Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle PDF
Preview Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle
Lecture Notes of the Unione Matematica Italiana Massimiliano Berti · Jean-Marc Delort Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle Lecture Notes of 24 the Unione Matematica Italiana Moreinformationaboutthisseriesathttp://www.springer.com/series/7172 Massimiliano Berti • Jean-Marc Delort Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle 123 MassimilianoBerti Jean-MarcDelort DepartmentofMathematics LAGA InternationalSchoolforAdvancedStudies SorbonneParis-Cité/UniversityParis13 SISSA Villetaneuse,France Trieste,Italy ISSN1862-9113 ISSN1862-9121 (electronic) LectureNotesoftheUnioneMatematicaItaliana ISBN978-3-319-99485-7 ISBN978-3-319-99486-4 (eBook) https://doi.org/10.1007/978-3-319-99486-4 LibraryofCongressControlNumber:2018952466 MathematicsSubjectClassification(2010):35A01,76B15,35Q35,35S50 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland TheUnioneMatematicaItaliana(UMI)has establishedabi-annualprize,sponsored bySpringer-Verlag,tohonoranexcellent, originalmonographpresentingthelatest developmentsinanactiveresearcharea ofmathematics,towhichtheauthormade importantcontributionsintherecent years. Theprize-winningmonographsarepublishedinthisseries. Detailsabouttheprizecanbefoundat: http://umi.dm.unibo.it/en/unione-matematica-italiana-prizes/book-prize-unione- matematica-italiana/ This book has been awarded the 2017 Book Prize of the Unione Matematica Italiana. Themembersofthescientificcommitteeofthe2017prizewere: FabrizioCatanese UniversityofBayreuth,Germany CiroCiliberto(PresidenteoftheUMI) UniversitàdegliStudidiRomaTorVergata,Italy VittorioCotiZelati UniversityofNaplesFedericoII,Italy SusannaTerracini UniversitàdegliStudidiTorino,Italy ValentinoTosatti NorthwesternUniversity,Evanston,USA Preface ThegoalofthismonographistoprovethatanysolutionoftheCauchyproblemfor thecapillary-gravitywaterwavesequations,inonespacedimension,withperiodic, even in space, initial data of small size (cid:2), is almost globally defined in time on Sobolev spaces; i.e. it exists on a time interval of length of magnitude (cid:2)−N for any N, as soon as the initial data are smooth enough, and the gravity-capillary parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal form procedure, in order to eliminate thosecontributionstotheSobolevenergythatareoflowerdegreeofhomogeneity inthesolution. Since the water waves equations are a quasi-linear system, usual normal form approaches would face the well-known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillary-gravitywater waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compen- sated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guaranteesakeycancellationwhichpreventsthegrowthoftheSobolevnormsofthe solutions. vii viii Preface Acknowledgements WethankWalterCraigandFabioPusateriforusefulcommentswhichledto animprovementofthemanuscript. Massimiliano Berti was partially supported by the program PRIN 2012 “Variational and perturbativeaspectsofnonlineardifferentialproblems”. Jean-Marc Delort was partially supported by the ANR project 13-BS01-0010-02 “Analyse asymptotiquedeséquationsauxdérivéespartiellesd’évolution”. Trieste,Italy MassimilianoBerti Villetaneuse,France Jean-MarcDelort July2018 Contents 1 Introduction .................................................................. 1 1.1 MainTheorem........................................................... 1 1.2 IntroductiontotheProof................................................ 6 1.3 ParadifferentialFormulationandGoodUnknown..................... 10 1.4 ReductiontoConstantCoefficients .................................... 16 1.5 NormalForms........................................................... 22 2 MainResult................................................................... 27 2.1 ThePeriodicCapillarity-GravityEquations ........................... 27 2.2 StatementoftheMainTheorem........................................ 28 3 ParadifferentialCalculus.................................................... 31 3.1 ClassesofSymbols ..................................................... 31 3.2 QuantizationofSymbols ............................................... 37 3.3 SymbolicCalculus...................................................... 54 3.4 CompositionTheorems................................................. 64 3.5 Paracomposition......................................................... 73 4 ComplexFormulationofthe EquationandDiagonalization oftheMatrixSymbol........................................................ 93 4.1 Reality,ParityandReversibilityProperties............................ 93 4.2 ComplexFormulationoftheCapillary-GravityWaterWaves Equations................................................................ 98 4.3 DiagonalizationoftheSystem.......................................... 100 5 Reduction to a Constant CoefficientsOperatorand Proof oftheMainTheorem ........................................................ 113 5.1 ReductiontoConstantCoefficientsoftheHighestOrderPart........ 113 5.2 ReductiontoConstantCoefficientSymbols........................... 119 5.3 NormalForms........................................................... 132 5.4 ProofofTheorem4.7................................................... 143 ix x Contents 6 TheDirichlet–NeumannParadifferentialProblem ...................... 157 6.1 ParadifferentialandPara-PoissonOperators........................... 157 6.2 ParametrixofDirichlet–NeumannProblem ........................... 189 6.3 SolvingtheDirichlet–NeumannProblem.............................. 199 7 Dirichlet–NeumannOperatorandtheGoodUnknown ................. 217 7.1 TheGoodUnknown .................................................... 217 7.2 ParalinearizationoftheWaterWavesSystem ......................... 235 7.3 TheCapillarity-GravityWaterWavesEquationsinComplex Coordinates.............................................................. 243 8 ProofofSomeAuxiliaryResults............................................ 253 8.1 Non-ResonanceCondition.............................................. 253 8.2 PreciseStructureoftheDirichlet–NeumannOperator................ 257 References......................................................................... 263 Index............................................................................... 267
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