An averaging theorem for quasilinear Hamiltonian P

AVERAGINGTHEOREMFORQUASILINEARHAMILTONIANPDEsINARBITRARYSPACEDIMENSIONSDarioBAMBUSIDipartimentodiMatematica,Universit`adeglistudidiMilanoViaSaldini50,20133MILANO,Italy.Abstract.WestudythedynamicsofquasilinearHamiltonianwaveequationswithDirich-letboundaryconditionsinann–dimensionalparallepided.Weproveanaveragingtheoremaccordingtowhichthesolutioncorrespondingtoanarbitrarysmallamplitudesmoothini-tialdatumremainsarbitrarilyclosetoafinitedimensionaltorusuptoverylongtimes.WeexpecttheresulttobevalidforaverygeneralclassofquasilinearHamiltonianequations.1.IntroductionInthispaperwestudythedynamicsofHamiltonianpartialdifferentialequationofquasilineartype(inthesenseofKato[1,2])inarbitraryspacedimension.Tobedefinitewewillconcentrateonanonlinearwaveequationinann–dimensionalparallelepipedRwithDirichletboundaryconditions(seeeq.(2.1)below):underanonresonanceconditionthatwillbeprovedtohaveprobability1wewillconstructinfinitelymanyapproximateintegralsofmotion,anddeducethatsolutionscorrespondingtosmallamplitudesmoothinitialdataremainO(ǫM)closetoafinitedimensionaltorusuptotimesO(ǫ−2),ǫbeingthenormoftheinitialdatumandManarbitraryinteger.TheproofisobtainedbygeneralizingtoPDEsaclassicalalgorithmofconstructionofintegralsofmotion[3,4,5,6],whichiscommonlyusedincelestialmechanics(wewillrecallitinsect.3).TogeneralizesuchanalgorithmtoPDEsweneedanonresonanceconditionforthefrequenciesofthelinearizedsystem.Herewewilluseanewdiophantinetypecondition(seeeq.(2.13)below)thatwillbeprovedtobefulfilledforalmostallvaluesoftheparametersofthesystem.Suchconditionallowstotreatsmalldenominatorsas“derivatives”(seelemma4.5below),inawaysimilartothatusedbyShatahin[7]andbyCraigin[8].Oncethisisdone,analmoststraightforwardapplicationoftheclassicalalgorithmallowstoconstructthecandidateapproximateintegralsofmotion,namelyfunctionswhoseLiederivativeissmallinaneighbourhoodoftheoriginofasufficientlysmoothSobolevspace.Thenwehavetoprovethat,uptothetimesweareinterestedin,suchfunctionsremainapproximately21.10.20022D.Bambusiconstant.Thisisobtainedbyshowingthatsolutionwithsufficientlysmallinitialdatumatmostdoubletheirsizeuptosuchtimes,andthereforetheydonotleavetheconsideredneighbourhoodoftheorigin.InturnthisisprovedbyusingKato’stheorywiththeadditionofexplicitestimatesontheexistencetimesandonthesizeofthesolution.Weexpectthetechniquesjustdescribedtobeapplicabletoaverygeneralclassofequations.WerecallthatwhilemanyresultshavebeenobtainedonthedynamicsofsemilinearHamiltonianPDEsinonespace[9,10,11,12,13,14,15,16,17],verylittleisknownforequationsinhigherspacedimensionsandofgeneralquasilineartype.Inparticular,inhigherdimensionalequationsonlypersistenceofperiodicorbits[18](seealso[19])hasbeenprovedinquitegeneralmodels,whilepersistenceoffinitedimensionalinvarianttoriisknownonlyinsomespecialcases[20,21].Forwhatconcernsothersolutions,averagingmethodshavebeenusedtostudythedynamicscorrespondingtoinitialdatainwhichtheenergyisinitiallyessentiallyconcentratedonfinitelymanylinearmodes[22].Howeverthemethodsof[22]givenoinformationsforgeneralinitialdataandforquasilinearequations.Wepointoutthatinthelattercase(wheretheperturbationinvolvesderivatives)onecannotuseGronwalllemmatoestimatethedifferencebetweenthetrueandtheapproximatesolution,andthisisthemaindifficultyrelatedtothiskindofequations.WepointoutthatalsoKAMtheoryallowstodealwithsomequasilinearequations[23,24],but,duetowellknownresultsontheformationofsingularitiesinquasilinearequations[25],onecannotexpectKAMtheorytobeapplicabletogeneralquasilinearmodels.Finallywerecallthepapers[26,27,28]whereaveragingtechniqueshavebeenusedto-getherwithaprioriestimatesinordertoconstructapproximatesolutionsofsomesemilinearPDEsbyamethodcloselyrelatedtothepresentone.Planofthepaper.Insect.2westatethemainresultsofthepaper.Insect.3werecalltheformalalgorithmofconstructionoftheintegralsofmotioninHamiltoniansystems.Insect.4weproofthemainresultofthepaper.Insect.5weshowthatthediophantineconditionweuseisfulfilledwithprobabilityone.FinallyintheappendixweuseKato’stheorytoestimatethesizeofthesolutionofthenonlinearwaveequation.2.StatementofthemainresultHavingfixedavectora=(a1,...,an)∈IRnfulfillingai0,considerthendimensionalparallelepipedRwithsidesoflengthπ/√ai,namelyR:=x≡(x1,...,xn)∈IRn:0xiπ√aiandthenonlinearwaveequationutt−Δu+mu−bij(u,∇u)∂i∂ju+g(u,∇u)=0,x∈Ru∂R=0,(2.1)21.10.2002AveragingtheoryforHamiltonianPDE’s.Preliminaryversion3whereweusedthesummationconventionfortheindexesi,j=1,...,n,wedenotedby∇u≡(∂1u,...,∂nu)thederivativesofuwithrespecttothespacevariables,andbij,garefunctionsofclassC∞inaneighbourhoodoftheorigin.MoreoverweassumethatthesystemisHamiltonian,i.e.thatthereexistsaC∞functionW=W(u,s1,...,sn)suchthatbij(u,s1,...,sn)=∂2W∂si∂sj(u,s1,...,sn),g=∂W∂u−∂2W∂u∂sk∂ku,sothattheHamiltonianfunctionofthesystemisgivenbyH(u,v)=ZRv22+k∇uk22+mu2+W(u,∇u)#dnx,(2.2)wherev=˙uisthemomentumconjugatedtou,andwedenotedbyk.ktheeuclideannormofavectorofIRn.FinallyweassumethatWhasazerooforderp+2attheorigin,namelythat|W(u,s)|≤C(|u|+ksk)p+2,p≥1,andthatthepotentialWisevenineachofthearguments,namelyW((−)c0u,(−