Spaces of Measures and their Applications to Structured Population Models

321 Pages·2022·3.465 MB·English

by  Christian Düll, Piotr Gwiazda, Anna Marciniak-Czochra, Jakub Skrzeczkowski

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36 SpacesofMeasuresandtheirApplications
to Structured Population Models
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Spaces of Measures and their Applications
to Structured Population Models
CHRISTIAN DU¨ LL
Universita¨tHeidelberg
PIOTR GWIAZDA
PolskaAkademiaNauk(PAN),Warsaw
ANNA MARCINIAK-CZOCHRA
Universita¨tHeidelberg
JAKUB SKRZECZKOWSKI
UniwersytetWarszawski,Poland
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Informationonthistitle:www.cambridge.org/9781316519103
DOI:10.1017/9781009004770
©ChristianDu¨ll,PiotrGwiazda,AnnaMarciniak-CzochraandJakubSkrzeczkowski2022
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2022
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Du¨ll,Christian,author.|Gwiazda,Piotr,author.|
Marciniak-Czochra,Anna,author.|Skrzeczkowski,Jakub,author.
Title:Spacesofmeasuresandtheirapplicationstostructuredpopulation
models/ChristianDu¨ll,PiotrGwiazda,AnnaMarciniak-Czochra,JakubSkrzeczkowski.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2022.|
Series:Cambridgemonographsonappliedandcomputationalmathematics;
36|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021029775(print)|LCCN2021029776(ebook)
|ISBN9781316519103(hardback)|ISBN9781009004770(epub)
Subjects:LCSH:Population–Mathematicalmodels.|Biology–Mathematical
models.|Radonmeasures.|Metricspaces.|Lipschitzspaces.|
Functionsofboundedvariation.|BISAC:MATHEMATICS/General
Classification:LCCHB849.51.D852022(print)|LCCHB849.51(ebook)|DDC304.601/51–dc23
LCrecordavailableathttps://lccn.loc.gov/2021029775
LCebookrecordavailableathttps://lccn.loc.gov/2021029776
ISBN978-1-316-51910-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pagevii
Notation ix
Introduction 1
1 AnalyticalSetting 10
1.1 Preliminaries 10
1.2 M(S)asaNormedSpace 15
1.3 ThePositiveConeM+(S) 26
1.4 RelationtotheWassersteinDistance 42
2 StructuredPopulationModelsontheStateSpaceR+ 58
2.1 IntroductionofaStructuredPopulationModel 58
2.2 AnalysisoftheLinearModel 63
2.3 TheNonlinearProblem 77
2.4 UniquenessofaMeasureSolution 80
2.5 SplittingOperatorApproach 82
2.6 MeasureSolutionswithSemigroupSplittingTechnique 90
2.7 FurtherNotesandReferences 99
3 StructuredPopulationModelsonProperSpaces 100
3.1 Two-ParameterFamiliesofHomeomorphisms 101
3.2 LinearStructuredPopulationModelonaMetricSpace 108
3.3 TheNonlinearModel 119
3.4 EquivalenceoftheGeneralisedModelandthePDEModel 128
3.5 Applications 136
4 NumericalMethodsforStructuredPopulationModels 148
4.1 FiniteDifferenceSchemes 148
4.2 AlgorithmstoComputeDistancesBetweenMeasures 150
4.3 TheEscalatorBoxcarTrainAlgorithm 155
v
vi Contents
4.4 Particle Method Combined with Splitting
OperatorTechnique 177
4.5 FurtherNotesandReferences 183
5 RecentDevelopmentsandFuturePerspectives 186
5.1 AsymptoticAnalysis 186
5.2 DifferentiabilityandOptimalControlProblems 195
5.3 IrregularTransport 203
5.4 FurtherNotesandReferences 207
AppendixA Topology,CompactnessandProperSpaces 209
AppendixB FunctionalAnalysis 215
AppendixC BoundedLipschitzandHölderFunctions 220
AppendixD ResultsonApproximationwithPolynomials 229
AppendixE DifferentialGeometry 233
AppendixF MeasureTheory 235
AppendixG WeakerTopologiesonSpacesofMeasures 242
AppendixH TheBochnerIntegral 269
AppendixI Semigroups 272
AppendixJ SupplementtoChapter2 278
AppendixK TechnicalProofsfromChapter3 282
References 289
Index 307
Preface
The idea of this book was developed over several years of joint research
and numerous discussions with colleagues whose focus is in functional
analysis, partial differential equations, transport processes, probability theory
ormathematicalmodelling.
The development of the mathematical framework for the analysis of
structured population models in measure spaces presented in this book was
supported by many projects, starting with the Emmy Noether Project funded
by the German Research Council (DFG) and by the European Research
Council Starting Grant for Anna Marciniak-Czochra and the International
PhD Programme “Mathematical Methods in Natural Sciences” (Foundation
forPolishScience)coordinatedbyPiotrGwiazda.ThelatterledtoseveralPhD
projectsthatacceleratedtheprogressofdevelopmentofanalyticandnumerical
theory.WeexpressthankstoAgnieszkaUlikowskaandJe˛drzejJabłoński,for
significantcontributions.
Many people have contributed to the development of this comprehensive
framework. The range of individuals involved is far too wide to be able to
list them all. In particular, however, we would like to thank our colleagues
andcollaboratorsJoséA.Carrillo,RinaldoColombo,KarolinaKropielnicka,
Sander C. Hille, Marie Doumic, Benoît Perthame, Agnieszka Świerczewska-
Gwiazda, Emil Wiedemann, Tomasz De˛biec, Horst R. Thieme, Azmy
S.Ackleh,NicolasSaintierandTomaszSzarek,whoinspiredusandhelpedin
clarifyinganalyticalconceptsandanchoringtheminrelevantareasofanalysis
andprobabilitytheory.
Many aspects of the book have been influenced by insightful questions
and suggestions from the students who worked through the book as a part
of the Master’s course “Spaces of Radon Measures: Functional Analysis and
Applications to PDEs” given at Heidelberg University in the summer term
vii
viii Preface
of 2020. The course was part of the research-oriented teaching programme
of the Heidelberg STRUCTURES Excellence Cluster supported by Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-
2181/1 - 390900948. It attracted mathematics and scientific computing
Master’s students interested in functional analysis and its applications to
partial differential equations, especially in the context of the modelling of
transport and growth processes. Teaching the subject helped significantly to
improve the quality of the book. The students detected and helped us to
removemanyerrors,askedinterestingquestions,madecarefulcommentsand
frequentlydemandedclarifyingexamples.Wewanttoexpressourgratitudefor
helptoJorisEdelmann,AlinaFeldmann,LukasHuber,SumetKhumphairan,
FinnMünnich,TuanTungNguyen,AchitaPrasertwaree,CamilloTissot,Kay
ZaradnyandMikulášZindulka.
The final shape of the book has profited from fruitful discussions with
Marek Kimmel and the constructive criticism and suggestions we received
from anonymous reviewers and colleagues from the Heidelberg Group on
AppliedAnalysisandModelinginBiosciences,especiallyMariaV.Barbarossa
and Johannes Kammerer. We also acknowledge the support of our friends
Zuzanna Szymańska and Błażej Miasojedow from Warsaw, who detected
glitches in our numerical pseudo-codes. During preparation of this book,
Piotr Gwiazda and Jakub Skrzeczkowski were supported by the National
Science Center, Poland, through projects nos. 2018/30/M/ST1/00423 and
2019/35/N/ST1/03459,respectively.
Keywords:structuredpopulationmodel,boundedLipschitzdistance,nonneg-
ativeRadonmeasures.
AMSSubjectClassification:26A16,28A33,35B30,35F10,35L50,35R02,
65M12.
Notation
GeneralNotation
Weusethefollowingnotationforsets.Let(S,d)beametricspace.Then
B(S),theBorelσ-algebraofS,i.e.theσ-algebrageneratedbytheopensets
ofS.
B (s) := {x ∈ S | d(x,s) ≤r},theopensetwithradiusr andcentres.
r
N= {1,2,3,...}.
N = {0,1,2,...}.
0
R+ =[0,∞).
R=R∪{+∞,−∞}.
Sd,theunitsphere{x ∈Rd+1 | (cid:6)x(cid:6) =1}.
Ingeneral,wewilluseasuperscript+toindicatenonnegativity.
SpecialFunctionsandSpaces
Herewepresentaselectionofspacesandfunctionsthatappearseveraltimes
throughout the book and whose definition the reader might want to look up
onceinawhile.Notethatspecialspacesthatoccuronlylocallyinonesection
arenotlistedhere.
1,constantfunction1(x) =1forall x.
(cid:2) (cid:3)
BCα,1 [0,T]×M+(R+);X , the space of X-valued, bounded functions
definedon[0,T]×M+(R+)thatareα-HöldercontinuousintandLipschitz
continuousinM+(R+),seeSection3.5.
BL(S),thespaceofboundedLipschitzfunctionsS →R.
BL(S)+∗,thespaceofpositivelinearfunctionalsonBL(S).
ix