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1、LocalandglobalanalysisofnonlineardispersiveandwaveequationsTerenceTaoDepartmentofMathematics,UCLA,LosAngeles,CA90095E-mailaddress:tao@math.ucla.edu1991MathematicsSubjectClassi?cation.Primary35Q53,35Q55,35L15TheauthorispartlysupportedbyagrantfromthePackar
2、dfoundation.ToLaura,forbeingsopatient.ContentsPrefaceixChapter1.Ordinarydi?erentialequations11.1.Generaltheory11.2.Gronwall’sinequality71.3.Bootstrapandcontinuityarguments141.4.Noether’stheorem181.5.Monotonicityformulae251.6.Linearandsemilinearequations2
3、81.7.Completelyintegrablesystems35Chapter2.Constantcoe?cientlineardispersiveequations412.1.TheFouriertransform472.2.Fundamentalsolution522.3.DispersionandStrichartzestimates562.4.ConservationlawsfortheSchr¨odingerequation612.5.Thewaveequationstress-energ
4、ytensor672.6.Xs,bspaces75Chapter3.Semilineardispersiveequations853.1.Onscalingandothersymmetries883.2.Whatisasolution?913.3.Localexistencetheory963.4.Ill-posednessresults1043.5.Conservationlawsandglobalexistence1043.6.Morawetzestimates1043.7.Scatteringth
5、eory1053.8.Perturbationtheory1053.9.Almostconservationlaws105Chapter4.KortewegdeVriestypeequations1054.1.Solitons1054.2.Localtheory1054.3.Correctionterms1054.4.Symplecticnon-squeezing1054.5.TheBenjamin-Onoequationandgaugetransformations106Chapter5.Energy
6、-criticalsemilineardispersiveequations1075.1.Theenergy-criticalNLW1075.2.Frequencyenvelopes1075.3.Bubblesofenergyconcentration107viiviiiCONTENTS5.4.Minimal-energyblowupsolutions1075.5.Localnon-concentrationofmass1075.6.Globalnon-concentrationofmass108Cha
7、pter6.Wavemaps1096.1.Localtheory1096.2.Gaugetransformations1096.3.Morawetzestimate1096.4.Heat?ow109Chapter7.Appendix:toolsfromharmonicanalysis111Chapter8.Appendix:constructionofgroundstates119Bibliography133PrefacePoliticsisforthepresent,butanequationiss
8、omethingforeternity.(AlbertEinstein)ThismonographisbasedonalectureseriesgivenattheNSF-CBMSregionalconferenceonnonlinearanddispersivewaveequationsatNewMexicoStateUni-versity,heldinJune2005.Theaimofthismonographistopresentso
