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
This page intentionally left blank
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
This page intentionally left blank
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) ,~~~<
Applications are occasionally made to closed sets, such as the cube. In this case, interpret as the cube interior. etc.
List of Symbols and Definitions
xvii
Symbol or Definition
Term (k 2 0) > 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 )
List of Symbols and Definitions
xviii
Symbol or Definition
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.
List of Symbols and Definitions
xix
Symbol or Definition
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
Boldface (for abstract and function spaces and subsets)
Special symbols Del Identity function Laplacian
id A:=V*V
Polynomials, degree Q L - 1
9(
(continued)
Not boldface if subset of Euclidean space.
List of Symbols and Definitions
xx
Term Sobolev projection Triangle (0 < s < t C T o ) Unit ball (closed) in W'.p Spectrum (of U) Strictly Coercive (bilinear form) Convex (norm)
Symbol or Definition
B(u, u ) 2 C I I U ~ ~ ~ A convex combination of distinct unit vectors has norm less than one.
Sum Algebraic Direct Operator Subspace Tensor Translation symbols
f, 7,
A@B
M, OM2 Displayed with subscripts to indicate rank f -uid
Q + Q,,
urn0 u Triangulation
U
Elements Maximal element diameter Symbol Uniform partition length Uniqueness (for) Multi-valued mapping H (of) Selection H(u)
e
Upper Lebesgue integral (off over J ) Vector
+ M2
M,
0
5,
'
= x,
+ hy
(Definition 4.1.2)
h A,, := { e ) atN,At
Induced by formulation or sequential convergence
J,*
f
n
Factorial
a! =
Format
Boldface
Length
la( =
Weakly asymptotic
T=(Y)
ai!
i= I
1 ai
(nonnegative integer entries)
i=l
z (identical 0 orders)
This book is devoted to the study of the initial-value and the mixed initial/ boundary-value problems. It is divided into two parts, which superficially appear to be entirely unrelated: Part I, a weak solution theory, global in time, for nonlinear and degenerate parabolic evolutions, and Part 11, a classical solution theory, local in time, for quasi-linear parabolic and hyperbolic evolutions. These theories are, in turn, motivated by models from the physical and biological sciences, whose formulations are described and succinctly developed, particularly in Chapter 1. These include phase transition through heat conduction, fluid filtration in porous media, reactiondiffusion systems, incompressible fluid dynamics, and general relativity. The basic question of the determination of existence-uniqueness classes for the solutions of these models leads to a consideration of both types of solutions as presented in Parts I and 11. For example, in the case of incompressible fluid dynamics, in spite of persistent current research efforts, such an existence-uniqueness class has not yet been determined for global solutions, though both properties have been proved for separate classes for viscous, incompressible fluids. One is then led to local solutions for the resolution of such basic well-posedness. On the other hand, very complicated analytical models such as the geometry of space-time have not yet been 1
Introduction
2
analyzed in any globally satisfactory way for the general initial-value problem. In fact, only fragmentary global theories exist for nonlinear hyperbolic problems in general. Some of the results obtained correlate the size of the initial datum, in some norm, to the length of the time interval on which existence of solutions is demonstrated. Such theories are clearly more closely related to local theories, in spirit, and are presented in Part 11. No attempt is made in this book to discuss recent global theories derived from applications of geometric measure theory to systems satisfying hyperbolic conservation laws, or applications of Nash-Mosert iteration to nonlinear wave equations. Nor do we discuss qualitative behavior of nonlinear evolutions such as shock formation, steady-state bifurcation, traveling wave solutions, finite support propagation in nonlinear “parabolic” equations, regularity of moving boundaries, or asymptotic analysis. In fact, the spirit of the book is closest to that of modern numerical analysis and approximation, although only one chapter is devoted to numerical methods as such, while approximation methods are somewhat more visible. This bias is accounted for by the method of proof. In Part I (see especially Chapter 5) existence is established by proving that certain consistent semidiscretizations in time lead to stable, and hence convergent, approximation schemes determined by the method of horizontal lines, introduced originally by Rothe (Chapter 5, Ref. [38]). This modus operandi is, of course, reminiscent of the Lax equivalence theorem in numerical analysis. In Part 11, existence is established by the contraction mapping principle, applied to a linear theory of evolution operators, resting upon a delicate product integration construction. Resolvent stability (or quasi-stability) plays a decisive role here. An alternative horizontal line analysis is presented as a by-product in Section 7.5. No attempt is made, however, to formulate a general global theory based on accretive operators, for example. In the writer’s viewpoint, this book represents a first, but a necessary, step in the quantitative analysis of complex nonlinear evolution models. Ironically, this perspective is revealed here in the introduction, although it represents the writer’s vantage point at the conclusion of this undertaking. What is needed is a reliable methodology, consisting of convergent algorithms for nonlinear problems in infinite-dimensional spaces, such that, upon interface with models such as those presented in this book, some semblance of optimality accrues for these algorithms with respect to known and possible computing processes. Indeed, there are at least two discernible levels at which this analysis can take place. One level is that of the reduction of complexity through a process roughly comparable to entropy determination. For example, one can fully discretize the two-phase Stefan problem (see
’ Not to be confused with Moser iteration in elliptic regularity theory.
Introduction
3
Section 1.1 for elaboration of this model), with a distribution equation describing the rate of change of (discontinuous) enthalpy
on a space-time domain 9,via a fully implicit method in time and a piecewise-linear, finite-element,Galerkin method in space and obtain a recursively generated sequence of well-posed finite-dimensional problems. While the first level of analysis computes the convergence of this scheme to the solution u, or solution pair [u,H(u)] and compares this rate to the optimal rate in finite-dimensional approximation of any finite-energy function u on 9 satisfying an energy norm bound, it stops short of the second level, actually devising an algorithmic strategy for the solution of the finite-dimensional problems. The latter are usually solved on modern parallel-processing computers by some combination of a constrained Newton method, in conjunction with iterative and sparse direct matrix routines for linear problems, for example multigrid methods. While this second level of algorithm formulation is not discussed in the sequel, we shall comment on both levels now in order to focus the ideas. It can be shown (see Chapter 2, Ref. [9]) that, under reasonable hypotheses, the fully discrete scheme in space and time described above gives rise to a sequence of finite-dimensional approximations in L2(9), convergent to u, O(hlln(l/h)l) when At = ch2; here h is the maximal finite-element cell diameter. The significance of this result, when h is related to the Euclidean dimension (n + 1)of 9 and the dimension N of the approximating L2(9) subspace according to the weakly asymptotict formula N w Ch-("+2),is that O ( N - ( " + ' ) )is the N-width of the unit ball of H'(9) as measured in L2(9). Thus, the complexity of the fully discrete method may be directly related to that of computing, via finite-dimensional approximations from N-dimensional subspaces, an arbitrary element from the unit ball of H'(9). The logarithm factor appears unavoidable; although u E H'(9), dH(u)/dt is essentially bounded in t as a regular Baire measure in x, and not necessarily a square-integrable function in (x, t), for both the Stefan problem and the porous-medium equation (for this measure property, see Proposition 5.1.5 and Theorem 5.2.1). We address these questions in Chapter 4. P r o p osition 4.3.4 derives the convergence result for the continuous-time, FaedoGalerkin method in the context of a general class of degenerate parabolic equations. The N-width results are developed in Chapter 4, Section 2 (see also Chapter 4 Introduction). Additional material is included involving mixed norms and also the relationship of the Weinstein-Aronszajn theory of intermediate eigenvalue problems to ellipsoidal N-widths.
' Two expressions are weakly asymptotic if they are of the same order (0).
4
Introduction
We comment briefly now on algorithm formulation and, to be consistent with the general theme of the book, we confine our attention to infinitedimensional spaces. The nonlinear elliptic theory developed in Chapter 3 for the steady-state and semidiscrete equations does not include the construction of solutions.’ It is not enough, as emphasized earlier, simpli to introduce complexity-reducing approximations, such as those defined by a Galerkin method. In lieu of this, we refer to the extremely promising work of Bank and Rose [ 3 ] on quasi-Newton methods for the solution of nonlinear elliptic boundary-value problems and operator equations more generally. This work extracts the essential features of gradient maps, for which Newton’s method is globally convergent, and produces, on an explicitly defined set S containing the solution and starting values, a globally convergent quasi-Newton method with controlled residuals. The basic hypotheses are a form of coerciveness, Lipschitz continuity of the Frechet derivative, and uniform boundedness of the derivative inverses on a set slightly larger than S. In certain cases, the last hypothesis simply reduces to the statement that the derivative (linear) maps A are injective. Interestingly, this last hypothesis applied to I-A is the major hypothesis employed by Sermange [8] in his construction of iterative methods, generalizing Picard successive-approximation convergence, as embodied in Kitchen’s theorem [ 6 ] . Whereas the latter theorem requires the spectral radius of the compact complexification A of the derivative of a given continuous operator T to be less than one at a prescribed fixed point of T, Sermange is able to construct a quadratic polynomial p(T), with fixed points containing those of T, such that p(T) satisfies the hypotheses of Kitchen’s theorem, although T need not. While the derivative hypothesis, in principle, need hold only at the fixed point, the latter is undetermined, so that it is essentially a neighborhood hypothesis. Moreover, the convergence result is local, not global, and, hence, must be deemed somewhat weaker computationally than the result of [3]. Still, these results indicate that iterative methods, in conjunction with quasiNewton methods, can yield effective algorithms down to the linear level where a new, highly nontrivial analysis must begin. Probably none of this will surprise the informed reader, who may realize that the proof of the implicit-function theorem in Banach space makes fundamental use of the contraction mapping principle, and that the generalization, embodied in Nash-Moser iteration, uses a quasi-Newton method involving approximate right inverses. Of course, all of this discussion has been begging a fundamental question : What is the effectiveness of these algorithms for degenerate problems, such as the two-phase Stefan problem, where differentiability requirements are not satisfied? While regularization, discussed in some detail in
’ See, however, the contractive method of Hartman and Stampacchia (Ref. [lS] in Chapter 3).
Introduction
5
Chapter 2, can prove helpful here, it may also be true that adjustments of standard methods, applied to the nonregularized problem, may be superior. Some evidence that this is the case is furnished by the remarkable success of Wheeler (Chapter 1 [25]) in his use of constrained-Newton methods for solving the finite-element equations arising in modeling the melting of permafrost prior to the construction of the trans-Alaska pipeline. Let us turn now to a synopsis of the key elements of the book. Many of the key ideas of Part I1 are due primarily to Kato and these results are taken from his papers (Chapter 6 [ l l , 123, and Chapter 7 [13, 17]), written either individually or with others. The reader will notice that some hypotheses have been strengthened for ease of presentation. The delicate part of this theory is the linear theory, or, more properly, the construction of the linear evolution operators and their invariance on a certain smooth space Y. Fundamental simplifications of the arguments by Dorroh are incorporated into the text. One of the most delicate aspects of Kato’s construction surrounds his notion of quasi-stability of the class {A(t)}of negative generators of strongly-continuous semigroups. This is suggested by the applications to quasi-linear systems, where (suppressing the nonlinear dependence) a similarity transformation on {A(t)}gives rise to a perturbed family {A(t)+ B(t)}. In general, it may be known only that IIB( .)[Ix is upper Lebesgue integrable, leading to the notion of quasi-stability for {A(t)+ B(t)} on the ground space X; this, in turn, leads to quasi-stability for {A(t)}on the smooth space Y. However, the quasi-linear systems considered in this book satisfy stability properties on X and Y appreciably stronger than the aforementioned quasistability. Although we present the details of the evolution operators’ construction in Chapter 6, through semigroup splicing in the case of stability, we refer the reader to Kato’s paper (Chapter 6 [12]) for the modifications of the details in the quasi-stable case. Here, uniform partitions of the time interval are no longer possible and delicate measure-theoretic questions enter, especially regarding approximation. Thus, Part I1 makes no pretense of being selfcontained with respect to proofs. We ask the reader’s indulgence in the genuine interest of brevity particularly since the more general result is not required for the application. Proofs are also omitted for some other results, such as the underlying lemma (see Lemma 7.3.4) for the commutator estimate required to obtain a uniform norm bound on the perturbations { B(t)}, in the applications to quasi-linear hyperbolic systems (see Section 7.3). This lemma is a variant of an earlier result of Calderon (Chapter 6 [4]), and plays a decisive role in both Sections 7.3 and 7.5. In the latter section, it forms the basis for an explicit estimate, derived by the author (Chapter 7 [14]), for an invariant time interval, on which the viscosity method produces convergent Navier-Stokes approximations to a solution of the Euler equations for an ideal fluid. Section 7.5 contains the only results in Part I1 due to
6
Introduction
the author. The other physical application of Part I1 involves the verification that consistent initial data can be evolved forward in time, subject to the vanishing of the Ricci curvature tensor, by the use of harmonic coordinates. These equations represent the vacuum-field equations of general relativity. The material on fluid dynamics in Part I is essentially well known, and is included for completeness and to serve as a reference point for the local result mentioned in the preceding paragraph. The analysis of the reactiondiffusion systems in Part I makes fundamental use of the idea of invariant regions with “inward-pointing” vector fields; however, while citing some of the fundamental literature here, we develop an independent approach based on semidiscretization. The analysis here also discriminates between concentration-type and potential-type variables, and the invariant regions are defined, via the confinement of the concentration variables, to a slab. This geometry is, of course, highly specialized. It is now known that bounded solutions of the two-phase Stefan problem and the porous-medium equation are continuous in the interior of 9.However, u E H’(9) is all that can be expected, in the sense of gross Sobolev space regularity for weak solutions, and this provided uo E H’(0) for the space domain 0, due to the degeneracies induced by the function H in Equation (1); H is discontinuous at zero in the Stefan problem, and may have an unbounded derivative near zero in the case of the porous-medium equation. This H’ regularity is embedded in our formulations of both models in Chapter 1. Although we consider each separately in Chapters 1 and 2, for the good reason that the (smoothing) regularization-convergence theories are distinct with respect to rates of convergence, the technique of pointwise a priori estimates permits a unified numerical analysis in Section 4.3, and a unified stability and existence analysis in Sections 5.1 and 5.2. Central to the analysis of these degenerate-parabolic evolutions are the equivalent abstract integral equations, holding pointwise on D, which are obtained through lifting the weak differential version from the dual of some subspace G of H’(0) to G, itself. The lifting, itself, is far from a novel idea, but perhaps our applications of this idea are not entirely evident. Among these we cite the following: (1) a simplified Lz stability analysis in Section 5.1; (2) a simplified uniqueness analysis in Section 1.5 ;(3) the semidiscrete error analysis in Section 4.4, in which the property that TH(u) is in H2(9) is decisive; (4) the regularization error analysis in Chapter 2; and, finally, ( 5 ) the quadrature formulas of Section 3.3, via the analogy with initial-value problems for ordinary differential equations. Properties (2), (3), and (4) are introduced in the author’s paper with Rose (Chapter 2 [9]). In his dissertation 4 [31], Rose made what was probably the first rigorous finite-element error analysis of a degenerate-parabolic model in his application of the lifting operators to the porous-medium equation. At that point in the late 1970s, the systematic
Introduction
7
use of these operators in the numerical analysis of linear and nonlinear parabolic evolutions was becoming widespread through the work of Bramble and his co-workers ([4], Chapter 2 [3]), as related to ( 5 ) above. However, the insight of (3) apparently originated in Chapter 2 [9], where it is applied in the context of fully-discrete approximations; the idea of (1) apparently originated in this text. Finally, if the lifting mapping is denoted by T, the estimate (for almost all t )
where T, = E, 0 T is the composition of the piecewise-linear finite-element energy projection E, with T, makes possible the continuous-time Galerkin estimates of Proposition 4.3.4’; here, C is independent of t. Note that (2) uses the essential boundedness of dH(u)/dt as a regular Baire measure in conjunction with the natural duality estimate, making use of recent sharp L“ estimates in the finite-element method. These latter estimates are discussed in Remark 4.3.10. For completeness, we also present the local approximation theory in Section 4.1, derived from the Sobolev integral representation theorem, which provides appropriate upper bounds for L2 finite-element estimates, and for mixed norm estimates, as well. The operators T also play roles, here, in terms of the Aubin-Nitsche duality “trick” of Proposition 4.3.1. The methods by which the stability estimates of Section 5.1 are derived, except for the L’ estimates of Proposition 5.1.5, depend upon an application of the discrete Gronwall inequality to the semidiscrete forms of these evolutions. We present several versions of the Gronwall inequality in Section 2.2, one of which was suggested by Veron [9]. The fundamental way in which the semidiscrete equations enter motivated the material presented in Chapter 3 on elliptic equations and inequalities. These are defined by mappings which are the direct sum of subgradient mappings and pseudomonotone mappings; in some of the applications the (multivalued) function H induces the appropriate subgradient mapping, whereas the pseudomonotone mapping is a convenient perturbation of the negative Laplace operator. In Chapter 3, we have tried to offer a reasonably compact presentation, adequately mixing the theory and applications; some proofs are omitted if they are readily reproduced elsewhere, or if they develop results not crucial to the sequel. The theory here was developed by many mathematicians during the 1960s, but the synthesis, which is preferred here, is due primarily to Brtzis (Chapter 3 [6]) and proceeds from the proximity mapping developed by Moreau (Chapter 3 [24]). Some balance is given by the Tarski fixed-point theorem
’
The estimates of this proposition are essentially best possible with respect to N-widths in H’(O), in contrast to the fully discrete estimates.
lntroduction
8
(Chapter 3 [28]), which is presented with proof and applied to quasi-variational inequalities. The reader will notice that we have downplayed the role of variational inequalities, particularly because of the appearance of the book by Kinderlehrer and Stampacchia (Chapter 1 [lo]). We close the introduction with a brief description of the development of the N-width as a measure of optimality in numerical analysis. Following the author’s doctoral dissertation [ S ] , written as a generalization to several variables of Kolmogorov’s original result on the N-widths of Lz classes defined by Sobolev seminorm bounds, two papers appeared simultaneously, that applied the asymptotic width computations to optimal Galerkin estimates. These were by Schultz [7] for the finite-element method and by the author (Chapter 4 [20]) for spectral approximations. Subsequently, Aubin, in his book [l], gave a systematic development of these ideas (see also Babuska [2]).
REFERENCES
PI c21
c31 [41
c51 C6l 171 C8l [91
J. P. Aubin, “Approximation of Elliptic Boundary Value Problems.” Wiley (Interscience), New York, 1972. I. Babuska, The rate of convergence for the finite element method, SIAM J. Numer. Anal. 8, 304-315 (1971). R. E. Bank and D. J. Rose, Global approximate Newton methods, Numer. Muth. 37, 279-295 (1981). J. H. Bramble, A. H. Schatz, V. Thomke, and L. B. Wahlbin, Some convergence estimates for semi-discrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14, 218-241 (1977). J. W. Jerome, On the L , n-Width of Certain Classes of Functions of Several Variables. Dissertation, Purdue Univ., Lafayette, Indiana (1966). J. W. Kitchen, Concerning the convergence of iterates to fixed points, Studiu Muth. 27, 247-249 (1966). M. H. Schultz, Multivariate spline functions and elliptic problems, in “Approximations with Special Emphasis on Spline Functions” (I. J. Schoenberg, ed.), pp. 279-347. Academic Press, New York, 1969. M. Sermange, Une mkthode numtrique en bifurcation-application a un probleme a frontiere libre de la Physique des Plasmas, Appl. Muth. Optim. 5, 127-151 (1979). L. Veron, personal communication, (1980).
GLOBAL WEAK SOLUTIONS
I
This page intentionally left blank
PROBLEM FORMULATIONS AND UNIQUENESS FOR DISSIPATIVE PARABOLIC MODELS
1.0
INTRODUCTION
Weak solution formulations are introduced for the two-phase Stefan problem, for the porous medium equation, for reaction-diffusion systems of equations, and for the Navier-Stokes equations for incompressible fluids, by means of divergence conservation principles in distribution form, in Sections 1.1-1.4. The initialrooundary-value problems are considered for these models with homogeneous Neumann, Dirichlet, Robin, and Dirichlet boundary conditions, respectively. Uniqueness theorems for the first three models are derived in Section 1.5, and a brief discussion of this property is presented for the fourth model. In the formulation of the reaction-diffusion systems, the variables are partitioned into those of concentration type and those of potential type, and the pointwise restriction of invariant region inclusion is imposed upon the concentration variables. Parts of the system are permitted to be degenerate with respect to diffusion. The Navier-Stokes system is placed in a generalized format of constrained systems, in which the usual transport ~ 0~ for solenoidal velocities u and v is relaxed. mechanism ((u V)v, v ) = The uniqueness results, as well as several convergence results in the sequel, depend upon a lifting of the weak solution relations from the dual spaces,
-
11
1.
12
Formulations for Parabolic Models
such as [H’(R)]* to H’(R), where they are realized as pointwise relations. This is achieved by means of standard linear inversion operators, which are introduced and utilized, following the definition of solution, in Sections 1.1-1.3.
1.1 HEAT CONDUCTION WITH CHANGE OF PHASE:
STEFAN PROBLEMS
Two-phase Stefan problems describe temperature variations in a substance undergoing a change of phase in a medium, in such a way that the temperature in both phases satisfies a diffusion equation. Strict conservation principles apply at the free boundary “interface”, corresponding to the nominal change-of-phase temperature. We may consider as an illustration the thawing of permafrost, which is any soil or rock which has existed at a temperature colder than zero degrees Celsius for two or more years. It is known that pure water, dispersed in a fine grained soil, freezes over a finitetemperature interval, rather than at precisely zero degrees Celsius. In fact, a fraction of the dispersed water, far from any soil surface, freezes at zero degrees Celsius, and the remaining fraction freezes at temperatures less than zero degrees Celsius. The latter is, thus, the nominal freezing point, that is the highest temperature at which ice can exist in a given soil, and represents a singular temperature for the various thermal coefficients. This problem from geotechnical engineering embodies the essential features of the model we now develop. Heat conduction with a change of state in an open region R can be formulated as the distribution equation for the temperature 8 given by
a
-(a at
+ s) - V - (kV8) + q(8) = 0
(1.1.1)
on 9 = gT,,= R x (0, To).Here a(8) = f! /3(<)d<,where /3 is the volumetric heat capacity, with units of energy per unit degree per unit volume; s is the latent energy content, with units of energy per unit volume and k is the thermal conductivity, with units chosen so that the flux, - k V8, has units of energy per unit area per unit time. The quantity q represents a source or sink, and is negatively signed for a source and positively signed for a sink. The formulation (1.1.1) contains considerable information as a distribution relation, specifically (1.1.6) below. To motivate (1.1.1) heuristically, start with the energy-balance relation, defined by an arbitrary open set W c R, with sufficiently smoothly oriented
1.1 Stefan Problems
13
boundary 833, (1.1.2) where v denotes the outward normal to d33. Equation (1.1.2) asserts that the net rate of heat flowing into 33, given by the right-hand side, increases the enthalpy, that is, the internal temperature and/or energy content, accordingly. We shall assume, for the present discussion, that fl, s, and k are piecewise-smooth functions of 8 only, with singularities restricted to the nominal change-of-phase temperature of zero; and that fl and k are positive, with k bounded away from zero. Thus, except for special s, it is essential to interpret (1.1.2) as a distribution on (0, To).We require further that, if (8 = 0} has positive measure in 9, then fl 0 8, s 0 8, and k 0 8 can be extended to measurable functions on this set. If (1.1.3a)
C = {(x,t) E 9 : 8 ( x ,t ) = O } ,
we write C = C, u C+ u C-,where C, is the (possibly empty) interior of C and C +, C - are possibly overlapping sets given by
C+ = (8 > 0} n C,
C- = ( 8 < 0 } n C.
(1.1.3b)
The computations to follow assume that C + and C - are sufficientlysmoothly oriented hypersurfaces. We have the following form of (1.1.2) -JOT"
Ja(a(8) + ~ ( 6 )34 ) dx dt at =
soT"ha
k(O) V 8 *
+
JOT"
sa
q(8)4 dx dt
v 4 da dt,
(1.1.4)
+
8- is the decomposition of 8 into its positive for 4 E C,"(O,To).If 8 = 8' = 33 x (0, To),we have, via integration by parts and negative parts and applied separately to the first and third terms of (1.1.4) in (0 > 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) < p < 00,
5 # 0,
(1.1.8a)
where H’(O+) and H’(0-) are assumed to exist. Normalization is chosen so that H ( 0 - ) = 0, (1.1.8b) and the jump condition takes the form H(O+)
=A
> 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.
1. Formulations for Parabolic Models
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.
1. Formulations for Parabolic Models
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)
1. Formulations for Parabolic Models
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
1. Formulations for Parabolic Models
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
1. Formulations for Parabolic Models
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
1.3 Reaction-Diffusion Systems
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.
1. Formulations for Parabolic Models
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
(1.3.8a)
1.3 Reaction-Diffusion Systems
29
and gi vanishes at 0, gi(0) = 0.
(1.3.8b)
Remark 1.3.2. The set Q is termed invariant, since initial datum u, with range in Q implies that the solution u, of the appropriate initial/boundaryvalue problem corresponding to (1.3.4), also has range in Q. The most physically meaningful boundary conditions corresponding to Example 1.3.1 are Robin boundary conditions; these are also important for the remaining two examples. Thus, if Di > 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
Formulations for Parabolic Models
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.
1.3 Reaction-Diffusion Systems
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
1. Formulations for Parabolic Models
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 < i < rn, be an index for which Di > 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
1. Formulations for Parabolic Models
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
Incompressible, Viscous Fluid Dynamics
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.
1. Formulations for Parabolic Models
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
Incompressible, Viscous Fluid Dynamics
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
Formulations for Parabolic Models
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
Definition 1.5.1. Let g be a monotone increasing locally Lipschitz continuous function, let h be a Lipschitz continuous function, and let H be strictly monotone and continuous, except possibly at zero, with H(0 -) = 0. Let G = H'(R) or G = HA(R), respectively, and let T:G* + G c H'(R) denote one of the operators N,, o > 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~)
1. Formulations for Parabolic Models
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).
1. Formulations for Parabolic Models
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
t2
E,
< E.
E
to
(2.1.3a) (2.1.3b)
Here, q, is the uniquely determined quadratic polynomial on [0, E ] , defined by (2.1.4) Finally, set H,(t) =
Remark 2.1.1.
J; h,(s)ds,
(2.1.5)
t E R'.
A routine calculation shows that qE is given explicitly by
q E ( l ) = 40.E
+ q l . E t + 42,Et27
< <
'7
(2.1.6a)
2.1 Regularization in the Stefan Problem
47
where q,,, =
e,,
ql,, = 2
q2,E= {(HI(&)-
~ - 32&e,~- &H'(&)I/&~,
oO)&- 2
~ - 32&e0 ~- &~1(&)1}/&3.
(2.1.6b)
Moreover, q, is concave on [ O , E ] , that is, q2,,< 0. To see this, we argue as follows. Using the inequalities 2A 680 + 36'
E d & * <
we conclude that c2ql,, - 4A 2 ~(66'0
If:(&) d 8,
+ 36/2,
+ 36) - 4 ~ 8 ,- 248, + 36/2) 2 0,
so that ql,,2 4A/&.Thus, 36
<2
001 - E q l , , which, in conjunction with E 2
qz.,
<&
< E*
--
4A -, &
d 8A/(36),yields
(F); 1
1
- (4A) d - (4A - 4A) = 0. &
This establishes the concavity of q,. In particular, q,(t) 2 min(e,,H'(E)) 2 i> 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
2. Regularizations and Pointwise Stability
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 ,
< p ~ This . gives
0 < J&(t)- J&(A)=
so that
H’(s)ds < A
J. J i ( T ) d T < J.(l/A)dT
0 < J&(t) - J ( t ) < /.LE/il
+
E
- J(t)
= (t - A)/A
< E(l + p/A),
< p&/il, (2.1.12)
if t > 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
for 0 < E < E * , where it is explicitly assumed that f, is defined by (1.1.26), and is Lipschitz continuous. If THe(~E)lt=o = TH(u)l,=,, then the estimates llH&(U3- H(u)llL-((O,To);F) G IIU&- 4 L 2 ( 9 T o )
cE1'2,
' < CE"2
(2.1.15a) (2.1.15b)
hold for some positive constant C , independent of E, where F = [H'(Q)]*. The constant C depends upon IRl and To explicitly.
Proof: Set u = H(u) and 'U = H,(u&).After subtraction of (1.1.25) from (2.1.14), followed by multiplication by (us - u), and integration over R, we have
-l _ d 2
dt
llUE
- ull:
+ (J&V) - J&(d, uE - U)L2(*)
= (JW - J&(4 lJc- &2(*) - V[f,
O
J&(O - f, J(4J V E - U)L2(*), O
(2.1.16)
where we have subtracted a term involving J,(u) from both sides of (2.1.19, and have used (1.5.5). The second term on the right-hand side of (2.1.16) may be estimated in analogy with (1.5.7):
I(f,
O
J(4,T(UC-U))LZ(*)I G 3CCvIIJ,(vE) + v - p-ullb] G c[vllJ&(4 + vllJ,W J,(U"-f
O
J(v)ll:2(*)
J&(41122(*)
J(u)l(t2(*)
+k-(p-ull;l.
(2.1.17)
where q is an arbitrary positive constant, and C = C , . The first term on the right-hand side of (2.1.16) is estimated, in the standard way, by
2. Regularizations and Pointwise Stability
50
The key terms in these two inequalities are the first terms on the right, which we shall see can be absorbed on the left-hand side of (2.1.16), with appropriate choices of ?. Indeed, the inequalities (J&(U&) -
(UE -
J,(4,J&(U?
- J,(U))LZ(R)
1
< 1(J&(U?
- J,(V),
UE
- U)LZ(R),
(2.1.19a)
u, UE - u)L2(n) = ( H , 0 J,(U&)- H , 0 J,(u), ue - u)Lz(n)
< -&Y (J,(o") - J,(u), uc which follow from J: < l / A and Hk tion of (2.1.16)as 1
T
d
- 011:
<
(: + :) -
< y/&, respectively, lead to the reformula-
+ 3(J,(u,) - J,(U),
llUE
-
(2.1.19b)
&2(n),
IIJ,(v) - J ( 4 l l Z Z ( * ,
UE -
+
h(n)
2c2
JluE- U I l L
(2.1.20)
provided the choices q = A/(4C) and q = ~/(2y)are made in (2.1.17) and (2.1.18),respectively. If (2.1.10)is applied to (2.1.20),we obtain the inequality
2zd 1
)IUE
- UllZ + 3(J,(u,) - J,(u), V E- k ( n )
If (2.1.21) is integrated from z = 0 to z = t, one obtains, for obvious choices of C, and C,, containing the factor IRlT,,, I(UE -
41: + J; (J,(u,) - J,(4ue - U)L2(n)dz
< C1&+ C2 J: llue - ull:dz,
(2.1.22)
for 0 < t < T o .By Gronwall's inequality applied to (2.1.22), we have
lIuE
- uIIZm((O,To);F)
+ ~oTo(J,(v') - J,(u), UE - U)L2(n)dz
< CIEeCzTo,
(2.1.23)
which implies (2.1.15a). To obtain (2.1.15b), we use the triangle inequality,
2.1
Regularization in the Stefan Problem
51
in conjunction with (2.1.10) and (2.1.19a), that is, IlJ&(UE)
- J(U)llt2(9)
< 2IlJ&(U?- J & ( 4 l l t 2 ( 9 ) + 211J&(U)< (2/4
so'"
(J,(u&)- J,(u), U~ - U
J(U)lltZ(9)
+
) ~ * ( ~ ) ~21ZZIT0 T
Inequality (2.1.15b)follows from (2.1.23) and (2.1.24).
Remark 2.1.3. The smoothing introduced in this section attains its full power when uo$H'(R), in which case, the property &EL'@) must be discarded. More precisely, the smoothing does not preserve H' initial data, with the result that we do not expect the regularity condition u: E L2(9)to hold. The regularization introduced here, then, more properly represents a transferral of singularities. Note that the lifting formalism employed here substitutes for u, E L2(9) the weaker property (TH(u)),E L'(9). There is, of course, an obvious way' to preserve H' initial data, namely, to require u; = uo, rather than u; = J, 0 H(uo) (see the initial condition in Proposition 2.1.2). Although this represents a more legitimate regularization, since uEE % in this case, it introduces the additional term IIHe(uo)- H(uo)lli on the right-hand side of (2.1.22). As noted by Rose and the author [9], this term is of order E, thus preserving the convergence rates of (2.1.15), provided the nondecreasing rearrangement A,, of uo satisfies on A,,(t) 2
- CE,
0 < t < E,
a
for sufficiently small E. Equivalently, p & b o ) - H(u0)lli
G CE,
(2.1.25a)
provided on the set 6, where uo 2 0, {X:uo(x) iE }
< CE.
(2.1.25b)
The condition (2.1.25b) already implies a certain degree of regularity, however, on the free boundary. Moreover, as stressed in the previous chapter, it is H(uo) which appears more physically significant, and it is this quantity, perhaps, which should dictate the initial datum for the regularization. An existence theory for the initialboundary-value problem corresponding to (2.1.14), with L2 initial datum, is presented in LadyZenskaja, Solonnikov, and Ural'ceva ([lo] pp. 465-475). A change of variable u, = H,(ue) is a necessary preliminary step to invoke the theory.
2. Regularizations and Pointwise Stability
52
2.2
SEMIDISCRETE REGULARIZATION AND MAXIMUM PRINCIPLES IN THE STEFAN PROBLEM
We begin with discrete inequalities of Gronwall type, which will be used repeatedly in the sequel. Proposition 2.2.1. Let a = to < t , < . . . < t , = b be a partition of [a, b], and suppose that 4 and $ are nonnegative step functions, with values 4,' and $k, respectively, on the intervals [ t k - 1, t k ) , k = 1, . . . , M. Suppose that, for some fixed p 2 1, there exists cr 2 0, such that
4: < d' +
C"'
k- 1
$$dt = d'
+ 1 $,,,$,,,(t,,, - t m - l )
(2.2.1)
m= 1
holds for each k = 1,. . . , M. Then the following bounds are valid: &'-'(t)
4( t )
< d'-' + (1 <
fJ
exp(1
-
!)
$(T)~T,
$(T)dT),
a
< t < b (p > 1); (2.2.2a)
a
< t < b (p = 1). (2.2.2b)
In particular, m= 1
$,,,(tm - t m - l ) ,
k
=
1,. . . ,M , (2.2.3a)
k
=
1,. . . , M , (2.2.3b)
for p > 1, and
for p
=
Proof:
1. Set
+
~ ( t=)d'
C
$(z)+(z)dz,
a
< t < b.
(2.2.4)
If o = 0 on [a, b],then 4 = 0 by (2.2.1)and, thus, (2.2.2)clearly holds. Suppose then that w ( t ) > 0 for t > ti,0 < i M. By a change of variable t' = t - ti, we may assume, without loss of generality, that i = 0, that is, o(t)> 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
o(t)
$(r)dr),
a < t < b,
(2.2.8)
since w(a) = CJ. Since 4 < w , we immediately obtain (2.2.2b) from (2.2.8).The relations (2.2.3) are immediate from (2.2.2), if we let t l t k - 1 . Remark 2.2.1. In the sequel, inequalities slightly different from (2.2.1) will also naturally arise, especially when implicit and explicit semidiscrete methods are mixed. Thus, for example, it will happen frequently that the inequality k
4; d CJ’ + 1 $,,,4,,-1(tm - tm-l), m= 1
can be derived. If the new partition
is introduced, then (2.2.9) may be rewritten as
and (2.2.3) assumes the form
q!g-l < [ C J ~+ +140(tl- r0)](p-l)’p
k
=
1,. . . , M
(2.2.9)
2. Regularizations and Pointwise Stability
54
for p > 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 $m(tm - tm-l),
k
=
m= 1
1,. . . ,M .
(2.2.12b)
This implication is a corollary of the following. Proposition 2.2.3. Suppose {ak}and {bk} are two sequences of nonnegative numbers, such that <$a:
k
+ 1 ajbj
for all k 2 0.
(2.2.13a)
j= 1
Then k
ak
< a, + 2 1 bj
for all k 2 0.
(2.2.13b)
j= 1
Proof:
Define the nonnegative sequence {ak} by & Z -l ZTa,
k
+ 1 ajbj j= 1
and note that ak
< ak,
for all k 2 0,
k 2 0.
(2.2.14)
(2.2.15)
2.2 Semidiscrete Regularization in the Stefan Problem
55
We deduce from (2.2.14) the quadratic recursion,
so that We obtain from the linear recursion (2.2.16), ak < a,
k
+ 2 1 bj, j= 1
k 2 0,
from which (2.2.13b) follows via (2.2.15). Remark 2.2.3. In the following chapter, we shall define a variety of semidiscrete approximations in time. However, in this chapter, we shall restrict attention to the fully-implicit, or backward-Euler method, combined with an explicit method, as applied to the Lipschitz part off. The justification for singling out this special semidiscrete method at this time is that it readily permits estimates on the L P norm of the approximate solutions at discrete time levels, in terms of the Lp norm of the initial datum. These estimates may be translated into L”((0, T o ) LP(Q)) ; estimates for the solution of the evolution equation in terms of Lp(Q)estimates of the initial datum. Our primary interest is in the cases p = 2 and p = 00. Definition 2.2.1. Let 0 = t: < ty < . . . < tE(,,,) = To denote an arbitrary sequence of partitions 9” of [O, To].We introduce the semidiscrete version of (1.1.9), based on the backward-Euler method, applied on {P”}, as a recursively generated finite sequence { u f , H(uf)}piY),for each N = 1,2, . . . , satisfying H ( u f ( x ) )E [O, A ] if uf(x) = 0, (2.2.17a) (2.2.17b)
H M ) E LZ(Q), uf
E
(2.2.17~)
H’(Q),
and the variational condition
of - tf- 1)- ‘([H(uf)- H ( 4 - ,)I,U)LZ(*) + (VUf, V&(*)
+ (g(uf) + h(uf-
1),
=0
for all u E H’(Q), (2.2.18)
for k = 1, . . . , M ( N ) . We require that (2.2.19) and that
(2.2.20)
2.
56
Regularizations and Pointwise Stability
for each N and 1 < k < M ( N ) .By the semidiscrete regularization of (2.2.17)(2.2.20) is meant a recursively generated finite sequence { U ; ~ } ~ J ? ) , for each N = 1, 2,. . . , satisfying (2.2.21)
u : ~E H'(R), and the variational condition
'([H,(U2") - W42N1)1, V)LZ(R) + (vu:", VU)L2(R) + (g(u2") + h(u;!'), u)~~(*)= 0 for all u E H'(R), (2.2.22)
0; - tk"- 1)for k
=
1 , . . . , M ( N ) .As above, we require that
H,(u2")
(2.2.23)
= Wuo),
and for each N and 1 < k
(2.2.24)
f M N E) L2(0),
< M(N).
+
Remark 2.2.4. Note that we have used the decomposition f = g h (see (1.1.8d)); current (implicit) values are used in the evaluation of g, whereas previous values are used for h. The existence and uniqueness of solutions are guaranteed by Proposition 3.3.1. We note this explicitly in Proposition 2.2.5. In the special case, the domain R possesses the property that N,LP(R) c W2*p(R),
1 < p < m,
(2.2.25)
then the solutions {u:} of (2.2.18) are progressively smoother, even if H(uo) E L'(R) is the assumption on the initial datum H(uo). In fact, an estimate can be given for ko, such that uf E L"(R), k 2 ko (all N), via Sobolev's inequality. If we define the (finite) sequence p.=
'
"Pi- 1 , n - 2pi-'
n-2pi-, >O,
p o = 1,
(2.2.26)
for i 2 1, then the sequence terminates at a value io, for which n - 2pi0< 0. Set n - 2pi0 < 0, ko = io' (2.2.27) io 1, n - 2pi0= 0,
{
+
and in the latter case let pko > pio be arbitrary. By Sobolev's inequality, uf
E W2*pko(R) c
L"(R),
k 2 ko.
(2.2.28)
We shall not attempt to exploit this fact to obtain the analog for evolution equations, which amounts to instantaneous smoothing, for t > 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
2. Regularizations and Pointwise Stability
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<-aq+-b', 4 r
a20,
1 1 b20, -+-=1, 4 r
(2.2.38)
(2.2.40)
2.2
Semidiscrete Regularization in the Stefan Problem
In particular,
~ e k =' ~J ,
E
0
59
Lq(0),and the estimate
IIH&(u;.N) IIL4(*) 6 IIH,(u$") IILP(R) + IWllQll'q To
+ (llhllL,/4
c k
i= 1
-
IIH&(uf"l)IIL4(R)(tN
6-
1)
holds for k = 1, . . . ,M ( N ) .By the version of the discrete Gronwall inequality contained in Corollary 2.2.2, we obtain (2.2.31). The result (2.2.32) is immediate from (2.2.31).
Proposition 2.2.5. The solution pair of (2.2.17)-(2.2.20) and that of (2.2.21)(2.2.24) exist and are unique. Iff is Lipschitz continuous, the estimates IIHE(~;~) - H(u;)l/F
p' k= 1
I I U y - u;11:2(n)(t;
- t;-
I"'
1)
< CE"', < CE"'
(2.2.41a) (2.2.41b)
hold for some constant C, independent of E, k and N , provided the local mesh ratios are bounded,
and provided
II@"NI is sufficiently small (cf. (2.2.47) below).
Proof: To establish the inequalities (2.2.41s b), we begin with the equations (t; - t;-
1)-
(t; - t;-
'T[H,(u:~) - H,(U;'!~)] 1)-
'T[H(u;) - H(u;-
I)]
+ u : ~+ Tf,(u;!'l) + U; + Tf,(u;-
1)
= 0,
(2.2.42a)
= 0,
(2.2.42b)
where T = Nu,0 < 0 < AT;' (see Definition 1.1.2) and f , is defined in (1.1.26). We find it advantageous to use the inverse formulation. Thus, set J, = Hep1,J = H - ' , and
u y = H,(u:N),
(2.2.43)
u; = H(u;).
Subtracting (2.2.42b) from (2.2.42a), multiplying by u;" - u;, and integrating over 0,we obtain (t; - t;-V_1)-1[IIV2N
- $1 ;
- (@!1
- Lf-1,
@N
- $)F]
+ ( J & ( V Z N ) - J&(#), G N- $)L2(R)
=
(J(4) - J,(& - (f,0 J,(u;!l)
02N - 4 L 2 ( R )
- f, J(u;0
I),
T[u:N - u;])L*(R).
(2.2.44)
2. Regularizations and Pointwise Stability
60
The estimation of the right-hand side of (2.2.44)parallels that in Proposition 2.1.2 (see (2.1.17)-(2.1.19)), adjusted by the fact that f, is evaluated explicitly. If we estimate the left-hand side of (2.2.44),we obtain, altogether, $(t: - t:-
1)-
‘“I,”
+ (J&(@”)
d IIJ&(v:)
- u:11; -
IIu;N1
- u:-
J&m, u;N &(a)
-
1 1:]
-
- J(~:)lJLz(sllu;N
+ COI(J&(@Nl) - 40:-
- u:llL2(a)
1)IIL2(RJ(u;N
- u:lF
(2.2.45) so that, after multiplication of(2.2.45)by t: - t:- 1, summation on k = 1, . . . ,I, and absorption of terms by the coercive resulting left-hand side, we obtain I
)IIu,,N
- u:ll;
+ $ 1 (J&(U;”)
- J&($), U k N - u:)L2(,)(t:
- t:-
1)
k= 1
(2.2.46) where ro is given by (2.2.41~).The Gronwall inequality of Proposition 2.2.1 may be applied (to 2.2.46), provided (IqNIdI p0 < W G r O ) .
(2.2.47)
Here C, may be defined, from (2.2.45),as the product of the Lipschitz norm off, and the norm of the injection L2(n)-,F. Under the assumption (2.2.47), the term k = 1 in (2.2.46) may be absorbed in the first term on the left-hand side, with resulting constant (1/2) - (2C;r,po)/L. We, thus, obtain (2.2.41a) for an appropriate choice of C. In fact, the Gronwall inequality also yields the estimate
2.2 Semidiscrete Regularization in the Stefan Problem
81
To obtain (2.2.41b), we use the triangle inequality in conjunction with (2.2.48),that is, M k= 1
(2.2.49) and (2.2.41b) follows from (2.2.48)and (2.2.49),with an adjustment in C. The existence and uniqueness are a consequence of Proposition 3.3.1. w
Corollary 2.2.6. Suppose the initial datum H(uo) is in LP(R), p 2 2, and suppose f is Lipschitz continuous. Then {H(Uf)}k,N satisfies the bound in (2.2.31), provided the partitions {9”}satisfy (2.2.41~)and (2.2.47). Proof: For 2 d p c co (resp p = a), there is a subsequence { H & J U ; ~ of *~)}~ { H , ( u ; . ~ ) } which , is weakly (resp weak-*) convergent in LP(R) to a function x: as j -,co and, by lower semicontinuity, x: satisfies the bound in (2.2.31). By composition of continuous embeddings, which preserve weak convergence, we conclude the weak convergence of this subsequence in F and, also, by Proposition 2.2.5, the convergence in F to H(u:). Uniqueness of weak limits yields x: = H(u:). H
Remark 2.2.6. Note that the Lipschitz continuity of f was used only in deducing the weak convergence of H , , ( u ~ *to ~ ) H(u:) in LP(Q)via Propas contained in osition 2.2.5. The actual a priori estimates for Proposition 2.2.4, do not require f to be Lipschitz continuous. However, we do observe that, if p = co and f is locally Lipschitz, the result of Corollary 2.2.6 holds, since f can be replaced by an appropriate extension f of the restriction off to a set containing the range of uf. This, of course, utilizes uniqueness properties. In Chapter 5, we shall have cause to consider more general H and more general smoothings. For this more general class, the estimates (2.2.31) and
2. Regularizations and Pointwise Stability
62
(2.2.32) will remain valid, since no properties beyond Definition 5.1.2 are required. However, Proposition 2.2.5 is expected to be valid only selectively for general classes of smoothings since it depends fundamentally upon (2.1.13). Finally, although we did not use both partition bounds, as defined by (2.2.41c), in the proof of Proposition 2.2.5, we shall have cause to make use of them in Chapter 5.
2.3
REGULARIZATION IN THE POROUS-MEDIUM EQUATION
The hypotheses on the function H of Section 1.2, which yields the moisture content or density in the case of the porous-medium equation, are given in Remark 1.2.1. In this section, we shall smooth H to obtain a strictly monotone increasing, continuously differentiable Holder function H E ,such that, on compact subsets of R1\{O}, H E and Hb agree with H and H for E sufficiently small. As in Section 2.1, this will be followed by a regularization theorem.
Delinition 2.3.1. Suppose 0 < E < 1. Define the linear function to be the uniquely determined function satisfying &(&) = HI(&),
so'
&) dt = H(E).
& on [O,E] (2.3.1)
The function f is given explicitly by &(t)= & E
+ e;.,t,
0
< E,
(2.3.2a)
where = [ ~ H ( E-) EH(E)]/E,
f l , E=
~[EH'(E) - H(E)]/E~. (2.3.2b)
< 0 follows from the decrease of the difference quotients for Note that fl,E the concave function H: H(E) H(E)- H(0) H(E)- H ( 6 ) * (1 -2 E
E-0
E - 6
-
+(&I
+7 E - 6 H(0) < H ( 4 , (2.3.3)
for 0 < 6 < E. Letting 6 t E immediately gives H(E)2 &HI(&).
(2.3.4)
2.3 Regularization in the Porous-Medium Equation
83
We also conclude that & & 2 H(E)/E. Let t, denote the even extension of
[ - E, E ] , and set
e?: to
(2.3.5) Finally, set (2.3.6) Proposition 2.3.1. For any compact subset K of R'\{O}, such that ( 0 , ~n) K = @ , H e and H: agree with H and H , respectively, on K. Moreover, H,(t)t 2 0 for t E R' and, for some c > 0, Hk(t) < C E ( ' / Y ) - ' ,
(2.3.7)
tER'.
In the notation of (1.2.14e), c is given explicitly by c = 2c,y. Moreover, the approximation IH,(t) - H(t)I d
t E R',
C&'/Y,
(2.3.8a)
holds for the same c. If J = H-' and J , = He-', then J, as well as J,, are continuously differentiable strictly monotone increasing functions and IJ,(t) - J(t)l d 2&,
(2.3.8b)
t E R'.
Proof: By construction, H, and Hk agree with H and H', respectively, on R'\( - E , E). For - E < t < E, we have, from the definition of l,, (2.3.9)
Hk(t) = l,(t) 6 2H(&)/&.
However,
H(&)< c,ye'/Y,
(2.3.10)
from the second bound in (1.2.14e). From the same bound, we deduce H'(t) < c,t(l/r)-
< Cz&(1/')-
,
1
t2&.
(2.3.11)
Altogether, from (2.3.9),(2.3.10),and (2.3.1 l), we deduce (2.3.7),with c = 2c,y. Now, (2.3.8a) is immediate if the difference is estimated crudely by the triangle inequality, where H, and H differ. Thus, since He(&)= H(E),by construction,
+
+
IH,(t) - H(t)I d H&(t) H ( t ) d H,(&) H(&) = 2H(&)d 2C2Y&1/Y, for 0 d t < E , hence, for --E < t < E, by the sign properties of odd functions. In a similar way, (2.3.8b) need only be estimated on [ - H(E), H(E)],and we
2. Regularizations and Pointwise Stability
64
find IJE(l) -
J(t)l <
JE(t)
+ J ( t )< J&(H(&))+ J ( H ( & ) )
+
= JE(HE(&))J ( H ( & ) = ) 2.5,
which is simply (2.3.8b).The remaining statements are clear.
Remark 2.3.1. As in Section 2.1, we may use H Eto define regularized parabolic problems, which are defined more conveniently in terms of the lifting (1.2.18).The next proposition describes the rate of convergence induced by the regularization. We use the solution class
Vl= C([O,T0];L2(SZ))nL'((0, To);Hh(SZ)). Proposition 2.3.2. Let T = Do be the Dirichlet inversion operator of Definition 1.2.2.If V1denotes the class defined above, suppose that u, uEE V,, respectively, satisfy (1.2.21)and aTHE(uE) + ue = 0, at
(2.3.12)
= TH(u)l,,,, then the estimate for 0 < E < 1. If THE(~E)It,O
llH&(u')- H(u)JIL-((O,To);H-l(R)) ' < C&"/2"1
+(l/Y))
(2.3.13)
holds for some positive constant C. If uc and u are assumed nonnegative and to satisfy the weak maximum principle (see Sections 2.4 and 5.2) IIUellL-(P)
(1
+ CO)IIUOIIL-(R),
llUIIL-(P)
< IIUOIIL-(*)7
O
then the additional estimate
]IuE
- UIlL2((0,To);L2(*)) ' < C&'/Y
(2.3.15)
holds. The constant C depends, in the first case, upon In(and T o(see (2.3.18)), and, in the second case, on )RI, To and (see (2.3.24)).
~Iu~II~~(~)
Proof: Setting u = H(u) and uE= HE(uC),we have, after subtraction of (1.2.21)from (2.3.12),multiplication by uE- u, and integration over 0, d -21 _ dl llUE - U l l A -
I(R)
= ( J ( u )- JE(u),
< +c&"'y'-'Ip&(u)
+ (J&(UE) - J,(U), UE
U E - U)LZ(R)
- ')L2(n)
- J(u)(1:2(n) + +&1-(1'7)
c - l (IuE-
U I ( ~ ~ ( ~ ) , (2.3.16)
2.3 Regularization in the Porous-Medium Equation
65
so that, for each t > 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 ${?llUE - UII&(n)
+ V-~IIH,(~ 'H(u~)II&)}, )
(2.3.19)
where the choice of q > 0 is at our disposal. In order to utilize the coercive properties of the second term in (2.3.19), we require the inequality (2.3.20)
=-
for y 2 0, z 2 0, y # z, and r 0, s > 0, r # s (see [l2], p. 85). The identifications r = 1, s = l/y, y = [H(u')]', z = [H(u)]' in (2.3.20), in conjunction with the assumed relations (2.3.14) and (1.2.14e) lead to, with K = (1 + c 0 y - y K Y p W ) - H(u)((cdJ)Y-lIlu 11 1 - ( l / r ) 0 Lyn) 2 y I ~ ( u &-) ~(u)I[max([~(u")]~, [H(u)]Y)]'-('/Y)
2 I[H(U"]Y
-
[H(U)]Yl 2
clue- UI,
(2.3.21)
where the latter inequality, with C = [c1yIy,follows from the fact that the composite function K ( t ) = [H(t)IY,
t20
(2.3.22a)
t 2 0.
(2.3.22b)
has derivative K', satisfying K'(t) 2 [c1yIy,
2. Regularizations and Pointwise Stability
66
If (2.3.21) is substituted into (2.3.19), we obtain, for uo # 0,
< +qIIU& - U1l:Zcn) + 41-‘c2JRle2’Y,
(2.3.23)
where we have used (2.3.8a). Multiplying (2.3.23) by for sufficiently small q (say, q = ( C / ( q ) ) ( c 2 y -Yl[uo[l~L,$~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)
2. Regularizations and Pointwise Stability
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 < i < i,, m
+C
g;lli(x)= fi(uci(x)ei
j# i j= 1
uf- l,j(x)ej),
x E R.
(2.5.8)
Now write the decomposition, for fixed k, i, N, as
R = R + u R - UR,,
where
R, R,
= {x E R:g$(x) =
> O},
(2.5.9a)
R - = {x E R:g&(x)< O},
{x E R:g&(x) = O}.
(2.5.9b)
Finally, introduce the notation
6 = {i:l < i
<m
and
Di > O}.
(2.5.10)
2. Regularizations and Pointwise Stability
70
Remark 2.5.2. Our analysis will proceed by showing that solutions uf of (2.5.1)-(2.5.4) necessarily satisfy uniform bounds in Lm(R;Rm)and satisfy (2.5.5). We shall do this in stages. The verification of (2.5.5) is complicated by the fact that the components of R, and R- need not have boundaries sufficiently regular for integration by parts to hold, thus necessitating component approximation. The correlative analysis makes fundamental use of (2.5.6). Proposition 2.5.1. Unique solutions of (2.5.1)-(2.5.4) exist in Lm(Q Rm), and satisfy the estimates (see (2.5.14) and (2.5.15)) IIULllLm(n)
< IIUO,illLm(n) + ( s u p l ~ l ) ~ O , 1 < i < io,
m
(2.5.11a)
m
C IIUj?IiIILm(fi) < CCiTo + (1 + Cz) iC= Iluo,ill~-(n)]exp(CzTo), (2.5.11b)
i= 1
1
for k = 1,. . . ,M ( N ) , and N 2 1. Moreover, for such k and N, u$
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)
2.5 Invariant Rectangles
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$(u[~ - ai)- dx 2 0 n o
(2.5.16)
Proof: Each component R, of R, and R- may be approximated from within, to any desired accuracy in the Hausdofl metric, by an open set R, ,with boundary sufficiently smooth to apply the integration-by-parts formula. This may be seen by taking a finite subcovei of balls of the compact boundary of R,, and smoothing the piecewise spherical boundary of the resulting finite cover. We may do this in such a way that an, n aR 3 aR, n an. In particular, the integration-by-parts formulae
(2.5.17a)
(2.5.17b)
2. Regularizations and Pointwise Stability
72
hold on the approximation domains a,. Note that, for R, a component of R- and for d(R,, a,) sufficiently small, it follows, from the restriction of the closed zero set off;. to the open region strictly between QOiand Qbi(see (2.5.6)), that on aR,\aR, n 8R.
(u$ - 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 < i < i,. The proof now proceeds as follows. By (1.3.6~)and (2.5.6), we conclude that uEi 2 ai holds almost everywhere in R or else meas(R -) > 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)
2.5 Invariant Rectangles
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
BIBLIOGRAPHICAL REMARKS
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)
3. Nonlinear Elliptic Equations and Inequalities
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 $ri is defined, in analogy with (3.1.9a), by (3.1.16b) then we suppose IIuroll < IIurill,i and (3.1f16a),
=
1, 2,. . . . Then we obtain, from (3.1.7b)
(3.1.17)
3.1 General Operator Results
where
81
and (T are defined by (3.1.10).Since (T E Range o,we have llUlill
d supw-'(a)
= c,
i 2 0,
(3.1.18)
so that {uri} c B(0,c) c S, and is, thus, a relatively compact sequence, with accumulation point u. Upon taking the limit supremum in (3.1.16a), we obtain (3.1.8) for u E B(0, c). If the sequences {ri} and {uri} cannot be chosen so that Iluri(l2 IIuroll,all i, then a sequence {uri}necessarily is contained in B(0, [/ur,,~~). An identical argument, as above, in conjunction with (3.1.18), yields (3.1.8). W Remark 3.1.3. The theory developed thus far gives sufficient conditions for the duality inequality (3.1.8) to possess a solution in a finite-dimensional space S. The formulation permits various choices of 4 to obtain, for example, variational inequalities and equations. Also covered is the important case of multivalued subdifferentials of convex functions. We briefly introduce some terminology to discuss this case. Definition 3.1.3. Let X be a normed linear space with dual X*, and let F : X + (- 03, a] be a proper, convex, lower-semicontinuous function. By the subdifferential aF, defined for u, such that F(u) is finite, is meant the set-valued function dF(u) = {f E X * : F ( o ) - F(u) 2 ( f , v - u )
for all u E X}. (3.1.19)
If X is a Hilbert space, it is common to identify the functionals of dF(u) with the set of representers or subgradients.
Remark 3.1.4. dF(u) is a weak-* closed, convex, but possibly empty, set for a given u E X. Suppose now that u is a solution of(3.1.8).Thenf - A(u) E &$(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
3. Nonlinear Elliptic Equations and Inequalities
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)
3.2 General Operator Results
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 ,< p . Proof: Since B(0,p) is bounded, the existence of a solution u of the variational inequality (A(u),u - u> 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.
3. Nonlinear Elliptic Equations and Inequalities
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.
3.1 General Operator Results
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)
3. Nonlinear Elliptic Equations and Inequalities
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
forall
U E ~ } .
(3.1.44a) (3.1.44b)
Then uo E W and uo E 9.We claim that W and 9 are inductive. Let ‘3 be a chain in W .Then, since the interval I = {w E E:uo < w < u o } is inductive, we may select a greatest lower bound a E I for V. We shall show that a E W . Indeed, a < c for all c E W so that, since A is increasing and c E “Y-, A(a) < A(c) < c. This shows that A(a) is a lower bound in I for ‘3 and, since the greatest lower bound is comparable with all lower bounds in I, we have A(a) < a, that is, a E V . Now let u E 42. Since c 2 u for all c E V, we deduce that u is a lower bound for V and, hence, a 2 u. Thus, a E W . The proofs for the other three cases are similar, and we conclude that W and 9l are inductive. It now follows from Zorn’s lemma (see Kelley [17] p. 33) that W (respectively 9 )possesses a minimal element ii (respectively maximal element u). These may be shown to satisfy the statement of the proposition. For example, ii 2 u for u E 42, so that, since A is increasing, A@) 2 A(u) 2 u
and, thus, A@) E W ;the inequality A@) < ii
(3.1.45)
3.2 Applications and Examples
89
is immediate from U E V .Thus, ti is a fixed point, since ti is of minimal type. Similar arguments show that ~fis a fixed point. Maximality and minimality are immediate. W
3.2
APPLICATIONS AND EXAMPLES A Dirichlet Problem in Semiconductor Modeling
Example 3.2.1.
The boundary-value problem -Au+e"-eP-"=q
in R,
(3.2.la)
on aR,
u=O
(3.2.1b)
where p is a nonnegative constant, arises in the study of semiconductors. We shall identify the weak form of (3.2.1) with the equation
4'b;9) = (4, g)LZ(n),
for all 9 E H W ) ,
(3.2.2)
for an appropriate proper, convex, and lower-semicontinuous function 4:L2(R)-,(- co,001, where &(u; -) : = d4(u). Here, $'(u;g) will be computed by the usual Ggteaux differentiation formula for g E L"(R) n to have the value given by (3.2.3) By a density argument, it can then be shown that @(u;- ) has the form given by (3.2.3) on all of Hh(R), so that a+(u) is indeed a singleton. Thus, set F(s) =
J:
e"do = 8 - 1,
G ( t ) = ep J i ( - e - ' ) d r
-co
= @(e-' -
-= s < co, l),
(3.2.4a)
- 00 < t c co, (3.2.4b)
and note that F and G are convex functions, satisfying F(0) = G(0) = 0. We define 4 by W )= a, f E L~(R)\H;(R), (3.2.5a)
4(f)= 00,
f E H@),
if Jn F ( f ( x ) ) d x= 00
or if JnG(f(x))dx
$(f)= 3
jn IVfl'
dx
+ Jn[ F ( f ( x ) )+ G(f(x))] dx,
(3.2.5b)
= 00,
otherwise. (3.2.5~)
3. Nonlinear Elliptic Equations and Inequalities
90
The function 4 is clearly proper and convex, 4(0)= 0, and satisfies the coerciveness condition (3.2.6) since for some C > 0 {Jn lvu12}1’2 2
F(s) 2 We shall show that that
CIML2(*)
- 1,
for u E W Q ) , G ( t )2 - e p .
(3.2.7a) (3.2.7b)
4 is lower semicontinuous. In particular, we must show B = {f E LZ(Q):4(f)< u } is closed.
Vu E R’,
(3.2.8)
Thus, iffk -+ f , f $ ( f k ) < a, we have by (3.2.5~)and (3.2.7b)that { f k } is bounded in HA@) and, hence, is weakly convergent (the entire sequence) in H&?) to f . We have lim inf k- m
Jn lvfk123 JQ
1vjl2,
(3.2.9)
and, by Fatou’s lemma (see (3.2.7)), lim inf k-
00
Jn
F ( f k ( x ) ) d x3 Jn lim inf F ( f k ( x )dx ) k- m
(3.2.10) via the lower semicontinuity of F , and the fact that lim inf,,, Similarly, lim inf Jn G ( f k ( X )dx ) 2 Jn G ( f ( x ) d) x . k-
00
Thus, by (3.2.9), (3.2.10), and (3.2.11),
= lim inf k- w
< a,
4(fk)
fk(x) = f ( x ) . (3.2.11)
3.2 Applications and Examples
91
which proves that B is closed. Corollary 3.1.7 is now applicable with X = Xo = L2(R), and A = 0. Note that (3.1.32) and the condition on IIqIIL2(n, follow from (3.2.6) and (3.2.7), with o a suitable affine function. We, thus, obtain a function u, such that q E d$(u), where q has been identified with its functional. Now select g E L“(R) n HA(R). By computing the standard difference quotients, for t > 0 and t < 0, defined by t - ‘ [ 4 ( u + tg) - 4 ( u ) ] , and letting t -,0, we see that (3.2.12) for every x E 84(u). Standard arguments show that eu and e-” are in L2(R) (cf. Example 3.2.2), so that (3.2.10) holds for all g E H;(R) by a standard density argument. This shows that d4(u) is a singleton, so that the definition @(u; .) := @(u) and (3.2.3) are valid. As noted above, the existence theorem contained in Corollary 3.1.7 guarantees (3.2.2). In fact, this example is a special case of the following.
Example 3.2.2.
Maximal Monotone Mappings and Perturbations
Let gr(a) be a maximal monotone graph in R’ x ( - co,m),+induced by a multivalued function We suppose that 0 is an interior point of the domain of p. If we define p : LZ(R)-, 2L2(n)by
a.
f~ p(u) if f , u E L2(R), and f ( x ) E p(u(x)) almost everywhere in R,
(3.2.13)
then the operator p is maximal monotone. We note that p may be realized as the subdifferential of a proper, convex, lower-semicontinuous functional. Indeed, if t is the unique convex function on (- oo,co], such that t ( 0 )= 0 and a t = (more precisely, the representers of d t ( t ) coincide with set
a(?)),
Yb) =
{k,t(u(x))dx,
if t 0 u E L’(R), otherwise.
(3.2.14)
Since t is convex and lower semicontinuous, t possesses an affine minorant, permitting the application of Fatou’s lemma to the functional y. In particular,
’ Note that we do not permit
to assume infinite values.
3. Nonlinear Elliptic Equations and Inequalities
92
y is lower semicontinuous. Moreover, gr@) c gr(dy) (More precisely, gr@)
is contained in the graph of the representers of dy). Since g is maximal monotone and d y is (maximal) monotone, 21 = d y , where we have identified functionals and subgradients. In summary, every maximal monotone graph induces an associated (multivalued) mapping on L2(Q), which is the subdifferential of a proper, convex, lower-semicontinuous function. As an illustration, consider the function H , arising in the Stefan problem (cf. Section 1.1).To obtain a maximal monotone graph, set (3.2.15) In this case, 8 is the primitive of H vanishing at 0. Iff: R' + R' is a continuous monotone function with primitive F vanishing at 0, then /3 may be taken equal to f,and 8 may be taken equal to F. Similar remarks apply to sums f + H,etc. To continue the development of the example, let L be a linear elliptic operator, defined in a weak sense on a closed linear subspace G of H'(Q), with range in G*. We shall suppose that L is an isomorphism, defined by a continuous, coercive bilinear form B( ., .) that is, for all u, u E G,
(Lu, u ) = B(u,u),
(3.2.16)
where B is a bilinear form satisfying for appropriate constants (3.2.17a) (3.2.17b) Special choices of L include the inverses of the operators No, Do, and R, of Sections 1.1, 1.2 and 1.3, where B ( . , - )is symmetric and C = 0. Now L is pseudomonotone. In fact L is continuous, and hence, bounded, and also weakly continuous. Now suppose ui
-
u
(weakly in G),
lim sup (Lui, ui - u ) G 0.
(3.2.18)
(in G),
(3.2.19)
i+ m
We shall show that ui + u
which clearly implies (Lu,u - u ) Now, ui
-
u
=
lim (Lui,ui - u ) .
i+ m
(weakly in G) = Lui -*Lu
(weak-* in G*).
(3.2.20) (3.2.21)
3.2 Applications and Examples
93
Moreover, by the compactness of the embedding G -+ LZ(Q),we conclude that ui -+ u (in I,'@)).
(3.2.22)
Thus, using this convergence relation, and (3.2.17b) and (3.2.21), we obtain 0 6 c lim inf llui - ull; 6 lim inf B(ui - u, ui - u ) i-+ m
i-. m
= lim inf (Lui - Lu, ui - u ) = lim inf (Lui, ui - u ) , i-. m
i+ m
so that lim inf (Lui, ui - u ) 2 0. i-m
(3.2.23)
Coupled with (3.2.18), this implies that lim (LUi, ui - u ) = 0,
i+ m
(3.2.24)
which implies (3.2.19), via (3.2.17b),(3.2.21) and (3.2.22). We have shown that L is pseudomonotone. In fact, similar arguments show that the operator L + f is pseudomonotone, i f f is any continuous, bounded operator from L2(Q) into LZ(Q).In particular, this includes the operator f induced by a Lipschitz continuous function r or an appropriate integral transform appearing in certain integrodifferential operator expressions. Suppose that /? = 88 is a maximal monotone function, 8(0) = 0, that y is defined by (3.2.14) and that
where C is given by (3.2.17b)and C1 2 0. We now draw a final conclusion for this case in a summarizing proposition.
Proposition 3.2.1. Let /? be a multivalued function defining a maximal monotone operator P:LZ(Q)-+ 2LZ(R), via (3.2.13), such that 0 E fi(0). Let G be a closed subspace of H'(R), and suppose L: G -+ G* is an isomorphism, satisfying (3.2.16)and (3.2.17).Let f be a bounded continuous transformation on Lz(R),satisfying (3.2.25),where = a y and y(0) = 0. Then, for any f E G*, there exists a function u E G , satisfying f(u) E LZ(Q),P(u) c LZ(Q),and the relation
f E Lu + i(u) + P(U),
(3.2.26)
3. Nonlinear Elliptic Equations and Inequalities
94
where we have identified T(u) + B(u) with the induced linear functionals. Moreover, there is an element fl E g(u), such that the relation (Lu, u )
+ (T(u) + fl, u)L2(n)= ( f , u ) ,
is valid. The solution is unique if (L(u - w),u - w)
for all v E G ,
+ (f(v) - T(w), u -
(3.2.27)
W)L2(R)
+ (PCu) - B(w), u - W)L2(R) 2 811w -
~11Z2(Q),
(3.2.28)
for some 6 > 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)
3. Nonlinear Elliptic Equations and Inequalities
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)
3. Nonlinear Elliptic Equations and Inequalities
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.
3. Nonlinear Elliptic Equations and Inequalities
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 -
3.3 Semidiscretizations Defined by Quadrature
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
<
!I
< t k (0 < 6 < 1). (3.3.14a)
If Q k ( U)
= [eu(tk)
+ (1 - e)u(tk-
l)](tk
- lk-
11,
(3.3.14b)
3. Nonlinear Elliptic Equations and lnequalities
102
then Y k , k = 6, Y j , k = 0, j # k - 1,j # k. (3.3.14~) 1 - 6, The choice 6 = 3 leads to the Crank-Nicolson method when applied to a j , k , if the latter are defined by (3.3.14~).
Y k- l,k
=
Example 3.3.2.
The semidiscretizations discussed in the previous example were induced by quadrature rules defined by the exact integration of step function approximations. If, instead, piecewise linear interpolation is employed to define the approximation ij of u, given by v(t) = (tk
- tk-
1)-'[(t
-
t k - 1)U(tk)
+
(tk
-
t)u(tk- l)],
tk- 1
< t < tk,
(3.3.15a)
then (3.3.11) yields Qk(U)
+
=t ( t k - tk- l)[u(tk- 1)
tk- 1
V(tk)],
< t < tk.
(3.3.15b)
This is precisely (3.3.14b) with 6 = f, leading once again to the CrankNicolson method. This suggests higher-order approximation properties of this method than might at first be apparent. The explicit Adams extrapolation method is obtained by employing a variant of (3.3.15b),in which Q k f l ( u )is defined by exact integration of the extension of the linear interpolant of v(t& 1 ) and u ( t k ) , that is, Qk+l(U)
:=
J:"
= $ctk
{(lk
- 'k-
- tk-l)u(tk)
- tk-l)-l[(t l)-l{(tk+
- (tk - tk+
1
1)2u(tk-
-
tk)(tk+ 1
+ (tk - t ) u ( t k - l ) ] } d t
f t k - 2tk- l)u(tk)
l)}.
(3.3.16 )
If (3.3.16) is employed for k = 1,. . . ,M - 1, then it must be used in conjunction with a quadrature rule Q 1 on [ t o , t l ] . Example 3.3.3.
Let y o , . . . ,y k be arbitrary real numbers. Then there is a unique natural cubic spline function s, defined by the properties: d4)(t) = 0,
s(ti) = yi,
ti-
< t < t i , i = 1, . . . , k,
i = 0,. . . , k,
(3.3.17a) (3.3.17b)
3.3 Sernidiscretizations Defined by Quadrature
103
(3.3.17~) (3.3.17d) If y i = hi,, then the corresponding spline s j is called a cardinal spline. We may uniquely express the general spline s as k
s=
1 yjsj.
(3.3.18a)
j= 0
Now given u, we may define
and &(U)
:=
Jt*_,
1 {U(tj) J1’” k
p ( f ) d f=
k-1
Sj(f)df
j=O
Although formula (3.3.18)would suggest that Qk(u)depends upon u(to), . . . , u ( t k ) , in reality only U ( t k - 3), u ( t k - z ) , U ( t & I), and u ( t k ) appear explicitly. This is due to local basis representations in terms of B-splines.
Remark 3.3.1. Various other rules could obviously be constructed.+ The formulae of Newton-Cotes type are those which utilize polynomial interpolation. Although spline interpolation appears attractive because of associated variational principles, there are, in fact, questions of stability for higher-order smooth spline interpolation, even in the case of initial-value problems for ordinary differential equations (see Bohmer [ 5 ] for discussion and references).Splines of Hermite type are generally free of such instabilities, though they are unsuitable in pristine form for partial differential equations, since (time)derivatives would be required at nodal points t i . The replacement of the derivatives by difference quotients is, of course, a possibility. We shall not pursue any of this, however.
’,
Remark 3.3.2. The interval of integration in (3.3.5)was chosen as [ t k - t k ] . One could easily choose an interval of the form [ t k - i , t k ] , which would lead to the replacement of ( t k - f k - l ) - ’ by ( t k - f k - i ) - ’ , and H ( u k - 1 ) by H ( u k - i ) in (3.3.9).In the event that the partition is uniform, one can achieve the same result by successive elimination of H ( u k - 1), H ( u k - z ) , etc. This is sometimes
’
This would include the implicit Runge-Kutta methods, not discussed here. The author is indebted to Kenneth Jackson for this observation.
3. Nonlinear Elliptic Equations and Inequalities
104
desirable in the use of spline quadrature rules, such as that for rules of Milne-Simpson type. For example, if i = 2, and one defines D on [ t k - 2 , t k ] by the unique quadratic interpolant of ~ ( t ~ -~ ~( t) ~, . and - ~ ) u(tk), , then, for ( t k - t k - ') = ( t k - 1 - t k - 2 ) = At, one obtains the standard formula Qk(U)
:= L:-2n(t)dt
At 3
+
=-[U(tk-2)
4U(tk-1)
+
U(tk)].
(3.3.19)
Remark 3.3.3. We shall note the application of Propositions 3.2.1 and 3.2.2 to the semidiscrete equations (3.3.9) for the models introduced in Chapter 1. For the applications of Sections 1.1, 1.2, and 1.3, we set L = c t k , k N ; l , L = a,$, and L = ', respectively, in (3.2.26), where n > 0 is required to satisfy only AT, > c t k , p . Each of these operators satisfies (3.2.17), with C = 0, provided ctk,k > 0. In the first case of the Stefan problem, we have - C l k , k A = L - c t k , k a I , SO that the function choices
',
'
+
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)
3.3 Semidiscretizations Defined by Quadrature
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
3. Nonlinear Elliptic Equations and Inequalities
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)
3.3 Semidiscretizations Defined by Quadrature
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)
3. Nonlinear Elliptic Equations and lnequalities
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
BIBLIOGRAPHICAL REMARKS
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. Nonlinear Elliptic Equations and lnequalities
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$(B) has integral value unity and is fixed, x 2 0.
Proposition 4.1.l.+ Suppose that f' > 0 is a specified integer, that k(x,y) =
jol s - " - ' ~ ( x+ s-'(y - x))ds,
x,y E R", x # y, (4.1.1a)
and for a a multi-integer, la( = 8, that the kernels k, are given by
x, y E R".
k,(x, Y) = (l/a!)(x - yYk(x,Y),
(4.1.1b)
Then k(x,.)is a continuous function on R"\{x}, is supported in the convex hull K, of {x} u support(X),and satisfies the estimate, for y E R",
(4x9Y)( G
((X((Lm(&(X
- ~(-"/n,
x E 0, x # Y.
(4.1.2)
In particular, k,(x;) is supported in K,, and
(ka(x,Y)J G Cf'(IXIILm(B)d"lX - ~ l - " + ' ~ l / n , x E R, x # y,
(4.1.3)
for a constant C depending only on n. When f E Cm(R), then a natural projection onto the polynomials of total degree less than f' on R is induced by the representation of Sobolev type
f(x) = PGf(X) + RGfb),
(4.1.4a)
where (4.1.4b) is a polynomial of degree less than 8, and (4.1.4c) In particular, the definition of Pdf depends only upon J B .
' Propositions (4.1.1-4.1.3)
and Corollary (4.1.4) reproduced with modifications from [13].
4. Numerical Optimality and Degenerate Equations
116
Proof: We note that the semi-infinite ray {x + s-l(z - x):O < s < l}, for z # x, intersects the support of x if and only if z is in K,. It follows that {z:k(x,z) > 0, x # z} c K, and, hence, by a simple closure argument, that support(k(x,.))c K,. To verify (4.1.2) we have, for Iy - XI < d, y # x, Ik(x,y)l =
;JI
X(X
+ s-'(y
- x))s-"-' ds
from which (4.1.2) follows for Iy - XI < d. Ofcourse, k(x, y) = 0 for Jy- XI 2 d. To verify (4.1.3)we , let C denote an equivalence constant, for which
(4.1.5) I(x - Y Y ~< IIx - Ylltm(an) la1 G C~X - yllal = Cllx and then use (4.1.2). Note that 1.1 has been used as a Euclidean norm on R' ~ll$'(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)
4.1 Representations of Sobolev-Type and Upper-Bound Estimates
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
(4.1.16)
Note that P, is a linear projection operator. In fact, the idempotent property of P, follows from the representation (4.1.4),since R, = 0 iff is a polynomial of degree less than e.
Proposition 4.1.2.
For f E Q'(R), the commutation result (4.1.17)
holds for la1 c e, for every positive integer 8.
118
4. Numerical Optimality and Degenerate Equations
Proof: It suffices to prove (4.1.17)forf E Cm(R),since this set is dense in 9'(R), and the composed operations in (4.1.17)are continuous on Q'(0). For f E C"(R), x E R, and y E B, set (4.1.18) the Taylor polynomial off. A proof by induction on t' shows that
Thus, differentiating the expression P6f(X)
=
jBX(y)TEf(x) dy
under the integral sign, and making use of (4.1.19),we are led to (4.1.17).
Definition 4.1.1. Let s be a real number, satisfying 0 < s < n. By the Riesz potentials are meant the convolutions I,f(x)
= jnlx
- YI-"+"fY)dY,
x E 0,
(4.1.20)
when the latter converge absolutely for almost every x.
Remark 4.1.2. The convolution (4.1.20) represents an adaptation of the classical Riesz potential, which is typically defined on all of R" with a constant multiplier to reflect the Fourier transform of a fractional power of the Laplacian. A fundamental question is whether the mapping f -,1,f is bounded from LP(R)to L4(R).This is discussed in the following.
Proposition 4.1.3.
For 0 < s < n, 1
< p < 00, 1 < q < 00
and (4.1.21)
the mapping I,:Lp(R)-P L4(R) is bounded if at least one of the two numbers q1 = (l/q) - (l/p) + (s/n) and q 2 = (1 - l/p)(l/q) is positive. For s 2 n, I, is uniformly bounded for all p and q.
Proof: Set (4.1.22a)
Representations of Sobolev-Type and Upper-Bound Estimates
4.1
119
and note the convolution identity iff is the extension off which is zero outside R. We consider first the most delicate case,+namely, (4.1.23)
=-
which corresponds to q 1 = 0, q2 0,O < s < n, and show that I, is bounded from LP(R)to L4(R).Construct the ray, in the (l/p), (l/q) plane, from (1, (l/q*)) to ((l/p),(l/q)), where l/q, = 1 - (s/n).If this ray of slope one is extended to a first quadrant point ((l/pl),(l/ql)),such that ((l/p),(l/q))is an interior point of the ray, then the Marcinkiewicz interpolation theorem (see Stein [38] p. 272) will yield the desired result, provided I, is of weak type (1, q*), (pl, ql). We are thus led to prove the following statement. The mapping I, is of weak type (P, 4) for (4.1.24) Thus, we show, in this case, that there exist positive constants AP,4,such that
I > 0, for all f
E
(4.1.25)
Lp(R).Clearly, if (4.1.25)holds for (If(lLP(R) = 1, it holds for general
f , upon division by (If)ILp(n), when f # 0. Similarly, an adjustment of Ap,q permits the replacement of I by 21 on the left-hand side of (4.1.25).Given ,u > 0, we introduce the kernels (4.1.26a) and (4.1.26b) and note that K, = Ks,p,l
+ K s , p , wFor . I > 0, define (4.1.27)
where (l/p') + (l/p) = 1, and on-1 is the measure of the unit sphere in R". As we shall see below, Ks,p,lE L1(R"),and Ks,p,wE Lp'(R"). Now, by the Proof adapted from [38], p. 119-121.
4. Numerical Optimality and Degenerate Equations
120
triangle inequality, mix E R:
IK,
* f(x)l>
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<s
(4.1.30)
4.1 Representations of Sobolev-Type and Upper-Bound Estimates
121
follows by a routine application of Young's inequality: IIIsfllLq(n,
= OKs
-
* f l I L W < IIKsllL~cna~)llfIILP(R),
(4.1.3la)
for 1 1 1 -=1--+->1--. r P 4
In this case, n
S
(4.1.31b)
n
+ (s - n)r > 0, and
(4.1.32) In the case s
n, Ks is bounded, so that
1IIsfl ILm(l-2)G dS- "[IfII L W , and the boundedness of Is for all p and 4, follows from Holder's inequality. Corollary 4.1.4. Let rn and d be integers, such that 0 G rn < 8, and suppose 1 < p < co, 1 < 4 < 00. Suppose that
_1 - - 1 +-t'-rn 2 0, 4
P
(4.1.33)
n
and, if equality holds in (4.1.33), that (1 - (l/p))(l/q) # 0. Then there are constants C(n,d , d, x, m,p, 4), such that
(If - P d f l ( W m . q ( n ) < a n , d,
4 x 9
112,
P, q ) ( f I w w n ) ,
(4.1.34)
for all f E W*p(n). Proof:
(&r(f
Fix p,
(PI < rn. By the commutation property of P,,
we have
- Pdf) = f@) - Pd-lBlf(B) = Rd-IBlf(B),
for f
E
Wd*P(n). Thus, by (4.1.3)and (4.1,4),
4. Numerical Optimality a n d Degenerate Equations
122
Is[,
and Proposition 4.1.3 applies, with s = t to yield (4.1.34), upon sum< m.Here, we use the inequality, for 1 < p < co,and Cp,n= mation over n p - (see Beckenbach and Bellman [S]),
Remark4.1.3. The conclusion of the previous corollary holds for more general domains than star-shaped domains, in particular for the connected, finite union of such domains Rj. Somewhat surprisingly, the polynomial Pdf may be defined with respect to any one of the balls B j . The extended corollary may, thus, be applied to connected bounded domains satisfying the restricted-cone condition, but also to other domains, such as slit domains. The proof of this fact is very easily demonstrated in the case R = R, u R,, S Z , n R, # Thus, setting pi = Pd,if,i = 1,2, we have
Ilf
a.
- PlJ(Wm*.7(n)< I(f- P111Wm.4(RI) + Ilf - P211Wm.'l(n2) + IlP1 - PZIlWm..7(R2)-
(4.1.36)
If the equivalence of norms on the finite-dimensionalspace Pdof polynomials of degree less than t is used in (4.1.36), with respect to l(*(lwm.q(n2) and 11 .11Wm.4(nl,n2), followed by the triangle inequality, we obtain
Ilf
- PlIIWm*4(*) G Clllf
- PlJIWm..7(RI) + Czllf - P211Wm*~(n2)
< C;lflWC.P(RI) + C;lflW(.P(R*) < (c;+ C ; ) ) f I W ~ . P ( * ) . A parallel argument holds for approximation by p , .
Remark 4.1.4. An immediate consequence of Corollary 4.1.4 is the equivalence of the norm Ilflld,p
= lllpdflll
+ IflW'.P(*)
(4.1.37)
with the standard Sobolev norm for 1 < p < 00 and t 2 1, where lllflll is any norm on Pd.Indeed, we have the one-sided estimate
< I I P d f l l W ~ * P ( R+) Ilf
((f(IWL.P(R)
G G
- PdfllW~.P(R)
+ [If - PdfllW"-'.P(R) + IflW"P(R) ~lllpdfll+ l C Z I f l W ~ * P ( R )+ I f l W W R ) CllllPdflll
< Cllf Ild,',
4.1 Representations of Sobolev-Type and Upper-Bound Estimates
123
in conjunction with the trivial estimate Ilfll8.p
G CllPGfIlw~-I.P(n)+ lflw"P(n) G CCl(lf(lWC-',P(n)+ I f l W W 2 ) c 2
IIf lIwc.P(n)
3
where we have used the continuity of P, as a linear mapping of W d - l * p ( Q ) onto 9, c W- ' * p ( Q ) .
Remark 4.1.5. Another consequence of Corollary 4.1.4is the inequality, for 1 < p < 00 and t 2 1,
If I W W R ) < I I Cf II I W w n ) , 5 5
I.Iwr,Pln)
G
cI f l w w n )
9
(4.1.38)
which shows the equivalence of and the standard norm on the quotient space WG,P(Q)/9,. The first inequality in (4.1.38) is trivial, whereas the second uses the estimate inf
PEPC
Ilf
-P
I I ~ ~ . ~ (<~ )\ I f
- PGfllWt.p(n) G ClfJW~.p(n)-(4.1.39)
Inequality (4.1.38) is the basis of the well-known Bramble-Hilbert lemma (see Bramble and Hilbert [7] p. 114),which derives the estimate JF(4
(4.1.40)
CI4W~.P(*),
for continuous linear functionals F on Wd*p(Q) which annihilate functions in 9,. Bramble and Hilbert did not claim (4.1.40)in the case p = 1, since their fundamental Lemma 2 ([7] p. 114)requires the weak compactness of the unit ball in Wd*p(Q). This point has been emphasized by Shapiro [37]. However, the approach sketched above yields (4.1.40) for p = 1.
Remark4.1.6. It is of interest to inquire whether other polynomial projection operators B, may be employed in (4.1.37). Interesting examples would include projections defined by relations, such as Jn D " ( W - f ) = 0,
14< f,
(4.1.41a)
c t.
(4.1.41b)
and Jn xa(~,f - f ) = 0,
la1
The key estimate permitting an affirmativeanswer, provided P, is continuous is from W d -' * P ( Q ) onto P,,
Ilf
- PGfllwt.P(n) G ClflWl.P(n).
(4.1.42)
4 . Numerical Optimality and Degenerate Equations
124
=-
However, (4.1.42) is verified only for p 1 in Morrey ([25] p. 85), where the special operator defined in (4.1.41a) is employed, though the argument is completely general. In particular, if s = (";!;I) and (A,}; is a set of continuous linear functionals on W d -'*"(a), which are linearly independent over Pd,then (4.1.43)
.IIW~.P(R)
for 1 < p is equivalent to 11 of (4.1.37) if Pd is defined by
< co. Note that (4.1.43)has the structure
(4.1.44) i = 1, . . . ,s. @,f = A i f , We shall now present an application of Corollary 4.1.4 to obtain upper bounds in the estimation of piecewise polynomial approximation on the unit hypercube. Definition 4.1.2. Let Q = {x E R":O < lxil < 1, i = 1,. . . , n} denote the unit hypercube in W, and, for k 2 1 a fixed integer, let Q , denote one of the k" subcubes of Q , defined by h = l/k and the relation for 0 < cci write
Q , = {x E Q : a i h < x i < (ai + l)h,
< k - 1, i = 1, . . . , n. Set x, x = za(Y) = X a
i = 1,. . . , n},
. . . , XJ
= (xal,
+ hy,
= (cc,h,
Y E Q,
(4.1.45)
. . . , cc,h), and (4.1.46a)
so that Qa
+ hQ = za(Q)*
= Xa
(4.1.46b)
Finally, for u E Wd*.(Q),define uau = u E WGsp(Qa) by u = 0,
Now, for t 2 1, 1 < p d co, 1 < q
0
u := u
< 00,
0
7,'.
(4.1.47)
such that
_l - _ l + -t 2 0 , q
p
n
and l i e if - - -+ - = 0 , q p n
(1-:)(:)>0 consider the idempotent operator
P:W d p p ( Q+) L4(Q),
(4.1.48a)
4.1 Representations of Sobolev-Type and Upper-Bound Estimates
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~ <
l}.
(4.2.16)
4.
130
Numerical Optimality and Degenerate Equations
Here Y is a normed linear space, continuously, though not necessarily densely, embedded in X. For subspaces Y, of Y, it is trivial that B, = BY n Y, satisfies the hypotheses of Proposition 4.2.3, although By,and even C1, By, need not have an interior in X. Since B , c U , : = { y ~ Y : l l y l l Y < l}cCl,B,,
(4.2.17)
dN(UY,X) inherits the lower bound (4.2.3). Since U, is, in some sense, more natural than By,we shall formulate our statements in terms of this set. Moreover, if XN+ and Y N + l c Y coincide as sets, but retain the norm structure induced by X and Y,respectively, then K = U, n Y N + l satisfies the hypotheses of Proposition 4.2.4 and, hence, (4.2.13)holds.
Remark4.2.4. Note that, if Propositions 4.2.3 and 4.2.4 are applied to U, and KN = Uy n YN+ 1, then we conclude that the N-width of KNdoes not decrease when K N is approximated in X. For X = Y, this is the famous Gohberg-Krein theorem (see Chapter 1 [14] p. 137). We shall emphasize these remarks in a summarizing proposition. Proposition 4.2.5. Let X and Y be normed linear spaces with Y continuously embedded in X, and set U, = { y E Y:\Iy\l, < 1). Let Y, be any fixed subspace of Y of dimension r > 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.
4.2 Lower-Bound Estimates and N-Widths
131
4
J
fI I \
with t - ( n / p ) + (n/q) > 0, and t - n/2 > 0 if 2 < p < q, t - n / p > 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<$,
- s)),
$ < s < 1,
otherwise.
(4.2.19b)
4 . Numerical Optimality
132
and Degenerate Equations
Further, define the piecewise polynomial function g on R" of degree mz in each variable xl, . . . , x, by n
g(x) = and define the 2"k" splines g,
E
JJ b(xi), i= 1
(4.2.20a)
C"-'(Q) by
Finally, set Mm= linearspan(g,), and observe that Mmc W;yP(Q), 1 < p < a.
Remark 4.2.6. The assertion b E Cm-'(R1) is an easy consequence of the observation that
[(3+ I' , 0,
=
i=l, j=O, ( i , j ) E (0, l} x (0,
. . . ,m - l}\(l,O).
(4.2.21)
u = CC,J,, we Note that support(g,) c Q,; in particular, for any u E Am, have
Proposition 4.2.6. There exist positive constants C1, C, ,and C3, such that, and m 2 t , for u E dm
< c h"((l") - (1'4)) \Iu IhQ)? l u l IWL*p(Q)< C2h-dllUIILP(Q) < c31 luIIWf,P(Q)
1 1UI
(Lp(Q)
1
.
3
In fact, C1, C, , and C3 may be chosen explicitly as
4
P,
1
< 00.
(4.2.23a) (4.2.23b)
4.2 Lower-Bound Estimates and N-Widths
133
Proof: The verification of (4.2.23a) proceeds by using (4.2.22) as a starting point, followed by the inequality II{ca}llGP
< ll{ca}llt4,
4
< P,
(4.2.24)
followed once again by (4.2.22), with p replaced by q. In this case, C , = IIgIILP(Q)/(Ig(ILP(Q).Similarly, the /uIwj.p(Q)= h-j+(n'p)l ~ l W ~ ~ p(4 ( ~ ~ IlcpT l l when combined with (4.2.22),yields (uIWJ-P(Q)
(4.2.25)
= Cjh-jl(UIILP(Q),
with c . = 1glwi,p(Q)/~~gII,~(Q!. This yields the first inequality in (4.2.23b), with C, = I~gllw',p(Q)/1~g(ILp(Q), since h - j < h - 8 for j < t'. If (4.2.25) is employed with j = t', we obtain
< c i lhc(l~IIwt.p(Q),
IIuIILP(Q) = C, lhCI~Iwdsp(Q)
which yields the second inequality in (4.2.23b), with C3 = C,c;
W
Remark4.2.7. It is worth mentioning that the constants may be chosen independent of p and q in (4.2.23). Indeed, the choices C , = IIgIILm(Q)/((gIlLl(Q), G = )lg((Wl.,(Q)/l(gl(L1(Q), and ~3 = Ilg1(W'.m(Q)/(g(WL.l(Q), respectively, suffice, since Q is of unit measure. The previous proposition, in conjunction with Proposition 4.2.5, yields lower bounds for the N-widths of the set UcPp = {U E W'*"(Q):IJUIIW,,~(Q)
< l},
t' 2 1, 1 < p
< 00,
(4.2.26)
when embedded in LP(Q).We state this as the following proposition.
Proposition 4.2.7. Suppose 1 < p
< 00, and define the sequence
Nk=2"kn-1, Then, for N
E
k = l , 2 ,... .
(4.2.27a)
{ N,} and C = C; given in (4.2.23), dN(ud.p,LP(Q))2 C (2lk)l
.
(4.2.27b)
Proof: The embedding WGpp(Q)+ LP(Q) is continuous for 1 < p < 00, though it fails to be dense for p = 00. Thus, Proposition 4.2.5 applies with r = d i m A c = N, + 1. In particular,
dN(UCsP, LP(Q))2 inf
2 (C;'hC), (4.2.28)
4. Numerical Optirnality and Degenerate Equations
134
upon use of (4.2.23). If h = 1/(2k) is substituted into (4.2.28), we immediately obtain (4.2.27b). We pass to the case 1 < p G q < 2. We first prove an auxiliary result in Hilbert space which will serve as a pivoting result.
Proposition 4.2.8. Let H be a Hilbert space, and let V be an r-dimensional subspace of H,with r > 0, such that {$i}; is a complete orthonormal system in V. Suppose that Y is a normed linear space, embedded in H and containing the space V as a subspace. Then,
Proof: Let N be any N-dimensional subspace of H, and set Z linear span(V u N). Write N=N1@Jv;, Then Z = V @ 4,say dim Z = r normal sequence in Z, such that linearspan{4i}y = Jv;, Then, for 1 < j
=
NlCVL, N Z C V .
+ M. Let {4i};+M be a complete ortholinearspan{4i}z+l = Nz.
< r,
Summing on j , and using the Fourier projection theorem, we obtain r
r
-M
+ E & J Y , N 1 ) 2 j=11 11$j\13,
(4.2.30)
and (4.2.29) is immediate from (4.2.30).
Corollary 4.2.9. Suppose 1 < p < 2, G 2 1, and ((1/2) - (I/P) + (//n))2 0. Define the sequence N, = 2"-'k".
(4.2.31a)
4.2 Lower-Bound Estimates and N-Widths
135
Then, for N E {Nk} and C = C ; given in (4.2.23b),
(
2;y-n((1/p)
d N ( U G . P , L2(Q))2 2- 1/2C -
- (1/2))
(4.2.31b)
It follows that, for general N = 1,2, . . . , d N ( u / . p , L ~ ( Q )2) c , - U~ n + ((UP) -
(4.2.32)
for some positive constant Co.
Proof:
Define the normalizing factor ha, so that ea
=
IlealIL2(Q)
(4.2.33)
= 1*
In particular, for V = 4, {ea) is a complete orthonormal system in the sense of LZ(Q).By (4.2.23b),we have
;{
<
~~ea~~k~~P(Q)C } 12 h ~ -2' { F
(4.2.34)
~~ea~~h'(Q)}1'2
If the inequality
[Iea11WQ)< hnUUp)-(l/q)) IleallLq(Q),
is applied to (4.2.34),with q
= 2, we
1
< q < 00,
(4.2.35)
obtain
If (4.2.36)is combined with (4.2.29),there results (4.2.31b).Note that we have used the embedding WGvp(Q) + L2(Q).We now verify (4.2.35).Note that the homogeneity property of norms permits the verification of the equivalent statement
[ I g 1)
a WQ)
< p((l/P)-(l/q))
1IgallLq(Q),
< p < < 00.
(4*2*37)
Thus, IlgallLP(Q)
= hn/PIIg(ILP(Q)
< hn/Pllgl(LP(Q) = hn((l'p)-(l'q))Iball W Q )
3
and (4.2.37)is verified. It remains to verify (4.2.32).For given N 2 0, select k 2 0, such that
2"-'kn < N < 2"-'(k
+ 1)".
(4.2.38)
Then, by the second inequality in (4.2.38),
(4.2.39)
4.
136
for Nk+
= 2"-'(k
Numerical 0ptimaJity.and Degenerate Equations
+ l)", and, by the first inequality,
(A)
1 ((2N)'I" + 2)'
(4.2.40)
Combining (4.2.39)and (4.2.40)yields (4.2.32). This is the Hilbert space result, permitting pivoting, to which we referred. Note that the lower bounds (4.2.31),(4.2.32)may be strengthened to the case where Ud*Pis replaced by U$l, the closed unit ball in WC,.p(Q)(see Remark 4.2.8), since A'$ c W$P(Q). The following remark discusses the relationship between N-widths under isomorphism.
Remark4.2.8. Let J be an isomorphism from a normed linear space X onto a normed linear space Z. Then, E,(JK, Jd) < IIJIIEdK,A),
(4.2.41)
for any set K c X and any finite dimensional subspace d c X. If the restriction j of J, to a subspace W continuously embedded in X, is also an isomorphism onto a subspace V continuously embedded in Z, then
dN(UV,Z) = dN(J
0
J-'Uv,JX) < JJJJJdN(J-'UV,X)
< ~~JllM~~j-'IpJw,x) = ((JJI IIJ- 'JldN(U,,X).
(4.2.42)
The principal application we shall make of this result is the case J - ' = J, is the Bessel potential operator (see Adams et al. [3], Bergh and Lofstrom [6], and El Kolli [lS)), which establishes an isomorphism
J,: WrSq(Q)+ Wr-s*q(Q),
s 3 0,
1 < q < CO,
(4.2.43)
between Sobolev spaces. Here the definition of J, depends only on s, and, for m 3 0, WF*q(Q)denotes the subspace of Wm*q(Q)of functions extendible to R" as periodic functions of period one in each variable, such that the extended functions are in W;ldc4(Rn). For rn < 0, W;.¶(Q) = W'"*¶(Q)= [W&'",4'(Q)]*.The isomorphism is proved in [15], by use of the Mihlin multiplier theorem. The classical results on Bessel potential operators establish the corresponding isomorphism result on R" (see Adams et al. [3], and Bergh and Lofstrom [6]). We can now handle the case 1 < p < q < 2.
+
Proposition 4.2.10. Suppose that 1 < p < q < 2,t' 2 1, and((l/q) - (l/p) (L'/n))3 0. Then, there is a constant C , depending on L', n, p , and q, such that, for N , = v" and N E {NY},
d,(U[.P, L¶(Q))3 Cv- d + " ( (
I/P)
- (I/'?))-
(4.2.44)
4.2 Lower-Bound Estimates and N-Widths
137
It follows that, for some constant C,, and N = 1 , 2 , . . . , dN(U8*P, Lq(Q)) 2 CON-8/n+((l/P)- ( 1/q)).
(4.2.45)
Lemma 4.2.11.+ Let Y be a normed linear space continuously embedded in a normed linear space X, and let A be a subset of Y. Then, d~~ + N ~ (x) A ~< dN,(A,Y)dN,(Uy, x),
N1 2 0, N , 2 0, (4.2.46)
when these numbers are finite. Proof of Lemma 4.2.11:
Suppose 6, > 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<, E X )A M i The result follows by letting d i tend to dNi(
+ M,) < 616,. a).
Proof of Proposition 4.2.10: Choose an integer j > 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
4 . Numerical Optimality and Degenerate Equations
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)
4.2 Lower-Bound Estimates and N-Widths
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
O}]-'
# 0}
= inf{(AQu,Qu)/(Qu,Qu):Pu = 0, u # = inf{(QAQu,u)/(u,u):Pu = 0, u #
0}
O}.
According to the Weinstein-Aronszajn theory of intermediate problems for eigenvalues (see Weinstein and Stenger [43] Theorem 1, p. 28) and Stenger [39]), the last infimum is the (N 1)st eigenvalue of Q A Q (note that 0 is an eigenvalue of multiplicity at least N) and is attained by an eigenvector of QAQ; moreover,
+
inf{(QAQu,u)/(u,u ) :Pu = 0, u # 0} = inf{(Au,u)/(u,u):P u = 0, u # O}. We, thus, have, by the sup-inf characterization of eigenvalues, E(d,A)-'
E(d,AN)-2= A N + 1 ,
so that (4.2.5 1) follows and the theorem is proved.
Remark 4.2.10. Proposition 4.2.12 can also be proved by using results of Golomb [17] on the approximation of ellipsoids of general positive selfadjoint operators. The class Y is far richer than the set of inverses of compact self-adjoint operators, and includes, for example, the Schrodinger operator in R3 for the potential induced by the helium atom (see [43] pp. 96-101). Two applications, one in general Hilbert space and the other to the computation of dN(Ud*2, L2(0)),with explicit asymptotic constants, are given now.
4.
140
Numerical Optimality and Degenerate Equations
Corollary 4.2.13. Let H be a Hilbert space, and let V be a dense linear subspace of H, such that V is a Hilbert space under the inner product (u, 4, = [u, v3
+ (u, U)",
(4.2.52)
where [.;I is a nonnegative, Hermitian bilinear form defined on V x V. Let V, be a linear manifold dense in V, that is, Cl,V, = V, and let Bo = ( u ~ V , : [ u , u ] < l}. If B = { u ~ V : [ u , u ] < l}, then C 1 , B 0 = C 1 , W = ~ and, hence, c 1 H B 0 = ClHa. Moreover, there exists a unique positive selfadjoint operator A, with ClvDA = V, satisfying [u, u]
= (Au, u)H
for all
u E DA,
u E V,
(4.2.53)
with D A characterized by (4.2.53). In particular, if d denotes the ellipsoid determined by A, then
d ~ ( dH), = d , ( B , H),
N 2 0.
(4.2.54)
Proof: The operator A is the well-known Friedrichs' operator (see Riesz and Sz-Nagy [30] pp. 332-333), and (4.2.54) follows from the earlier closure results, upon identifying DAwith V, and d with B,. The closure statement, ClvBo = B, follows from normalizing any sequence convergent to a nonzero element of B, and the closure statement in H follows from c1H
= cl,(clV
BO)= c 1 H B o ,
where the last equality results from the continuity of the embedding of V into H. The N-widths are of course invariant under closure. We have the following sharp theorem concerning the L2 N-width of the unit ball of WdS2(R).Note the adjustment in the definition of the norm of W"'(R) in the following theorem to reflect the number of times a particular derivative appears in a tensor product reformulation (see List of Symbols/ Definitions). Theorem 4.2.14. Let R be a domain with C" boundary. Then
-
N
dN(UCv2,L2(R))cN-"",
---f
a,
(4.2.55a)
where c has the value c = (2n)-"p, meas(Q),
(4.2.55b)
and pn is the volume of the unit ball in R",given by Pn
+
= 7 ~ / 1 - ( ( ~ / 2 ) 1).
(4.2.55~)
4.2 Lower-Bound Estimates and N-Widths
141
Lemma 4.2.15. If the self-adjoint Friedrichs’ operator A in L2(a)is defined by (u, u)wc.2(n) = (Au, for all u E WG*2(L?), (4.2.56) then, DAc
W2‘p2
(a).
(4.2.57)
Here, t >, 1. Proof of Lemma 4.2.25:
Define self-adjoint operators Aj on L2(Q)by
((u,u))wj,z(n) = (Aju, v)Lqn),
for all
E wj32(Q),
forj = 1, . . . ,t, where the double bracket notation is defined by 2
((u,U))wJ*Z(n) = IUlWj.2(*).
An elementary argument shows that Cg(Q) t DAj, and
Aj4 = (- l)jAj4,
4 E C;(n).
It follows from this and the self-adjointness of Aj that for all
(Aju,+)L2(n) = (u,(- 1)jAj4),2,,),
4 E c;(a),
for u E DA.Thus, u is a weak solution of (- 1)jAh = Aju = f E L2(0),
and, hence (see Agmon [l] Theorem 11.10, p. 165), DAj c W2jp2 (a).It now follows easily that
A.=A{=(-l)jA’, J and, hence, DAj3 DA,+,,j = 1 , . operator
A=
j = 1 , . . . , t,
. . ,t - 1.
In particular, the self-adjoint
G
1 Aj + I,
j= 1
DA= DAC,
satisfies Di, t W2‘s2(a),
(4.2.58)
and also satisfies the relation (4.2.56)defining A. However, by the uniqueness in Wc92(i2)of A, it follows that A = A, and (4.2.57)is immediate from (4.2.58).
rn
Proof of Theorem 4.2.14: Setting V = WG.2(sZ),[.,*I= (-,.)wc,2(n)and H = L2(n),we see that Corollary 4.2.13 is applicable. In particular, the N-widths
142
4. Numerical Optimality and Degenerate Equations
of Uds2are given by Ii:i2, that is, in terms of the eigenvalues of the operator A satisfying (4.2.56).It is known (see Agmon [l] Theorem 14.6, p. 250) that the asymptotic distribution of the eigenvalues of such self-adjoint operators is given by
N ( I ) = cI"/2d+ O(A"/26),
I
-+
where N(A) is the number of eigenvalues < I and c = (2n)-"Jn w ( x ) d x ,
{
w ( x ) = meas c:O <
(4.2.59)
co,
(11
<;
Y
I
< 1 . (4.2.60)
Since N(Aj+1) = N(Aj)(l + O(1)) (see the proof of Lemma 3.2 of Jerome [20]), a routine argument shows that
I j = (j/c)Zd/"
+ o(j 2 / / " ),
j-+m.
(4.2.61)
Equations (4.2.51)and (4.2.61)immediately give (4.2.55a). Equation (4.2.55b) follows directly from (4.2.60).
4.3
CONVERGENCE RATES FOR THE CONTINUOUS GALERKIN METHOD
We shall describe briefly the linear elliptic theory of finite-element approximations, in a fairly general setting, prior to nonlinear applications to degenerate parabolic problems. Thus, let G be a Hilbert space with dual G* and suppose a(.,-) is a continuous, strictly coercive bilinear form defined on G x G. If T is the isomorphism of G* onto G, defined by
a(Tt,u)= ( t , u ) ,
for all u E G,
(4.3.1)
and guaranteed by the Lax-Milgram theorem, we seek approximations for u = T t and estimates of that approximation. The operator T is, of course, a generalization of the operators Do, N, (a 2 0), and R, of Chapter 1. We shall explicitly assume that a( *, is symmetric so that T is self-adjoint. 9 )
Definition 4.3.1. For m 2 1, consider a family G, of Hilbert spaces for s = -m , - m + 1, . . . ,2rn - 1, 2m, decreasing as functions of s with dense, continuous inclusions. We identify G with G,. These spaces are to have the property G: = G + ,
0
< m,
(4.3.2)
4.3 Rates
for the Continuous Galerkin Method
143
and the restriction mappings are required to satisfy T:Gk
Gk+zm,
-m
< k < 0,
(4.3.3)
and to be isomorphisms. Assume that {sh}h is a family of finite-dimensional subspaces of G,, and let Eh be the projection onto s h , defined by a(U
- EhU, V ) = 0,
for all
V
E Sh.
(4.3.4)
Here, h is a positive parameter related to dim Sh by a simple rational function. We say that Sh is of degree k - 1 for Eh, for some integer m c k < 2m,if sh C
Gk-1,
and if, for each s = 0,. . . , rn - 1, there exists a positive constant such that
(4.3.5a) Cm,k,s,
for u E G z , - , . The approximation order of u - Ehu in G,, s c m,is now described. Proposition 4.3.1.' Let f E G k - 2 mbe given, and suppose u E Gk satisfies u = Tf. Then, for s = 0,. . . , m - 1, there are constants Cm,k,s,such that
if S h is of degree k - 1 for Eh.
and I(g, u - Ehu) I = la(u l.4 - Ehu)l G c((u - Ehu((Grn((U - Eh'(lGrn*
(4.3.8)
Applying (4.3.5b), for the specified s, to I u - Ehu((G,,and applying the same inequality, with s = 2m - k, to IIu - EhuIG, we deduce, from (4.3.8),
' This is a generalized form of the Nitsche-Aubin
lemma.
144
4.
Numerical Optimality and Degenerate Equations
Using the isomorphism property of T, we obtain, from (4.3.9),
and (4.3.6) is immediate from (4.3.7) and (4.3.10).
Remark 4.3.1. It is often useful to have the parallel concept that s h is of degree k - 1 for P,, where P, is the orthogonal projection in G o . In this case we require, for s = 0,. . . , 2 m ,
for u E G2,-, as the appropriate modification of (4.3.5b). The result is the following. For - m < s < 0 and 0 < j < 2m, there exist constants Cm,j,k,s, such that
for u E G 2 m - jThe . proof is analogous, Notice that u E Go is essential for this result, since it is derived under the supposition that ( u , w ) = (u, wlc0 for u, w E G o ,that is, under the supposition that Go is a pivot space.
Remark 4.3.2. The primary application of Proposition 4.3.1 is to the case where G, = Hs((SZ)or G, = Hi@), and T is one of the operators No,R,, or Do. Here, the spaces Sh are typically C k - 2spaces of piecewise polynomials of degree at least k - 1, for k 2 2, defined by means of underlying meshes or triangulations of the domain. Although the reader might expect such trial spaces automatically to satisfy (4.3.5b), in the light of the results of Section 4.1, in fact, additional arguments are needed. This is due to the fact that EhUI, need not be the best approximation of uI, from ShI, in H"(e), for e an element of a given finite-element triangulation, although E,u is the best approximation, up to equivalent norms, from S h in Hm(Ue). The verification of (4.3.5b) in Sobolev space for finite-element spaces is achieved by the construction of an interpolation operator, or, in the case rn < (n/2), the composition of a smoothing operator and an interpolation operator (see Strang [40] and Strang and Fix [41] for complete details), and employs the notion of a nodal basis spanning s h . The argument does use, however, the decomposition u = Pk + R in each element e, where Pk is the polynomial of degree k - 1, defined, say, by the Sobolev representation theorem on e. The
4.3
Rates for the Continuous Galerkin Method
145
interpolation operator u --+ u, reproduces pk, so that the final estimation of u - u, can, somewhat crudely, estimate R and its interpolation polynomial R,, separately, on the element e. Of course, the estimation of R was already
carried out in Section 4.1 and yields (RIHye)< chk-mlulHk(e), for example, for u E H2"(SZ). The estimation of R, is a simple application of Sobolev's inequality, applied to terms such as DPR(zj),when RI is represented, via a nodal basis { $ j } , in terms of the values of R and its derivatives at nodal points z j E R; here, zj = zi' is possible only in the case where derivatives of R appear in the expansion of R,. For example, for u E H2"(O) and distinct z j , we have, under uniform basis assumptions on { $ j } ,
ID=R,(x)~
G
lUIHk(e),
= m, x
E
e9
which also gives IRIIHm(e) < Chk-m/t&k(e). These computations are described in Strang and Fix [41] p. 146.
Definition 4.3.2. Given three Hilbert spaces G - , , G o ,and G,, with m 2 1, G: = G - , , and G, c Go c G - , , where the latter embeddings are dense and continuous, and Gois a pivot space, let a( ., be a symmetric, continuous, positive definite bilinear form on G,. Denote by T the isomorphism of G: onto G,, defined by (4.3.1). We agree to identify a ( . ; ) notationally with the inner product on G,, and we select, for the inner product on G - , , a )
(f,g)c-, = (f9Tg).
(4.3.13)
Let S h be a finite-dimensional subspace of G , , where the dimension of s h is a simple rational function of h. We say that Sh satisfies the inverse hypothesis in G, if there exists a constant C, such that (4.3.14)
We denote by Ehthe orthogonal projection of G, onto sh, defined by (4.3.4, and, finally, we set Th = EhT.
Remark 4.3.3. The mappings Do,N, (a 2 0) and R,, introduced in Chapter 1, are examples of the mapping T. From the discussion of those examples, which carries over to the general case, we see that T is positive definite and self-adjoint on the pivot space G o ,and (4.3.13) determines a norm on G - , equivalent to the duality norm. Moreover, Th may be characterized by iff
(f,x> = (Thf, x)C, , for x S h, (4.3.15) is given in G - , . The mapping T, is self-adjoint and nonnegative definite
4. Numerical Optimality and Degenerate Equations
146
on G o ,and positive definite on s h . Note that the invertibility of T and the = lIuIIG_, permit the usual pivot space inequality identity IITu)IGm
Gm.
\ ~ ~ ~ \ G m ~ ~ ~ ~ U[ G , ~- mE ,
I(u,u)GoJ
(4.3.16)
(4.3.17) and extend this inner product on s h to a semi-inner product on G - , , by permitting 4, J/ in (4.3.17) to be arbitrary elements of G - , .
Remark4.3.4. If, as earlier, onto s h , then, for 4, $ E G o ,
(4,@)C-,.,,h
= (Th4,
Ph
$)Go
denotes the orthogonal projection in Go = (Th4,
Ph$)Go
= (ThPh4, Ph@)Co.
(4.3*18)
Note also the natural duality, 1(4,$)Gol
G
~ ~ ~ [ [ ~ - m , h \ ~ @4 9~$ ~ G ‘ m h, ~
(4.3.19)
which follows, since ThlSh is invertible.
Proposition 4.3.2.+ The inequality IIxllG-m.h G l \ x \ l G - m -
x
for
sh,
(4.3.20a)
is valid. If S h is of degree 2m - 1 for E h , and if the inverse hypothesis (4.3.14) holds, then there is a constant C, not depending on h, such that
Iklk-m
Proof:
CllxllG-m,~~
for all
x E Sh.
(4.3.20b)
We first establish (4.3.20a). Thus, for x E s h , Ilxllc-m,h
= (ThX,X)kkz = (lThXl(Gm
= IIEhTx(IGm
G
llTxllCm
= (TXAk62 = Ilxllc-ml
where we have used (4.3.1), (4.3.13), and (4.3.17), as well as the fact that llEhll G 1 as an operator on G,. To establish (4.3.20b), we first note that
‘
C
llxllG0
’
IkllG-m.h’
(4.3.21)
Propositions 4.3.2-4.3.4 are adapted from the author’s paper with Rose (see Chapter 2, Ref. [9]).
4.3 Rates for the Continuous Galerkin Method
which establishes (4.3.21). We now estimate ~
~ ~ ~ ~ c - m :
X)Co + ((I - Eh)Tx, - Th)xIICoIIXIICo
llxlli-m= (Tx, x)Co = (EhTx,
' '
lklli-m,h
+
llXl\i-m,h
+ Ch2"llxl(&
1kIk-m.h
+ c~lklli-m,h9
147
x)Co
where we have applied (4.3.21) and the inequality
(I(T
- Th)fllCo G C h Z m ( l f l I G o ,
which follows from (4.3.6), with k = 2m and u = Tf. This concludes the verification of (4.3.20b). We introduce now the specific finite-element space of piecewise linear trial functions. Definition 4.3.4. Let {Ah} be a family of triangulations or simplicia1 decompositions of R. Thus, Ah consists of closed simplicia1 elements e. The positive parameter h will be specified shortly. The boundary elements are permitted to have a curvilinear edge coinciding with 8R, and it is required e. For each e E Ah, we define p(e) (respectively a(e))to that = Rh = UeEbh be the radius of the smallest ball containing e (respectively the largest ball contained in e). Set and
(4.3.22)
h = sup{p(e):e E Ah}, Mh = {x E C(sZ):x,, is affine,
for all e E Ah}.
(4.3.23)
For simplicity, we assume h < 1 in the sequel. Remark 4.3.5.
We shall assume that Mhhas the approximation property
E Mh} < Chd-jlJullHtcn), for all u E HC(R), (4.3.24) inf{)lu- x)IHjcn,:x
4 . Numerical Optimality a n d Degenerate Equations
148
for 8 = 1,2 and j = 0, . . . , 8 - 1. This implies that M, is of degree 1 for E, and P h (see (4.3.5b) and (4.3.11)).In the framework of the earlier abstract structure, we are taking rn = 1, k = 2, G - = [H'(R)]*, and G j = H'(R) for j 2 0. We have, thus, chosen to consider explicitly only the Neumann and Robin boundary-value approximations. However, trivial modifications, for Q a convex, polyhedral domain permit the handling of the Dirichlet problem, where the trial functions in Mhare required to vanish on the boundary of R. The Dirichlet problem in the case of general (smooth) boundary geometry requires interior and exterior approximation by polyhedral domains to order hZ. Even when such approximation is carried out, the convergence rate deteriorates (see Strang and Fix [41] pp. 192-196) due to boundarylayer phenomena (see, however, [33]).
Remark4.3.6. As noted in Remark 4.3.2, the estimate (4.3.24) is valid whenever the elements e E Ah permit a polynomial projection locally in e, defined by the Sobolev representation formula (4.1.4) for 8 = 1,2. This places a restriction on the boundary elements e of Ah, such as e is the union of star-shaped domains. However (see Remark 4.1.3), this covers a broad class ofdomains. In order to utilize (4.3.24),via Proposition 4.3.1 and Remark 4.3.1, it is necessary that the appropriate smoothing operator T satisfy T:Lz(Q) + Hz(R),
(4.3.25)
as an isomorphism. In order to utilize Proposition 4.3.2, we shall require the analog of (4.3.14): 11X11H1(f2) G
C
I(x11L2(Q),
for
x
(4.3.26)
The inverse hypothesis (4.3.26) is known to be implied by the hypothesis that the underlying triangulations are quasi-uniform, that is, for all h > 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 .
4. Numerical Optimality and Degenerate Equations
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.
4.3 Rates for the Continuous Galerkin Method
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
4.3 Rates for the Continuous Galerkin Method
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)
4 . Numerical Optimality and Degenerate Equations
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<-++--,
P
4
1
1
P
4
~ 2 0 , b20, -+-=I.
(4.4.9)
Applying (4.4.3e) and (4.4.8) to (4.4.7), and summing over k = 1,. . . ,m, we obtain
4.4 Rates for Semidiscrete Approximations
157
after multiplication by ( t t - $- l). To estimate this right-hand side, we have
(4.4.11) Applying (4.4.1 1) to (4.4.10), we obtain (4.4.4). Combining (4.3.34) and the pointwise boundedness hypotheses, we may begin with (4.4.7) and proceed in a similar manner to obtain (4.4.5). An immediate corollary is the following.
Corollary 4.4.2. Let H ( . ) and uo be given, as in Section 1.2, for the case of the porous-medium equation. Then (4.4.4) and (4.4.5) hold, with G = Hi@). Remark 4.4.1. Note that the rates in LY+'(9)and L2(9), given by (4.4.4) and (4.4.5) for the step-function approximation of H(u) and u, respectively, are I I 9 " \ l 1 l y and IIPNIl(l/Z)(l +('I7)). Note that Proposition 4.4.1 does not apply directly to the Stefan problem because (4.4.3e), with y = 1, fails. It does hold, however, if H is replaced by the smoothed function H , of Section 2.1, and, in this case, the constant C of (4.4.3e) satisfies C = &/A and y = 1. If the line method is applied to the smoothed problem with ub = uo (see Remark 2.1.3), then estimate (4.4.5) yields (under hypothesis (2.1.25b)) IIU"N - UEIIL2(9) CE-1/211~Nll lIu1cIIL2(9)
(4.4.12)
is made in (4.4.12), and it for the step function U"'. If the choice E = )(SNII is noted that IIt&2(9) may be bounded, independently of E, by standard arguments (see Chapter 2 [ S ] ) , there results the estimate IlUE7" - UEIIL2(9) collgN
II
1/2
.
(4.4.13)
If (4.4.13) is combined with (2.1.15b) (under hypothesis (2.1.25b)), we have the estimate, for some constant C, I)U"N- U(IL2(9) G
clipN )I1/2,
for the convergence of the fully implicit method.
E =
IIgNIJ,
(4.4.14)
4. Numerical Optimality and Degenerate Equations
158
4.5
BIBLIOGRAPHICAL REMARKS
The exposition of Section 4.1 is based upon the paper of Dupont and Scott [13]. In the proof of Proposition 4.1.3, the case of equality in (4.1.21) rests upon arguments found in Stein [38, pp. 120-1211. In Dupont and Scott [13], the authors are interested in versions of Proposition 4.1.3, in which the bound does not depend upon p and 4.Moreover, elsewhere in the paper, they prove theorems where approximation by complete multidimensional polynomials is relaxed to situations including tensor products and other situations involving mixed indices. An extremely readable account of an alternative proof of KaSin's result, together with companion results on the linear N-width and entropy, has been given by Hollig [19]. The linear N-width is defined with respect to linear operators only. It is larger than the Kolmogorov N-width in certain cases, and perhaps represents a fairer yardstick of comparison for linear approximation processes. A tableau for the linear N-widths differs in two of the four regions from the tableau displayed in Remark 4.2.5,namely, when p < 4 and 4 2 2. For example, in the range 2 < p < q, the linear N-width is of order N-(dln)f(llp)-(llq). Though this differs from the Kolmogorov Nwidth, it corresponds to the upper bound determined by the Sobolev representation formula for piecewise polynomial approximation. For additional details see Hollig [19]. Schumaker [35, Section 6.61 gives independent proofs for the sharpness of the order of piecewise polynomial approximation. Proposition 4.2.4 is due to Brown [9], who has also shown [lo] that the Borsuk theorem, used in the lower-bound estimates of Proposition 4.2.3, is logically equivalent to these estimates. Lemma 4.2.11 has been proved by El Kolli [l5]. A variant of Proposition 4.2.8 has been used by Scholz [34], who also basically used the spline space introduced in Definition 4.2.3. The arguments here are the author's, however. The author is indebted to David Foxt for the argument used in the proof of Proposition 4.2.12. The key result of Section 4.3, Proposition 4.3.4, is proved in the author's paper with Rose (Chapter 2 [9]). The author is indebted to Scott for bringing the paper of Johnson and Thomee [22] to his attention, and to Lars Wahlbin for sketching the argument of Remark 4.3.10. Various forms of Proposition 4.3.1 are due to Aubin [4] and Nitsche [27]. Rannacher and Scott [29] show that the logarithm term is not present in L" first-derivative estimation, if piecewise-linear elements are used. A version of Proposition 4.4.1 was proved by the author [21]. A general linear theory for the convergence of the line method for the generators of +
Private communication.
References
169
C, semigroups and groups has been given by Hersh and Kato [18], and Brenner and Thomee [8]. We close with a very brief description of previous numerical work on the two-phase Stefan problem and the porous-medium equation. For the former, we refer to the existence proof of Kamenomostskaja (Chapter 5 [28]), the paper of SamarskG and Moiseenko [32], and the paper of Meyer [24] as being representative, though far from exhaustive (see the bibliography of (Chapter 2 [9]) for additional references). For the porous-medium equation, we refer to the paper of Rose [31]. The unified error analysis of Proposition 4.3.4, applicable to both the two-phase Stefan problem and the porousmedium equation (see Remark 4.3.8),appears to be a new observation.
REFERENCES S. Agmon, “Lectures on Elliptic Boundary Value Problems.” Van Nostrand-Reinhold, New York, 1965. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12,623-727 (1959). R. A. Adams, N. Aronszajn, and K. T. Smith, Theory of Bessel potentials, part 11, Ann. Znst. Fourier (Grenoble) 17, 1-135 (1967). J.-P. Aubin, Approximation des espaces de distributions et des operateurs differtntiels, Bull. SOC.Math. France Suppl. MPm. 12 (1967). E. F. Beckenbach and R. Bellman, “Inequalities.” Cambridge Univ. Press, London and New York, 1961. J. Bergh and J. Lofstrom, “Interpolation Spaces.” Springer-Verlag, Berlin and New York, 1976. J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. A n d . 7, 112-124 (1970). P. Brenner and V. Thomte, On rational approximations of semigroups, SIAM J. Numer. Anal. 16, 683-694 (1979); On rational approximations of groups of operators, &id. 17, 119-125 (1980). A. Brown, Best n-dimensional approximation to sets of functions, Proc. London Math. SOC.14,577-594 (1964). A. Brown, The Borsuk-Ulam theorem and orthogonality in normed linear spaces, Amer. Math. Monthly 86, 766-767 (1979). A. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88, 85-139 (1952). J. Douglas, T. Dupont, and L. Wahlbin, Optimal L, error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29, 475-483 (1975). T. Dupont and R. Scott, Polynomial approximations of functions in Sobolev spaces, Math. Comp. 3 4 , 4 4 - 4 6 3 (1980).
160
4. Numerical Optimality and Degenerate Equations R. E. Edwards, “Functional Analysis, Theory and Applications.” Holt, New York, 1965. A. El Kolli, Nieme epaisseur dans les espaces de Sobolev, J . Approx. Theory 10, 268-294 (1974). J. Frehse and R. Rannacher, Eine L’-Fehlerabschatzung fur diskrete Grundlosungen in der Methode der finiten Elemente, in Finite Elemente,” pp. 92-1 14. Tagung, Inst. fur Angew. Math., Univ. Bonn, 1975. M. Golomb, Optimal approximating manifolds in L,-spaces, J. Math. Anal. Appl. 12, 505 -5 12 ( 1965). R. Hersh and T. Kato, High-accuracy stable difference schemes for well-posed initialvalue problems, SIAM J. Numer. Anal. 16, 670-682 (1979). K. Hollig, Approximationszahlen von Sobolev-Einbettungen, Math. Ann. 242,273-281 (1979); Diameters of classes of smooth functions, in “Quantitative Approximation,” pp. 163-175. Academic Press, New York, 1980. J. Jerome, On n-widths in Sobolev spaces with applications to elliptic boundary value problems, J . Math. Anal. Appl. 29,201-215 (1970). J. Jerome, Horizontal line analysis of the multidimensional porous medium equation, in “Numerical Analysis” (G. A. Watson, ed.), Lecture Series in Mathematics 773, pp. 64-82. Springer-Verlag, Berlin and New York, 1980. C. Johnson and V. Thomee, Error estimates for some mixed finite element methods for parabolic type problems, R.A.I.R.O. Numer. Anal. 15,41-78 (1981). B. S. KaSin, The widths of certain finite dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR 41, 334-351 (1977); Math. USSR Izv. 11, 317-333 (1977). G. H. Meyer, Multidimensional Stefan problems, SIAM J. Numer. Anal. 10, 522-538 (1973). C. B. Morrey, “Multiple Integrals in the Calculus of Variations.” Springer-Verlag, Berlin and New York, 1966. L. Nirenberg, “Topics in Nonlinear Functional Analysis.” Courant Institute of Mathematical Sciences, New York Univ., New York, 1973-74. J. Nitsche, Ein Kriterium fur die Quasi-Optimalitat des Ritzschen Verfahrens, Numer. Math. 11, 346-348 (1968). J. Nitsche, L, convergence of finite element approximation, Rennes ConJ Finite Elements, 2nd, Rennes, France (1975);“Lecture Notes in Mathematics 606,” pp. 261 -274. Springer-Verlag, Berlin and New York, 1977. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Marh. Comp. 38,437-445 (1982). F. Riesz and B. Sz-Nagy, “Functional Analysis.” Ungar, New York, 1955. M. Rose, Numerical methods for flows through porous media. Report ANL-78-80, Argonne National Laboratory, 1978. A. A. Sarnarski; and B. D. Moiseenko, An efficient scheme for (the) thorough computation in a many dimensional Stefan problem, 2. VyEisl. Mat. i Mat. Fiz. 5 , 816-827 (1965) (Russian). A. H. Schatz and L. B. Wahlbin, On the quasi-optimality in L, of the fi’ projection into finite element spaces, Math. Comp. 38, 1-22 (1982). R. Scholz, Abschatzungen h e a r e r Durchmesser in Sobolev und Besov Raumen, Manuscripta Math. 11, 1-14 (1974). L. L. Schumaker, “Spline Functions: Basic Theory.” Wiley (Interscience), New York, 1980. R. Scott, Optimal L“ estimates for the finite element method on irregular meshes, Math. Comp. 30,681-697 (1976).
References [37] [38] [39] [40] [41]
[42] [43]
161
H. S. Shapiro, On some Fourier and distribution theoretic methods in approximation theory, in “Approximation Theory 111” (E. W. Cheney, ed.), pp. 87-124. Academic Press, New York, 1980. E. M. Stein, “Singular Integrals and Differentiability Properties of Functions.” Princeton University Press, Princeton, New Jersey, 1970. W. Stenger, On the variational principles for eigenvalues for a class of unbounded operators, J . Math. Mech. 17,64-648 (1968). G. Strang, Approximation in the finite element method, Numer. Math. 19,81-98 (1972). G. Strang and G. Fix, “An Analysis of the Finite Element Method.” Prentice-Hall, Englewood Cliffs, New Jersey, 1973. A. E. Taylor, “Introduction to Functional Analysis.” Wiley, New York, 1961. A. Weinstein and W. Stenger, “Methods of Intermediate Problems for Eigenvalues.” Academic Press, New York, 1972.
EXISTENCE ANALYSIS VIA THE STABILITY OF CONSISTENT SEMIDISCRETE APPROXIMATIONS
5.0
INTRODUCTION
In Chapter 2, we examined the pointwise stability of semidiscrete approximation schemes, fully implicit in the monotone part, and fully explicit in the Lipschitz perturbation. These pointwise estimates permit a unified existence analysis (see Theorem 5.2.1) for the two-phase Stefan problem and the porous-medium equation which is carried out in Sections 5.1 and 5.2, via the method of horizontal lines. The appropriate auxiliary estimates are obtained in Section 5.1. For the two-phase Stefan problem, an existence analysis is also presented (see Theorem 5.2.9) in the case when the initial datum, H(u,), is not necessarily essentially bounded. In fact, convergence of semidiscrete schemes is carried out in Theorem 5.2.9, in the case of convex linear combinations of the fully implicit and fully explicit schemes; the Crank-Nicolson scheme is included as a special case. Pointwise estimates are neither derived nor utilized for this result. One of the key estimates of this chapter is contained in Proposition 5.1.5. The proof is an original one, though the result itself is a restatement of known L' contraction principles in a somewhat different format. This estimate leads directly to the properties of aH(u)/at in terms of measures. In Chapter 2, we had occasion to introduce regularizations for the two-phase 162
5.1 Stability in Sobolev Norms
163
Stefan problem involving initial data u;, not necessarily in H'. Although the existence theory for such problems is well known, an independent proof could be assembled along the lines of the proof of Proposition 5.2.12, where uo E L Z ( Qis assumed. Sections 5.3 and 5.4 are sparing in detail. These contain, respectively, the existence analysis for reaction-diffusion systems, and the Navier-Stokes system for incompressible viscous fluids. The only major technical hurdle in Section 5.3 involves the relaxation of the vector field conditions on the invariant region boundary from strict inequality to conditions which permit equality as well. We mention, finally, that the existence theory of Section 5.2 is sufficiently general and flexible so as to cover models not explicitly described in C h a p ter 1. Thus, for example, porous media horizontal water flow and certain chemical engineering problems involve formulations for which H' may vanish. Theorem 5.2.1 to follow covers cases such as these and other degenerate cases involving a mixing of various types of singular behavior. We do not discuss the additional regularity of H(u) for 1 < y < 3.
5.1
STABILITY IN SOBOLEV NORMS FOR SEMIDISCRETIZATIONS OF DEGENERATE PARABOLIC EQUATIONS
In this section we shall derive the stability properties required for an existence theory and an approximation theory for solutions of a class of degenerate parabolic equations, including the two-phase Stefan problem and the porous-medium equation. We shall find it advantageous to correlate, as precisely as possible, the derived stability properties, with the associated features of the degenerate parabolic class, including the regularity of the initial datum, and the particular semidiscrete method chosen for analysis. The existence theory of the next section requires only a stability analysis for one (mixed) implicit method, and, in this case, the results of Chapter 2 lead to the conclusion that the respective analyses of the two-phase Stefan problem and the porous-medium equation essentially merge, except for technical questions associated with the multivalued enthalpy function in the former case. The reason for this merger is that the pointwise stability of the (mixed) implicit scheme leads, in the latter case, to the conclusion that J = H - ' is a de fucto globally Lipschitz continuous monotone function, despite the fact that, for the porous-medium equation, J has superlinear growth at infinity. Here J represents a left inverse.
5. Existence via Stability and Consistency
164
We shall, in fact, examine those semidiscrete methods which are convex combinations of the fully implicit method and the (purely) explicit method, including the Crank-Nicolson method. For these methods, we prove stability in familiar energy, or Sobolev norms for both Lz and H' initial data, although we do not prove pointwise stability, as in Chapter 2. For the estimates of the divided differences in time, however, we require global Lipschitz properties of J , or an equivalent substitute, in terms of pointwise stability, together with H' initial data. Note that, for most of this section, we assume that the function f, in equation (5.1.1) to follow, is Lipschitz continuous. We later relax this assumption and sketch the modified estimates.
Definition 5.1.1. 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'. Suppose, also, that the induced maximal monotone operator on L Z ( Q(Example 3.2.2) is bounded. Let T denote one of the operators N,(o > 0) or Do. Let f be a Lipschitz continuous function, and let uo and H(uo) be given functions in L2(R). Consider the model equation
'
~aH(u) at
AU + f(u) = 0
on 9 = R x (0,To),
(5.1.1)
in distribution form, with (weakly) prescribed homogeneous Dirichlet or Neumann boundary conditions and initial data in terms of uo and H(uo). In the lifted formalism, we write
aTH(u) at
+ u + Tf,(u) = 0
in 9,
(5.1.2a)
=-
where f, = f - oid if T = Nu (o 0), and f, = f if T = Do,and, for the moment, we understand (5.1.2a) in the distribution differentiated sense. For
4 G el G 1,
o G 8,
G 1,
(5.1.3)
a sequence of partitions of [0, To],we define the admissible semiand (9") discretization schemes of(5.1.2) (and, hence, of(5.1.1)) by H(u:) = H(uo),and
'T[H(u;) - ~ ( u ; -,)I @zT',(uF) + (1 - O,)Tj,(UF-
(t; - t;-
+
+ elu; + (1 - el)& 1)
= 0.
(5.1.4)
5.1 Stability in Sobolev Norms
165
The following properties are explicitly assumed. If (5.1.5b)to follow holds, then (5.1.5a) (1 - t:- I ) - 3 e2cll.rllLiP + a).
+w:
for some fixed
E,
0 < E d 1; for if 8,IlfbllLip # 0, H ( . ) is assumed to satisfy
[ H ( O - H ( r l ) ] ( t- rl) 3 4 t - r l ) 2 ,
1 > 0,
(5.1.5b)
for all <, q E R'. In (5.1.5b). we have identified sets and elements.
Remark 5.1.1. It follows from Proposition 3.2.1 that unique solutions u[ of (5.1.4) exist in D,-l c H1(R), with H(u:) E L2(R), under assumption (5.1.5a). Note that H(u:) is uniquely determined. Remark 5.1.2. The stability analysis proceeds by showing, first, that (.:)k,N and, hence, ( H ( u : ) } lies in a fixed ball of L2(R). A significant technical obstacle is overcome by using the format of (5.1.4), rather than (the weak form of) the equivalent elliptic boundary-value problem, to derive these latter estimates on ( H ( u : ) } . We shall use the elliptic problems to obtain the gradient estimates, however. The reason for the technical simplification revolves around the fact that H ( w ) H1(R),in general, if w E H'(S2). Thus, the standard technique of theoretical numerical analysis, of obtaining an estimate on H(u:) by using this function as a test function in the weak formulation, breaks down. Of course, the standard resolution of this dilemma is a smoothing of H , such as introduced in Chapter 2, and realized for general H by the Yosida approximation. This smoothing technique gives satisfactory bounds for { H ( u r ) } in the case of the fully implicit scheme d1 = 1, but apparently not otherwise without further restrictions on H . The use of the lifted format, thus, provides greater generality as well as greater simplicity.
+
Proposition 5.1.1. Consider the semidiscrete scheme defined by (5.1.4). Under the hypotheses of Definition 5.1.1, solutions ur exist in D,- c H1(R), and are stable in L2(R).More precisely, there is a constant C, not depending on k or N , such that IIu;llLz(n) < C ,
k
=
1 , . . . ,M ( N ) ,
all N .
(5.1.6a)
If 8 , > 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 : ) ~ ~<~ C, Z(~) k
=
1,. . . ,M ( N ) ,
all N.
(5.1.14)
Remark 5.1.3. Suppose f is permitted the more general representation f = g h, where h is a Lipschitz continuous function, and g is a monotone Lipschitz continuous function, vanishing at 0 and satisfying
+
[s(5)- s(1)1[5- 13 3 4 5 - 112,
c > 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<( C,( ~ ~k =~1,.~. .,,M ( N ) ,
all N.
(5.1.17)
5. Existence via Stability and Consistency
168
Proof: We use the weak differential equation formulation corresponding to (5.1.4): H ( u t ) = H(uo),and ct: - t:-
1)-
~ u : ) ~(u:-
+ (1 - el)(vu:-
+ em:, vu)Lz(n) + f 3 2 ( f ( 4 ) , U)LZ(fl) + (1 - 0 2 ) ( f ( 4 -
U)Lz(R)
1 3 VV)LZ(*)
= 0,
l ) r U)L"(n)
(5.1.18)
for all u E DT-1. Setting u = u: - u:- in (5.1.18), and using an analog of (5.1.7) and (5.1.10a), we obtain, in analogy with (5.1.1l),
2 +p4-lIIL2(Q) + (w: - cl/2)114 - 4lIIL(R) (5.1.19) 6 cct: - cl l l t z ( n ) + If(o)121fill,
tlpJ:IItZ(R)
-
1)-
1~~llflltipll~:-
where now C has the value C = (As)-'. Note that f, is replaced by f in the nonlifted formalism, and that a weaker form of (5.1.5a) has been used. If (5.1.19) is summed from k = 1 to k = m, the right-hand side is seen to be bounded, independent of m and N , by (5.1.6), and there is obtained immediately the gradient estimate IIvU:lltz(n)
6
Ipolltz(n)
+ c,
(5.1.20)
for some positive constant C . The gradient estimate (5.1.20), in conjunction with the L2 estimate (5.1.6), yields (5.1.17).
Corollary 5.1.4. Under the hypotheses of Definition 5.1.1 and the additional hypothesis uo E D T - ~there , exists a constant C, such that
for all N . Proof: Use (5.1.19) as a starting point, and sum from k = 1 to k = m, as in the proof of the preceding proposition. The left-hand side of (5.1.20) is then augmented by a constant multiple 4 2 of the left-hand side of (5.1.21).
Remark 5.1.4. The content of Remark 5.1.3 applies to the results of Proposition 5.1.3 and Corollary 5.1.4, if (5.1.18)is replaced by ct: - -:c
,I- Y H ( ~ : ) wu:-
+ (1 - el)(vu:-
+ (1 - 0,)(h(up-
1, v 4 L q R )
u)L2(R)+ el(vu:,
VO)~~(,,
+ ( g ( 4 ) ,d L Z ( R ) + 02(&4:),
u)Lz(n) = 0,
for all u E DT- I .
h ( R )
(5.1.22)
5.1 Stability in Sobolev Norms
169
Specifically, the estimates (5.1.17) and (5.1.21) hold for the solutions of (5.1.22), with the same provisions as in Remark 5.1.3. There is an analog of (5.1.21) in the L' norm, applied to difference quotients involving H( .). For simplicity, we shall derive these somewhat complicated estimates only in the case of the fully implicit scheme. We first make an important preliminary remark.
Remark 5.1.5. In Chapter 2, we proved the pointwise stability of the mixed scheme corresponding to (5.1.18), with 8, = 1 and 8, = 0, as adjusted by (5.1.22). The proofs given in the special cases described in Sections 2.2 and 2.4 may be unified into a single proof, valid for equations described in Definition 5.1.1. Thus, in the following proposition, we shall assume that there exists a constant c = c(lluollLmcn,), such that, for 8, = 1 and 8, = 0,
ll~:llL-(*)
(5.1.23)
c,
k = 1 , . . . , M ( N ) , N 2 1. Now, (5.1.5b) may be relaxed, in this case, to the local condition IIJp,(I-c,cl)llLip < l/A, described by
[ H ( 5 ) - H(rl)1(5- rl) 2 4
5 - rlI2?
151,111 < C?
(5.1.24)
where c is prescribed by (5.1.23), and J = H - ' . This has the important implication, noted at the outset of this section, that (5.1.21) holds, when (5.1.5b) is replaced by (5.1.24). Thus, under (5.1.24), Proposition 5.1.5, to follow, applies to semidiscretizations of both the two-phase Stefan problem and the porous-medium equation. Some additional regularity is required of uo in the following proposition, viz., uo E W2'1(sZ).We also require H(uo)to be the L2 limit of admissible smoothings to preclude further assumptions on uo. Proposition 5.1.5. Consider the equation (5.1.2), and consider the corresponding semidiscretization (5.1.4), with 8, = 1 and 8, = 0. Suppose the assumptions of Definition 5.1.1, excluding (5.1.5b), hold, with the following additional assumptions : Hypotheses (5.1.23)and (5.1.24) hold;
(5.1.25a)
uo E D,-l n W2'1(sZ);
(5.1.25b)
H(uo) E L"(0) and H(uo) = (L') lim H,(uo), for an admissible &+O
smoothing H , (see Definition 5.1.2); (t: - r f - ' ) = 6tN , k = 1 , . . . , M ( N ) (uniform partition).
(5.1.25~) (5.1.25d)
5. Existence via Stability and Consistency
170
Then, there is a constant C, such that
- WLl)IIL'(R) G
(st")-'JIH(u;)
for all N and k = 1,. . ., M ( N ) .
c,
(5.1.26)
Proof: Define, for 1 < k < M ( N ) , (5.1.27) Certain technical aspects of the proof require the assumptions that co; has an essential pointwise bound, independent of k and N , and Auo E Lpo(R), some p o > 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 ~ .<~ (+~I )I ~ o l l D T - l n W z , l ( n ) .
For maximum clarity, we shall simply prove the result in the unsmoothed case, when there exists an upper bound A, for of, and when Auo E L2(R), i.e., in the cases (5.1.30a) 1 < o: G 1 1 , (5.1.30b)
Auo E L2(R),
and we shall derive the estimate (5.1.23), with C independent of A,. We may extend the definition of to all of R, by defining of to have the value 1 on {uf = With the assumption (5.1.30), the proof naturally divides into two parts. In the first part, we prove that the verification of (5.1.26) reduces to the verification of the case k = 1. For this part of the proof, we find it convenient to introduce the auxiliary problems, for k = M ( N ) - 1,. . . , 1, given by
or
o f ( 6 t N ) '(Cf-
, - Cf)
- ACf-
= 0,
Cf-
,E D,-
(5.1.31a)
which are fully implicit discretizations of a backward, or dual, parabolic problem, and are understood in the weak sense. We specify the terminal value L;M(N) by G ( N ) = sgn(ulv,(N)-1 - U L ) . (5.1.31b) Prior to utilizing the auxiliary problems (5.1.31), we note the key a priori estimate llCI?IIILm(n) < 1, k = 1,. . . , M ( N ) , N 2 1. (5.1.32) Inequality (5.1.32)follows, as in the proof of Proposition 2.2.4, via the choice u = Oj,q-,(Cf- ,) in the weak formulation corresponding to (5.1.31). The analog of (2.2.40) is the inequality
ll4xf- lIILS(*)
(5.1.33)
G IIWfCfllLs(n).
where q is an arbitrary even integer. Now, using the bounds (5.1.30a) in (5.1.33), and letting q -+ co,we obtain
IK-lIILm(R) G IlL;:llLm(n) <
llClv,(N)lIL-(n)
=
1,
which implies (5.1.32). +
In the case of the Yosida approximation, a sharper lower bound of I holds.
(5.1.34)
5. Existence via Stability and Consistency
172
For notational convenience, we introduce the backward difference operator l]k
which maps { ( k } ; onto takes the form L
1
= 6 k < k =
(5.1.35)
ck-19
The familiar summation-by-parts formula
{l]k}f.
1
j=L
L
(6kak)bk
+1
(6kbk)ak- 1
k= 1
k= 1
= aLbL- aobo:= ajbj
so that, with L = M ( N ) - 1, ak
N
= ck+
bk
1,
= H(uf) -
, (5.1.36)
j=O
H(ur+I ) ,
we obtain from (5.1.36)the identity M ( N )- 1
1
=
( d 6 k ( 6 r + l)r
k= 1
M ( N )- 1
+ 1
(6k(H(ur)
ur - u;+
1)Lz(f2)
- H(ur+l)hc;)L2(f2).
(5.1.37)
k= 1
If (5.1.31a) is applied to the first term on the right-hand side of (5.1.37), and the defining equation (5.1.4) is applied to the second term (0, = 1,02 = 0), we obtain
so that
k= 1
The second term on the right-hand side of (5.1.39) may be estimated by M ( N )- 1
M ( N )- 1
1
k= 1
118kf(Ur)llL1(f2)
G
(If(lLip
1
k= 1
G ~ l l f l l L i p {T O
I16kUrIIL1W)
+ Inl
M ( N )- 1
1
k= 1
(st",-'~~6ku~~~~z(f2)}~
(5.1.40) If(5.1.21) is applied to (5.1.40), this sum is seen to be bounded by a constant independent of k, N , and A l , although dependent on A. Altogether, we have
5.1 Stability in Sobolev Norms
verified the case k
= M ( N ) of
173
the inequality
(6tN)-'IIH(d'+1) - H(u:)ll~l(n)< (6tN)-'llH(u7) - H(u:)IILI(~)+ C . (5.1.41) The verification for other values of k involves the obvious modifications in (5.1.31b)and in the choice of L (see (5.1.36)).The derivation of (5.1.41), thus, completes the reduction of the first part of the proof. We introduce two transformations to facilitate the derivation of the estimate (6tN)-111H(UI;J) - H(u:)llLi(n)
I;,
< C,
N 2 1.
(5.1.42)
For 6 < 1, and for g E L'(Q), set S i g = - S,'( -g), where g(x),
S,'(g)(x) =
For 1
-= p < 2,and for u
E
g(x)
1 2 3'
(5.1.43) otherwise.
Lp(Q),u # 0, set
(5.1.44) The operator I, maps the set LP(R)\{O}into the unit sphere in Lp'(Q),where (l/p) (l/p')= 1. Moreover,' I , + sgn(.), p 1 1, in the following sense. For u E L2(Q),I,(u) + sgn(u)pointwise, with Ip(u)dominated by an L2(Q)function. Indeed, the domination is evident from the inequality
+
Moreover, I, o S i is a mapping of H'(Q) into itself as induced by the composition of two Lipschitz continuous functions, with an element of H'(R) and a constant multiplier. Note that, here, we use the fact that the function < ~ < 1 < 1 ~ - ~ , 151 2 1/6,is Lipschitz continuous for p < 2. Also, the crucial inequality v u * V(1,O
s;
0
(5.1.45)
u) 2 0
holds for u E H'(Q), 6 2 1, and 1 < p G 2. We are now ready to verify (5.1.42).We begin with (5.1.18),with k = 1, O1 = 1, and O2 = 0:
(6tN)-'(H(u7)- H(uo),u)Lz(n)
+ (Vu7, vu),2(n) + (f(uo),
V)LZ(R) = 0.
(5.1.46)
' We have used the Italic symbol here, since I , clearly acts as a function in this statement.
5. Existence via Stability and Consistency
174
neglect the energy term, via (5.1.45), We select v = I, S i ( u 7 - uo) in (5.1.46), after addition and subtraction of uo, and apply Holder's inequality, in conjunction with integration by parts, to obtain, for 1 < p G 2, 0
(atN)-' ( H ( 4 ) - W
O ) ,
G p u o + f ( U O ) , I,
where C
=
llfllLip
-<
I, O
S&Y - UO))LZ(R) S i b 7 - UO))L2(R)l
O
+ CIIUOIILP(R) + c1,
IlA~OIILP(R)
and C1 = If(0)l lQl'/p. Letting 6 + 0 gives
( 6 t N ) - ' ( H ( 4 )- H(uo), I,bY
- UO)*)LZ(R)
G
IIAUOIILP(R)
+ CI(UO(ILP(R) + c1(5.1.47)
-1 1 give - ~ ( ~ O ) l * I l L ~ (GR )I I A ~ O l l L l ( R )+ CII~OIIL'(R)+ If(0)l IQI,
Multiplying (5.1.47)by 101-''P and letting p
(6tN)-'II[H(u7)
(5.1.48)
IQI.
after multiplying by Note that we have used, here, the Lebesgue dominated-convergence theorem, and the fact that u: - uo and H(u7) H(uo) have the same sign. Note that (5.1.42)follows directly from (5.1.48). Estimate (5.1.26)follows directly from (5.1.41)and (5.1.42). We shall now derive gradient estimates for the case €I1 = 1, €I2 = 0,under the strong monotonicity condition (5.1.5b), where the condition uo E D,-I c H'(Q) is relaxed to uo E L2(Q). Since smoothing is an indispensible tool in this analysis, we introduce a class of admissible smoothings.
Definition 5.1.2. Let H ( .) be a multivalued surjective function, defining a maximal monotone graph on R' x R', with continuous left inverse J, so that J H = id. We shall say that { H E ) O < E s is E *an admissible smoothing for H , provided the following conditions hold. 0
(1) H e and J, = H;' are monotone Lipschitz continuous functions with H&G C/E, c > 0. (2) For all u E L2(Q),J,(v) + J ( v ) in L2(Q)as E + 0. (3) If H satisfies (5.1.5b), then H : 2 A12 for E sufficiently small. (4) If 0 E int(Dom H ) and 0 E H(O), then H,(O) = 0. ( 5 ) For each U E L2(Q), there exists a constant C = C(v), such that 1IH&(v)IIL2(R)G (1 + C)IIH(v)lILz(n).
-
Lemma 5.1.6. Under the hypotheses of Definition 5.1.2, suppose that ' + u in L2(0) (strongly), and H,(uE) x in L2(Q) (weakly). Then x = H(u) U for an admissible L2(Q) selection H(u).
5.1 Stability in Sobolev Norms
175
Proof: Since H defines a maximal monotone operator H t in L2(Q),it suffices to show that
(x - H(u), u - u ) ~ ~2 (0,~ ) for all (u, H ( u ) ) E graph (H), (5.1.49) which shows that (u, x) is an element of the graph of H. To verify (5.1.49),we write
(x - H(4, u - U)L2(R) = (x - H,(u”, u - h ( R ) + W&(UC)- H(4, u - UE)L’(R) + W & ( 4- H(4, J , + ( H , ( U E ) - H(4, J ,
2
O
H,(u&)- J ,
O
H(4 - J
O
O
W4)LqR)
WU))L2(R)
(x - H & ( U E )u, - d L 2 ( R ) + W & ( U E )- Mu), u - UE)LZ(R) + W & ( 4- H(4, J ,
H(4 - J H ( 4 ) L Z ( R ) ? for all E > 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.
5. Existence via Stability and Consistency
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)
5. Existence via Stability and Consistency
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
5. Existence via Stability and Consistency
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)
5.2 Existence of Weak Solutions
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)
5. Existence via Stability and Consistency
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
5.2 Existence of Weak Solutions
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<
t
0 6 k < M - 1.
(5.2.22b)
5. Existence via Stability and Consistency
184
Suppose that {w:} c I/, and satisfies M
1 ( t : - t:- ~)-'[IW:
- w:- Ill:z(fi)
k= 1
< C,
N 2 1,
(5.2.23)
for some constant C . Then, if wN
we conclude that
GN
provided llSNll-, 0.
-
w
(weakly in L2(9)),
(5.2.24a)
w
(weakly in L2(9)),
(5.2.24b)
Proof: It is routine to see that the boundedness of {w"}in L 2 ( 9 )implies the corresponding boundedness of {EN}.This does not require (5.2.23). It is now enough to show that every weakly convergent subsequence of { E N } has limit w. Suppose that such a subsequence, labeled again by N , has limit G. Thus, if $ E Cm(B),$( ., t ) E V, t E [0, To],and t,hN is defined as in the previous lemma, we have, since the product of a weakly convergent and convergent sequence is weakly convergent on the class of bounded functions, (w,$ ) L 2 ( 9 )
=
lim
N-+ m
( W N ,ICIN)L2(9) M
Here, we have used the inequality I
IM-1
which clearly tends to zero as N follow similarly.
+
co by (5.2.23). The remaining statements
Proof of Theorem 5.2.1: Let $ E Cm(9),$(., t ) E V, t E (0,To).In order to obtain a solution pair [u, u ] satisfying (5.2.3(a-e)), we first select a sequence
5.2 Existence of Weak Solutions
185
9" of partitions satisfying (5.2.15), (2.2.41c), and II9"II
+ 0, and we define [uf, H(uf)] via (5.2.7); we then construct the corresponding step function sequences, as defined by (5.2.22a), which we label {u;} and { u f } , respectively. We shall also have need of the corresponding piecewise linear (in time) sequences { E N } and { E N } , defined as in (5.2.22b).Here, uz = uo and u z = H(uo). Using Proposition 5.1.3 and Corollary 5.1.4, as modified by Remark 5.1.4, and Lemmas 5.2.2, 5.2.3, 5.2.4, and 5.2.6, we may obtain functions' u E % (see (5.2.3d)) and u = H(u), satisfying (5.2.3a), and subsequences {u"}, { E N t } , {u"}, and { E N S } ,satisfying
-
uNL * u
(weak * in L"(9)
EN' - u
(weakly in H'([O, To];L2(R))),
(5.2.25b)
uNt -+ u
(in L2(9)),
(5.2.25~)
(weak * in Lm(9)),
(5.2.25d)
(weakly in H'( [0, T o ] V*)). ;
(5.2.25e)
UNm
L.*u
ENt
u
-
and
L"( (0,T o ) V)), ; (5.2.25a)
Moreover, if uo E W2,'(R), and the partitions are selected to be uniform, then, by Proposition 5.1.5, we may select u, so that (au/at)E L"((0, To);M(Q)), and { i j N Z }so , that acN8 ~
at
2 .
av at
*-
(weak * in L"((0, To);M(fi))).
(5.2.25f)
Note that the boundedness estimate needed for (5.2.25e) is obtained directly from the discrete equations in terms of other quantities already estimated. Moreover, a modified form of Lemma 5.2.6, in which L2(Q) is replaced by V*, is employed to equate the limits in (5.2.25d) and (5.2.25e). We are now ready to proceed to the verification of (5.2.3~).Using (5.2.7) as a starting point, we set 4 = sum on k, and apply summation by parts to obtain
+,;
The equality of the appropriate limits in (5.2.25) follows from the results quoted above. Note that (5.2.25~) depends upon (5.2.23) or, alternatively, upon the weaker estimate (5.1.6b). These estimates relate the convergence of {uNc}and {."').
5. Existence via Stability and Consistency
186
Replacing the discrete format (5.2.26) by the corresponding continuous format gives, in the notation of Lemma 5.2.5,
j( (uN,
SN)LZ(R) dt
-
+ (&
9
$%Z(R)
- (003$Y)I,Z(R)
+ j ; ' - ' ( V ~ ~ , V $ ~ ) d+t j ~ - ' ( g ( ~ ~ ) , $ ~ ) ~ z ( ~ ) d t
+ J?'
- 1
(h(uz),+
+ (h(u,),$ ~ ) ~ z ( =~ )0,t ~
N ) ~ ~ dt ( ~ )
(5.2.27)
where we have used the notation 4 to designate the same translation relation to uN as xN bears to $ N . Now, the following limit relations are valid.
since the product, when defined, of a weakly convergent and a convergent sequence in L2(9)is weakly convergent. Also, lim
N,+m
&:
(
M(N,)-I
since { u N i }is bounded in L2(9),and
~
~
c1 N, i ) L z ( n )dt
= 0,
{cNi}is pointwise bounded. Thus,
In a similar way,
and
the latter relation following, since the a priori estimates permit g to be assumed Lipschitz. By difference quotient estimates, we have { u N i- uzi} + 0 as N i+ co,so that
The final term in (5.2.27) clearly tends to 0, and the third term tends to - (uo, $( *, O))L2(n), by the Lebesgue dominated convergence theorem. It remains to analyze the second term in (5.2.27). By the fundamental theorem
5.2 Existence of Weak Solutions
187
of calculus in reflexive Banach spaces,
and the weak limit in V may be taken in this relation, by (5.2.25e). This shows that UM N i ( N i ) - u ( - , To) (weakly in V). (5.2.32) These remarks, together with (5.2.28)-(5.2.31), permit us to let N tend to infinity through values Ni in (5.2.27), thereby obtaining (5.2.3~). It remains to show that (5.2.3~)remains valid for all $ E V,. If an interpolation sequence is defined, built upon a sequence {P"}, IIPNII-,0, of partitions of [0, To], each member of the sequence can be approximated arbitrarily closely in Vo by a function in Cm(9)n W0. The convergence of the interpolation sequence in Vothen yields the existence result. Uniqueness follows from Proposition 1.5.1.
Corollary 5.2.7. Under the hypotheses of Section 1.1, a unique weak solution pair [u, H(u)] exists for the two-phase Stefan problem, satisfying (1.1.10) and (1.1.11). Pointwise bounds for u and H(u) are given by c and l c , respectively, where c is defined by (5.2.1) (C = 0). If uo E W2*'(0), then aH(u)/at E L"( (0, T o ) ;M(fi)). Corollary 5.2.8. Under the hypotheses of Section 1.2,a unique nonnegative weak solution u exists, satisfying (1.2.15) and (1.2.16). A pointwise bound for the solution is given by iluOIILm(n). If uo E W2*' (R), then aH(u)/dt E Li%(O?m);M(Q).
Proof: The additional arguments revolve about the fact that the class V in (1.2.16a) involves (vector) functions, defined on the time interval (0, a). Corresponding adjustments are necessary in the redefinition of the partitions, which now contain countably (infinite) many points, and in the redefinition of 9 as 9 = 0 x (0, m). Note that V = Hb(R) now. In the new notation, the sequences selected in (5.2.25) are selected as before, except that (5.2.25e,f) need only hold locally on (0, m), and one additional condition is specified. Corresponding estimates are required, and we address these now. P r o p osition 2.4.1 directly yields a pointwise estimate, and the estimate IlVUrllL2(R)
IlV~OIIL2(R)
is routinely derived by setting u = ur in (2.4.2). Thus, we may select sequences satisfying (5.2.25a), (5.2.25c), and (5.2.25d). The estimate
5. Existence via Stability and Consistency
188
(5.1.21), for any M, holds, with C given by ~ / ( ~ A ) ~ ~ here, V U A~ satisfies ~ ~ ~ ~ ( ~ ) ; (5.1.24), with c = IIuOIILm(n),and may be chosen explicitly, via (2.3.21), as c l ( c l / c z ) y -l / ~ ~ u o ~ [ ~It l ~remains $ y ) . to obtain an estimate of the form
This and the preceding estimate discussed permit the choice of sequences, satisfying (5.2.25b) and the additional condition
-
uNi
u
(weakly in L2(( 0 , ~ HA@)). );
Equation (5.2.33) is routinely obtained by selecting u in (2.4.2), as described by (2.4.7), with / = 1 and j sufficiently large, in which case O,,,(H(uF))= [H(u:)]'. The simple inequality y[H(t)]Y-' H ' ( t ) 2 yyc:
shows that (2.4.8) can be strengthened to ( V 4 , vu)Lz(*) 2 YYC:(lV~:lliqn), which implies (5.2.33), if the correspondingly strengthened form of (2.4.9) is used.
Remark 5.2.2. The following result will achieve two objectives. First, the approximations, defined via the more general format of (5.1.18), will be shown to converge in Lz(.9) to the unique solution of (5.2.3~).Second, we relax the requirement that H(u,) is essentially bounded. Note that uo E V is retained, however. Actually, minor modifications of the proof of Theorem 5.2.1 permit the hypothesis uo E V to be relaxed to uo E H'(R), that is compatibility between the initial datum and the boundary datum need not hold. For simplicity, however, we retain uo E V. Theorem 5.2.9. We assume that the hypotheses stated in Definition 5.2.1 hold with the following qualifications. The function H(uo) is required to belong to L2(R), rather than L"(R), c = 00 in (5.2.la), and f in (5.2.2a) is assumed Lipschitz continuous. Suppose a sequence of partitions {P"} is specified, satisfying (2.2.41~)and (5.2.15), and IIY"I( --* 0 as N + 00. Let O1 and 0, be specified, with 3 < 8, d 1 and 0 < 8, < 1, and define (u:} as the recursively generated semidiscrete solutions of (5.1.18). Let {u"'} and {EN) be the step function sequence and piecewise linear (in time) sequence, as defined in Lemma 5.2.6. Then, both sequences converge in L2(9) to a function u. If u: = H(u:) are defined by (5.1.18), and {u"} and {EN} denote the corre-
5.2 Existence of Weak Solutions
189
sponding sequences, then both sequences converge weakly in L2(9) to a function u = H(u). The pair [u, u] is a solution of (5.2.3c,e), subject to E
L"((0, To);v)n H'([O, T0];L2(fi)),
u E L2(9) n HI( [0, To]; V*).
(5.2.34a) (5.2.34b)
Proof: N o additional arguments are required. Thus, by the estimates of Corollary 5.1.2, Proposition 5.1.3, and Corollary 5.1.4, together with Lemma 5.2.3, Lemma 5.2.6, and the Rellich compactness property of Lemma 5.2.2, we conclude that subsequences can be found satisfying
-, -
*u
(weak* in L"( (0, To);V),
(5.2.35a)
u
(weakly in H'([O, T0];L2(Q)),
(5.2.35b)
uN,
u
(in L2(9)),
(52.3%)
uN1
u
(weakly in L2(9)),
(5.2.35d)
ENz
u
(weakly in H'([O, T o ] ; V * ) ,
(5.2.35e)
uNi U"",
where u and u = H(u) satisfy (5.2.34). The remainder of the proof proceeds as Theorem 5.2.1. Note that the uniqueness of the solution pair [u, u] implies that the subsequences displayed in (5.2.35) may be selected to be the entire sequences.
Remark 5.2.3. Note that Theorem 5.2.9 does not cover the porous-medium equation, since (5.2.la) does not hold with c = co.Note also that f is required to be Lipschitz continuous, whereas Theorem 5.2.1 permitted a locally Lipschitz monotone increasing perturbation. Moreover, it is worth making explicit the convergence theory used to prove Theorem 5.2.1. We prefer to view the following result as a corollary of both theorems. Corollary 5.2.10. We assume the hypotheses of Definition 5.2.1 and a sequence of partitions, as defined in Theorem 5.2.9. If { u f } are defined via (5.2.7), and {d"},{ii"}, { u N } , { E N } have the meaning of Theorem 5.2.9, then (5.2.35) holds, with {Ni} = { N } . In particular, the solution pair, described in Theorem 5.2.1, is realized as the unique limit of this construction. Our final result relaxes the requirement that uo E V. We also relax the pointwise boundedness hypothesis as well. However, to achieve this, a stronger compactness criterion than that of Lemma 5.2.2 is needed. We shall state the result in the form necessary for our purposes (see Lemma 5.4.3, to follow).
5. Existence via Stability and Consistency
190
Lemma 5.2.11. A set bounded in both H'([O, To];V*) and L'((0, To);V) is relatively compact in Lz(9). Proposition 5.2.12. We assume the hypotheses of Definition 5.2.1 with the following qualifications. The function uo is not required to belong to V(uo E L2(C2) only), and f is required to be Lipschitz continuous. Then a unique solution pair [u, u ] , u = H(u), exists for (5.2.3c,e), satisfying 24 E
LZ((0,To);V) n H'([O, To];V*),
u E L z ( 9 ) n H'([O, To]; V*).
(5.2.36a) (5.2.36b)
Proof: We select a sequence of partitions, as in Theorem 5.2.9, and define the sequences of functions, as in Corollary 5.2.10. For the estimates, we use Propositions 5.1.7 and 5.1.8. We also use Lemmas 5.2.3, 5.2.4, and 5.2.11. In addition we use a modified version of Lemma 5.2.6 with L'(C2) replaced by V*. Altogether, we may select subsequences, satisfying
EN' Pi EN' uNi
u
(weakly in Lz((0, To);V)),
u
(weakly in H'( [0, To];V*) and Lz((0, To);V)), (5.2.37b) (in L2(9)),
(5.2.37~)
u
(weakly in L2(9)),
(5.2.37d)
u
(weakly in H'([O, T o ] ;V*)).
(5.2.37e)
+u
uNi
(5.2.37a)
The remainder of the proof follows that of Theorem 5.2.1 once it has been shown that (5.2.37~)implies the L2(9) convergence of the step function sequence. This follows, as has been observed earlier, from estimate (5.1.6b).
5.3
EXISTENCE FOR REACTION-DIFFUSION SYSTEMS
The task of this section is to obtain gradient estimates and divided difference estimates in time for the concentration and potential variables, and to relax the strict inequalities in (2.5.6) to simple inequalities. We note that pointwise estimates have been obtained in Section 2.5 (see Proposition 2.5.1) and do not depend upon (2.5.6).They do require that f;. be a bounded, locally Lipschitz continuous function for i = 1,. . . , io, and that g i be a locally Lipschitz continuous, monotone increasing function for i = io + 1,. . . ,m,
5.3 Existence for Reaction-Diffusion Systems
191
with hi Lipschitz continuous. The pointwise estimates permit us to assume that these functions are Lipschitz continuous. Proposition 5.3.1.
There exists a constant C, such that, for i = 1,. . . , m,
(IVU;~~~&)< C,
for all N , k
=
1,. . . , M ( N ) ,
(5.3.la)
and M(N)
1 llt&
k= 1
- ~:-~,~ll&,(t:
- t:-l)-l
< C,
for all N . (5.3.lb)
Here, {uf} denote the (vector)solutions of (2.5.2),guaranteed by Proposition 3.3.3.
(5.3.2) where the constants cjSiare defined from m
The corresponding inequality, for i = io
+CTIIU;i(lt2(,)(tf-t:-
+ 1, . . . ,m, is
(5.3.3)
11,
where the constants dj,i and ci are defined from (hi(Y1,. . ., Ym) - hi(x1, . . ., xm)l Q
m
1 dj,ilYj - xj(,
j= 1
(gi(t)- gi(s)l < tilt - ~ 1 . By summing (5.3.2) and (5.3.3) from k = 1 to k = I , and noticing that the right-hand sides are bounded for each r = 1,. . . , M ( N ) , we obtain (5.3.lb)
5. Existence via Stability and Consistency
192
and (5.3.la) for those i for which Di # 0. A separate argument, which we now give, is required for those indices i for which Di = 0. For such indices, the corresponding functional equations hold in a pointwise sense almost everywhere in 0.By computing the gradient of these pointwise equations, followed by an L2(0) inner product of the resultant with Vu&, we obtain, after some estimation,
+1
3llvU?il(Z2(n)
G
j# i
- +llV4c;,iIlVuf-
+ (ci,i +
2
l,illL2(n)
l,jllt2(n,(tf
-
C- 1)
y)
IIVucillZ2(n)(tf
-
(5.3.4)
C- 11,
for i = i, . . . , io ( D i = 0), and
3I(VU:illt2(n)
-
3IlV4'- 1 ,i(lLZ(n) 2
+
for i = io 1 , . . . , m (Di = 0). Summing (5.3.4) from k = 1 to k = r, we obtain, after transpositions from the right-hand side of (5.3.4) to the left, and vice versa, IIVUflillZ2(R)
2 +2 ~ 1 0 cj2,illVu~,jIIL2(~)
21lVuo,i11Z~(~)
+2
j# i
r- 1
1 1
k = 1 j:Dj#O
cj2,i(lv4,jlltz(n)
provided
+
[ 2 ~ , , ~(m- 1)](tf - tf-,) G
3,
k
=
1, . . . , M ( N ) .
(5.3.6)
By the previous part of the proof, we see that r- 1
An estimate similar to (5.3.7) holds, based upon (5.33, if i = io If we sum over i, such that Di = 0, we have, altogether,
1
i:Di = 0
IIVufIilliZ(n)
G mC
+ 2 (i,j:l 1
):I:1
Cj2.i
1
i:Di=O IIVU:illZ2(n)(tf
+ 1,. . . ,m. -
rf-
5.3 Existence for Reaction-Diffusion Systems
193
and an application of the discrete Gronwall inequality (cf. Proposition 2.2.1) yields the remaining conclusion, since only a finite number of partitions fail to satisfy (5.3.6). W
Remark 5.3.1. The proof of Proposition 5.3.1 reveals that the dependence of the constant C in (5.3.1)upon J (and its components g, and hi)is a multivariable polynomial dependence upon ess sup(J1 and ess sup1(dJ/duj)l. In particular, if fi is perturbed by a parameter-dependent expression in such a way that the perturbation is bounded, with (essentially) bounded partial derivatives, then the estimates of Propositions 2.5.1 and 5.3.1 continue to remain valid for the semidiscrete solutions of the perturbed system. Similarly, if fi is redefined over part of its domain, which can also be characterized as a perturbation, similar statements hold. Such a redefinition will be necessary to remove the assumption of strict inequality in (2.5.6). Definition 5.3.1. Suppose 1 d i d i,, and let 0 < E < (b, - ai)/2 be specified. We define the Lipschitz continuous function f ; as follows. If q E Q a i , then, as a function of qi, set
ff(~11 ,...,qi-lrqf,~li+l,...,qrn) a, d q: d a,
+
E,
where tfis the linear function, with value - E + J ( q ) at q: = ai,and value J ( q l , . . . , qi- 1, a, + E , qi+ . . . , q m ) at qj = ai+ E . A similar construction is carried out on [(ai+ bi)/2, b,] for q E Qbi,where tfis determined to have value f ( q l , . . . , qi- 1, b, - E , qi+ 1, . . . ,q m ) at b, - E , and E J(q) at b,.
+
Remark 5.3.2. The quantity f fis a Lipschitz continuous function and the Lipschitz constants are bounded independently of E . Important for our purposes is the fact that f f satisfies (2.5.6).Moreover, f fsatisfies the ordering of sign-regions property (1.3.6d), and f f is uniformly convergent to fi as E -+ 0 on R", if appropriately extended outside Q. Lemma 5.3.2. Proposition 2.5.3 remains true if strict inequality in (2.5.6) is relaxed to inequality as in (1.3.6b).
Proof: We consider the corresponding solutions ui;y of the semidiscrete system (2.5.2),when fi is replaced by f fin (2.5.2a).By the estimates obtained
5. Existence via Stability and Consistency
194
in Proposition 5.3.1, together with Remark 5.3.1, it is possible to obtain sequences, satisfying
-
w :
+: w
(weakly in H'(R; Rm)),
(5.3.8a)
(in L2(R;Rm)),
(5.3.8b)
with limits w : in H'(R; Rm)n L"(R; Rm).However, the limits
obtained via the triangle inequality, the uniform convergence of fi, (5.3.8b), and the Lebesgue dominated convergence theorem, together with (5.3.8), : are also solutions of (2.5.2a). A similar statement is true show that the w for (2.5.2b). Hence, wf = u:, by uniqueness of solutions of these equations. Since, by Proposition 2.5.3, Range(upsN)c Q, for each v in (5.3.8b), it follows that this property is preserved in passing to the limits. W We are now ready for the major theorem of this section. Theorem 5.3.3. Under the hypotheses of Section 1.3, there exists a unique solution of (1.3.1l), with Range(u) c Q.
Proof: The identity (1.3.1la) follows from the intermediate identity
J.[ $ + DiVui V$, + fi(u)lLi1dxdt -ui
*
for all $ E H'(i2) 0 C"([O, T o ] ) upon , integration by parts in the first term in (5.3.10), and use of the denseness of C"([O, T o ] )in L2(0,To).The verification of (5.3.10), however, follows exactly the lines of the proof of Theorem 5.2.1, with the following qualifications. We treat the terms as a unit, since this expression represents an equivalent inner product on H'(R) (see (1.3.14)). Also, it is evident that only sequences {u"} and {ii"} need be considered, in the notation of Lemma 5.2.6, satisfying
-
(weak* in L"(9; Rm) and L"((0, T,);H'(R; R'"))),
(5.3.11a)
uNr + u
(in L 2 ( 9 ;Rm)),
(5.3.1lb)
iiNi
(weakly in H'( [0, T o ] L2(R; ; Rm))),
(5.3.1lc)
uNi
-
*u
u
5.4
Existence for the Navier-Stokes Equations
195
where we have used the estimates of Propositions 2.5.1 and 5.3.1, and the Rellich compactness principle. Since, by Lemma 5.3.2, Range(uNi)c Q, this property is inherited by the limit in (5.3.1 lb). Thus, using (2.5.2) as a starting point, we may proceed, as in the proof of Theorem 5.2.1, to obtain (5.3.10) and, thus, (1.3.11a). Note that membership in (1.3.11b) follows from (5.3.11). This uniqueness was demonstrated in Proposition 1.5.3.
Remark 5.3.3. The pointwise estimates of Proposition 2.5.1, made possible by the assumption of L" initial data, considerably simplify the existence analysis by permitting us to assume that the componentsA off are Lipschitz continuous. In the analysis of the FitzHugh-Nagumo system, given by the writer in Jerome [25], this assumption was not made and a weaker form of (5.3.lb) was necessitated. This, in turn, required L' limit principles for the rapidly growing components and a substitute for the Rellich compactness criterion.
5.4
EXISTENCE FOR THE GENERALIZED FORM OF THE NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLUIDS
This section will be relatively brief since the key estimate and essential ideas are already in place. Also, we state the result in function space in the format of Remark 1.4.3, rather than in the more general setting suggested by Example 3.2.3.
Definition 5.4.1. Let V and V, be defined as in (1.4.20), (1.4.21), and let a ( . ; ; ) be a continuous trilinear form on V x V'ic V,, satisfying (1.4.22b) and (1.4.24). The notion of a weak solution, as defined in Definition 1.4.1, is retained. Let (P"} denote a sequence of partitions of [0, To], satisfying (5.2.15), (2.2.41c), and llBNll-,0, and let {uf} c V denote the recursively generated solutions of (3.3.32), understood as an identity on V,*,and guaranteed to exist by Proposition 3.3.4. We assume 3 c 8 < 1. The step function sequence {u"} and piecewise linear in time sequence {a") are defined as in Lemma 5.2.6. We also assume (5.4.2) (see (1.4.14), valid for n 2 2).
Theorem 5.4.1. A subsequence of {u"} (respectively {a")) is convergent in L 2 ( 9 ) to a weak solution of the generalized Navier-Stokes system (1.4.27) and (1.4.28). This assumes s 2 n/2 for the case (1.4.11).
5. Existence via Stability and Consistency
196
Proof: The boundedness of {u"} and {a"} in L2((0,To);V) is a direct consequence of Lemma 3.3.5. Moreover, the estimate M
1 I[('$
k= 1
-
'$- 1 ) -
l N (uk
N
- uk-
2 N I)IIV:(tk
-
N t k - 1)
G
c,
(5.4.1)
which implies that {iiN} is bounded in H'([O, T o ] ;V:), follows by directly estimating the norm of (3.3.32),and using the fact that
Il.llv:
Ila(w,w, 9llv: G
(5.4.2)
CIIWIIVIIWII",
in conjunction with (3.3.35). In particular, we conclude that {EN} is a relatively compact subset of L2((0,To);H), by Lemma 5.4.2, to follow. Altogether, then, we may select subsequences {uN1}and {ii"}, and a function u E (see (1.4.28a)), satisfying the following properties: %j
UN,
3"
-
(weakly in L2((0,T o ) V)), ;
u
(5.4.3a)
(weakly in L2((0,T o ) V) ; and H'([O, ToI;Vf)), (5.4.3b)
A
uNt -+ u,
6"
-,u
(in ~ ~ w)). ( 9 ;
(5.4.3c)
Some comments are in order concerning (5.4.3). That the limit in (5.4.3b) is the same for both spaces and coincides with the second limit in (5.4.3~)is standard and has already been discussed. Note the application of relative compactness. That the first limit in (5.4.3~)is a strong limit and coincides with the second limit is a consequence of Lemma 3.3.5. Indeed, by direct estimation, one sees that IIuNi- iiNillL2(9;Rn) + 0 if (3.3.33) holds. Finally, it is then immediate that the limit in (5.4.3a) and the first limit in (5.4.3~) coincide. Given a test function in the class Cm([0, T o ] ;V ) ,where -Ir is defined by (1.4.20c),we define the averages 4: as in (5.2.11). In the weak formulation, corresponding to (3.3.32), we select 4: for the test function, and apply summation by parts as in the proof of Theorem 5.2.1. Analogous to (5.2.27), we obtain
+
soT'
' (uN,t$N)V dt
+ so''
4N)dt = 0,
- ' a(uN,uN,
(5.4.4)
where CN is the vector analog of (5.2.14), and the inner product (.;)" is defined as suggested by (3.2.40a). Using Lemma 5.2.5, and the easily verified addendum that
-
4N 4
(strongly in L2((0,T,J;V, )),
(5.4.5)
5.5 Bibliographical Remarks
197
and using the convergence results (5.4.3),we may take limits in (5.4.4),much as in the proof of Theorem 5.2.1. Note that a relation analogous to (5.2.32) must be employed to conclude that
-
u Z ( ~ , ) * u(., To)
(weak *
in
V:).
(5.4.6)
Of course, (1.4.24b,c) must be used, together with the a priori estimates of (3.3.33). The generalization to the test function class W0 of (1.4.28b) follows the proof of Theorem 5.2.1, and depends upon the characterization of V, given by (1.4.21), and V given by (1.4.20a). W We close the section with the compactness property mentioned previously and a statement of the general principle.
Lemma 5.4.2. A set bounded in L'((0, T o ) ;V) and in H'([O, To]; V,*) is relatively compact in L2((0,T o ) H). ; This is a special case of the following (Aubin) lemma (see Lions [33] p. 58, Chapter 1 [23] p. 271). The original reference is C. R . Acad. Sci. Paris 256, 5042-5044 (1963). Lemma 5.4.3. Let X,, X, and X , be three Banach spaces, such that X, and X , are reflexive, and
x, c x c x,, where the embeddings are continuous and dense, with X, compactly embedded in X. Define, for p , and p1 greater than 1, and To > 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
BIBLIOGRAPHICAL REMARKS
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
5. Existence via Stability and Consistency
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.
5. Existence via Stability and Consistency
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
(see Serrin [41]; also Lions and Prodi [34]). Classical regularity results were obtained by Serrin [40]. In the case R = R", these results guarantee that u is C" in all variables, if u E Lp*' for 2 n -+-
It is, however, a multidimensional principle.
* J . Math. Pures er Appl. XI1 1-82, (1933);&id., XI11 331-418,(1934).
References
201
This implies that the Kiselev-LadyZhenskaja solution is C“ in all variables for R = R3. A fundamental point of view has recently been adopted by Bona and Di Benedetto [4], who have studied local properties of solutions without reference to initial and boundary data. These authors have studied C“ regularity in x for almost all t, and have used the equations themselves as the regularizing tool. These investigations were carried out in the case n = 2. We now make some final remarks. The reader will recall the fundamental role played by the lifting operators in Section 5.1 (see especially Proposition 5.1.1). We have made some historical comments concerning these operators in Chapter 2. Recent use of these operators to define suitable renormings has been made by Showalter [42], in applying Hilbert space methods to partial differential equations. We also call attention to very recent work of Alt and Luckhaus+ on quasi-linear elliptic and parabolic equations, including degenerate equations, such as the porous-medium equation and the equation describing the two-phase Stefan problem. Mixing of elliptic and parabolic regions is discussed for these general problems. Complicating the qualitative behavior of solutions of equations, such as the porous-medium equation, is the finite propagation speed associated with such equations, due to the fact that H(.) is not Lipschitz continuous. Thus, compact support of initial data is not instantaneously globally diffused as in truly parabolic problems. The reader is referred to this paper and to [2] for further details. REFERENCES [11 H. Amann, Periodic solutions of semilinearparabolic equations, in “Nonlinear Analysis” (L. Cesari, R. Kannan, and H. Weinberger, eds.), pp. 1-29. Academic Press, New York, [2] [3] [4]
[S] [6]
1978. D. G. Aronson, Regularity properties of flows through porous media. SIAM J . Appl. Math. 17, 461-467 (1969); A counterexample, Ibid. 19, 299-307 (1970); The interface, Arch. Rational Mech. Anal. 37, 1-10 (1970). P. Benilan, Solutions inttgrales d’equations d’evolution dans un espace de Banach, C. R . Acad. Sci. Paris Skr A-B 274, A47-A50 (1972). J. Bona and E. Di Benedetto, On the interior regularity of weak solutions of the NavierStokes equations in dimension 2, manuscript. H. Brtzis, On some degenerate nonlinear parabolic equations, in “Nonlinear Functional Analysis” (F. Browder, ed.) Part I, pp. 28-38 (Proc. Symp. Pure Math. 18). American Mathematical Society, Providence, Rhode Island, 1970. L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Dzflerential Equations 5,427-448 (1980).
’ Unpublished manuscript.
5. Existence via Stability and Consistency L. A. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana Uniu. Math. J . 28, 53-70 (1979); Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. SOC.252, 99-113 (1979); Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Uniu.Math. J . 29, 361-391 (1980). L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Rational Mech. Anal. (to appear). A. J. Chorin, T. J. R. Hughes, M. F. McCracken, and J. E. Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math. 31, 205-256 (1978). M. G . Crandall and L. C. Evans, On the relation of the operator ?ids + ?/as to evolution governed by accretive operators, Israel J. Math. 21, 261-278 (1975). M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93, 265-293 (1972). M. G. Crandall and A. Pazy, Semigroups of nonlinear contractions and dissipative sets, J. Functional Anal. 3, 376-418 (1969). A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equations 2. 1017-1044 (1977). E. Di Benedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mar. Puru App’pl. 130, 131-177 (1982). Continuity of weak solutions to a general porous media equation, Indiana Uniu.Math. J . , 1983, to appear. L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, in “Nonlinear Evolution Equations” (M. G. Crandall, ed.), pp. 163- 188. Academic Press, New York, 1978. A. Friedman, The Stefan problem in several space variables, Trans. Amer. Math. SOC. 133,51-87 (1968); One dimensional Stefan problems with nonmonotone free boundary, Ibid. 133, 89-114(1968). A. Friedman. “Variational Principles and Free Boundary Problems.” Wiley, New York, 1982. A. Friedman and D. Kinderlehrer, A one phase Stefan problem, Indiana Uniu. Math. J . 24, 1005-1035 (1975). H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal. 16,269-315 (1964). B. H. Gilding and L. A. Peletier, The Cauchy problem for an equation in the theory of infiltration, Arch. Rational Mech. Anal. 61, 126-152 (1976). R. Glowinski, J. Lions and R. Tremolieres, “Analyse Numerique Des Inequations Variationnelles,” Vols. I and 11. Dunod, Paris, 1976. E-I. Hanzawa, Classical solutions of the Stefan problem, T6hoku Math. J. 33, 297-335 (1981). E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundleichungen, Math. Nachr. 4, 213-231 (1951). J. W. Jerome, Nonlinear equations of evolution and a generalized Stefan problem, J. Differential Equations 26, 240-261 (1977). J. W. Jerome, Convergence of successive iterative semidiscretizations for FitzHughNagumo reaction diffusion systems, SIAM J. Numer. Anal. 17, 192-206 (1980). J. W. Jerome, Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities, Math. Meth. Appl. Sci. 2, 149-167 (1980). A. S. Kalashnikov, Formation of singularities in the solutions of the equation of nonstationary filtration, 2. VyEist. Mat. i. Mat. Fiz. 7 , 440-444 (1967) (Russian). [28] S. L. Kamenomostskaja, On the Stefan problem, Mat. Sb. 53,489-514 (1961)(Russian).
References
203
A. Kiselev and 0. Ladyihenskaja, On existence and uniqueness of the solution of the
nonstationary problem for a viscous incompressible fluid, Izo. Akad. Nauk SSSR 21, 655-680 (1957) (Russian). S. N. Kruzhkov, Results on the character of the regularity of solutions of parabolic equations and some of their applications, Mar. Zarnetki 6, 97-108 (1969) (Russian). S. N. Kruzhkov, First order quasilinear equations in several independent variables, Marh. USSR Sb. 10,217-243 (1970). 0. A. Ladyienskaja, The solution in the large of the first boundary-value problem for quasilinear parabolic equations, Trudy Moskoo. Mat. ObSf. 7 , 149-177 (1958) (Russian). J . L. Lions, “Quelques Methodes de Resolution des Problemes aux Limites non Lineaires.” Dunod, Paris, 1969. J. L. Lions and G. Prodi, Un thtoreme d’existence et d’unicite dans les equations de Navier-Stokes en dimension 2, C . R . Acad. Sci. Paris 248, 3519-3521 (1959). 0. A. Olehik, A method of solution of the general Stefan problem, Sooier Math. Dokl. 1, 1350-1354 (1961). 0. A. Olehik, A. S. Kalashnikov and C. Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, Izo. Akad. Nauk SSSR Ser. Math. 22, 667-704 (1958) (Russian). J. Rauch, Global existence for the FitzHugh-Nagumo equations, Cornrn. Partial Diferenrial Equarions 1, 609-621 (1976). E. H. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Marh. Ann. 102, 650-670 (1930); Uber die Wiirmeleitungsgleichung mit nicht-konstante Koeffizienten im raumlichen Falle, Erste Mitteilung, ibid. 104, 340-354 (1931); Ibid., Zweite Mitteilung, 355-362. E. S. Sabinina, On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Dokl. Akad. Nauk SSSR 136, 1034-1037 (1961) (Russian). J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Rational Mech. Anal. 9, 187-195 (1962). J. Serrin, The initial-value problem for the Navier-Stokes equations, in “Nonlinear Problems” (R. Langer, ed.), pp. 69-98. Univ. of Wisconsin Press, Madison, Wisconsin, 1963. R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal. 6,25-42 (1975); “Hilbert Space Methods for Partial Differential Equations.” Pitman, London, 1977.
This page intentionally left blank
LOCAL SMOOTH SOLUTIONS
I1
205
This page intentionally left blank
LINEAR EVOLUTION OPERATORS
6
6.0
INTRODUCTION
In this chapter, we begin with a given family { - A(t)} of closed generators of strongly continuous semigroups on a Banach space X, and construct the associated evolution operators { U(t, s)}, under an assumption of quasistability on { A(t)}, certain continuity assumptions, as well as the similarity of this family to the perturbation family {A(t)+ B(t)}, for an appropriately measurable family { B(t)} of bounded operators. The summarizing theorem containing the major result is Theorem 6.3.7. It is sufficiently strong to cover the nonlinear applications of the next chapter where {A(t)} is typically stable on X and also stable on an appropriate dense smooth subspace Y. In order to avoid a needlessly technical development, we carry out the details of the construction in Section 6.2 in the case where {A(t)}is stable on X and Y. The modifications required for quasi-stability are technical considerations of integration theory, and, for these, the reader is referred to the appropriate literature. In this chapter, X and Y are not assumed reflexive. The properties developed in Section 6.2 for the operators {U(t,s)} are not sufficiently strong to classify them as evolution operators. The remaining property is the invariant action of U(t, s) on a specified smooth space Y c DA(,).This is achieved in Section 6.3 by a similarity relation between U(t,s) 207
6. Linear Evolution Operators
208
and W(t,S) where the latter are defined as bounded operators on X via a Volterra integral equation. The convolution structure associated with solutions of such integral equations is compatible with the quasi-stability assumption on {A(t)}and the measurability of {B(t)}. For the reader's convenience, we have presented, without proof, in Section 6.1, a brief development of strongly continuous semigroups, particularly those properties required in the sequel. In the final section we prove the validity of the formal representation of solutions of abstract linear Cauchy problems, and use the classical example of a linear symmetric hyperbolic system to illustrate the concepts of the linear theory. This application, however, requires only the stability of { A ( t ) } ,not the more general notion of quasi-stability. Finally, it is possible to weaken the norm continuity assumption of this chapter to strong continuity. The reader is referred to the bibliographical comments.
6.1
SEMIGROUP PRELIMINARIES
We shall summarize the basic facts concerning semigroups of linear operators in this section. For proofs and further amplification, the reader is invited to consult the book of Butzer and Berens [3] and the classic treatise of Hille and Phillips [ 101. The semigroup notion may be thought of as an appropriate generalization of the separation-of-variables solution T(t)u, = u ( t ) =
1
ake-"'Uk
k
(6.1.1)
of the equation u, = -Au, where A is an appropriate positive-definite, self-adjoint Hilbert space operator, with eigenvalues { & } and normalized eigenfunctions { U k ] , and ak = (uo,t+.). The family {T(t)}tBOis called the semigroup generated by -A. The generalization applies equally to operators A, not possessing compact inverses, where (6.1.l) is replaced by an appropriate convolution. In general, the semigroup satisfies the properties T(t
+
(6.1.2a)
S) = T(t)T(s),
T(0) = I, lim T(t)f = f,
r+o+
where {T(t)},
(6.1.2b)
f EX,
,is defined on the normed linear space X.
(6.1.2~)
6.1 Semigroup Preliminaries
209
Definition 6.1.1. Designate by B[X] the Banach space of continuous linear transformations of a given Banach space X into itself. Then,
T(t):[0, 00)
+
B[X]
is called a semigroup of operators (in B[X]) if (6.1.2a,b) hold. The family { T ( C ) ) ~is~said , to be strongly continuous (of class ( C , ) ) if (6.1.2~)holds. Remark 6.1.1. All semigroups considered will be of class (C,). Proposition 6.1.l.+ (1) IIT(t)llis bounded on every finite subinterval of [0, 00). (2) For each f E X, the function
T(t)f:[O, co)+ X is (strongly) continuous. (3) If o,= inf{(f-)lnllT(t)ll:t t
=- 0
(6.1.3a)
o, = lim ((k)lnllT(t)[l)
< co.
(6.1.3b)
then t+ m
(4) For each w > w o ,there exists a constant M u , such that IIT(t)ll < Mueat,
t 2 0.
(6.1.4)
Definition 6.1.2. The infinitesimal generator U of the semigroup {T(t)}t20 is defined by 1 (6.1.5) U f = lim U,f, U, = [T(T)- I], z r+O+
whenever the limit exists. The domain of U, D,, is the set of elements f for which this limit exists.
'
Propositions 6.1.1, 6.1.2, 6.1.4 and Theorems 6.1.6, 6.1.7 are adapted from Butzer and Berens [3].
6. Linear Evolution Operators
210
Proposition 6.1.2. (1) The set D, is a linear manifold and U is a linear operator. (2) The family T(t) leaves D, invariant for t 2 0, and, for f E D, and t 2 0, d (T(r)f) = UT(t)f dt
-
T(t)f - f
=
(6.1.6a)
= T(t)Uf,
ji T(u)Ufdu.
(6.1.6b)
(3) The manifold D, is dense in X, and U is a closed linear operator. Definition 6.1.3. Let X be a Banach space, and let U be a linear operator with domain and range in X : D , c X, R, c X. Set
u, = A1 - u = A - u .
The spectrum of U is defined as the set a(U) = {A E C:U, does not have a continuous, densely defined inverse}. The resolvent set p(U) of U is the complement of o(U) in C. Thus, A E p(U) if and only if U;' exists as a bounded operator with dense domain. Set R(A, U) = U;
',
A E p(U).
(6.1.7)
The operator R( .,U) is called the resolvent of U. Proposition 6.1.3. If U is closed, then R,, R(I, U) transforms X one-to-one onto D,, i.e.,
(A1 - U)R(A, U)f
= f,
R(A, U)(AI - U)g = g,
if
A E p(U) and, thus,
f E X,
(6.1.8a)
g E Du.
(6.1.8b)
=X
Proposition 6.1.4. Let {T(t)}[>,be a semigroup of class (C,) in B[X], with infinitesimal generator U. If oo is given by (6.1.3), and Re2 > oo,then, A E p(U), and R(A,U)f
=
JOm
e-"T(t)fdt,
f E X.
(6.1.9)
Furthermore, for each f E X, (6.1.10)
6.1 Semigroup Preliminaries
211
The resolvent is an analytic function on p(U):
Remark 6.1.2.
m
R(I, U) =
1 (A0 -
- U)-k-
',
k=O
(6.1.11)
I&, - 11 < lIR(Io, U)ll- '. Thus, for such I , m
Ip(AU)ll G 1 p o - AlkpWo,U)(lk+'. k=O
(6.1.12)
It follows from (6.1.11) that
[R(A U)]"+' f
(6.1.13)
=
and, if (6.1.13)is applied to (6.1.9), there results 1
[R(I,U)]"+'f= 7 (- 1)" JOm n.
(- rye-"T(t)fdt.
In particular, if (6.1.14a)
IIT(t)ll G MeW', then
,"I
M
U)l'Il
o = R e I > w, r 2 1.
-7
(6.1.14b)
Proposition 6.1.5. Let U be a closed linear operator with dense domain and range in a Banach space X. Suppose there exist real numbers M and w, such that (6.1.14b)holds. For I > w, set B, = 12R(I, U) - I1 = IUR(I, U),
(6.1.15a)
and S,(t)
= exp(tB,) = e-'I
1[R(I, U)Ik. k! (I2t)k
k=o
(6.1.15b)
Then, lim BJ = Uf,
1- m
f E D,.
Moreover, the limit lim S,(t)f
I-
= T(t)f,
f
E
X,
(6.1.16)
a,
,~~ exists uniformly for t in any finite interval [O,b].The family { T ( C ) }defines
6. Linear Evolution Operators
212
a (C,) semigroup, satisfying IIT(t)ll d Meu',
t 2 0,
with infinitesimal generator U. Remark 6.1.3. The estimate (6.1.14b) directly implies (6.1.17) =M
exp [tAw(A-
'1,
0)-
and (6.1.14a) follows, if A + co in (6.1.17). Thus, (6.1.14a) and (6.1.14b) are seen to be equivalent, via the Laplace transform (6.1.9) and the exponential formula (6.1.16). Theorem6.1.6. A necessary and sufficient condition for a closed linear operator U, with domain dense in a Banach space X and range in X, to generate a semigroup (T(t):O d t < a}of class (C,) is that there exist real numbers M and w, such that A E p(U) for every real A > 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 <
-0,
(6.1.22b)
then A E G(X, 1,m) (see Bellini-Morante [2] p. 142).
6.2 THE LINEAR EVOLUTION EQUATION AND
EVOLUTION OPERATORS
We shall be motivated by the desire to represent, in terms of evolution operators, the solution of the abstract Cauchy problem for linear equations du
-
dt
+ A(t)u = F(t),
0
< To,
(6.2.1)
in a Banach space X. Equation (6.2.1) is of hyperbolic type, in the sense that the linear operators - A(t) are the infinitesimal generators of (C,) semigroups on X.The subspace DA(r) will not be required to be constant in t, but rather to contain a dense linear manifold Y for all t. The associated semigroups will be required to satisfy a certain stability or quasi-stability condition. Definition 6.2.1. Let X be a Banach space, and suppose that a family A(t) E G(X) of linear operators is given on 0 < t < T o .The family {A(t)} is said to be stable if there are (stability) constants M and 0,such that (6.2.2a) j= 1
6. Linear Evolution Operators
214
’,
for any finite family {ti};= with 0 < t, < . . . < t k < To, k = 1,2, . . . .Moreover, is time-ordered, i.e., [A(tj) + A]-’ is to the left of [A(t,.) + A]-’ if j > j ’ . More generally, {A(t)}is said to be quasi-stable if there exists a constant M and a real-valued, upper Lebesgue integrable function o,defined almost everywhere on [0, To], such that
n
[I fi
j= 1
(A(tj)
+
n k
A)-l/I
< M j = 1 ( I - o(cj))-’,
I > 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<
Mexp{w(s,
+ . . . + sk)}
resp. M e x p x s,w(tj) j
(6.2.3)
(6.2.4) Propositions 6.2.1-6.2.4, Theorem 6.2.5, and Lemmas 6.2.6, 6.2.7 are adapted from Kato [ l l , 123.
6.2 Linear Evolution Equation and Evolution Operators
215
for A j > 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)
6.2 Linear Evolution Equation and Evolution Operators
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,
6.2 Linear Evolution Equation and Evolution Operators
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
< s < t.
For example, if
then set s.,
=((i
- l)/n)To,and define
U,,(t,r)= exp{(s, - t)A[(v)To]}exp{(r
- s,)A[(?)T,]].
Now, U,,(t,s) clearly satisfies (c) and (d), if s # (j/n)To.1 < j
d
U,@,slg = -A,(WJ,(t, s)g9
-
dt
g
E
< n - 1, and
Y,
(6.2.9)
if t # (j/n)To;it is necessary to note here that U,,(t,s)Yc Y by (2). By the stability of {A(t)}, {A@)}, and (6.2.3),we have IIU,,(t,s)(lx < M exp{w(t - s)}
Ilu,,(t,s)lly < fi exp{W - s)} (6.2.10)
We shall show that lim U,,(t,s): = U(t,s)
n+ m
(6.2.11)
exists strongly (X)uniformly in t,s, t 2 s. It will then be clear that U(t,s) satisfies (a) and (b) by inheriting the corresponding properties of { U,,(t,s)}.By the first inequality in (6.2.10),it suffices to show that U,(t, r)g exists in X, for each g E Y, uniformly in t, r. Now, differentiating U,,(t,s)U,(s, r)g with respect to s, by using (d) and (6.2.9),and then integrating in s, we obtain
6. Linear Evolution Operators
220
the identity
U,(t, s)[A,(s) - A,(s)]U,(s,
= -
r)gds, (6.2.12a)
which immediately gives the estimate IIU,(t, r)g - U,(t, r)gllx d MfieY('-') llglly Jrt IIAn(s) - Arn(s)Ily,X ds, (6.2.12b) where y = max(w, 0). By (6.2.8) and the triangle inequality, we have IIAn(s)Arn(s)lly,X .+ 0, n, m -+ co uniformly in s, so that {U,(t, r ) g } is Cauchy and, hence, strongly convergent. We are now ready to prove (c) and (d). Fix 0 < r < T o ,and set A'($ = A(r) for s E [0, To]. We may identify this family with that described in Lemma 6.2.6, to follow. In particular, the right member of (6.2.14) is o(t - r) as t 1 r. On the left, we have ~ ' ( tr)g , =
- (f - r)A(r)
9,
which has the right derivative -A(r)g at t = r. Hence, division of (6.2.14) by ( t - r) yields (c) upon letting t 1 r. By fixing 0 d t < T o ,and setting A'(s) = A(t) for s E [O, To], we obtain for the left-hand derivative [D;U(t, s)91,=, = A(t)g.
(6.2.13)
Property (d) is now proved by reductions to (c) and (6.2.13). Thus, if s < t, we have D:U(t,s)g
+ h)g - U(t,s)g]} = lim {u(t,s + h)h-'[g - U(s + h,s)g]} = lim h10
{h-'[U(t,s
h10
= U(t, s)A(s)g,
by (c) and the strong continuity (X) of U(t, s). Again, for s d t, D;U(t,s)g = lim{h-'[U(t,s)g - U(t,s - h)g]) h10
= lim{U(t,s)h-'[g h10
=
U(s,s - h)g])
U(t, s)A(s)g,
by (6.2.13). This proves (d). To verify uniqueness, suppose {V(t,s)} satisfies (a) and (d). Then, differentiation of V(t, s)U,(s, r)g in the variable s, by use of (d), and (6.2.9) gives,
6.2 Linear Evolution Equation and Evolution Operators
221
after integration,
for g E Y. Using (a), (6.2.8),and (6.2.10),we find, as above, that V(r, r)g = lim Un(r,r)g n+
m
= U(t, r)g,
hence, V(t, r) = U(t, r). We cite now the lemma referred to in the proof of Theorem 6.2.5. We omit the proof, which follows the verification of (6.2.12), with an additional limiting process.
Lemma 6.2.6. Let {A’@)}be another family, satisfying the assumptions of Theorem 6.2.5, with stability constants M’, of,k’, 6’. Let {U’(t,s)} be constructed from {A’(t)}, as in the construction (6.2.11). Then, r)g - U(t, r)gJJ,< 1 M ! k e y ( t - r ) 1JgJlv IJA’(4- A(S)llY,XdS, (6.2.14) JJU’(4
where y = max(o’,6).
Remark 6.2.4. In the more general case of quasi-stability, considerable care must be exercised in the construction of both the partitions and the step-function sequence, itself. Fundamental to these constructions is the approximation of measurable functions, with values in a Banach space, by Riemann step functions. These step functions, and the associated partitions, provide the basis for the definition of A,,(t). The reader is referred to the paper of Kato [12], especially the Appendix, for details. We note here that the additional modifications involve the replacement of (6.2.10)by
and the replacement of Lemma 6.2.6 by the more special application required to prove the differentiability. The following proposition gives a sufficient condition for hypothesis (2) of Theorem 6.2.5. Proposition 6.2.7. Hypothesis (2) of Theorem 6.2.5 is implied by: (2’)There is a family { S ( t ) } of isomorphisms of Y onto X,such that
S(t)A(t)S(t)-’ = A,(t) E G(X),
0< t
< To,
(6.2.16)
6. Linear Evolution Operators
222
and that {Al(t)} is a quasi-stable family on X, say with index ( M l , w l ) . Furthermore, there is a constant c, such that IIS(t)llY,xd c, IlS(t)- lIJx,y < c, and the map t H S ( t ) is of bounded variation in B(Y, X) norm. If {Al(t)}is stable, then a stronger form of hypothesis (2) holds in which {&t)} is stable on Y.
Proof: We give the details only in the case of stability. From Proposition 6.2.4, we conclude that Y is A(t)-admissible. Let &t) E G(Y) be the part of A(t) in Y.To show that {&)) is stable, we write, by (6.2.7) and Proposition 6.2.3, (&t) A)-' = S(t)-'(A,(t) A)-'S(r),
+
+
so that (6.2.17) Setting Pj = [S(tj) - S(tj_l)]S(tj-
so that S(tj)S(tj_1)-
=I
1)- l ,
(6.2.18)
+ Pj,
we have, for (6.2.17),
x
fi (Al(tj)+ A)-',
j=1
where k > j l > j 2 > . . . > j r > 0. If (6.2.2a) is used to estimate the product resolvent blocks, and the estimator polynomial is then recombined, we obtain
6.3 Regularity of Evolution Operators
223
it follows that C!=2llPjlIx < cV, where V denotes the total variation of S ( . ) . The right-hand side of (6.2.20) is, thus, bounded from above by - w,)-keCM1".
Remark 6.2.5.
If (2') holds, then
(M1c2eCM1", w l).
6.3
{A(t)}is
quasi-stable on Y with index
PERTURBATIONS OF GENERATORS AND REGULARITY OF EVOLUTION OPERATORS
We begin with a standard type perturbation result. Proposition 6.3.1. (1) If A E G(X,M,w) and B E B[X], then A + B E G(X, M , w llBllM). (2) Let {A(t)} be quasi-stable on [0, To],with index (M,o), and let B(t) E B[X] be a family defined almost everywhere on [O,To], such that [lB(.)llis upper Lebesgue integrable. Then {A(t) + B(t)} forms a quasistable family on [0, To],with index ( M , w + MllB(.)11).The expression {A(t) + B(t)} is stable, with index ( M , w + MP), if {A(t)J is stable and IIB(t)ll ,< /3 for all t E [0, To].
+
Proof: We follow [lo, Theorem 13.2.11 with minor modifications. Suppose the hypotheses of (1) hold. Then, the resolvent identity (when R(1, - A - B) is defined), R(1, - A - B) - R(1, - A )
= R(1, - A - B)(- B)R(1, -A),
leads to a,
R(1, - A - B) =
1 R(1, -A)[
- BR(1, -A)]',
(6.3.1)
j= 0
if [IBR(A-A)[I < 1. In particular, the latter holds, if
1 > w1 =
+ MIIBII,
(6.3.2)
6. Linear Evolution Operators
224
since, in this case,
Now (1) and (2) are verified by the estimates
< M(I - o , ) - k ,
and
(6.3.3a)
respectively. We shall derive the first estimate, since the second is proved in a parallel manner. A typical term containing L' of the B s in the expansion within 11.11 in (6.3.3a)is of the form
[R(I, -A)]"(-B)[R(I, where Eft: ri = k norm, by
-A)]"(-B).
. . [R(I, -A)]"(-B)[R(I,
-A)]"+',
+ L', and ri > 0. Such a term is estimated from above, in
M ( I - O)-~'IIBIIM(A- o)-"IIBII . . . M(A - o ) - ~ ' ~ ~ B ~ ~-Mo () -I~ ' + ' - Mc?++' 11B 11 c?( I - O ) - ( k + c?). Now, the number of terms containing / of the B s is the binomial coefficient C; in (1 - x ) - ~= XF= Czx'. Thus,
Remark 6.3.1. The second topic in this section deals with the regularity of the evolution operators, or, more precisely, the invariance of Y under U(t, s), and the differentiability of the latter with respect to the first argument. To
6.3 Regularity of
Evolution Operators
225
achieve this, we shall introduce an operator convolution structure, and examine the solution of certain Volterra integral equations. We prepare for this with a definition and technical lemma. Definition6.3.1. Set J=[O,T,], and A={(t,s):Ogsgr
These are sometimes abbreviated to Ilf I(m,X and Similar notation holds for functions on A to X. For an operator-valued function F: J + B(X, Y) almost everywhere or F: A + B(X, Y) almost everywhere we use the obvious symbols, sometimes abbreviated to IIFllm,x,yand ~ ~ F ~ \ l ,respectively. x,y, We even suppress X, Y without ambiguity on occasion.
llflll,x.
Lemma 6.3.2. Suppose that X‘, Y’, X”, and Y” are given Banach spaces. Let G’:A + B(X’,Y’) and G“:A + B(X”, Y ” )be strongly continuous. Let F:J + B(Y’,X”) almost everywhere be strongly measurable with llFlll < 00. Then, there is G : A -,B(X’, Y”), denoted by G = G”FG’, such that G(t, r ) f
=
s’ G”(t,s)F(s)G’(s,r ) f ds,
(6.3.4)
( t , r ) E A, for each f E X ’ . The function G is strongly continuous on A to B(X’,Y”), and
IlGfllm
< IIG”II~ l l ~ l l ~ l l ~ ~ l l ~ ~
IIGllm
IIG”l1~IlFII~lIG’Il~~
(6.3.5a) (6.3.5b)
Proof: We omit the proof of this important technical result and refer the reader to Kato [12, p. 6521.
Corollary 6.3.3. Let D : J 4 B[X] almost everywhere be strongly measurable, with IIDlI, < 00,and let {U(t,s)}be the evolution operators ofTheorem 6.2.5. Then, for each integer p 2 0, the p-fold convolution estimate II(UD)PUII,,x < Mell”lllMPllDll:/P!
(6.3.6)
holds on A. Thus, the Neumann series m
Z
=
1 (-UD)W
p=o
(6.3.7)
6.
226
Linear Evolution Operators
converges uniformly on A in B[X] norm; Z satisfies the integral equation
z(t,r)f
= W t , r)f -
z(t,s)D(s)U(s,r)fds,
(6.3.8)
for f E X and ( t , r ) E A.
Proof: To verify (6.3.6), let 4 : J -+ [0, co)be Lebesgue integrable, with MIID( .)/Ix d 4. Then, by (6.3.5), if t = t,, and X' = X" = Y' = Y" = X,
Jz"
4(t,). . . J: 4 ( t , ) d t , . . . dt,
II[(UD)PU](t,r)ll d
where @(u)=
J; 4(s)ds.
Note that is the cumulative exponential part of the evolution operator estimates. Relations (6.3.6)and, thus, (6.3.7) are now immediate, upon taking an infimum over 4 and a supremum over ( t ,I ) . The solution of (6.3.8) may be characterized by Z ( t , r ) f = (1 + *(.))-'u(t,r)f,
where the operation
(6.3.9a)
* is defined by * ~ ' ( rt),f
=
J ~ ' ( st ), ~ ( s ) ~ r( )sf ,.
(6.3.9b)
The expansion of (6.3.9)is just (6.3.7), which shows the equivalence of (6.3.7) and (6.3.8). We return now to the notational framework of the previous section. The use of the Volterra equation follows in the key identities (6.3.12) and (6.3.13). We require one further technical result which affords a transition between the operators { S ( t ) } , introduced in Proposition 6.2.7, and the hypotheses of the previous corollary.
Lemma6.3.4. Let S : J -B(Y,X) be an indefinite strong integral of a strongly measurable function S : J -,B(Y, X), with ]IS(.)IIy,x upper Lebesgue integrable on J . Let S-'(.) exist and, hence, be bounded on J . Then S ( . ) and S - '( .) are absolutely continuous in operator norm on J , and (d/ds)S- '( .) exists strongly almost everywhere on J , with (d/ds)[S(s)-'f]
=
-s(s)-'S(s)s(s)-'f,
f
E
x.
(6.3.10)
6.3 Regularity of Evolution Operators
227
Proof: The absolute continuity of S( .) in operator norm is immediate from
I(W)- S(S)ll d
JTt) IIS(r)ll dr,
0 <s
< To.
The boundedness of S - ' ( * ) on J is due to the fact that the image of the closed unit ball in Y, under S(t), contains a fixed ball in X, independent of t, due to the continuity of S( .) on the compact set J . The absolute continuity of S-'(.) follows from s(t)- - S(s)-
- S(t)]S(s)-',
= s(t)-'[S(S)
the absolute continuity of S( .), and the boundedness of S- '( .). The strong differentiability of S - ' ( * ) and (6.3.10) follow from s(t)-' - S(s)-' t - S
= s(t)-
(-
S(t)
+ S(s))s(s)-
1,
t - s
the strong differentiability of S( .), and the uniform boundedness of S- '( -).
Theorem 6.3.5. Suppose that hypotheses (1) and (3) of Theorem 6.2.5 are satisfied together with the following hypothesis. (2") There is a family { S ( t ) } of isomorphisms of Y onto X, such that S(t)A(t)S(t)-' = A(t) + B(t),
B(t) E B[X],
(6.3.11)
almost everywhere on [0, To], where B( * ) is strongly measurable, with l\B(*)llxupper Lebesgue integrable on [0, To]. Furthermore, there is a strongly measurable function S :[0, To] + B[Y, XI, with (IS(.)IIy,x upper integrable on [0, To ] ,such that S is an indefinite strong integral of Then (2) of Theorem 6.2.5 holds, hence, the evolution operators {U(t,s)} exist. Moreover, if {W(t,s)} c B[X] are uniquely defined by
s.
W(t,r)f
for f
E
= U(t,r)f -
X and 0 < r
fW(t,s)[B(s)
- C(s)]U(s,r)fds,
< t < T o ,where C(s) = S(s)S(s)-', U(t,s) = s(t)-'w(t,s)S(s).
(6.3.12)
then
(6.3.13)
In particular, the following additional conditions hold. 0 < s < t < To. (e) U(t,s)Y c Y, (f) The operator function U(t,s) is strongly continuous (Y), jointly in s and t.
6. Linear Evolution Operators
228
(g) For each fixed g E Y, and 0 d s < T o ,
for s d t d T o ,and this derivative is continuous (X). Proof: The implication (2") * (2) is a consequence of Propositions 6.2.7 and 6.3.1,and Lemma 6.3.4.The existence of a unique solution W(t, s) E B[X] of(6.3.12),strongly continuous jointly in s, t , is guaranteed by Corollary 6.3.3. Note that D = B - C is strongly measurable, llDlll c co, by (2") and Lemma 6.3.4. The verification of (6.3.13) is the heart of the theorem, since (e) and (f) follow directly; (g) is then seen to follow, via the right and left difference quotient relationships U(t
+ h,s)g - U(t,s)g -- (U(t + h , t ) - I)U(t,s)g h
h
9
and (a)-(f). We now verify (6.3.13).+Define Q(t,s) = U(t,s)S(s)-'. It is enough to show Q(t,S)
=
(6.3.15)
S ( t ) - W(t, s).
It follows from (6.3.10)and (d) that, for each f ' X, ~ and almost all s (d/ds)Q(t,s)f = U(t,s)A(s)S(s)-'f- U(t,s)S(s)-'S(s)S(s)-'f' = U(r,s)A(s)S(s)-'f' - Q(t, s)C(s)f. Since A(s)S(s)-'Y c Y by (2"), we then have A(s)S(s)-' g
= S(s)-
'S(s)A(s)S(s)-' 9 = S(s)- '[A(s)
for each y E Y. Thus, for g E Y and 0 Q s Q f s on this interval,
+ B(s)]g,
< T o ,we have, for almost all
(d/ds)Q(t,s)g = Q(t, s)[A(s) + B(s) - C(s)]g.
(6.3.16)
If { U,(t, s)) are the approximating evolution operators constructed in the proof of Theorem 6.2.5, or the modified operators constructed in [12],
' Adapted from Dorroh [ 6 ] . This paper represents a significant simplification of the earlier development of the theory.
6.3 Regularity of Evolution Operators
229
then, by (6.2.9) and (6.3.16), (dlds)Q(t,s)U,(s,
=
Q(t3s)[A(s) + Ws) -
- A,(s)]U,(s, r)g,
(6.3.17)
for each g E Y and for almost all s. Integrating (6.3.17) from r to t, and utilizing the absolute continuity of the integrand, we obtain s(t)-'U,(t, r ) g - Q ( t , r)g =
c
Q(t, s)[A(s)
+ B(s) - C(s) - A,(s)IU,(s, r)gds,
for g E Y. Using (6.2.8) and (6.2.10) (or (6.2.15)), and letting n obtain Q(t,r ) g = S
( K ' U k r)g -
-, 00,
we
fQ ( t , s)[B(s) - C(s)]U(s, r)gds, (6.3.18)
for each g E Y; by continuity, (6.3.18) holds on all of X. Thus, Q(t,s) = U(t, s)S(s)- ' may be characterized as the unique solution of(6.3.18). However, by (6.3.12), S ( t ) - 'W(t, r ) f
= S ( t ) - 'U(t,
r)f
-
'
S ( t ) - 'W(t, s)[B(s) - C(s)]U(s, r ) f d s ,
where we have interchanged S ( t ) - and of (6.3.18) and (6.3.19)gives (6.3.15). W
(6.3.19) .)fds by continuity. Comparison
Corollary 6.3.6. The estimate on Y, IlU(tAIl,,Y
IISllm,Y,XMexP{ll~II1 + MllB - cIIl,x~IIS-'llm,x,Y~ (6.3.20)
holds for the operators satisfying the hypotheses of Theorem 6.3.5. Proof: B-C.
This follows directly from (6.3.6), (6.3.12), and (6.3.13), with D =
We summarize the discussion of these two sections with the following theorem. Theorem 6.3.7. Assume that hypotheses (1) and (3) of Theorem 6.2.5, and (2") of Theorem 6.3.5 hold. Then, there exists a unique family {U(t,s)}, defined on A:O < s Q t Q T o ,with the following properties. (a') The operator function U is strongly continuous on A to B[X] with U(s,s) = I.
6. Linear Evolution Operators
U(t,s)U(s, r) = I. U(t,s)Y c Y, and U is strongly continuous on A to B[Y]. dU(t,s)/dt = - A(t)U(t,s), dU(t,s)/ds = U(t,s)A(s), which exist in the strong sense in B[Y, XI, and are strongly continuous on A to B[Y, XI.
THE INHOMOGENEOUS PROBLEM AND AN APPLICATION TO LINEAR SYMMETRIC HYPERBOLIC SYSTEMS
Let F : J + X be given. We consider the linear Cauchy problem
du - + A(t)u(t) = F(t), dt
0d t d To,
(6.4.1a) (6.4.lb)
40)= uo, in the Banach space X. The solution is formally given by
+ Ji U(t,s)F(s)ds,
u(t) = U(t,O)uo
0 < t d To,
(6.4.2)
where {U(t,s)} satisfy Theorem 6.3.7. Proposition 6.4.l.+ Let u be given by (6.4.2). (1) If uo E X and F *
E L'(J;X),
then u E C(J;X) and I1UIIw.x(ll~ollx+ I I F l l L X ) .
IIUllm,x
(6.4.3)
(2) If uo E Y and F E L'(J;Y), then u E C(J;Y) and II4lm.y
d
+
I I ~ l l ~ . y ( l l ~ o l l llFll1.y). y
(6.4.4)
(3) If uo E Y and F E C ( J ; X ) n L'(J;Y), then u E C(J;Y) n C'(J;X), and u satisfies (6.4.1)and the estimate
(6-4.5) Ild~/dtllm,x< IIFllm,x + I I A l l w , Y , X l l U ( l o o , Y ( ~ ~ ~+O IIFllLY) ll~ holds.
Proof: Results (1) and (2) follow routinely from the properties of {U(t,s)}; (6.4.5) follows from (6.4.4) and (6.4.1). To prove the latter, we set up the
' Reproduced from Kato [12];
used with permission.
-
6.4
Inhomogeneous Problem and an Application
231
difference quotient
+
Ji U(t,s)F(s)ds
-
U(t,s)F(s)ds
t-r (6.4.6)
The second term on the right-hand side of (6.4.6) tends to the limit U(r,r)F(r)= F(r) as t -+ r, by the property F E C ( J ;X), and the third term tends to - A(r)U(r,s)F(s)ds =
-A(r)
U ( r ,s)F(s)ds,
by the property F E L'(J; Y). The first term, of course, tends to - A(r)U(r,O)u,. This verifies (6.4.1). The regularity of u follows from (6.4.1) and (6.4.2), and the properties of { U ( t , s ) }and { A ( t ) } . We now give an application to symmetric hyperbolic systems. Definition 6.4.1.
Consider the linear Cauchy problem, au
-
at
+ 1 a j ( x ,t ) au + b ( x ,t)u = F(x, t), n
-
j=l
axj
u(x,0)= u,(x),
(6.4.7a) (6.4.7b)
x E R",0 < t < T o . Here, u = ( u l , . . . , urn), a j , and b are rn x rn matrix functions, and the a j are assumed to be Hermitian symmetric. Regarding the regularity of the aj and b, we assume: The functions aj and b are in C ( J ;C,'(R")) as mappings of J into C,'(W). Here, C,'(R") is the set of all rn x rn matrix-valued functions a, such that a and aa/axj, 1 < j < n, are in the space C,(Rn) of continuous, bounded functions on R".The supremum norm for C,(R") and Ci(R") is taken on the Euclidean norm of a. In the remainder of this section, we make the identifications X = L2(R"4.Rm) and Y = H'(R"; Rm). The following lemma, proved in Friedrichs [8], permits the recasting of (6.4.7) into (6.4.1).
Lemma 6.4.2.
Let Ao(t)be defined by (6.4.8)
6. Linear Evolution Operators
232
for 4 E D A , = C;(W). Then, the linear operator A,(t), defined by the formal adjoint, or weak, relation, (u, A;(t)4)LZ(IW") = (v, 4)LZ(IW")
for all
4 E C;(R"),
(6.4.9)
where v = A,(t)u, u E D A , ( t ) , is identical to the closure of Ao(t) in L2(R",Rm), sometimes called the strong extension A,(t) of Ao(t). Definition 6.4.2. We denote by A(t) the common operator A,(t) = A,(t). The formal adjoint, as usual, is defined by
Remark 6.4.1. t E J.
The space H'(R"; Rm)is continuously embedded in
Lemma 6.4.3.
The energy inequality (A(t)u, U)L'(IW") 2 - 4 u , U)LZ(IW")
DA([),
(6.4.10)
holds for all u E D A ( [ ) , where
Proof: It suffices to prove (6.4.10) for u = 4 E D A o , since A(t) is closed. Writing
6.4 Inhomogeneous Problem and an Application
233
Lemma 6.4.4. The relation {P:ReP < - w,) = P(A(t))
(6.4.13)
holds for each t E J .
Proof: The operator A,(t) = A(t)
+ (w, + 2)1
(6.4.14)
is clearly injective for 2 5 0 by (6.4.10), and has dense range by (6.4.9). The (uniform) continuity of [A,(t)]- on the range of A&) follows from (6.4.10). In particular, 0 E p(A,(t)). The conclusion (6.4.13) is now standard (see Agmon [I] Theorem 12.8). Remark 6.4.2. Lemmas 6.4.3 and 6.4.4, taken together, imply A(t) E G(X,l , ~ , )via , the Hille-Yosida theorem, as expressed in Remark 6.1.5. In particular, (A(t)} is stable, with stability constants M = 1 and w = sup{w,:t E J } . Definition 6.4.3. We now define S ( t ) = S = ( I - A)"',
(6.4.15 )
which is an isomorphism of H'(R"; Rm)onto L'(R"; Rm).Here, we may use the definition, which makes S an isometry,
where denotes the Fourier transform, defined by A
for g in the Schwartz class 9 of rapidly decreasing functions at infinity. With this notation, as usual, if S is understood to induce an action on each vector component,
S = (I
- A)'/' = F-'(l
S-' = ( I -A)-'/'
+ lc1')'''FI
= F-'(1
+ ~~~')-'~'F.
Remark6.4.3. The derivative a/dxj commutes with S on 9, and 1.
Il(a/axj)s-'II,z,Rn)
6. Linear Evolution Operators
234
Theorem 6.4.5. The operator S of (6.4.15) satisfies (2") of Theorem 6.3.5. In particular, the evolution operators { U(t,s)} exist for (6.4.7), satisfying (a')-(d) of Theorem 6.3.7.
Proof:
The only condition to be verified is that SA(t)S-'
= A(t)
+ B(t),
(6.4.16)
where B(t) E B[X] is, say, strongly continuous (X) as a mapping from J into B[X]. In fact, we shall show that the mapping is norm continuous. One computes, for u E 9,+ SA(t)S-Iu
n
=
1 Saj
j= 1
=
i
j= 1
+
ajS(&)S-lu
il
n
= A(t)u
+ 1 [S,aj]
S-'u
[&aj]
S-'u
j= 1
+ bu + [S,b]S-'u
+ [S,b]S-'u.
According to a result of Calder6n [4], [S, a( .)] can be extended to a bounded operator from Lz(Rn;R") to L2(W;R"), if a~Cbl(R"),with bound < ~ l l a ( ( , - ~ ( ~Since .,. [s,aj(.,t)]
-
[S,aj(.,t')]
=
[S,aj(.,t) - aj(.,t')],
it follows from Remark 6.4.3 that
with a similar statement for [S, b1S-l. Thus, SA(t)S-'u
= A(t)u
+ B(t)u,
u
E
9,
(6.4.17)
where B(.) E C(J;B[X]). It remains to prove that (6.4.17) holds on DA(*). Thus, let v E DA(f), and let A(t)Uk + A(t)V
Uk --* V,
with {uk} c 9. Then, B(L)Uk -+ B(t)V, s- 'Uk A(t)S-'uk
=
in L2@";[W"), --*
S-'V, and
s--'(A(t) + B(t))Uk+ S-'(A(t) 4- B(t))v.
Since A(t) is closed, it follows that A(t)S-'v exists and equals S-'(A(t) B(t))v. Thus, SA(t)S-' 3 A(t) B(t).
+
' The symbol [aj] is used to emphasize operator action.
+
References
It follows that
235
+
S(A(t) l , ) - ’ S - ’
+ B(t) + A)-’,
I> (A(t)
(6.4.18)
for sufficiently large A. Since the right-hand side of (6.4.18) has domain Lz(R“;Rm), it follows that (6.4.18) represents an equality on L2(R”;Rm). Taking inverses leads to (6.4.16).
Remark 6.4.4. A comprehensive analysis of linear symmetric hyperbolic systems was given by Friedrichs [9] for the pure Cauchy problem, as well as the mixed initial/boundary-value problem.
6.5
BIBLIOGRAPHICAL REMARKS
We have followed the development of Kato in his two fundamental papers [ l l , 121, and have incorporated the logical simplification, due to Dorroh [6], concerning the invariant action of U(t, s) on the smooth space Y. The requirement that A( .) be continuous from [0, To] to B(Y, X) may be appropriately relaxed (see Kobayashi [13]) to strong continuity. Recent alternative treatments have been given by Dorroh and Graff [7], and by Dollard and Friedman [ 5 ] , and the latter exposition, based on product integration, serves as a nice introduction to the present development. It should be recognized that we have presented this particular development not only for its generality, but also because it is sufficiently robust to serve as a basis for the nonlinear theory developed in the next chapter.
REFERENCES [13 S. Agmon, “Elliptic Boundary Value Problems.” Van Nostrand-Reinhold, New York, 1965. [2] A. Bellini-Morante, “Applied Semigroups and Evolution Equations.” Oxford Univ. Press (Clarendon), London and New York, 1979. [3] P. L. Butzer and H. Berens, “Semigroups of Operators and Approximation.” SpringerVerlag, Berlin and New York, 1967. [4] A. P. Calderh, Commutators of singular integral operators, Proc. Nut. Acad. Sci. 53 1092-1099 (1965). [ 5 ] J. Dollard and C. Friedman, On strong product integration, J . Functional Anal. 28 309-354 (1978).
236
6. Linear Evolution Operators
[6]
J. R. Dorroh, A simplified proof of a theorem of Kato on linear evolution equations, J . Math. SOC.Japan 27,474-478 (1975).
[7]
J. R. Dorroh and R. A. Graff, Integral equations in Banach spaces; a general approach to the linear Cauchy problem and applications to the nonlinear problem, J. Integral Equations 1, 309-359 (1979). K. Friedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. SOC.55, 132-151 (1944). K. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 , 345-392 (1954); Symmetric positive linear differential equations, ibid. 11, 333-418 (1958). E. Hille and R. S. Phillips, “Functional Analysis and Semigroups.” American Mathematical Society Col. Publ. 31, Providence, Rhode Island, 1957. T. Kato, Linear equations of “hyperbolic” type, J. Fac. Sc. Uniu. Tokyo 17, 241 -258 (1970). T. Kato, Linear evolution equations of “hyperbolic” type 11, J . Math. SOC.Japan 25, 648-666 (1973). K. Kobayashi, On a theorem for linear evolution equations of hyperbolic type, J . Math Soc. Japan 31,647-654 (1979). K. Yosida, “Functional Analysis,” 5th. ed. Springer-Verlag, Berlin and New York, 1978.
[8] [9] [lo] [ l 11 [12] [13]
[ 141
QUASI-LINEAR EQUATIONS OF EVOLUTION
7
7.0
INTRODUCTION
An existence, uniqueness, and well-posedness theory, which is local in time, is presented in Section 7.2 for quasi-linear equations of evolution in separable reflexive Banach spaces (see Theorem 7.2.4); the preliminaries for this theory are presented in the first section. The robustness of the theory is demonstrated in Section 7.3, where it is shown that a significant class of quasi-linear second-order hyperbolic systems is covered by the abstract theory (see Theorem 7.3.6). A nontrivial application is presented in Section 7.4, where the vacuum field equations of general relativity are discussed. The standard reduction afforded by harmonic coordinates is utilized. In the final section, an invariant time interval is determined on which the solutions u" of the Navier-Stokes system for an incompressible viscous fluid tend to the solution of the Euler equations for an ideal fluid as v --t 0. This result (see Theorem 7.5.4) is discussed in the format of much more general results, including a stability analysis for the horizontal-line method. The basic ideas could be used, if desired, to present an alternative approach to that of Section 7.2. The advantage here is that the evolution operators, and the attendant measure theoretic questions concerning their construction, are totally bypassed. In addition, an implicit estimate for the length of the time interval is presented. 237
7. Quasi-linear Equations of Evolution
238
7.1
PERTURBATION OF THE LINEAR PROBLEM AND NONLINEAR PRELIMINARIES
Suppose we have another equation of the same type as (6.4.1), say, dU’
-
dt
+ A’(t)u’ = F’(t),
0
< To,
~’(0)= ub,
(7.1.1)
in the (same) Banach space X. For {A’(t)}, we make the same basic assumptions (l), (2”) and (3) (cf. Theorem 6.3.7), and use corresponding notation S’(t), B‘(t),etc. The space Y c X is assumed common to the two systems. For this section, we assume only that X and Y are Banach spaces, with Y densely and continuously embedded in X.
Proposition 7.1.l.+ Let uo E Y, F E L’(J; Y), and ub E X, F’ E L’(J; X). Let u and u’ be given by (6.4.2), written u = U(6 @ uo @ F), and U’ = U‘(6@ ub 0 F‘). Then, llu’ - uIIm,x
< K”1l .b
- uollx
+ [IF’- Fll1.x + “(A’ - A ) ~ l ~ l . X l (7.1.2) 7
where K’ depends only on A’ and A‘ - A is regarded as a multiplication operator from C ( J ;Y) to L’(J; X). The quantity K‘, in fact, may be taken equal to IIUllm,x.
Proof: By differentiating the expression U(t, s)U(s,r)g with respect to s, and integrating from r to t, we obtain the identity, for g E Y, U’(t,r)g
- U(t, r)g =
- J U’(t,s)[A(s) - A(s)]U(s,r)g ds.
(7.1.3)
Setting I = 0 and g = uo in (7.1.3) gives
(U’ - U)(6 @ uO) = - U[A’
- A]U(6 @ uO),
where the notation of Lemma 6.3.2 has been used on the right-hand side. Setting g = F(r), and integrating (7.1.3),with resp&t to r, gives
( U - U)F =
- [U’[A
- A]U]F.
Hence, addition of these two equations gives
(U’ - U)(6@ U O @ F ) = - U’[A’ - A]U(6 @ U O Q F ) = -U‘[A - A]u.
’ Reproduced from Kato (Chapter 6 [12]), with permission.
7.1 Nonlinear Preliminaries
Definition 7.1.1. such that
239
Denote by E = EL the metric space of functions u : J + X, Ilu(t') - u(t)llx
< Llt' - tl,
t, t'
E
J,
(7.1.4)
for some fixed positive constant L , with metric d(u, u) = IIu - UII,,~.Let {A(t,w)}, {B(t, w)}, and { S ( t , w)} be given, for 0 < t < T o , and w in some subset W of X at our disposal, satisfying, generically, the following relations:
CAI I W '
(A)'
IIA(t, w') - A(t, w)ll G - wIIx, for A(t, w) E B(Y, X), with t I+ A(t, w) E B(Y, X) continuous in norm for each w;
(S)'
IIS(t', w') - S(t9 W)l)Y,X
< %(It' - tl + IIW' - wllx)9
for S ( t , w) an isomorphism of Y onto X ;
(B)'
S ( t , w)A(t,w)S(t,w)-
' = A(t, W) + B(t, w),
for t H B(t, w) E B[X] bounded, independently of w. Denote by A"(t), B"(t), and S"(t) the expressions A(t, u ( t ) ) , B(t, u ( t ) ) , and S ( t , u ( t ) ) , respectively, for u in a prescribed subset Eo of E with Range(u) c W for u E Eo. Assume, finally, (S-')' t H S"(t)-' E B(X,Y) is norm continuous.
Proposition 721.2. The mappings t H A"(t) E B(Y, X) and t H S"(t) E B(Y, X) are continuous in norm. Proof:
The first statement is immediate from
7. Quasi-linear Equations of Evolution
240
The following lemma is a preparation for the strong measurability of the mapping t H B"(t).
Lemma 7.1.3.' (1) Any closed, convex and bounded subset X of Y is closed in X if Y is reflexive. (2) If a function G: J --* Y is bounded in Y-norm and continuous in X-norm, then G is weakly continuous as a Y-valued function if Y is reflexive. In particular, G is strongly measurable (Y) in this case. Proof: (1) The subset X is weakly compact in Y since Y is assumed reflexive. Since the inclusion map of Y into X is continuous and, hence, weakly continuous, we conclude that X is weakly compact in X. It follows that X is closed in X. (2) Let G be a linear, continuous (real-valued) functional on Y. We must show that m = f G : J -, R' is continuous. It is enough to show that, if t , -,to in J, every subsequence of {m(t,)}has a subsequence convergent to m(to).Since G is Y-bounded, every subsequence of {m(t,)} has a subsequence convergent, say, to m,; however, since G is X-continuous and Y is reflexive, m , = m(to).The final assertion is a consequence of the Dunford-Pettis theorem (see Chapter 6 [14], p. 131),since G ( J ) c X is compact and, hence, separable inXandY.
Proposition 7.1.4. The map t H B"(t)E B[X] is weakly continuous on X if Y is reflexive. If, in addition X is separable, the mapping is strongly measurable (X) and, in fact, strongly integrable. Proof: By (S)' and (S-I)', the map t H B"(t)f E X is weakly continuous on X if and only if the map t H S"(t)-'B"(t)f E Y is weakly continuous on Y. Identify the latter map with the map G of the previous lemma. If g E Y, then, by (BY? S"(t)-'B"(t)g = A"(t)S"(t)-'g - S"(t)-'A"(t)g. (7.1.5) Since t H S"(t)-'g E Y is continuous, it follows, from (7.1.5) and from Proposition 7.1.2, that t H S"(t)-'B"(t)g is X-continuous for g E Y. Since IJs"(t)-'B"(t)llx,Y6 Adapted from Kato [17].
cu,
(7.1.6)
7.2 The Quasi-linear Cauchy Problem in Banach Space
241
and since Y is dense in X, it follows that t H S'(t)-'B"(t)f is X-continuous for f E X. The weak continuity is now immediate from (7.1.6) and Lemma 7.1.3. The strong measurability follows from the Dunford-Pettis theorem and the X integrability from the Bochner theorem (see Chapter 6 [141 p. 133). Note that llB"( . ) f l l x is lower semicontinuous, hence integrable. W
Proposition 7.1.5. Let X be a reflexive Banach space, and let F : [0, To] +X' be absolutely continuous. Then F is strongly differentiable almost everywhere, and the fundamental theorem of calculus holds: (7.1.7) (see Komura [20]).
Proposition 7.1.6. Let { F(t, w ) } be given, for 0 < t W of X,satisfying
< Toand w in some subset
(F)' p ( t , w') - F(t, w)llx < CFIIW' - WIIX, and t H F(t,w) E Y is uniformly continuous in the X-norm and bounded in the Y-norm. Then, the mapping t H F"(t)is continuous in the X-norm and, if Y is reflexive, weakly continuous (hence, strongly measurable if Y is also separable) in the Y-norm. In this case, F" E L'(J; Y).
Proof: Continuity in the X-norm follows the proof of Proposition 7.1.2. The weak continuity follows from Lemma 7.1.3, the strong measurability from the Dunford-Pettis theorem and the (Y) integrability from the theorem of Bochner, via the lower semicontinuity of llF"(-)lly.W
7.2
THE' QUASI-LINEAR CAUCHY PROBLEM IN BANACH SPACE
In this section, we consider the abstract Cauchy problem (7.2.la) (7.2.lb)
242
7. Quasi-linear Equations of Evolution
where the unknown u takes values in a Banach space, and A(t, u) is a linear, possibly unbounded, operator depending on t and u. We shall start from four real Banach spaces
Y c x c v c z,
(7.2.2)
with each of the spaces reflexive and separable, and the inclusions continuous and dense. We shall split the roles of X,as developed in the linear theory, and assign them to the three spaces X, V, and Z, so that u(t) E X,ti(*) is strongly continuous on V and - A(t, u) generates a (C,)semigroup on Z; this semigroup is assumed to be quasi-contractive (see Proposition 7.2.1),with respect to an equivalent norm N(t, u) on Z. The dependence on u is made precise in the sequel. For the moment, u(t) E Eo c E,where E is defined in Definition 7.1.1.
Definition 7.2.1. Let N(Z) = {I[ .I[, } denote the set of all norms in Z equivalent to the given norm, with metric
Remark 7.2.1.
The triangle inequality follows from the following estimates:
Proposition 7.2.1. Suppose { N ( t ,w)} c N(Z) is given for 0 < t in some subset W of Y, satisfying d(N"(t'),NU@))< clt' - tJ,d(N"(t),l[.l(z)< c1,
< Toand w (7.2.4)
7.2 The Q u a d i n e a r Cauchy Problem in Banach Space
243
for 0 G t’, t < To and all u E E,. Suppose that, for all t and w, A(t, w ) E G(ZN(r,w), l,o), where ZN(r,w) denotes the space Z normed by the equivalent norm N ( t , w). Then, {A”(t)} c G(Z) is stable on any subinterval J’ of [0, To], with stability constants o and
M
= exp(2{cl
+ +‘I}).
In particular, (7.2.4) holds, with c1 = I N ,c = (1 + L ) p N ,if { N ( t ,w ) } satisfies (N) N(t, w ) E N ( Z ) ,
with
d ( N ( t ,w), 11 * I [ Z ) IN, d(N(t’,w‘),N(t, w ) ) < pN( It’ - tl
+ llw’ -
WllX).
Proof: By the proof of Proposition 6.2.2 and Remark 7.2.2, we have, for J‘ = [O, To],
so that {A”(t)} c G ( Z )is stable as stated. The second statement is immediate.
rn
Definition 7.2.2. Let W be an open subset of Y, let yo E W, and let R = dist(y,,Y\W). We introduce families {A@,w ) } , {S(t, w ) } , {B(t,w ) } , { F ( t ,w ) } , and { N ( t ,w ) } , such that, for t, t’ E [0, To]and w, w’ E W, the following conditions hold: (N) (S)
See (N) of Proposition 7.2.1 ; S(t, w ) is an isomorphism of Y onto Z , with
.
IlS(t, W)((Y.ZG I,, IIS(t‘, w’) -
IlS(t, w ) -
so,W)(IY,Z < &(It’
-
G
lIJZ,Y
&9
tl + IJW’ - wllx);
A(t, w, G(ZN(r,w), l,u); (A2) S(t, w)A(t,w)S(t, w ) - = A(t, w ) + B(t, w), where
’
B(t, w ) E “1,
(A31 A(t, w ) E B(Y, X),with IIA(t, W
) [ I ~ ,<~ I,,
pet, w)llz < 2,; and
pet, w’) - A(t, W)I)Y,V < P A W ’ - wllv,
with t H A(t, w ) E B(Y, V) continuous in norm for each w ;
7. Quasi-linear Equations of Evolution
244
and with t
H
F(t, w)E V strongly continuous for each w.
Definition 7.2.3. For 0 < 7',< T o , L' > 0 to be specified below, let E, denote the (complete) metric space of functions u : [0, Tb] -, Y, such that Q 2R, Ilu(t') - u(t)llx Q L'lt' - tl, I ( u ( t )- YOllY
(7.2.5a) (7.2.5b)
with metric given by (7.2.5~)
Remark7.2.3. The set E, is a complete metric space. Indeed, the set E satisfying (7.2.5b,c) is routinely complete. Since W, = { w E Y: IIw - yell Q ($)R} is closed in Z (cf. Lemma 7.1.3), it follows that E,
=
n
I
OQtQTb
iU E E : u ( t )E w,}
is closed in E and, hence, complete.
Remark 7.2.4. The hypotheses (N)-(F) imply that Propositions 7.1.2,7.1.4, 7.1.5, and 7.1.6 are valid for t E J', u E E,. and X replaced by V, respectively, Z for statements concerning {A"(.)} and { F"(*)}, respectively, { S"(.)} and {B"(.)}. Moreover, Su(- ) is Lipschitz continuous for u E E,, i.e., IlS"(t') - S"(t)l(y,z< k ( 1
+ Ult' - tl.
In particular by Proposition 7.1.5, S"(.)is a strong indefinite integral of a strongly integrable function S"(.), such that
+
almost everywhere on [0, Tb]. (7.2.6) ps(l L') Moreover, B"(.) is strongly measurable (Z) b< Proposition 7.1.4, and F"( .) is integrable (Y) by Proposition 7.1.6. Since ~~Su(~)~~y,z and llB"(.)\Iz are upper integrable, it follows that (2") ofTheorem 6.3.5 holds for {A"(t)},{Bu(t)}, and {S"(t)}. Thus, by Proposition 7.2.1 and Theorem 6.3.7, the evolution operators {U"(t,s)}c B[Z] n BLY] exist. It is of interest that {A"(t)} is stable both on Z and (when properly restricted) on Y, by Propositions 6.2.7 and 6.3.1, so that the construction of the evolution operators given in Chapter 6 is sufficient. IISu(t)lly,z Q
This hypothesis can be shown to be redundant (see Kato [ 191).
7.2 The Quasi-linear Cauchy Problem in Banach Space
245
Lemma 7.2.2. Suppose the initial datum u, E Y satisfies lluo - yolly < e-2aNR/(2AsX~) = p' (
(7.2.8)
such that the mapping u H u = @(u), Then, there exist constants L' and To, given by
+ Ji U"(t, s)F"(s)ds,
~ ( t=) @(u)(t) = Uu(t,O)uo
(7.2.9)
0 < t < Tb, is a mapping of E, into itself. The function u satisfies (6.4.1), with A = A" and F = F", and t H ( du/ dt ) ( t )E V is continuous.
Proof: We first represent
u ( t ) - yo
u(t) - yo = U"(t, O)(u, - yo)
by
+ Ji U"(t, s)[F"(s)- A"(s)yo]ds.
This gives Ilu(t) - yolly
< &A$e2AN+Y2Tb [IlUO
- YOllY
+
+ AO)TO1*
(7.2*10)
By Proposition 6.4.1, (F),and Remark 7.2.4,
.
du
dt = F"(t) - A"(t)u(t),
and the right-hand side of this equation is continuous into V by (A3) and (F).Direct estimation of this differential equation gives
(7.2.12)
7. Quasi-linear Equations of Evolution
246
By (7.2.8), (7.2.1l), and (7.2.12), we see that Il(du/dt)(O)ll, G C/2, so that, by (7.2.10) and (7.2.1l),
for Tosufficiently small, and independent of u. In particular, u E E,.
Lemma 7.2.3. If u, satisfies (7.2.8), and L‘ is chosen by (7.2.12),there exists Tosufficientlysmall, and independent of uo, such that the mapping @ is a (strict) contraction of E, into itself. The contraction factor y does not depend on u o . Proof: By (7.2.9) and Proposition 7.1.1 (cf. (7.1.2)), with X replaced by Z, we have, using (7.2.7a), (F),and (A3), d(@w,@v) = I @ w -@vII,,Z G Tbexp{2&
where llyolly (7.2.13).
+ y,Tb)
* [PF
+ P”(IJY0lIY + R)I d ( W , V ) ,
(7.2.13)
+ R is an estimate for IIu(t)llm,Y.The result is immediate from
Let the hypotheses of Definition 7.2.2 be satisfied. Then, there are positive constants p’ and To< T o , such that, if uo E Y, with lluo - yell G p‘, (7.2.1) has a unique solution u on [0, To] = J’,with
Theorem 7.2.4.’
u E C(J’; W),
(7.2.14a)
du C(J’;V) n L“(J’;X). dt
(7.2.14b)
-E
When uo varies in Y, subject to lluo - yolly < p’, the map u, continuous in the Z-norm, uniformly in t E [0, To].
H
u(t)is Lipschitz
1
Proof: The unique fixed point u of @, guaranteed by Lemma 7.2.3, and the contraction mapping principle, satisfies u(t) = uyt,0)uo
+ J; uyt,s)F(s,u(s))ds.
(7.2.15 )
By the final statement of Lemma 7.2.2, we conclude that u satisfies (7.2.1) and (duldt) E C(J’;V). Since l~(du/dt)~lm,x < L’, (duldt) E L“(J; X). Of course, u E C(J’;Y) follows from Proposition 6.4.1. The uniqueness of solutions of (7.2.1) is equivalent to the uniqueness of fixed points of 0.
’This theorem and its proof are adapted from Hughes, Kato, and Marsden [13].
7.3 Quasi-linear Second-Order Hyperbolic Systems
247
To prove the Lipschitz continuity, we let ub be another initial datum, defining a map W and a solution u'. Then,
+
d(u, u') = d(Ou, WU') < d ( h , W U ) d(Wu,WU') SUP r
IIUU(t,O)llzlluo- ubllz
+ yd(u, u'),
by (7.2.9) and the contractive property of 0';note that y does not depend on the mapping CD. The result follows. W
Remark 7.2.6. Set K = e-2AN"/23.s3.3.By (7.2.8), we may select p' to be exactly equal to KR.Now, suppose uo is given, and yo is undetermined. If there is a dense subset of W satisfying (A4), then select an admissible yo as follows. Let R,, = dist(u,, Y\W), and choose yo to satisfy
By the triangle inequality, R = dist(yo,Y\W) 3 Thus, R,,
< (1 + K)R,and [(YO
- uolly
:
-
(1
JRuo-
< K R = P',
as required. \
7.3
QUASI-LINEAR SECOND-ORDER HYPERBOLIC SYSTEMS
We shall consider quasi-linear second-order hyperbolic systems of the form
where the unknown # = ( t + b l ,.. . ,$,,) is an rn-vector valued function of t E [O,TO] and of x = ( x l , . . . ,x,) E R", where { u i j : i , j= 0,1,. . . ,n} is a collection of (rn x +matrix valued functions of the (suppressed) arguments t, x, *,a*/&, V*, and b is a function of these same arguments. The formal definition of solution of(7.3.1)is given in the following definition and remark.
7. Quasi-linear Equations of Evolution
248
Definition 7.3.1. Let R c R" x R" x R"" be an open set containing the origin which is contractible to the origin, and let aij:i,j = 0, 1, . . . , n, and b be defined on [0, To] x R" x 0.These variables will be denoted by (t, x, p) E [0, To] x R" x R. Let C;(R" x R; RN) denote the class of N-component functionsof class Ckin x and p, whose x-derivatives up to order k are bounded. Regarding the functions aij and b, we make the following hypotheses (s is specified below). (all aij€ Lip([O,T,]; c;+'(R" x R; R"')), for i, j = 0, 1, . . ., n, replaced by Ctl+ if [s] # s), b~ Lip([O,T,];C",+'(R" x R;Rm)), (C;" replaced by CF]+' if [s] # s), b(O;,O) E H"(R";R"); (a2) aij = a;; (a3) (Hyperbolicity)There is an E > 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,. . . , <,,) E R".
For s > n/2 + 1 (s > n/2 if the aijdo not depend on the derivatives of #), set
X = H"(R";R") x HS-'(R";R") = V, Y = H"+'(R";R") x Hs(R";R"), Z = H'(R";R") x Ho(R";R"). Adjoined to (7.3.1) are the initial conditions
#(O,*)= #o
E
HS+'(R";R"),
(7.3.2a) (7.3.2b)
where it is assumed that (#o(x), $,,(x), v#,(x)) E R,
for all x E R".
(7.3.3)
We suppose that W c Y is a ball, centered at uo = (#o, $o), with radius small enough, so that u E W satisfies (7.3.3).This is possible by Sobolev's inequality. We denote the general element in W by w = (a,&).
7.3 Quasi-linear Second-Order Hyperbolic Systems
249
Remark7.3.1. We may reduce (7.3.1) to a first-order system in t by the following standard device. For w E W, define A(t, w), formally (see Proposition 7.3.1 for a precise statement), by
r
I
0
1
Then, for (*, $) E Y(a core), A(t, w)(+,$) E X is given by
so that d dt
- (*,
*
$1
+ A(t, (9,$))(*,
$) = (0, b(t,., +, $, V$)),
(7-3.6)
if and only if satisfies (7.3.1), 0 < t < To, To< T o ,with $ = d*/dt. By a solution of (7.3.1)-(7.3.2), we shall understand a pair ($, &) satisfying (7.3.6) and (7.3.2).
Remark 7.3.2. Later in this section, we shall discuss the ring properties of aij. For the moment, we need only observe that aij are pointwise bounded, together with (a/ax,)aij, by our assumptions. Definition 7.3.2. Define the bilinear forms (7.3.7a) and
(7.3.7b) I = l
\
7. Quasi-linear Equations of Evolution
250
where do is a constant, independent oft and w, specified by Proposition 7.3.1, to follow.
Remark 7.3.3. The equivalence of the norms, defined by N(t,w) and ll*llz on Z, follows from Proposition 7.3.1 (see (7.3.9)) and (a3). In fact, the existence of the constant AN of hypothesis (N) follows from (7.3.9). Proposition 7.3.1. There are constants c, co, do > 0, such that (7.3.9a)
IB(l, w;@19@2)l G CII@lIIH'(W")11@211Hl(W"),
B(t,w;@,@I 2
COIJ@IIi'(R")
- dOll@llZ2(W"),
(7.3.9b)
for all @,@',yj2 E H'(R"; Rm),t E [0, To], and w E W. Similar constants c', cb, and db exist, so that C ( *, . ;., .) satisfies the continuity condition (7.3.9a) and the Carding inequality (7.3.9b). If C(t, w) is the closed linear operator in L2(R";Rm),specified by the Lax-Milgram lemma, according to C(t,w;@ 1 9 @I = (@1, C(t, W)@)LZ(R")
9
(7.3.10)
= H2(R";Rm),and for all @ E DC(t,w)rE H'(R"; Rm),then DC(t,w)
In particular, the operator A(t, w), defined formally by (7.3.4), is a closed linear operator in Z, with domain DA(t,w) = H2(R"; Rm)x H1(R";Rm).
Proof: Inequality (7.3.9b) is Garding's inequality for uniformly elliptic systems on R", and may be adapted from the bounded domain case (see the bounded domain case in Morrey (Chapter 4 [25], Section 6.5), while (7.3.9a)is immediate from the Cauchy- chwarz inequality. By (7.3.7b) and the Cauchy-Schwarz inequality, we have
c(t,w;@?@)2 B(t?w;@,@) - CIII@IIHl(R")ll@llLZ(R"),
(7.3.12)
for some constant cl, independent of t, w, and @. Hence, the Garding inequality for C is immediate from (7.3.9b), if we use (7.3.12) and
= H2(R"; Rm)is If C(t,w) is defined by (7.3.10), the domain assertion DC(t,w) the classical statement that weak and strong solutions coincide (Chapter 4 [25]) and, hence, (7.3.11)follows upon use of(a2).The fact A(t, w) is closed in Z
7.3 Quasi-linear Second-Order Hyperbolic Systems
251
and its domain assertion follow from the identity (see (7.3.5))
(7.3.14) This concludes the proof, W Lemma 7.3.2. There exists a constant w, such that, for t E [0, To],w E W, and v E Z, the energy inequality
(A(t,W)v,V)N(t,w)3 - o l \ v l l ~ ( t . w )
(7.3.15)
holds, where w E R'. In particular, A(t, w) satisfies (Al).
Proof: Let v = (*,$), with (7.3.8), (A(t, w)v, V ) N ( t , W ) = N t , w ; -$, *)
*,$
E C$(Rn). Then, by
+ do( - 4, *)Lz(Rn)
( 7 . 3 3 , (7.3.7), and
7.Quasi-linear Equations of Evolution
252
from which (7.3.15) follows. The fact that A(t, w) E G(ZN,,,,,, 1,o)follows from a spectral analysis, as in the example of symmetric linear hyperbolic systems of Section 6.4 (see Lemma 6.4.4). The format (7.3.6)and (7.3.2)is now clearly related to the abstract Cauchy problem (7.2.1). The spaces Y, X, V, and Z have been identified in Definition 7.3.1, and the family { N ( t ,w)>is contained in N(Z) (see Definition 7.2.1), by Remark 7.3.3. Moreover, by the previous lemma, A(t, w) E G(ZN,,,,,, 1,o) for some fixed constant o.We shall begin systematically to verify the hypotheses (N)-(F) of Definition 7.2.2. Fundamental to the various continuity, invariance, and boundedness concepts of these definitions are the ring properties, satisfied by the Sobolev and uniformly local Sobolev spaces. We develop these now. Definition 7.3.3. For P a fixed Hilbert space, and 1 < p < 00, let L;AR"; P) denote the set of all (equivalence classes of) P-valued strongly measurable functions u on R", such that (7.3.16) The space Lid is called a uniformly local LP-space. For each integer s 2 0, we denote by Hid the set of u E L:d(R";P), such that the distribution derivatives D"u of order la1 < s are in L;!([W";P). The norm in Edis given by
Remark 7.3.4. The spaces L;{ and H t 4 are Banach spaces. Interpolation yields the spaces Hid for nonintegral s. The following properties will be necessary (see Kato [18] and Hughes, Kato, and Marsden [13]). (1)
If s > 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 < I < s. If s > 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~)
7. Quasi-linear Equations of Evolution
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.
7. Quasi-linear Equations of Evolution
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).
7. Quasi-linear Equations of Evolution
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.
7. Quasi-linear Equations of Evolution
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
< TO.
(7.5.4)
The solutions u = uv of the "parabolic" systems are required to converge in L'((0, Tb) x R") as v 1 0 to the solution u = uo of the "hyperbolic" system. Space limitations prevent us from supplying full details of these results, which are appearing elsewhere (see Jerome [143). However, we shall state some of the results and present the core arguments. At the basis is a stability analysis of the horizontal-line method applied to (7.5.2), in which ideas of the previous chapter play a decisive role. We shall develop the general theory for this, and return to the applications at the end of the section. However, for the reader who wishes to proceed no further, we briefly describe the choice of Tb and r'. Suppose, for r > 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
7. Quasi-linear Equations of Evolution
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)
7. Quasi-linear Equations of Evolution
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
<(1+6)M'
< (K) 1 - 0 (A) I - 0 , cllu - w(Ix The final statement follows from the contraction mapping principle applied to the complete metric subspace Wo of X; note that, here, we use the fact that Y is reflexive. Corollary 7.5.2. Suppose the hypotheses of Proposition 7.5.1 are satisfied, including (7.5.16) and (7.5.17). Suppose that there is a uniform bound C, for (A(t, w)} as bounded linear mappings from Y into X, for 0 < t < To and w E Wo. Then, the solutions of (7.5.9) uniquely exist and satisfy
k = 0, . . . , N , N 2 N o , max(IIufIIY,IlufllX) < r, lluf - uf-lllx < C1r& N 2 No,
(7.5.24a) (7.5.24b)
where N o is defined via (7.5.17). Proof: The first of these inequalities has been proved already. The second results from a simple estimation of (7.5.9). It is possible to use the previous corollary as a starting point for a parallel development of quasi-linear equations of evolution in Banach spaces. However, this would needlessly repeat the results of Section 7.2. We shall, instead, return to the ideas introduced at the beginning of this section. In order to emphasize that the convergence of the viscosity method is routine, once the invariant time interval has been established, we prove the former result first under a dissipation hypothesis. We now define this concept and present the hypotheses upon a j , b and P.
Remark 7.5.3.
7. Quasi-linear Equations of Evolution
268
Definition 7.5.3. We shall call the system (7.5.1)-(7.5.3) dissipative on (0, Tb) if, for 0 < t Tb and w E PHs(R"; R"),
-=
(E(t,W ) V , V ) ,2 , ~0,( ~ ~ )for all v E Y,.
(7.5.25)
The coefficients aj and b are assumed to belong to the class (C;"
Lip([O, To]; C;+l(Rn x R; R"'))
replaced by C11+2if s # [s])
for every relatively compact open subset R of R". The L2 orthogonal projection P is explicitly assumed to satisfy PH"(R"; R") c H"(R"; R"),
0
2 1,
(7.5.26b)
APHZ(R";R") c PL2(R";R"), P(I - A)ai2 = (I - A)""P,
(7.5.26a)
0
2 1.
(7.5.26~)
Finally, we assume that E(t, w) E G(PL2,l,w(r, w)) for all w E PHs(R").
Remark 7.5.4. The Navier-Stokes/Euler system projection satisfies (7.5.26) and the generator property on E(t,w) (see Kato [17]). The first hypothesis implies that PH" is closed in H", and the third hypothesis permits a resolvent estimation on H". Note that (I - A)a/', here, has a diagonal operator interpretation, as does A in (7.5.26b). Proposition 7.5.3. Consider the system (7.51)-(7.5.2), with the associated properties (7.5.3) and (7.5.4) and the dissipation property (7.5.25). There exist constants C , and C , , such that, if u1 and u, are solutions of (7.5.2), with v = v, and v = v , , respectively, the estimate
Il(ul holds for 0 < t
< Tb.
< Cl(v, - v2)2eC2Tb
- UZ)(r)lltZ(Rn)
(7.5.27)
7.5 Invariant Time Intervals
269
where we have used the dissipation property (7.5.25) and the Lipschitz property
lIE(t,
w, - E(t, W')I(HS(R"),L2(Rn)
< cllw- W'IILz(R")r
(7.5.28)
for w,w' E B(0, r') c H'(l%";Rm)(see Remark 7.3.4). By the classical Gronwall inequality applied, after integration in t, we have (7.5.27), where C , 2 IIAulll&cRn,,C2 2 2C11u211L2(Rn)1 are constants independent of v1 and v2, since u1 and u2 are assumed to satisfy (7.5.4).
+
Remark 7.5.5. The method of selecting the pair (To, r') may now be implemented, as described prior to Definition 7.5.1. We note that -vA E G(PL2,1,O) for v > 0. The perturbation E(t, w) is relatively bounded, with respect to - vA, by a standard interpolation, with bound arbitrarily small, and by assumption is in G(PL2,l,w), where an upper bound for w can frequently be determined. Thus, A(t, w ) E G(PL2,1,w) (see Kato [15] pp. 499 and 500). In the particular case of the Navier-Stokes/Euler system, the operator A(t, W)
=
-vA
+ P(w
*
V)
(7.5.29)
is unconditionally stable on L2(Rn),in the sense that A(t, w) E G(L2,1,O) (see Kato [17]). In this case, the choice w(r) = 0 may be made. The estimation of wl(r) uses the similarity relation (7.5.8a), with A, = A + B. Earlier results (see Proposition 6.3.1) reveal that IIBIILz(Rn),LZ(Rm) < B 3 w1 = w + /3 is an acceptable choice. We have already discussed the choice of /3 in the context of second-order quasi-linear hyperbolic systems. The present analysis is similar. Thus, a simple computation gives SA(t,w)S-' = A(t,w)
+P
"
j= 1
[S,aj]A1-'
A-'
+ [S,b]Al-'A-'
1
,
hold for s > 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
BIBLIOGRAPHICAL REMARKS
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
7. Quasi-linear Equations of Evolution
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).
This page intentionally left blank
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