Evolutionary Equations: Picard's Theorem for Partial Differential Equations, and Applications

321 Pages·2022·3.657 MB·English

by  Christian Seifert, Sascha Trostorff, Marcus Waurick

Operator Theory
Advances and Applications
287
Christian Seifert
Sascha Trostorff
Marcus Waurick
Evolutionary
Equations
Picard's Theorem for Partial
Differential Equations, and
Applications

OperatorTheory: Advances andApplications
Volume287
Foundedin1979byIsraelGohberg
Editors:
JosephA.Ball(Blacksburg,VA,USA)
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LinearOperatorsandLinearSystems
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Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/4850
Christian Seifert (cid:129) Sascha Trostorff (cid:129)
Marcus Waurick
Evolutionary Equations
Picard’s Theorem for Partial Differential
Equations, and Applications
ChristianSeifert SaschaTrostorff
InstitutfürMathematik MathematischesSeminar
TechnischeUniversitätHamburg Christian-Albrechts-UniversitätzuKiel
Hamburg,Germany Kiel,Germany
MarcusWaurick
InstitutfürAngewandteAnalysis
TUBergakademieFreiberg
Freiberg,Germany
ISSN0255-0156 ISSN2296-4878 (electronic)
OperatorTheory:AdvancesandApplications
ISBN978-3-030-89396-5 ISBN978-3-030-89397-2 (eBook)
https://doi.org/10.1007/978-3-030-89397-2
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Preface
The theory of evolutionary equations has its origins in the seminal paper [82] by
RainerPicard,workingattheTechnischeUniversitätDresden,Germany.Allthree
of us were students at this university at the time. Thus, we were lucky enough to
learnthetheoryofevolutionaryequationsfromitsearlydayson.Wetookandstill
taketheopportunitytobepartofthecontinuouslygrowinggroupofpeopleactively
developingthe theory further. In fact, both the PhD and the habilitation theses of
S.T. and M.W. are concerned with generalisations of the initial theory as well as
opening up new directions of research. It is also an aim of these lecture notes to
presentsomeoftheselatestresultsinacoherenttext.
In generalterms, the theoryof evolutionaryequationsprovidesa Hilbertspace
method to understand differential equations. It comprises a unified approach
to solving both ordinary and partial differential equations as well as to show
generalwell-posednessresultsfor both stationaryand nonstationary,thatis, time-
dependentproblems.Besideswell-posednesstheoremsforlargeclassesofdifferen-
tialequations(includingnonlinearproblems),thetheoryaddressesquantitativeand
qualitativequestionsrelatedtoexponentialstability,homogenisationandregularity.
Thislistisboundtogetlongerinfuture.Thegeneralapproach,furthermore,allows
for either a comparison or unification (depending on the context) of approaches
initially tailored for particular types of equations, such as parabolic, hyperbolic
or elliptic. In particular, mixed type equations can be considered and understood
withthepresentedperspective.Thus,manyfundamentalequationsofmathematical
physics such as the heat equation, wave equation, Maxwell’s equations and the
equationsofelasticitytheorycanbetreatedusingthismethod.
The abovementioned equations fitting into a general solution theory posed a
surprising fact (at least for us). Even more so as the general problem class of
evolutionaryequationsbases on fourratherelementaryobservationsbeingshortly
summarisedasfollows:
(cid:129) the (distributional, time) derivative can be realised as a boundedly invertible,
normaloperatorinexponentiallyweightedL -spaces,
2
v
vi Preface
(cid:129) manyequationsofmathematicalphysics(includingtheabove)canbewrittenas
asumoftwounboundedoperators:oneoftheminvolvingfirstorderdifferential
operatorsinspaceandtheotheroneafirstorderdifferentialoperatorintime,
(cid:129) theintroductionofabstractso-called‘materiallaws’or‘materiallawoperators’
as coefficients of the time derivative describes both heterogeneous media and
couplingeffects,
(cid:129) the solution mechanism is based on monotonicity of both the sum of the
mentionedunboundedoperatorstogetherwithitsadjointcomputedinthespace-
timeHilbertspace;inmanycases,thismonotonicityreadilyfollowsfromtheone
ofthetimederivativesmultipliedwiththemateriallawoperator.
The last observationis particularlystriking in as much as the monotonicityof the
timederivativemultipliedwiththemateriallawoperatorisrathereasilyobtainedin
manyapplications.Thisprovidesawell-posednesscriterionthatisbothelementary
andgeneral,oftenleadingtogeneralisationsofknownsolutioncriteriaforparticular
situations.Fromanappliedperspective,thesecriteriacanoftenbeverifiedwithout
divingintotheintricaciesofmoreinvolvedsolutionmethodsand,thus,theexisting
numericalmethodsforevolutionaryequationscanbeusedtonumericallysolvethe
consideredequationathand.
In the context of time-dependent equations and related topics, there is a well-
established format of introducingvarious subjects to advancedmaster or diploma
students as well as PhD students, namely the Internet Seminar on Evolution
Equations. Since 1997, it has been organised by various groups from Germany,
Hungary, Italy, the UK and the Netherlands, providing virtual lectures as well
as supervised student projects. In the academic year 2019–2020, we organised
the InternetSeminar focussing on evolutionaryequations. The present book is an
extendedversionofthelecturenotesforthevirtuallectures.Assuch,itpresentsa
thoroughintroductiontothetheoryofevolutionaryequationsandthecorresponding
solution theory and provides many properties, different classes of examples and
properties of solutions, taking the reader from the early beginning of Picard’s
theoremto(almost)thestate-of-the-artinthistheory.
As the text is based on weekly virtual lectures, each chapter of the book is
intendedto(roughly)compriseaselectionofmaterialthatcovers4hoflecturesand
2hofexerciseclasses.Hence,thisbookcoversmaterialforoneortwosemesters.It
isintendedformasterordiplomastudentsaswellasPhDstudentsandresearchers
and requires only basic knowledge on functional analysis, foundations in Hilbert
space theory and complex analysis in one variable. The needed amount of these
is similar to the ones providedin basic courses on these topics. Apart from these
prerequisites,thematerialofthebookisself-contained.Attheendofeachchapter,
we appended7 exercisesof varyingdifficulties from easy to challengingand also
wecommentedonfurtherreadingand/oronthewidercontextofthecontentsofthe
chapter.
We are indebted to Rainer Picard for introducing this theory to us more than
a decade ago and for his past and ongoing support in many areas. We are very
gratefultotheparticipantsofthe23rdInternetSeminarforreadingthemanuscript,
Preface vii
working with the material and thus checking large parts of the present text. In
particular, we cordially thank Jürgen Voigt, Hendrik Vogt and Michael Doherty
fortheirvaluablecomments,whichledtomanyimprovements.M.W.thanksJussi
Behrndt for the invitation on a guest professorship at the TU Graz at the end of
2020andthebeginningof2021.Thisguestappointmentledtothepresentationof
the course at TU Graz with many interested students, in particular, Julia Hauser,
PeterSchlosser,GeorgStenzelandRaphaelWatschinger,studyingthematerialand
providingusefulfeedbackthathelpedtoprofoundlyimprovethetext.Wethankthe
anonymousreferees for their comments that led to further improvements. All the
remainingmistakesareourown.
We thank Christiane Tretter, Editor of the Operator Theory series, for her
encouragementandguidance.Moreover,wethankDorothyMazlumforhersupport
during the earlier stages of the manuscript (and its submission) as well as Daniel
Jagadisan for the completion and final submission process. Last but not the least,
we thank the TU Bergakademie Freiberg for providing the open access costs for
thismanuscript,thusmakingthefinalversionoftheselecturenoteseasilyavailable
aroundtheworldwithoutfurthercosts.
Hamburg,Germany ChristianSeifert
Kiel,Germany SaschaTrostorff
Freiberg,Germany MarcusWaurick
August2021
Contents
1 Introduction................................................................. 1
1.1 FromODEstoPDEs................................................. 1
1.2 Time-independentProblems......................................... 4
1.3 EvolutionaryEquations.............................................. 5
1.4 ParticularExamplesandtheChangeofPerspective................ 6
1.5 ABriefOutlineoftheCourse ....................................... 9
1.6 Comments ............................................................ 11
Exercises ..................................................................... 12
References.................................................................... 12
2 UnboundedOperators ..................................................... 15
2.1 OperatorsinBanachSpaces ......................................... 15
2.2 OperatorsinHilbertSpaces.......................................... 18
2.3 ComputingtheAdjoint............................................... 21
2.4 TheSpectrumandResolventSet.................................... 23
2.5 Comments ............................................................ 29
Exercises ..................................................................... 29
References.................................................................... 29
3 TheTimeDerivative........................................................ 31
3.1 Bochner–LebesgueSpaces........................................... 31
3.2 TheTimeDerivativeasaNormalOperator......................... 42
3.3 Comments ............................................................ 47
Exercises ..................................................................... 48
References.................................................................... 49
4 OrdinaryDifferentialEquations.......................................... 51
4.1 TheDomainof∂ andtheSobolevEmbeddingTheorem........ 51
t,ν
4.2 ThePicard–LindelöfTheorem....................................... 54
4.3 DelayDifferentialEquations ........................................ 60
4.4 Comments ............................................................ 64
ix
x Contents
Exercises ..................................................................... 64
References.................................................................... 66
5 TheFourier–LaplaceTransformationandMaterialLaw
Operators.................................................................... 67
5.1 TheFourierTransformation ......................................... 67
5.2 TheFourier–LaplaceTransformationandItsRelation
totheTimeDerivative................................................ 72
5.3 MaterialLawOperators.............................................. 74
5.4 Comments ............................................................ 80
Exercises ..................................................................... 81
References.................................................................... 83
6 SolutionTheoryforEvolutionaryEquations............................ 85
6.1 FirstOrderSobolevSpaces.......................................... 85
6.2 Well-PosednessofEvolutionaryEquationsandApplications ..... 88
6.3 ProofofPicard’sTheorem........................................... 97
6.4 Comments ............................................................ 99
Exercises ..................................................................... 100
References.................................................................... 101
7 ExamplesofEvolutionaryEquations..................................... 103
7.1 Poro-ElasticDeformations........................................... 103
7.2 FractionalElasticity.................................................. 107
7.3 TheHeatEquationwithDelay....................................... 110
7.4 DualPhaseLagHeatConduction ................................... 111
7.5 Comments ............................................................ 113
Exercises ..................................................................... 114
References.................................................................... 116
8 CausalityandaTheoremofPaleyandWiener ......................... 119
8.1 ATheoremofPaleyandWiener .................................... 119
8.2 ARepresentationResult ............................................. 125
8.3 Comments ............................................................ 128
Exercises ..................................................................... 128
References.................................................................... 129
9 InitialValueProblemsandExtrapolationSpaces ...................... 131
9.1 WhatareInitialValues?.............................................. 131
9.2 ExtrapolatingOperators.............................................. 133
9.3 EvolutionaryEquationsinDistributionSpaces..................... 138
9.4 InitialValueProblemsforEvolutionaryEquations................. 140
9.5 Comments ............................................................ 147
Exercises ..................................................................... 147
References.................................................................... 148