A KAM Theorem for Hamiltonian Partial Differential

DigitalObjectIdentifier(DOI)10.1007/s00220-005-1497-0Commun.Math.Phys.262,343–372(2006)CommunicationsinMathematicalPhysicsAKAMTheoremforHamiltonianPartialDifferentialEquationsinHigherDimensionalSpaces
JianshengGeng,JiangongYouDepartmentofMathematics,NanjingUniversity,Nanjing210093,P.R.China.E-mail:[e-mail]:12November2004/Accepted:5September2005Publishedonline:20December2005–©Springer-Verlag2005Abstract:Inthispaper,wegiveaKAMtheoremforaclassofinfinitedimensionalnearlyintegrableHamiltoniansystems.ThetheoremcanbeappliedtosomeHamilto-nianpartialdifferentialequationsinhigherdimensionalspaceswithperiodicboundaryconditionstoconstructlinearlystablequasi–periodicsolutionsanditslocalBirkhoffnormalform.TheapplicationstothehigherdimensionalbeamequationsandthehigherdimensionalSchr¨odingerequationswithnonlocalsmoothnonlinearityarealsogiveninthispaper.1.IntroductionInlate1980’s,motivatedbytheconstructionofquasi-periodicsolutionsforHamiltonianpartialdifferentialequations,thecelebratedKAMtheorywassuccessfullygeneralizedtoinfinitedimensionalsettingsbyKuksin[14]andWayne[20],seealso[15–18],whichappliesto,astypicalexamples,one-dimensionalsemi-linearSchr¨odingerequationsiut−uxx+mu=f(u),andwaveequationsutt−uxx+mu=f(u),withDirichletboundaryconditions.WhentryingtofurthergeneralizetheKAMtheorysoastoapplytotheone-dimensionalwaveequationswithperiodicboundaryconditionsandhigherdimensionalHamiltonianpartialdifferentialequations,themultiplicityoftheeigenvaluesbecomesanobstacle.Especially,themultiplicitygoesasymptoticallytoinfinityinthehigherdimensionalcase.Ononehand,themultiplicitymakestheunper-turbedpartmorecomplicatedatsucceedingKAMsteps,asaconsequencesolvingthe
TheworkwassupportedbytheNationalNaturalScienceFoundationofChina(10531050)344J.Geng,J.Youlinearizedequationsbecomesverycomplicated;ontheotherhand,itmakesthemeasureestimationverydifficultsincetherearesomanynon-resonanceconditionstobesatis-fied.Forthosereasons,thereisnoKAMtheoremforhigherdimensionalHamiltonianpartialdifferentialequationssofar.Toovercomethisdifficulty,CraigandWayneretrievedtheoriginationoftheKAMmethod—NewtonianiterationmethodtogetherwiththeLiapunov-Schmidtdecom-positionwhichinvolvestheGreen’sfunctionanalysisandthecontroloftheinverseofinfinitematriceswithsmalleigenvalues.Theysucceededinconstructingperiodicsolutionsoftheone-dimensionalsemi-linearwaveequationswithperiodicboundaryconditions.BourgainfurtherdevelopedtheCraig–Wayne’smethodandprovedtheexis-tenceofquasi-periodicsolutionsofpartialdifferentialequationsinhigherdimensionalspaceswithDirichletboundaryconditionsorperiodicboundaryconditions.WepointoutthattheCraig-Wayne-Bourgain’smethodallowsonetoavoidexplicitlyusingtheHamiltonianstructureofthesystems.Wewillnotintroducetheirapproachesindetail.ThereaderisreferredtoCraig–Wayne[9],Bourgain[3–7].ComparingwithCraig-Wayne-Bourgain’sapproach,theKAMapproachhasitsownadvantages.Besidesobtainingtheexistenceresultsitallowsonetoconstructalocalnormalforminaneighborhoodoftheobtainedsolutions,andthisisusefulforbetterunderstandingofthedynamics.Forexample,onecanobtainthelinearstabilityandzeroLiapunovexponents.Thequestionis:IsthereaKAMtheoremwhichcanbeappliedtoHamiltonianpartialdifferentialequationsinhigherdimensionalspaces?Thispaperismotivatedbythisquestion.Inthispaper,wegiveaKAMtheoremwhichappliestosomeHamiltonianpartialdifferentialequationsinhigherdimensionalspaces.Weusethetheoremtoconstructthequasi-periodicsolutionsandprovetheirlinearstability.TheKAMtheoremcanbeappliedtosomeHamiltonianpartialdifferentialequationsnotexplicitlycontainingthespacevariablesandtimevariable,includingthehigherdimensionalbeamequationsutt+(−
+m)2u+f(u)=0,x∈TdandthehigherdimensionalSchr¨odingerequationswithnonlocalsmoothnonlinearities(seeSect.3fordetails)iut+Au+N(u)=0,x∈Td,aswellasone-dimensionalwaveequationsundertheperiodicboundaryconditions.Differentfromthefinitedimensionalcase,theKAMtheoremmaynotbetrueforinfinitedimensionalnearlyintegrableHamiltoniansystems.Onehastoimposefurtherrestrictionsbothontheunperturbedpartandontheperturbationbesidessmallness.IntheexistentinfinitedimensionalKAMtheorems,e.g.,Kuksin[14],Wayne[20]andP¨oschel[18],someassumptionsonthegrowthofthenormalfrequenciesandtheregu-larityoftheperturbationarerequired(see(A1)–(A3)inthenextsection).Inthispaper,weadditionallyassumethattheperturbationhasaspecialformdefinedin(A4)inthenextsection.Ourproofbenefitsalotfromsuchspecialityoftheperturbation.Withthespecialityoftheformoftheperturbation,wecanprovethatthenormalformpartoftheHamiltonianremainssimpleduringtheiteration.Actually,thenormalvariablesinthenormalformpartarealwaysuncoupledalongtheKAMiteration.Thismakesthemeasureestimateaseasyastheone-dimensionalcase.ComparedwiththeproofoftheexistentKAMtheorems,anadditionaljobdoneinthispaperistoprovethatperturbationalwayshasthespecialformdefinedin(A4)alongtheKAMiteration.KAMTheoremforHamiltonianPDEsinHigherDimensionalSpaces345Weremarkthatalthoughtheassumption(A4)looksartificial,theHamiltoniansystemsderivingfromtheHamiltonianpartialdifferentialequationsinTdno