Mathematics of Wave Phenomena PDF
Preview Mathematics of Wave Phenomena
Trends in Mathematics Willy Dörfler · Marlis Hochbruck Dirk Hundertmark Wolfgang Reichel · Andreas Rieder Roland Schnaubelt Birgit Schörkhuber · Editors Mathematics of Wave Phenomena Trends in Mathematics TrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarisingfrom conferences and lecture series focusing on a particular topic from any area of mathematics.Itsaimistomakecurrentdevelopmentsavailabletothecommunityas rapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreference. ProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmission Formatourwebsitewww.birkhauser-science.com. Materialsubmittedforpublicationmustbescreenedandpreparedasfollows: Allcontributionsshouldundergoareviewingprocesssimilartothatcarriedoutby journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, shouldberejected.Highqualitysurveypapers,however,arewelcome. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/4961 ¨ Willy Dorfler • Marlis Hochbruck • Dirk Hundertmark • Wolfgang Reichel • Andreas Rieder • Roland Schnaubelt • ¨ Birgit Schorkhuber Editors Mathematics of Wave Phenomena Editors WillyDo¨rfler MarlisHochbruck KarlsruheInstituteofTechnology KarlsruheInstituteofTechnology Baden-Wu¨rttemberg Baden-Wu¨rttemberg Karlsruhe,Germany Karlsruhe,Germany DirkHundertmark WolfgangReichel KarlsruheInstituteofTechnology KarlsruheInstituteofTechnology Baden-Wu¨rttemberg Baden-Wu¨rttemberg Karlsruhe,Germany Karlsruhe,Germany AndreasRieder RolandSchnaubelt KarlsruheInstituteofTechnology KarlsruheInstituteofTechnology Baden-Wu¨rttemberg Baden-Wu¨rttemberg Karlsruhe,Germany Karlsruhe,Germany BirgitScho¨rkhuber KarlsruheInstituteofTechnology Baden-Wu¨rttemberg Karlsruhe,Germany ISSN2297-0215 ISSN2297-024X (electronic) TrendsinMathematics ISBN978-3-030-47173-6 ISBN978-3-030-47174-3 (eBook) https://doi.org/10.1007/978-3-030-47174-3 MathematicsSubjectClassification:35-06,35Lxx,35Qxx,65-06,65Mxx,65Nxx,78-06 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The mathematical modeling, simulation,and analysis ofwave phenomena entaila plethora of fascinating and challenging problems both in analysis and numerical mathematics.Duringthepastdecades,thesechallengeshaveinspiredanumberof importantapproaches,developments,andresultsaboutwave-typeequationsinboth fieldsofmathematics. WeorganizedtheConferenceonMathematicsofWavePhenomenaatKarlsruhe, July23–27,2018,tobringtogetherinternationalexpertsinthisfieldwithdifferent backgrounds.Theconferencewasagreatsuccessattractingabout250participants, among them many leading scientists and promising young researchers working in analysis and numerics of wave-type problems and their applications. In their lectures,theypresentedrecenthigh-levelresearchinawiderangeoftopicswhich stimulatedthetransferofideas,results,andtechniqueswithinthisexcitingarea. This volume reflects the contents of the conference, although many of the contributions contain material that goes beyond the results presented in talks. The papers treat various types of nonlinear Schrödinger and wave-type systems, as well as water-wave problems, Helmholtz equations, and hyperbolic systems. Among the main subjects covered by the contributions are well-posedness and stability,constructionofsolitonsolutions,dispersiveestimates,invariantmeasures, inverse scattering, error analysis of space and time discretizations, and efficient implementationofnumericalschemes. v vi Preface WegratefullyacknowledgethefinancialsupportbytheDeutscheForschungsge- meinschaft(DFG,GermanResearchFoundation)throughthecollaborativeresearch center“WavePhenomena:AnalysisandNumerics”(Project-ID258734477–SFB 1173)whichenabledustoorganizethissuccessfulconference. Karlsruhe,Germany WillyDörfler March2020 MarlisHochbruck DirkHundertmark WolfgangReichel AndreasRieder RolandSchnaubelt BirgitSchörkhuber Contents MorawetzInequalitiesforWaterWaves ...................................... 1 ThomasAlazard,MihaelaIfrim,andDanielTataru NumericalStudyofGalerkin–CollocationApproximationinTime fortheWaveEquation........................................................... 15 MathiasAnselmannandMarkusBause EffectiveNumericalSimulationoftheKlein–Gordon–Zakharov SystemintheZakharovLimit.................................................. 37 SimonBaumstark,GuidoSchneider,andKatharinaSchratz ExponentialDichotomiesforEllipticPDEonRadialDomains............. 49 MargaretBeck,GrahamCox,ChristopherJones,YuriLatushkin, andAlimSukhtayev Stability of Slow Blow-Up Solutions for the Critical Focussing NonlinearWaveEquationonR3+1............................................. 69 StefanoBurzio LocalWell-PosednessfortheNonlinearSchrödingerEquationin theIntersectionofModulationSpacesMps,q(Rd)∩M∞,1(Rd).............. 89 LeonidChaichenets,DirkHundertmark,PeerChristianKunstmann, andNikolaosPattakos FEM-BEMCouplingofWave-TypeEquations:FromtheAcoustic totheElasticWaveEquation ................................................... 109 SarahEberle OnHyperbolicInitial-BoundaryValueProblemswithaStrictly DissipativeBoundaryCondition................................................ 125 MatthiasEller vii viii Contents On the Spectral Stability of Standing Waves of Nonlocal PT SymmetricSystems .............................................................. 145 WenFengandMilenaStanislavova SparseRegularizationofInverseProblemsbyOperator-Adapted FrameThresholding............................................................. 163 JürgenFrikelandMarkusHaltmeier SolitonSolutionsfortheLugiato–LefeverEquationbyAnalytical andNumericalContinuationMethods ........................................ 179 JaninaGärtnerandWolfgangReichel ErrorAnalysisofDiscontinuousGalerkinDiscretizationsofaClass ofLinearWave-typeProblems ................................................. 197 MarlisHochbruckandJonasKöhler Ill-posednessoftheThirdOrderNLSwithRamanScatteringTerm inGevreySpaces ................................................................. 219 NobuKishimotoandYoshioTsutsumi InvariantMeasuresfortheDNLSEquation.................................. 235 RenatoLucà AGlobaldiv-curl-LemmaforMixedBoundaryConditionsinWeak LipschitzDomains ............................................................... 243 DirkPauly Existence and Stability of Klein–Gordon Breathers in the Small-AmplitudeLimit.......................................................... 251 DmitryE.Pelinovsky,T.Penati,andS.Paleari OnStrichartzEstimatesfrom(cid:2)2-DecouplingandApplications ............ 279 RobertSchippa OnaLimitingAbsorptionPrincipleforSesquilinearFormswithan ApplicationtotheHelmholtzEquationinaWaveguide..................... 291 BenSchweizerandMaikUrban Some Inverse Scattering Problems for Perturbations of the BiharmonicOperator............................................................ 309 ValerySerov Morawetz Inequalities for Water Waves ThomasAlazard,MihaelaIfrim,andDanielTataru Abstract Morawetz estimates capture the long time local decay properties for various linear and nonlinear dispersive flows. In these notes we provide a brief overview of recent and ongoing work concerning Morawetz estimates for water wavesintwospacedimensions. 1 Introduction Thewaterwaveequationsdescribethemotionofthefreesurfaceofthewaterunder the action of gravity, capillarity and other physical forces. Many of these models admitanatural,conservedenergy.TheclassicalMorawetzestimates,firstpioneered byMorawetz[30]inthecontextofthelinearwaveequation,describethedecayin timeofthelocalenergy,i.e.theenergyrestrictedtoafixedspatialregion.Theaim of recent and ongoing work of the authors has been to explore whether Morawetz estimates,andthuslocalenergydecay,holdforthewaterwaveequation. Precisely, in our work so far we consider gravity water waves in two space dimensions, with finite or infinite depth, and with or without surface tension. AssumingsomeuniformscaleinvariantSobolevboundsforthesolutions,weprove localenergydecay(Morawetz)estimatesgloballyintime.Ourresultsareuniform intheinfinitedepthlimit. The article is structured as follows. In the next section we describe the water waveequationsintheEulerianformulation.Afterthat,inSect.3,weprovideabrief T.Alazard CNRSandCMLA,ÉcoleNormaleSupérieuredeParis-Saclay,Cachan,France e-mail:[email protected] M.Ifrim UniversityofWisconsin,Madison,WI,USA e-mail:[email protected] D.Tataru((cid:2)) UniversityofCalifornia,Berkeley,CA,USA e-mail:[email protected] ©SpringerNatureSwitzerlandAG2020 1 W.Dörfleretal.(eds.),MathematicsofWavePhenomena,TrendsinMathematics, https://doi.org/10.1007/978-3-030-47174-3_1
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