Table Of ContentTEUBNER-TEXTE zur Mathematik
Band 140
Oliver Caps
Evolution Equations
in Scales of Banach Spaces
TEUBNER-TEXTE zur Mathematik
Herausgegeben von:
Prof. Dr. Jochen BrOning, Berlin
Prof. Dr. Herbert Gajewski, Berlin
Prof. Dr. Herbert Kurke, Berlin
Prof. Dr. Hans Triebel, Jena
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Oliver Caps
Evolution Equations
in Scales of Banach Spaces
1m
Teubner
B. G. Teubner Stuttgart· Leipzig· Wiesbaden
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Ein Titeldatensatz fUr diese Publikation ist bei
der Deutschen Bibliothek erhaltlich.
Dr. rer. nat. Oliver Caps
Born in 1972 in Mainz (Germany). From 1991 to 1996 he studied mathematics, physics and computer
sciences at the Johannes Gutenberg-Universitat Mainz. After a stay at the Universita degli Studi of Bo
logna (Italy) in spring 1997 he returned to the Johannes Gutenberg-Universitat Mainz and received his
Dr. rer. nat. in 2000.
From summer 1997 to summer 2001 he held a position as scientific assistant at the University of Mainz.
Currently he is working on financial mathematics at a financial institute in Germany.
1. Auflage Juli 2002
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ISBN-13:978-3-519-00376-2 e-ISBN-13:978-3-322-80039-8
001: 10.1007/978-3-322 -80039-8
Preface
The book provides a new functional-analytic approach to evolution equations
by considering the abstract Cauchy problem in a scale of Banach spaces. The
usual functional analytic methods for studying evolution equations are formu
lated within the setting of unbounded, closed operators in one Banach space.
This setting is not adapted very well to the study of many pseudo differential
and differential equations because these operators are naturally not given as
closed, unbounded operators in one Banach space but as continuous opera
tors in a scale of function spaces. Thus, applications within the setting of
unbounded, closed operators require a considerable amount of additional work
because one has to construct suitable closed realizations of these operators.
This choice of closed realizations is technically complicated even for simple
applications.
The main feature of the new functional analytic approach of the book is to
study the operators in scales of Banach spaces that are constructed by simple
reference operators. This is a natural setting for many operators acting in
scales of function spaces. The operators are only expected to respect the
scale and to satisfy certain inequalities but we can avoid completely the choice
of any closed realization of these operators which is of great importance in
applications. We use the mapping properties of the reference operators to
prove sufficient conditions for well-posedness of linear and quasilinear Cauchy
problems. In the linear, time-dependent case these conditions are shown to
characterize well-posedness. A similar result in the standard setting (i.e., a
time-dependent generalization of the Hille-Yosida/Lumer-Phillips theorem) is
still an open problem.
The generality of the new functional analytic approach of the book is demon
strated by many applications to several mathematical and physical fields. One
of the most important applications is a simultaneous treatment of some par
abolic and hyperbolic equations. In the standard approach this is not pos
sible, and hyperbolic and parabolic equations have to be treated by differ
ent and incompatible methods. In particular, the approach of the book can
be used for applications to (strongly) degenerate parabolic equations appear-
6 Preface
ing in connection with some physical and probabilistic problems. Classical
results on hyperbolic and parabolic equations are special cases of these re
sults. A further important example of equations of that type is the parabolic
N avier-Stokes equations, which degenerates hyperbolically to the Euler equa
tion. Hence, in contrast to the standard approach, with the new methods of
the book we obtain results on Navier-Stokes equations degenerating to Euler
equations in some parts of the space. Further results of this book include con
ditions on symbols for essential selfadjointness of pseudodifferential operators
and well-posedness of Schrodinger equations, linear and quasilinear evolution
equations in U-Sobolev spaces, and spaces of continuously differentiable func
tions, degenerate-elliptic boundary value problems, evolutions equations on
networks, and a unified approach to both types of Kadomtsev-Petviashvili
equations with periodic boundary conditions.
The book contains 5 chapters. Chapter 1 provides some functional analytic
methods. The abstract theory of linear evolution equations in scales of Ba
nach spaces is developed in chapter 2 and the abstract theory of quasilinear
equations in chapter 3. Applications of the abstract methods to linear equa
tions are given in chapter 4 and to quasilinear equations in chapter 5. The
abstract part of the book, i.e. chapter 1-3, is kept completely self-contained.
Assuming only basic knowledge on functional analysis of bounded, linear op
erators in Banach spaces, all functional analytic methods needed for further
reading are proved in chapter 1. Readers experienced in functional analysis
may skip chapter 1, start reading directly chapter 2, and go back to chapter 1
only occasionally. Whereas all auxiliary results in the abstract part are proved
in detail in the book, we cannot continue this way of presentation in chapter
4 and 5 because detailed proofs of analytic results and methods necessary for
applications would exceed the . limit of this book. Therefore, results of this
type are formulated in a self-contained way and for proofs we give references
to standard monographs treating these topics.
There are several people to thank. First, I wish to thank B. Gramsch for sug
gesting to study evolution equations in scales of Banach spaces, many helpful
discussions, and valuable support.
I am grateful to G. Schleinkofer for proposing to apply the abstract results
to Kadomtsev-Petviashvili equations, W. Arendt, M. Hieber, and J. Voigt for
some references, and F. Ali Mehmeti for several remarks. Moreover, I profited
from numerous discussions with F. Baldus, R. Lauter, and J. Lutgen.
Finally, I wish to thank J. WeiB from Teubner Verlag for strongly supporting
the publication of the present manuscript.
Mainz, April 2002 Oliver Caps
Contents
List of Symbols 9
Conventions 12
Introduction 13
1 Tools from functional analysis 27
1.1 A brief introduction into the theory of semigroups 27
1.2 Selfadjoint operators ................ . 38
1.3 Generators of analytic semigroups and their powers. 45
1.4 Fractional Powers of operators of positive type 53
1.5 Complex interpolation spaces ....... . 59
1.6 Time-dependent, linear evolution equations .. 66
2 Well-posedness of the time-dependent linear Cauchy problem 78
2.1 Properties of well-posed linear Cauchy problems in scales of Ba-
nach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79
2.2 Scales of Banach spaces generated by families of closed operators 89
2.3 Commutator estimates and scales of Banach spaces . . . . . . . 100
2.4 Characterization of well-posedness of the Cauchy problem . . . 109
2.5 Sufficient conditions for well-posedness of the Cauchy problem. 113
3 Quasilinear Evolution Equations 130
3.1 Semilinear Evolution Equations . . . . . . . . . . . . . . . . . . 130
3.2 Commutator estimates and quasilinear evolution equations 140
3.3 A local existence and uniqueness result for quasilinear evolution
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.4 Regularity for quasilinear evolution equations in scales of Ba-
nach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4 Applications to linear, time-dependent evolution equations 166
4.1 Pseudo differential operators and weighted Sobolev spaces .. . 167
4.2 Pseudo differential evolution equations in scales of weighted
Sobolev spaces ....................... 186
4.3 Essential selfadjointness of pseudo differential operators. . . . . 199
8 Contents
4.4 Evolution equations in Co(Rn) and Feller semi groups . . . . . . 212
4.5 Evolution equations in scales of Lq-Sobolev spaces ....... 221
4.6 An application to a degenerate-elliptic boundary value problem 227
4.7 Evolution equations on networks . . . . . . . . . . . . . . . . . 234
5 Applications to quasilinear evolution equations 245
5.1 Estimates of Nash-Moser type for differential operators. 246
5.2 Quasilinear evolution equations in Sobolev spaces . 265
5.3 Degenerate Navier-Stokes equations ........... 271
5.4 The generalized Kadomtsev-Petviashvili equation. . . . 278
5.5 Quasilinear evolution equations in scales of Lq-Sobolev spaces 290
5.6 First order hyperbolic evolution equations in the C~-scale 292
Bibliography 295
Index 307
List of Symbols
General Notation
Y continuous embedding JL_(XO,Xl) p. 60
a.e. almost everywhere (Xo, Xl)lI,q p. 63
signum ( a ) sign of a [Xo, Xl]1I p. 61
linh linear hull dist p. 231
[s] := min{l E 7L: l 2: s} p. 105 supp(J) support of f
1 := [to, tl] (x) p. 167
/::,. := {(t , s) E 12 : t 2: s} aD< p. 167
~II p.45 (J *g)(x):= IlRnf(y)g(x - y)dx
'YR,<p p. 45 convolution
{ el, ... , en} canonical base of IRn u(~) := JL[u](~) = JLx--+du(x)](~)
EEl direct sum p. 167
x V y := sup(x, y) p. 212 u(~) := JL-l[U](~) p. 167
x A y := inf(x, y) p. 212 de<" .'-- 2d7E1"; p. 167
x+ := x V 0 p. 212 div x divergence with respect to x
x- := (-x) V 0 p. 212 grad gradient with respect to x
x
Ixl := x+ + x- = x V (-x) p. 212 /::"x Laplacian with respect to x
~ := {z E ( : 0 < Re z < I} p. 59 e J.L,v (x , y) .' = .2.7l1"e iJ.Lxeivy p • 278
~ := {z E ( : 0 :s; Re z :s; I} p. 59 u(/-L,v):= (U,eJ.L,nu)p(T2) p.278
Xo +Xl p.60 o
ov p. 235
JL(XO,Xl) p. 60 A-I
p. 96
Spaces
IN natural numbers ( complex numbers
No := IN U {O} Mm(lR) m x m matrices over IR
7L integers Xc := X EEl iX p. 84
R real numbers 2'(Y, X) p. 27
10 List of Symbols
£,(X) := £,(X, X) p.27 XCO p.79
YyX p.27 (xt 11·llk) p. 89
X* p.27 E p. 167
\lJk
(·,·h*,x p.27 p. 107
c
(Xkh p.79 \lJco p. 108
c
Operators
AcB p. 27 At[ p.84
A p. 27 Za. p. 93
R(A) p.27 AZ p. 55
N(A) p. 27 [P,Q]:= PQ - QP p. 168
p(A) p.27 Z = {Zl' ... , Z p. 89
M }
O"(A) p. 27 Za. p. 93
A* p. 27 J p.94
l
ReA= HA+A*) p. 186 ad] (A) (A) p. 103
1m A:= ~(A - A*) p. 186 Oz p. 99
A ~ cId p. 186 Pu p. 272
etA = exp(tA) p.28
Function spaces
c(n, E) p. 167 Bk(Rn) p. 176
c(n) := c(n, ([) Bco(Rn) p. 225
ck(n) p. 167 Bco,l(Rn) p.274
c~(n) :={f E ck(n) :supp(f) Bk (IRn) p. 176
compact} (LP(n, E), 11·llu(ll,E)) p. 167
cg(Rn,IR) p. 176 LP(n) := LP(n, ([) p. 167
Ck(n) p. 227 Wk(n),II'llwk(ll) p. 227
Cw(I, X) p. 148 Wt(n) p. 227
V(n) p. 227 Hr,r'[E] ,
p. 170
P,,,(
V'(n) p. 227 Hr,r' [E] := Hr,r' [E] p. 170
(', ')v'(ll),'D(ll) p. 227 P p,(.)
Hr,r'[E] := H;,r'[E] p. 170
Y(Rn, E) p. 167
Y'(Rn, E) p. 167 H;[E] := H;,O[E] p. 170
Y(Rn) := Y(Rn, ([) p. 167 Hr[E] := Hr,O[E] p. 170
Y'(Rn) := Y'(Rn, ([) p. 167 Hrp,,,r,(' := HrP,,,r,(' [([] p. 170
