Table Of ContentLecture Notes in Mathematics 0751
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.A Dold, Heidelberg
.B Eckmann, Ztirich
E Takens, Groningen
hplaR slefnebuaLed
Existence Families,
Functional Calculi
and Evolution Equations
galreV-regnirpS
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
Budapest
Author
Ralph deLaubenfels
Mathematics Department
Ohio University
315 B Morton Hall
Athens, Ohio 45701-2979, USA
Mathematics Subject Classification (1991): 47D05, 47D 10, 47A60, 34G 10, 35G 10,
47A10, 47A12, 47F05
ISBN'3-540-57703-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-57703-3 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. DeLaubenfels, Ralph,
1951- . Existence families, functional calculi, and evolution equations / Ralph
deLaubenfels, p. cm. - (Lecture notes in mathematics; 1570) Includes bibliographi-
cal references and index. ISBN 0-387-57703-3
.1 Evolution equations. 2. Linear operators. 3. Functional analysis. I. Title. II. Series:
Lecture notes in mathematics (Springer-Verlag); 1570. QA3.L28 no.
1570 QA377 510 s-dc20 515'.353 93-47576
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(cid:14)9 Springer-Verlag Berlin Heidelberg 1994
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To Karen, David, Michael
and Evan
TABLE OF CONTENTS
.0 Introduction IX
I. Intuition and Elementary Examples 1
II. Existence Families 7
III. Regularized Semigroups 31
IV. The Solution Space of an Operator and
Automatic Well-Posedness 42
V. Exponentially Bounded (Banach) Solution Space 83
VI. Well-posedness on a Larger Space; Generalized Solutions 55
VII. Entire Vectors and Entire Existence Families 06
VIII. Reversibility of Parabolic Problems 66
IX. The Cauchy Problem for the Laplace Equation 17
X. Boundary Values of Holomorphic Semigroups 37
XI. The SchrSdinger Equation 67
XII. Functional Calculus for Commuting Generators of Bounded
Strongly Continuous Groups 97
XIII. Petrovsky Correct Matrices of Generators of Bounded
Strongly Continuous Groups 68
XIV. Arbitrary Matrices of Generators of Bounded Strongly
Continuous Groups 29
XV. More Examples of Regularized Semigroups 49
XVI. Existence and Uniqueness Families 79
vii
XVII. C-resolvents and Hille-Yosida Type Theorems 401
XVIII. Relationship to Integrated Semigroups 011
XIX. Perturbations 311
XX. Type of an Operator 521
XXI. Holomorphic C-existence Families 821
XXII. Unbounded Holomorphic Functional Calculus for
Operators with Polynomially Bounded Resolvents 331
XXIII. Spectral Conditions Guaranteeing Solutions
of the Abstract Cauchy Problem 451
XXIV. Polynomials of Generators 851
XXV. Iterated Abstract Cauchy Problems 461
XXVI. Equipartition of Energy 571
XXVII. Simultaneous Solution Space 871
XXVIII. Exponentially Bounded Simultaneous Solution
Space 381
XXIX. Simultaneous Existence Families 781
XXX. Simultaneous Existence Families for Matrices
of Operators 191
XXXI. Time Dependent Evolution Equations 691
Notes 102
Bibliography 812
Subject Index 822
VIII
0. INTRODUCTION
In this book, we wish to present the basic theory, and the more inter-
esting developments, of existence families for the abstract Cauchy problem
d
-~u(t,x) = A(u(t,x)) (t _> 0), u(O,x)= x, (0.1)
and regularized semigroups (these have been recently appearing in the
literature as "C-semigroups"). These are generalizations of strongly con-
tinuous semigroups, that can be applied directly to the many differential
and integral equations that may be modeled as an abstract Cauchy prob-
lem on a Banach space, where strongly continuous semigroups cannot be
applied directly, that is, when the operator A in (0.1) does not generate
a strongly continuous semigroup. Examples of this are the backwards
heat equation, the SchrSdinger equation on LP,p ~ 2, the Cauchy prob-
lem for the Laplace equation and Petrowsky correct systems of constant
coefficient partial differential equations.
These families of operators unify well-posed and ill-posed problems,
cover the entire spectrum of ill-posedness and unify and simplify results
about ill-posed problems, just as strongly continuous semigroups do with
well-posed problems.
Of particular interest in this book will be the interplay between func-
tional, or operational, calculus constructions, existence families and evolu-
tion equations. We do not attempt to give detailed physical applications.
The examples of evolution equations are meant primarily to illustrate the
great scope and broad, yet simple and intuitive, applicability of existence
families and regularized semigroups. We shall see that regularized semi-
groups may also be used as a powerful tool for constructing functional
calculi for unbounded operators.
More complete descriptions of how physical problems may be modeled
as an abstract Cauchy problem (0.1) may be found in any of the references
for strongly continuous semigroups.
Another central feature of this book is what is called the solution space,
for an arbitrary closed operator. This is a unifying concept for all the gen-
eralizations of strongly continuous semigroups that have appeared, besides
the existence families presented here. Even for the strongly continuous
case, the introduction of the solution space greatly simplifies the proof of
the fundamental relationship between the abstract Cauchy problem and
semigroups.
IX
In this book, we have tried to avoid those aspects of C-regularized
semigroups that are obvious generalizations of strongly continuous semi-
groups, that is, where one takes a proof for strongly continuous semi-
groups and places a C everywhere. It is gratifying to find that many of
these results may be immediately reduced to the corresponding result for
strongly continuous semigroups with the solution space. In this book, we
have included primarily results about existence families and regularized
semigroups that would seem surprising to anyone familiar with strongly
continuous semigroups. The existence of entire C-regularized groups with
unbounded generators is an example. Another example is immense sim-
plications and clarifications, both in the statement of results and their
proofs, of old results, involving, for example, the SchrSdinger operator on
p,PL ~ 2, and Petrowsky correct systems of differential equations.
Many of the most fundamental results in this book, including Chapters
II, V, VII and XII, are new, that is, have not appeared, at least in their
present exposition, and are not intended to appear, in research journals.
We have also organized and simplified a great deal of basic material that
is currently scattered throughout many different papers.
Some familiarity with the basic results of strongly continuous semi-
groups would be helpful in reading, or at least appreciating, this book,
but is not necessary for most of the material, since these results will be a
corollary of our results here.
The theory of strongly continuous semigroups of operators has been
extremely successful in dealing with the many situations that may be
modeled as an abstract Cauchy problem (0.1). When A is closed, gener-
ating a strongly continuous semigroup corresponds to the abstract Cauchy
problem having a unique mild solution, for all initial data x.
What should be done when the abstract Cauchy problem (0.1) is not
well-posed or does not have a mild solution for all initial data? It does
not seem constructive to merely say it is not well-posed, throw up our
hands with despair and refuse to deal with it. Ill-posed problems arise
naturally (see Pay); the initial formulation of (0.1)is often made out of
convenience or familiarity. The supremum norm and the L 1 norm, for
example, are very natural choices for Banach space norms, but in many
cases, such as the SchrSdinger equation, do not yield well-posedness.
There are two approaches one can take. One can look for initial data
in the original space that yield solutions. Or one can renorm a subspace
in such a way that A, when restricted to that space, generates a strongly
continuous semigroup. We shall see that existence families and regularized
semigroups provide a simple yet powerful tool for either approach. For
the first approach, we shall see that, for any bounded operator C, having
X
a mild solution of the abstract Cauchy problem, for all initial data in the
image of C, corresponds to having a mild C-existence family. We present
in this book an intuitive method for choosing such C and constructing the
desired existence family. For the second approach, we show that a mild
C-existence family gives us a straightforward method for constructing
and approximating a subspace of the original space, containing all mild
solutions of the abstract Cauchy problem, on which A generates a strongly
continuous semigroup. We call this the solution space for A (see Chapters
IV and V).
Besides existence and well-posedness, it is also important to have meth-
ods for representing or constructing solutions. We feel the essential con-
cept here is an operator theoretic one of a good deal of independent inter-
est, that of an JC-functional calculus for A. For bounded operators, this
is a continuous algebra homomorphism, from the Banach algebra of func-
tions ~- into the Banach algebra of bounded linear operators, mapping
the constant function fo(s) = 1 to the identity operator, and mapping
fl(s) = s to A. Some modifications are required to adapt this concept
to unbounded operators, where the subject is relatively unformed (except
for spectral operators--see Du-S2). Informally, one is replacing the com-
plex variable z by A, and making sense out of f(A). Again informally, the
solution of the abstract Cauchy problem is u(t, x) = etAx, that is, ft(A)x,
where ft(z) - e tz. Many of the most important results about semigroups
of operators can be best understood with this perspective, and many new
results, presented in this book, can be created and understood similarly.
We may summarize the functional calculus approach to evolution equa-
tions, via the abstract Cauchy problem (0.1), as follows. We think of a
strongly continuous semigroup generated by A as being etA. We think
of a C-regularized semigroup generated by A as being etAg(A), for some
g such that the complex-valued function z ~ etZg(z) decays sufficiently
rapidly on the spectrum of A; here C is g(A). Finally, we think of a
solution of (0.1) as being u(t,x) _= etAx. Throughout this book, we will
try to make literal sense out of these exponentials.
To conclude our general introduction, we should make it clear that
we do not consider regularized semigroups or existence families to be
any sort of revolutionary replacement of strongly continuous semigroups.
We view it more as a completion of the theory of strongly continuous
semigroups, extending and finishing a beautiful and fundamental theory
that strongly continuous semigroups began. For example, when there
are solutions of the abstract Cauchy problem, it is natural to ask when
they are reversible (see Chapter VIII). When {e~A}n~(z)>O is a strongly
continuous holomorphic semigroup of angle yr , it is natural to ask in what
X1
sense we can let Re(z) ~ 0; that is, can we make sense out of boundary
values e isA (see Chapter X)? As with strongly continuous semigroups,
an approach that allows us to deal with bounded operators has the best
chance of being successful.
Now we will give a rough chapter-by-chapter introduction.
For an arbitrary bounded operator C, and an unbounded operator A,
C-existence families for A, which we think of etAC, are introduced in
as
Chapter II. We must quibble about whether we are obtaining mild solu-
tions or strong solutions (Definition 2.1) of the abstract Cauchy problem
(0.1), thus we have mild existence families, existence families and strong
existence families.
We introduce regularized semigroups in Chapter III. These are essen-
tially existence families e~AC, where e~AC : Ce tA. This gives us an
algebraic definition, W(t)W(s) = Wit § s)C, analogous to strongly con-
tinuous semigroups (/-regularized semigroups). To enable us to define the
generator, A, we insist that C be injective (so that we may differentiate
Ce tA at t -- 0, then apply C -1).
In Chapter IV, for an arbitrary closed operator A on a Frechet space,
we introduce its solution space, Z. This is the set of all initial data,
x, for which the abstract Cauchy problem has a mild solution. When
all solutions are unique, we may topologize Z in such a way that Z is
a Frechet space, and A restricted to Z generates a strongly continuous,
locally equicontinuous semigroup. This is the only chapter where X is a
general Frechet space, rather than a Banach space.
By restricting ourselves to exponentially bounded mild solutions of the
abstract Cauchy problem, we obtain maximal Banach spaces, on which a
closed operator A generates a strongly continuous semigroup, in Chapter
V. We present numerous methods for testing whether a vector x is in
one of these exponentially bounded solution spaces, and constructing the
corresponding mild solution of the abstract Cauchy problem; some of these
tests may be considered "pointwise Hille-Yosida theorems."
In both Chapters IV and V, we see how regularized semigroups and
existence families provide a way to approximate the solution spaces and
their topologies.
Chapter VI presents a strategy opposite to that of Chapters IV and V.
Rather than shrink the original space, we enlarge it, to produce a space,
W, in which the original space is continuously embedded, and an operator
B, that generates a strongly continuous semigroup, such that the original
operator A equals the restriction of B to X.
In Chapter VII, a concept that has been used a great deal for strongly
continuous semigroups, that of an entire vector, is used to give a simple
XII
