Table Of ContentLecture Notes in Mathematics 2158
CIME Foundation Subseries
Adrian Constantin
Joachim Escher
Robin Stanley Johnson
Gabriele Villari
Nonlinear
Water Waves
Cetraro, Italy 2013
Adrian Constantin Editor
Lecture Notes in Mathematics 2158
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis,Zürich
MariodiBernardo,Bristol
AlessioFigalli,Austin
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GaborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,ParisandNewYork
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AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Adrian Constantin (cid:129) Joachim Escher (cid:129)
Robin Stanley Johnson (cid:129) Gabriele Villari
Nonlinear Water Waves
Cetraro, Italy 2013
Adrian Constantin
Editor
123
Authors
AdrianConstantin JoachimEscher
FacultyofMathematics Inst.forAppliedMathematics
UniversityofVienna GottfriedWilhelmLeibnizUniversity
Vienna,Austria Niedersachsen
Hannover,Germany
RobinStanleyJohnson GabrieleVillari
SchoolofMathematicsandStatistics DepartmentofMathematics“UlisseDini”
UniversityofNewcastle UniversityofFlorence
NewcastleuponTyne,UnitedKingdom Florence,Italy
Editor
AdrianConstantin
FacultyofMathematics
UniversityofVienna
Vienna,Austria
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-31461-7 ISBN978-3-319-31462-4 (eBook)
DOI10.1007/978-3-319-31462-4
LibraryofCongressControlNumber:2016941322
MathematicsSubjectClassification(2010):76B15,35Q35,34C05
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Preface
Thestudyofwaterwavesinvolvesvariousdisciplinessuchasmathematics,physics
and engineering—to name the obvious—and within this, there are many specific
areasofdirectorassociatedinterestsuchaspuremathematics,appliedmathematics,
modelling, numerical simulation, laboratory experiments, data collection in the
field, the design and construction of ships, harbours and offshore platforms, the
prediction of natural disasters (e.g. tsunamis), climate studies and so on. We are
allfamiliarwith,andprobablyexcitedby,theexperienceofseeingwavesinlakes,
rivers, oceans and even baths and sinks; they are often beautiful, but sometimes
terrifying.Theyarealso mathematicallyintriguingandsusceptibletoa numberof
different,butveryparticular,theoreticalapproaches.Allthesevariousstudieshelp
ustoimprove,inonewayoranother,ourunderstandingofwavepropagationwhich,
inturn,inevitablyleadstobetterphysicsandengineeringasweworkwith,anddeal
withtheeffectsof,wavesonwater.
The meeting held in Cetraro, Italy, June 24–28, 2013, under the auspice of,
and supported by, the Centro Internazionale Matematico Estivo (C.I.M.E., the
InternationalMathematicalSummer Centre) aimed to presentsome of the current
mathematical research in this area. The summer school provided a vehicle for a
selectionofthemainmathematicalavenuestobepresentedviaaseriesoflectures;
in addition, there were short presentations and much discussion, covering other
related topics, such as numerical methods and modelling. This volume brings
togetherthe fourmain lecture courses. The intention, throughthe lectures, was to
presentquitearangeofmathematicalideas,primarilytoshowwhatispossibleand
what,currently,isofparticularinterest.Thegeneralbackgroundtothemathematical
formulation of the classical water-wave problem, and the interplay between what
is observed and how we model this using a robust mathematical formulation,
appears in ‘Asymptotic methods for weakly nonlinear and other water waves’ by
R.S. Johnson. These lectures also covered some aspects of the construction and
generalisation of soliton-type equations (including a brief introduction to ‘soliton
theory’) and, of particular current interest, the rôle that background vorticity can
playintheevolutionofthe wavesanditseffectsupontheirproperties.Inorderto
showthewealthofpossibilitiesusinganasymptoticapproach,periodicwaveswith
v
vi Preface
vorticity and edge waves are also discussed. The lectures givenby A. Constantin,
entitled ‘Exact travelling periodic water waves in two-dimensional irrotational
flows’,alsoexplaintheconnectionbetweenwhatisobserved(bothinthelaboratory
and in the field) and what we can describe and predict using a mathematical
approach. The exact solutions are described by, for example, the properties of
the associated particle pathsbased on harmonicanalysis and the theoryof elliptic
partialdifferentialequations.Thustheverypracticalessenceofthewavesandvery
powerfulandrigoroustechniquesaremouldedtoproduceacomprehensivepicture
of the types of flows associated with classical water waves. The theme of particle
paths is taken up in ‘A survival kit in phase plane analysis: some basic models
and problems’ by G. Villari, but the approach here is to use another fundamental
mathematicaltool:themethodofphase-planeanalysis.Again,thethrustistoshow
howasophisticatedandfamiliarbranchofmathematicscanrelateto,andusefully
describe,thedetailsofthecomplexflowpatternsthatareobserved.Thefinalseries
oflecturesmadeuseoftheverypowerfulideasthatunderpinthemoderntechniques
offunctionalanalysis.J.Escherdiscussedthenatureofwavebreakingasitapplies,
mainly,tothesolutionsoftheCamassa-Holmequationin‘Breakingwaterwaves’.
One of the exciting properties of this model equation is that it captures both the
non-breakingandbreakingwavephenomenaofclassicalwaterwaves.Somedetails
thathelptoexplaintherôleoftheinitialdatainpredictingthefinaldevelopmentof
thewaveareprovided,producingsomeimportantestimates.
These four lectures provide a useful source for those who want to begin to
investigate how mathematics can be used to improve our understanding of this
rapidly developing classical research area. In addition, some of the material can
be used by those who are already familiar with one branch of the study of water
waves, to learn more about other areas. We therefore commend this collection of
lecturestoboththenoviceandtheexpert.
Vienna,Austria A.Constantin
Hannover,Germany J.Escher
NewcastleuponTyne,UK R.S.Johnson
Florence,Italy G.Villari
Acknowledgements
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
- INdAM(IstitutoNazionalediAltaMatematica)
- MIUR(Ministerodell’Istruzione,dell’UniversitàedellaRicerca)
- EnteCassadiRisparmiodiFirenze
Contents
ExactTravellingPeriodicWaterWavesinTwo-Dimensional
IrrotationalFlows................................................................ 1
AdrianConstantin
BreakingWaterWaves .......................................................... 83
JoachimEscher
AsymptoticMethodsforWeaklyNonlinearandOtherWaterWaves ..... 121
RobinStanleyJohnson
ASurvivalKitinPhasePlaneAnalysis:SomeBasicModels
andProblems..................................................................... 197
GabrieleVillari
vii
Exact Travelling Periodic Water Waves
in Two-Dimensional Irrotational Flows
AdrianConstantin
...fluiddynamicistsweredividedintohydraulicengineerswho
observedthingsthatcouldnotbeexplainedandmathematicians
whoexplainedthingsthatcouldnotbeobserved.
SirJamesLighthill(1924–1998)
Abstract Mostofthewavesthatareobservedonthesurfaceoftheworld’soceans,
seas and lakes are wind generated. Once initiated, these water waves propagate
substantial distances before their energy is dissipated—propagation distances in
excessofhundredsorthousandstimesawavelengthareneededfortheoccurrence
of a significant energyloss. We address some fundamentalaspects of water-wave
propagationoncewaveshavebeengenerated,withintheframeworkofinviscidtwo-
dimensionalflowtheoryandintheabsenceofunderlyingcurrents.Theemphasisis
placeduponperiodictravellinggravitywaterwavesoflargeamplitude.Thesewave
patterns can only be fully understood in terms of nonlinear effects, linear theory
being not adequate for their description. An in-depth mathematicalstudy is made
possible by uncovering the rich structure of the hydrodynamical free-boundary
problem under investigation, taking advantage of insights from physical observa-
tion,experimentalevidenceandnumericalsimulations.Theinterdisciplinarynature
of this classical research subject is also reflected in the fact that its theoretical
investigationreliesonaninterplaybetweenmethodsandtechniquesfromdynamical
systems,complexanalysis,functionalanalysis,topology,harmonicanalysis,andthe
calculusofvariations.
1 Introduction
MathematicsisthebasiclanguageofPhysicsandEngineeringbutthethreesubjects
emphasizedifferentapproachestoaspecificprobleminfluidmechanics,evenifthe
respectiveboundariesoverlapconsiderably.Mathematicaltechniquesandphysical
A.Constantin((cid:2))
DepartmentofMathematics,King’sCollegeLondon,Strand,LondonWC2L2RS,UK
FacultyofMathematics,UniversityofVienna,Oskar-Morgenstern-Platz1,1090Vienna,Austria
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2016 1
A.Constantin(ed.),NonlinearWaterWaves,LectureNotesinMathematics2158,
DOI10.1007/978-3-319-31462-4_1
2 A.Constantin
principles aim to explain and predict natural phenomena, while engineers mostly
useexperimentaltoolstoprobethesephenomena.Byadheringtoahighstandardof
rigour,mathematiciansworkinginfluidmechanicsprimarilyexpandandelucidate
physical arguments that are more heuristic or intuitive, thus contributing to the
growth of our understanding. Often this process reveals interesting features that
were overlooked,and sometimes it even permits the discovery of new facets that
were not within reach of less advanced mathematical techniques. However, while
thecapacityto grasp,manipulateanddevelopconceptsandtoolspossiblysuffices
to define the mathematicalvalue of an approach,it is notenoughto validate real-
worldapplications.Acoherentexplanationofanaturalphenomenoninvolvescarein
selectingrelevantexplanatoryfactorsandback-upbyempiricaltestsforthetheory.
Thishighlightstheimportanceofengineeringexpertiseinthecontextofwater-wave
studies.A significantadvancein waterwavesrequiresthecombinationofabstract
ideas and techniques with an understanding of the physical reality. While our
approachreliesonrigorousmathematicalconsiderations,throughouttheselectures
we will try to support our theoretical claims with field data. There are also a few
aspects contingent to our considerations where a mathematical proof appears to
remain elusive, in which case we will present some numerical evidence pointing
towardsalikelyconclusion.
2 Preliminaries
Natural phenomena are more complex than any model that anyone can make, so
that one must necessarily accept less than exhaustive descriptions. An efficient
mathematicalmodelofanaturalphenomenonusessmallamountsofinformationto
produceexperimentallyvalidatedconclusions.To captureall factorsis impossible
andnotevendesirablesinceonecanalwaysexpandtheproblemtothepointwhere
it could not be answered (at the currentstate-of-the-art).Even if the mathematics
involvedin the study is highlysophisticated, one shouldalways be aware that the
modelis a simplifiedidealisationofthe realworldphenomenon.Theadequacyof
the model depends on how well it represents the key factors and on how reliable
its predictions are (in the physical regime in which it is legitimate to apply it).
The degree of allowed complexity should capture the essential physical factors
and permit the pursuit of an in-depth analysis that leads to qualitative as well as
quantitativepredictions.
2.1 PeriodicTravelling Waves
We willstudythemostregularwaterwavepatterns:periodictravellingwavesthat
propagateonthesurfaceofwater in a givendirectionandatconstantspeed.Such
waves,termedswellinoceanography,arenotgeneratedbythelocalwindbutbya
