Studies in Phase Space Analysis with Applications to PDEs

390 Pages·2013·3.195 MB·English

by  Massimo Cicognani, Ferruccio Colombini, Daniele Del Santo

Progress in Nonlinear Differential Equations
and Their Applications
84
Massimo Cicognani
Ferruccio Colombini
Daniele Del Santo
Editors
Studies in Phase
Space Analysis with
Applications to PDEs

Progress in Nonlinear Differential Equations
and Their Applications
Volume84
Editor
HaimBrezis
UniversitéPierreetMarieCurie
Paris,France
and
RutgersUniversity
NewBrunswick,NJ,USA
EditorialBoard
AntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,
Trieste,Italy
A.Bahri,RutgersUniversity,NewBrunswick,NJ,USA
FelixBrowder,RutgersUniversity,NewBrunswick,NJ,USA
LuisCaffarelli,TheUniversityofTexas,Austin,TX,USA
LawrenceC.Evans,UniversityofCalifornia,Berkeley,CA,USA
MarianoGiaquinta,UniversityofPisa,Pisa,Italy
DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh,PA,USA
SergiuKlainerman,PrincetonUniversity,NJ,USA
RobertKohn,NewYorkUniversity,NY,USA
P.L.Lions,UniversityofParisIX,Paris,France
JeanMawhin,UniversitéCatholiquedeLouvain,Louvain-la-Neuve,Belgium
LouisNirenberg,NewYorkUniversity,NY,USA
LambertusPeletier,UniversityofLeiden,Leiden,Netherlands
PaulRabinowitz,UniversityofWisconsin,Madison,WI,USA
JohnToland,UniversityofBath,Bath,England
Forfurthervolumes:
http://www.springer.com/series/4889
Massimo Cicognani • Ferruccio Colombini
Daniele Del Santo
Editors
Studies in Phase Space
Analysis with Applications
to PDEs
Editors
MassimoCicognani FerruccioColombini
DipartimentodiMatematica DipartimentodiMatematica
UniversitàdiBologna UniversitàdiPisa
Bologna,Italy Pisa,Italy
DanieleDelSanto
DipartimentodiMatematicae
Geoscienze
UniversitàdiTrieste
Trieste,Italy
ISBN978-1-4614-6347-4 ISBN978-1-4614-6348-1(ebook)
DOI10.1007/978-1-4614-6348-1
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013931265
Mathematics Subject Classification (2010): 26D10, 35A23, 35A30, 35B30, 35B40, 35B45, 35B60,
35B65,35J10,35J75,35K05,35K10,35L15,35L50,35L99,35P25,35Q30,35Q40,35Q41,35Q53,
35Q55,35R45,35S05,35S30,42C15,46E35,47A40,47A60,47G30,76D03,81Q05,81U40
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Maisoùsontles neigesd’antan?
F. Villon

Preface
This volume is a collection of papersmainly concerningphase space analysis, or
microlocalanalysis, and its applicationsto the theoryof partialdifferentialequa-
tions(PDEs).
Anumberofpaperscomposingthisvolume,allwrittenbyleadingexpertsintheir
respectivefields,areexpandedversionoftalksgivenatameetingheldinSeptember
2011 at the Centro Residenziale Universitario (CEUB) of Bertinoro, on the hills
surroundingCesena,Italy.
The Bertinoro workshop was the occasion to fix the state of the art in many
different aspects of phase space analysis. In fact the results collected here con-
cerngeneraltheoryofpseudodifferentialoperators,Hardy-typeinequalities,linear
and non-linear hyperbolic equations and systems, water-waves equations, Euler–
PoissonandNavier–Stokesequations,Schrödingerequationsandheatandparabolic
equations.
We wouldliketoseize thisoccasiontothankallthecontributorsaswellasthe
peoplewhotookpartintheworkshop.
A numberof institutionshavemadeitpossibleto holdthe Bertinoroworkshop
through their financial support. We’d like to list them here: the Italian Ministero
dell’Istruzione,dell’UniversitàedellaRicercabymeansofthePRIN2008project
“Phase Space Analysis of PDE’s”, the Istituto Nazionale di Alta Matematica, the
University of Bordeaux 1, the University of Pisa and the Fondazione Cassa di
RisparmiodiCesena.Wethankallofthemfortheirgenerosity.
Bologna,Italy MassimoCicognani
Pisa,Italy FerruccioColombini
Trieste,Italy DanieleDelSanto
vii

Contents
1 TheWater-WaveEquations:FromZakharovtoEuler............. 1
ThomasAlazard,NicolasBurq,andClaudeZuily
1.1 Introduction.............................................. 1
1.2 LowRegularityCauchyTheory ............................. 4
1.3 FromZakharovtoEuler.................................... 6
1.3.1 TheVariationalTheory............................. 6
1.3.2 TheMainResult .................................. 7
1.3.3 StraighteningtheFreeBoundary..................... 8
1.3.4 TheDirichlet-NeumannOperator .................... 9
1.3.5 Preliminaries ..................................... 10
1.3.6 TheRegularityResults ............................. 12
1.3.7 ProofoftheLemmas............................... 13
References..................................................... 20
2 On the Characterization of Pseudodifferential Operators
(OldandNew) .............................................. 21
Jean-MichelBony
2.1 Introduction.............................................. 21
2.2 Weyl–HörmanderCalculus ................................. 22
2.2.1 AdmissibleMetrics................................ 23
2.2.2 SymbolicCalculus ................................ 24
2.2.3 Can One Use Only Metrics WithoutSymplectic
Eccentricity? ..................................... 25
2.3 CharacterizationofPseudodifferentialOperators ............... 26
2.3.1 GeodesicTemperance.............................. 26
2.3.2 Characterization................................... 27
2.3.3 FunctionalCalculus ............................... 29
2.4 SufficientConditionsfortheGeodesicTemperance............. 30
2.4.1 ProofofTheorem2.5(i)............................ 31
2.4.2 ProofofTheorem2.5(ii) ........................... 33
References..................................................... 34
ix