Nonlinear Oscillations of Hamiltonian PDEs PDF
Preview Nonlinear Oscillations of Hamiltonian PDEs
Progress in Nonlinear Differential Equations and Their Applications Volume74 Editor HaimBrezis Universite´ PierreetMarieCurie Paris and RutgersUniversity NewBrunswick,N.J. EditorialBoard AntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,Trieste A.Bahri,RutgersUniversity,NewBrunswick FelixBrowder,RutgersUniversity,NewBrunswick LuisCaffarelli,TheUniversityofTexas,Austin LawrenceC.Evans,UniversityofCalifornia,Berkeley MarianoGiaquinta,UniversityofPisa DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh SergiuKlainerman,PrincetonUniversity RobertKohn,NewYorkUniversity P.L.Lions,UniversityofParisIX JeanMawhin,Universite´ CatholiquedeLouvain LouisNirenberg,NewYorkUniversity LambertusPeletier,UniversityofLeiden PaulRabinowitz,UniversityofWisconsin,Madison JohnToland,UniversityofBath Massimiliano Berti Nonlinear Oscillations of Hamiltonian PDEs Birkha¨user Boston • Basel • Berlin MassimilianoBerti DipartimentodiMatematicaeApplicazioni“R.Caccioppoli” Universita`FedericoII,Napoli viaCintia,MonteS.Angelo I-80126Naples,Italy [email protected] MathematicsSubjectClassification(2000):35L05,37K50,58E05 LibraryofCongressControlNumber:2007932361 ISBN-13:978-0-8176-4680-6 e-ISBN-13:978-0-8176-4681-3 Printedonacid-freepaper. (cid:1)c2007Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com (BT/SB) Preface Manypartialdifferentialequations(PDEs)arisinginphysicscanbeseenasinfinite- dimensionalHamiltoniansystems ∂ u J( H)(u,t), u , (0.1) t u = ∇ ∈H wheretheHamiltonianfunction . H. R R H× → isdefinedonaninfinite-dimensionalHilbertspace and J isanondegenerateanti- H symmetricoperator. ThemainexamplesofHamiltonianPDEsare: (a) thenonlinearwaveequation u (cid:19)u f(x,u) 0, tt − + = (b) thenonlinearSchro¨dingerequation iu (cid:19)u f(x, u 2)u 0, t − + | | = (c) thebeamequation u u f(x,u) 0, tt xxxx + + = andthehigher-dimensionalmembraneequation u (cid:19)2u f(x,u) 0, tt + + = (d) theEulerequationsofhydrodynamicsdescribingtheevolutionofaperfect,in- compressible,irrotationalfluidundertheactionofgravityandsurfacetension, aswellasitsapproximatemodelsliketheKorteweg–deVries(KdV)equation, theBoussinesqequation,andtheKadomtsev–Petviashviliequation. ForexamplethecelebratedKdVequation u uu u 0 t x xxx − + = wasintroducedtodescribethemotionofthefreesurfaceofafluidinthesmall- amplitude,long-waveregime. vi Preface WestartbydescribingtheHamiltonianstructureofthe1-dimensionalnonlinear waveequationwithDirichletboundaryconditions u u f(t,x,u) 0, tt xx (NLW) − + = (cid:11)u(t,0) u(t,π) 0, = = orperiodicboundaryconditionsu(t,x 2π) u(t,x).Inthisbookweshallcon- + = siderboththeautonomous nonlinear wave(f independentoftime)andthenonau- tonomousnonlinearwave(forcedcase)equations. Equation(NLW)canbeseenasasecond-orderLagrangianPDE u ( W)(u,t) (0.2) tt =− ∇L2 definedontheinfinite-dimensional“configurationspace” H1(0,π),where 0 W(,t)..H1(0,π) R · 0 → isthe“potentialenergy” π u2 W(u,t).. x F(t,x,u) dx = 2 + 0 (cid:61) (cid:114) (cid:115) . and(∂ F)(t,x,u) . f(t,x,u).Indeed,differentiating W andintegratingbyparts u = yields π d W(u,t)[h] u h f(t,x,u)h dx u x x = + 0 (cid:61) π(cid:114) (cid:115) u f(t,x,u) hdx xx = − + 0 (cid:61) (cid:114) (cid:115) bytheDirichletboundaryconditions,sothatthe L2-gradientofW withrespecttou is ( W)(u,t) u f(t,x,u), L2 xx ∇ =− + and(NLW)canbewrittenas(0.2). TheLagrangianfunctionof(NLW)is,asusual,thedifferencebetweenthekinetic andthepotentialenergy: π u2 π u2 u2 (u,u ,t).. t dx W(u,t) t x F(t,x,u) dx. t L = 2 − = 2 − 2 − 0 0 (cid:61) (cid:61) (cid:114) (cid:115) Thenonlinearwaveequation(0.2)canbewrittenalsoinHamiltonianform u p, t = (cid:11)pt ( L2W)(u,t) uxx f(t,x,u), =− ∇ = − andsointheform(0.1)withHamiltonianfunction H..H1 L2 R R, 0 × × → Preface vii π p2 π p2 u2 H(u,p,t).. dx W(u,t) x F(t,x,u) dx, = 2 + = 2 + 2 + 0 0 (cid:61) (cid:61) (cid:114) (cid:115) definedontheinfinite-dimensional“phasespace” .. H1 L2,andtheoperator H = 0 × J 0 I represents a symplectic structure: the variables (u, p) are “Darboux = I 0 coordin−ates.” (cid:98) (cid:99) Similarlywecandealwiththenonlinearwaveequationunderperiodicboundary conditions,choosingasconfigurationspace H1(T),whereT.. R/2πZ. = TheHamiltonianstructureofthe nonlinear Schro¨dinger equation(b), the beam and membrane equation (c), the KdV and the Euler equations (d) are presented in theappendix. In this book we shall adopt the point of view of regarding the nonlinear wave equation(0.2)asaninfinite-dimensionalHamiltoniansystem,focusingonthesearch forinvariantsetsfortheflow. The analysis of the principal structures of an infinite-dimensional phase space suchasequilibria,periodicorbits,embeddedinvarianttori,centermanifolds,aswell as stable and unstable manifolds, is an essential change of paradigm in the study of hyperbolic equations with respect to the traditional pursuit of the initial value problem(mainlystudiedfordispersiveequations). Sucha“dynamicalsystemsphilosophy”hasledtomanynewresults,knownfor finite-dimensionalHamiltoniansystems,forHamiltonianPDEs. Inthestudyofacomplexdynamicalsystem,thesimplestinvariantmanifoldsto lookforaretheequilibriaand,next,theperiodicorbits. The relevance of periodic solutions for understanding the dynamics of a finite- dimensionalHamiltoniansystemwasfirsthighlightedbyPoincare´. In 1899 in “Les me´thodes nouvelles de la Me´canique Ce´leste” Poincare´ wrote (aproposperiodicsolutionsinthethree-bodyproblem) D’ailleurs,cequinousrendcessolutionspe´riodiquessipre´cieuses,c’estqu’elles sont,pourainsidire,laseulebre`cheparou` nouspuissonsessayerdepe´ne´trer dansuneplacejusqu’icire´pute´einabordable. Atfirstglance,knowledgeofthesedoesnotseemreallyinterestingbecausethe periodicorbitsformazero-measuresetinthephasespace: Eneffet,ilyauneprobabilite´nullepourquelesconditionsinitialesdumouvement soientpre´cise´mentcellesquicorrespondenta` unesolutionpe´riodique. But,asconjecturedbyPoincare´,theirknowledgeisnotirrelevant: voici un fait que je n’ai pu de´montrer rigoureusement, mais qui me paraˆıt pour- tanttre`svraisemblable.E´tantdonne´esdese´quationsdelaformede´finiedans len.13[Hamilton’sequations]etunesolutionparticulie`requelconquedeces e´quations, on peut toujours trouver une solution pe´riodique (dont la pe´riode peut, il est vrai, eˆtre tre`s longue), telle que la diffe´rence entre les deux solu- tionssoitaussipetitequ’onleveut,pendantuntempsaussilongqu’onleveut. ([108],Tome1,ch.III,a.36). viii Preface Thisconjecturewasthemainmotivationforthesystematicstudyofperiodicor- bitsthatbeganwithPoincare´andwasthencontinuedbyLyapunov,Birkhoff,Moser, Weinstein,andmanyothers. Thefirstexistenceresults,basedonthePoincare´ continuationmethod,regarded bifurcationofperiodicorbitsfromelliptic(linearlystable)equilibriaundernonreso- nanceconditionsontheeigenfrequenciesofthesmalloscillations(Lyapunovcenter theorem[90]).Theseperiodicsolutionsarethenonlinearcontinuationsofthelinear normalmodeshavingperiodsclosetotheperiodsofthelinearizedequation. Subsequently, more in the direction inspired by the conjecture of Poincare´, Birkhoff,andLewis[32]–[34]provedthatfor“sufficientlynonlinear”Hamiltonian systems,thereexistinfinitelymanyperiodicsolutionswithlargeminimalperiodin anyneighborhoodofanellipticperiodicorbit. Great progress in understanding the very complex orbit structure of a Hamil- tonian system was made by Kolmogorov [80], Arnold [8], and Moser [94] (KAM theory)inthe1960s:accordingtothistheory,atleastforsufficientlysmoothnearly integrablesystems,asetofpositivemeasureofthephasespaceisfilledbyquasiperi- odicsolutions(KAMtori).Themaindifficultyintheirexistenceproofisthepresence ofarbitrarily“smalldenominators”intheirapproximateexpansionseries. From a dynamical point of view, these “small denominators” had already been highlighted by Poincare´ as the main source of chaotic dynamics due to very com- plex resonance-type phenomena. Nowadays, this is the object of study of “Arnold diffusion”[9]. Returning to periodic orbits, in the 1970s, Weinstein [129], Moser [98], and Fadell–Rabinowitz[61]succeeded,withtheaidofvariationalandtopologicalmeth- ods in bifurcation theory, to extend the Lyapunov center theorem without nonreso- nancerestrictionsonthelineareigenfrequencies(resonantcentertheorems). AboutthepreviousconjectureofPoincare´,someprogresswasachievedbyCon- leyandZehnder[46],whoprovedthattheKAMtorilieintheclosureofthesetofthe periodicsolutions(having,itistrue,verylongminimalperiod).Asaconsequence, theclosureof theperiodicorbits haspositiveLebesguemeasure.RecentBirkhoff– Lewis-typeresultsconcerninglower-dimensionalKAMtori—andapplicationstothe threebodyproblem—havebeenobtainedin[23]. The conjecture of Poincare´ was positively answered only in a generic sense by Pugh and Robinson [113] in the early 1980s: for a generic Hamiltonian (in the C2 topology)theperiodicorbitsaredenseonacompactandregularenergysurface.On theotherhand,forspecificsystems,thepossibilityofapproximatinganymotionwith periodicorbitsremainsanopenproblem. Finally, the search for periodic solutions in Hamiltonian systems gave rise to many new ideas and techniques in critical point theory, especially after the break- throughofRabinowitz[117]andWeinstein[130]ontheexistenceofaperiodicso- lution on each compact and strictly convex energy hypersurface; see Ekeland [56], Hofer–Zehnder[72],andreferencestherein. Preface ix Buildingontheexperiencegainedfromthequalitativestudyoffinite-dimensional dynamicalsystems,thesearchforperiodicsolutionswasregardedasafirststepto- wardbetterunderstandingofthecomplicatedflowevolutionofHamiltonianPDEs. Inthisdirection,Rabinowitz[118]andBrezis–Coron–Nirenberg[42]provedthe existenceofperiodicsolutionsfornonlinearwaveequationsviaminimaxvariational methods (see Theorem C.1). Interestingly, these proofs work only to find periodic orbits with a rational frequency, the reason being that other periods give rise to a “smalldenominators”problemtype(RemarkC.4). Independentlyoftheseglobalresults,inthelate1980sandearly1990sthelocal bifurcation theory of periodic and quasiperiodic solutions began to be extended to “nonresonant”HamiltonianPDEssuchas u u a (x)u a (x)u2 a (x)u3 (0.3) tt xx 1 2 3 − + = + +··· byKuksin,Wayne,Craig,Bourgain,andPo¨eschel,followedbymanyothers. Dothereexistperiodicandquasiperiodicsolutionsof(0.3)closetotheinfinite- dimensional“elliptic”equilibriumu 0? = It turns out that in infinite dimension, the searchfor periodic solutions also ex- hibits a “small divisors difficulty,” whereas in finite dimensions it arises only for findingquasiperiodicsolutions. The first existence proofs of small-amplitude periodic and quasiperiodic solu- tions wereobtainedbyKuksin[81]andWayne[128].Theseresults werebasedon KAMtheoryandwereinitiallylimitedtoDirichletboundaryconditions,becauseof thenearcoincidenceofthelineareigenfrequenciesunderperiodicspatialboundary conditions. InanefforttoavoidtheserestrictionsCraig–Wayne[51]introducedtheLyapu- nov–SchmidtreductionmethodforHamiltonianPDEs,extendingtheLyapunovcen- tertheoreminthepresenceofperiodicboundaryconditions.Later,thismethodwas generalized by Bourgain [35], [36] to construct quasiperiodic solutions. For subse- quent developments of the KAM approach, see [109], [45], [84], [82], [59], [133], andreferencestherein. All the previous results concerned “nonresonant” PDEs, such as the nonlinear waveequation(0.3)witha (x) 0.Theterma (x)allowsonetoverify,asinthe 1 1 (cid:47)≡ Lyapunov center theorem, suitablenonresonanceconditions on the linear eigenfre- quenciesofthesmalloscillations,andthebifurcationequationisfinite-dimensional.1 Theaimofthisbookistopresentrecentbifurcationresultsof“nonlinearoscil- lationsofHamiltonianPDEs,”especiallyfor“completelyresonant”nonlinearwave equations(0.3)with a (x) 0 1 ≡ in which infinite-dimensional bifurcation phenomena appear jointly with small- divisor difficulties. We shall deal with both autonomous (time-independent) and 1FornonresonantPDEs,anextensionoftheBirkhoff–Lewisperiodicorbittheoryhasalso beenobtained;see[14],[31].
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