Table Of Content53
Texts in Applied Mathematics
Editors
J.E. Marsden
L. Sirovich
S.S. Antman
Advisors
G. Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton
Texts in Applied Mathematics
1. Sirovich: Introduction to Applied Mathematics.
2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos.
3. Hale/Koçak: Dynamics and Bifurcations.
4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed.
5. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary
Differential Equations.
6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems,
2nd ed.
7. Perko: Differential Equations and Dynamical Systems, 3rd ed.
8. Seaborn: Hypergeometric Functions and Their Applications.
9. Pipkin: A Course on Integral Equations.
10. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd ed.
11. Braun: Differential Equations and Their Applications, 4th ed.
12. Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed.
13. Renardy/Rogers: An Introduction to Partial Differential Equations.
14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications.
15. Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed.
16. Van de Velde: Concurrent Scientifi c Computing.
17. Marsden/Ratiu: Introduction to Mechanics and Symmetry, 2nd ed.
18. Hubbard/West: Differential Equations: A Dynamical Systems Approach:
Higher-Dimensional Systems.
19. Kaplan/Glass: Understanding Nonlinear Dynamics.
20. Holmes: Introduction to Perturbation Methods.
21. Curtain/Zwart: An Introduction to Infi nite-Dimensional Linear Systems Theory.
22. Thomas: Numerical Partial Differential Equations: Finitc Difference Methods.
23. Taylor: Partial Differential Equations: Basic Theory.
24. Merkin: Introduction to the Theory of Stability of Motion.
25. Naber: Topology, Geometry, and Gauge Fields: Foundations.
26. Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach.
27. Reddy: Introductory Functional Analysis with Applications to Boundary-Value
Problems and Finite Elements.
28. Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering
Mathematics.
29. Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach.
30. Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical
Computation, Wavelets.
(continued after index)
Grigorios A. Pavliotis • Andrew M. Stuart
Multiscale Methods
Averaging and Homogenization
Grigorios A. Pavliotis Andrew M. Stuart
Department of Mathematics Mathematics Institute
Imperial College London University of Warwick
London SW7 2AZ Coventry CV4 7AL
United Kingdom United Kingdom
[email protected] [email protected]
Series Editors
J.E. Marsden L. Sirovich
Control and Dynamical Systems, 107–81 Division of Applied Mathematics
California Institute of Technology Brown University
Pasadena, CA 91125 Providence, RI 02912
USA USA
[email protected] [email protected]
S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA
[email protected]
ISBN: 978-0-387-73828-4 e-ISBN: 978-0-387-73829-1
DOI: 10.1007/978-0-387-73829-1
Library of Congress Control Number: 2007941385
Mathematics Subject Classifi cation (2000): 34, 35, 60
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Series Preface
Mathematics is playing an ever more important role in the physical and biological
sciences, provoking a blurring of boundaries between scientific disciplines and a
resurgence of interest in the modern as well as the classical techniques of applied
mathematics.Thisrenewalofinterest,bothinresearchandteaching,hasledtothe
establishmentoftheseriesTextsinAppliedMathematics(TAM).
The development of new courses is a natural consequence of a high level of
excitementontheresearchfrontierasnewertechniques,suchasnumericalandsym-
boliccomputersystems,dynamicalsystems,andchaos,mixwithandreinforcethe
traditional methods of applied mathematics. Thus, the purpose of this textbook se-
riesisto meet thecurrent and futureneeds of theseadvances and to encourage the
teachingofnewcouses.
TAMwillpublishtextbookssuitableforuseinadvancedundergraduateandbe-
ginninggraduatecourses,andwillcomplementtheAppliedMathematicalSciences
(AMS) series, which will focus on advanced textbooks and research-level mono-
graphs.
Pasadena,California J.E.Marsden
NewYork,NewYork L.Sirovich
CollegePark,Maryland S.S.Antman
TomyparentsAργυρηandΣoυλταναandtomybrotherΓιωργo.
CarryHome.Γρηγoρης.
FormychildrenNatalie,Sebastian,andIsobel.
AMS.
Preface
The aim of these notes is to describe, in a unified fashion, a set of methods for the
simplification of a wide variety of problems that all share the common feature of
possessingmultiplescales.1 Themathematicalmethodswestudyareoftenreferred
toasthemethodsofaveragingandofhomogenization.Themethodsapplytopar-
tialdifferentialequations(PDEs),stochasticdifferentialequations(SDEs),ordinary
differential equations (ODEs),and Markov chains. The unifying principle underly-
ing the collection of techniques described here is the approximation of singularly
perturbed linear equations. The unity of the subject is most clearly visible in the
applicationofperturbationexpansionstotheapproximationofthesesingularpertur-
bation problems. A significant portion of the notes is devoted to such perturbation
expansions.InthiscontextweusethetermResulttodescribetheconclusionsofa
formalperturbationargument.Thisenablesustoderiveimportantapproximationre-
sultswithouttheburdenofrigorousproof,whichcansometimesobfuscatethemain
ideas.However,wewillalsostudyavarietyoftoolsfromanalysisandprobability,
used to place the approximations derived on a rigorous footing. The resulting the-
orems are proved using a range of methods, tailored to different settings. There is
lessunitytothispartofthesubject.Asaconsequence,considerablebackgroundis
requiredtoabsorbtheentirerigoroussideofthesubject,andwedevoteasignificant
partofthebooktothisbackgroundmaterial.
The first part of the notes is devoted to the Background; the second to the
Perturbation Expansions, which provide the unity of the subject matter; and the
third to the Theory justifying these perturbative techniques. We do not necessarily
recommend that the reader covers the material in this order. A natural way to get
an overview of the subject is to read through Part II of the book on perturbation
1In this book we will apply the general methodology to problems with two widely sepa-
ratedcharacteristicscales.Theextensiontosystemswithmanyseparatedscalesisfairly
straightforwardandwillbediscussedinanumberoftheDiscussionandBibliographysec-
tions,whichconcludeeachchapter.Inallcases,theimportantassumptionwillbethatof
scaleseparation.
X Preface
expansions,referringtothebackgroundmaterialasneeded.Thetheorycanthenbe
studied,aftertheformoftheapproximationsisunderstood,onacase-by-casebasis.
PartI(Background)containstheelementsofthetheoryofanalysis,probability,
and stochastic processes, as required for the material in these notes, together with
basicintroductorymaterialonODEs,Markovchains,SDEs,andPDEs.PartII(Per-
turbationExpansions)illustratestheuseofideasfromaveragingandhomogenization
tostudyODEs,Markovchains,SDEs,andPDEsofelliptic,parabolic,andtransport
type;invariantmanifoldsarealsodiscussedandareviewedasaspecialcaseofav-
eraging.PartIII(Theory)containsillustrationsoftherigorousmethodsthatmaybe
employedtoestablishthevalidityoftheperturbationexpansionsderivedinPartII.
ThechaptersinPartIIIrelatetothoseinPartIIinaone-to-onefashion.Itispossible
topickparticularthemesfromthisbookandcoversubsetsofchaptersdevotedonly
to those themes. The reader interested primarily in SDEs should cover Chapters 6,
10,11,17,and18.MarkovchainsarecoveredinChapters5,9,and16.Thesubject
ofhomogenizationforellipticPDEsiscoveredinChapters12and19.Homogeniza-
tionandaveragingforparabolicandtransportequationsarecoveredinChapters13,
14,20,and21.
The subject matter in this set of notes has, for the most part, been known for
severaldecades.However,theparticularpresentationofthematerialhereis,webe-
lieve,particularlysuitedtothepedagogicalgoalofcommunicatingthesubjecttothe
widerange ofmathematicians, scientists,andengineers whoarecurrentlyengaged
in the use of these tools to tackle the enormous range of applications that require
them.Inparticularwehavechosenasettingthatdemonstratesquiteclearlythewide
applicability of the techniques to PDEs, SDEs, ODEs, and Markov chains, as well
ashighlightingtheunityoftheapproach.Suchawide-rangingsettingisnotunder-
taken, we believe, in existing books, or is done so less explicitly than in this text.
WehavechosentousethephrasingMultiscaleMethodsinthetitleofthebookbe-
causethematerialpresentedhereformsthebackboneofasignificantportionofthe
amorphousfieldthatnowgoesbythatname.However,werecognizethatthereare
vastpartsofthefieldwedonotcover.Inparticular,scaleseparationisafundamen-
tal requirement in all of the perturbation techniques presented in this book. Many
applications,however,possessacontinuumofscales,withnoclearseparation.Fur-
thermore,manyoftheproblemsarisinginmultiscaleanalysisareconcernedwiththe
interfacingofdifferentmathematicalmodelsappropriateatdifferentscales(suchas
quantum, molecular, and continuum); the tools presented in these notes do not di-
rectlyaddressproblemsarisinginsuchapplications,asourstartingpointisasingle
mathematicalmodelinwhichscaleseparationispresent.
These notes are meant to be an introduction, aimed primarily toward graduate
students. Part I of the book (where we lay the theoretical foundations) and Part III
(where we state and prove theorems concerning simplified versions of the models
studiedinPartII)arenecessarilyterse;otherwiseitwouldbeimpossibletopresent
thewiderangeofapplicationsoftheideasandillustratetheirunity.Extensionsand
generalizationsoftheresultspresentedinthesenotes,aswellasreferencestothelit-
erature,aregivenintheDiscussionandBibliographysectionattheendofeachchap-
ter. With the exception of Chapter 1, all chapters are supplemented with exercises.
Preface XI
Wehopethattheformatofthebookwillmakeitappropriateforusebothasatext-
bookandforself-study.
Acknowledgments
We are especially grateful to Konstantinos Zygalakis who read and commented on
much of the manuscript, typed parts of it, and helped to create some of the fig-
ures. Special thanks also are due to Martin Hairer, Valeriy Slastikov, Endre Su¨li,
and Roger Tribe, all of whom read portions of the book in great detail, resulting
in many constructive suggestions for improvement. We are also grateful to Niklas
Branstrom,MikhailCherdanstev,DrorGivon,WeinanE,DavidEpstein,LiamJones,
Raz Kupferman, Brenda Quinn, Florian Theil, and David White for many helpful
comments that substantially improved these lecture notes. The book grew out of
courses on multiscale methods and on homogenization that we taught at Imperial
CollegeLondonandatWarwickUniversityduringthelastfouryears;wethankall
thestudentswhoattendedthesecourses.Someofthismaterial,inturn,grewoutof
thereviewarticle[125],andwearegratefultoDrorGivonandRazKupfermanfor
theircollaborationinthisarea.
Partsofthebookwereusedasabasisfortwoshortcoursesonmultiscalemethods
thatwetaughtinApril2007attheMathematicalSciencesResearchInstitute(MSRI),
Berkeley, CA, and the Mathematics Research Centre (MRC), Warwick University.
Wethankalltheparticipantsatthesecourses,togetherwiththeMSRI,MRC,Lon-
donMathematicalSociety,andtheUKEngineeringandPhysicalSciencesResearch
Council(EPSRC)foradministrativeandfinancialsupportfortheseshortcourses.
GP has received financial support from the EPSRC, and AMS has received fi-
nancialsupportfromtheEPSRCandfromtheU.S.OfficeofNavalResearchduring
thewritingofthisbook.Thisfundedresearchhashelpedshapemuchofthematerial
presentedinthebook,andtheauthorsaregratefulforthefinancialsupport.
GAPavliotisandAMStuart,August2007
Contents
SeriesPreface...................................................... V
Preface............................................................ IX
1 Introduction................................................... 1
1.1 Overview ................................................. 1
1.2 MotivatingExamples ....................................... 1
1.2.1 ExampleI:SteadyHeatConductioninaCompositeMaterial 2
1.2.2 ExampleII:HomogenizationforAdvection–Diffusion
Equations........................................... 3
1.2.3 ExampleIII:Averaging,Homogenization,andDynamics... 4
1.2.4 ExampleIV:DimensionReductioninDynamicalSystems.. 5
1.3 AveragingVersusHomogenization ............................ 6
1.3.1 AveragingforSystemsofLinearEquations .............. 7
1.3.2 HomogenizationforSystemsofLinearEquations ......... 8
1.4 DiscussionandBibliography ................................. 9
PartI Background
2 Analysis....................................................... 13
2.1 Setup..................................................... 13
2.2 Notation .................................................. 14
2.3 BanachandHilbertSpaces................................... 16
2.3.1 BanachSpaces ...................................... 16
2.3.2 HilbertSpaces....................................... 18
2.4 FunctionSpaces............................................ 18
2.4.1 SpacesofContinuousFunctions........................ 18
2.4.2 LpSpaces .......................................... 19
2.4.3 SobolevSpaces...................................... 21
2.4.4 BanachSpace–ValuedSpaces.......................... 22
2.4.5 SobolevSpacesofPeriodicFunctions ................... 23
