Lectures on Nonlinear Evolution Equations: Initial Value Problems

315 Pages·2015·2.905 MB·English

by  Reinhard Racke (auth.)

Reinhard Racke
Lectures
on Nonlinear
Evolution
Equations
Initial Value Problems
Second Edition

Reinhard Racke
Lectures on Nonlinear
Evolution Equations
Initial Value Problems
Second Edition
Reinhard Racke
Department of Mathematics and Statistics
University of Konstanz
Konstanz, Germany
ISBN 978-3-319-21872-4 ISBN 978-3-319-21873-1 (eBook)
DOI 10.1007/978-3-319-21873-1

Library of Congress Control Number: 2015948864

Mathematics Subject Classification (2010): 35B40, 35K05, 35K55, 35L05, 35L45, 35L70, 35Q99, 36Q60,
35Q74, 74F05

Springer Cham Heidelberg New York Dordrecht London
First edition published by Vieweg, Wiesbaden, 1992
© Springer International Publishing Switzerland 2015
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Preface
This book is the second edition of the book Lectures on nonlinear evolution equations.
Initial value problems [150] from 1992. Additionally, it now includes a new Chapter 13
on initial-boundary value problems for waveguides, addressing more advanced students
and researchers.
Several people contributed helpful comments on the first edition and on the new Chap-
ter 13. In particular I would like to thank Dipl.-Math. Karin Borgmeyer, Dr. Michael
Pokojovy, Dipl.-Math. Marco Ritter, and Dipl.-Math. Alexander Sch¨owe. For typing
Chapter 13 I thank Gerda Baumann. I am obliged to Birkh¨auser, in particular to
Clemens Heine, for the interest in publishing this book.
Konstanz, April 2015 Reinhard Racke
Preface to the first edition:
The book in hand is based on lectures which were given at the University of Bonn in
the winter semesters of 1989/90 and 1990/91. The aim of the lectures was to present
an elementary, self-contained introduction into some important aspects of the theory of
global, small, smooth solutions to initial value problems for nonlinear evolution equa-
tions. The addressed audience included graduate students of both mathematics and
physics who were only assumed to have a basic knowledge of linear partial differential
equations. Thus, in the spirit of the underlying series, this book is intended to serve as
a detailed basis for lectures on the subject as well as for self-studies for students or for
other newcomers to this field.
Thepresentationofthetheoryismadeusingtheclassicalmethodofcontinuationoflocal
solutionswiththehelp ofaprioriestimatesobtainedforsmall data. The corresponding
globalexistence theoremshavebeenprovedmainlyinthelastdecade,focussingonfully
nonlinear systems. Related questions concerning large data problems, the existence of
weak solutions or the analysis of shock waves are not discussed. Also the question of
optimal regularity assumptions on the coefficients is beyond the scope of the book and
is touched only in part and exemplarily.
Most of the material presented here has only been previously published in original pa-
pers, and some of the material has never been published until now. Therefore, I hope
thatboththeinterestedbeginnerinthefieldandtheexpertwillbenefitfromreadingthe
book. Inaddition,alonglistofreferenceshasbeenincluded,althoughitisnotintended
v
vi Preface
to be exhaustive. Of course the selection of the material follows personal interests and
tastes.
Severalcolleaguesandstudentshelpedmewiththeircommentsonearlierversionsofthis
book. In particular I would like to thank R. Arlt, S. Jiang, S. Noelle, P. P. Schirmer,
R. P. Spindler, M. Stoth and F. Willems. Special thanks are due to R. Leis who also
suggested writing first lecture notes in 1989 (SFB 256 Vorlesungsreihe Nr. 13, Univer-
sit¨at Bonn (1990), in German). I am obliged to the Verlag Vieweg and to the editor of
the “Aspects of Mathematics”, K. Diederich, for including the book in this series. The
major part of typing the manuscript was done by R. Mu¨ller and A. Thiedemann whom
I thank for their expert work. Last, but not least, I would like to thank the Deutsche
Forschungsgemeinschaft, Sonderforschungsbereich 256,forgenerousandcontinuoussup-
port.
Bonn, August 1991 Reinhard Racke
Contents
Introduction 1
1 Global solutions to wave equations — existence theorems 7
2 Lp–Lq-decay estimates for the linear wave equation 15
3 Linear symmetric hyperbolic systems 21
3.1 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 A global existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Remarks on other methods . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Some inequalities 34
5 Local existence for quasilinear symmetric hyperbolic systems 57
6 High energy estimates 78
7 Weighted a priori estimates for small data 82
8 Global solutions to wave equations — proofs 89
8.1 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.2 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9 Other methods 104
10 Development of singularities 108
11 More evolution equations 112
11.1 Equations of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.1.1 Initially isotropic media in IR3 . . . . . . . . . . . . . . . . . . . . 114
11.1.2 Initially cubic media in IR2 . . . . . . . . . . . . . . . . . . . . . . 121
11.2 Heat equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.3 Equations of thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . 144
11.4 Schr¨odinger equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.5 Klein–Gordon equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.6 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.7 Plate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12 Further aspects and questions 207
vii
viii Contents
13 Evolution equations in waveguides 221
13.1 Nonlinear wave equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 222
13.1.1 Linear part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
13.1.2 Nonlinear part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.2 Schr¨odinger equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
13.3 Equations of elasticity and Maxwell equations . . . . . . . . . . . . . . . 243
13.4 General waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Appendix 271
A Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
B The Theorem of Cauchy–Kowalevsky . . . . . . . . . . . . . . . . . . . . 278
C A local existence theorem for hyperbolic-parabolic systems . . . . . . . . 282
References 289
Notation 302
Index 304
Introduction
Many problems arising in the applied sciences lead to nonlinear initial value problems
(nonlinear Cauchy problems) of the following type
V +AV =F(V,...,∇βV), V(t=0)=V0.
t
Here V =V(t,x) is a vector-valued function taking values in IRk (or Ck), where t≥ 0,
x ∈ IRn,andAisagivenlineardifferentialoperatorofordermwithk, n, m∈IN. F is
a given nonlinear function of V and its derivatives up to order |β|≤m, and ∇ denotes
the gradient with respect to x, while V0 is a given initial value. In particular the case
|β|=m, i.e. the case of fully nonlinear initial value problems, is of interest.
Animportantexamplefrommathematicalphysicsisthewaveequationdescribinganin-
finitevibratingstring(membrane,soundwave,respectively)inIR1(IR2, IR3,respectively;
generalized: IRn). The second-order differential equation for the elongation y = y(t,x)
at time t and position x is the following:
∇y
y −∇(cid:2)(cid:2) =0,
tt
1+|∇y|2
where ∇(cid:2) denotes the divergence. This can also be written as
∇y
y −Δy =∇(cid:2)(cid:2) −Δy=:f(∇y,∇2y).
tt
1+|∇y|2
We notice that f has the following property:
f(W)=O(|W|3) as|W|→0.
Additionally one has prescribed initial values
y(t=0)=y0, yt(t=0)=y1.
ThetransformationdefinedbyV :=(y,∇y)turnsthenonlinearwaveequationforyinto
t
a first-order system for V as described above. The investigation of such nonlinear evo-
lution equations hasfound an increasing interest in the last years, in particular because
of their application to the typical partial differential equations arising in mathematical
physics.
We are interested in the existence and uniqueness of global solutions, i.e. solutions
V =V(t,x) which are defined for all values of the time parameter t. The solutions will
be smooth solutions, e.g. C1-functionswith respect tot taking values in Sobolev spaces
ofsufficientlyhighorderofdifferentiability. Inparticulartheywillbeclassicalsolutions.
Moreover we wish to describe the asymptotic behavior of the solutions as t→∞.
© Springer International Publishing Switzerland 2015 1
R. Racke, Lectures on Nonlinear Evolution Equations,
DOI 10.1007/978-3-319-21873-1_1