(gT0),which may be replaced by u E C([O,To];LZ(0)).Thus, uniqueness holds over this broader class. We pass now to uniqueness for reaction-diffusion systems. u, E

Proposition 1.5.4. The solution of (1.3.11) is unique within the class V. Proof: Let u and w be solutions as described by Definition 1.3.2. Then u, w E C([O, T o ] ; L"(R)), hence, have range in a bounded subset of R". Thus, there is no loss of generality in assuming that f is Lipschitz, since it is assumed C' and, hence, locally Lipschitz. Set v ( - , z ) = u - w in (1.3.11).


42

For a fixed i = 1 , . . . ,rn, we have, after subtracting the two relations and integrating from z = 0 to z = t,

+Di

Ji 11 f i ( u i

- Wi)lltZ(dn) d~ = -

Ji (fi(u)

- fi(w),

ui

- Wi)LZ(n) dz,

(1.5.11)

and (1.5.11) holds for all 0 < t c T o . If the Lipschitz property off, is used, and (1.5.11) is summed on i, we obtain, for each 0 < t < T o , m

1 1

1

i= 1

c JO i 1 llui - W i l l t z ( n ) d z . ,l m

IIui

- Willi2(n) Q

(1.5.12)

Here we have used the pointwise Lipschitz estimate Jfi(u)- fi(w)J

< mc (U- W J = -

for some C > 0, and the nonnegativity of the second and third terms in (1.5.11). An application of Gronwall’s inequality to (1.5.12) immediately yields llui - will&) = 0 and, hence, u = w.

17=

Remark 1.5.2. Uniqueness for the generalized model (1.4.27)-(1.4.28) and, indeed, for the Navier-Stokes equations, themselves, is a major open problem. For the latter, only the case n = 2 is satisfactorily understood. The reader is referred to the book of LadyZenskaja [111 for a detailed discussion. Roughly, the class within which global existence can be demonstrated is larger than the corresponding uniqueness class. Within this class, only local existence results can be demonstrated. This type of skewness of existence and uniqueness classes is not unusual in mathematical physics, and accounts for both parts of this book, that is, the local and global theory, are required for an understanding of this model.

1.6

BIBLIOGRAPHICAL REMARKS

Our format in (1.1.1) follows that of Wheeler [25], where the specific problem of thaw-bulb development, near a pipeline submerged in permafrost, is studied. This study was carried out prior to the construction of the trans-Alaska pipeline. Supporting physical discussion is given by Anderson

References

43

and Morgenstern [2], and in the articles edited by Andersland and Anderson [11. Other developments of Stefan-type problems are presented by Rubinstein [19], and in the articles edited by Ockendon and Hodgkins [17]. Crystal growth, in particular, is a rich source of phase-change problems (see Ueda and Mullin [24]). The use of the Kirchhoff transformation to recast such problems is a well-established technique. The physics of liquid filtration in a porous medium is summarized by Philip [lS], and by Bear [4], where it is observed that Darcy’s law is valid only for sufficiently small values of k@. Gas filtration is described by Scheidigger [20]. Our source for the material on chemically-reactingsystems is Aris [3], while Nagumo, Arimoto, and Yoshizawa [16] describe various analog realizations and simulations of the FitzHugh-Nagumo system (see also the survey article of Hastings [7]). Predator-prey models are described by De Mottoni and Rothe [S] (see also Hassell [6]). Another source of reactiondiffusion models is enzyme activation (see Kernevez [9]). Elucidation of the physical models of fluid dynamics can be found in Landau and Lifshitz [12] and in Serrin [Zl]. Our derivation has followed the former. An axiomatic approach, with discussion of the nonslip condition u = 0 on an, is given in Meyer [lS]. The relevant mathematical facts concerning the Navier-Stokes models for incompressible fluids presented in this chapter are based on the book by Temam [23]. Although this model is a common one in applied mathematics, it has been noted by Serrin ([21], p. 179) that viscosity and heat conduction arise from similar mechanisms in kinetic theory and, hence, are of comparable magnitude. Temperature variations, however, influence density, so that the model becomes considerably more complicated in this case, leading to the common suppression of such effects. We note, finally, that authentic physical models often involve the combined effects of the simplified models described herein. Some of these, for example, are described in Rubinstein [19] and the other references noted above.

REFERENCES [l] [2] [3]

0. B. Andersland and D. M. Anderson (eds.), “Geotechnical Engineering for Cold Regions.” McGraw-Hill, New York, 1978. D. M. Anderson and N. R. Morgenstern, Physics, chemistry and mechanics of frozen ground, in Proceedings, North American Permafrost Second International Conference, 2nd pp. 257-295 National Academy of Sciences, Washington, D.C., 1973. R. Aris, “The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,” Vol. I. Oxford Univ. Press (Clarendon), London and New York, 1975.

44

[4] [5] [6] [7] [8] [9] [lo]

[I I] [I21

[ 131 [I41 [ 151 [I61 [I71 [I81

[I91 [20] [21] [22] [23] [24] [25]

1. Formulations for Parabolic Models J. Bear, “Dynamics of Fluids in Porous Media.” American Elsevier, New York, 1972. P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math. 37, 648-663 ( 1979). M. P. Hassell, “The Dynamics of Anthropod Predator-Prey Systems.” Princeton Univ. Press, Princeton, New Jersey, 1978. S. P. Hastings, Some mathematical problems from neurobiology, Amer. Math. Monthly 82,881-895 (1975). A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol. 117, 500-544 (1952). J. P. Kernevez, “Enzyme Mathematics.” North-Holland Publ., Amsterdam, 1980. D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications.” Academic Press, New York, 1980. 0. A. Ladyienskaja, “The Mathematical Theory of Viscous Incompressible Flow,” 2nd English ed. Gordon and Breach, New York, 1969. L. D. Landau and E. M. Lifshitz, “Fluid Mechanics.” Pergamon, Oxford, 1959. Distributed by Addison- Wesley, Reading, Massachusetts. J. L. Lions and E. Magenes, “Nonhomogeneous Boundary Value Problems and Applications.” Springer-Verlag, Berlin and New York, 1972. G. G. Lorentz, “Approximation Theory.” Holt, New York, 1966. R. E. Meyer, “Introduction to Mathematical Fluid Dynamics.” Wiley (Interscience), New York, 1971. J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating (a) nerve axon, Proc. Inst. Radio Eng. 50, 2061-2070 (1962); Bistable transmission lines, IEEE Trans. Circuit Theory 12,400-412 (1965). J. R. Ockendon and W. R.Hodgkins (eds.), “Moving Boundary Problems in Heat Flow and Diffusion.” Oxford Univ. Press (Clarendon), London and New York, 1975. J. R. Philip, The infiltration equation and its solution, Soil Sci. 83, 345-357 (1957); The profile at infinity, ibid. 83,435-448 (1957); Moisture profiles and relation to experiment, ibid. 84, 163-178 (1957); Sorptivity and algebraic infiltration equations, ibid. 84, 257-264 (1957); The influence of the initial moisture content, ibid. 84, 329-339 (1957); Effect of water depth over soil, ibid. 85,278-286 (1958). L. I. Rubinstein, “The Stefan Problem,” Transl. Math. Monographs 27. American Mathematical Society, Providence, Rhode Island, 1971. A. E. Scheidigger, “The Physics of Flow Through Porous Media,” 3rd ed. Univ. of Toronto Press, Toronto, 1974. J. Serrin, Mathematical principles of classical fluid dynamics, “Encyclopedia of Physics,” Vol. VIII, p. 1. Springer-Verlag, Berlin and New York, 1959. S. L. Sobolev, “Applications of Functional Analysis to Mathematical Physics,” Transl. Math. Monthly 7. American Mathematical Society, Providence, Rhode Island, 1963. R. Temam, “Navier-Stokes Equations” (rev. ed.). North-Holland Publ., Amsterdam, 1979. R. Ueda and J. B. Mullin (eds.), “Crystal Growth and Characterization.” NorthHolland Publ.-American Elsevier, Amsterdam and New York, 1975. J. A. Wheeler, Simulation of heat transfer from a warm pipeline buried in permafrost, Exxon Production Research Company Report, Houston, Texas. Presented at American Institute Chemical Engineers Nat. Meet., 74th 1973; Permafrost thermal design for the trans-Alaska pipeline, in “Moving Boundary Problems in Heat Flow and Diffusion” (J. R. Ockendon and W. R. Hodgkins, Eds.), pp. 267-284. Oxford Univ. Press (Clarendon), London and New York, 1975.

CONVERGENT REGULARIZATIONS

AND POINTWISE STABILITY OF IMPLICIT SCHEMES

21

2.0

INTRODUCTION

Regularizations, or smoothings, are defined for the two-phase Stefan problem in Section 2.1, and for the porous-medium equation in Section 2.3, in terms of approximations H , of H for the time-differentiated expressions aH(u)/at, and rates of convergence are derived for each of these two cases, under the assumption that (unique) solutions exist for both the model and regularized problems. The associated existence theory is developed in Chapter 5 for the nonregularized problem (see Theorem 5.2.1; also Corollaries 5.2.7 and 5.2.8). The backward-Euler, or fully-implicit, method, appropriately combined with an explicit method, is introduced for the purpose of deriving, in Sections 2.2, 2.4, and 2.5, semidiscrete maximum principles. These are expressed, via pointwise L"(l2) bounds at subsequent discrete times, in terms of corresponding bounds of prescribed initial data. In fact, L P bounds for p 2 2 are derived, in the process, for the Stefan problem. Such bounds are used in the sequel to deduce uniform bounds on space-time domains, in terms of initial data, for the solutions of the evolution models introduced in Sections 1.1, 1.2, and 1.3. The associated existence theory for the semidiscrete equations, for general classes of time approximations, including the fully-implicit 45

2. Regularizations and Pointwise Stability

46

method as a special case, is developed in Chapter 3. In the case of the semidiscrete approximations for the reaction-diffusion systems, it is shown in Section 2.5 that the solutions continue to have range within the invariant region Q at subsequent discrete times.

2.1

REGULARIZATION IN THE STEFAN PROBLEM

The hypotheses on the (discontinuous) enthalpy-type function H are stated in Remark 1.1.1. In this section, we shall smooth H to obtain a strictly monotone increasing, continuously differentiable, Lipschitz function HE, such that H Eand H: approximate H and H' on compact subsets of R1\{O}. This will be followed by a regularization theorem. The construction follows. In the sequel, H- denotes a left inverse for H.

Definition 2.1.1. Set 6, = IH'(O+) 0 < E, < min(A, I), such that

and define 6 = 1 + 6,. Select

- H'(O-)I

~H'(E) - H'(O+)~ < 6/2,

if

If 6 , = H'(0-) and

+

o < E < E,.

"1

E1=min[ 66, 2A 36' 36 '

(2.1.1)

(2.1.2)

set E , = min(e, ,E ~ ) .Henceforth, we restrict the smoothing parameter the interval 0 < E < E,. For fixed E, define the derivative h, of H e by h,(t) = H'(t),

t < 0,

h,(t) = q,(t),

0 0,

0d t

< E,

(2.1.7)

where 2 is the lower bound of H' assumed in Remark 1.1.1.

Proposition 2.1.1. H , and H : converge uniformly to H and H', respectively, on compact subsets K of R1\{O}. In fact, the estimate,

0 < H ( t ) - H,(t) d

,UE

for t 2 E

(2.1.8)

holds, where p is given in (1.1.8a). Moreover, H,(t)t 2 0 for t E R' and

y/& 2 HL(t) 2 i> 0,

t E R',

(2.1.9)

for some positive constant y not depending on E. If J = H - ' and J , = He-', then J is a locally absolutely continuous, monotone increasing function, and IJ,(t) - J(t)l < (1

+

/A/&,

t E R'.

(2.1.10)

Proof: Since H : = H' and H , = H on (- m,O) we may assume, for the convergence assertions concerning HL and H , , that K c (0,a).Now let E


48

be given and set t 2 E . We have

=

Ji H’(s)ds < p ~ ,

which verifies (2.1.8)and the assertion concerning K. The inequality H : 2 il is immediate from (2.1.7), and the definition of H E , whereas H : = O ( ~ / E ) follows from (2.1.6).That H,(t) has the sign of t is immediate from (2.1.5). In order to verify (2.1.10),first note that J ( t ) = 0, 0 < t < A, J’ = l/(H’ J) on R’\[O, A], hence, J is locally absolutely continuous. Also, (2.1.4) and (2.1.5)imply that HE(&) = A, so that J,(A) = E. Now J ( t ) = J,(t) for t < 0 and, for 0 < t < A, we have 0

J&(t)- J ( t ) = J&(t)< J&(A)= E.

(2.1.11)

For t > A, there are two cases, depending upon whether J ( t ) < E or J ( t ) 2 E. In the former case, we have t < H ( E )= A

so that t - A

+

+Jipds =A +~ p ,

 A and J ( t ) < E. For J ( t ) 2 E , select s such that J,(s) = J(t). We have, by the mean-value theorem, J&(t) - J ( t )= J&W- J&W= Ji(t)(t - 4, for 5 on the open interval determined by t and s. However, 1s - t( = IH& J ( t ) - H 0

0

J(t)l < pE,

by (2.13).Combining the two previous relations gives IJ&(t) - J(t)l < p@,

(2.1.13)

for t > A and J ( t ) 2 E. The relation (2.1.10)is now immediate from (2.1.11), (2.1.12),and (2.1.13). W

Remark2.1.2. It is natural to use the approximation H e to define regularized parabolic initial value problems with homogeneous Neumann boundary conditions. The next proposition describes the rate of convergence of

2.1 Regularization in the Stefan Problem

49

the solutions uE of these regularized problems to the solution satisfying (1.1.11). However, we find it more economical to use the equivalent formulation, described by Proposition 1.1.1, in formulating these results. Recall that Proposition 1.5.1 guarantees uniqueness over the class Wl

=

C"0, T01;L2(R))n L2((0,TJ;H1(Q)).

Proposition 2.1.2. Let T = N u , o > 0, be the Neumann inversion operator defined in Definition 1.1.2 and, if denotes the class defined above, suppose that u, u' E W 1, respectively, satisfy (1.1.25) and aTH,( uE) +uE+Tf,(uE)=O almost everywhere in R, O 0 for t > ti,0 0 for a < t < b. Note that o is an absolutely continuous function of t, and its derivative exists and is equal to $(t)4(t)at each point of continuity of $4,

-=

2.2 Semidiscrete Regularization in the Stefan Problem

53

i.e.,fortE(tk-l,tk),k= 1, . . . , M.Thus,by(2.2.1), w‘ =

*+ < *wl/P,

(2.2.5)

and we obtain on (a, b) the formal differential inequality w - ‘Ip dw

< $ dt.

(2.2.6)

Integration of (2.2.6) yields, for p > 1,

since o ( a ) = oP.Since 4 p< w , we immediately obtain (2.2.2a) from (2.2.7). t,b(z)d.r} to (2.2.5), If p = 1, we apply a standard integrating factor exp{ and obtain

-c

1, and

(2.2.11b) for p = 1. We summarize this result as

Corollary 2.2.2. If (2.2.9) holds for a given partition 9 of [0, To] and some

p, 1 G p < 00, then (2.2.11) is valid.

Remark2.2.2. A little reflection shows that the preceding results are sharply formulated for p = 1, but not so for p 1, in the following sense. It is possible to weaken hypothesis (2.2.1) to permit integration up to t,, at the cost of slightly weakening conclusion (2.2.3a),with respect to the constant multiplier and, of course, subsequent summation to k in the bound. We illustrate this for the case p = 2. Suppose, then, that

=-

4; < 0'

k

+ 11 $,,,4,,,(tm- t m - l ) ,

k

=

1,. . . , M .

(2.2.12a)

m=

Then

4 k < fl +

k

1 $m(tm - tm-l),

k

=

m= 1

1,. . . ,M .

(2.2.12b)

This implication is a corollary of the following. Proposition 2.2.3. Suppose {ak}and {bk} are two sequences of nonnegative numbers, such that 0, of L'(R) initial data. Note that modifications apply if (2.2.25) fails for p = 1.

2.2 Semidiscrete Regularization in the Stefan Problem

57

Prior to the statement and proof of the next proposition, we shall introduce the truncation operators which provide a way of avoiding the hypothesis (2.2.25). Definition 2.2.2. We define a sequence of bounded Lipschitz continuous, cutoff functions of the identity by

(2.2.29a) and the corresponding truncation operators O j :L’(0) -P L“(R) by OjZ

=

oj

0

z,

z E L’(R).

(2.2.29b)

The “product” truncations Oj,q:L’(R)+ L“(R) are then given by

o , ,=~(ej~

z)q,

E

L ~ R ) , 2 1,

(2.2.30)

when this exponentiation is well defined. Here j is a positive integer. Remark 2.2.5. It is readily seen that Oj:qmaps H’(R) into itself. This “ring” property does not hold for H’(R) functions themselves except for n = 1. Proposition 2.2.4. Suppose that the initial datum satisfies H(uo) E Lp(R) for 2 < p < 00, and suppose that {@} is a semidiscrete regularization as in Definition 2.2.1. Then

Proof: We shall obtain the estimates (2.2.31)-(2.2.32) for the cases q = 21/(2m - 1) 2 2, where 1 and m are positive integers. Since such rationals are dense in [2,00) via tertiary expansions, for example, the result for general p follows by letting q 7 p. The essential property of such q, which is used in


58

this proof, is the fact that

q P Z ( t 2) 0,

t E W.

(2.2.33) Now, set u = O j , q - l ( H , ( u 2 N )in ) (2.2.18), and assume, inductively, that uipNE Lq(zZ)for i < k. Note that q - l ( t ) t 2 0,

u y := J, H,(u"d) = J,(H(uo)) 0

(2.2.34)

has L4 norm G ( 1 / 4 l p ( U O ) I I L q n ) . Since g and g- H , have the same sign as t, then 0

(g(UY",

@j,q-

~(H,(U;~N)))L~(~) 2 0,

(2.2.35)

and since

(2.2.36a) then, by the monotonicity of H , , and by (2.2.33), (VU:N,

VOj,,- 1 ( H , ( U : N ) ) ) L 2 ( n ) 2 0.

(2.2.36b)

Hence, we obtain from (2.2.18),after using the fact that the identity dominates O j , in the sense of absolute values, Jn

loj

H , ( U : ~ ) I ~G

1

Jn

lej

+ 41 Jn p,(U:-Nl) - h(U;~l)(t; - t;- l)p

~,(Uy)p

-

(2.2.37) where we have used the inequality 1 1 ab 0, 1 f

d

1'

+ ~ ( J , ( u ' -) J,(u), U& -

- ~ll&-i(fi)

< 2clRle'

~)Lz(n)

+(l/y).

(2.3.17)

Here, we have used (2.3.8b), and the inequality

(uE- u, U E -

= ( H , 0 .I& - H@ &0 J&(U),uc & - u)LZ(R) )

< C E ( l / Y ) - l (J&W) - J,(U),

uE- V)L2(R),

which follows from (2.3.7).The inequality (2.3.13) follows from (2.3.17),upon integration from 7 = 0 to z = t, that is, llVE - UllA- '(Q)

+ J)J&(U&)

- J,(u),

U E - V)L2(&

d 4 c p ( T o & ' + ( ' ~ ~ )0, < t < T o .

(2.3.18)

To obtain (2.3.15), we repeat the steps which led to (2.3.16). This gives

rl -dtd( p & ( u e-) w ) l l & - l ( n )+ be- u, W U E )- H(u))L'(n) = (UE - u, H(UE)- H&(U&))L2(n)

6 ${?llUE - UII&(n)

+ V-~IIH,(~ 'H(u~)II&)}, )

(2.3.19)

where the choice of q > 0 is at our disposal. In order to utilize the coercive properties of the second term in (2.3.19), we require the inequality (2.3.20)

=-

for y 2 0, z 2 0, y # z, and r 0, s > 0, r # s (see [l2], p. 85). The identifications r = 1, s = l/y, y = [H(u')]', z = [H(u)]' in (2.3.20), in conjunction with the assumed relations (2.3.14) and (1.2.14e) lead to, with K = (1 + c 0 y - y K Y p W ) - H(u)((cdJ)Y-lIlu 11 1 - ( l / r ) 0 Lyn) 2 y I ~ ( u &-) ~(u)I[max([~(u")]~, [H(u)]Y)]'-('/Y)

2 I[H(U"]Y

-

[H(U)]Yl 2

clue- UI,

(2.3.21)

where the latter inequality, with C = [c1yIy,follows from the fact that the composite function K ( t ) = [H(t)IY,

t20

(2.3.22a)

t 2 0.

(2.3.22b)

has derivative K', satisfying K'(t) 2 [c1yIy,


66

If (2.3.21) is substituted into (2.3.19), we obtain, for uo # 0,

< +qIIU& - U1l:Zcn) + 41-‘c2JRle2’Y,

(2.3.23)

where we have used (2.3.8a). Multiplying (2.3.23) by for sufficiently small q (say, q = ( C / ( q ) ) ( c 2 y -Yl[uo[l~L,$~j )’ I),

soTo

(Id- uIl&,,,dt

< C,l(u0l12(1-(1’y))e2’y, Lyn)

we obtain,

0 < t < T o , (2.3.24a)

where

The inequality (2.3.15) follows immediately from (2.3.24). H

Remark 2.3.2. Uniqueness of solutions of (1.2.21) and (2.3.12) follows from Proposition 1.5.1. Although it is certainly possible to parallel the development of the previous section in developing semidiscrete regularizations, we shall consider the implicit method for the nonregularized problem only. A major reason for this is that it is possible to derive the maximum principles, and, in fact, the entire existence theory, directly from the nonregularized formulation. The rate of convergence in (2.3.13) and (2.3.15) is preserved under the hypothesis (2.1.25b) if, instead of the specification u: = J, 0 H(uo), made in the regularization of Proposition 2.3.2, one specifiesthe initial datum ui = uo. The latter choice of the initial datum leads to u: E L2(R x (0,co)) if uo E H;(R). In (2.1.25b), the zero set of uo may be ignored. 2.4

NONNEGATIVE SEMIDISCRETE SOLUTIONS

OF THE POROUS-MEDIUM EQUATION AND MAXIMUM PRINCIPLES

We begin by defining the semidiscrete approximations.

Definition 2.4.1. Let 0 = t: < t y < * < t&,, = T o denote an arbitrary sequence of partitions BN of [0, To].By the implicit semidiscrete solution of the initial/homogeneous Dirichlet boundary-value problem corresponding to (1.2.13) is meant a recursively generated finite sequence { u ~ } ~ Afor ~ )each , 1

2.4 Semidiscrete Porous-Media Solutions

N

=

67

1, 2, . . . ,satisfying

4E H m ,

(2.4.1)

and the variational conditions

- H ( u ~ i)], - ~)Lz(n)+ (Vuf, Vu),z(n) for all u E Hh(R), for k = 1, . . . , M ( N ) . We require that (tf

- tf-

1)-

'([H(uf)

u: = uo,

all N,

= 0,

(2.4.2) (2.4.3a)

and uo 2 0,

uo E

L"(R).

(2.4.3b)

Proposition 2.4.1. Solutions of (2.4.1)-(2.4.3) exist and are uniquely determined and nonnegative on R. They satisfy the weak maximum principle ll4lL=(n) G

114-lIILm(n)G IIUOII'=(n),

(2.4.4)

for all N 2 1, k = 1, . . . , M ( N ) .

Proof: The existence and uniqueness of uf follows from Proposition 3.3.2. Suppose, for k 2 1, that uf- 2 0. Using the decomposition of uf into positive and negative parts, we obtain from (2.4.2), since (uf)- E HA(R),

(2.4.5) so that

and it follows that (uf)- = 0. Thus, uf 2 0. The verification of (2.4.4) also proceeds by induction on k, and utilizes the truncation operators introduced in Definition 2.2.2. Thus, for I2 1, set = @ j , l Y ( H ( 4 ' ) )= @ j , l ( [ H ( 4 ) ] Y )

(2.4.7)

in (2.4.2). Since HYis a Lipschitz function vanishing at zero, it follows that

u E HA(R), and such a choice is permissible. In analogy with (2.2.36), we have

(Vuf, VV)LZ(*, 2 0,

(2.4.8)


88

where we have used the monotonicity of H and the nonnegativity of uf. We, thus, conclude from (2.4.2), by the use of (2.4.8) and (2.2.38), that

+

for q = l y 1 and r = q/(q - 1). Here, as in the derivation of (2.2.37), we have used the domination of 6, by the identity, and we have set a = H(u[- 1) and b = [ O j 0 H(u[)]'-'. The inductive proof is concluded by applying the equality (l/q) (l/r) = 1, taking the qth roots and letting l = ( q - l)/y tend to infinity. W

+

Remark 2.4.1. Strictly speaking, the proof has shown that I I H ( u ~ ) ( I ~ d ~ ( ~ ) llH(uf- l)IILm(n, ,< ~ ( H ( U ~ ) ~ However, ~ ~ ~ ( ~ )we . obtain (2.4.4) directly by utilizing the continuity and monotonicity of H. Note that similar estimates are valid for the solutions uf of the semidiscrete regularized problems, which we have not formally introduced. One might expect a sharper version of (2.4.4) on the basis of recent decay estimates for the parabolic problem (see Aronson and Peletiert).

2.5

INVARIANT RECTANGLES AND MAXIMUM PRINCIPLES FOR REACTION-DIFFUSION SYSTEMS IN SEMIDISCRETE FORM

We begin by defining the implicit semidiscretization for the reactiondiffusion systems. We maintain the hypotheses of Section 1.3. Definition 2.5.1. Given a sequence 9" of partitions of [0, T o ] ,the semidiscrete solution of the initial/Robin boundary-value problem corresponding to (1.3.4)and (1.3.9) is a recursively generated finite sequence {uf}rAy),for each N = 1, . . . ,satisfying the condition that uf

E

(2.5.1)

H'(R; Rm),

the variational conditions, for v E H'(R; Rm), (tf - tf- 1 ) - ' ( ( 4 , i -

+ (fi(& + Di(miu,N,

1, 1 9

*

4-I,i),Vi)L2(R) + D i ( V u c i , VVi)L2(R)

..

9

N uk- 1, i - 1 9 $i,

ui)LZ(aR) =

0,

J . Differential Eqs. 39, 378-412 (1981).

N uk- 1 , i + 1 >

...

9

uf- 1,mh Ui)L2(R)

(2.5.2a)

2.5 Invariant Rectangles

69

i = 1,. . . , i,, and

i = i,

+ 1,. . . ,m.We require, in addition, u; = u,,

(2.5.3)

f(uf) E LZ(i-2;W),

(2.5.4)

= Q,

Range($)

(2.5.5)

all N >, 1, k = 1, . . . , M ( N ) . The partitions are required to satisfy (3.3.29) and (3.3.30)to follow. Remark 2.5.1. Throughout this section, we shall assume that the region Q is actually contractive, that is, strict inequality of the vector field holds on the faces of Q. More precisely, (1.3.6b) is strengthened to

fib9 < 0,

fi(v) > 0,

v E Qai,

v E Qbi,

(2.5.6)

i = 1, . . . , io. This restriction will be removed in Section 5.3. We shall also assume that the Robin inversion operators Rm, possess the property that RmiLP(Q)c W2*p(R),

1 < p < 00,

(2.5.7)

which, of course, is a restriction upon R. Definition 2.5.2. Let ei denote the unit vector in R", whosejth component is hi,, and set, for 1 O},

(2.5.9a)

R - = {x E R:g&(x)< O},

{x E R:g&(x) = O}.

(2.5.9b)

Finally, introduce the notation

6 = {i:l O}.

(2.5.10)


70

Remark 2.5.2. Our analysis will proceed by showing that solutions uf of (2.5.1)-(2.5.4) necessarily satisfy uniform bounds in Lm(R;Rm)and satisfy (2.5.5). We shall do this in stages. The verification of (2.5.5) is complicated by the fact that the components of R, and R- need not have boundaries sufficiently regular for integration by parts to hold, thus necessitating component approximation. The correlative analysis makes fundamental use of (2.5.6). Proposition 2.5.1. Unique solutions of (2.5.1)-(2.5.4) exist in Lm(Q Rm), and satisfy the estimates (see (2.5.14) and (2.5.15)) IIULllLm(n)

< IIUO,illLm(n) + ( s u p l ~ l ) ~ O , 1 < i < io,

m

(2.5.11a)

m

C IIUj?IiIILm(fi) < CCiTo + (1 + Cz) iC= Iluo,ill~-(n)]exp(CzTo), (2.5.11b)

i= 1

1

for k = 1,. . . ,M ( N ) , and N 2 1. Moreover, for such k and N, u$

E W2*p(R)

for 1 < p < co

and i E 6.

(2.5.12)

Proof: The existence and uniqueness of solutions of (2.5.1)-(2.5.4) follow from Proposition 3.3.3. The hypothesis (2.5.7), in conjunction with a bootstrapping sequence of applications of Sobolev's inequality, yields (2.5.12), via induction on k. To outline the verification of (2.5.11), we note that, in analogy with (2.2.39), we have, for a sequence of even integers q, 1 < i < io,

IIUEillLq(n)

< 114-'l.illLq(n) + (su~\fil)(tf- C- 11,

IIU&llLq(n)

< lluf-l,illLq(Q) + Ilhil(Lip 1IIUk-l,IIILq(n)(tf - lf-

m

/

for

m

\

k

m

1)

(2.5.13a)


71

By applying to (2.5.14) the version of the discrete Gronwall inequality contained in Corollary 2.2.2, and letting q -+ 00, we obtain (2.5.11b), with (2.5.15a) and Cz

m

=

1

(2.5.15b)

IIhilJLip.

i=io+l

Note that (2.5.11a)is an immediate consequence of (2.5.13a).

Remark 2.5.3. The functions uEi are continuous, hence, R + and R - are open subsets of R with compact boundaries. For i E 6, the Lipschitz continuity (in fact, ClSA property) follows from (2.5.12). For i # 6, the Lipschitz continuity follows from the arguments preceding Proposition 3.3.3 and from the Lipschitz assumption on uo. Proposition 2.5.2. For i E 6, 1 d i d io, let R, be a component of R, (respectively R-). Then Jn, - Auti(uZi- bi)+dx 2 0

respectively

).

- Au$(u[~ - ai)- dx 2 0 n o

(2.5.16)

Proof: Each component R, of R, and R- may be approximated from within, to any desired accuracy in the Hausdofl metric, by an open set R, ,with boundary sufficiently smooth to apply the integration-by-parts formula. This may be seen by taking a finite subcovei of balls of the compact boundary of R,, and smoothing the piecewise spherical boundary of the resulting finite cover. We may do this in such a way that an, n aR 3 aR, n an. In particular, the integration-by-parts formulae

(2.5.17a)

(2.5.17b)


72

hold on the approximation domains a,. Note that, for R, a component of R- and for d(R,, a,) sufficiently small, it follows, from the restriction of the closed zero set off;. to the open region strictly between QOiand Qbi(see (2.5.6)), that on aR,\aR, n 8R.

(u$ - a,)- = 0

(2.5.18)

In particular,

(2.5.19) where we have used the Robin boundary condition and wiu;i(u[i

- a,)- = Oi[Ui

+ (UE,- ui)](u;i - a,)- 2 0.

In a similar way, we deduce

(2.5.20) for R, a component of R + and d(R,,R,) sufficiently small. Thus, taking limits in (2.5.19) and (2.5.20) (as d(R,,R,) + 0), we obtain (2.5.16).

Proposition 2.5.3. in Q.

If {u:}~L~)satisfies (2.5.1)-(2.5.4), then the range of :u is

Proof: We use the pointwise relations (t; - rF- l)-l(u:i -

uF-

l,i)

- D,Auci + g t i = 0,

(2.5.21a)

which hold almost everywhere on R for i E 6 and 1 ,< i ,< i,, and (t: - q - J - l ( u t i - u:-

+ g& = 0,

(2.5.21 b)

for i # 6 and 1 0, and these are not necessarily mutually exclusive alternatives. We shall show that the second of these implies the first. Multiplication of (2.5.21) by (uk,i - ui)-, and integration over 0, c R - yields, after addition and subtraction of a,, 0 < Jn,(u& - U ~ ) ( U ;~ ail- dx

< jn, (uF- l , i - ui)(uci- ail- dx < 0, (2.5.22)


73

where we have used g;, < 0 in R-, (2.5.16), and the inductive hypothesis. We conclude from (2.5.22) that u;, 2 a, in R-, and, using (1.3.6d), that uzi 2 aiin R. A parallel argument shows that u;, < biin R or else meas(R+)>0, and t hat the second of these alternatives actually implies the first, upon multiplication of (2.5.21) by (u:, - bi)+,and integration over R, c R,. In particular, the range of ur lies in Q. W

2.6


The 'negative' norms introduced in Chapter 1, as equivalent dual space norms, are variations of those introduced by Lax [ll] into the theory of partial differential equations (see also Leray [13]). Their use in numerical analysis dates from the work of Bramble and Osborn [3] on eigenvalue problems. The convergence results contained in Propositions 2.1.2, 2.2.5, and 2.3.2 depend upon the systematic use of these norms. They will be employed again, particularly in Chapter 4. The explicit smoothing of Section 2.1 was introduced in [8], whereas the smoothing of Section 2.3 is new to the writer. The idea of using the lifted equations as primary pointwise relations, defined by the operators Nu,Do,and R,, appeared in the paper of the writer and Rose [9], where Proposition 2.1.2 was derived, as well as the uniqueness of solutions in the two-phase Stefan problem. The operators Nu, CJ > 0, make possible the derivation of uniqueness results for models involving nonlinear terms f with monotone components, not necessarily Lipschitz. This is due to the fact that Nu is (pointwise) nonnegative for o > 0. This remark also applies to Do and R,. The writer is indebted to Veron (Introduction [9]) for the statement and proof of Proposition 2.2.3. The discrete Gronwall inequalities contained in Proposition 2.2.1 and Corollary 2.2.2 are familiar to numerical analysts, although the formulations and proofs are our own. The derivation of the semidiscrete stability relations in L", for the case of the porous-medium equation, is closely tied to the existence of u, as an L2 function in this problem (see the introductory part of Section 5.1 for details). We could, of course, have derived comparable Lp estimates, but their limit-case validity for the solution of the porous-medium equation would necessitate a relaxation in the regularity prescribed in (1.2.16a), with associated changes elsewhere. The character of the results on invariant regions is due to the earlier investigations of Weinberger [15], and Chueh et al. [4], although the approach we have developed here is independent of these investigations. Also, the writer has not observed a mixing of the concentration and potential

2.

74

Regularizations and Pointwise Stability

variables in any analysis of reaction-diffusion systems. The explicit determination of invariant regions for a specific reaction-diffusion system with given vector field is a major step in the application of these results. Some examples, including nonrectangular invariant regions, are discussed in Chueh et al. [4]. The FitzHugh-Nagumo system exclusively is considered by Rauch and Smoller in [141, where critical lower bounds and upper bounds on size are obtained for contracting rectangles. A study of some specific models, such as predator-prey models, reveals the inadequacy of attempts at global existence proofs based upon a priori estimates insensitive to the location and/or size of the initial datum. Some type of comparison or weak maximum principle appears necessary even to obtain existence in such cases. Sections 2.2, 2.4, and 2.5 derive various formulations of weak maximum principles at discrete times within space-time cylinders. (see also some alternative discrete time investigations of Weinberger [16]). A qualitative study of asymptotic and stationary behavior of solutions of reaction-diffusion systems and associated properties, such as bifurcation of solutions, is outside the scope of this analysis. For this, we refer the reader to the associated literature, for example, Aronson and Weinberger [11, Auchmuty and Nicolis [2], Fife [S], Fife and McLeod [6], Greenberg et al. [7], as well as to the bibliographies of these references. A treatment alternative to invariant rectangles is given by Lopes [ J . Diflerential Equations 44, 400-413 (1982)l. REFERENCES D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in “Partial Differential Equations and Related Topics” (J. A. Goldstein, ed.), Lect. Notes Math. 446,pp. 5-49. Springer-Verlag, Berlin and New York, 1975; Multidimensional nonlinear diffusions arising in population genetics, Adu. Math. 30,33-76 (1978). J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations: I. Evolution equations and the steady state solutions, Bull. Math. Biol. 37, 323-365 (1975); 111. Chemical oscillations, ibid. 38, 325-349 (1975). J. H. Bramble and J. E. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators, in “The Mathematical Foundations of the Finite Element Method” (I. Babuska and A. K. Aziz, eds.), pp. 387-408. Academic Press, New York, 1972; Rate of convergence estimates for non self-adjoint eigenvalue approximations, Math. Comp. 27, 525-549 (1973). K. Chueh, C. Conley, and J. Smoller, Positively invariant regions for systems of nonlinear parabolic equations, Indiana Univ. Math. J . 26, 373-392 (1977). P. C. Fife, Stationary patterns for reaction-diffusion equations, in “Nonlinear Diffusion” (W. E. Fitzgibbon and H. F. Walker, eds.), Res. Notes Math. 14, pp. 81-121. Pitman, London, 1977.

References

75

[61 P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal. 65, 335-361 (1977). c71 J. M. Greenberg, B. D. Hassard, and S.P. Hastings, Pattern formation and periodic structures in systems modeled by reaction-diffusion equations, Bull. Amer. Math. SOC. 84, 1296-1327 (1978). J. Jerome, Existence and approximation of weak solutions of nonlinear Dirichlet problems with discontinuous coefficients, SIAM J . Math. Anal. 9, 730-742 (1978). J. Jerome and M. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), in press. 0. A. Ladyienskaja, V. A. Solonnikov, and N. N. Ural’ceva, “Linear and Quasilinear Equations of Parabolic Type,” Trans. Math. Monographs 23. American Mathematical Society, Providence, Rhode Island, 1968. P. Lax, On Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8, 615-633 (1955). E. B. Leach and M. Sholander, Extended mean values, Amer. Math. Monthly 85,84-90 (1978). J. Leray, “Hyperbolic Differential Equations.” Princeton Univ. Press, Princeton, New Jersey, 1952. J. Rauch and J. Smoller, Qualitative theory of the FitzHugh-Nagumo Equations, Ado. Math. 27, 12-44 (1978). H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. Univ. Roma 8,295-310 (1975). H. F. Weinberger, Asymptotic behavior of a class of discrete-time models in population genetics, in “Applied Nonlinear Analysis” (V. Lakshmikantham, ed.), pp. 407-422. Academic Press, New York, 1979.

NONLINEAR ELLIPTIC EQUATIONS AND INEQUALITIES

3

3.0

INTRODUCTION

The method of semidiscretization in time, when applied to nonlinear parabolic evolutions, gives rise to elliptic equations and/or inequalities. The same is true of various steady-state theories. In this chapter, we provide the required theoretical foundations for studying such elliptic problems. The principal results of Section 3.1 are Tarski’s fixed-point theorem (see Proposition 3.1.10) and an existence theorem for “equations” of the form f E 84(u) + A(u), where 4 is proper, convex, and lower-semicontinuous, and A is pseudomonotone (see Corollary 3.1.7). Here, 84 is the multivalued subdifferential mapping with range in an appropriate dual space. In Section 3.2, four applications are presented. Perhaps the second, which specializes 8 4 ( u ) + A(u) to those applications met in Chapter 1, and the third, related to the Navier-Stokes equations, are the most pertinent. An application of the Tarski fixed-point theorem to a quasi-variational inequality is also presented. In Section 3.3, standard interpolation and extrapolation methods are discussed for the purpose of defining semidiscretizations for nonlinear evolution equations. Although these methods are quite routine in the theory of initial-value problems for ordinary differential equations, the novelty 76

3.1

General Operator Results

77

here is that they arise naturally from quadrature formulae for the lifted versions of parabolic evolutions. This demonstrates the theoretical interest of the lifting operators, over and above their technical use as demonstrated in the uniqueness and error analysis of the previous two chapters.

3.1

GENERAL OPERATOR RESULTS IN BANACH SPACES AND ORDERED SPACES

The development of this section will proceed to general Banach space results, after a fairly thorough treatment of the finite-dimensional case, which proves pivotal. The initial concept is that of the proximity mapping, which generalizes the notion of a projection onto a closed, convex set in Hilbert space.

Definition 3.1.1. Let V be a Hilbert space and let +:V -P ( - 00, a] satisfy f 00, that is is proper. We suppose is convex and lower-semicontinuous on V. By the proximity mapping, prox,: V -P V, is meant the mapping

+

+

+

x -P prox,(x) = u,

(3.1.1a)

F(u, x) = min F(u, x),

(3.1.1b)

where u (uniquely) satisfies UEV

and F( ., .) is given by F(u, x) = t l ( u - xl12

+ +(u).

(3.1.1c)

Remark 3.1.1. The existence of a minimum of F is now a standard result (see Ekeland and Temam [13] p. 35). Use is made of the weak lowersemicontinuity and coerciveness of F. Note that the assumed lowersemicontinuity of is preserved in passing from the strong to the weak topology. The coerciveness property, F(u) + co as llull+ co, follows from the fact that is bounded from below by a continuous affine function. The uniqueness follows from the strict convexity of 11 .[I2, hence of F. Just as the standard projection in Hilbert space has a familiar variational inequality characterization, the same is true of the generalized proximity mapping. For convenience, we quote the result (see [131 Sections 2.2 and 2.3).

+

+

3. Nonlinear Elliptic Equations and Inequalities

78

Lemma 3.1.1. The element u = prox4x is characterized by either of the following equivalent conditions : (u - x,

u - u ) + $(u)

( u - x, v - u)

- $(u) 2

0,

+ $(u) - $(u) 2 0,

for all u E V,

(3.1.2a)

for all u E V.

(3.1.2b)

for all x, y E V.

(3.1.3)

Moreover, prox4 is nonexpansive, that is Ilprox4x - prox4 Yll G 1Ix - Yll,

Remark 3.1.2. Let $ be a proper, convex, lower-semicontinuous function on V as above. Let Dom $ = { u E V:$(u) < co}.

(3.1.4)

Then Dom $ is a convex subset of V and prox4 has range in Dom 6. Example 3.1.1. Let K be a nonempty, closed convex subset of V. If $K: K -,(- co,001 is a given proper, convex, lower-semicontinuous function, then so is the extension,

(3.1.5) to V. An application of the previous lemma and remark gives the following characterization of u = prox4x = p r o x G .

+ &(u) - X, u - U) + &(u)

0,

for all u E K,

(3.1.6a)

2 0,

for all u E K.

(3.1.6b)

(U - X, u - U)

- &(u) 2

(U

- &(u)

Definition 3.1.2. Let S be a finite-dimensional normed linear space with dual S*, and let $ :S -+ (- co,co] be a proper, convex, lower-semicontinuous function. Let A: S -,S* satisfy A is continuous,

(3.1.7a)

and, if Dom $ is unbounded,

where o:[p, co)-+ [0, co) is a monotone increasing function with w - ' ( t ) bounded for each t. Here, uo is assumed to be a fixed element of S with 0 G lluoll < p and p is a fixed positive number.

3.2 General Operator Results

79

Proposition 3.1.2.t Let S, A, 4, a,uo, and p be given as in the previous definition. If Dom 4 is bounded in S, then the inequality (A(u) - f , u - u )

+ 4 ( u ) - +(u) 2 0,

for all u E S,

has a solution u E Dom 4 for f E S*. If Dom 4 is unbounded, let the function, with Dom 4robounded, given by

+,,

(3.1.8) denote (3.1.9a)

for ro > p fixed, and let u,, be a solution of the inequality (A(u,,) - f , u - u,,)

+ 4,,(u) - 4,,(ur0) 2 0,

for all u E S. (3.1.9b)

If ur0 # 0, and

then theinequality(3.1.8)hasasolutionuEB(O,c),wherec = sup{t:o(t) = u}. If (3.1.9b) has only the solution u,, = 0 for all ro > p , then u = 0 is a solution of (3.1.8). In order to indicate the usefulness of Proposition 3.1.2, we delay its proof to present two corollaries.

Corollary 3.1.3. If o ( l [ u [ (+ ) co as [IuI(-+ 00, then (3.1.10) is necessarily satisfied for every solution u,, # 0 of (3.1.9b),and (3.1.8) has a solution for each f E S* under the hypotheses of Definition 3.1.2. Corollary 3.1.4. Let K be a nonempty, closed, convex subset of S containing 0. Then the inequality (A(u) - f , u - u ) 2 0,

for all u E K,

if llfll has a solution u E C1 B(0, sup o-'([If/l)), equation

E

(3.1.11)

Range o.In this case, the

N u )=f

(3.1.12)

holds if u is an interior point (in KO),or if 0 is an interior point and (A(w) - f , w ) 2 0,

for all w E dK.

' This is a slight generalization of Theorem 3.1 (p. 41) of Ref. 1133.

(3.1.13)


80

Proof of Corollary 3.1.4: We choose uo = 0 and function of K, that is,

0, co,

4 to be the indicator

XEK, x$K.

(3.1.14)

The choice cro = cr = llfll in (3.1.10) is now valid by hypothesis and the definition of uo and 4. Hence, the proposition applies. If u is an interior point, (3.1.12) is immediate upon choosing u = u & EW, w E S. If u E aK, then (A(#) - f,u) 2 0 for all u E K, which implies (3.1.12), since 0 is an assumed interior point in this case. Proof of Proposition 3.1.2: If Dom 4 is bounded, then the mapping u H prox,(u

+ f - A(u)),

(3.1.15)

obtained by identifying f and A(u) with elements of S, is a continuous mapping of the closed, convex, bounded set Dom 4 into itself by Lemma 3.1.1 and Remark 3.1.2. Hence, by the Brouwer fixed-point theorem (see Chapter 4 [26] pp. 18 and 19) this mapping has a fixed point u E Dom 4 which satisfies (3.1.8). In particular, u E Dom 4. Now suppose that Dom 4 is unbounded. By the first part of the proposition, (3.1.9b) has a solution for each positive r o , in particular for ro > p. If (3.1.9b) has only the zero solution for all ro > p, then taking limits in (3.1.9b) as ro -,co shows that u = 0 is a solution of (3.1.8). Otherwise, let uro satisfy (3.1.9b), with 0 # uro and ro > p, and select a sequence {ri}zl, satisfying ro < ri -+ co. If the uri solve -

f,u - uri) + 4ri(V) - 4ri(uri) 2 0,

(3.1.16a)

where $ri is defined, in analogy with (3.1.9a), by (3.1.16b) then we suppose IIuroll < IIurill,i and (3.1f16a),

=

1, 2,. . . . Then we obtain, from (3.1.7b)

(3.1.17)


where

81

and (T are defined by (3.1.10).Since (T E Range o,we have llUlill

d supw-'(a)

= c,

i 2 0,

(3.1.18)

so that {uri} c B(0,c) c S, and is, thus, a relatively compact sequence, with accumulation point u. Upon taking the limit supremum in (3.1.16a), we obtain (3.1.8) for u E B(0, c). If the sequences {ri} and {uri} cannot be chosen so that Iluri(l2 IIuroll,all i, then a sequence {uri}necessarily is contained in B(0, [/ur,,~~). An identical argument, as above, in conjunction with (3.1.18), yields (3.1.8). W Remark 3.1.3. The theory developed thus far gives sufficient conditions for the duality inequality (3.1.8) to possess a solution in a finite-dimensional space S. The formulation permits various choices of 4 to obtain, for example, variational inequalities and equations. Also covered is the important case of multivalued subdifferentials of convex functions. We briefly introduce some terminology to discuss this case. Definition 3.1.3. Let X be a normed linear space with dual X*, and let F : X + (- 03, a] be a proper, convex, lower-semicontinuous function. By the subdifferential aF, defined for u, such that F(u) is finite, is meant the set-valued function dF(u) = {f E X * : F ( o ) - F(u) 2 ( f , v - u )

for all u E X}. (3.1.19)

If X is a Hilbert space, it is common to identify the functionals of dF(u) with the set of representers or subgradients.

Remark 3.1.4. dF(u) is a weak-* closed, convex, but possibly empty, set for a given u E X. Suppose now that u is a solution of(3.1.8).Thenf - A(u) E &$(ti) and, of course, the converse holds. Equivalently,

f E a4m + A(u)

(3.1.20)

characterizes solutions u of (3.1.8). If 4 is Giiteaux differentiable at u, with derivative @(u;.), then (3.1.20) is, in fact, the equation

f = @(u; .) + A(u).

(3.1.21)

Remark 3.1.5. We shall introduce additional properties required of the mapping A, which will permit us to pass from the finite-dimensional formulation to the infinite-dimensional formulation. A natural class of such


82

operators is the class of pseudomonotone operators, which we now describe. Note that, whereas for the previous results, we required A( .) to be defined on the entire space, we now relax this requirement. Definition3.1.4. Let K be a convex subset of a normed linear space X. Suppose A:K + X* satisfies, for all u E X, (A(u), u - u )

< lim inf (A(ui), ui - u ) , i+

(3.1.22a)

ou

whenever ui-u (weakly)

and

limsup(A(ui),ui- u )

< 0,

(3.1.22b)

for ui, u E K. Suppose also that A(K,)

is bounded in X*,

(3.1.22c)

for every bounded subset Kb of K. Then A is called a pseudomonotone operator. Note that the weak convergence ui- u is specified as sequential convergence, rather than the more general convergence of nets; (3.1.22~)is separate in [6] and [12].

Remark 3.1.6. Pseudomonotone operators, with domains consisting of open subsets of finite-dimensional spaces, are characterized, in an elementary way, by continuity (see Brezis [6] p. 132). In particular, Proposition 3.1.2 is valid for a pseudomonotone operator A, defined on K = S, since A is, thus, continuous on S. A more subtle problem arises when K, say, is simply a compact, convex subset of S and Dom 4 c K. In this case, a standard reduction to KO = (Dom 4)' # 0 is possible, and, by expressing K O as a union of compact, convex sets and employing the continuity of A on these sets, Brezis obtains a solution on KO, by passing to a limit, via pseudomonotone methods. The transition to K is standard. We refer the reader to Brezis [6], pp. 134-136, for details. This passage to the limit, however, is quite similar to that in passing from the finite- to the infinite-dimensional case, which we discuss in detail. Note that, if K is not bounded, some coerciveness condition, such as (3.1.7b) and (3.1.10), is required. Rather than state the finite-dimensional result, we avoid repetition and proceed directly to the Banach space result. Proposition 3.1.5. Let K be a closed, convex subset of a reflexive Banach space X, with 0 E K, and suppose that A:K + X* is pseudomonotone, and 4 : K + ( - 00, a] is a proper, convex, lower-semicontinuous function satisfying 4(0) = 0. Consider the inequality, for the unknown u E K, (A(u) - f,u - u )

+ 4 ( u ) - 4 ( u ) 2 0,

for all u E K,

(3.1.23)


83

for f specified in X*. Then (3.1.23) has a solution u E K, if K is bounded. On the contrary, suppose K is unbounded. If

with w and p specified in Definition 3.1.2, and

llfll

(3.1.25)

d CJ E Rangew, then (3.1.23) has a solution u ~ C B(0, 1 sup o - ' ( o ) ) .

Proof: Let z E K satisfy 4(z) < co, and let % denote the family of finitelet K F = K n F, dimensional subspaces F of X, such that z E F. For F E 5, and let AF and $F denote the restrictions of A and 4 to KF. Applying the finite-dimensional result of [6] (see Remark 3.1.6) to KF, we conclude that the set SF= {UF

E KF A

B(0,C):C = SUpO-'(CJ)

(A(uF) - f , u - uF)

+ 4 ( ~-)

and

~ ( u F )b

0,

for all u E KF} (3.1.26)

is nonempty. In the event K is bounded, we replace K F n B(0, c) by K F . For each F E 9, denote by VF c K the weak closure of {SF*:Fc F } . The family

u

(3.1.27)

Y ' = {VF:FE .F}

of weakly closed subsets of the weakly compact set K n B(0,c) has the finite-intersection property, and, hence, has a nonvoid intersection, say, u . 0 { V F : F E ~We J . shall demonstrate that u satisfies (3.1.23). Let u E K and let F E 9 be chosen so that u and u are in F; in particular, u and u are in KF. Since u is an accumulation or adherance point, in the weak topology, of the relatively weakly compact subset {S,:F c F'} of the reflexive Banach space X, it follows that u is a weak sequential limit point (see Browder [12] p. 81) of this set. Thus, there is a sequence uj = uF;E SF;, such that u j - u , where F c F;. We, thus, have, for eachj,

u

(A(uj)

- f , uj - 0 )

< 4(0)- $ ( ~ j ) ,

(3.1.28)

so that, taking the limit infimum and limit supremum of left- and right-hand

sides gives

lim inf(A(uj), u j - u ) d lim sup [$(u)- 4 ( u j ) ] j- w

I-

=

+ (f,u - u )

Jc

4 ( v ) - lim inf +(uj) + ( f , u - u ) j- m

(3.1.29)

3. Nonlinear Elliptic Equations and lnequalities

84

If we can show that lim sup (A(uj),u j - u ) d 0,

(3.1.30)

j - co

then the pseudomonotone property applied to (3.1.29)yields (A(u), - u> Q

4w - 4(4 + (f,

- u>,

which is (3.1.24). To establish (3.1.30), we choose u = u in (3.1.28), and take the limit supremum of both sides. This gives lim sup (A(uj), u j - u ) d lim sup [ 4 ( u ) - 4(uj)] j- w

I-

=

4(u) - lim inf &uj) j- w

d 4 ( 4 - 4 ( 4 = 0, which is (3.1.30). 4 Corollary 3.1.6. Let X be a reflexive Banach space, and let A be a pseudomonotone mapping defined on B(0,p ) . Suppose that g(x) >, 0 for u E dB(0,p ) , where g(4 = ( 4 4 ,u>,

u#

0;

(3.1.31)

then the equation A(u) = 0 has a solution llull , 3 0,

llull Q p

is guaranteed in B(0, p) by Proposition 3.1.5. By the hypotheses, we conclude (A(u),u ) 3 0 for all u, llull < p . This yields A(u) = 0. Remark 3.1.7. The previous result admits ofgeneralization to closed convex sets K. In particular, if g 3 0 outside the relative interior KY of a compact, convex subset K 1 of K, with 0 E KY, then A(u) = 0 has a solution in K (see [6] p. 137 for details). We now record another corollary for use in the sequel. Corollary 3.1.7. Let X be a reflexive Banach space, let X, be a closed linear subspace of X, and suppose that A: X, -+ X* is a pseudomonotone mapping and that 4 :X + ( - co,co] is a proper, convex, lower-semicontinuous function with 4(0) = 0. Suppose that w : [ p ,co) + [0, co) is a monotone increasing

3.1

General Operator Results

85

function with w - ' ( t ) bounded for each t and p > 0, such that (3.1.32) If llfll d cr E Range o,then there exists u E X, such that llull d s u p o -

f

E

'(4,

(3.1.33a)

d4b) + A(4.

(3.1.33b)

Remark 3.1.8. The obvious extension of Corollary 3.1.4 carries over if A:K c X + X* is pseudomonotone, and X is a reflexive Banach space. An important subclass of the class of pseudomonotone operators is the class of monotone. hemicontinuous mappings. These are defined later in this section (see Definition 3.1.6). Aside from the important surjectivity property described by (3.1.33b), the pseudomonotone operators enjoy the following property when M is bounded : B=A

+M

is pseudomonotone,

(3.1.34)

if A is pseudomonotone and M is hemicontinuous and monotone. Property (3.1.34) is stated precisely later (see Proposition 3.1.9). The class of pseudomonotone operators defined on X is, in fact, a subclass of a still larger class introduced by Brtzis [6] for the purpose of describing an existence theory for the equation A(u) = f . The operators of this class are termed operators of type M. It is not our intent to discuss these operators at length. However, we shall present their definition and the basic surjectivity result (see also (3.1.33b)) quoted with some modifications from BrCzis [6], pp. 124-128. Definition 3.1.5. Let X be a normed linear space. Then A:X + X* is said to be of type M if The restrictions of A to finite-dimensional subspaces of X arecontinuous; and (3.1.35a) A(u) = j '

whenever

lim sup (A(ui), ui) j- x

ui

- u, Aui

f ,+

and

< ( f , u).

(3.1.35b)

These operators enjoy the following surjectivity property. Proposition 3.1.8. Let X be a reflexive Banach space, and suppose A: X+X* is of type M. Suppose the coerciveness hypothesis llfll d cr E Range o holds

' The weak star convergence in the dual space, implied by the weak convergence in the dual space, usually suffices.


86

for o,described in Definition 3.1.2 and satisfying (3.1.36) Then the equation =

f

(3.1.37)

has a solution u, JIu(I< supo-'(a). The definition of a monotone, hemicontinuous operator is now given. Definition 3.1.6. Let K be a convex subset of a normed linear space X. A mapping A : K -.+ X* is said to be hemicontinuous if, for any x E K , y E X, and any sequence (ti} of positive real numbers, such that x + tiy E K, i = 1,2, . . . , then A(x

+ tiy)

-

A(x),

i + 00.

(3.1.38)

3 0,

(3.1.39)

The mapping A is said to be monotone if

- A(Y), x

- Y>

for all x, y E K . Further, A is strictly monotone if A is monotone and equality in (3.1.39) implies x = y.

Remark 3.1.9. It is an easy exercise to show that solutions of (3.1.23) are uniquely determined if A is strictly monotone. The following proposition is proved by straightforward arguments in Brezis [6], p. 133. Proposition 3.1.9. Let K be a convex subset of a normed linear space X. Let A : K + X* and M : K + X* be, respectively, a pseudomonotone and a monotone, hemicontinuous, bounded mapping. Then the mapping B = A + M is pseudomonotone. In particular, B = M is pseudomonotone. We briefly state now the connection of the previously developed ideas with the topic of multivalued maximal monotone operators. Definition 3.1.7. Let X be a normed linear space, and let K be a subset of X. A (multivalued) mapping A : K --+ 2' is said to be monotone if the graph of A,? gr(A) = {(x, Y):x

'

E

K, Y E A(x)},

(3.1.Ma)

The conventional ordering is preserved. It is standard to reverse this order in the duality pairing.


87

is monotone, that is, (3.1.40b) - Y , , x1 - xz> 2 0, for y , E A(x,), y , E A(x,). The mapping A is maximal monotone if the graph of A has no proper monotone extension to X x X*. (Y1

Remark 3.1.10. An example of a maximal monotone mapping is any hemicontinuous, monotone mapping defined on all of a Banach space X (see Opial [25], p. 77). Another example is the subdifferential A

= L@:DomA-+ 2* '

(3.1.41)

of a proper, convex, lower-semicontinuous function (see Barbu [3] Chapter 2, Section 13). Here Dom A = { u E Dom 4:&$(u) # Equation (3.1.33b) may be paraphrased to say that the coercive sum of the pseudomonotone operator A and the maximal monotone operator B = 84 is surjective. In fact, this result remains valid (see Brezis [7] Theoreme 1) if B is any maximal monotone operator and A is pseudomonotone on K B . Additional examples will be discussed in the next section. The reader, interested in what subclass of maximal monotone mappings is characterized by (3.1.41), is referred to the cyclically monotone mappings of [8]. If X = R', the classes coincide. Related ideas are discussed by Rockafellar [26].

a}.

Remark3.1.11. It is sometimes useful to have fixed-point theorems for monotone (increasing) mappings, which do not depend on topological properties, such as continuity. We shall present this result, sometimes called Tarski's fixed-point theorem, after an initial definition, which describes the ordering with respect to which the monotonicity is defined. The term isotone is often applied in the literature to such monotone mappings. Definition 3.1.8. A set d is said to be (partially) ordered if there exists a reflexive, antisymmetric, transitive relation defined by a subset of d x d. A chain in d is a subset V,such that x, y E V imply x < y or y < x. If W is a subset of d,then z E d is an upper bound for W, if x < z for all x E W , and z is a least upper bound for 93 if, in addition, z < y for all upper bounds y of W . Lower bounds and greatest lower bounds are defined similarly. The set d is said to be a lattice if each pair of elements possesses a greatest lower bound and a least upper bound. A lattice a2 is inductive if every chain V in d has a greatest lower bound a in d and a least upper bound b in d . A lattice is complete and, in particular, inductive, if every subset W of d has a greatest lower bound a in d and a least upper bound b in d .Finally, an element x in a lattice d is a minimal element (respectively maximal element)


88

if y < x implies y = x (respectively x < y implies y isotone (increasing) if x < y A(x) < A(y).

= x) and

A: d

-,d

is

Proposition 3.1.10. Let E be a partially ordered set, and suppose uo < uo are elements of E, with the property that the interval I = { u E E: uo < u < u o } is an inductive lattice. Suppose that A is an isotone mapping of I into E, such that ~0

< A(uo),

A(uo) < ~

0 .

(3.1.42)

Then the set of fixed points u of A satisfying uo < u < uo is nonempty and possesses a minimal element 11 and a maximal element ii.

Proof: We make the initial observation that A maps I into itself; this follows from the increasing property of A in conjunction with (3.1.42). Now set x 2 y ify < x, and set 42

= {U E

E:uo < u < uo, u < A(u)},

V = { u E E:uo < u

< uo, A(u) < u } .

(3.1.43a) (3.1.43b)

Then uo E 42 and uo E “Y-. Set W = { w ~ V : w a u forall u s % } ,

9={y~42:y 0 and all u, w E G . Proof:

The coercive relation as

(3.2.29)

is immediate from (3.2.25) and (3.2.17b). We have seen that L + f is pseudomonotone, and that y is a proper, convex, lower-semicontinuous function on G . The existence of a solution u satisfying (3.2.26) now follows from Corollary 3.1.7, if we set X = X, = G, A(u) = Lu + T(u), and 4 ( v ) = y(u). Note that B(u) c L2(!2) by the definition of p. Equation (3.2.27) is simply a restatement of (3.2.26). The uniqueness follows directly from (3.2.28). Example 3.2.3.

Abstract Stationary Navier-Stokes Theory

This example is related to the stationary Navier-Stokes theory or, more properly, to the generalization introduced in Section 1.4. The general situation considered here involves a separable, reflexive Banach space X, a dense linear, separable, subspace Y continuously embedded in X and a mapping A:X + Y*.The mapping A is defined explicitly by A(u) = LU

+ U(U,U, *),

(3.2.30)

where L:X --+ X* is a linear isomorphism, satisfying (3.2.16) and (3.2.17),with C = 0; note that in (3.2.17) we have identified G with X and L2 with a space W to be described shortly. Also a( ., ’,.) is a continuous trilinear form on X x X x Y,satisfying a(u,u,u) 2 0,

if u,

-

a(u,, urn,u ) + a(u, u, u),

for all u E Y,

(3.2.31a)

for all u E Y,

(3.2.31b)

u (weakly in X) and urn+ u (strongly in W). Here, X is densely and

3.2 Applications and Examples

95

compactly embedded in the Banach space W. The reader correlating this with Section 1.4 should set X = V, Y = V,, W = H, and L = -A. The mapping A is clearly continuous, yet, because Y* is, in general, strictly larger than X*, the infinite-dimensional theories of the previous section do not apply directly for the equation A(u) = f,even iff E X*, which will be required for this example. More precisely, given f E X*, we shall solve A(u) = AY. Suppose then that {Yk} is a sequence of increasing finite-dimensional subspaces of Y, such that k u Y k is dense in Y. Consider the finitedimensional problem of determining u k E Y,, such that A(Uk)

(3.2.32a)

= f;Yk

or, equivalently, such that (Luk,u)

+ a(uk,uk,u) = (f,v>,

for all u E Yk.

(3.2.32b)

To verify the existence of a solution u k of (3.2.32), with the property that {uk} lies in a bounded subset of X, we first impose upon Yk the norm induced by X.Then, by (3.2.17b),with C = 0, and (3.2.31a),we deduce the coerciveness property (3.2.33) In particular, there is an r, namely, r such thatt

=

Ilfllx./c, in the notation of (3.2.17b), (3.2.34)

for IIullyk = r. Since A:Yk + Y t is continuous, and hence, pseudomonotone, on the finite-dimensional space Yk, Corollary 3.1.6 applies to yield a solution u k E Y k of (3.2.32), with the property that (3.2.35) Since X is assumed reflexive, and X is compactly embedded into W, we may extract a subsequence ukj, and an element u E X, such that

-

uk, u (weakly in X),

uk, + u

(strongly in W).

(3.2.36)

Now, let w E UYk. Then w E Yko,for some k,,, and taking limits in (3.2.32b) yields ( L U , w>

+ 4%4 w ) = (f,w>,

Recall that the induced X norm is used in (3.2.34).

(3.2.37)


96

where we have used (3.2.31b).Finally, if u E Y, then letting wj -, u in Y yields (Lu, 0)

+ 44 u, 4 = (f,u),

(3.2.38)

where we have used the continuous injection of Y c X. Note that (3.2.38) is just the statement that AbJ) =

(3.2.39a)

fiY9

and (3.2.35)directly implies (3.2.39b)

llullx G r*

We shall summarize these results insofar as they apply to the case of Section 1.4.

Proposition 3.2.2. Let V and V, be defined as in (1.4.20) and (1.4.21), and let a( ., ., .) be a continuous trilinear form on V x V x V,, satisfying (1.4.22) and (1.4.24a). Let f = (fl, . . . ,f,)E V*. Suppose the norm in V is defined by (equivalently)

11~11:

= rl

JQv v

*

vv

(rl > 0)

(3.2.40a)

(see (1.4.26) for dot notation). Then there exists a u E V, satisfying llullv G V(Vui,Vui)L2(n)

for v = ( u l , . . . , u,)

E

+ a(ui,ui,Ui) = (f;:,ui>,

(3.2.40~)

V,, and i = 1,. . . ,n.

Proof: Set L = -qA, X = V, Y hence, r = H

IlfIlv..

Example 3.2.4.

(3.2.40b)

IlfllV*)

= V,,

and W = H. Note that c = 1 and,

A Quasi-Variational Inequality

Let X be a reflexive Banach space with a strictly convex dual, and suppose that X possesses a lattice structure. In particular, u = u+ + u - , with u + = max(u,O)and u- = min(u,O).Set X + = {u E X:u 2 0},and suppose J : X + X* is the uniquely defined duality map, satisfying (J(u),u) = \lul\i,

\ [ J ( ~ ) \ \ x=* I\ullx,

for all u E X,

(3.2.41)

and the lattice compatibility condition (J(w2) - J(w,), u - )

< 0,

for all u E X,

(3.2.42)

3.2 Applications and Examples

97

and w2 2 w,. Suppose that M : X + -,X + satisfies M(0) 2 0, and that A:X -, X* is a pseudomonotone operator, satisfying (3.2.43) for all u E X,where A1 2 0,A2 > 0. Suppose also that

(A(u) - A(v), (U - u ) * )

+ A,(J(u)

- J(u), (U - u)*) 2

0,

(3.2.44a)

and

+

(A(u) - A(u),(U - u)*) A1 (J(u) - J(u),(u - u)*) = 0 o (u - u)+ = 0 (respectively(u - u)- = 0) if the positive/negative parts are chosen. (3.2.44b) Suppose, moreover, that f E X* is given, with the property that a solution uo E X + exists for the equation A(v0) = f.

(3.2.45)

Suppose that M is increasing on { u E X:O < u < u o } , that is, M(u) - M(u) 2 0 if u - u 2 0. Finally, we set

A,

=A

+ AIJ,

(3.2.46)

and observe that A, is strictly monotone by (3.2.44).Moreover, A, is pseudomonotone as the sum of a pseudomonotone and a bounded, continuous, monotone mapping (see Proposition 3.1.9). Also, by (3.2.43), (3.2.47) In particular, we conclude the existence of a unique solution u = S(w) of the variational inequality (see Proposition 3.1.5),

( M u ) - A,J(w)

-

f,v - u ) + m - 444 2 0,

(3.2.48)

for all u E X, where is the indicator function of the closed (see Birkhoff [4]), convex, nonempty set

K(w) = {U E X + : u < M(w)},

(3.2.49)

and 0 < w < uo. The inequality (3.2.48)may be rewritten as u

< M(w),

( A , ( ~ ) , u- U) 2 (A,J(w)

(3.2.50a)

+ f,u - u),

u

< M(w).

(3.2.50b)


98

The implication S(0) 3 0 is immediate from S(0) E X'. The implication S(uo) < uo follows from setting w = u o , u = S(u0),and u = S(uo) - ( S ( u o )- u,,)' in (3.2.50b), and adding the resulting inequality to (-A1(~0)9(S(~o)- u o ) + )

= (-~1J(~o)-f,(S(Uo)-

u,)+)

to obtain

Combining (3.2.51) with (3.2.44) gives (S(uo) - uo)+ = 0, that is S(uo) Q u o . If we can prove that S is increasing on the interval 0 < u d uo, then it will follow from Proposition 3.1.10 that S has a fixed point on this interval and, hence, that u < M(u) and (Al(u),u - u ) 2 (AIJ(u)

+ f,u - u),

for all u Q M(u), (3.2.52)

provided the interval is inductive in the sense of Definition 3.1.10. In light of (3.2.46), we see that (3.2.52) is equivalent to solving on X + (A(u),u - u ) 3 (f, u - u ) ,

for all u Q M(u),

(3.2.53)

for u < M(u). To prove that S is increasing, let 0 < w 1 Q w2 < u o . Then u1 = S(w,) and u2 = S(w,) satisfy u1 Q M(wl) and u2 Q M(w2), and

we may set u = u in (3.2.54b). With the choice u = u1

+ (u2 - u l ) -

in (3.2.54a),we have, after addition of the two inequalities 0 Q (Al(U2) - A,(u,),

Q 0,

(u2

- u1)-)

< 4(J(w2) - J(Wl),

(u2

-%-)

(3.2.55)

where we have used (3.2.42) and (3.2.44a). By (3.2.44b), we conclude (u2 - ul)- = 0, so that S is increasing. We summarize this result in the following proposition.

3.3 Semidiscretizations Defined by Quadrature

99

Proposition 3.2.3.' Suppose that X,M, and A satisfy the hypotheses stated above. Then the quasi-variational inequality (3.2.53) has a solution u, satisfying u < M(u) c X + .

3.3

SEMIDISCRETIZATIONS DEFINED BY QUADRATURE

Suppose we are considering a partial differential equation: d [H(u)] + Lu dt

-

+ f ( u ) = 0,

0 < t < To,

(3.3.1)

for an appropriate elliptic operator L, and specified functions H and f , with imposed boundary conditions of homogeneous Dirichlet, Neumann, or Robin type. Although more general frameworks are possible, and sometimes desirable, we conceive of L as being defined on a linear subspace of H'(Q). More precisely, let G be a closed subspace of H'(Q), and let an inversion operator T:G* --* G c H'(Q)

(3.3.2)

be specified, where T is a continuous linear mapping onto a closed linear subspace G of H'(Q), satisfying

L0T

= identity.

(3.3.3)

The operator L, thus, has domain G and range G*. Hence, as observed in Example 3.2.2, the homogeneous boundary conditions are embedded in G, via G = T(G*). The special cases of interest were discussed in detail in Chapter 1. Formally, we transform (3.3.1) into d [TH(u)] dt

-

+ u + T ~ ( u=) 0,

(3.3.4)

and we find, integrating from t = t k - 1 to t = t,, within a given partition = To),

9 = (0 = to < t , < . . . < t M TH(u(t,))

+

s'"

fk- 1

u(t)dt

' This generalizes Theorem 4.3 of [19],

+

fk

fk- 1

T f ( u ( t ) ) d t= T H ( u ( t k - l ) ) . (3.3.5)

and its proof, contained in pp. 170-173.

* We anticipate uniqueness and require single-valuedness for H(u); the x-dependence is

implicit.


100

Suppose a quadrature rule for u: [0, To] + G as follows:

ji;-l u(t)dt is specified for functions

where Y j , k are specified constants. Note that all values v(to), . . . , u ( t k ) may be used in (3.3.6). Such a quadrature rule may be used to define semidiscretizations of (3.3.4) by direct application to (3.3.5). In the literature on initial-value problems for ordinary differential equations, the application of quadrature formulae, such as (3.3.6),to the integrated form of a differential equation induces what is known as a multistep method. If Y j , k = 0 for j c k - 1, the method is called, not surprisingly, a single-step method. It is advantageous to permit the application of different quadrature rules to the integrals appearing in (3.3.5). With the applications in mind, let us set f = g + h, in the notation of Chapters 1 and 2, and rewrite (3.3.5) as TH(U(tk))

+ s'"

tk-1

[u(t)

+ T g ( u ( t ) ) dt] +

Jk

fk-1

Th(u(t))dt= T H ( U ( t k - I ) ) . (3.3.7)

If (3.3.6) is applied, with Y j , k = ctj,k and Y j , k = p j , k , respectively, to (3.3.7), we obtain a corresponding formal approximation equation. We shall display this equation in the following definition. Definition 3.3.1. Given a partition 9 of [0, T o ] ,and a pair of quadrature rules Q k , = ( * ) and Q k , p ( * ) , defined by (3.3.6), for k = 1, . . . , M , with a = ( c t j , k ) , fl = ( B j , k ) , we define the semidiscretization of the equation

d [H(u)] dt

-

+ Lu + g(u) + h(u) = 0,

(3.3.8)

induced by the quadrature rules, to consist of the recursive sequence (lk

- lk-

l)-lH(Uk)

+ crk,k[Luk + g ( u k ) ] + P k , k h ( U k )

k- 1

= -

j=O

{ctj,k[Luj

+ g ( u j ) ] + flj.kh(Uj)} + (tk - tk-

l)-lH(uk-

l),

(3*3*9)

1,. . . , M . For each k, it is required that the unknown functions have already been computed, and that u k is determined by (3.3.9). The functions { u k } are approximations to U ( t k ) . We refer to the lifted

for k

uo,..

=

. ,u k -


101

form of (3.3.9) as the recursive sequence (lk

- tk-

+ CLk,k[Uk + T g ( u k ) ] + f i k , k T h ( u k )

l)-lTH(uk) k- 1

=-

1

j=O

+ T g ( u j ) ] + fij,kTh(uj)} + (tk - t k -

{aj,k[uj

l)-'TH(uk-

I)),

(3.3.10) for k fik, k

=

1,. . . , M. The method is said to be implicit if either

# 0, and explicit otherwise.

Mk,k

# 0 or

Example 3.3.1. In this example, we shall define Qk(U)

:=

for appropriate combinations i7, of t. If

J:-

I

and

v(tk- 1)

-

U ( l ) = U(tk-1),

(3.3.11)

c(t)dt,

tk-1

u(tk), which

< t < tk,

are constant in (3.3.12a)

then Qk ( U)

= u(tk-

i)(tk

- tk-

(3.3.12b)

1).

In this case, Y k - 1.k = 1 and Y j , k = 0, j # k - 1. If C L and ~ , ~f i j , k are defined by these choices of Y j , k , there results the well-known explicit or forward Euler semidiscretization. On the other hand, if -

u(t) = u ( t k ) ,

tk- 1

< t < tk,

(3.3.13a)

t k - I),

(3.3.13b)

then Qk(V)

= V(tk)(tk -

and the fully implicit or backward Euler method results. In this case, yj,k = ~ 5 ~ Of , ~ course, . a mixing of the methods may be desirable (see Chapter 2), that is, defined by (3.3.13) and f i j , k defined by (3.3.12). The step functions defined by (3.3.12a) and (3.3.13a) are the special cases 8 = 0 and e=iof D(t) = e v ( t k )

+ (1 - e ) U ( t k -

I),

tk- 1

'

+

B = (tk - t k - l ) - ' H

ctk,kg

- Uk,kCid + Bk,kh,

r

= 0,

lead to the operator identity -ak,kA

+

(tk

- t k ) - 'H

+

+

ak,kg

Bk,kh

=

L

+ p,

and (3.2.25) is satisfied, with

G = [H'(Q)l0,

c = 0,

and

cl = 1&kl

Ih(0)lln(1/2/n1/2,

provided

- lk-

(3.3.20) We, thus, have, as an immediate consequence of Proposition 3.2.1, the following. 1)-

- 'k,kn1 2

IPk,kl llhlIL*P.

Proposition 3.3.1. Let H, g , h, and il be given as in Section 1.1, with the initial datum H(uo), and consider the formal semidiscretization (3.3.9) of (1.1.9), rewritten here as (lk

-

t k - 1)- I H ( % )

- %,k

k- 1

= -

1

j=O

{aj,k[-Auj

Auk

+ clk,kg(uk) + Pk,kh(Uk)

+ g ( u j ) ] + flj,kh(uj)} + (tk

-

lk- l)-'H(uk-l)?

(3.3.21a) with Neumann boundary condition (3.3.21 b)


105

k = 1,. . . , M, where a k . k > 0 is assumed. Suppose that - tk-

11-l

> IBk,kl Ilh)lLip.

(3.3.22)

Then (3.3.21) has a unique weak solution pair ( U k , H(uk)),such that H(Uk(X))

[o, A],

H(Uk)

if

u k ( x ) = 0,

(3.3.23b)

L2(R),

'(n),

uk

(3.3.23a) (3.3.23~)

and (3.3.21)holds, when interpreted as an identity in [H'(n)]*. Equivalently, if c > 0 is chosen to satisfy (3.3.20), then (3.3.21) may be understood in the lifted sense, when N, is applied. Proof: For a given k, select c, such that (3.3.20) holds. With the identifications of Remark 3.3.3, the result follows from Proposition 3.2.1, in conjunction with an argument employing J = H - ' , which shows that H ( U k ) is also uniquely determined, via the norm in [H'(R)]*, and standard lifting by means of N,.

The application to the case of the porous-medium equation is similar. We state this explicitly as follows. Note that no condition on the partition is required.

Proposition 3.3.2. Let H be given as in Section 1.2, with the initial datum uo, and consider the formal semidiscretization (3.3.9) of (1.2.13),rewritten as k- 1 (tk-

tk-l)-'H(Uk)-ak,kAUk=

1

j=O

aj,kAUj+(tk

-tk-l)-lH(Uk-l)r

(3.3.24a)

with Dirichlet boundary condition u k = O

on 82,

(3.3.24b)

k = 1,. . . , M, where ak,k> 0 is assumed. Then (3.3.24) has a unique weak

solution u k E HA(R), and (3.3.24) holds, when interpreted as an identity in H-'(R). Equivalently, (3.3.24) may be understood in the lifted sense, when Do is applied. The reader will recall from Section 2.5 that the semidiscretization defined there had the property of maintaining the range of the solution in the specified invariant region Q, under the corresponding supposition on uo and certain hypotheses on the vector field f. Parallel theorems for other semidiscretizations would be extremely useful, but we shall not attempt them here. We analyze now the existence and uniqueness theory for (2.5.1)-(2.5.4). For


106

those equations for which Di # 0, Proposition 3.2.1 is applicable. However, if Di = 0, direct arguments are required. We present these now. If i is the index of a concentration variable, then it is required to determine an H'(R) solution, satisfying u[i

N

= Gi(uk-

1,1,

...

9

u:-

1, i - 1

9

u[i,

N

uk- 1, i + 1 7

* *

. > u:-

1,m 7

u:-

l,i)?

(3*3.25a)

where Gi(x1, . . . , x,+

1) = fi(x1,

. . . , X,)(tk N

-

N

tk- 1)

f x,+

1.

(3.3.25b)

If i is the index of a potential variable, then Gi is written as Gi(x1,.

..

9

xm+1)

- {gi(xi)

+ hi(xl

9

. . . ,x i -

1 9

x m + 1, x i +

1 9

...

9

Xm)}(tk

N

N

- tk-

1)

+ xm+ 1

*

(3.3.25~)

We make here the obvious induction hypothesis that II:-~ is determined. If it can be shown that, for fixed x' = (xl, . . . , xi- x i +1 , . . . , x,+ J, the equation R(x1, . . ., X i - 1 ,xi+1, . . .

7

X , + 1 ) = Gi(x1, . .

. , xi-

19

R( * . .),xi+1, . . . x m + 1) (3.3.26) 9

has a solution which is Lipschitz in the variables denoted by x', then we E H'(R; W), define may proceed as follows. Given II[by (3.3.25). Since

n H'(R) m

R:

-+

H'(R),

1

we conclude that uci E H'(R) for each fixed i = 1,. . . , rn, such that Di = 0. We shall now determine a fixed compact interval K , such that the mapping

5 H G i ( x 1 , . . . > x i - 1 , S , x i + l , . ..

7

Xm+1)

(3.3.27)

has domain and range in K for x' suitably restricted, and with restrictions on (t; - t:-J; these will guarantee (3.3.27) is a strict contraction. Let us now use the a priori bounds derived in Proposition 2.5.1. Thus, by the latter, let 0 denote half the length of a side of a closed hypercube K , , centered at 0 in R", within which the range of IIF-~ is constrained to lie by the bounds d 0. (2.5.11). The interval K will now be determined independently of x', Let upper bounds be defined by (3.3.28a)


Then K

=

107

[ - 2a, 201 is invariant under (3.3.27),provided

+

(Ifi(0)l ~ ~ 2 a ) ( t-, "t,"- 1) < a,

+

(lhi(O)(+ ( I h i ( ( ~ i ~ Mi2a)(t," a - t,"-

< a,

1 < i < i,,

(3.3.29a)

i,

(3.3.29b)

+ 1 < i < m,

and moreover the mapping is a contraction. We emphasize that x' is constrained to lie in K,. The assumptions (3.3.29) not only guarantee that (3.3.26)has a solution, but that R is Lipschitz in x' E K,. This completes the verification of existence if Di = 0. The uniqueness follows from the a priori estimates of Proposition 2.5.1 and the contractive property of (3.3.26). Proposition 3.3.3. Let the system (1.3.4) be given, with the initial datum u,, and let a denote the maximum of the upper bounds in (2.5.11). Then unique solutions exist satisfying (2.5.1)-(2.5.4), provided (3.3.29) is satisfied when Di = 0, and provided (3.3.30) where Di > 0 and 1 < i < i,. The existence of unique solutions of (2.5.2b) corresponding to the potential variables follows directly from Proposition 3.2.1, with L = R; ', g = (t," - t,"- 1)-11 + gi, and f = 0, provided Di # 0. A similar statement holds, under hypothesis (3.3.30), for the system (2.5.2a), with L = R;', g = (t," - t,"-J-'I fi(u,"-l,l, . . . ,u,"-', i - 1, ., . . . ' , u , " - ~ , and ~ ) , f = 0. For those equations, such that Di = 0, it has already been demonstrated that unique solutions exist under hypothesis (3.3.29). W Proof:

+

Our final statement of this section is an application of Proposition 3.2.2 to the semidiscretization of the generalized Navier-Stokes system (1.4.25). Proposition 3.3.4. Let a ( - , - , -be ) a continuous trilinearform on V x V x V,, satisfying (1.4.22b)and (1.4.24a), where V and V, are defined by (1.4.20) and (1.4.21) in terms of an orthogonal projection P. Consider a formal semidiscretization of (1.4.25) given by (tk

- tk-l)-luk =

- ak.k?Auk

+

Pk,ka(Uk,Uk,')

k- 1 j=O

{(aj,k?

Auj) - f i j , k a ( u j , u j , * ) }

+

(tk

-

tk- l ) - l U k -

1

(3.3.31a)


108

and Puk=

0;

uk

=o

on 80,

(3.3.31b)

k = 1,. . . , M , for a given initial datum uo, where a k , k > 0 and P k , k 2 0 are assumed. Then a solution u k E V exists for (3.3.31), when the latter is interpreted as an identity in V:. Proof: The continuous trilinear form a( ., ., .) of Proposition 3.2.2 is realized here as ((‘k

- tk-

‘)L2(R)

+ P k , k a ( U , U , $1,

when applied to u E V. The properties (1.4.22b) and (1.4.24) clearly remain intact. W

Remark 3.3.4. An important special case of (3.3.31)is the semidiscretization (tk-

tk-l)-l(Uk-uk-i)-e?AUk-(l

-@?Auk-1

+a(uk,uk,.)=O,

(3.3.32a) Puk=

0;

uk

=o

on 8 0 ,

(3.3.32b)

where 3 < 0 < 1. An observation necessary for the existence analysis of the evolution equation is the fact thatt M

1

k= 1

M

Iluk

- Uk-1))i2(R;Rn)

+ k1 ~ I U k ~ ~ : ( tk tk-1) = 1

c?

(3.303~)

for some constant C, not depending on the partition selected for [0, T o ] . For the most part, we have relegated such estimates to Section 5.1. However, we shall discuss the derivation of (3.3.33) here, since the technique requires a preliminary finite-dimensional estimate and subsequent passage to the limit, as discussed in Example 3.2.3. Thus, with an obvious change of notation, we obtain, from Proposition 3.2.2 and its proof, a sequence {ur) of solutions of the equations Ak,N(uf)

=

fklYN,

uf

YN,

(3.3.34a)

where {Y”} is a sequence of increasing finite-dimensional subspaces of IJN Y” is dense in v,. Here A k , N and f k are defined through

v,, such that

’

A similar estimate is obtained by Temam; for example, see Chapter 1 [23] (Lemma 4.4, p. 325).

3.4 BibJiographicaJ Remarks

109

the relation

in an evident recursive manner for k = 1 , . . . ,M . For uf, we may conveniently choose the orthogonal projection in V of uo onto YN.The proof of Proposition 3.2.2 permits a passage to the limit in (3.3.34),from which (3.3.32) results. Moreover, standard lower semicontinuity guarantees (3.3.33) if the corresponding a priori estimate can be obtained, independent of N , for the solutions of (3.3.34). By selecting $I = uf in (3.3.34b),and using (1.4.22b) and (2.2.38) ( p = q = 2), we obtain, upon summing from k = 1 to k = M , $IIUClltZ(n;Rn)+

+ 1 [Iu? M

k= 1

N

- uk-

+ (2e - ')

111t2(n;lw")

M k= 1

[lufll:(tk

- lk-

1)

(3.3.35) where the norm of (3.2.40a) has been used, together with the identity (u9v) = +[llul12 - l l V ( l 2

+ IIU - vl121

(3.3.36)

in L2(!2;R"). Now the right-hand side of (3.3.35) is bounded by a constant times lluoll$.We summarize this result in Lemma 3.3.5.

+

Lemma 3.3.5. If < 8 < 1, then the estimate (3.3.33) holds for the semidisCrete solutions of (3.3.32). Here, C depends upon ( ( u & ,and T o , but not upon the partition selected for [0, To]. 3.4


The proximity mapping corresponding to a given proper, convex, lowersemicontinuous function was introduced by Moreau [24], and is a natural device with which to construct a theory of variational inequalities. The origin of this idea is attributed to Brezis [6] by Ekeland and Temam [13], who describe a slightly less general version of our finite-dimensional result contained in Proposition 3.1.2. The latter result attempts to match the coerciveness hypothesis directly to the given f E X*.We have essentially followed Brtzis [6] in the development of Proposition 3.1.5 and Corollaries 3.1.6 and

110


3.1.7, via pseudomonotone operators. For simplicity, however, we have developed the theory in reflexive Banach spaces in terms of sequential weak convergence. Nonetheless, it is not possible to avoid transfinite induction in the proof of Proposition 3.1.5, and we have here adapted the corresponding proof of Opial[25], pp. 98 and 99. We have not presented here an alternative constructive approach, via contraction mappings, which is possible when A is a monotone, hemicontinuous operator. This idea is probably due to Brezis [6]. Pseudomonotone operators represent a precise generalization of continuous mappings defined on open subsets of finite-dimensional spaces X,, with range in X,*.Continuity properties for more restricted classes were discovered by Browder [9] and Kato [16]. The origin of monotonicity methods appears to be found in the study of integral equations. Minty [23], pp. 67 and 68 credits Golomb [141 with perhaps the earliest ideas. Perhaps the first systematic identification and development are due to Zarantonello [31] and Vainberg [30]. The reader is referred to the lecture notes of Opial [25] for elaboration and historical development through 1966. Early work of Minty [21], Browder [lo], and Hartman and Stampacchia [l5] was decisive. In addition, Minty [22] drew the important connection between convex functions and their subgradients. The early applications to partial differential equations were recognized by Browder [113. The books of Br6zis [8] and Barbu [3] give further elaboration, as does the book-length article of Browder [12]. These three references are concerned with evolution equations in Hilbert and/or Banach spaces. In an attempt to obtain regular solutions via nonexpansive semigroup methods, the mappings are defined from X to X, rather than from X to X*.In this context, it is the accretive mappings, with monotone realizations in Hilbert space, which prove decisive. The reader is referred to these sources for details. The Tarski fixed-point theorem, described as Proposition 3.1.10, is set forth by Birkhoff [4], p. 115 (see also Tarski [28]). Our proof proceeds under the apparently weaker hypothesis that I is inductive rather than complete. However, we do not consider commuting families of increasing functions as did Tarski originally. Interesting applications are given by Amann [l]. The application given in Example 3.2.4 was motivated by a similar application in Lions [19], pp. 169-173 (see also Tartar [29]). It is now known that quasi-variational inequalities model many physical and probabilistic phenomena (see Kikuchi and Oden [18] for applications to seepage problems). Moreover, mathematical analogies exist which identify the solutions of certain optimal stopping-time problems with Stefan problems. In these cases, the primary formulation is a quasi-variational inequality. Some surprising facts concerning stability emerged when spline functions were employed to obtain quadratures for initial-value problems. One of the

References

111

earliest studies was carried out by Loscalzo and Talbot [20] (see also Bohmer [S]). It is now known that a number of traditional methods, such as the Milne-Simpson method, may be represented by such quadratures. A comprehensive analysis of discrete methods for initial-value problems has been given by Stetter [27]. We note, finally, that the numerical solution of the elliptic boundary-value problem in Example 3.2.1 has been achieved by descent methods by Bank and Rose [2]. REFERENCES H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18,620-709 (1976). R. Bank and D. Rose, Parameter selection for Newton-like methods applicable to nonlinear partial differential equations, SIAM J. Numer. Anal. 17, 806-822 (1980). V. Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces.” Noordhoff, Leyden, 1976. G. Birkhoff, “Lattice Theory,” 3rd ed. American Mathematical Society Colloq. Publ. 25, Providence, Rhode Island, 1973. K. Bohmer, “Spline-Funktionen.” Teubner, Stuttgart, 1974. H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst. Fourier (Grenoble) 18, 115-175 (1968). H. Brezis, Perturbation non lintaire d’opkrateurs maximaux monotones, C. R. Acad. sci. Paris Sir. A-B 269, 566-569 (1969). H. Brezis, “Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert.” North-Holland Publ., Amsterdam and American Elsevier, New York, 1973. F. E. Browder, Continuity properties of monotone nonlinear operators in Banach spaces, Bull. Amer. Math. Soc. 70, 551-553 (1964). F. E. Browder, Nonlinear monotone operators and convex sets in Banach space, Bull. Amer. Math. SOC.71, 176-183 (1965). F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Amer. Math. SOC.Proc. Symp. Appl. Math. 17, pp. 24-49 (1965). F. E. Browder, “Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces,” Amer. Math. SOC.Proc. Symp. Pure Math. 18 (Part 2). Providence, Rhode Island, 1976. c131 I. Ekeland and R. Temam, “Convex Analysis and Variational Problems.” NorthHolland Publ., Amsterdam and American Elsevier, New York, 1976. ~ 4 1M. Golomb, Zur Theorie der nichtlinearen Integralgleichungen, Integralgleichungsysteme und allgemeinen Funktionalgleichungen, Math. 2.39,45-75 (1934). P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, A m Math. 115, 271-310 (1966). T.Kato, Demicontinuity, hemicontinuity and monotonicity, Bull. Amer. Math. SOC.70, 548-550 (1964). J. L. Kelley, “General Topology.” Van Nostrand-Reinhold, New York, 1955. N. Kikuchi and J. T. Oden, Theory of variational inequalities with applications to problems of flow through porous media, Internur. J. Eng. Sci. 18, 1173-1284 (1980).

112

3. Nonlinear Elliptic Equations and Inequalities J. L. Lions, “Sur Quelques Questions d’Analyse de Mechanique et de ContrBle Optimal.” Univ. of Montreal Press, Montreal, 1976. F. Loscalzo and T. D. Talbot, Spline function approximation for solutions of ordinary differential equations, SIAM J. Numer. Anal. 4,433-445 (1967). G . Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29,341-346 (1 962). G. Minty, On the monotonicity of the gradient of a convex function, Pacific J . Math. 14, 243 -247 (1964). G. Minty, On some aspects of the theory of monotone operators, in Proc. NATO Ado. Study Inst. Theory and Appl. Monotone Operators. Edizioni Oderisi, Gubbio, Italy, 1969. J. J. Moreau, Proximite et dualite dans un espace hilbertian, Bull. SOC.Math. France 93,273-299 (1965). Z. Opial, “Nonexpansive and Monotone Mappings in Banach Spaces,” Lecture Notes, Division of Applied Mathematics, Brown Univ., 1967. R. T. Rockafellar. On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. SOC.149,75-88 (1970). J. J. Stetter, “Analysis of Discretization Methods for Ordinary Differential Equations.” Springer-Verlag, Berlin, and New York, 1973. A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific J . Math. 5, 285-309 (1955). L. Tartar, Inequations quasi-variationelles abstraites, C . R . Acad. Sci. Paris 278, 1193-1196 (1974). M. M. Vainberg, On the convergence of the method of steepest descent, Sibirsk. Mat. 2. 2,201-220 (1961) E. H. Zarantonello, “Solving Functional Equations by Contractive Averaging, MRC Technical Summary Rep. 160. Univ. of Wisconsin, Madison, Wisconsin (1960).

NUMERICAL OPTIMALITY AND THE APPROXIMATE SOLUTION OF DEGENERATE PARABOLIC EQUATIONS

41

4.0

INTRODUCTION

If a square-integrable function u, on a bounded space-time domain 9 of dimension n + 1, possesses a square-integrable gradient and time derivative, then the theory of approximation asserts that the optimal asymptotic approximation order in L2(9), by subspaces of dimension N, is N-"("+'), provided the estimate is to be uniform over the unit ball of H'(9). The approximation concept, which expresses this property, is known as the (Kolmogorov) N-width or N-dimensional diameter, which is developed in Section 4.2. It follows that any numerical procedure for approximating the solution of a degenerate parabolic equation, for example, the two-phase Stefan problem or porous-medium equation, for which the solution u is known to belong only to H'(9), should be constructed with this optimal order in mind. In Proposition 4.3.4, we prove that the continuous-time Galerkin (Faedo-Galerkin) procedure, based on piecewise linear finite elements of diameter h, is convergent in L2(9), with order hlln(l/h)r(")/2. Here, d(n) = 0 if n = 1, and d(n) = 2 if n > 1. This includes both the case of the two-phase Stefan problem and the porous-medium equation. One 113

114

4. Numerical Optimality and Degenerate Equations

sees, intuitively, that this order is essentially optimal if the continuous-time Galerkin approximation is replaced by its piecewise-linear interpolant based on t, = kh. The approximation order is unchanged, and the resultant finite-dimensional spaces are of dimension N x ch-("+').The presence of the logarithmic factor is accounted for by the occurrence in the equation of the term aH(u)/at,which is not, in general, square integrable. It is, however, essentially bounded in time as a regular Baire measure, which permits the application of the recent L" finite-element estimates, for linear elliptic problems, via duality. The plan of this chapter is actually more ambitious than described in the previous paragraph. For example, the exposition on N-widths gives a fairly complete discussion of the general abstract problem, and describes, as a major application, the determination of the widths in Lq(R) of the unit ball of Wd*p(R),when the embedding is compact. The general result is stated in Remark 4.2.5. A complete proof for all of the subcases would simply be too lengthy here. We present our own proofs for the subcases p = q and 1 < p < q < 2. These are variationally based, and lead naturally into the case p = q = 2, where very precise results are obtainable (see Theorem 4.2.14). The upper N-width bounds for p = q and 1 < p < q < 2 are obtained conveniently in Section 4.1 by means of the Sobolev integralrepresentation formula, so that the first two sections mesh nicely. However, the standard linear finite-element estimates depend upon the results of Section 4.1, and we use them in Section 4.3, where a very general version of the Aubin-Nitsche result is obtained in Proposition 4.3.1. Only minor modifications are required to extend this Hilbert space result to a Banach space duality result. We believe the reader is entitled to an explanation of why the case p # q is considered in such detail in this chapter. It is precisely because mixed norms enter in such a fundamental way in nonlinear partial differential equations. For example, in Proposition 4.4.1, where the convergence of the horizontal-line method is analyzed, a natural choice is q = 2 and p = 1 (l/y) for the expression of the estimates (see (4.4.5)). We mention, finally, that a really complete mixed-norm analysis, extending the first two sections, would provide for different orders of differentiation in the various independent variables, together with different Lp norms of the respective variables. This is the situation which actually occurs in nonlinear partial differential equations. The symmetric subcase of a uniform order t and uniform LP norm, hence a standard Wd*Pspace, actually impinges on the relatively innocuous linear equation -Au = f , f E Lz. By the Sobolev embedding theorem, u E Lq, q > 2. Thus, even this simple linear problem has a natural mixed-norm interpretation with p = 2 and q > 2.

+

4.1 Representations of Sobolev-Type and Upper-Bound Estimates

4.1

115

REPRESENTATIONS OF SOBOLEV-TYPE AND UPPER-BOUND ESTIMATES

We suppose that R is a bounded open domain in R", with diameter d, which is star-shaped, with respect to every point in an open ball B c R, and that x E C$(B) has integral value unity and is fixed, x 2 0.

Proposition 4.1.l.+ Suppose that f' > 0 is a specified integer, that k(x,y) =

jol s - " - ' ~ ( x+ s-'(y - x))ds,

x,y E R", x # y, (4.1.1a)

and for a a multi-integer, la( = 8, that the kernels k, are given by

x, y E R".

k,(x, Y) = (l/a!)(x - yYk(x,Y),

(4.1.1b)

Then k(x,.)is a continuous function on R"\{x}, is supported in the convex hull K, of {x} u support(X),and satisfies the estimate, for y E R",

(4x9Y)( G

((X((Lm(&(X

- ~(-"/n,

x E 0, x # Y.

(4.1.2)

In particular, k,(x;) is supported in K,, and

(ka(x,Y)J G Cf'(IXIILm(B)d"lX - ~ l - " + ' ~ l / n , x E R, x # y,

(4.1.3)

for a constant C depending only on n. When f E Cm(R), then a natural projection onto the polynomials of total degree less than f' on R is induced by the representation of Sobolev type

f(x) = PGf(X) + RGfb),

(4.1.4a)

where (4.1.4b) is a polynomial of degree less than 8, and (4.1.4c) In particular, the definition of Pdf depends only upon J B .

' Propositions (4.1.1-4.1.3)

and Corollary (4.1.4) reproduced with modifications from [13].


116

Proof: We note that the semi-infinite ray {x + s-l(z - x):O < s < l}, for z # x, intersects the support of x if and only if z is in K,. It follows that {z:k(x,z) > 0, x # z} c K, and, hence, by a simple closure argument, that support(k(x,.))c K,. To verify (4.1.2) we have, for Iy - XI < d, y # x, Ik(x,y)l =

;JI

X(X

+ s-'(y

- x))s-"-' ds

from which (4.1.2) follows for Iy - XI < d. Ofcourse, k(x, y) = 0 for Jy- XI 2 d. To verify (4.1.3)we , let C denote an equivalence constant, for which

(4.1.5) I(x - Y Y ~< IIx - Ylltm(an) la1 G C~X - yllal = Cllx and then use (4.1.2). Note that 1.1 has been used as a Euclidean norm on R' ~ll$'(Rn)'

and R", respectively, in (4.1.5). Now suppose x E 51 and y E B. By Taylor's theorem,

(4.1.6) and multiplication by ~ ( y ) followed , by integration with respect to y, yield

f(x) = P t f ( X ) + e

c a1! J x(y)(x

lal=d

-

B

- y)" Jolsd- ' f ( @ ) ( X

+ s(y - x)) ds dy. (4.1.7)

We shall analyze the latter integrals. By (4.1.3), the iterated integrals a! ~ ( aX ), = 7 JKx ka(x, z)f(a)(z) dz

(4.1.8)

are finite for la( = t, and, by the Fubini theorem, we have equality with the product integral expressed by I(a,x)=Jol ~ K x { ~ ( x + s ~ ' ( z - x ) ) ( x - z ) " s ~ ' f ~ a ~ ( zdsdz. ) s ~ " } (4.1.9)

For s E (0,1]and z E K,, consider the mapping +(s, z) = (s,y) = (s,x

+ s-l(z - x)) E (0,1]x R".

(4.1.10)


117

One easily sees that 4 is injective onto its range, since 4(s1, zl) = 4(s2, z2) implies s1 = s2 and, hence, z1 = z2. Now, by the Lebesgue-Radon-Nikodym theorem ([14] p. 230) the substitution of (4.1.10) is permitted in (4.1.9), yielding Z(a,x) =

SRnnge(&) -

+

~ ( y ) ( x y ) a ~ ' - ~ f ( ~ ) ( xs(y - x))dsdy.

(4.1.11)

However, Range(4) = (0913 x support(x),

(4.1.12)

and x vanishes for y # support(X). It follows that Z(a,x)= Jol

SB~ ( y ) ( x- ~ ) " s ' - ~ f ( ~ )+( xs(y - x))dsdy.

(4.1.13)

If the Fubini theorem is applied to (4.1.13),and the resultant substituted into (4.1.7), we obtain (4.1.4). Note that here we have used the facts that support(k,(x, .)) c K , and that the latter set is contained in the star-shaped region 0.

Remark 4.1.1.

For x # y, we may generalize (4.1.2) and (4.1.3) to

and

An elementary integration by parts shows that

which shows that P, may be defined for arbitrary distributions f on R by Pef(x) =

1 f(")(~(.)(x- *)"/a!).

lal

21)

< m{xER:l K,,,,l*f(x)l >A} + m{xER:(K,,,,,

*f(x)l > A}.

(4.1.28)

We shall show that the second term on the right-hand side of (4.1.28) is zero. Indeed, by Young's inequality, IFs.p.m

* fIILm(n)

IIKs.r,mIILP'(IW")IIfII~p(IWn)

Note that this computation has used the relation (-n

+ s)p' + n = -np'/q

< 0,

which follows from (4.1.24), and has used the definition (4.1.27). It remains to estimate the first term. Thus,

where Ap,qis given explicitly by (4.1.29)

This completes the verification of (4.1.25) and, hence, the boundedness in the case (4.1.23). The case q1 > 0,O < s < n, that is, 1 1 s - - - +->O,

q

p

n

O<s0 consider the idempotent operator

P:W d p p ( Q+) L4(Q),

(4.1.48a)


125

Remark4.1.7. The following change of scale result is a routine computation: IUIWm.p(Q,) = h(n'p)-mlUIWm,P(Q) (m2 (4.1.49) where u = U,U.

Proposition 4.1.5. Let p , q, d , and k be specified as in Definition 4.1.2. Then there is a constant C = C(n,d , p , q), such that < Ck - d - n ( ( l / d - ( l / P ) )(U(Wl.P(Q), p < 4. (4.1.50) IIv - P v I I L ~ ( Q )1

Here P is defined by (4.1.48).

Proof: By the change of scale result (4.1.49), we have 1(u - puI\L.(&) =

h"/qI U

- pduIILS(Q),

(4.1.51)

where u and u are related by (4.1.47). The latter quantity is estimated, via Corollary 4.1.4, by

(Iu

- pduI(LS(Q)

< CIU(W'.p(Q).

(4.1.52)

A change of scale now yields (4.1.53)

Coalescing (4.1.51)-(4.1.53), with h = l/k, gives IJV

- PV\ILS(&)

"+Nl/d - ( U p ) )

-

IuIW/*p(Q,)*

(4.1.54)

The argument to this point has not required the hypothesis p < q, delineating (4.1.50). For the completion of the argument, we take qth powers of both sides of (4.1.54), sum over a, and then extract qth roots. Inequality (4.1.50) follows upon application of

which is an elementary version of Jensen's inequality.

Remark 4.1.8. The range of the projection P is the subspace & of Lq(Q), with dimension N = k"s, such that o E & satisfies olQa E PAQ,), each Q,.

4.

126

Numerical Optimality and Degenerate Equations

Here s = ( “ ; d ; l ) is the dimension of the space of polynomials of degree not exceeding t - 1. Thus, in terms of N , the order in (4.1.50) is N - ( d ’ ” ) f ( l ’ p ) - ( l ’ q ) for p < q. Note that Holder’s inequality in the form

implies the estimate

110

- Pvl\Lqo)< Ck-‘l~)w~.p(p), r G p , (4.1.57)

when taken in conjunction with (4.1.50) with q = p. It is reasonable to ask whether other operators onto possibly different finite-dimensional spaces yield superior asymptotic estimates of approximation with respect to N for given values of p and q. The answer is negative, except in the range p c 2 < q. This point is amplified in the bibliographical remarks. 4.2

LOWER-BOUND ESTIMATES AND N-WIDTHS

We shall introduce the notion of the dispersion of a set from a linear manifold, followed by the notion of N-width or N-dimensional diameter. This will be followed by some abstract lower-bound estimates obtained (necessarily) by the Borsuk antipodal theorem. Specific applications to Sobolev classes will follow.

Definition 4.2.1. For a set B in a normed linear space X,the dispersion of B from a linear manifold d is given by the (extended) real number E ( B , d ) = E , ( B , d ) = sup inf IIu - 011, usB u e A

(4.2.la)

and the (Kolmogorov) N-width by the (extended) real number

d,(B, X) = inf(E(B, d): A c X,dim A = N},

(4.2.lb)

for N = 0,1,. . . . The following result is a superficialgeneralization of the Borsuk antipodal theorem.

Proposition 4.2.1. Let X, and X, be subspaces of a normed linear space X, with dimensions n and N, respectively,where 0 < N c n. Let rZ, be a bounded open subset of X,, symmetric about the origin, such that 0 E n, and let II/ :an, 4X, be a continuous, odd mapping. Then there exists x E an,, such that I&) = 0.

4.2 Lower-Bound Estimates and N-Widths

127

Proof: The usual Borsuk theorem is the special case X = R" = X, and X, = RN (see Nirenberg [26] p. 25). A simple reduction to this case is afforded by the map of IXnan,,into R",given by IJ =

I,,

0

*

0

I&

+ RN are (linear) isomorphisms. W where Ixm:Xn+ R" and IXN:XN

Remark4.2.1. The classical Borsuk theorem is itself a consequence of degree theory. For a continuous map +:aR, c R" + R"\{O}, the degree of $ may be defined, for any (connected) component onof R"\{+(aR,,)}, by

p # 0. Here, where p is any smooth n-form in R", with support in onand jRn we may choose any continuous extension of I) to It is then an elementary fact that deg(ll/,R,,o,) = 0 if onc R"\{+(a,)}. This may be used, in conjunction with a basic fact about odd mappings, to prove the classical Borsuk theorem. We state this basic fact for the reader's interest in the form of a lemma' (for the proofs see Nirenberg [26] pp. 21-25).

a,.

Lemma 4.2.2. Let R, be a bounded, open subset of R",symmetric about the origin, such that 0 E R,, and let t,h:i?R,,+ R"\{O} be a continuous odd mapping. Then, if on is that component of R"\{$(dR,)} containing 0, deg(*, R,, a,,) is odd. Proposition 4.2.1 has an important consequence in terms of lower bounds for N-widths. Proposition 4.2.3. Let X be a normed linear space, and suppose X, c X, dimX, = r > 0. Let B be a subset of X, such that the set B, = B n X, is a bounded open subset of X,, symmetric about 0, with 0 E B,. Then, for each 0 < N < r, and each subspace .M of X of dimension N, there is an element x = x ( A ) E aB,, satisfying (4.2.2)

llxll = inf{llx - u(I:uE &}.

In particular, d,(B,X) = d,(B,X)

> inf{llull:u E aB,},

0 < N < r.

(4.2.3)

Lemma 4.2.2 is referred to by Nirenberg [26] as the Borsuk theorem. Following what is possibly more common usage, we have preferred the formulation of Proposition 4.2.1.

128

4 . Numerical Optimality and Degenerate Equations

Proof: Select Z, of dimension not exceeding r + N, such that Z = linear span(X,, A).Assume first that Z is strictly convex (see Chapter 1 [14] p. 17). Define a mapping $ :dB, -, A by $ ( y ) = z,

Ily -

zJI = inf{Jly- uJI:uE A}.

(4.2.4)

Here, we have used the strict convexity of Z, which guarantees that z is uniquely determined. Now the mapping [:Z + R', (4.2.5)

[ ( y ) = inf{lly - u(I:uE A},

is continuous from the inequalities, for y , and y, in Z, These hold irrespective of the strict convexity of Z. It follows that $ is continuous. Since $ is odd, Proposition 4.2.1 applies to give x E dB,, for which $(x) = 0. Thus, (4.2.4) implies (4.2.2) for y = x. If Z is not strictly convex, its norm can be approximated by strictly convex norms E > 0, satisfying (see Chapter 1 [14] p. 138)

IJ.IIE,

llxll G llXllEG (1 + E)IIXII,

x

E

z.

(4.2.6)

If E, -,0 is chosen, and the preceding argument is applied for each m, a sequence {x,} c dB, is obtained, with accumulation point x E dB,, such that I(xm1I

(1 + E r n K ( X m ) ,

(4.2.7)

where [(.) is defined in (4.2.5) with respect to the fixed norm 11. 1 1. Letting E, -,0 in (4.2.7), we find that llxll 6 [(x), hence, llxll = [(x). To verify (4.2.3), let JZ be a subspace of X of dimension 0 G N < r, and let x satisfy (4.2.2). We have E ( B , J Z ) 2 E ( B r , A )2 5(x) = llxll 2 inf{lluII:u E dB,>.

(4.2.8)

Taking the infimum over JZ c X,dim A = N, gives (4.2.3). The relation (4.2.2) also suggests the crude upper bound for d,(B,, X) of sup{llull:u E dB,}. A much more precise estimate is possible, however, for certain sets when N = r - 1. We discuss this now.

Definition 4.2.2. A set K in a linear topological space X is said to be absorbing if, for each x E X, there exists 0 < y = y(x), such that x E AK for [A[2 y. K is balanced if I K c K for [A/6 1.

4.2

Lower-Bound Estimates and N-Widths

129

Remark 4.2.2. The class of closed, convex, balanced, absorbing sets K, for which 0 is an interior point, is in one-to-one correspondence with the class of continuous seminorms p on X , via the correspondence K = {X

E X:p(x)

< l}.

(4.2.9)

Given K, p is termed the Minkowski functional of K. Moreover, intK = { x E X : p ( x ) < l},

aK = { x E X : p ( x ) = l}

(4.2.10)

(see Taylor [42] Chapter 3). Moreover, to each point x o E dK, there can be associated a continuous linear functional F, defined on X , such that

IlFll = 1,

F(x0) = sup{lF(x)(:x E K}

(4.2.11)

(see [42] p. 145).

Proposition 4.2.4. Let X N + be a normed linear space of finite dimension N + 1. Let K c X N + be a closed, balanced, convex, absorbing set, for which 0 is an interior point. Set

(4.2.12)

p(K) = inf{lluII:u E as}, where it is explicitly assumed that aK # fa. Then dN(K, x N

(4.2.13)

+ 1) = p(K)*

Proof: That p(K) is a lower bound is an immediate consequence of Proposition 4.2.3.We shall establish that p(K) is an upper bound for the N-width. Thus, choose x o E aK, such that llxoll = inf{lluII:u E aK} = p(K),

(4.2.14)

and select the linear functional F, satisfying (4.2.11).Then, defining H = {xEXN+1:F(X)=O},

we have dimH = N, and, by the formula for the distance of a point to a closed subspace, we have, for x E K, llxoll 2 E ( { x o } , H )= F(x0) 2 IF(x)l = E ( { x ) , H ) .

The result follows by taking the supremum over x

E

(4.2.15)

K in (4.2.15).

Remark4.2.3. The major application we shall make of the preceding propositions is to the case where

B = By = { J J

E Y:IIJJ\I~
0. Then, for N < r, dN(UY

= dN(clX

uY

2 inf(()ull,:u E Y,, llully = I}. For r = N uY

+ 1, the

(4.2.18a)

final quantity characterizes the N-width of KN =

yN+l:

dN(KN,X)= inf{llulIx:uE YN+I,I ( U ~ ~=Y 1).

(4.2.18b)

We now pass on to the specific application where Y is a Sobolev space and X a Lebesgue space. We state the complete result in a preliminary remark prior to a detailed discussion of certain special cases. The symbol a ( N ) z b ( N ) below means that a ( N ) = O ( b ( N ) )and b ( N ) = O ( a ( N ) ) .

Remark4.2.5. If we denote by Ud.P the closed unit ball of WGsp(Q)(see (4.2.26)) on the unit hypercube in R", then the numbers dN(UGrP, L*(Q))may be displayed asymptotically in a tableaut as follows, where 1 < p, q < 00.

' Reproduced with permission from [19], p. 171.


131

4

J

fI I \

with t - ( n / p ) + (n/q) > 0, and t - n/2 > 0 if 2 0 if p c 2 c q. This striking result was obtained by KaSin [23]. The reader will see that only in the ranges 1 < p < q < 2 and q < p does the asymptotic estimate match the upper bound given by the projection method of Section 4.1. The fault does not lie entirely with the Sobolev projection, since the values of the Kolmogorov N-width do not correspond asymptotically in all ranges of p and 4 to the so-called linear N-width defined by operators (see the bibliographical remarks). We shall give detailed proofs only for the cases 1 < p = q < co and 1 < p < q < 2. For these cases, the estimates in the tableau follow by combining Proposition 4.1.5 (upper estimates) with Propositions 4.2.7 and 4.2.10 (lower estimates).

Definition 4.2.3. Let k 2 1 be specified, set h = 1/(2k) and construct a uniform partition of Q into (2k)" cubes Q,, defined by (4.1.45), with 0 < ui < 2k - 1, i = 1, . . . , n. Define the polynomial of degree m2,m 2 1, a(s) =

1 - (1 - s"')"',

0 < s < 1,

(4.2.19a)

and the (deficient)spline b E C"'-'(R1), O<S 0 and 6, > 0 are such that

< 61, dN2(uY,x) < 6 2 , and choose Mi, dim Mi = Ni,such that dN~(A~Y)

Ey(A,M,) < 61,

Ex(Uy,M,) < 6 , .

Then, given x E A, there exists y E M,, such that IIx - ylly < a,, and there exists z E M,, such that "((x - y)/6,) - zllx < 6,. Altogether, Since y

+ 6,z

IIX

E M,

- (Y

+ W l l x < 616,.

+ M,, dim(M, + M,) < N1 + N.,, we have

~ N ~ + N ~ ( A 0, such that ((1/2) (l/q) ( j / n ) ) > 0; in particular, WJ.q(Q)is continuously embedded in LZ(Q). If we apply Lemma 4.2.11, with A = Ut+j*p, Y = Wiq(Q),and X = L:(Q), we obtain

+

Now, choose an integer p, such that p" 2 2j, and set N, = v"(":!;'), N , + N , = 2"-lp"v".This gives

and

Thus, applying Proposition 4.1.5,with k = v, to the denominator of (4.2.47), and a strengthened form of Corollary 4.2.9,with k = pv and Ud+j*P replaced

' Due to El Kolli [15].

138


by U f + i , pto, the numerator, we obtain d N ,(uJ + j.p wj.q(Q)) 3 cv- - i- n(( 1 / 2 ) - (UP)) ,,j+ n(( 112)- ( 1 / q ) ) n 7 n for some C > 0, from which it follows that dN,(U:+ j.P, w?(Q))2 cv- J + n( ( 1/P)- ( 1/ 4 )). An application of the inequality (4.2.42), with W = W$P(Q), V = Wd,'jgp(Q), X = L4,(Q), Z = W?(Q), and J = J,: yields d,,(U$P, L4,(Q))2 Cv- 8 + n(( l/P)- ( 1I S ) )

',

for some C > 0, and the result (4.2.44) follows by the monotonicity properties of N-widths. Inequality (4.2.45) follows in analogy with (4.2.32). W Remark 4.2.9. Note that we have now completed the details for the N-width orders in the cases 1 < p = q < co and 1 < p < q < 2 (see Remark 4.2.5.) In the case p = q, Proposition 4.2.7 suggests that the largest constant C in (4.2.27b) may be obtained from a nonlinear extremal problem of the form = suP{llgllLP(Q)/(1gIIw6.P(Q):o

f 9E

WgP(Q)}

= [inf{llgllw6.p(p)/ll911,,(0,:O # 9 E WOG.P(Q)}l-l,

(4.2.48)

which, when p = 2, is the reciprocal square root of the smallest (positive) eigenvalue of the Dirichlet problem on Q, when the appropriate norm is defined on W$"(Q). The reader may justly inquire whether this value of C gives the exact value of dN in (4.2.27b). This is, in fact, the case, and we shall give the proof using the famous Weinstein-Aronszajn theory for intermediate eigenvalue problems. We shall actually present a much more general result. Definition4.2.4. Let H be a real or complex Hilbert space with norm [lull = (u, We shall denote by Y the class of self-adjoint linear operators A which are positive, that is, (Au,u) 2 0 on the domain D = D, of A, such that the lower part of the spectrum of A consists of a finite or infinite number of isolated eigenvalues0 < A1 d A2 d * ' . repeated according to their finite multiplicity. Thus, lj< 1, for these eigenvalues, where 1, is the lowest point in the essential spectrum (if nonthe (extended) empty) of A, or 1, = co otherwise. We define, for A E 9, integer

M

= sup{j:Aj < A,},

(4.2.49)


139

and the ellipsoid d = { u ~ D , : ( A u , u ) < l}.

Proposition 4.2.12. given by

(4.2.50)

If 0 < N < M, then the N-width of the ellipsoid d is d ~ ( dH) , = E ( d , AN) = Ai:':.

(4.2.51)

, = 00 Here, ANis the linear span of the first N eigenvectors of A, and d N ( d H) if AN+ 1 = 0. Proof: By taking orthogonal complements, we may assume Al > 0. Let A be any N-dimensional subspace ofH. Then, if P is the orthogonal projector onto A, and Q = I - P is its complement, E ( ~ , A ) -=~[ S U P { ~ ~ Q U ~ ( A 0,

{"s' }

inf - : " € A h

2 yo>O,

all h.

(4.3.27)

Thus, the results of Propositions 4.3.1 and 4.3.2 and Remark 4.3.1 are valid under the hypotheses associated with (4.3.24),(4.3.25),and (4.3.27).

Remark 4.3.7. One of the remarkable properties of the projection E,, taken in Hilbert space, is its stability, or near stability, in the pointwise norm spaces. For our purposes, this simply means that, for piecewise linear trial functions, an estimate of the form

I( T

- Th)fllLm(f2)

G

ChZ(1n(1/h))d'"'lIfIILmcf2)

(4.3.28)

4.3 Rates for the Continuous Galerkin Method

149

can be proved, where d(n) depends upon the Euclidean dimension n. It is known that d(1) = 0 and d(n) = 2, n >, 2, for the standard Dirichlet and Neumann homogeneous boundary-value problems (see Douglas et al. [123 and Schatz and Wahlbin [33] and Remark 4.3.10 to follow). Many other results similar to (4.3.28), in which is replaced by various secondorder norm differential expressions of Tf, e.g., 11TfIlw2,m(n), have been obtained by Scott [36], Nitsche [28], and others. However, for the applications of this section, the estimate (4.3.28) is the crucial one. Frehse and Rannacher [16], obtained a result slightly weaker than that of (4.3.28), with higher powers of the logarithm appearing. It is of interest that, for many higher-order differential problems, involving piecewise polynomials at least of quadratic degree, the exponent d(n) is zero, at least when (4.3.28)is interpreted in the relaxed sense, in which is replaced by a more comprehensive expression involving f (see Nitsche [28] and Scott [36]), and, of course, in which the exponent of h is accordingly increased.

Il f l L mcn)

\ l f l L mcn,

Definition4.3.5. We denote by M(a) the space of finite regular Baire measures on normed with the total variation norm. It is well known that M(0) = [C(n)]*. We make use of the norm-preserving Lebesgue extension p of p, so that

a,

ljnfdfilG Ilf(ILm(n)llPllM(fi), for f Lm(n), (4*3*29) holds for each p E M(a),and we identify p with p. Finally, T is an isomorphism

of F = [H'(Q)]* onto H'(Q), defined, via (4.3.1), by a symmetric, positive definite bilinear form a( ., .), T, = E,T, where E, is defined by (4.3.4, with s h = M,, and F, is the (semi) inner product space described by (4.3.17), that is, Fh = G-1.h.

Proposition 4.3.3. Suppose that T and Th satisfy (4.3.28). Then, for p [H'(R)]* n M(Q, the estimate

E

(4.3.30) holds for the same constant C .


150

and (4.3.30) is immediate. Note that we have used the self-adjointness of T - T, with respect to the duality pairing. W We turn now to an application of these ideas to nonlinear parabolic evolutions.

Definition 4.3.6.

Consider the differential equation in distribution form, av

- - AU at

+ f o ( ~ )= 0,

u = H(u)+,

(4.3.31)

on a space-time domain 9 = R x (0,To),with initial datum u l t = , E L"(R) specified. It is explicitly assumed that u E L2((0, T o ) ;H'(R)) n H'( [0, T o ] ;L2(Cl))n La@), u E L"(g),

~t

(4.3.32a) (4.3.32b)

E L"((0, To);M(Q)),

fo is Lipschitz continuous,

(4.3.32~)

H(* ) defines a surjective, maximal, monotone graph in R2 with bounded sections.

(4.3.32d)

We also assume that (4.3.31) has a pointwise lifting, via an isomorphic, self-adjoint mapping T onto H'(R), so that the pair u, u satisfies, for f = fo - aid, some 0 >/ 0, aTu

+ u + Tf(u)= 0,

at

almost everywhere in 0, 0 < t < T o , (4.3.33)

and that there exist positive monotone increasing functions c1 and c2, defined on [0, GO), such that, for all wl,w2 E L"(R),

I

I

IH(wl(x))- H(w2(x)11 2 [c,( IIw1 JLm(n))cz( )w~J(L-(R))] - Jwi(x) - WAX)( (4.3.34) for almost all x E R. This model includes, with homogeneous Neumann or Robin boundary conditions, the two-phase Stefan problem, the porousmedium equation (see (2.3.21)), and the nondegenerate reaction-diffusion system, if (4.3.31) is understood in vector format. However, the latter can be

Remark 4.3.8.

' Uniqueness is anticipated here.


151

analyzed directly in a more straightforward manner and with a sharper estimate, with v = u, once the maximum principle u E L"(9) is utilized to permit the assumption (4.3.32~).The assumption u, E L"( (0, To);M(0)) is the really distinguishing featuret in (4.3.32); this was not directly required in defining the solution classes in Sections 1.1 and 1.2, but will be demonstrated in Chapter 5 (see Theorem 5.2.1).

Definition 4.3.7. We shall define the finite-element approximation [O, To]+ M, as the solution of the differential equation

Uh:

for all x E M,, 0 c t c T o , subject to the initial condition Uh(0)E M,, defined by PhH(Uh(o))

= PhUll,o*

(4.3.35b)

Note that the system (4.3.35a), of dim M, ordinary differential equations in t, must be understood in the sense of distributions on (0, To),and that Ph is the orthogonal projection in LZ(Q)onto M,. Concerning the regularity of u h , which must be considered in tandem with H( Uh), we require Uh E L2((o,TO);H'(Q))n H'([O, To];L2(n)),

(4.3.35c)

c([O,

(4.3.35d)

and H(Uh)

L2(n))*

Remark 4.3.9. The existence of a solution pair satisfying (4.3.35) follows from the existence theory developed in Chapter 5. The uniqueness is a consequence of the pointwise lifting relation, wheref = fo - aid, as in Definition 4.3.6,

a

+ + Thf(

-ThH(Uh) at

almost everywhere in R, 0 < t c T o , (4.3.36) which follows from (4.3.35a), by setting x = Th$, $ E C;(R), and employing the self-adjointness of T,. The relations (4.3.33) and (4.3.36) are the central relations for the convergence estimates to follow. Uh

u h )=

0,

This analytical property makes possible the application of L" finite-element estimates via extended duality.

4.

152

Numerical Optimality and Degenerate Equations

Proposition 4.3.4. Suppose that the continuous Galerkin approximations u h , satisfying (4.3.35)and (4.3.36),are bounded, say, IIuhIILm(9)

G

cO,

(4.3.37)

for the system satisfying Definition 4.3.6. Then, under the assumption that (4.3.30)holds, there exists a positive constant C, such that

If (4.3.24),(4.3.25),and (4.3.27)hold, then,

Here Fh = F as a set, with semi-norm defined by (4.3.17).

Proof: Subtracting (4.3.36) from (4.3.33), multiplying the resultant by H(u) - H( u h ) , and integrating over 0,we obtain, for 0 -= t -= T o ,

+ ((Th - T)f(u), H(u) - H(uh))L2(n) + ( T h ( f ( uh) - f(u)H(u) - H( uh) 19

)Lz(n).

(4.3.40)

Suppose the first term on the left-hand side of (4.3.40)is rewritten in Fh seminorm differentiated terms, and the final term on the right-hand side is estimated by the Cauchy-Schwarz inequality in Fh,and by the domination of the Fhseminorm by the F norm as I(Th(f(uh)

- f(u)), H(u) - H ( u h ) ) L 2 ( n ) l

(If(uh)

< @qIIuh

for

f i the product

- f(u)llF(IH(u)

- H(uh)(lFh

+ $q-'[(ff(u)- H ( u h ) [ l : k ?

- u11?2(n)

(4.3.41)

of the Lipschitz constant off and the embedding of

L2(n)into F, and where q is chosen satisfying rl =

c- "cl(lJUJILm(D))C2(CO)I-

(4.3.42)

with Co specified in assumption (4.3.37).The estimate (4.3.41) is valid for almost all t, 0 < t < T o , by virtue of (4.3.32a,b)and the continuity of u h on 9. Here, the functions c1 and c2 are given by (4.3.34).Then, we may


153

rewrite (4.3.40), by the use of (4.3.41) and (4.3.34), to obtain

l d

(IH(u)

- H(Uh)llzh

+ 3(H(u)- H(Uh),

u-

Uh)L2(R)

(4.3.44) where we have applied (4.3.32a,b). Combining (4.3.44) with (4.3.34) and (4.3.37),we obtain

\IH(')

- H(Uh)l(im((O,To);Fh) + ~ c 1 ~ ~ ~ u ( ~ L m ( 0 ) ~-c '12' ~ c-O ~'hl(i'(0) ~ (4.3.45)

(4.3.46)


154

where we have used (4.3.32a) and (4.3.37). Combining the estimate for IIPh[H(u) - H(uh)]ll&, given by (4.3.38), with (4.3.20b) and (4.3.46), we obtain (4.3.39).

Remark 4.3.10. We shall elaborate here on the derivation of estimate (4.3.28),+as given in Schatz and Wahlbin [33], for n 2 2. If p < co is arbitrary, the estimate - Th)fllLm(n)

6)

G C1n -

hz-("'p)IITfl(W2.P(n)

(4.3.47)

holds, by combining standard approximation theory with the estimate,

which has been obtained in Schatz and Wahlbin [33] for Dirichlet problems, and in Scott [36] for Neumann problems. Now the Calderbn-Zygmund lemma [ll], as employed by Agmon et al. [2], gives, for C, independent of p,

ll(T

- Th)fllLm(n)

G Cl(ph-'"'P')hZ In

(3 -

I(fIILP(R),

(4.3-48)

where the explicit linear dependence on p was noted by Johnson and Thomke [22]. A simple calculus minimization of g ( p ) = ph-("IP), for p 2 1 and 0 < h < 1, yields the unique minimum, en ln(l/h) at p = nln(l/h), from which (4.3.28) easily follows, with d(n) = 2, n 2 2.

4.4

CONVERGENCE RATES FOR SEMIDISCRETE APPROXIMATIONS

We shall derive convergence rates for the fully implicit method applied to a class of degenerate parabolic equations of the form au

_ - AU = 0, at

u = H(u),

(4.4.1)

in a format including the porous-medium equation and certain forms of the two-phase Stefan problem.

' The existence of this estimate was brought to the author's attention by Ridgway Scott. * The convergence of the horizontal line method is discussed here.

4.4 Rates for Semidiscrete Approximations

155

Definition 4.4.1. Consider the differential equation (4.4.1) in distribution form on a space-time domain 9 = R x (0,To).We shall assume that (4.4.1) has a pointwise lifting induced by an isomorphic self-adjoint map, T: G* + G, onto G c H'(R): aTu -+u=o,

almost everywhere in R, 0 < t < T o ,

at

(4.4.2)

where we assume explicitly that there is a pair (u,u) satisfying (4.4.2), with u = H(u),+and t.4 E

L2((0,7-0);H ' m ) n H'([O,ToI; L2(Q)),

(4.4.3a)

u : [0, To] + L2(R) weakly continuous,

(4.4.3b)

H defines a surjective maximal monotone graph with continuous function left inverse,

(4.4.3c)

Tult=,is specified in the range of T,

(4.4.3d)

u E L2(9),

Y 2 1; here the inequality holds for all elements in H ( < , ) , H ( t 2 ) ,and all 151

- r 2 l 2 CIH(t,) - H(t2)IY,

(4.4.3e)

r,, r2

E

R'.

Finally, let {9"}be a sequence of partitions of [0, To],and define the fully implicit approximations {u:, H(#)}fi?) in analogy with (2.4.1)-(2.4.2), with H($) = ult=,. It is required that H(u:) E L2(R). Proposition 4.4.1. the estimate

Let d: = u ( { )

- u:

and

4 = H(u)(t:) - H(u[). Then,

(4.4.4) holds for some positive constant C,, and C given by (4.4.3e). Under the additional restrictions that u, u E L"(9), u l t = , E L"(R), and that the pointwise hold, as well as the inequality (4.3.34), estimates IIu:IILmcn, < IIHthen we have the inequality

l~lt=ol(Lm(n)

The quantity 1IBNII,above, denotes the maximal mesh spacing, and the constants C, and C2 contain C-('lY)as a factor.

*

Again, uniqueness is utilized.

* The latter statement has particular relevance for the Stefan problem.

4. Numerical Optirnality and Degenerate Equations

156

Proof: Using the lifted format, we obtain for the implicit approximations and the evolution equation, respectively, for k = 1, . . . ,M ( N ) ,

+4 =0,

T[H(u,") - H(u,"- I)] (t,"- t,"- 1)-

(4.4.6a)

l)](t,"-t,"I ) - ' +u(t,")

[TH(u)(tk)-TH(u)(tk-

= [TH(U)(tk) - TH(U)(tk- I)] (t,"- t,"-

= (t,"- t,"- 1) -

N s'k

tP-

I

(tf-

1

- s)

a'TH(u) ~

at'

-

1)-

aTH(u) ~

at

(t,")

(s) ds.

(4.4.6b)

Subtracting (4.4.6a) from (4.4.6b),multiplying by 4,and integrating over 0, we obtain (t," - ~ , " - l ) - l ~ ~ e, "(t," ~ ~&t,"-l)-1(4,4-l)G* + (4,d,")Lz(n)

(4.4.7)

= (4,f,")LZ(R),

where we have noted that (au)/(at) = -(a'/at')TH(u), and where f," is defined by (4.4.7). Now, if C is the constant described in (4.4.3v), then I(.,",f,")Lz(n)l

N ~ + l +c-(l/y)-

G y+ - 1lhIILv+l(n)

7 N 1+(1/y) y + 1 l l f k llLl + ( l / v ) ( Q ) ,

(4*4.8)

which follows from the Holder inequality with p = y + 1 and q = 1 + (l/y), followed by the inequality up bq ab 0,

(5.1.5b)

for all J, then the estimate k= 1

holds for some C .

(5.1.6b)

5. Existence via Stability and Consistency

166

=

- (Tf,(U:-

117

4 - 4-l)LZ(n).

(5.1.7)

The fourth term may be seen to dominate the term N

N

& l ( t ; - $- l)-’(T(U: - $-1 1 9 u k - u k -

(5.1.8)

l)LZ(R),

if the nonnegativity of T is combined with the monotonicity of H ( with (5.1.5a), when 8,f, # 0. If the inequality I(u, w)l

6

6-

< j llq + 2 11u112,

a),

and

(5.1.9)

6 > 0,

is applied to the third term in (5.1.7), with 6 = 1, and if the dominance of (5.1.8) by the fourth term is utilized, we obtain

w:

- $114-11122(R) + - t:- 1)- ‘ ( T ( 4 - 41 ) , 4 - 4-1)LZ(R) (5.1.10a) G 4 - 4-l)LZ(n)(, when O,f, # 0. By formally setting E = 0, we see that the corresponding inequality holds when O 2 = 0 andf, # 0. Also, we have the simpler inequality $llu:llZ.(n)

pm4-

119

411u:11Z~(n)

- 4114- 11122(n)G 0,

(5.1.1Ob)

when f, = 0. If (5.1.9) is applied to the right-hand side of (5.1.10a), in the = (To, u)&), with the choice 6- = h ( t f 1)we “negative” norm, obtain

fr-

1 01

-

fll~:IIZ2(n)

G at:

+ ( M t : - C- 1)- ‘/2)(T(u:

tllu:-

i11Zz(n)

- t:-

l)[/lfullZipll4-

lIIZZ(R)

- u:-

I), U:

+ lfu(o)121~ll~

’,

- u:- 1)Lz(n) (5.1.11)

where C = (AE)-’C~ and C , is the norm of the injection of L’(i2) into the (“negatively” normed) dual space. Summing (5.1.11) over k = 1,. . . ,rn, we obtain

+

where C, = 2CTolf,(0)lZl!21 I(u,(l&, and Cz = 2C(lf,lJ&,,for rn = 1, . . . , M ( N ) . If the version of the discrete Gronwall inequality contained in Remark 2.2.1 is applied to (5.1.12)(cf. (2.2.11b)), we obtain

ll~:llZz(n) < (Cl + CZIIUoll:z(n)To)

exp(C,T,),

(5.1.13)

5.1 Stability in Sobolev Norms

167

for k = 1,. . . ,M ( N ) and all N . This yields (5.1.6a) iff, # 0. Iff, = 0, we sum (5.1.10b),instead of (5.1.11), and the result (5.1.6a) is clearly immediate with C = The proof of (5.1.6b) proceeds similarly. In this case, the left-hand side of (5.1.10) is augmented by the term

IIu~(I~~(~).

The hypothesis that H defines a bounded operator on L2(n),in conjunction with the previous proposition, yields the following.

Corollary 5.1.2. Under the hypotheses of Definition 5.1.1, there exists a constant C, such that ( I H ( U : ) ~ ~ 0.

(5.1.15)

The analog of (5.1.4) is now given by

+

(t: - t:- 1)- 1 ~ [ ~ ( ~ : ) H(& ,)I e,u: (I - el)& Tg(uf) O,Th,(ur) (1 - O2)Th,(uC- 1) = 0,

+

+

+

+

(5.1.16)

where we may conveniently choose o in (5.1.16)to have the value c in (5.1.15). The estimates (5.1.6) and (5.1.14) are valid in this case also, provided ((f((Lip CT in the statement of (5.1.5a) is replaced by ((h((Lip, and provided is retained in the estimates (5.1.5b) holds when 0211hllLIp# 0, though (5.1.1 l), etc. The motivation for (5.1.16) is the semidiscretization (2.2.18), which is the differential equation corresponding to (5.1.16) with O1 = 1 and O2 = 0. Although g was only required to be locally Lipschitz continuous in the earlier format, the case of essentially bounded initial data leads, via the estimates of Section 2.2, to a de fucto Lipschitz continuous function. The next step is the derivation of gradient estimates.

+

l fUl Lip

Proposition 5.1.3. Under the hypotheses of Definition 5.1.1 and the additional hypothesis wo E D,- 1, there is a constant C, such that, for the solutions u r of the semidiscrete scheme (5.1.4),

( ( ~ r 1. Since there is no a priori reason why the former must be the case, and since we wish to minimize the assumptions on Auo, we are led to a smoothing argument, in which uo is approximated in D,- n W2*'(R)by a function u$ in Cm(a),and H is replaced by a smoothing, say, its Yosida approximation 1 1 (5.1.28) H , = - ( I - K , ) = - [ I - ( I + E H ) -'3, E > 0, &

&

in which we may unambiguously interpret the inversion in (5.1.28) either as the inversion of a function, or the inversion of an Lz-operator. The function H , is an admissible approximation in the sense of Definition 5.1.2, to follow, and is monotone and Lipschitz continuous, with Lipschitz constant 1/&.It is now a standard fact (see Lions [33] or Lemma 5.1.6, to follow) that, if u;*" denote the solutions of (5.1.4) (0, = 1, O2 = 0), with H replaced by H e and uo by u$, then sequences {u?."} and {HEV(upN)} exist, such that u?*" .+ u; (in L2(R)) and H,,(uEV") H ( u f ) (weakly in L2(R)) as E , -,0. Clearly, the latter sequence also converges weakly in L1(R). Strictly speaking, we have complicated the usual argument by introducing ui. However, this smoothing sequence can be chosen, so that Hz(ui) H(uo) in L2(R), by the Lipschitz property of H E (choose llu$ < E') and the assumed property H,(uo) .+ H(uo) in L2(R) (see Brezis, Chapter 3 [8] in the case of the Yosida approximation), where H(uo), here, must be defined uniquely in the case of the Yosida approximation by choosing the real number of minimal modulus in the closed convex set H(uo(x)),x E R. As noted in (5.1.25c), other choices of admissible smoothings correspondingly determine H(uo).It follows, finally, that, if estimates of the form

-

.+

u ~ I ~ ~ ~ ( ~ )

(st")-

'IIH,(u;"N)

- H,(U;!1)IIL'(R)

d

c

(5.1.29)

can be derived, in which C is independent of E , k, and N , then the lower semicontinuity of the L' norm, with respect to weak convergence, implies (5.1.26), when the limit infimum of (5.1.29) is taken. Note that C is actually

5.1 Stability in Sobolev Norms

171

permitted to depend upon quantities, such as ui, which are strongly convergent as E + 0, since slight adjustments yield a constant independent of E. The point of introducing the smoothing is the obvious inequality o;N< 1 / ~ for all k, N , and the relation Aut E L2(R). For E sufficiently small, one sees that cotN2 A/2,+ where 1is given in (5.1.24), and lluo

- u E O I I D ~ - ' ~ W ~ . 0. Letting E + 0 gives (5.1.49), if we use (2) of Definition 5.1.2, together with the convergence properties of uEand H,(u‘)). Note that He(uc) is necessarily bounded in L2(Q). O

O

Remark 5.1.6. The Yosida approximation of (5.1.25) is admissible (see Crandall and Pazy [12]); so also is the special approximation of Section 2.1. Definition 5.1.3. We shall consider a function H ( . ) and an operator T, satisfying the hypotheses of Definition 5.1.1. Independent of U 2 , we assume (5.1.5b) or a substitute, such as (5.1.24). Specify the semidiscretization of (5.1.1)( U , = 1, U 2 = 0), given in weak form by H ( u t ) = H(uo), and ct; - r;-

W u r ) - wu;- l), U)LZ(R) + (f(u;- 1), u ) ~ ~= (0,~ ) for all 1)-

+ (W, WLZ(Cl) u E DT-I .

(5.1S O )

Here, f is assumed to be a Lipschitz continuous function, and uo and H(uo) are assumed to be in L2(Q).

Proposition 5.1.7. Under the assumptions of Definition 5.1.3, there exists a constant C , such that (5.1.51)

for all N , where the { u ; } are defined by (5.1.50). Proof: By employing an admissible smoothing, with H,(u;”) = H(uo), combined with Lemma 5.1.6 and standard lower semicontinuity arguments, it suffices to obtain (5.1.51) for a differentiable H (say H E ) ,where C is not to For notational convenience. we conceive of H as defining function composition rather than operator composition for most of this proof.


176

depend upon E . More precisely, we shall obtain the estimate (5.1.51), under the assumption (5.1.52) A < H' < A l , with C independent of I,. Using (5.1.50) as a starting point, with H now understood to satisfy (5.1.52), we may set u = H(u:), sum on k = 1,. . . , in, and obtain by standard methods, for llBNll< T o ,

m- 1

1

m-1

(5.1.53) where C is a Lipschitz constant for f . Note that the inequalities (5.1.9) and (5.1.52) have been used here. If the inequality 2

N

2

IIH(u~IIZ~(~, 3 2 llumllL2(fi)

is used in (5.1.53), followed by the discrete Gronwall inequality of Remark 2.2.1, we obtain the estimate (5.1.51). The final estimate of this section deals with the appropriate weakening of (5.1.21). Proposition 5.1.8. Under the hypotheses of Definition 5.1.3, there is a constant C , such that (5.1.54) for all N . Here, D, is the dual space with norm 11 .(IDs = (T(.), ( .))Litfi) on L2(Q). Proof: This result is a corollary of the proof of Proposition 5.1.1. Indeed, the hypothesis that (5.1.5b)holds permits us to use (5.1.11)as a starting point. If we proceed, as in the proof of Proposition 5.1.1, carrying the left-hand side of (5.1.54) from (5.1.11) to (5.1.12), we are led directly to (5.1.54).

Remark 5.1.7. Estimates (5.1.26) and (5.1.51) remain intact if f as in Remark 5.1.3, and (5.1.50) is extended by

+

t h,

+

(t: - t:- ~ ) - ' ( H ( U : ) H($- I ) , ~ ) L q f i ) (Vu:,v~)L~(fi, (g(u:), u ) ~ ~ ((h(u:~ ) u ) ~ ~ ( ~ for ) , all v E D,-

+

=g

I .

(5. .55)

5.2 Existence of Weak Solutions 5.2

177

EXISTENCE OF WEAK SOLUTIONS FOR THE STEFAN PROBLEM AND THE POROUSMEDIUM EQUATION AND APPROXIMATION RESULTS

The stability estimates of the preceding section permit us to prove the existence of weak solutions of degenerate parabolic equations, including the two-phase Stefan problem and the porous-medium equation as special cases. Definition 5.2.1. Let H ( . ) be a surjective multivalued function on R', defined so that a unique left inverse J = H - exists and is continuous, with 0 E int(Dom H ) and 0 E H(O), such that H(* ) defines a maximal monotone graph in R' x R', and such that the induced maximal monotone operators on L2(52)and L2(R x (0, T o ) )are bounded. Let f = g + h, where g is a locally Lipschitz continuous, monotone increasing function on R vanishing at zero, and h is a Lipschitz continuous function on R', and let uo and H(uo) be specified functions in L"(R), where H(uo) is the L2 limit of an admissible smoothing (see Definition 5.1.2). We assume that there exists a number A > 0, such that

',

(x - Y ) ( 5 - rl) 2 4 for every x E H ( 5 ) and y

E

5 - 112,

151, lrll

< c,

(5.2.1a)

H(q), where 0 < c < co is defined by

(5.2.1b) where C is the constant of Definition 5.1.2(5). Note that (5.2.1) imposes a Lipschitz condition on J = H - ' restricted to the interval H [ - c, c ] . Denote by V either the space HA(52) or H'(52) and assume that uo E V. By a weak solution u of the formal nonlinear distribution equation of evolution ~aH(u)

at

AU + f(u) = 0

in 9 = 52 x (0, To),

(5.2.2b)

H(u( .. 0)) = H(u,), u ( . , t ) E V,

au

av

-=

0

for almost all t on 852, if

(5.2.2a)

E

(0, To),

V = H'(R),

(5.2.2~) (5.2.2d)


178

is meant a pair u, u, satisfying the following properties: u E L"(9) n H'([O, T,];V*) u E V,

and

u is a selection of H ( u ) ; (5.2.3a)

(5.2.3b)

f(u) E L"(9);

and the relation

(5.2.3~) holds for all $ E V, , where V

= L"((O,T,);V) n H'([O,

(5.2.3d)

T,];L2(Q)) n L"(9),

and

w 0= C([O,T,];V)

(5.2.3e)

n H'([O, To]; L2(Q)).

Note that the second integral in (5.2.3~)has a genuine interpretation as a distribution since u E C([0, To]; V * ) is guaranteed. However, (see Chapter 1 [23] p. 263) under the regularity condition (5.2.3a),it follows that u is weakly continuous from [O, T,] into L2(Q).

Theorem 5.2.1. Suppose that the hypotheses of Definition 5.2.1 hold. Then, there exists a unique weak solution pair u, u of (5.2.3). If u, E W2*'(Q),then u = H(u) satisfies the property that &/at E L"((0, To);M(n)), where M(0) denotes the finite regular Baire measures on The proof will follow a sequence of preliminary lemmas. The first of these is a slight variant of the Rellich compactness theorem. We quote it in the form useful for our applications. The generalized version, to be applied again later in this section (see Lemma 5.2.1l), and in Section 5.4, is known as the Aubin lemma (see Lions [33] p. 58, Chapter 1 [23] p. 271]), and its statement is given in Lemma 5.4.3.

n.

Lemma 5.2.2. A set bounded in both is relatively compact in L2(9). Lemma 5.2.3.

L2((0, T o )V) ;

Let { w k } be a sequence of functions on 9,satisfying wk H(Wk)

Then, x

HI( [0, T o ] L2(Q)) ; and

= H(w) for

w

-x

+

(in L2(9)),

(5.2.4a)

(weakly in L2(9)).

(5.2.4b)

an admissible selection.

5.2 Existence of Weak Solutions

179

(5.2.6) where C denotes an upper bound for {IIH(wk)- H(4)llLl(Q,}.The proof is now completed by applying (5.2.4a) to the right-hand side of (5.2.6).

Lemma 5.2.4. Consider the semidiscretization, defined by (2.2.1 8), rewritten as

0: - c- 11- ' ( [ H ( u 3- wu:-

I)], 4)L2(R)

+ (g(ui3,4)L'(R) + (Nu;- l), 4)L'(R)

+ ( V 4 , V4)L2(R)

= 0,

for all

4 E v.

(5.2.7)

Let { H E }denote an admissible smoothing (see Definition 5.1.2) of H , and let {u;."} satisfy u2" = uo and, for 1 d k d M ( N ) ,

0 :

tr-

'(CH&(U5")- H & ( U 2 9 1 , 4 ) L 2 ( R )+ (VU;.", V4)L%2) + (g(U"k", 4)LZ(R) + ( h ( U i 2 1 ) , 4)L"R) = 0, for all 4 E v.

-

11-

(5.2.8)

Then, { u : ~ }and {HE(@")} lie in fixed balls of LO0@), with radii c and AC, respectively, where c is defined by (5.2.lb). Moreover, { u r } and {H(up)} are similarly bounded, with the same constants, and the locally Lipschitz function g may be assumed Lipschitz continuous. Finally, subsequences {up*"}, and { H E v ( ~ p . N exist, ) } Ysuch that for fixed N , k (respectively, k - 1) U2.N

+ up

-

H , , ( u ~ * " ) H(up)

(in LZ(Q)),

(5.2.9a)

(weakly in L2(Q)).

(5.2.9b)

Proof: Although the estimates (2.2.31) and (2.2.32) were derived under the formal assumption that H satisfies (1.1.8), as described in Remark 1.1.1, with H E defined via (2.1.5), the estimates remain valid for H described in


180

Definition 5.2.1, provided the constant C is inserted in (5.2.lb). This reflects the fact that (2.2.23)is replaced by u:" = u o . In fact, C = 0 for the smoothing defined by (1.1.8). More generally, (5.2.1b) holds because an admissible smoothing (see Definition 5.1.2) exists for H , and no further properties were used in deriving (2.2.31) and (2.2.32). Thus, we have verified the fact that {u:"} and {He(u2")}lie in fixed closed balls of L"(R), with radii c and Ac, respectively. Note that (5.2.7) and (5.2.8) correspond to (2.2.18) and (2.2.22),respectively, and the existence and uniqueness theory of Proposition 3.2.1 applies. We may obtain a gradient estimate on {u;"} in standard fashion by setting 6 = u2" - u:tl in (5.2.8), and using the pointwise estimates already derived. Applying this estimate with Lemma 5.1.6, we obtain (5.2.9), though, as yet, we are not able to identify the limits with the solution pair of (5.2.7). We shall take limits in the lifted version of (5.2.8), i.e., (tf -

+ u;" + Tg(U;") + Th,(u;!'l)

rf- l)-lTIHe(~;N)- H,(u;!J

= 0.

(5.2.10)

Here, T represents one of the operators Nu (a > 0) or Do, and h, = h - aid. Of course, g may be assumed Lipschitz continuous by extending it as such outside the interval [ - c, c]. Note that we use the compactness of T as an operator on L2(n).Altogether, the net result is the lifted version of (5.2.7), namely : (tf -

cf- J- 'T[H(ur)

,)I + uf + Tg(ur) + Th,(uf-

-H(u~-

1)

= 0.

Finally, we use the facts that the limit in (5.2.9a) lies in the closed ball of radius c in L"(n), and that the limit in (5.2.9b) lies in the closed ball of radius Lc in L"(R). H

Remark 5.2.1.

For most admissible smoothings, the property e+O

holds, so that the limit functions { u r } and { H ( u f ) } actually are expected to lie in (contracted) balls, for which C in (5.2.lb) has value zero.

Lemma 5.2.5.

For $ E C"(b), $( ., t ) E V, 0 < t G T o , set

$f(x) = ( t r - rF-

and define the step function

1)-1

IN'

k-I

$(x, t ) d t ,

k

=

1,. . . , M ,

(5.2.11)


181

where o,N(t) =

1, 0,

t:-l

< t < t:,

otherwise,

(5.2.12b)

for k = 1, . . . ,M. Similarly, define the translate

where (5.2.13b) Finally, define the difference quotient

Proof: By the Schwarz inequality, (5.2.17a)


182

for k = 1,. . . , M . It follows that the sequence {t+hN} is pointwise dominated by a function in Lz((0, To);V). The limit (5.2.16a) will follow from the Lebesgue dominated convergence theorem, if we can establish the pointwise estimates (5.2.18) almost everywhere in 9.However, by the mean value theorem for integrals, if t C 1 < t < tr, and i = 1,. . . , n,

for some t , = t,(x) lying on the interval (tf- 1, tr). Since a similar representation holds for I C / N ( ~ , t), we conclude that (5.2.18) and, hence, (5.2.16a), holds. The limit (5.2.16b) holds, since {x” - $”} is convergent to 0 in Lz((0,T o);V), as a consequence of (5.2.19) and (5.2.17). In order to verify (5.2.16c), we must argue somewhat more delicately, since the partitions are not assumed uniform, or even asymptotically uniform. First, we prove that {cN} is uniformly pointwise bounded. Indeed, by two applications of Taylor’s theorem, if t E [t:- 1, t r ] for 1 < k < M - 1,

1

+ o( (s - if)’)

ds - (t: - tr- 1)-

a* (x, tf- l)(s +at

a*at

- -(x, tr-

1

- tr-

1)

r,!’ k-1

[$(x, tr- 1)

+ o((s - tr- l)z)

I1 ds

;I

+ o((tr+1 - t k )

N Z

N (tk

- tr-l)-’

+ o (( $ - tr-1))

(5.2.20)

holds, and this is clearly uniformly bounded pointwise if the bounded mesh ratio hypothesis (2.2.41~)holds. The constants depend only upon II/ and the


183

mesh ratio. The proof will be completed if we can show that

cN - -2 at

(5.2.21)

(weakly in L'(9)).

[r,

Indeed, if p denotes the piecewise linear (in time) interpolant of then Lemma 5.2.2 shows that an L2(9)gradient estimate on and an estimate of the form (5.2.23) on the first differences of [: (i.e., an L2(9) estimate on has a the time derivative of 4") will imply that every subsequence of subsequence convergent in L2(9). By Lemma 5.2.6, to follow, these limits must uniquely coincide with that defined by (5.2.21). Now, the gradient estimate is trivial from the properties of $, whereas an even stronger estimate than (5.2.23) holds, viz., the uniform boundedness of the difference quotients. This latter property follows directly from (5.2.20) and the properties of $. To obtain (5.2.21), we construct the functions

p,

p

whose restrictions to [t?, t z -

1] coincide

with the piecewise linear functions

perturbed by an expression of order O(ll9"II). By (5.2.16a) and Lemma 5.2.6, to follow, which depends upon (5.2.16a,b) only, the sequence {e"}is weakly We immediately conclude that (8") is (strongly) convergent to $ in L2 (9). convergent to I) in L 2 ( 9 ) by Lemma 5.2.2, since {e"} is appropriately bounded in the proper topology. The convergence of {g"} is transferred to {ON},since ON - 0" + 0 in L2(9). Finally, (dON/dt)and (aON/axi)are pointwise bounded, and hence, bounded in L2(9), so that (aON/dt) is weakly convergent in L 2 ( 9 ) to (a$/&), i.e., (5.2.21) holds. Note that the pointwise boundedness of (dON/axi)asserted above follows from obvious modifications of (5.2.20).

Lemma 5.2.6. Let { w"} and { S N }denote step function and piecewise linear sequences, defined by M

(5.2.22a)

where {(or) are defined by (5.2.12b), and

tk
0, the space

Y

= { u :u E

Lpo((O,To);X,)

and

u, E Lp'((0,T o ) ;X,))

(5.4.7a)

with norm

+

IlUllY = I I ~ I I L P o u O . T l l ) ; X O ) II~tIILPI((0. ,rll);xl).

(5.4.7b)

Then, Y is compactly embedded in LP"((0,To);X). 5.5


The notion of a weak solution for the two-phase Stefan problem was introduced by Olehik [35] and Kamenomostskaja [25]. These authors treated the one-dimensional and multidimensional problems, respectively, the latter involving a technique of explicit finite differences. Both of these authors obtained existence results by using the Kirchhoff transformation

198


and the resulting enthalpy formulation. The results of [28] were refined by Friedman [161, who employed a smoothing method suggested by the analysis of [35]. Subsequent results were obtained via the Faedo-Galerkin method by Lions [33], and by the use of maximal monotone operators by Brezis [5]. Uniqueness is standard when the formulation is monotone. Damlamian [131 obtained uniqueness for Lipschitz perturbations by introducing an L' contraction theory. This contrasts with the method of lifting, used in Section 1.5. For the porous-medium equation, existence of continuous weak solutions of the pure initial-value problem in one space variable was demonstrated by Oleinik, Kalashnikov, and Y ui-Lin [36], and global regularity properties were derived by Kruzhkov [30] and Aronson [2], who also investigated properties of the free boundary; see also Kalashnikov [27]. Models containing convective effects have been investigated by Gilding and Peletier [20]. The multidimensional pure initial-value problem was considered by Sabinina [39], who demonstrated existence of unique weak solutions. Caffarelli and Friedman [7] have established continuity of this weak solution and certain regularity properties of the free boundary. They have also deduced continuity ofthe solution ofthe one-phase Stefan problem. Short-time classical solutions of the one-phase Stefan problem have been demonstrated by Hanzawa [22] by use of the Nash-Moser implicit function theorem. Friedman and Kinderlehrer [181 have established regularity of the free boundary for star-shaped regions in the one-phase problem. For constructive approaches to these inequalities, see [26] and the two-volume work of Glowinski, Lions, and Tremolieres [211. For the two-phase Stefan problem, continuity has been established by Caffarelliand Evans [8] and by Di Benedetto [14], who has also successfully studied the porous-medium equation by similar methods, originally suggested by the Di Georgi-Nash-Moser theory. There is a substantial literature on boundary regularity in one space dimension for the two-phase Stefan problem, which we shall not attempt to summarize here. However, the book of Friedman [17] is quite comprehensive in this respect (see Chapter 5). One of the most powerful tools for analyzing regularity of free boundaries, related to variational inequalities, is a C'-stability result due to Caffarelli [6] (see Friedman also [171 for contextual development). BrCzis [5] observed that the initialhoundary-value problem for the porous-medium equation can be treated in H- by the theory of maximal monotone operators. Analysis on L' by accretive operators is also possible. A survey of such applications is given by Evans [151. The theory of accretive operators, the associated generation of nonlinear semigroups, and the underlying differential equation are related by the fully implicit method employed in this section. In fact, this method of proving existence is due to

5.5 Bibliographical Remarks

199

Rothe [38], and was effectively carried on by the Russian school of mathematicians (see e.g., LadyZenskaja [32]). A summary of the method, as applied to a linear parabolic initialboundary-value problem is given in the book of Ladyienskaja, Solonnikov, and Ural'ceva (Chapter 2 [10, pp. 241-2521). The historical antecedents and the effectiveness of the method were pointed out to the writer in 1975 by Gunter Meyert. It was Hille, with his famous exponential formula, lying at the core of the Hille-Yosida theorem (see Section 6.1),who first drew the rigorous connection between the line method and semigroup generation. A nice account of this for accretive operators is given by Crandall and Evans [ 10) (see also Crandall and Liggett [111). In particular, it is emphasized in Crandall and Evans [lo] that semigroup generation, in conjunction with semigroup differentiability, are equivalent to a solution of the differential equation. The differentiability requirement does present a serious impediment in the application of this theory to specific problems. On the other hand, the convergence of the difference scheme is consistent with certain generalized notions of solution, such as that proposed by Benilan [3]. It is likely that a rich abstract theory of generalized solutions is latent here in this circle of ideas. The author developed the methods used for the horizontal-line analysis of this book in a series of papers, beginning with [24]. Rather than use classical maximum principles to deduce stability, we have used methods familiar to numerical analysts. Also, the theorems discussed here are various nonlinear realizations of the famous Lax equivalence principle, which characterizes convergence in terms of consistency and stability. For further discussion, with interesting perspectives, the reader is referred to the paper of Chorin, Hughes, McCracken, and Marsden [9]. We repeat again an earlier statement of Chapter 2 that we have not attempted an L' initial data theory, although the methods are flexible enough to handle this. Our point of view, for the most part, has been to require that u, be a square integrable function, and this has led to the specification of H' initial data (see, however, Proposition 5.2.12).One of the very nice features of the horizontal-line method is that the divided difference estimate in time, expressed via spatial norms, leads directly to the property that u, is square integrable. The more delicate estimate of Proposition 5.1.5, leading to the result that dH(u)/dr is essentially bounded on (O,T,), with range in M@), uses a semidiscrete version of an idea due to Kruzhkov [31]. Kruzhkov's idea was used by the writer and Rose (Chapter 2 [9]), where a different proof of this result is given. This estimate is fundamental for the error estimates of Chapter 4. We believe that the semidiscrete version of invariant regions, obtained in Chapter 2, and used as the basis for the corresponding parabolic principle +

Private communication.


200

here, is a new result. However, the parabolic principle, as stated here, is certainly less general in certain respects than that contained in the references of Chapter 2, e.g., Chapter 2 [4]+. The reader may compare the difference in approach presented here with that of Rauch and Smoller (Chapter 2 [14]) and Rauch [37]. Periodicity properties of solutions of parabolic systems have been investigated by Amann [13. The results we have presented in this chapter concerning weak solutions of the Navier-Stokes system do not differ appreciably from those presented by Temam (Chapter 1 [23]), though some superficial generalizations have been made. Actually, we have presented these results as background for the results of Chapter 7 concerning classical solutions. As preparation, we shall give a brief literature survey concerning this problem. Following the pioneering work of Lerayt early writers, including Hopf [23], Kiselev and Ladyihenskaja [29], Serrin [40,411, and Ladyihenskaja (Chapter 1 [ 1 l]), introduced notions of global weak solutions and local classical solutions for this problem. Specifically, Hopf considered solenoidal (distribution-wise) L2 initial velocities and Kiselev and LadyZhenskaja considered H2 initial solenoidal velocities. Whereas Hopf's weak solution is global in time, but possibly nonunique, the latter authors obtain a local time result within an existence-uniqueness class. It should be noted that the difficult problem here is for n >, 3; the case n = 2 is well-understood (see Lions [33]). Several authors generalized the results of Kiselev and LadyZhenskaja [29], obtained for n = 3, to n > 3, e.g., Serrin [41] for n = 4 and H2 solenoidal velocities, and Fujita and Kato [I91 for initial datum in an interpolation space roughly corresponding to H''2. We have quoted these results for situations neglecting external effects, such as gravitation, or assuming these effects are derivable from a potential. The result for global weak solutions, given by Temam (Chapter 1 [23]), assumes only L2 solenoidal initial velocities. The proof of Section 5.4 does not differ, in essential respects, from that of Temam (Chapter 1 [23]), and could readily be modified to relax the assumption uo E V to uo E H. Concerning uniqueness, it is known that Lp((0, T o ) L'(R)) ; (abbreviated LP.') is a uniqueness class, if 2 n -+-=1, P

r

n o,and, moreover, I\[R(A,U)lrll < M(A - w ) - ~ ,

r 2 1.

(6.1.18)

In this case, IIT(t)ll d Me"", t 2 0. Corollary 6.1.7. (Hille-Yosida). If U is a closed linear operator with domain dense in a Banach space X and range in X, and if R(I, U) exists for all real I larger than some real number w, and satisfies the inequality IIR(AU)II d (2 - a)-',

J.

> 0,

(6.1.19a)

then U is the infinitesimal generator of a semigroup of class (C,), such that IIT(t)ll d e"', Proposition 6.1.8. (Trotter-Kato).

t 2 0.

(6.1.19b)

The strong limit

T(t)f = lim (I - tU/n)-"j, n+ a,

t 2 0,

(6.1.20)

holds for every f E X, if U is the infinitesimal generator of the (C,) semigroup T(t). Remark 6.1.4. One can prove that, if t E [0, To], (([T(t)- (1 - tU/n)-"]fll< M~,,n-lt~11U~fll,

(6.1.21)

6.2 Linear Evolution Equation and Evolution Operators

213

for f E DU2 (see Yosida [14] p. 269). Thus, (6.1.20) follows from (6.1.21), by the uniform boundedness of

(I

-

tU/n)-"

=

[nR(n, tU)]",

and the denseness of DU2in X. Definition 6.1.4. The set { - U : U is closed and satisfies (6.1.18)} will be denoted by G(X, M , m). Write

IJ

G(X,M) =

G(X, M,w),

G(X) =

u

G(X,M).

M>O

-m<WCm

Remark 6.1.5. There is an equivalent formulation of the hypothesis of the Hille-Yosida theorem, if X is a (real) Hilbert space. In particular, if

( A f A2 and

(6.1.22a)

- mllflli?

I E ~ ( A ) forall A
w ( t j ) , (6.2.2b)

for any finite increasing family {rj} taken almost everywhere on [O, To].In this case, (M,o) is a stability index for {A(t)},and the domain of definition of A( may omit a set of measure zero on [0, To]. In the notation G(X, M, 0): introduced in Definition 6.1.4, we permit only constant values of o. a )

Remark 6.2.1. Quasi-stability is preserved, with only a change in M, under equivalence of norms. An upper Lebesgue integrable function i,b is, of course, dominated, pointwise, by an integrable function 4, and the value, $, of the upper integral is the infimum f 4. For the nonlinear theory of the next chapter, we shall not require the result of Theorem 6.2.5 in the full generality of quasi-stability, thus we shall give the complete details of the construction of the evolution operators, described therein, only in the stable case, referring the reader to the literature for the required technical modifications. For the other results of this chapter, we shall prepare the statements, for the most part, in the more general format, but give the details of proof only in the stable case for ease of reading. Here, the required changes are essentially “cosmetic.”

s*

Proposition6.2.1.’ The stability (respectively, quasi-stability) condition (6.2.2) is equivalent to each of the following conditions, in which { t j } r = varies over all admissible finite families, as in Definition 6.2.1 : k

exp{ -sjA(tj)}

I
w and A j > w(tj), respectively. Here, we have written e-IA for the (C,) semigroup generated by - A .

Proof: We prove, in the case of stability, (6.2.2)* (6.2.3).

(6.2.3)3 (6.2.4) By (6.1.9),we have

where we have used (6.2.3).The evaluation of the preceding integral is k

M

JJ

j= 1

( l j

- w)-lllfll~

-

so that we have proved (6.2.3) * (6.2.4).The implication (6.2.4)* (6.2.2)is immediate upon taking l j = A, j = 1, . . . , k. Now, by (strong) continuity, we note that (6.2.2) (6.2.3)certainly holds, if it can be established whenever s j is a rational number. By selecting nonnegative integers m j , and a rational number s, satisfying s j = mjs,

j = 1,.

. . ,k,

for given rationals s j , we see that it suffices to prove the implication for s j = mjs. Finally, by absorbing the repetition of s into the equivalent repetition of t j , we see that it suffices to verify (6.2.2) G. (6.2.3)for s j (6.1.20)and (6.2.2),

= M lim ((1 - ws/m)-hllfII} m-l m

= Meoksllfll,

so that (6.2.3)holds. W

= s. Thus,

by

6. Linear Evolution Operators

216

Remark 6.2.2. It is difficult, in general, to decide whether a given family {A(t)}is quasi-stable. A criterion is given by the following.

ll.llt

Proposition 6.2.2. For each t E [0, To],let be a new norm in X, let X, = (X, 11 .I[,), and suppose there is a real number c, such that l l f ~ ~ t /G ~ ~eclt-sl f ~ ~1 s

0#f€

x,

s, t E [O, To].

(6.2.5)

Suppose A(t) E G(X,, l,w(r)), where w is an upper Lebesgue integrable function on [0, To],for almost all t. Then {A(t)} is quasi-stable, with stability index (exp(2cT0),w), with respect to 11 for almost every t E [0, To].

.[If

Proof: We consider only the case where w is constant, assuming the stable case. Applying (6.2.5), with t = T o and s = t k , gives

where we have used A(tk)E G(Xrk,1,w). Inductive application of (6.2.5) and the hypothesis give

= (A - w)-kecTollfIJO

< (A - w)-keZcTollfllTo, where we have used (6.2.5) to obtain the last inequality. The proof for arbitrary t E [0, To]is similar. H Definition 6.2.2. Let Y be a Banach space densely and continuously embedded in a Banach space X. If Q is a linear operator on X, then the part of Q in Y, ?, is the restriction of Q to DQ={~EY~D~:QJEY}. Let A E G(X, M , w). The space Y is said to be A-admissible if {e-'*} leaves Y invariant, and forms a semigroup of class (C,) on Y. Proposition 6.2.3. ciently large A,

For A E G(X), Y is A-admissible if and only if, for suffi-

(A + A)-'Y c Y,

"(A

+ A)-"lly < fi(A - &)-",

n

=

1,2,. . ., (6.2.6a)


for constants

aand

(3,

217

and

(A + A)-'Y

is dense in Y.

(6.2.6b)

A

In this case, the part - of - A in Y generates the part of {e-rA} in Y, the part of(A A)-' in Y is just (A + A)-', and

+

Ile-rAlly

< fieGt.

(6.2.6~)

Proof: Suppose Y is A-admissible and let -A be the generator of the part of {e-rA} in Y, so that A E G(Y,fi,G). Let g E Y. Since e-fAg = e-"g,

t 2 0,

the Laplace transform (6.1.9)shows that

+ 2)- ' g = (A + A)- ' g E Y, A > max(o, (3). Thus, (A + 2)- 'Y C Y, and (A + A)- ' E B[_Y] is exactly the part of (A + A)- ' (A

in Y. In particular, (6.2.6a) follows from A E G(Y, follows from Propositions 6.1.2, 6.1.3, and Dn = (A + A)-'Y

= (A

fi,6).Note that (6.2.6b)

+ A)-'Y.

Suppose, conversely, that (6.2.6a,b) hold, and let For g E Y, let f

= (A

+ A)-'g,

A be the part of A in Y.

2 sufficiently large.

Then, f E Y by hypothesis, and, by Proposition 6.1.3,f E D A , Af = g - Af E Y. Hence, f~ D i , with Af = Af, and

+

g = (A

+ A)f

=

(A + A ) f .

Thus, (A A)- ' is defined on Y and coincides with the part of (A + 2)on Y. Thus, by (6.2.6a),

II(A + A ) - " ( ( ~Q

- GI-",

'

n = 1,2,. . . .

Since A is densely defined by (6.2.6b), we have A E G(Y, 6l, 6)by Theorem 6.1.6. The limit characterization (6.1.20) shows that e-rAg = e-rAgE Y for g E Y. In particular, Y is A-admissible and (6.2.6~)holds.

Proposition 6.2.4. Let S be an isomorphism of Y onto X, where X and Y are described in Definition 6.2.2. For A E G(X, M,o), Y is A-admissible if and only if A, = SAS-' is in G(X). In this case, Se-IAS-' = eCrA1, t 2 0 . Here, DA,= { f E x:s- 'fE DA,AS-'f E Y}.

6. Linear Evolution Operators

218

Proof: Since A ,

+ A = S(A + A)S-', it follows that (Al + A)-' = S(A + A ) - ' S - ' , A > w.

(6.2.7)

Now suppose that Y is A-admissible. By (6.2.7) and the previous proposition, (A, ,+ A)-' E B[X], A > w, so that A , is closed, since (A, + A)-' is similar to (A + 2)- E B[Y], with E G(Y). Moreover, D A , is dense in X, SO that, by direct estimation of (6.2.7) and Theorem 6.1.6, it follows that A , E G(X). The converse is similar. The relation e-rA1 = Se-'AS-l follows from (6.1.20).

Theorem 6.2.5. Let X and Y be Banach spaces, such that Y is densely and continuously embedded in X. Let A(t) E G(X), 0 < t < T o , and assume the following. (1) The family {A(t)} is quasi-stable on X, say with stability index ( M , w ) . (2) The space Y is A(t)-admissible for each t. If &t) E G(Y) is the part of A(t) in Y, is quasi-stable on Y, say with stability index (2,G). (3) The space Y c D A ( , ) , and the map A(.):[O, To] -,B(Y,X), defined by t I+ A(t), is continuous.

{A@)}

Under these conditions, there exists a unique family of operators U(t, s) E B[X], defined for 0 Q s < t < T o ,with the following properties. (a) The operator function U(t, s) is strongly continuous (X) jointly in t and s, with U(s,s) = I, and ~ ~ U ( r , . sQ ) ~M ~ xexp{j$,,, IwI dz}. (b) U(t,r) = U(t,s)U(s,r), r < s < t. (c) [D,+U(t,s)g],=, = -A(s)g, g E Y, 0 Q s < To. (d) (d/ds)U(t,s)g = U(t,s)A(s)g, g E Y, 0 < s < t < To. Here, D+ denotes right derivative in the strong sense and d/ds the corresponding two-sided derivative in the strong sense. When s = t , the latter is simply a left derivative.

Remark 6.2.3. The family {U(t,s)} will be referred to as the evolution operators for the family {A@)}. Proof: We give the details only in the case where {A(t)} and {&t)} are stable. We define a sequence {A,( .)} of step-function approximations to 4 . 1 by

A,(t)

= A(To[nt/To]/n),

0 Q t Q To,


219

where [s] denotes the greatest integer less than or equal to s. Since A( .) is norm continuous (Y, X), we have IIA.(t) - AWlly,x

+

0,

n

(6.2.8)

+

uniformly for t E [O, To]. Moreover, {A,,@)}and {A,,(t)}are stable, with constants M ,w and 65, respectively, independent of n. We define {U,,(t,s)}as follows. If s < t, and s and t belong to the closure of an interval, in which A,,(.) = constant = A, then set U,,(t,s)= e-(*-')*. For other values of s < t, U,(t,s) is determined by conditions (a) and (b). Explicitly,

a,

U,,(t,r ) = U,(t, s)U,(s, r),

0 0, such that the inequalities

'

a o o k x, P)

(cs,+'

2 &I

and

hold, in the sense of matrix entry comparisons, for all (t, x, p) E [0, To] x R" x R, and all (11,. . . , n/2 + k, k a nonnegative integer, then

HS(R";Rm)c H",(R"; W') c CE(W; Rm), (2)

and the inclusions are continuous. If s > 4 2 , then pointwise multiplication induces continuous bilinear maps

H"-'([W";Rm)x Hk+'(R"; R')

-+

Hk(R";[W"'),

H",T(R"; Rm)x Hk+'(R"; R')

-+

Hk(R";R"'),

and for 0 < L

< s, 0 < k < s - L.

7.3 Quasi-linear Second-Order Hyperbolic Systems

253

(3a) Assume that R, c RN is open, s > n/2, and u, E Hs(R";RN) takes its values in R,. Suppose Wo c Hs(R";RN)is a ball, centered at u,, with radius chosen small enough so any u E Wo takes values in a compact set K c R,, and that G:R" x R, -,R4 is in CS, (CF1+' if [s] # s). Then, there is a constant C1,such that, for all u E W,, I(G(',u('))llH;l(Rn) < Cl(1 + IIUII"H(Rn))

(3b) If G is of class CS,' (CV1+2if [s]

f s),

(7.3.17a)

there is a constant C , , such that

IIG(.,u(*)) - G(',v('))(lHr(Rn) G C2llu -

vIIH~(R~),

(7.3.17b)

for u , v W ~ a n d 0 G r < s. (3c) If, in addition, 0 E R,, if 0, is contractible to 0, and if G(.,O)E H'(R"; RN),then G ( * , u ( - ) )E H'(R"; RN)for u E W,.

Remark 7.3.5. We mention some direct conclusions of the facts cited in the previous remark as applied to the given hypotheses. From (al) and (7.3.17a,b),we conclude that aij - a;j := Uij(t, .,a,b, Va) - Uij(t', .,a',b', Va') and

b - b := b(t, .,a, b, VU) - b(t', .,a',&,Va') are in H'(R") for 0 < r Pl and P2,

< s and i , j = 0, 1, . . . , n, and satisfy, for constants

< Pl(lt - t'l + IIw - w ' l l H r + l ( R n ) ~ H r ( R n ) ) , ,lib - b ' l l H y R n ) < P2(lt - t'l + IIW - W ' I I H r + I ( R n ) x H ' ( R n ) ) ,

(laij

for t, t'

E

(7.3.18)

- a:jllHqR.)

[0, To]and w

u;:uij

E

= (a,b),w' = (a',b').From

(7.3.17a),we have

is uniformly bounded,

H",R"; Rm2)

(7.3.19) (7.3.20)

for w E W and t E [0, To].Thus, since, in this case, HiG. HS-' c Hs-',t we have

1 IA(t,w)(*, $1 I < Cl(1l1d.1 IX

IHs(Rn)

+

+

n

1

i= 1

~~'&l(uOi

< C2ll(*9$)IlY.

n

1 O;'lI

i,j= 1

I

aij IHG[(Rn)

II*\ IH'

+

'(a")

+ aiO)llHGc(Rn)II$IIH.clw.,) (7.3.21)

The second space in this inclusion consists of scalar functions (see (2) in Remark 7.3.4). This convention is maintained throughout this section.

7. Quasi-linear Equations of Evolution

254

-

From (7.3.18), we obtain, by first using Hs-' H' c H' (if s > n/2 Il(A(t, w, - A(t', w'))(*,

G

c 1

1

iJ= 1

+ c1 G

$)llH'+

+ l),

1(R") x H'(W")

lla&laij - ( a , - , ' a i j ~ l l H ~ ( R ~ ) ~ ~ * ~ ~'(Rn) H~+

n

i= 1

IlaiOl(aOi

cz(lt - t'l + I ( w

+ aiO)

- [aoa'(aOi

+ aiO)]'llHr(wn)ll~llHa(~n)

- w'lIHr+l(Rn)XHr(Rn~)lI(*,~)II~,

(7.3.22)

for 0 4 2 , and aij does not depend on the derivatives of w, the same inequality holds, by use of H"-' H'+l c H' and a strengthened version of (7.3.18), with H' replaced by H'+l on the left-hand side. We shall now verify the hypotheses of Definition 7.2.2 in two stages, leaving the definition of S and the verification of the associated properties (S) and (A2) until the conclusion of the following lemma and its proof.

Lemma 7.3.3. The hypotheses (N), (A3), and (F) hold if s > (n/2) + 1. Hypothesis (A4) holds for a dense subset of W. Proof: (A3) A(t, w) E B(Y,X), with LA = Cz,follows from (7.3.21). The remaining continuity properties follow directly from (7.3.22), with r = s - 1 ; note here the choice V = X. (A4) An easy calculation shows that such a dense set is given by

W n (HS+Z(Rn; Rm)x Hs+'(Rn;Rm)), (F)

The constant Lo depends only on y o . The continuity properties of F = (0, b) follow from (7.3.19). Since b is not obtained via a multiplier, it is necessary to conclude that F(t,w) E Y, by using (3c) of Remark 7.3.4, in conjunction with the assumption on b(0, 0). The Y-range boundedness property is now immediate from (7.3.17b). Using H"-' . Ho c Ho and (7.3.18), with r = s - 1, we obtain (for s > n/2 1) a,

(N)

+

pW,w;*,*) - W,w';*,*)I G c l ( l t - t'l + IIw - W'IIH'(R")xH'-1(R"))

ll(*,@)ll~,

(7.3*23)

< Alt - t'l + (IW - W'I(X)(IUIIH,

(7.3.24)

'

which leads to

I IIUll;(t,n) for t, t'

E

- IIUll;(t,,rv,)l

[0, T o ] and w, w' E W, u E Z. The same inequality (7.3.24)

7.3 Quasi-linear Second-Order Hyperbolic Systems

255

holds for s > 1112, if aij does not depend on derivatives, by use of H"H0 c Ho. Now write

for IIuIIN(~.~) 3 ~ ~ u ~ ~The N ( other ~ ~ , ~ inequality ~ ) . is parallel. For x 2 0, ln(1 x ) < x. Thus, by (7.3.24),

+

< pAit(lt - t'( + IIw - W'IIX), where the existence of 1, follows from Remark 7.3.3. Hypothesis (N) is now verified with pN = &/2.

Lemma 7.3.4. Let A = (I - A)''', with domain H'(Rn; Rm) and range L2(Rn;Rm).Then, for a E Lm(Rn;R'), with grad a E H"-'(R"; R"), we have, for some constant c,

< cIJgradallw- '(w).

J1[~S,a]~'-S(1L2(,~).L~(IW~)

(7.3.25)

Proof: See Kato [17], p. 66. Note the diagonal operator interpretation of A here and throughout the section. Note also the operator connotation of a. Remark 7.3.6. We select the operator

1

A simple computation gives SA(t,w)S-' = A(t, w)

+ B(t, w),

(7.3.26)

where B(t, W)=

[

0 [AS,A,]A-S

O

I

[As,A2]A-s '

(7.3.27a)

and n

A,(t,w) = - i ,C u;aijj= 1

c n

A207 w ) =

- i=1 a;;(aoi

and [ , 3 denotes the commutator.

a2

(7.3.27b)

axi axj'

+

UiO)

a

-, axi

(7.3.27~)


256

Lemma 7.3.5. Properties (A2) and (S) hold if s > n/2 Proof: By the commuting property of A-", $) E

(@9

z,

C n

[A", A1]A-'@

= -

i.j= 1

D

+ 1. we have, for

> 0, and

[As, u&,~u~~]A' -"

az

axi axj A- '$,

~

(7.3.28)

By Lemma 7.3.4, and property (2) of Remark 7.3.4, applied with (7.3.20) to the expanded gradient, and

[][As,ai;aij]A'

-'IILz(w)J-.z(w)

ll[As,G k ( a o i + aio)]A' -sllL2(Iwm),L2(Iwn)

< cllgrad a i ; a i j l l H s -

'(an) < C,

< llgrad a,-,'(aoi + aio)lles-*p) < C.

When these inequalities are applied to the Lz estimations of (7.3.28) and (7.3.29),there results the estimate p ( ~ A ( @ ? $ ) lG l z &ll(@~$)IlZ?

(7.3.30)

where A, does not depend on t E [0, To]and w E W. Property (A2) follows directly. Property (S), of course, is immediate from the time independence of S( .) and the continuity of the embedding Z -,X. W

Theorem 7.3.6.+ Assume s > n/2 + 1 and that (a1)-(a3) hold (if the aij do not depend on the derivatives of @, s > n/2 need only be assumed). Then, the Cauchy problem (7.3.1)-(7.3.2) is well-posed ip the following sense. Given (@,,$,), satisfying (7.3.2)-(7.3.3), there is a neighborhood Yo of (@o, $o) in HS+'(Rn)x H"(Rn),and a positive number To< T o , such that, for any initial condition in Yo, (7.3.1)-(7.3.2) has a unique solution @(l,.) for t E [0, Tb],satisfying (7.3.3): (@(t,x),

d@ (t, x), V @ ( t x)) , E 52,

for all x E R".

(7.3.31)

Moreover,@E C'([O, T ~ ] ; H " ~ ' ~ r ( I W n ; [ W " ) ) , O < r ~ s , a n d t h e m a p ( @ O , $ o ) ~

(@(t;),(d@/dt)(t,.)) is continuous in the topology of H' x Ho, uniformly in t E [O, To].

Proof: The C' continuity of @ for r > 1 follows from the cases r = 0 and r = 1 by direct use of(7.3.1).The remaining conclusions follow from Theorem +

This theorem and its proof are adapted from Hughes, Kato, and Marsden [ 131.

7.4 Vacuum Field Equations of General Relativity

257

7.2.4. The hypotheses of this theorem are verified by Lemmas 7.3.2, 7.3.3, and 7.3.5. H

Remark 7.3.7. It can be shown that (see Hughes et al. [13])

Mode) H ( W 7 9 d ( t A ) x H", uniformly in t.

is continuous in H""

7.4

THE VACUUM FIELD EQUATIONS OF GENERAL RELATIVITY

The field equations of general relativity may be viewed as an analog of Poisson's equation of the Newtonian theory. In fact, suppose we begin with the metric tensor coefficients g i j of the gravitational potential, under the assumption of a well-defined semi-Riemannian, space-time manifold, with ds2 = gijdxidxj.+Suppose a symmetric tensor expression G i j in the g i j is sought, satisfying the following conditions as described in Einstein [5].

c:j=o

(1) The tensor Gij may contain no differential coefficients of the g i j higher than the second order. (2) The tensor G i j must be linear and homogeneous in these second-order coefficients. Then, the expression must be of the form

+

I

+

G i j = Rij agijR bgij, where Rij denotes the Ricci curvature tensor, R I.J . = RklkJ9.

(7.4.1a) (7.4.1b)

of second order, R is the scalar curvature,

R

= g'jR.. I J = Rk k

((d')= ( g i j ) - ' h

(7.4.1c)

and a and b are constants. Here, we have adopted the Einstein summation convention. The quantity {R$kd}is the Riemann curvature tensor. A coordinate-free definition of this tensor is given in differential geometry as the unique (1,3) tensor field R, such that R(w,z , x,Y ) = w(R,,Z),

'

(7.4.2)

This includes Lorentz manifolds as a special case. These are the solution manifolds of interest.


258

for all 1-forms o,and for all vector fields X, Y, 2. Here, Rxy is the curvature operator which assigns, to each ordered pair X, Y of vector fields on the space-time manifold, the operator Rxy on vector fields, defined by RxyZ = DxDyZ - DyDxZ - D,x,y,Z,

(7.4.3)

where D, denotes the covariant derivative in the direction X, and [ ., .] denotes the Lie bracket. The quantity D is referred to as the Levi-Civita, or semi-Riemannian, connection, and, in particular satisfies the torsion-free property (7.4.4) DXY - DyX = [X, Y]. If rfj denote the connection coefficients for dual local bases {mi} and {Xi}, then DXi(xj) = rfjxk, (7.4.5)

1 k

and if {Xi}is chosen, so that rfj = (that is, [Xi, Xj] = 0), then we deduce from (7.4.2) and (7.4.3) the standard format, with the notation f k = xk(f),

R ; ~= , ~ ( oxij,, x k ,x,) = r 5 t . k

- rjk,,

+ rLkq- rL,r;.

(7.4.6)

Moreover, the metric coefficients gij = (Xi,Xj)

((.:)

:=metric tensor)

(7.4.7)

define the connection coefficients, via the formula

rijk --l kTgd

(gdi,j+ gtj,i - gij.0,

(7.4.8)

if [X,, X,] = 0, all m,n. Note that (7.4.8) is a direct consequence of the conjunction of (7.4.5)and (7.4.7), with Xi(X,, Xj)

+ Xj(X,,

Xi) - X,(Xi, XJ) = 2(DxiXj, X,).

The latter is an immediate consequence of the torsion-free property (7.4.4) and the identity Z(X, Y > = (DzX, Y >

+ (X, DzY),

both of which hold for semi-Riemannian connections. Returning to the derivation of the field equations, we may require, on the basis of conservation principles, the following. (3) The divergence of Gij must vanish identically. This imposes the condition a = -f in (7.4.la). If, in addition, we require (4) Gij = 0 in flat space-time (that is, when RiU = 0), then b = 0. Thus, under the assumptions (1)-(4), the Einstein tensor Gij has the form GIj. . = Rt.j . - L29ijR

3

(7.4.9)

7.4

Vacuum Field Equations of General Relativity

259

and the generalization of Poisson’s equation is

G.. I J = -8nGT.. V’

(7.4.10)

where Tij is the stress-energy tensor of matter, and G is Newton’s gravitational constant, and units are normalized by a speed of light of unity. In this section, we discuss the general existence theory only for the vacuum case, T i j= 0. If a coordinate base field Xi = a/axi is selected at each point of the manifold, then the Lie bracket of any two basis vectors is zero and (7.4.6) holds, where the connection coefficients are described by (7.4.8). In both equations, commas denote partial differentiation. Thus, the Einstein field equations in a vacuum are completely described by the system

Remark 7.4.1.

0 < i , j < 3,

Gij = 0,

(7.4.11)

of partial differential equations in the metric tensor coefficients, where Gij is given by (7.4.9), (7.4.lb,c), (7.4.6), and (7.4.8). If the constraint (4) is discarded, then b # 0 is permitted in (7.4.1). This (universal) constant b is the much-heralded cosmological constant, originally introduced by Einstein to obtain a static universe solution to the “dust” field equations of the form ds2 = c2 dt’ - R2

{$

1-P

+ p2(d02 + sin’ 8 d B 2 ) } ,

(7.4.12)

where b = 1/R2 is the constant curvature of the associated 3-space and p, 8, and 4 are spherical coordinates. The solution is static, since p is necessarily constant. In the case of “dust”, an average density po is assumed, leading to Tij = dir)g(O,O,0, po) and b = 4aGp0. Although Einstein later abandoned the cosmological constant in the light of red-shifted spectral evidence of an expanding universe, cosmologists have not. A nonzero value of b is, of course, consistent with expanding universe models. In fact, the Friedmann models are those of the form ds’ = c 2 d t 2 - R’(t)

{

~

dp’

1 - kp2

+ p2(d02 + sin’ 0 d B 2 )

(7.4.1 3)

for k = 0, 1 and associated 3-space curvature k/R2(t)(cf. also the equivalent form given in [23, p. 2101). The form of R ( . ) is determined by the “dust” model field equations, and the classification schemes permit b = 0 and b # 0. The metric (7.4.13) describes a universe satisfying large-scale isotropy and homogeneity, and some of the special solutions are associated with various names (cf. Rindler [23] Section 9.10).


260

Remark 7.4.2. Before we can address the Cauchy problem for Gij = 0, we must make several remarks related to the transformation and/or reduction of the system. First, we make the standard observation that Gij = 0 o Rij = 0 (as systems). If we set

rk = gijrk. Y'

(7.4.14a)

and (7.4.14b) = 0, subject to the condition that then we shall solve the reduced system &, the local coordinates xi on the initial manifold be harmonic coordinates, defined by the condition Tk(O,xi)= 0. The reason for the term harmonic is the general identity

where 0 is the generalized d'Alembertian, defined by

n4

=

i a J-S axi { f i g i ,

~

-

g},

(g = det(gij)).

Here we explicitly assume a Lorentzian manifold with g < 0. Provided the initial data satisfy the necessary conditions referred to in the next paragraph, the evolution k,, = 0 preserves the harmonic property of the coordinates (see Foures-Bruhat [3]), so that k,, = 0 o Rij = 0. It can be shown that kij= 0 has the form (7.4.15 ) where H i , is an explicit rational function of& and is homogeneous quadratic in its first derivatives (see Fock [8] p. 423). The reduction to (7.4.15)achieves an uncoupling of the second derivatives. The initial data are not completely arbitrary, but must satisfy certain constraints, which are related to the second fundamental form of the embedding of the initial 3-manifold into the space-time manifold (see Fischer and Marsden [7]). The remarkable fact accrues that, given initial data satisfying the constraints in arbitrary coordinates, there exists a sufficiently regular transformation to harmonic coordinates, so that the initial data continue to satisfy the embedding constraints in the new coordinates. The evolution (7.4.15) preserves the harmonic coordinates (locally in time), as well as the embedding constraints.

7.4 Vacuum Field Equations of General Relativity

261

Still another technical difficulty is that the metric coefficients gij(t,.) need not be in Hs(R3;R'). In the elementary case of the Minkowski metric, g i j = diag(1, - 1, - 1, - 1) and the diagonal entries are clearly not L2 functions on R3, though their derivatives surely are. However, for a wide class g i j of space-time metrics, including the asymptotically flat metrics, with perturbations of order O ( r - ' ) at spatial infinity, it is to be expected that the variables (7.4.16) +.. EJ = 9tJ. .- g?. tJ are Sobolev class variables. Accordingly, we recast the reduced system (7.4.15) into the format of (7.3.1), with = ++OOz, = -L+ijz , ( i , j ) # (O,O), (7.4.174 1J and Hij+

1

'

a2gioi -

k,l= 0

(7.4.17b)

ij

with n = 3 and m = 10. Note that we discard redundancy due to the symmetry of the g i j . Here +kdis the entry with index (k,L') in the (inverse)matrix (+ij g$'. The generality of the previous section, which required only that the aij multipliers be in a uniformly local Sobolev space, is clearly necessary here.

+

Remark 7.4.3. The Cauchy problem to be considered, then, is defined by the quasi-linear system (7.3.1),(7.4.17)in the 10-variable I(lij, subject to initial data = 0 and I&. We shall assume that the initial data satisfy certain regularity conditions and shall prescribe the domain R of "hyperbolicity" as follows. The quantity s is assumed to satisfy s > 3.

+;

(gl)'

I

g; E

c;+3(R3;

@ axi

R'),

g; E HS(R3; R'), 0 < i, j < 3

HS(R3; R'),

(Ci+3replaced by Ci+4if [s] # s).

(g2)

R c R'O x R'O x R30 is chosen as follows: R

= R, x

R'O x

R30,

(7.4.18)

where R, is a ball centered at 0, such that, if I(lij(x)= g i j ( x )- g;(x) E Ro, then g i j ( x )is of Lorentz signature (+, -, -, -). This has the standard + C;'

can be replaced by C;+z if one verifies (F) of Section 7.2 directly.


262

meaning that the matrix (gij)is unitarily similar to diag(1, - 1, - 1, - 1). Two immediate implications are that the two requirements of (a3) of Definition 7.3.1 are satisfied. It may, of course, be necessary to replace Ro with an open subset with compact closure in R,. It is straightforward to check that (gl) implies (al). It follows that, when (gl) and (g2) are satisfied, the Cauchy problem, stated at the beginning of this remark, has a solution locally in time in the sense of Theorem 7.3.6. Rather than state our result in terms of the artificial variables $ i j , we present the natural formulation in terms of the g i j . Theorem 7.4.1. Let (gl) and (g2) hold.+ Then, for s > 4 and initial data in a neighborhood of (g;,g;) in H Z t x Hh#, equations (7.4.15) have a unique solution in the same space for a time interval [0, To], To> 0. Here, H& is the space of u, such that u - uo E H', with correspondingly induced topology.

7.5

INVARIANT TIME INTERVALS FOR THE ARTIFICIAL VISCOSITY METHOD

Suppose A(t,w) = - v A

+ E(t,w),

(7.5.1a)

v 3 0,

where A has the interpretation of a diagonal operator and the first-order operator E(t, w) is given formally by j= 1

a axj + b ( t , . ,W)

aj(t;,w) -

1

,

(7.5.1b)

and P is an orthogonal projection in L2(R";R"). We shall suppose, for simplicity, that aj and b are defined on [0, To] x [w" x R" with range in R"', and that uo E PHs(R";R") is given for s > n/2 1. The problem addressed in this section is the following. We seek to determine a pair (r', Tb), not depending on v, and a solution u of the Cauchy problem

+

du dt

-

+ A(t,u)u = 0, u(0;)

0 G t G Tb,

= uo,

'Our hypotheses are stronger than those of Hughes, Kato, and Marsden [13].

(7.5.2a) (7.5.2b)

7.5 lnvariant Time lntervals

263

such that u E Y, = W'*m([O,T0];H"(')(R";Rm))n L'((0, Tb);PH"(R";Rm)), (7.5.3)

with a(v) = s - 1 - sgn(v), and Ilu(t9 *)IIHS(Wm)

< r',

0 0, such that uo E B(0,r) c H"(R";Rm), that A(t,w)E G(PL', l,o(r)) for 0 < t < To, < r, and a restriction of A satisfies &t, w)E G(PHs,l,G(r)) for I I W I I ~ . ( ~ ~ ) < r and 0 < t < To. Then, for y = 1 + max(w,G),and Tb satisfying (see (7.5.33))

IIwII~~(~~)

the interval [0, To] is an invariant interval. If Tb is fixed, then I' = r may be chosen to maximize the function f(r) = re-y(r)Tbwhen this occurs. Explicit solutions are possible in the case of the Navier-Stokes/Euler system, for example, and the maximization may be employed to determine the critical value To = sup(To). We now begin the basic development; we require a more stringent notion of stability than introduced in Definition 6.2.1.

Definition7.5.1. Let X be a Banach space, and W a closed subset of X. Suppose that a family A(t,u) E G(X) is given for 0 < t < To and u E W.The family {A(t,u)} is said to be stable if there are (stability) constants M and W , such that

A> n [ A ( t j , ~ +j ) A]-' < M(A for any finite families {tj}j"= and {uj}j"= W,with 0 < < k 1,2,. . . . Moreover, n is time-ordered. k

c

(7.5.5)

w,

-u)-~,

tl

*

*

< tk < T o ,

=

Remark 7.5.1. It can be shown (see Proposition 6.2.1) that (7.5.5) can be replaced by the equivalent, but superficially stronger, condition


284

Note also that a family A(t, u) E G(X, 1,w ) is automatically stable, with constants 1,o.

Definition 7.5.2. Let Y be a reflexive Banach space densely and continuously embedded in a Banach space X,such that the norm of Y is determined by an isomorphism S of Y onto X, i.e., IlUllY

(7.5.7)

= IIsullx.

Let W be a closed ball in Y, centered at 0, and let {A(t,u)}, 0 < t < To, u E W,be stable, with constants M and o.Suppose that A,(t,u) E G(X) is defined by A,(t, U) = SA(t, u)S-

’,

(7.5.8a) (7.5.8b)

DA,(f,U) = { V E X:A(t,u)S-’v E Y},

and that {A,(t, u ) } is also stable, with constants M , and 0,. It is explicitly assumed that Y is contained in the domain of A(t, u) for each (t,u).

Remark 7.5.2. The restriction of A(t,u) to S-lDA,(t,U) is in G(Y), with stability constants o1and M (see Proposition 6.2.4). Now, let N 2 1 be given, and set At =’ T O I N ,To< To.Given uo E W,we are interested in the recursive solution of

for t k = k At, k 7.5.2, set

= 0,

1, . . . , N . Under the format of Definitions 7.5.1 and

M‘ = max(M,M,),

o’= max(w,w,),

+

y = o’ 1,

(7.5.10)

where M‘ 2 1 is assumed. Suppose that W (closed in X and Y) is given explicitly by

< r, llwllx < I } ,

(7.5.11)

0 < M’max(lluolly,Iluollx)eYTb< r.

(7.5.12)

VV

= {w E Y:IIwlly

and that Toand r satisfy Select p, such that

7.5 Invariant Time Intervals

265

and define 6 and (T by

(M'max(~~uo~~Y,~~uo~~x)) (7.5.14b) Then, clearly, W oc W,where

W o = {. Proposition 7.5.1.

E

y:IIUIlY

G

~ I I U O I I Y llullx ,

(7.5.15)

~lI~ollx>*

Suppose that there exists a constant C,such that IIAct, u) -

U')llY,X

G

cllu - U'IIX?

(7.5.16)

where Tb and W oare described in Definition for 0 G t G Tb and u, u' E Wo, 7.5.2 and Remark 7.5.2. For (7.5.17a) the { u [ } [ = ~are characterized formally as fixed points of mappings Q::Wo -,Wo (see (7.5.20)).If, in addition, pz satisfies 1 1' p z - 1 - w

(7.5.17b)

then {Q:} are strict contractions on X,with contraction constants given by the right-hand side of (7.5.17b) as functions of N . In particular, the recursive equations (7.5.9) have unique solutions in this case.

Proof: For fixed N , assume, inductively, that (7.5.9) possesses a solution < m,where m 2 1, such that

u[ for k

n

k- 1

=

pzR(pZ,-A(tj,uY))uO,

j= 1

p2 =

(k).

(7.5.18)

Rewriting (7.5.9), we see that u: may be displayed as a fixed point of the mapping Qu = -R(pZ - 1, - A ( t k , U ) ) U

+ pzR(pz - 1, -A(fkrU))U:-l.

(7.5.19)

To prove that Q = Q[ maps W ointo itself, we combine (7.5.18) and (7.5.19)


266

to obtain, for u E Wo, QU

=

- R(p2 - 1, - A(tk, U))U k- 1

+ p2R(p2- 1, -A(tk,U)) fl p2R(/A2,-A(tj,ur))Uo,

(7.5.20)

j= 1

and application of (7.5.5) yields

By the choice of N , M

p2-1-w

< S/(l + d),

and, by choice of 0 and N,

In particular, IIQullx

< oIIuollx.Now, by (7.5.8) and (6.2.7), R(A, -Al(t,u))

= SR(A, -A(t,u))S-’.

(7.5.21)

Applying S to (7.5.20), and using (7.5.21) yields SQu

=

-R(p2 - 1, -A,(tk,

V))su k- 1

and the estimation of llSQullx proceeds much like that of llQullx. In particular,

so that llSQullx < ollSuollx. We conclude that QWo c Wo. The contractive property of Q reduces to proving the estimate

IIR(A, -A(t,

0))-

- A(t, w))(lx

< C1110

- wllx/[(A - w)(A- 0111, (7.5.22)

where C1 = MMIC. Indeed, an application of (7.5.22) to Q u - Qu, where Q is given by (7.5.20), leads to

+ ~ l l l ~ o l l x (2

P2 p -1-0,

)(*))u

p -w

- WIIX,

7.5 Invariant Time Intervals

267

and the right-hand side of this expression is bounded, via the right-hand side of (7.5.17b), since M1/(p2- 1 - ol) < 6/(1 + a), and since P2 p2 - 1 - 0 ,

+1

0,

+

(1 + 6 - ' ) M , '

(A) N

d

n/2 + 1. Since a/dxjA-' and A - ' are bounded on L2(R";Rm) by 1, the choice of and, hence, w1 is possible.

Remark 7.5.6. In the case of the operator given by (7.5.29), a careful study of Kato's proof reveals that P(r) in (7.5.30) may be chosen as the linear

7.

270

function

Quasi-linear Equations of Evolution

B(r) = n3/2s(c1+ c2)r = cr,

(7.5.31a)

where c1 is the least constant in the ring inequality

IIfg I

\Ha

- (W")

c1

1If 1IHs- (W")l 1 I

\Ha- '(W"),

(7.5.31b)

for real-valued functions f and g, and c2 is the Sobolev constant in the inequality C2llfllH.-'(W"),

IIflIL-(R")

(7.5.31~)

for real-valued functions f. Here we have adjusted B(r) for the fact that b = 0 for (7.5.29). Now, let To = sup Tb represent an as yet undetermined critical value (actual or spurious). Maximize the function f ( r ) = re-y(')*O,

where y(r) = B(r) + 1, and by elementary calculus as

r > 0,

B is given by (7.5.31a). The solution is rendered (7.5.32a)

giving the relation (7.5.32b) as the defining relation for To, when taken in conjunction with (7.5.33) to follow. The quantity Tb may be chosen as any value satisfying To< T o . It is time to bring all of this to a close. We end with a summarizing statement, which may be proved with the aid of Corollary 7.5.2 and the standard convergence techniques for the method of horizontal lines.

Theorem 7.5.4. Let A(t, w) be given as in (7.5.1). Suppose v 2 0 and r' > 0 under the hypotheses on P and E(t, w) of Definition 7.5.3. Then there exists a positive constant y, not depending on v, such that, if uo E B(0,r') c Hs(R";Rm),s > n/2 + 1, and uo and Tb satisfy the relation IIuOIJHS(Rn)eYTb < r',

(7.5.33)

there exists a solution of (7.5.1)-(7.5.2), satisfying the properties (7.5.3)(7.5.4). If the system is dissipative, the convergence result (7.5.27) holds and all solutions are unique. In the particular case of the (dissipative) NavierStokes/Euler system, I' and Toare given explicitly via (7.5.31) and (7.5.32), and a unique solution of the Cauchy problem (7.5.2), satisfying (7.5.3) and of (7.5.29) on [0, To]. (7.5.4), exists for the operator A( a,

a )

7 .6 Bibliographical Remarks 7.6

271


The development of Sections 7.2 and 7.3 largely follows that of Hughes, Kato, and Marsden [13] in conceptual content, although the hypotheses here differ in certain respects. Not only is the linear theory, developed in the previous chapter, indispensable for this development, but certain associated ideas, developed by Kato in earlier papers (see Kato [17, 18]), some of which are presented in Section 7.1, are also necessary. Other ideas which prove decisive, and which were developed earlier by Kato, include the role of the uniformly local Sobolev spaces in a generalized ring theory (see Remark 7.3.4), and the commutator estimate of Lemma 7.3.4, which implies the uniform boundedness and, hence the upper Lebesgue integrability of the family {B"(.)}.The hypotheses of Sections 7.2 and 7.3 are sufficiently strong to guarantee that {A"(t)} is stable both on the ground space and the smooth space Y,so that the full strength of the linear theory of Chapter 6 is not required. Many sources have been used for the example of Section 7.4. Of course, the basic source is [13] for the existence theorem. For the physics, we have been guided by the eminently readable monograph of Einstein [5] and the ambitious treatise of Hawking and Ellis [113, where existence theorems are also proved. Other valuable ideas were derived from the expository book of Rindler [23]. The mathematical development of the section was assisted by reference to Bishop and Goldberg [13, Hicks [123, and Sachs and Wu [24]. A clear account of the role of harmonic coordinates in the analytic reduction is presented by Fock [9]. This reduction was employed by Choquet-Bruhat [3] in the first existence proof for the field equations, where it is noted that the evolution preserves the harmonic coordinates. An informative mathematical analysis of the elliptic constraint equations satisfied by the initial data has been given by Cantor [2] in the general context of tensor field decomposition. An alternative approach, making use of the Hamiltonian formalism for general relativity, via the lapse and shift functions, has been given by Fischer and Marsden [8] (see also Marsden C211). The final section follows Jerome [14]. The viscosity method in fluid dynamics was examined by Golovkin [lo] for planar flow, and by Swann [25] and Kato [16], among others, for three-dimensional flows. The estimate for the time interval given by (7.5.31)-(7.5.32) in this case appears more explicit than that given in the two previous references. However, it depends directly upon the commutator estimates obtained by Kato and referred to above. The reader may have already conjectured that the semidiscrete analysis can probably spawn a semidiscrete viscosity convergence result.

272


This proves correct, and details may be found in Jerome [14]. A study of the viscosity method, employing global analysis, has been carried out by Ebin and Marsden [4]. There are, of course, other approaches to the theory of nonlinear Cauchy problems in Banach spaces. An approach which constructs the evolution operators directly for the nonlinear system has been given by Evans [6] for accretive systems. These systems, though more restricted, permit global solutions in time. Though the hypotheses that Y,X, V, and Z are separable and reflexive are more stringent than necessary, we have used these hypotheses to invoke the Dunford-Pettis theorem and the fundamental theorem of calculus in Banach spaces in our analysis of the similarity relations embodied, say, in (A2). Stronger hypotheses here would yield results valid in general Banach spaces. We note, finally, that Theorem 7.5.4 serves as a useful counterpoint to the global weak existence theory of Chapter 5, particularly for models such as the Navier-Stokes model, where uniqueness has not been demonstrated for the general class of weak solutions. Other applications of the theory of this chapter are possible, for example, to elastodynamics. We refer to the monograph of Marsden and Hughes [22] for an account of this. We also refer the reader to Kato [19] for a discussion of the redundancy of condition (A4) of Definition 7.2.2.

REFERENCES c13 R. L. Bishop and S. I. Goldberg, “Tensor Analysis on Manifolds.” MacMillan, New York, 1968. P I M. Cantor, Elliptic operators and the decomposition of tensor fields, Bull. Amer. Math. SOC.5, 235-262 (1982). [31 Y. Choquet-Bruhat (Y. Foures-Bruhat), Theoreme d’existence pour certain systemes d‘bquations aux derivees partielles nonlinkaires, Acta Math. 88, 141-225 (1952). c41 D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. ofMath. 92, 102-163 (1970). 51 A. Einstein, “The Meaning of Relativity,” 3rd. ed. Princeton Univ. Press, Princeton, New Jersey, 1950. C6l L. C. Evans, Nonlinear evolution equations in an arbitrary Banach Space, Israel J . Math. 26, 1-42 (1977). 171 A. Fischer and J. Marsden, The Einstein evolution equations as a first order quasilinear symmetric hyperbolic system, I, Comm. Math. Phys. 28, 1-38 (1972). C81 A. Fischer and J. Marsden, The initial value problem and the dynamical formulation of general relativity, in, “General Relativity and Einstein’s Centenary Survey” (S. Hawking and W. Israel, eds.), Chapter 4. Cambridge Univ. Press, London and New York, 1979.

r

References

273

V. Fock, “Theory of Space, Time and Gravitation.” MacMillan, New York, 1964. K. Golovkin, Vanishing viscosity in Cauchy’s problem for hydromechanics, Trudy Mat. Inst. Steklov 92, 31-49 (1966); and Proc. Steklov Inst. Math. 92, 33-53 (1966). S. W. Hawking and G. F. R. Ellis, “The Large Scale Structure of Space-Time,” Cambridge Univ. Press, London and New York, 1973. N. J. Hicks, “Notes on Differential Geometry.” Van Nostrand, Princeton, New Jersey, 1965. T. Hughes, T. Kato, and J. Marsden, Well-posed, quasi-linear, second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273-294, (1977). J. W. Jerome, Quasilinear hyperbolic and parabolic systems: contractive semidiscretizations and convergence of the discrete viscosity method, J. Math. Anal. Appl. 90,185-206 (1982). T. Kato, “Perturbation Theory for Linear Operators.” Springer-Verlag, Berlin and New York, 1966. T. Kato, Nonstationary flows of viscous and ideal fluids in R’, J . Functional Anal. 9, 296-305 (1972). T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations. Springer Lecture Notes in Mathematics 448, pp. 25-70. Springer-Verlag, Berlin and New York, 1975. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58, 181-205 (1975). T. Kato, “Linear and Quasilinear Equations of Hyperbolic Type,” Bressanone Lectures. Centro Inter. Mat. Estivo, Rome, 1977. Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan 19, 473-507 (1967). J. E. Marsden, “Lectures on Geometric Methods in Mathematical Physics.” SIAM, Philadelphia, Pennsylvania, 1981. J. E. Marsden and T. J. R. Hughes, “Topics in the Mathematical Foundations of Elasticity.” Prentice-Hall, Englewood Cliffs, New Jersey, 1982. W. Rindler, “Essential Relativity,” 2nd ed. Springer-Verlag, Berlin and New York, 1977. R. K. Sachs and H. Wu, “General Relativity for Mathematicians.’’ Springer-Verlag, Berlin and New York, 1977. H. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3, Trans. Amer. Math. Soc. 157, 373-397 (1971).


Bochner theorem, 241 Borsuk antipodal theorem, 126, 127 Boundary conditions, 11, 15, 16, 18, 24, 29, 30, 32, 34,99, 148, 149, 150, 154, 164 Dirichlet, 11, 15, 24, 34, 99, 148, 149, 154, 164 Neumann, 11, 15, 16, 18, 24, 99, 148, 149, 150, 154, 164 Robin, 11, 29, 30, 32, 99, 148, 150 Bramble-Hilbert lemma, 123 Brouwer fixed-point theorem, 80

A

A-admissible space, 216, 217, 218, 222 Absorbing set, 128, 129 Accretive mapping, 110, 198 Adams extrapolation method, 102 Approximation, degree, 143, 146, 147, 148 Artificial viscosity method, see viscosity method Asymptotic estimates, 49, 59, 64, 148, 152, 155, 157 formulas, 131, 142 Asymptotically flat metric, 261 Aubin lemma, 178, 197; see also Rellich compactness property Aubin-Nitsche lemma, 7, 114, 143, 144, 158

C

Calder6n-Zygmund lemma, 154 Cauchy problem, 208, 213, 230, 231, 235, 237, 238, 239, 241-247, 248, 249, 252, 256, 260, 261, 262, 267, 270, 272; see also initial-value problem; linear evolution; quasi-linear evolution Chain, 87, 88 Chemical systems, 26, 30 Coercive, see List of Symbols and Definitions

B Backward Euler, see fully implicit method Baire measure, 3, 7, 20, 114, 149-153, 178 Balanced set, 128, 129 Bessel potential operator, 136 275

Subject lndex

276

Commutator estimate, 5 , 234, 255, 269, 271 Conductivity hydraulic, 21 thermal, 12, 26 Connection, 258, 259 Contractible to zero, 253 Contraction mapping principle, 2, 4, 107, 162, 246, 247, 265, 267 Contractive rectangle, 69; see also invariant region Convolution operator, 225 Core of domain, 249 Cosmological constant, 259 Covariant derivative, 258 Crank-Nicolson method, 102, 162, 164 Critical value, 263, 270 Curvature, 257-259; see also Ricci curvature tensor; Riemann curvature tensor operator, 258 scalor, 257, 259

D d’Alembertian, 260 Darcy’s law, 21, 43 Degenerate parabolic evolution equation, 1, 6, 150-157, 163-190, see also Stefan two-phase model; porous medium model convergence theory Galerkin, 150-154 horizontal line, 154-157 existence theory, 177-190 stability, 163-176 Degree of mapping, 127 DiGeorgi-Nash-Moser theory, 198 Dirichlet, see boundary conditions, Dirichlet Dispersion of set, 126 Dissipative, 268, 269, 270 Dual parabolic problem, 171 Dunford-Pettis theorem, 240, 241, 272

E Einstein field equations Cauchy problem, 261, 262, 271 derivation, 257-259 vacuum, 259 tensor, 258

Ellipsoid, 139, 140 Energy inequality, 232, 251 Enthalpy, 3, 13, 14, 15, 26, 46, 163, 198 Euler equations for ideal fluid, 5 , 32, 33, 237, 263, 268-270 derivation, 32, 33 existence, 263, 268-270 Evolution equation, 213, 237, 267; see also Cauchy problem; initial-value problem; initial/boundary-value problem; linear evolution; quasi-linear evolution Evolution operators, 2, 5 , 207, 213, 218-221, 224-231, 234, 237, 244 Existence-uniqueness class, 1, 42, 200 Explicit method, 5 5 , 101, 162, 164

F Fick’s (first) law, 26 Finite-element method, 3, 6, 7, 113, 114, 142-154, 158; see also Galerkin method Finite propagation speed, 201 Finite support propagation, 2 FitzHugh-Nagumo equations, 26, 27, 19, 30, 74 Fluid filtration in porous medium, see porous medium model Fourier transform, 233 Fdchet derivative, 4 Free boundary, 22, 198 Friedmann models, 259 Friedrichs’ extension, 140, 141 Fully implicit method, 5 5 , 56, 101, 162, 164, 198; see also horizontal line method Fundamental theorem of calculus in Banach space, 186, 187, 241, 272

G

Galerkin method, 3, 7, 113, 151-154, 198 Girding inequality, 250 Gateaux differentiable, 8 1, 89 Gohberg-Krein theorem, 130 Greatest integer function, 218, 219 Gronwall inequality classical, 41, 42, 153, 269 discrete, 7, 52-55, 59, 60, 73, 166, 176, 193

Subject lndex

277

H Harmonic coordinates, 6, 237, 260, 271 Heat capacity, volumetric, 12, 26 Heat of fusion, 15 Hemicontinuous, 85, 86 Hermite-type splines, 103 Hille-Yosida theorem, 199, 212, 213, 233 Horizontal line method, 2, 199, 237, 263, 270; see also fully implicit method; quadrature Hyperbolic evolution, 1, 2, 5, 208, 231-235,

inversion operator; lifting mapping; Riesz mapping Isotone mapping, 87, 88; see also increasing mapping

K Kirchhoff transformation, 14, 22, 197 Kitchen’s theorem, 4

237, 247-257, 262, 269, 270

system linear, 208, 231-235, 252 quasi-linear, 5, 237, 247-257, 262, 269, 270

I Implicit function theorem, 4 Incompressible fluid dynamics, see Euler equations for ideal fluid; Navier-Stokes model Increasing mapping, 97, 98, 110; see also isotone mapping Indicator function, 80, 97 Infinitesimal generator, see semigroup, generator Initial/boundary-value problem, 1, 11, 16, 23, 29, 38, 51, 66, 177, 178, 198, 199; see also parabolic evolution Initial-value problem, 1, 48, 198, 213, 230-235, 241-257; see also Cauchy

problem; initiallboundary-value problem; linear evolution; quasi-linear evolution linear, 208, 213, 230-235 quasi-linear, 241-257 Invariant region, 6, 29, 46, 68, 70, 72-74,

Latent energy content, 12 Lattice, 87, 96 complete, 87 inductive, 87 Lax equivalence theorem, 2, 199 Lax-Milgram theorem, 142, 250 Lebesgue extension, 149 Left inverse of maximal monotone function,

15, 46, 47, 59, 105, 155, 163, 164, 174, 177 Lie bracket, 258, 259 Lifted weak equation, 6, 19, 25, 32, 39, 49, 64, 73, 77, 100, 101, 105, 150, 151, 155, 156, 164-167 Lifting mapping, 6, 7, 11, 17, 24, 32, 49, 64, 73, 77, 105, 164, 201; see also inversion

operator; isomorphism; Riesz mapping Linear evolution in Banach space, 213, 230, 231, 238, 239 for systems, 231-235 Lorentz manifold, 257, 260, 261 signature, 261 Lower-bound estimation, 126-138

M

105, 194, 195

semidiscrete, 68, 70, 72 Invariant time interval, 5, 237, 263, 270 Inverse hypothesis, 145, 146, 148 Inversion operator, 12, 17, 49, 64, 69, 99, 142, 146; see also isomorphism; lifting mapping; Riesz mapping Inward-pointing vector field, 6 Isomorphism, linear, 92, 93, 94, 118, 127,

136, 142, 143, 145, 148, 155, 217, 221, 227, 233, 239, 243, 264; see also

Marcinkiewicz interpolation theorem, 119 Maximal element, 87-89 Mean derivative estimate, 7, 20, 162, 169- 174

Measurable, strongly, 221, 225, 226, 240, 241, 244, 252

Mesh ratio, 59, 62, 181, 182, 183 Metric tensor, 257, 258, 261 Mihlin multiplier theorem, 136

Subject Index

2 78

Minimal element, 87-89 Minkowski metric, 261 Momentum flux tensor Euler equations, 26 Navier-Stokes equations, 27 Monotone mapping, 85-87, 91-94, 97, 150, 155, 164, 170, 174, 175, 177, 179, 198 maximal, 87, 91-94, 150, 155, 164, 174, 175, 177, 198 strictly, 87,97 Moving boundary, 2

N Nash-Moser iteration, 2,4, 198 Navier-Stokes model, 1, 5, 6, 32-39, 42, 43, 76, 94-96, 107-109, 150, 163, 195-197, 200, 201, 237, 263, 268, 269, 270, 272 Navier-Stokes conservation equations, 33-34 De Rham decomposition, 34-35 derivation, 32-34 existence classical solutions, 263, 268-270 weak solutions, 195-197 semidiscrete existence theory, 107-108 formulation, 107- 108 stability, 108-109 stationary theory, 94-96 weak solution, 38-39 Neumann, see boundary conditions, Neumann Newton iteration, 3, 4, 5 Nodal basis, 144, 145 Nonexpansive, 78 Numerical complexity, 2, 3, 4 N-width, 3, 7, 8, 113, 114, 126-142, 158 0 Ordering of sign regions property, 28, 193

P Parabolic evolution, 1, 2, 77, 262-270 equation, see degenerate parabolic equation; Navier-Stokes model; porous medium

model; reaction-diffusion model; Stefan two-phase model quasi-linear system, 1, 262-270 Partially ordered set, 87, 88 Permafrost, 5, 12, 15 Perturbation of generator, 5, 223, 269 Phase change, see Stefan problem Piecewise linear sequence, 183-190, 194-196 trial functions, 3, 7, 113, 147, 148 Porous medium model: horizontal flow, 1, 3, 6, 11, 20-25, 41, 43, 62-68, 105, 113, 150, 154, 157, 162, 163, 169, 187, 188, 198 continuity, 198 existence, 187, 188 lifting for, 25 mass balance, 21 regularization convergence, 64-66 properties, 62-64 semidiscrete convergence, 157 existence theory, 105 formulation, 66-67 maximum principles, 67-68 uniqueness for, 41 weak solution, 23-24 Predator-prey model, 27, 30, 74 Proper mapping, definition, 77 Proximity mapping, 7, 77, 78, 80, 109 Pseudomonotone mapping, 7, 76, 82, 84, 85, 86, 87, 92, 97, 110 defhtian, 82

Q Quadrature, 99-109, 110, 111 explicit, 101 implicit, 101 Quasi-contractive, 242 Quasi-linear evolution, 1, 237, 241-247, 247-257, 262, 263, 264-267 in Banach space, 241-247, 264-267 for systems, 247-257, 262, 263, 268-270 Quasi-stability, operator, 2, 5, 207, 213, 214, 216, 218, 221, 222, 223; see also stability, operator Quasi-uniform triangulation, 148 Quasi-variational inequality, 8, 96-99, 110

Subject Index

279

R Reaction-diffusion model, 1, 11, 25-32, 41, 42, 43,68-74, 105-107, 150, 162, 190-195, 199 existence, 190-195 lifting for, 32 semidiscrete existence theory, 105-107 formulation, 68-69 maximum principles, 70-71 sign regions, 28, 193 uniqueness for, 41 variable concentration, 6, 28, 74 potential, 6, 28, 74 weak solution, 29 Regularity of evolution operators, 224 Relatively bounded perturbation, 269 Rellich compactness property, 178, 189, 195; see also Aubin lemma Resolvent, 2, 210, 211, 223, 267, 268 operator, 210, 211, 223, 267, 268 set, 210 Ricci curvature tensor, 6, 257 Riemann curvature tensor, 257 Riesz mapping, 17-20, 24, 25, 31, 32, 39-41, 49, 59, 64, 69, 73; see also inversion operator; isomorphism; lifting mapping Dirichlet, 24-25, 41, 64,73 negative norm, 18, 24, 31, 73, 146-147, 152- 156 Neumann, 17-20, 41, 49, 59, 73 nonnegativity, 19, 24, 32, 40,73 Robin, 31-32, 69 Riesz potential bounded mapping, 118 definition, 118 Ring property, 252, 270, 271 Robin, see boundary condition, Robin

S Schwartz class, 233 Semigroup, 5, 199,207-213, 216, 217, 242, 263, 264, 269, generator, 5, 207, 208, 209, 210, 212, 213, 217, 242, 263, 264, 269

strongly continuous, 5, 208, 209 Similarity transformation, 5 Simplicial decomposition, 147; see also triangulation Smoothing, 46, 51.62, 73, 169, 170, 171, 174, 175, 177, 179, 180 admissible, 169, 170, 171, 174, 175, 177, 179, 180 Sobolev integral representation, 114-126, 131, 144, 145, 148 projection operator, 115-126, 131, 144, 145, 148 Spectral radius, 4 Stability, operator, 2, 5, 207, 213, 214, 215, 216, 218, 219, 221, 222, 223, 233, 243, 244, 263, 264, 269; see also quasi-stability, operator Stability, solution, 6, 7, 52-62, 66-73, 108-109, 163-176, 183, 187, 190, 199, 267 divided difference, 164, 168, 170-174, 176, 183, 185, 199, 267 energy, 108-109, 164, 167, 168, 175, 267 mean square, 6, 52-62, 165-167 pointwise, 52-62, 66-73, 163, 164, 169, 187, 190 Star-shaped domain, 115, 117, 122, 198 Stefan model, 2, 3, 6, 11, 12-20, 25, 41, 42, 43, 46-51, 55-62, 104-105, 113, 150, 154, 157, 162, 163, 169, 187, 197, 198, 20 1 Stefan problem continuity, 198 energy balance, 12-13 existence theorem, 187 lifting for, 19 regu1arization convergence, 49-51 properties, 46-48 semidiscrete convergence, 157 existence theory, 104-105 formulation, 55-56 maximum principles, 57-61 uniqueness for, 41 weak solution, 15-16 Step function, sequence, 52, 180-190, 194-197, 218-221, 228 Stress-energy tensor, 259 Strong extension, 232 Subadditive, 23

Subject Index

280

Subdifferential mapping, 76, 81, 85, 89, 91-93; see also subgradient definition, 81 Subgradient, 7, 81, 92, 110

T Tableau, 131 Tarski fixed-point theorem, 76, 87-89, 98, 110

Taylor polynomial, 118 Torsion-free property, 258 Trace operator, 30 Triangulation, 144, 147, 148; see also simplicia1decomposition Truncation operators, 57, 58, 67 Type M,mapping, 85, 86

U Uniformly local multiplier, 252, 261, 271 Upper-bound estimation, 114, 115-126, 158; see also asymptotic estimation Upper Lebesgue integrable, 5, 214, 216, 223, 244

v Variational inequality, 8, 78-84 Vector field, on manifold, 258 Viscosity method, 5, 263, 268-270, 271, 272 Viscosity, stress tensor, 34 Volterra operator equation, 208, 225, 226, 227

W Weakly asymptotic, 3 Weinstein-Aronszajn theory, 138, 139

Y Yosida approximation, 165, 170, 175

2 Zero set, of smooth function, 16; see also moving boundary

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