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Journal
of
Mathematical
Analysis
and
Applications
www.elsevier.com/locate/jmaa
On
stabilization
of
small
solutions
in
the
nonlinear
Dirac
equation
with
a
trapping
potential
Scipio Cuccagnaa,∗,Mirko Tarullib
a
DepartmentofMathematicsandGeosciences,UniversityofTrieste,viaValerio12/1,Trieste34127, Italy
b DepartmentofMathematics,UniversityofPisa,LargoBrunoPontecorvo5, Pisa56127, Italy
a r t i c l e i n f o a b s t r a c t
Article history:
Received 12 March 2015
Available online 29 December 2015 Submitted by H.R. Parks
Keywords:
Nonlinear Dirac equation Standing waves
We consider a Dirac operator with short range potential and with eigenvalues. We add a nonlinear term and we show that the small standing waves of the correspondingnonlinearDirac equation (NLD) are attractorsfor small solutions of the NLD. Thisextends to the NLD results alreadyknown for the Nonlinear SchrödingerEquation(NLS).
© 2016ElsevierInc.All rights reserved.
1. Introduction
Weconsider
iut− Hu + g(uu)βu = 0, with (t, x) ∈ R × R3,
u(0, x) = u0(x), (1.1)
where forM > 0 we haveforapotential V (x),
H = DM + V, (1.2)
where DM =−i3j=1αj∂xj +Mβ, withfor j = 1, 2,3,
αj = 0 σj σj 0 , β = IC2 0 0 −IC2 , σ1= 0 1 1 0 , σ2= 0 i −i 0 , σ3= 1 0 0 −1 . * Corresponding author.
E-mailaddresses:scuccagna@units.it(S. Cuccagna), tarulli@mail.dm.unipi.it(M. Tarulli).
http://dx.doi.org/10.1016/j.jmaa.2015.12.049
The unknown u is C4-valued. Given two vectors of C4, uv := u · v is the inner product in C4, v∗ is the
complexconjugate, u · v∗ isthehermitianproduct inC4,whichwewriteas uv∗= u· v∗.Weset u := βu∗, sothat uu = u · βu∗.
WeintroducetheJapanesebracketx:=1 +|x|2 andthespacesdefinedbythefollowing norms: Lp,s(R3,C4) defined withuLp,s(R3,C4):=xsuLp(R3,C4);
Hk(R3,C4) defined withuHk(R3,C4):=xkF(u)L2(R3,C4), whereF is the classical Fourier
transform (see for instance [25]);
Hk,s(R3,C4) defined withuHk,s(R3,C4):=xsuHk(R3,C4);
Σk := L2,k(R3,C4)∩ Hk(R3,C4) withu2Σk=u
2
Hk(R3,C4)+u2L2,k(R3,C4). (1.3)
Forf ,g∈ L2(R3, C4) consider thebilinear map f, g = R3 f (x)g(x)dx = R3 f (x)· g(x)dx. (1.4)
Weassumethefollowing.
(H1) g(0)= 0, g∈ C∞(R,R).
(H2) V ∈ S(R3, S4(C)) with S4(C) thesetofself-adjoint4× 4 matricesandS(R3, E) thespaceofSchwartz functionsfromR3 toE,withthelatteraBanachspaceonC.
(H3) σp(H) = {e1 < e2 < e3· · · < en} ⊂ (−M, M). Here we assume that all the eigenvalues have
multiplicity1.Eachpoint τ =±M isneitheraneigenvaluenoraresonance(thatis,if(DM+V )u= τ u with u ∈ C∞ and|u(x)|≤ C|x|−1 forafixed C, then u = 0).
(H4) Thereisan N ∈ N with N > (M+|e1|)(min{ei− ej : i > j})−1suchthatif μ · e:= μ1e1+· · · + μnen
then
μ∈ Zn with|μ| ≤ 4N + 6 ⇒ |μ · e| =M, (1.5) (μ− ν) · e = 0 and |μ| = |ν| ≤ 2N + 3 ⇒ μ = ν. (1.6) (H5) Considerthe set Mmin definedin (2.5)and for any (μ, ν) ∈ Mmin we consider the function Gμν(x)
(seetheproof ofLemma 5.11or alsolater intheintroductiontheeffective hamiltonian),Gμν(ξ) the
distorted Fouriertransformassociated to H (see (5.64) andAppendix A formoredetails)of Gμν(x)
andthesphere Sμν ={ξ ∈ R3:|ξ|2+M2=|(ν −μ)·e|2}.Thenweassumethatforany(μ,ν) ∈ Mmin
therestrictionofGμν onthesphere Sμν isGμν|Sμν = 0.
Toeach ej weassociateaneigenfunction φj.Wechoosethem suchthatReφj, φ∗k= δjk.Toeach φj we
associatenonlinearboundstates.
Proposition 1.1 (Bound states). Fix j ∈ {1,· · · , n}. Then ∃a0 > 0 such that ∀zj ∈ BC(0,a0), there is a unique Qjzj ∈ S(R 3, C4):=∩ t≥0Σt(R3, C4), such that HQjzj+ g(QjzjQjzj)βQjzj = EjzjQjzj, Qjzj = zjφj+ qjzj, qjzj, φ ∗ j = 0, (1.7)
(1) (qjzj, Ejzj) ∈ C∞(BC(0,a0),Σr × R); we have qjzj = zjqj(|zj| 2), with q j(t2) = t2 qj(t2), qj(t) ∈ C∞((−a02, a 02),Σr(R3, C4)) and Ejzj = Ej(|zj| 2) with E j(t)∈ C∞((−a02, a02),R); (2) ∃ C > 0 such that qjzjΣr ≤ C|zj| 3, |E jzj− ej| < C|zj| 2.
For the Σr see(1.3). The onlynon-elementary point inProposition 1.1is theindependence of a0 with
respect of r (which strictly speaking is not necessary in this paper), which can be proved with routine arguments asintheAppendixof[21].
Definition 1.2.Let b0> 0 be sufficientlysmallsothatfor zj ∈ BCn(0,b0), Qjzj existsforall j ∈ {1,· · · ,n}. Set zj = zjR+ izjI,for zjR, zjI ∈ R and DjA:= ∂z∂jA,for A = R,I. Weset
Hc[z] = Hc[z1· · · , zn] where z := (z1, . . . , zn) :=η∈ L2(R3,C4); Re i η∗, DjRQjzj = Re i η∗, DjIQjzj = 0 for all j . (1.8) Inparticular wehave Hc[0] = η∈ L2(R3,C4); η∗, φj = 0 for all j . (1.9)
Wedenote by Pc theorthogonalprojectionof L2(R3, C4) ontoHc[0].
Wewill provethefollowingtheorem.
Theorem 1.3. Assume (H1)–(H5). Then there exist 0> 0 and C > 0 such that for =u(0)H4 < 0 then
the solution u(t) of (1.1)exists for all times and can be written uniquely for all times as
u(t) =
n
j=1
Qjzj(t)+ η(t), with η(t)∈ Hc[z(t)], (1.10)
such that there exist a unique j0, a ρ+ ∈ [0,∞)n with ρ+j = 0 for j = j0 such that |ρ+|≤ Cu(0) H4, an
η+∈ H4 with η+L∞ ≤ Cu(0)H4 and such that we have
lim t→+∞η(t) − e −itDMη+ H4(R3,C4)= 0, lim t→+∞|zj(t)| = ρ+j. (1.11)
Furthermore we have η = η + A(t,x) such that,
for preassigned p0> 2, for all p≥ p0 and for
2 p= 3 2 1−2 q , (1.12) we have zL∞t (R+)+ η Lpt([0,∞),B 4− 2p q,2 (R3,C4) ≤ Cu(0)H4(R3,C4) , ˙zj+ iejzjL∞t (R+)≤ Cu(0) 2 H4(R3,C4) (1.13)
(for the Besov spaces Bk
p,q see Sect.2) and such that A(t, ·)∈ Σ4 for all t ≥ 0, with
lim
Remark1.4.In H4(R3, C4) thefunctional u → g(uu)βu islocallyLipschitzand (1.1)islocallywell posed,
seepp.293–294involume III[51].
Remark1.5. There isno attempthereto getasharp resultwith aminimum amountof regularity onthe initialdatum u0.Theneedof H4 comesup(5.39).
Theorem 1.3 states thatsmall solutions of the NLD are asymptotically equalto exactly onestanding wave(possiblythevacuum) upto radiationwhichscattersand upto aphasefactor. Asweexplainbelow, this happens becauseof the naturaltendency of radiation to scatter because of linear scattering, and of energy to leak out ofall discrete modes except at most for onebecauseof nonlinear interactionwith the continuous modes whichproduces aform offriction onmost discretemodes. Theorem 1.3 doesnotsay if someofthestandingwavesarestable orunstable(thisremainsanopenproblem).
Theasymptoticbehaviorofsolutionswhichstartclosetoequilibriarepresentsafundamentalproblemfor nonlinearsystems.Hereweareinterestedtohamiltoniansystemswhichareperturbationsoflinearsystems admittingamixtureofdiscreteandcontinuouscomponents.Decayphenomenaimposedbytheperturbation onthediscretemodesanddecayof metastable states areaclassicaltopicofphysicalrelevanceinthestudy ofradiationmatterinteraction,[15].Thereisasubstantialliteraturefocusedonthecaseoflinearsystems, see[17] forarecentreferenceandthereinforsomeof theliterature.
TheNLShasbeenexploredinpaperssuchas[30,31,42,43,45–47,53–56].AnanalogueofTheorem 1.3for theNLSisprovedin[21].ThecaseofthenonlinearKleinGordonequation(NLKG),inthecontext ofreal valued solutions,where allsmall solutions scatter to 0,has been initiated in[47] inspecialcase and to a largedegreesolved inageneralway in[2].ForcomplexvaluedsolutionsoftheNLKGsee [22].
Inthispaper wefocus ontheNLD.There aremanypapersonglobal wellposednessand dispersionfor solutionsbelongingto muchlargerspacesthanwhatconsideredhere,forexamplesee [3,11,12,24,27,34,59]
and references therein. However, these papers do not addressthe asymptotic behavior in the presence of discretemodes. For asurvey onthe1Dcasewe referalsoto [39]. TheessentialindefinitenessoftheDirac operators (1.2) forbids the use of the stability theory of standing waves developed in [13,29,57], which involves conditionally positive Lyapunov functionals. See [50] for an attempt to apply this theory to the Diracequation, incombination to numericalcomputations.Asuccessful useofrather elaborateLyapunov functionalsinthespecialcaseofanintegrableNLDisin[40].Sinceenergymethodscannotbeusedtoprove stability then anotheroptionis to provestability bymeans oflineardispersion. Thisis aproblem arising also in other settings, see for example [38]. Thus on a strictly technical standpoint the NLD is a rather interestingclassofhamiltoniansystems,especiallybecauseithasbeenexploredmuchlessthansystemslike theNLS.
Weturntothespecificproblemconsideredinthispaper.InthecaseofthelinearDiracequationiut= Hu
weknowthatthereareinvariantcomplexlinesformedbystandingwavesincorrespondencetotheeigenstates of H. Proposition 1.1 describes the well known fact that the NLD (1.1), at least at small energies, has invariantdisksformedbystandingwaves.Thequestionarisesthenonwhatshouldbethebehaviorofother initiallysmallsolutionsoftheNLD.InthecaseoftheNLStheanswergivenin[21]isthattheunionofthese invariantdisks isan attractorfor allsmallsolutions. Thisis bynomeansobvious inviewofthefactthat inthelinearequationthesituationisdifferent.Infactinthelinearcasediscretemodesandthecontinuous partare all independentfrom each other so thatsolutionsof the linearequation contain aquasiperiodic component.There are examplesofnonlinear systemswhere theselinear likepatterns persist. Notablythe NLSwhere x ∈ Z,whichhassmallquasi-periodic solutions,seefor examplein[35] andreferencestherein. TherearealsoequationsinRn whichadmitfamiliesofquasiperiodicsolutionscontainingsolitarywavesas
thestabilizationdisplayedin[21]isaveryslowphenomenon,non-detectable easilynumerically:see[33]for comments on a somewhat different but related context. So results such as those in[21], from the earlier papers[37,47,48,53–55] whichtreatspecialcases,tothesolutionsingenericsetting containedin[2,21],are farfrom obvious.
Here wetranspose partially to the NLD theresult provedfor theNLS in[21].No particular deepnew insight is neededsince theproof for theNLD issimilar to thatfor theNLS, with somedifference related to thedispersiontheory of radiation. However, resultssuch as Theorem 1.3 areimportant given thatthe literaturegivesaveryfragmentarypictureoftheasymptoticanalysis ofsolutionsoftheNLD.
While prior to [21] there had been a large body of work for the NLS, very little has been written for the NLD about our problem,that is the analysis of the NLD with a mixture of discrete and continuous components.It isworthcomparingtheknownresultsintheliteraturewithTheorem 1.3.
One study is[6]. [6]contains anumberof usefulresults onthedispersion forthe groupeitH which are
used here.Intermsof analysisofsmall solutionsoftheNLDwithalinearpotential,[6]considersthecase of σp(H)={e1< e2} with e2− e1< min(m − e1, e1+ m) andthenprovestheexistenceofahypersurfaceof
initial dataperpendicular attheorigintotheeigenspace of e2, andwhosecorresponding solutionsof(1.1)
converge tothe1stfamilyofstandingwaves Q1z1.
Anotherstudyis[41].[41]provestheasymptoticstabilityofthestandingwaves Q1z1when σp(H)={e1},
that is an analogue of [42,46]. [41], like [42,46], does notexamine a whole neighborhood of 0 but proves thatifaninitial datumstartsveryclosetoa Q1z1 then,inasensesimilarto Theorem 1.3,itwillconverge
to a nearby Q1z1. Notice thatwhile here we don’t provethat the Q1z1 are stable, under the hypothesis
σp(H) ={e1} it canbe provedto be stable also inour3 D set up. Infact this is muchsimpler to prove
than whatwedohere.
In[8]isdiscussedtheasymptoticstability ofstandingwavesoftheNLDinadifferent context,whichis somewhat closerinspirit toTheorem 1.3sincetheemphasis isonthecasewhenmanydiscretemodes are present. Foranotherresult,seealso[16].
Here wegivemorecomprehensiveconclusionsthanin[6,41]becausewetreatallsmallsolutionsof(1.1). Maybe we could have analyzed the stability of specific standing waves, applying the theory in [8] which providessometoolscharacterizestandingwavesofthe(1.1)throughananalysisofthelinearizationof(1.1)
at thestandingwave,butwedon’tdothishere.Sounfortunatelywedonotidentifystablestandingwaves and we don’t produce acriterion to define ground states for(1.1). Recall thatin[21] it is provedfor the NLS thatwhile the Q1z1 arestable(whichwaswell knownsince[43]),the Qjzj for j≥ 2 areunstable.No similar analysisisdonehere.
AfteraddingtotheDiracequationanonlinearitysuchasin(1.1)aswehavealreadymentionedthelines ofstandingwavespersist,onlyintheformoftopologicaldisks.Herewebrieflyexplainhowtheasymptotics claimed inTheorem 1.3comes about.Themechanism isthe sameof theNLS andto be seenrequiresthe identification of an effective hamiltonian. In an appropriate system of coordinates, this turns out to be, heuristically,oftheform
H(z, η) = n j=1 ej|zj|2+Hη, η∗ + |(μ−ν)·e|>M (zμzνGμν, η + zμzνG∗μν, η∗), with e = (e1, . . . . , en),
where the 2nd summation is on an appropriate finite set of multi-indexes. The coordinates (z, η), with
z = (z1, . . . , zn),representingthediscretemodesand η∈ Hc[0] representingtheradiation,arecanonical(or
Darboux) coordinates(but notsotheinitialcoordinatesinLemma 2.9).Thisinparticularmeansthatthe equation for η is
Then,succinctly,
ηStrichartz≤ +zμ+ν
L2, (1.16)
forappropriateStrichartznormsof η. These estimatesarelesswellknownthanthecorrespondingones for the NLS, butnonetheless are known, see Sect. 5.1 where they are proven using [6–8]. There are various other sets of Strichartz like estimates forDirac potentials inthe literature, see forexample [9,11,23], but theydonotapplytoourcasesincetheyinvolveDiracoperatorswithouteigenvalues.Theequationsforthe discretemodes are,heuristically,
i ˙zj = ejzj+ |(μ−ν)·e|>M νj zμzν zj η, G μν + |(μ−ν)·e|>M μj zνzμ zj η ∗, G∗ μν. (1.17)
Byanargument introducedin[10,48]wewrite
η = g− Y, where Y :=
|(α−β)·e|>M
zαzβR+H(e· (β − α))G∗αβ. (1.18)
Weobserve that(1.18) isadecompositionof η into −Y ,whichis thepartof η that mostlyaffects thez’s,
and g which is small. Substituting in(1.17) and ignoring the very small g we get an autonomous system in z.Ignoringsmallerterms,wehave
d dt j |zj|2=−2π |(μ−ν)·e|>M |zμzν|2δ(H − (ν − μ) · e)G∗ μν, Gμν. (1.19)
Ther.h.s.isnegative, seeLemma A.1 intheAppendix. Noticethatin[10,47]thestructure ofther.h.s.is asclearas (1.19)onlyunderveryrestrictivehypothesesonthediscretespectrum.
Weassumethatforanappropriatesetofpairs(μ, ν) in (1.19)wehaveδ(H − (ν − μ) · e)G∗μν, Gμν > 0.
Thisisthecontentofhypothesis(H5)andisanexactanalogueofinequality(1.8)in[47].Thenintegrating
(1.19)weobtain j |zj(t)|2+ |(μ−ν)·e|>M t 0 |zμzν|2≤ C j |zj(0)|2≤ C 2, (1.20)
forafixed C. Thisallowstoclose(1.16)andtoprovethat η scatters. From(1.20)andthefactthatthe ˙z(t) isuniformlybounded,wegetthatthelimt→+∞zμ+ν(t)= 0 forthe(μ, ν) in (1.20).Thisallowstoconclude
thatlimt→+∞zj(t) = 0 for all except for at mostone j, becausethe pairsof multi-indexes (μ,ν) involve
casessuchas zμj
j z νk
k forany pair j = k (whilecasesoftheform z μj
j z νj
j arenotallowed).
Whathashappenedisthatwhenwehavesubstituted(1.18)inside(1.17)andignored g, wegotasystem on the z’s which has some degree of friction, ultimately due to the coupling of the discrete modes with radiation. This can be proved thanksto thesquare structure of ther.h.s. of (1.19) where whatis crucial is the fact that each Gμν is paired with its complex conjugate G∗μν. This comes about and can be seen
transparentlybecausewehaveahamiltoniansysteminDarbouxcoordinates,whichtellsusthatthe G∗μν’s inther.h.s.of(1.15)arethesameofthecoefficientsinthe(1.17).
There are various papers, such as [18,20], that discuss how to first produce Darboux coordinates and how to find the effective hamiltonian in an appropriate way that guarantees that the system remains a semilinear Dirac equation. In fact this part of the proof is here the same of [21], since in terms of the hamiltonian formalism, the NLS and the NLD are the same. So Darboux coordinates and search of the effectivehamiltonian bymeansofBirkhoffnormalformsare accomplishedin[21].
TheonlypartoftheproofwherethereisdifferencebetweenNLDandNLSisinthestudyofdispersion. TheNLDrequiresitsownsetoftechnicalmachineryaboutlineardispersiontheory,Strichartzand smooth-ing estimates, whichis notthe sameofthe NLSand which issomewhat tricker andless well understood. Nonetheless, all thelineartheory we needherehasbeen already developedintheliterature, inparticular in[6–8].
Since themainideasonthehamiltonian structureandcoordinatechangescomefrom [21], wewillrefer to [21]foranextensivediscussionofthemainissues.Therestoftheproofconsistsinprovingdispersionof thecontinuous componentofthesolution(andhereweusethetechnologyofDiracoperators in[6,7]),i.e.
(1.16), and theso called NonlinearFermi GoldenRule, i.e. (1.19). This latterpart of theproof is similar to [21], but requires some modification in the spirit of [19] because of the possible presence of pairs of eigenvalueswithdifferentsigns.
Hereas in[21]wedonotprove,asdonein[2]inaneasiersetting,thathypothesis(H5)holdsforgeneric pairs(V,g(uu)). Whilein[21]whatwasmissingtorepeattheargumentin[2]wasameaningfulmassterm, herewehavemassM butthedependence ofthelinearoperator H on M requiresnewideas.
2. Furthernotationandcoordinates
2.1. Notation
For k ∈ R and 1≤ p,q ≤ ∞, the Besov space Bk
p,q(R3, Cd) is thespace of all tempered distributions
f ∈ S(R3, Cd) suchthat fBk p,q = ( j∈N 2jkqϕj∗ fqp) 1 q < +∞,
withF(ϕ)∈ C0∞(Rn\ {0}) suchthatj∈ZF(ϕ)(2−jξ) = 1 forall ξ∈ R3\ {0},F(ϕj)(ξ)=F(ϕ)(2−jξ) for
all j ∈ N∗and forall ξ∈ R3,andF(ϕ0)= 1−
j∈N∗F(ϕj).ItisendowedwiththenormfBk p,q.
• WedenotebyN={1,2, . . .} theset ofnaturalnumbersandset N0=N∪ {0}.
• GivenaBanachspace X, v∈ X and δ > 0 we set
BX(v, δ) :={x ∈ X | v − xX< δ}.
• Wedenote z = (z1, . . . , zn),|z|:=
n j=1|zj|2.
• Weset ∂l:= ∂zl and ∂l:= ∂zl.Hereas customary ∂zl=
1
2(DlR− iDlI) and ∂zl=
1
2(DlR+ iDlI).
• Occasionallyweuseasingleindex = j, j.Todefine we usetheconvention j = j. Wewillalsowrite
zj= zj.
• We will consider vectors z = (z1, . . . , zn) ∈ Cn and for vectors μ, ν ∈ (N∪ {0})n we set zμzν :=
zμ1
1 ...znμnzν11...zνnn. Wewill set|μ|=
jμj.
• Wehave dzj = dzjR+ idzjI, dzj= dzjR− idzjI.
Remark2.1. We draw the attentionof the readerto the factthat thecomplex conjugateof v ∈ C4 is v∗
with v = βv∗ while for ζ ∈ C thecomplexconjugateis ζ which is moreconvenientnotation insomelater formulasthanwriting ζ∗.
2.2. Coordinates
Thefirstthingweneedisthefollowingwell standardansatz,see forexampleinLemma2.6[21].
Lemma 2.2. There exists c0 > 0 such that there exists a C > 0 such that for all u ∈ H1 with uH1 < c0,
there exists a unique pair (z,Θ)∈ Cn× (H1∩ Hc[z]) such that
u =
n
j=1
Qjzj + Θ with|z| + ΘH4 ≤ CuH4. (2.1)
Finally, the map u → (z,Θ) is C∞(BH4(0,c0),Cn× H4) and satisfies the gauge property
z(eiϑu) = eiϑz(u) and Θ(eiϑu) = eiϑΘ(u). 2 (2.2) Wenowrecallfrom [21]thefollowingdefinitions.
Definition 2.3. Given z ∈ Cn, we denote by Z the vector with entries (z
izj) with i, j ∈ [1,n] ordered in
lexicographic order(that iswe write zizj before zizj ifeither i < j or, when i = j,if j < j).Wedenote
by Z thevector with entries (zizj) withi, j ∈ [1, n] ordered inlexicographic order but onlywith pairs of
indexeswith i = j.HereZ∈ L with L the subspaceofCn0 ={(a
i,j)i,j=1,...,n: i= j} where n0= n(n− 1),
with(ai,j)∈ L iff ai,j= aj,iforall i, j. Foramultiindexm ={mij ∈ N0: i= j} wesetZm=(zizj)mij
and|m|:=i,jmij.
Definition2.4. Considerthesetofmultiindexesm asinDefinition 2.3.Considerforany k∈ {1,. . . , n} the
set Mk(r) ={m : n i=1 n j=1 mij(ei− ej)− ek >M and|m| ≤ r}, M0(r) ={m : n i=1 n j=1 mij(ei− ej) = 0 and|m| ≤ r}. (2.3) Setnow Mk(r) ={(μ, ν) ∈ Nn0× Nn0 :∃m ∈ Mk(r) such that zμzν= zkZm}, M (r) =∪nk=1Mk(r). (2.4)
Wealsoset M = M (2N + 4) and
Mmin={(μ, ν) ∈ M : (α, β) ∈ M with αj ≤ μj and βj≤ νj ∀ j ⇒ (α, β) = (μ, ν)}. (2.5)
Thefollowingsimplelemmaisused inLemma 5.17.
(1) If (μ,ν) ∈ Mmin then for any j we have μjνj= 0.
(2) Suppose that (μ,ν) ∈ Mmin, (α, β) ∈ Mmin and (μ− ν)· e= (α− β)· e. Then (μ, ν) = (α,β).
Proof. Firstofallitiseasytoshowthat(μ, ν) ∈ M(r) ifandonlyif|ν|=|μ|+1,|μ|≤ r and|(μ−ν)·e| >M. Supposethat μj≥ 1 and νj ≥ 1.Thenconsider (α, β)
αk = μ k for k= j, μj− 1 for k = j βk = ν k for k= j, νj− 1 for k = j.
Since|ν|=|μ|+ 1 wehave|β|=|α|+ 1.Furthermore(μ− ν)· e= (α− β)· e.Thisimplies(α,β) ∈ M and
so (μ,ν) /∈ Mmin.Thisprovesclaim(1).
By(μ−ν)·e= (α−β)·e and(H4)weobtain μ −ν = α−β,whichbyclaim(1)yields(μ,ν) = (α,β). 2
Lemma 2.6. Assuming (H4) then the following properties are fulfilled.
(1) For Zm = zμzν, then m ∈ M0(2N + 4) implies μ = ν. In particular m ∈ M0(2N + 4) implies
Zm=|z1|2l1. . .|z
n|2ln for some (l1, . . . , ln)∈ Nn0.
(2) For |m|≤ 2N + 3 and any j we have a,b(ea− eb)mab− ej = 0.
Proof. The following proof is in[21]. If μ = ν then zμzν =|z1|2μ1...|z
n|2μn.So thefirst sentenceinclaim
(1) impliesthesecond sentenceinclaim(1).Wehave
Zm = n i,l=1 (zizl)mil= n i=1 z n l=1mil i z n l=1mli i = zμzν.
Thepair(μ, ν) satisfies |μ|=|ν|≤ 2N + 4 by
|μ| = l μl= i,l mil=|ν|. Wehave(μ− ν)· e= 0 bym∈ M0(2N + 4) and i μiei− l νlel= i,l mil(ei− el) = 0.
Weconcludeby(H4)that μ − ν = 0.Thisproves the1stsentenceofclaim(1). Theproofof claim(2)issimilar.Set
Zmzj = n i,l=1 (zizl)milzj= n i=1 z n l=1mil i z n l=1mli i zj= zμzν. Wehave (μ− ν) · e = i μiei− l νlel= i,l mil(ei− el)− ej,
inadditionwehavealso
If(μ− ν)· e= 0 thenby|μ− ν|≤ 4N + 5 andby(H4)wewouldhave μ = ν,impossibleby(2.6). 2 ThefollowingelementarylemmaisverysimilartoLemma2.4[21]andisusedtoboundthe2ndand3rd terminthe1st lineof(5.2)intermsofthe zμzν with(μ,ν) ∈ M
min and η.
Lemma2.7. We have the following facts.
(1) Consider a vector m = (mij)∈ Nn00 such that
i<jmij > N for N >M(min{ej− ei: j > i})−1, see
(H4). Then for any eigenvalue ek we have
i<j
mij(ei− ej)− ek<−M. (2.7)
(2) Consider m ∈ Nn0
0 and the monomial zjZm. Suppose |m|≥ 2N + 3. Then there are a,b∈ Nn00 such that we have i<j aij = N + 1 = i<j bij, aij= bij= 0 for all i > j,
aij+ bij ≤ mij+ mji for all (i, j) (2.8)
and moreover there are two indexes (k,l) such that
i<j aij(ei− ej)− ek <−M, i<j bij(ei− ej)− el<−M (2.9)
and such that for |z|≤ 1
|zjZm| ≤ |zj| |zkZa| |zlZb|. (2.10)
(3) For m with|m|≥ 2N + 3 there exist (k,l) and a∈ Mk and b∈ Ml such that (2.10) holds.
Proof. Theproofis verysimilar toLemma2.4in[21].Forexample,(2.7)follows immediatelyfrom
i<j
mij(ei− ej)− ek≤ − min{ej− ei: j > i}N − e1<−M,
with the latter inequality due to the definition of N . All the other claims can be proved like the rest of Lemma2.4in[21].
Given a,b∈ Nn0
0 satisfying(2.8), byclaim(1) theysatisfy(2.9)forany pairof indexes(k, l). Consider
nowthemonomial zjZm.Since|m|≥ 2N + 3,therearevectorsc,d∈ Nn00 suchthat|c|=|d|= N + 1 with cij+ dij ≤ mij forall(i, j). Furthermorewehave
zjZm= zjzμzνZcZdwith|μ| > 0 and |ν| > 0. (2.11)
So,for zk afactorof zμ and zl afactorof zν,andfor
aij= cij+ cji for i < j 0 for i > j, bij= dij+ dji for i < j 0 for i > j, (2.12)
|zjZm| ≤ |zj| |zkZc| |zlZd| = |zj| |zkZa| |zlZb|.
Furthermore, (2.8)is satisfied.
Since our(a,b) satisfya∈ Mk andb∈ Ml,claim(3)isaconsequenceof claim(2). 2
Since (z,Θ) in(2.1)are notasystemofindependent coordinateswe needthefollowing,seeLemma2.5
[21].
Lemma 2.8. There exists d0> 0 such that for all z∈ C with|z| < d0 there exists R[z] :Hc[0]→ Hc[z] such
that Pc|Hc[z]= R[z]−1, with Pc the orthogonal projection of L2 onto Hc[0], see Definition 1.2. Furthermore,
for |z| < d0 and η ∈ Hc[0], we have the following properties.
(1) R[z]∈ C∞(BCn(0,δ0),B(Hr, Hr)), for any r∈ R. (2) For any r > 0, we have (R[z]− 1)ηΣr ≤ cr|z|
2ηΣ
−r for a fixed cr.
(3) We have the covariance property R[eiϑz] = eiϑR[z]e−iϑ.
(4) We have, summing on repeated indexes,
R[z]η = η + (αj[z]η)φj, with αj[z]η =Bj(z), η + Cj(z), η∗, (2.13)
where, for Z as in Definition 2.3, we have Bj(z) = Bj( Z) and Cj(z) = zizCij( Z), for Bj and Cij
smooth in Z with values in Σr.
(5) We have, for r∈ R,with Z as in Definition 2.3 Bj(z) + ∂zjq
∗
jzjΣr+Cj(z)− ∂zjqjzjΣr ≤ cr|Z|
2. 2 (2.14)
ThenLemma 2.6givesusasystemofcoordinatesneartheoriginin H4.Thesimpleproofisthesameof
Lemma2.6[21].
Lemma 2.9. For the d0> 0 of Lemma 2.8 the map (z, η) → u defined by u = n j=1 Qjzj + R[z]η, for (z, η)∈ BCn(0, d0)× (H 4∩ H c[0]), (2.15)
is with values in H4 and is C∞. Furthermore, there is a d1 > 0 such that for (z, η) ∈ BCn(0, d1)×
(BH4(0, d1)∩ Hc[0]), the above map is a diffeomorphism and
|z| + ηH4 ∼ uH4. (2.16)
Finally, we have the gauge properties u(eiϑz, eiϑη) = eiϑu(z, η) and
z(eiϑu) = eiϑz(u) and η(eiϑu) = eiϑη(u). 2 (2.17) Weendthissectionexploitingthenotationintroducedinclaim(5)ofLemma 2.8tointroducetwoclasses of functions.Firstofallnotice thatthelinearmaps η→ η, φ∗j extendinto boundedlinearmapsΣr→ R
forany r∈ R.Weset
Σcr:=η∈ Σr:
Definition 2.10. Wewill say thatF (t, z, Z, η) ∈ CM(I× A,R), with I a neighborhood of0 inR andA a
neighborhoodof0inCn×L×Σc−K is F =Ri,jK,M(t, z, Z,η), ifthereexist a C > 0 and asmallerneighborhood
A of0suchthat
|F (t, z, Z, η)| ≤ C(ηΣ−K+|Z|)j(ηΣ−K+|Z| + |z|)i in I× A. (2.19) Wewillspecify F =Ri,jK,M(t,z, Z) if
|F (t, z, Z, η)| ≤ C|Z|j|z|i (2.20)
and F =Ri,jK,M(t, z, η) if
|F (t, z, Z, η)| ≤ Cηj
Σ−K(ηΣ−K+|z|)i. (2.21) Wewillomit t if thereisnodependenceonsuchvariable.Wewrite F =Ri,jK,∞if F =Ri,jK,m forall m ≥ M.
Wewrite F =Ri,j∞,M ifforall k≥ K theabove F is therestrictionofan F (t, z, η) ∈ CM(I× A
k, R) with
Ak aneighborhoodof0inCn× L× Σc−kandwhichis F =Ri,jk,M.Finallywewrite F =Ri,j∞,∞ if F =R i,j k,∞
forall k.
Definition 2.11. We will saythat an T (t, z, η) ∈ CM(I× A,ΣK(R3, C)),with the abovenotation, is T = Si,jK,M(t, z, Z,η), ifthereexists a C > 0 and asmallerneighborhoodA of0suchthat
T (t, z, Z, η)ΣK ≤ C(ηΣ−K+|Z|)j(ηΣ−K+|Z| + |z|)i in I× A. (2.22) WeusenotationsSi,jK,M(t,z, Z),Si,jK,M(t, z, η) etc. asabove.
Remark 2.12. For given functions F (t, z, η) and T (t, z, η) we write F (t, z, η) = Ri,jK,M(t,z, Z, η) and T (t, z, η) = Si,jK,M(t, z, Z,η) when theyarerestrictionsto theset of vectorsZ∈ {(zizj)i,j=1,...,n: i= j} of
functionssatisfyingthetwo abovedefinitions.
Furthermorelater,whenwe writeRi,jK,M andSi,jK,M,we meanRi,jK,M(z,Z, η) and Si,jK,M(z,Z, η).
Noticethat F =Ri,jK,M(z,Z) or S = SK,Mi,j (z,Z) donotmeanindependencebythevariable η.
3. Invariants
Equation(1.1)admits theenergyandmassinvariants,definedasfollowsfor G(0) = 0 and G(s)= g(s):
E(u) := EK(u) + EP(u), where EK(u) :=DMu, u∗ and
EP(u) =
1 2
R3
G(uu)dx; Q(u) :=u, u∗. (3.1)
Wehave E∈ C∞(H4(R3, C),R) and Q ∈ C∞(L2(R3, C),R).Wedenoteby dE the Frechétderivativeof E.
We define ∇E ∈ C∞(H4(R3, C),H4(R3, C)) bydE(X) = Re∇E,X∗, for any X ∈ H4. We define also ∇uE and ∇u∗E by
dEX =∇uE, X + ∇u∗E, X∗, that is ∇uE = 2−1(∇E)∗ and∇u∗E = 2−1∇E.
i ˙u =∇u∗E(u). (3.2)
We recallthatnormal forms argumentsconsist inmaking Taylorexpansionsof thehamiltonian and in thecancellation ofthe non-resonant terms oftheexpansion.Thefollowingpropositionidentifiesthekindof expansionwehaveinmind.Weshouldthinkof(3.3)asanexpansioninthevariables z, η and theauxiliary variable Z.Eventuallytheeffective hamiltonianwill containtermsinther.h.s.suchas the1st and2ndin the1st lineandthetermsofthe2ndline.Thecancellations willoccurlaterinthe2ndline.
Proposition 3.1. We have the following expansion of the energy for any preassigned r0∈ N:
E(u) = n j=1 E(Qjzj) +Hη, η∗ + R 1,2 r0,∞(z, η) +R 0,2N +5 r0,∞ (z, Z) + n j=1 2N +3 l=0 |m|=l+1 Zmajm(|zj|2) + n j,k=1 2N +3 l=0 |m|=l (zjZmGjkm(|zk|2), η + c.c.) + ReS0,2N +4r0,∞ (z, Z), η∗ + i+j=2 |m|≤1 ZmG2mij(z), η⊗i⊗ (η∗)⊗j + 3 d=2 d i=1 R0,3−d r0,∞(z, η) R3 Gdi(x, z, η, η(x))η⊗i(x)⊗ (η∗(x))⊗(d−i)dx + EP(η), (3.3)
where c.c. is the complex conjugate of the term right before in the ( ) andwhere:
(1) (ajm, Gjkm, G2mij)∈ C∞(BR(0, d0),C× Σr0(R
3, C)× Σ
r0(R
3, Bi+j(C4, C)));
(2) Gdi(·, z, η, ζ) ∈ C∞(BCn(0, d0)× Σ−r0(R3, C)× C4, Σr0(R3, Bd(C4, C)))); (3) for |m|= 0 we have G20ij(0)= 0 and
i+j=2 G20ij(z), η⊗i(η∗)⊗j = 2−1 n j=1 g(QjzjQjzj)η, η ∗ + n j=1 Reg(QjzjQjzj) Re(Qjzjη)βQjzj, η ∗. (3.4)
Inorderto proveProposition 3.1weset
K(z, η) := E( n j=1 Qjzj + R[z]η) = KK(z, η) + KP(z, η), with KK(z, η) := EK( n j=1 Qjzj+ R[z]η), KP(z, η) := EP( n j=1 Qjzj + R[z]η).
ByTaylorexpansion,wewrite
Weexpand K3(z, η) = K3(0, η) + RP(z, η), where K3(0, η) = KP(0, η) = EP(η) (3.7) and RP(z, η) = 1 0 ∂zK3(tz, η)z dt := 1 0 n j=1 A=R,I DjAK3(tz, η)zjAdt = n j=1 1 0 ∂jK3(tz, η)zj+ ∂¯jK3(tz, η)¯zj dt. (3.8)
ToproveProposition 3.1we computethetermsof
K(z, η) = K(z, 0) + Re∂ηK(z, 0), η∗ +
1 2Re
∂2ηK(z, 0)η, η∗ + EP(η) + RP(z, η), (3.9)
startingwith ∂ηK, ∂η2K and ∂η3K.
Lemma3.2. Set u = u(z, η) =j=1n Qjzj + R[z]η. We have the following equalities:
∂ηKK(z, η) = 2R[z]∗Hu, ∂η2KK(z, η) = 2R[z]∗HR[z], ∂η3KK(z, η) = 0; (3.10) ∂ηKP(z, η) = R[z]∗(g(uu)βu) , ∂η2KP(z, η)ν = R[z]∗g(uu)βR[z]ν + 2R[z]∗g(uu)Re u R[z]νβu, ∂η3KP(z, η)ν
ν = 6R[z]∗g(uu)ReuR[z]νβR[z]ν + 4R[z]∗g(uu)ReuR[z]ν2βu. (3.11)
KP(z, η + εν) = 2−1
G((u(z, η) + εR[z]ν)(u(z, η) + εR[z]ν)) dx,
and by
G((u(z, η) + εR[z]ν)(u(z, η) + εR[z]ν))
= Gu(z, η)u(z, η) + 2εReu(z, η)R[z]ν+ ε2R[z]νR[z]ν
= o(ε3) + G(u(z, η)u(z, η)) + 2εg(u(z, η)u(z, η))Re (βu(z, η)(R[z]ν)∗)
+ ε2
g(u(z, η)u(z, η))R[z]νR[z]ν + 2g(u(z, η)u(z, η))
Re
u(z, η)R[z]ν
2
+ ε3 2g(u(z, η)u(z, η))Re
u(z, η)R[z]ν
R[z]νR[z]ν +4
3g
(u(z, η)u(z, η)) Reu(z, η)R[z]ν3 . 2
Wenow examinether.h.s.of(3.9).
Lemma 3.3. Consider the first two terms in the r.h.s. of (3.9). We then have
K(z, 0) = n j=1 E(Qjzj) + n j=1 2N +3 l=0 |m|=l+1 Zmajm(|zj|2) +R0,2N +5∞,∞ (z, Z), (3.12) Re∂ηK(z, 0), η∗ = n j,k=1 2N +3 l=0 |m|=l (zjZmGjkm(|zk|2), η + c.c.) + ReS0,2N +4∞,∞ (z, Z), η∗, (3.13)
where the coefficients in the r.h.s.’s have the properties listed in claim (1) in Proposition 3.1.
Proof. Firstofall,bothl.h.s.’sof(3.12)–(3.13)aregaugeinvariant.Then(3.13)isanimmediateconsequence of claim(3)ofLemma 3.4below.
Wehave K(z, 0) = E( n j=1 Qjzj) = E(Q1z1) + E( j>1 Qjzj) + [0,1]2 ∂2 ∂s∂tE(sQ1z1+ t j>1 Qjzj)dtds = n k=1 E(Qkzk) + αk(z) with αk(z) := n k=1 [0,1]2 ∂2 ∂s∂tE(sQkzk+ t j>k Qjzj)dtds.
The αk(z) are gaugeinvariant,so thatwecanapplyto themclaim (2)ofLemma 3.4below.Furthermore,
since αk(z)= O(|Z|) weconcludethatintheexpansion(3.14)for αk(z) wehaveequalities bj0(|zj|2)= 0. 2
Lemma 3.4. The following facts hold:
(1) For a(ζ) smooth from BC(0,δ) to R such that a(eiθζ) = a(ζ) for any θ∈ R there exists α∈ C∞([0, δ2);R)
such that α(|ζ|2)= a(ζ).
(2) Let a ∈ C∞(BCn(0, δ), R) satisfy a(eiθz1, · · · ,eiθzn)= a(z1, · · · , zn) for all θ ∈ R and a(0) = 0. Then
for any M > 0 there exist smooth bj0 such that bj0(|zj|2)= a(0,· · · ,0,zj, 0,· · · ,0) and
a(z1,· · · , zn) =
|m|≤M−1
(3) Let a ∈ C∞(BCn(0,δ), Σr)∀r ∈ R such that a(eiθz1, · · · , eiθzn)= eiθa(z1, · · · , zn). Then for any M > 0
∃Gjm such that Gjm∈ C∞(BCn(0, δ), Σr)∀r,zjGj0(|zj|2)= a(0,· · · ,0,zj, 0,· · · ,0) and
a(z1,· · · , zn) = n j=1 |m|≤M−1 zjZmGjm(|zj|2) +S∞,∞1,M (z, Z). (3.15)
Proof. Thiselementarylemma isprovedin[21]. 2
The3rdterminther.h.s.of(3.9)isdealtbythefollowinglemma.
Lemma3.5. There exist G2mi(2−i)(z) as in the statement of Proposition 3.1such that (3.4) holds and
Re ∂η2K(z, 0)η, η∗ =Hη, η∗ + R1,2∞,∞(z, η) + 2 i=0 |m|≤1
ZmG2mi(2−i)(z), η⊗i(η∗)⊗(2−i). (3.16)
Proof. ByLemma 3.2andbyclaim(2)inLemma 2.8wehave
1 2Re ∂η2K(z, 0)η, η =HR[z]η, (R[z]η)∗ + g( j,k QjzjQjzj)R[z]η, (R[z]η) ∗ + 2 Reg( j,k QjzjQjzj)Re ⎛ ⎝n j=1 QjzjR[z]η ⎞ ⎠n j=1 Qjzj, (R[z]η) ∗ =Hη, η∗ + R1,2∞,∞(z, η) +g( j,k QjzjQkzk)η, η ∗ + 2 Reg( j,k QjzjQkzk)Re ⎛ ⎝n j=1 Qjzjη ⎞ ⎠n j=1 Qjzj, η ∗.
Thelast lineyields thelastterm of(3.16)and itisstraightforwardto seethatithastherequired proper-ties. 2
To finish the discussion of the r.h.s. of (3.9) we need to compute RP. We consider first apreparatory
lemma.
Lemma3.6. Let l = 1, · · · , n and A = R,I. Then we have (DlAR[z])η =nj=1R1,1∞,∞(z,η)φj.
Proof. One has R[z]η = η +nj=1(αj[z]η)φj. So DlAR[z]η = nj=1((DlAαj[z])η) φj. Now, DlAα[z]η =
DlABj(z)η dx+
DlACj(z)η∗dx, where B and C are given inLemma 2.8andwehave DlABj(z)=S∞,∞1,0
and DlACj(z)=S∞,∞1,0 .Thisyieldsthelemma. 2
We set u := nj=1Qjzj + R[z]η.Then, we have DlAu = DlAQlzl+ (DlAR[z])η. For l = 1, · · · , n and
A = 2g(uu)ReDlAu ReuR[z]ηβR[z]η, (R[z]η)∗ +g(uu)ReDlAuR[z]η βR[z]η, (R[z]η)∗ + g(uu)Reu(DlAR[z])η βR[z]η, (R[z]η)∗ + 2g(uu)ReuR[z]ηβ(DlAR[z])η, (R[z]η)∗, B= 2g(uu)Re(uDlAu) ReuR[z]η2βu, (R[z]η)∗
+ 2g(uu)ReuR[z]ηReDlAuR[z]η
βu, (R[z]η)∗
+ 2g(uu)ReuR[z]ηReu(DlAR[z])η
u, (R[z]η)∗
+g(uu)ReuR[z]η2βDlAu, (R[z]η)∗ + g(uu)
ReuR[z]η2βu, ((DlAR[z])η)∗.
AllthetermsintheformulasforA and Bcanbe expressedas
d=2,3
i+j=d
Gdij(z, η), η⊗i⊗ (η∗)⊗jR∞,∞2,3−d(z, η) +R2,2∞,∞(z, η). (3.17)
Therefore RP admitsanexpansionoftheform(3.17), whichis absorbedintermsofther.h.s.of(3.3).
Proof ofProposition 3.1. Wehavejustseenthat RP isabsorbedinther.h.s.of (3.3).Theother termsof
ther.h.s.of(3.9)aretreatedbyLemmas 3.3 and3.5. 2 4. EffectiveHamiltonian
In thissection we applythe theoryof Sect.4and 5in[21]which yields aneffectiveHamiltonian inan appropriatecoordinatesystem,whichinturn willbeusedtoproveTheorem 5.1whichyieldsTheorem 1.3. Finding an effective Hamiltonian entails canceling as many terms as possible from the 2nd line of (3.3)
throughappropriatechangesofvariables.ThisprocessiscalledBirkhoffnormalformargumentandisdone by means of a recursive procedure where each time we need to cancel a term from the hamiltonian we findanappropriatecoordinatechangebyfirstsolvinganequation,the homological equation. Itiseasierto implement thisprocedureusing coordinateswhichareDarboux.Inafinite dimensionalsetting this would mean that the symplectic form is equal to a simple model, like ω0 := jidzj ∧ dzj. This corresponds
to diagonalizingthe homologicalequations. Furthermore, itisimportant thatthenewcoordinatesremain Darboux. This means thatthe change of coordinatesshould leave ω0 invariant. Oneway to do this is to make changesof coordinates using flows of hamiltonian vector fields. Seefor example Sect.1.8 [32] fora generalintroductiontothesubject.
Thesystem(3.2)isHamiltonianwithrespect tothesymplecticform
Ω(X, Y ) :=−2 ImX, Y∗. (4.1) ThefirstthingtonoticeisthatthecoordinatesinLemma 2.9,initiallythemostnaturalcoordinatesinour problem,donotformasystemofDarbouxcoordinatesfor(4.1)inanyreasonablesense.IndeedΩ israther complicated inthis coordinatesystem.
Weconsider asalocalmodelthesymplecticform
Ω0:= i n j=1 (1 + γj(|zj|2))dzj∧ dzj+ idη, dη∗ − i dη∗, dη , where γj(|zj|2) :=− qj(|zj|2),q∗j(|zj|2) + 2|zj|2Re q∗ j(|zj|2),qj(|zj|2) , (4.2) with qj(t)= d
dtqj(t).ByProposition 1.1andDefinition 2.10we have γj(|zj|
2)=R2,0
Remark4.1.Ω0 is thesamelocalmodelsymplecticform of[21].Wedonotknow ifProposition 4.2below
holds whenchoosing γj ≡ 0 becausewe donotknow ifin(4.6)belowwe wouldstill haveSj =R1,1r,∞ and
Sη = S1,1r,∞,which iscrucial.Notice, incidentally, thattheDarboux Theorem is anabstract result. Butin [21] it is provedwith an ad hoc argument exactlybecause wewant this changeof coordinates, as well as all theother coordinateschanges inthepaper, to have thiscrucial property.The factthatall coordinate changeshavethispropertyguaranteesthatthelimits(1.10)inonecoordinatesystemimplythesamelimit inanyothercoordinatesystem.
WhileΩ0wouldbesimplerif γj≡ 0,nonethelessitissimpleenoughforanormalformargument.
InSect.4[21]thefollowingpropositionisproved.
Proposition4.2 (Darboux Theorem). Fix any r∈ N. There exists a δ0∈ (0, d0) such that the following facts hold.
(1) Thereexists a gauge invariant 1-form Γ= ΓjAdzjA+Γη, dη+Γη∗, dη∗, with
ΓjA=R1,1∞,∞(z, Z, η) and Γξ= S1,1∞,∞(z, Z, η) for ξ = η, η∗, (4.3)
such that dΓ = Ω− Ω0.
(2) For any (t, z, η) ∈ (−4,4)× BCn(0, δ0)× BΣc
−r(0,δ0) there exists exactly one solution X
t(z, η) ∈ L2 of the equation iXtΩt = −Γ. Furthermore, Xt(z, η) is gauge invariant, Xt(z,η) ∈ Σr and if we
set Xt
jA(z,η) = dzjAXt(z,η) and Xηt(z,η) = dηXt(z, η), we have XjAt (z,η) = R1,1r,∞(t, z, Z,η) and
Xt
η(z, η) = S1,1r,∞(t, z, Z,η).
(3) Consider the following system in (t, z, η) ∈ (−4,4)× BCn(0, δ0)× BΣc
k(0,δ0) for all k∈ Z∩ [−r,r]: ˙zj =Xjt(z, η) and ˙η =Xηt(z, η). (4.4)
Then the following facts hold for the corresponding flow Ft.
(3.1) For δ1∈ (0,δ0) sufficientlysmall we have
Ft∈ C∞((−2, 2) × BCn(0, δ1)× BΣc k(0, δ1), BCn(0, δ0)× BΣck(0, δ0)) for all k∈ Z ∩ [−r, r] Ft∈ C∞((−2, 2) × BCn(0, δ1)× BH1∩Hc[0](0, δ1), BCn(0, δ0)× BH1∩Hc[0](0, δ0)). (4.5) In particular we have ztj= zj+ Sj(t, z, η) and ηt= η + Sη(t, z, η), (4.6) with Sj(t, z, η) =R1,1r,∞(t, z, Z,η) and Sη(t,z, η) = S1,1r,∞(t,z, Z, η).
(3.2) The map F= F1is a local diffeomorphism of H1into itself near the origin, and we have F∗Ω= Ω
0.
(3.3) We have Sj(t, eiϑz, eiϑη) = eiϑSj(t,z, η), Sη(t, eiϑz, eiϑη) = eiϑSη(t,z, η). 2
Wenowconsider thepullback K := E◦ F.
Lemma 4.3. Consider the δ1 > 0 and δ0 > 0 of Proposition 4.2 and set r0 = r with r the index in Proposition 3.1. Then we have
F(BCn(0, δ1)× (BH1(0, δ1)∩ Hc[0]))⊂ BCn(0, δ0)× (BH1(0, δ0)∩ Hc[0]) (4.7)
and F|BCn(0,δ1)×(BH1(0,δ1)∩Hc[0]) is a diffeomorphism between domain and an open neighborhood of the origin
in Cn× (H1∩ H
K(z, η) = H2(z, η) + j=1,...,n λj(|zj|2) + 2N +4 l=1 |m|=l+1 Zma(1)m(|z1|2, . . . ,|zn|2) + n j=1 2N +3 l=1 |m|=l (zjZmG(1)jm(|zj|2), η + c.c.) +R1,2r1,∞(z, η) +R0,2N +5r1,∞ (z, Z, η) + ReS0,2N +4r1,∞ (z, Z, η), η∗ + 3 d=2 d i=1 R0,3−d r0,∞(z, η) R3 G(1)di (x, z, η, η(x))η⊗i(x)⊗ (η∗(x))⊗(d−i)dx + i+j=2 |m|≤1 ZmG(1)2mij(z), η⊗i⊗ (η∗)⊗j + EP(η), (4.8) where H2(z, η) = n j=1 ej|zj|2+Hη, η∗
and where: G(1)jm, G(1)2mij are S0,0
r1,∞; a
(1)
m(|z1|2, . . . , |zn|2) = R0,0∞,∞(z); c.c. means complex conjugate;
λj(|zj|2)=R2,0∞,∞(|zj|2); G(1)di (·, z, η, ζ) ∈ C∞(BCn(0, δ1)× Σ−r 1(R 3,C) × C4, Σ r1(R 3, Bd(C4,C)))). For |m|= 0, G(1)2mij(z,η) = G2mij(z) is the same of (3.4). Finally, we have the invariance R1,2r1,∞(e
iϑz, eiϑη) ≡ R1,2
r1,∞(z, η).
Proof. The proof oftheabovestatementwith possiblynonzerotermsalsocorresponding to l = 0 in both summationsinthe2nd lineof(4.8)iselementary,see Lemma4.10in[21] andLemma4.3 [20].
The key factthatin the2nd line of (4.8)both summationsstart from l = 1 and there are no l = 0 is
provedintheCancellation Lemma4.11in[21]. 2
ConsidernowthesymplecticformΩ0in(4.2).Weintroduceanindex = j, j,for j = j with j = 1, . . . , n.
We write ∂j = ∂zj and ∂j = ∂zj, zj = zj. Given F ∈ C1(U,C) with U an open subset of Cn× Σcr, its
Hamiltonian vectorfield XF isdefinedby iXFΩ0= dF .Wehavesumming on j
iXFΩ0= i(1 + γj(|zj|
2))((X
F)jdzj− (XF)jdzj) + i(XF)η, dη − i (XF)η, dη
= ∂jF dzj+ ∂jF dzj+∇ηF, dη + ∇η∗F, dη∗ , (4.9) where (4.9)isusedalsotodefine ∇ξF for ξ = η, η∗.
Comparing thecomponentsof thetwo sidesof (4.9)we getfor 1+ j(|zj|2)= (1+ γj(|zj|2))−1 where
j(|zj|2)=R2,0∞,∞(|zj|2):
(XF)j=−i(1 + j(|zj|2))∂jF , (XF)j= i(1 + j(|zj|2))∂jF,
(XF)η=−i∇η∗F , (XF)η∗= i∇ηF. (4.10)
Given G ∈ C1(U,C) and F ∈ C1(U,E), withE aBanachspace,we set{F,G}:= dF X
Definition4.4 (Normal Forms). RecallDefinition 2.4and(2.3).Fix r∈ N0.Arealvaluedfunction Z(z, η) is
innormalformif Z = Z0+ Z1with Z0 and Z1finitesumsofthefollowingtype,forl≥ 1:for Gjm(|zj|2)=
Sr,∞0,0 (|zj|2),c.c.thecomplexconjugateand am(|z1|2, . . . , |zn|2)=Rr,∞0,0 (|z1|2, . . . , |zn|2),
Z1(z, Z, η) = n j=1 |m|=l m∈Mj(l) zjZmGjm(|zj|2), η + c.c. , (4.11) Z0(z, Z, η) = |m|=l+1 m∈M0(l+1) Zmam(|z1|2, . . . ,|zn|2). (4.12)
Remark 4.5. By Hypothesis (H4), in particular by (1.6), for any m ∈ M0(2N + 4) we have Zm = |z1|2m1...|z
n|2mnforan m ∈ Nn0 with2|m|=|m|.Similarlyby(H4),inparticularby(1.5),for|m|≤ 2N + 4
wehave|a,b(ea− eb)mab− ej|= M.
Forl≤ 2N +4 wewillconsiderflowsassociatedtoHamiltonianvectorfields Xχwithrealvaluedfunctions
χ of thefollowingform,with bm=R0,0r,∞(|z1|2, . . . , |zn|2) and Bjm= S0,0r,∞(|zj|2) forsomer∈ N definedin
BCn(0,d) forsomed > 0: χ = |m|=l+1 m /∈M0(l+1) Zmbm(|z1|2, . . . ,|zn|2) + n j=1 |m|=l m /∈Mj(l) (zjZmBjm(|zj|2), η + c.c.). (4.13)
Thefollowingresultisprovedin[21].
Proposition 4.6 (Birkhoff normal forms). For any ι ∈ N∩ [2,2N + 4] there are a δι > 0, a polynomial χι
as in (4.13) with l = ι, d= δι and r= rι = r0− 2(ι+ 1) suchthat for all k∈ Z∩ [−r(ι),r(ι)] we have for each χι a flow (for δ1> 0 the constant in Proposition 4.2)
φtι∈ C∞((−2, 2) × BCn(0, δι)× BΣc
k(0, δι), BCn(0, δι−1)× BΣck(0, δι−1)),
φtι∈ C∞((−2, 2) × BCn(0, δι)× BH1∩Hc[0](0, δι), BCn(0, δι−1)× BH1∩Hc[0](0, δι−1)) (4.14)
and such that, if we set F(ι):= F◦φ2◦· · ·◦φι, with F the transformation in Proposition 4.2and the φj= φ1ι,
then for (z,η) ∈ BCn(0, δι)× (BH1(0,δι)∩ Hc[0]) we have the following expansion
H(ι)(z, η) := E◦ F(ι)(z, η) = H2(z, η) + n j=1 λj(|zj|2) + Z(ι)(z, Z, η) + 2N +3 l=ι |m|=l+1 Zma(ι)m(|z1|2, . . . ,|zn|2) + n j=1 2N +3 l=ι |m|=l (zjZmG(ι)jm(|zj|2), η + c.c.) +R1,2rι,∞(z, η) +R0,2N +5rι,∞ (z, Z, η) + ReS0,2N +4rι,∞ (z, Z, η), η∗ + 3 d=2 d i=1 R0,3−d r0,∞(z, η) R3 G(ι)di(x, z, η, η(x))η⊗i(x)⊗ (η∗(x))⊗(d−i)dx + i+j=2 |m|≤1 ZmG(ι)2mij(z), η⊗i⊗ (η∗)⊗j + EP(η), (4.15)
Z(ι)= m∈M0(ι) Zmam(|z1|2, . . . ,|zn|2) + n j=1 ( m∈Mj(ι−1) zjZmGjm(|zj|2), η + c.c.). (4.16)
We have R1,2rι,∞=R1,2r2,∞ and Rr1,22,∞(eiϑz, eiϑη) ≡ R1,2r2,∞(z,η).
In particular we have for δf := δ2N +4and for the δ0 in Proposition 4.2,
F(2N +4)(B
Cn(0, δf)× (BH1(0, δf)∩ Hc[0]))⊂ BCn(0, δ0)× (BH1(0, δ0)∩ Hc[0]) (4.17)
with F|BCn(0,δf)×(BH1(0,δf)∩Hc[0]) a diffeomorphism between its domain and an open neighborhood of the
origin in Cn× (H1∩ H
c[0]).
Furthermore, for r = r0− 4N − 10 there is a pair R1,1r,∞ and S1,1r,∞ such that for (z, η)=F(2N +4)(z,η)
we have
z = z +R1,1r,∞(z, Z, η), η = η + S1,1r,∞(z, Z, η). (4.18)
Furthermore, by taking all the δι > 0 sufficiently small, we can assume that all the symbols in the proof,
i.e. the symbols in (4.18)and the symbols in the expansions (4.15), satisfy the estimates of Definitions 2.10 and 2.11for |z| < δι and ηΣr(ι) < δι for their respective ι’s. 2
5. Dispersion
Weapply Proposition 4.6, setH = H(2N +4) so thatforsome r∈ N whichwecantakearbitrarilylarge,
H(z, η) = H2(z, η) + n j=1 λj(|zj|2) +Z(z, Z, η) + R, (5.1) with Z(z,Z,η) =Z(2N +4)(z,Z,η) and R = R1,2 rι,∞(z, η) +R 0,2N +5 rι,∞ (z, Z, η) + ReS 0,2N +4 rι,∞ (z, Z, η), η ∗ + 3 d=2 d i=1 R0,3−d r0,∞(z, η) R3 G(ι)di(x, z, η, η(x))η⊗i(x)⊗ (η∗(x))⊗(d−i)dx + i+j=2 |m|≤1 ZmG(ι)2mij(z), η⊗i⊗ (η∗)⊗j + EP(η). (5.2)
Ourambientspaceis H4(R3, C4).Sounder(H1)thefunctional u → g(uu)βu islocallyLipschitzand(1.1),
(3.2) andthe equivalentsystemwith Hamiltonian H(z,η) and symplecticform Ω0,are locally well posed,
see pp.293–294involumeIII[51].
Bystandardarguments,see[21], Theorem 5.1belowimpliesTheorem 1.3.
Theorem5.1 (Main Estimates). Consider the of Theorem 1.3. Then there exist 0> 0 and a C0> 0 such that if < 0 then for I = [0, ∞) we have the following inequalities:
η
Lpt(I,B
4− 2p q,2 (R3,C4))
≤ C , for all the pairs (p, q) as of (1.12), (5.3)
ηL2
t(I,H4,−10(R3,C4))≤ C , (5.4)
ηL2
t(I,L∞(R3,C4))≤ C , (5.5)
zjZmL2
t(I)≤ C , for all (j, m) with m ∈ Mj(2N + 4), (5.6)
Furthermore, there exists ρ+ ∈ [0,∞)n such that there exists a j0 with ρ+j = 0 for j = j0 and there exists η+ ∈ L∞ such that |ρ+|≤ C andη+L∞ ≤ C such that
lim t→+∞η(t, x) − e −itDMη +(x)L∞x(R3,C4)= 0, (5.8) lim t→+∞|zj(t)| = ρ+j. (5.9)
Byanelementary continuationargument (see[8,21]or [49],end of theproof ofTheorem 2.1, Sect.II), theestimates(5.3)–(5.7)for I = [0, ∞) areaconsequenceofthefollowing proposition.
Proposition5.2. There exists a constant c0> 0 such that for any C0> c0 there is a value 0= 0(C0) such that if the inequalities (5.3)–(5.7)hold for I = [0, T ] for some T > 0, for C = C0 and for 0 < < 0, then in fact for I = [0, T ] the inequalities (5.3)–(5.7) hold for C = C0/2.
Proof. ByLemma 5.11,thereexistsafixed c1> 0 such thatgivenany C0 if 0> 0 is smallenoughwehave foralladmissiblepairs(p,q), forthe M of Definition 2.4andforapreassigned τ0> 1,
η Lpt([0,T ],B 4− 2p q,2 )∩L2t([0,T ],H 4,−τ0 x )∩L2t([0,T ],L∞x) ≤ c1 + c1 (μ,ν)∈M |zμzν| L2 t(0,T ). (5.10)
Theaimof Sect.5.1isto provethatthere existsafixed c2> 0 such thatif 0> 0 for anygiven any C0 if
0> 0 is small enoughwehave
j zj2L∞t (0,T )+ (μ,ν)∈M zμ+ν2 L2(0,T )≤ c2 2+ c2C0 2. (5.11)
This implies that we canreplace C0 with C1 with C1 =
c2(1 + C0) ≤ 2 √
c2C0. We have C1 ≤ C0/2 if C0≥ c0:= 16c2.Theproofisnow completed. 2
5.1. Completion of the proof of Proposition 5.2 5.1.1. Bounds on the continuous modes
Westartthissectionbylistingsomeresults knownintheliterature,thenwe provesomeauxiliarytools requiredfortheproofofProposition 5.2.Thefollowingtheorem isTheorem1.1[6].
Theorem5.3. Under hypotheses (H2)–(H3) for s > 5/2 and any k∈ R we have eiHtP
cu0Hk,−s(R3,C4)≤ Cs,kt− 3 2P
cu0Hk,s(R3,C4). 2 (5.12)
ThesubsequentisTheorem1.1in[7].
