Nonlinear Water Waves: Cetraro, Italy 2013 PDF
Preview Nonlinear Water Waves: Cetraro, Italy 2013
Lecture 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 CatharinaStroppel,Bonn 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 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland 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
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