Table Of ContentLecture 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
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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
