Quasi-periodic Solutions for the Derivative Nonlinear Schr¨odinger Equation with Finitely Differentiable Nonlinearities?

Meina GAO Kangkang ZHANG

1 Introduction

In this paper we prove the existence of quasi-periodic solutions of the derivative nonlinear Schr¨odinger(DNLS for short)equation

subject to Dirichlet boundary conditions u(t,0)=0=u(t,π),?∞

The same as in[18],introducing the inner product in a suitable phase space,for example,the usual Sobolev space([0,π])

then(1.1)can be written in the Hamiltonian form

with the “Reality” conditionwhere the gradient of H is de fined with respect to ?·,·?and

KAM theory is a powerful tool to deal with the existence of periodic,quasi-periodic or almost periodic solutions of partial differential equations(PDEs for short)under small perturbations.The first KAM results for PDEs have been obtained for 1-d semi-linear Schr¨odinger and wave equations by Kuksin[14],Craig-Wayne[10,29],see the references therein.For PDEs in higher space dimension,the theory has been more recently extended by Bourgain[8],Eliasson-Kuksin[11],Berti-Bolle[5],and Geng-Xu-You[12].For unbounded perturbations,the first KAM results have been proved by Kuksin[15–16]and Kappeler-P¨oschel[13]for KdV equation(see also[7]),and more recently by Liu-Yuan[17–19],Zhang-Gao-Yuan[32]for derivative NLS equation,Baldi-Berti-Montalto[1]for the Hamiltonian quasi-linear perturbations of the KdV equation,and Berti-Biasco-Procesi[2–3]for derivative NLW equation.

However,the results mentioned above require the analyticity of the perturbations of the PDEs to overcome the well-known “l(fā)oss of regularity”problem.By shrinking the width of the angle variables,one can estimate the solutions of the homological equations and obtain the convergence of the KAM iterative procedure.For dynamical systems with differentiable perturbations,it is clear that after finitely many steps all derivatives are exhausted which leads to the failure of the KAM iteration.To cope with this difficulty,the primary approach is due to Moser[20–21],which extended the classical KAM theory for nearly integrable Hamiltonian systems under real-analytic perturbations,to smooth category.The main idea exploited by Moser is to use a smoothing operator,and re-insert enough regularity into the problem at every Newton iterative step in order to compensate the loss of regularity.A closely related approach was given by Nash[24]researching the embedding problem of compact Riemannian manifolds.In[21],Moser first proved the existence of the invariant curves for area preserving annulus mappings satisfying the monotone twist property which corresponds to the Hamiltonian system case in “one and a half” degrees of freedom.The number of derivative of the perturbation is required to be ?>333,which was later reduced by R¨ussmann to ?>5 in[26].For the Hamiltonian case we refer to[23,25].

The KAM theory in Moser[20–21]dealt with the persistence of maximal-dimensional invariant tori in the context of smooth category.It is natural to ask whether lower-dimensional tori can be persisted or not.By exploiting a technique following[23],Chierchia-Qian[9]considered the existence of lower-dimensional elliptic tori of any dimension between one and the number of degrees of freedom for the nearly integrable Hamiltonian system with finitely differentiable perturbation.The framework of this method is mainly based on an approximation of the differentiable functions with analytic ones.Zhang[31]proved the existence of the lowerdimensional invariant tori for the reversible system with finite degrees of freedom under finitely differentiable perturbation.For in finite dimensional Hamiltonian systems,the research just began in the last few years,the main results were given by Berti-Bolle-Procesi[4–6].By using a Nash-Moser iterative scheme in scales of the Sobolev functions space,they got the existence of quasi-periodic solutions which has Sobolev regularity both in time and space for PDEs with bounded perturbations,such as the NLS and NLW for any spatial dimension.

The perturbation of(1.1)is finitely differentiable.What is more important,the answer is unbounded.Thus(1.1)is excluded by the above approach.The aim of the present paper is to construct a large amount of quasi-periodic solutions of small amplitude for the derivative NLS equation(1.1).More precisely,in the following we consider a class of“vector” derivative NLS equations:

subject to Dirichlet boundary condition u(t,0)=0=u(t,π),where the nonlinearities are quasiperiodic in time with frequency ω ∈ Rnandfor somelarge enough and the second equation is the formal complex conjugation of the first one.

We have the following theorem.

Theorem 1.1Suppose that the nonlinearities f and are finitely differentiable with >100(1+ ρ)(3n+2τ+1)+3+p,where ρ,p and τ are positive constants which will be de fined below,and Π?Rnis a compact set of positive Lebesgue measure.Then there exists a small constant ?? >0 such that for|?|< ??,a Cantor set Π? ? Π with Meas(Π Π?) → 0 as ?→ 0,and for arbitrary ω ∈ Π?,(1.4)possesses a quasi-periodic solution of frequency ω with small amplitude.

The main ideas of our proof consist of a smoothing technique elaborated in[23]and Newton iterative scheme.Note that(1.4)is a Hamiltonian PDE,however,we do not use its Hamiltonian structure explicitly.Instead,we write(1.4)into an abstract nonlinear equation

(see Theorem 2.1),of which we try to construct the quasi-periodic solutions by Newton’s method.The essential element of Newton’s method is to find the approximate solution by solving the linearized equation of the original equation.Therefore,at each Newton iteration,we solve the linearized equation of(1.5)

Moreover,we need to prove the convergence of the iterative process.In order to solve(1.6),we need to estimate the inverse of(Λ+P(ωt))?1,which is “big” due to the unboundedness of the perturbation.Hence we do not solve the linearized equation(1.6)directly but do the KAM type reduction first.Luckily,as we discuss(1.4)under Dirichlet boundary condition,the frequencies are simple,the reduction process is feasible(see Lemma 3.2 for the details).The Hamiltonian structure guarantees the reality of Λ which is necessary in Theorem 1.4 in Liu-Yuan[17].In fact we get a new system after the reduction

whereis much smaller and can be treated as perturbation.Thus we just need to find the solution for the linearized equation

wheredenotes a diagonal matrix close to Λ in some sense(seeLemma4.1 for the details).

We remark that P(ωt),F(ωt)andare only differentiable with respect to t.The main difficulty during the whole procedure is the phenomenon of“l(fā)oss of regularity”.To overcome this,we use an approximation theorem which is closely related to the classical theorem due to Jackson,of which the fundamental observation is that the qualitative property of differentiability of a function can be characterized in terms of quantitative estimates for an approximating sequence of analytic functions.Then we can solve the linearized equation in analytic category with good estimates to guarantee the convergence of the Newton iterative process.

This paper is organized as follows:In Section 2,we rewrite the derivative NLS equation(1.4)in in finite coordinates,and this new equation will be our starting point for the following discussion.What’s more,we list a similar theory to the approximation theory of Jackson,Moser and Zehnder,which will be used as the basis of our smoothing technique.In Section 3,a KAM type reduction lemma will be proven with finitely differentiable unbounded perturbation.In Section 4,we describe the solving procedure of the linearized equation at each step and the iterative process in details.Finally some technical lemmas are exhibited in Section 5.

2 The Basic Modes

We study(1.4)on some suitable phase space,for example,the usual Sobolev spaceWe rewrite it in in finitely many coordinates by making the ansatz

The coordinates are taken from the Hilbert space ?pof all complex-valued sequences q=We fix p>later.Then(1.4)can be written as

wherewithIn the following we considerto be independent,and(1.4)equals to(2.2)when the bar means the complex conjugate.We investigate the regularity of the nonlinear vector field first.In fact,we have the following observation.

Lemma 2.1The nonlinear vector fieldde fines a finitely differentiablemap fromintowhere denotes some small neighborhood of the originin ?p × ?p.To be more precise,for anywith

where T means the transpose of a vector.

ProofIntroduce a map F de fined on Hp×Tnwith

We will prove that there exists some neighborhood U of the origin in Hpsuch that for anyThen(2.3)follows directly.

From the assumption thatwe can find some N being the neighborhood of the origin in C2such that for a+b≤ p?1,all the derivativesare bounded on N×Tn.Then we set U denoting the neighborhood of the origin in Hpof which the elementhas graphs lying in N.We will show that for any ξ∈ U,? ∈ Tn,Using the chain rule we can write

where ? represents i1+···+ia+j1+···+jb+m=p?1.For ξ∈ U,we haveandwhich is due to the fact thatThus we can get the estimate

Then by using the interpolation estimate in Lemma 5.1 in the Appendix,we can get

Consequently we haveThe estimate corresponding to F2can be obtained similarly,and we omit the details here.Hence we obtain

Then the conclusion thatde fines a map from U into Hp?1for any ? ∈ Tnfollows.

Now we investigate the first order Fr`echet derivative ofwith respect to ξ.When|k|= ?,then nothing remains to be done.Hence in the following we assumeHp,we get

with a different constant C depending on p. Thus,and the same estimate can be obtained for F2.Therefore,for any(ξ,?)∈U×Tn,the first Fr`echet derivativede fines a bounded linear operator from Hpintoas the set of bounded linear operators from Hpinto Hp?1,then we can obtainfor all ? ∈ Tn.For any other derivative of high order,we can handle in the same way.By now we have finished the proof.

We set some notations and de finitions for the sake of convenience.Set Pp= ?p× ?p,andrepresent the set of bounded linear operators fromand the corresponding norm respectively.Here we just focus our attention on α∈N for simplicity,and hence inLemma2.1 we choose ?,|k|∈ N also.Set X0:=Pp?1.For a given vector R(ζ,?)=(f(ζ,?),g(ζ,?))T,de fine

For the original system(2.2),setThen based onLemma2.1,we conclude that for anyestimate

holds true.Then we have the following theorem.

Theorem 2.1Consider the system

which ful fills the following hypotheses:

where

with λi=i2.

(A2)The perturbationis finitely differen-tiable with respect to ζ and ? with

In addition,

(A3)The first order Fr`echet derivativefunction from O × Tninto the space of bounded operators from ?pinto ?p?1.Moreover,

forand ?∈Tn.

Then for any

where ρ and τ are two fixed positive constants with 0< ρ n+3,there exists ?? >0 and a Cantor subset Π?? Π with Meas(ΠΠ?)→ 0 as ?→ 0,such that for ?< ?? and ω ∈ Π?,(2.11)has a quasi-periodic solution with frequency ω.

From(1.3)in Section 1,we can find a Hamiltonian perturbation P(q,,?)such thatThen taking the “Reality” condition of P into account,we can easily check the assumption(A3).Obviously,we can directly get the conclusion in Theorem 1.1 from Theorem 2.1,thus we will discuss the proof of Theorem 2.1 in the following.In the remaining part of the present section we list a well known and fundamental approximation result.Starting from the following lemma,we can set up a sequence of analytic functions which approximate to the original finitely differential one.

Lemma 2.2(Jackson,Moser,Zehnder)Let X be a Banach space and f ∈ C?(Rn;X)for some ?>0 with finite C?norm over Rn.Let ? be a radial-symmetric,C∞ function with support being the closure of the unit ball centered at the origin,where ? is completely flat and takes value 1,and let K=b? be its Fourier transform.For all σ>0,de fine fσ(x):=Then there exists a constant C≥1 depending only on ?and n such that the following holds:For any σ >0,the function fσ(x)is a real-analytic function from Cnto X such that if denotes the n-dimensional complex strip of width σ,then for all α ∈ Nnsuch that|α|≤ ?,one has

and for all 0≤s≤σ,

Here Xαis the Banach space of bounded operatorswith the norm

The function fσpreserves periodicity(i.e.,if f is T-periodic in any of its variable xj,so is fσ).Finally,if f depends on some parameter ξ∈ Π ? Rnand if the Lipschitz-norm of f and its x-derivatives are uniformly bounded by then all the above estimates hold with ∥ ·∥replaced by ∥ ·∥L.

This lemma is similar to the approximation theory obtained by Jackson,Moser and Zehnder,and the only difference is that we extend the applied range from C?(Rn;Cn)to C?(Rn;X).The proof of this lemma consists in a direct check which is based on standard tools from calculus and complex analysis,for details see[27–28]and the references therein.

Fix a sequence of fast decreasing numbers sν↓0,ν ≥0,and s0≤for F(?)∈ C?(Tn;X)we can construct a sequence of analytic and quasi-periodic functions F(ν)(?)such that the following conclusions holds:

(1)F(ν)(?)is analytic on the complex stripof the width sνaround Tn.

(2)The sequence of functions F(ν)(?)satis fies the bounds:

where C denotes a constant depending only on n and ?.

(3)The first approximate F(0)is “small” with the perturbation F.Precisely speaking,for arbitrary ?∈we have

where constant C is independent of s0,and the last inequality holds true due to s0≤

(4)From(2.19),we have the equality below.For arbitrary ?∈Tn,

3 Reduction Lemma

In this section,we will give an iterative lemma,which is the key part of our proof.Let m≥0 be the m-th step,we introduce some recursive parameters.

(1) ?0=C?,Cdenotes a positive constant depending only on n and

(2) ?m=whereis a small constant satisfying 0<<

(3)sm=with 0<<ρ

(4)σm=which acts as a bridge from smto sm+1,

(5)andwhich denote the length of the truncation of Fourier series,

(6)which dominates the measure parameters excluded in the m-th iteration step,

(7)hence

(8)hence

(9)hencewhere C1is a positive constant depending only on n and τ,and τ is a fixed real number greater than n+3.

Before we give the iterative lemma,we list some notations.

(1)Suppose thatis the set of bounded linear operators from ?qintoand we de fine the operator norm of its element byAccordingly,we denote the set of bounded linear operators from Pqintobyand the operator norm of its element byHence from the above section,we can conclude that|∥ ·∥|p,p?1= ∥ ·∥X1.

(2)Letbe the complexi fication of Tn,and de fine={?∈:|Im?|=|Im?i|

Furthermore,if f has an additional(Lipschitz-continuous)dependence on ω ∈ Π,we de fine the Lipschitz norm

(3)Choose ? and α0such that

(4)In what follows we use the notations ab to represent that there exists a constant C independent of m,? and α0but may depending on n,τ and ? such that a

(5)Let

Apparently we know that Meas(Π Π′)l α0.

Lemma 3.1Assume that at the m-th iteration step,we have a system as follows:

with∈Pp,ω∈Πm,m≥1 which satis fies the following hypotheses:

where

with

Moreover,λi,mis Lipschitz-continuous in ω and ful fills the estimate

(?,ω): × Πm → C is real analytic in ?,Lipschitz-continuous in ω and of zero average,It also ful fills the following estimates in×Πm:

wheredenotes the k-th Fouriercoefficient ofμ.

(H2)is real analytic with respect to ?,Lipschitz-continuous in ω and satis fies the estimate:

Furthermore,for ? ∈ Tnand ω ∈ Πm,

(H3)For any ω ∈ Πm,

Then there exist Πm+1? Πmwith Meas(ΠmΠm+1)αm+1and Bm(?,ω): ×Πm →Bp,p ∩which is real analytic with respect to ? ∈ ,Lipschitz-continuous in ω∈Πmand satis fies

such that for any ω ∈ Πm,by the transformation(3.2)can be changed into

Moreover,Λm+1,Pm+1ful fill(H1)–(H3)with m replaced by m+1.

The above result is similar to the iteration Lemma 3.2 in[17],and the key part of the proof is to find a suitable estimate for the solutions of the homological equations with large variable coefficients.Hence the process is parallel except the following two points:(1)The width of the angle variable smrelies on ?m,hence the system(3.2)in our paper has weaker regularity,(2)is controlled byinstead of ?min[17].Consequently,we concentrate our attention on these two aspects in the following.

ProofNow we include our system into a more general framework.Abbreviate the notations Λm,Am,Pm,λi,m,μi,m,Bm,and ΓKmby Λ,A,P,λi,μi,B and ΓK,and Λm+1,Am+1,Pm+1,λi,m+1and μi,m+1by Λ+,A+,P+,andrespectively.Following the procedure in the proof of Lemma 3.2 in[17],we set χ =eB(?)φ,and pluging into(3.2)yields(3.9),where

Thus we need to solve the homological equation for the unknown B:

whereTo this end,Bi,1≤i≤4 should satisfy the homological equations:

We only give the details of solving(3.13)in the following,and(3.14)–(3.16)can be handled in the same way.In view of the proof of Lemma 3.2 in[17],we can obtain the estimates of the elements of B1.Precisely speaking,we have the following statements:

(1)B1,ii=0.

(2)For(i,j)with 0<|λi? λj|<2Km,

(3)For(i,j)with|λi? λj|≥ 2Km,

By the assumptionswe can obtain

It follows from(3.1)thatConsequently,we have

Taking(3.18)and(3.20)into account and using Lemma 5.2 in the Appendix,we get the estimate of B1=(B1,ij)i,j≥1:

In view of(3.1),we can setThen by the de finition of αmand ?m,we get

Taking the de finition of σminto account,together with(3.21)and(3.22),we have the following estimate:

From the assumption

we can obtainConsequently,we have

For the other terms of B,i.e.,B2,B3,B4,the same results can be obtained.Thus,we finally get the estimate for B:

The remaining estimates for λ+, μ+and the new perturbed term P+can be handled in the classical way,and we do not give the proof here.For the detail we can refer to[30],and we just need to verify(3.5)for P+.

From(3.13)–(3.16),it is easy to see that for real ?,i.e.,? ∈ Tn,

Then by a direct calculation,we can obtain that R satis fies(3.5).Furthermore,from Lemma 5.3 in Appendix,together with the fact that Λ satis fies(3.5)and

we can conclude that P+satis fies(3.5).Till now we complete the proof of this lemma.

Remark 3.1In the following,for a block matrixif the components Pi,1≤i ≤ 4 satisfy(3.5)for real ?,we call P(?)satis fies(3.5).Similarly,for a given operatorif(3.27)can be ful filled for any real ?,we call B(?)satis fies(3.27).

We can obtain a more general reduction lemma below by applying Lemma 3.1.

Lemma 3.2Suppose that at the m-th iteration step,we have a system as follows:

with Λ in(2.12),and Πmdescribed by(H3)in Lemma 3.1.Moreover,Pm(?,ω):Tn×Πm → X1has the formPi(?,ω).In addition,Pi(?,ω)can be written as

and for ν ≥ i,(?,ω):×Πm → X1is real analytic in ? ∈ ,Lipschitz-continuous in ω and satis fies(3.5).Moreover,we have the following estimates:

Then there exists Um(?,ω)de fined on Tsm?4σm × Πmwith Um(?,ω)=eB1(?,ω)···eBm(?,ω),where Bi(?,ω)satis fies(3.27)and the following estimates:

such that by the transformation η =Um(ωt,ω)φ,(3.28)can be changed into

where Λm+1ful fills(H1)in Lemma 3.1 with m replaced by m+1,and Qm+1(?,ω):Tn×Πm+1 →X1can be written as

with(?,ω),(?,ω)being real analytic in ? ∈ ,Lipschitz-continuous in ω ∈ Πm+1 and satisfying(3.5)as well as the estimates:

ProofWe proof this lemma by induction.When m=1,we have

with

Consider the system

and it is easy to check the hypotheses(H1)–(H3)in Lemma 3.1 are satis fied.Applying Lemma 3.1 to(3.37),we can find a set Π2? Π1and a linear transformation η=eB1φ such that(3.37)is changed into

where Λ2andsatisfy(H1)–(H3)in Lemma 3.1 with m=2.Moreover,the operator B1(?,ω):× Π1→ Bp,p∩Bp?1,p?1satis fies the estimate:

Hence by the same transformation,in view of the expansion(3.35),the original equation(3.28)for m=1 can be changed into

where

which means that the lemma is true for m=1.

Now suppose that the lemma is true for m?1.Atthe m-th step,rewrite the system(3.28)as

By induction,we can find a coordinate transformation η =Um?1χ with Um?1=eB1···eBm?1and Πm? Πm?1changing the equation

into

where

and

Differentiating the transformation η=Um?1χ with respect to t,we have=Inserting it into(3.28)and in view of(3.29),we can obtain the new system:

where

Consequently based on Lemma 3.1,there exist Πm+1? Πmwith Meas(ΠmΠm+1)αm+1,and Bm(?,ω):× Πm→ Bp,p∩Bp?1,p?1satisfying

By the transformationwe can change(3.44)into

with

Set Um:=Um?1·eBm,then the lemma is true for m,and we finish the proof.

4 Iteration Process and Proof of Theorem 2.1

Lemma 4.1Let us consider the system

de fined on× Πm+1,where Λm+1and Πm+1ful fill the hypotheses(H1)and(H3)in Lemma 3.1 with m replaced by m+1,and the vector field(?,ω)=((?,ω),(?,ω))T: × Πm+1 → Pp?1is real analytic in ? ∈,Lipschitz-continuous in Πm+1andsatis fies the estimate

Furthermore,we assumeexists a quasi-periodic solution φm+1(?,ω)=(ym+1(?,ω),(?,ω))Twhich is real analyticin ? ∈and Lipschitz-continuous in ω ∈ Πm+1such that

with rm=()and

In addition,for any ?∈Tn,we have

ProofAbbreviate the notationsand ΓL,respectively.Then(4.3)can be written as

In the following we just find the solution to the first equation,and the second one can be solved similarly.Set r=(r1,r2)and let r1be an in finite vector with elements

where ΓLis the truncation operatorThen yjsatis fies equations as follows:

(1)For j with 0<|λj|<2Lm,

(2)For j with|λj|≥ 2Lm,

Now we solve the homological equations(4.8)–(4.9)with large variable coefficient.First let us consider(4.8).By(3.3)and(3.6),we haveThen applying Theorem 1.4 in[17]to(4.8),we have

In view ofhave

Thus by(3.1)we can obtain

Hence from(4.10)–(4.12),we conclude that

Next we consider(4.9).By(3.3),we have

In view ofLemma 2.6 in[17]to(4.9),we have

Taking(4.13)and(4.15)into account,using Lemma 5.2 in Appendix,we can get the estimate of the ?p-norm of y=(yj)j≥1:

For the estimate of the Lipschitz norm,we proceed as follows.Given a function B of ω,set△B=B(ω)? B(ω′).Applying △ to(4.8)–(4.9),we obtain the following assertions:

(1)For 0<|λj|<2Lm,

(2)For|λj|≥ 2Lm,

Applying Theorem 1.4 in[17]to(4.18),for 0<|λj|<2Lm,we obtain

Then applying Lemma 2.6 in[17]to(4.19),for|λj|≥ 2Lm,we have

Then from(4.20)–(4.21),using Lemma 5.2 in Appendix,we get

Divided by|△ω|,together with(4.17),we obtain

The estimate for ey is the same as y in(4.24).Hence we have

In view of the de finition of σmand(3.22),we have

where the last inequality follows from the assumption on ? in(2.16).

We handle with the estimate of the remaining part r1as in[17].First,we divide r1into three parts,that is r1=++,whereandhave the vector elements as follows:

andis the truncation ofthat is=(1?ΓL)From(4.16),(4.22)and

we get

by Lemma 5.2 in Appendix.Since

applying Lemma 5.2 in Appendix,we obtain

Applying again Lemma 5.2 in Appendix to △=(ω)?(ω′),we getFoewe obtain

By the same method,we can obtain the estimate with respect to r2,and consequently we have

In view ofwe can get(4.5)by a direct calculation.Now we finish the proof.

Remark 4.1(1)For a given vector F(?)=(F1(?),F2(?))T,if for any ? ∈ Tn,we have

then we say that the vector F satis fies(4.33).

(2)Given an operator P(?)ful filling(3.5),and ζ(?)=(y(?), ey(?))T,we assume that ey=y for real ?.Then it is easy to conclude that the vector P(?)ζ(?)satis fies(4.33).

(3)Givensuch that for any real ?,

then we call that B(?)satis fies(4.34).It is easy to check that the operator eB(?)ful fills(4.34)if B(?)satis fies(4.34).

(4)Given an operator B(?)satisfying(4.34)and a vector F(?)satisfying(4.33),we can obtain that the vector B(?)F(?)satis fies(4.33).

Now with the above preparation work at hand,we begin to make the Newton iteration process clear.Setting F(ζ)=? Λζ+F(ζ,?),we have the following lemma.

Lemma 4.2(Iteration Lemma)Assume that for m≥1,at the m-th iteration step,we have a solution ζm(?,ω)=(qm(?,ω),(?,ω))T:×Πm→O ful filling the following hypotheses:

(T1) ζm(?,ω)has an expansion of the form ζm(?,ω)=ηi(?,ω),where for any 1 ≤ i ≤ m, ηi(?,ω)=(vi(?,ω),(?,ω))Tis real analytic in ? ∈ ,Lipschitzcontinuous in ω ∈ Πi,and satis fies the estimate

Moreover,=for any real ?.

(T2) ζmis an ?m-approximate solution to the system(2.11),that is,In addition,we have

which satis fies the following assumptions:

(a)hm(?,ω)=((?,ω),(?,ω))Tis real analytic in ? ∈ and ful fills the estimate

In addition,hm(?,ω)satis fies(4.33)for any ω ∈ Πm.

(b)X1,ν ≥m are real analytic in ? ∈Lipschitz-continuous in ω and satis fies(3.5).Moreover,we have the following estimates:

(c)For 1≤i≤m?1 and ν≥m?1,(?,ω):×Πm→X1is real analytic in ?∈,Lipschitz-continuous in ω and ful fills(3.5)as well as the estimates:

(d)For 0 ≤ i≤ m ? 1 and ν≥ m ? 1,(?,ω):× Πm→X0is real analytic in ? ∈Lipschitz-continuous in ω and ful fills(4.33)and the estimates:

(e)For m≥1,

Then there exists Πm+1? Πmwith Meas(ΠmΠm+1)αm+1and a quasi-periodic solution ζm+1(?,ω)de fined on× Πm+1such that(T1)and(T2)hold true with m replaced by m+1.

ProofSet ζm+1= ζm+ ηm+1,then

First let us investigate the higher order term gm.For convenience,we set

withF(ζ,?)=A(ζ,?),(?,ω)=ζm?1(?,ω)+sηm(?,ω).Then for a fixed constant s,we can obtain

Observe that for 0≤s≤1,

then in view of i+k≤??3,we have

Here the norm is de fined as the operator norm fromto X2.Thus we have

For the estimate of the Lipschitz norm,we proceed as follows:As what mentioned before,for a given function T(ω),set △T=T(ω)? T(ω′).Applying △ to(4.42)we can obtain

Observe that

then in view of i+k+1≤??2,we have

where the norm is de fined as the operator norm fromto X.Thus

2

Divided by|△ω|,together with(4.43)and(4.45),we can obtain

Applying Cauchy estimate with respect to ηiontogether with the de finition of sm,we get for 0≤j≤??3,1≤i≤m,

thus

where ? means the admissible index set with t indices ji1,···jitlying in ?.Then?

By integrating with respect to s,we have

withNow we apply Lemma 2.2 to Nm,then for{sν}ν≥m,in view of(2.21),? ∈,Lipschitz-continuous in ω ∈ Πmand satis fies

Then from(4.40),together with(4.35),we get

and for ν≥m,in view of the assumption that 0<ρ<,we have

SetThen Fm(?,ω)is real analytic in ? ∈Lipschitz-continuous in ω ∈ Πmand satis fies(4.33)and the estimate

In addition

In the following we get down to find the solution ηm+1of the homological equation

Now let us give some more analysis concerning the operator Pm(?):=DζF(ζm,?)de fined in(4.36).Using Taylor’s formula,we have

then by induction,we get

ForThus by the same analysis as above,we have

Hence,by applying Lemma 2.2,we can obtainwith

Accordingly,for 2≤i≤m,

Taking(4.35)into account,we have

For i=1,we have

whereand the second term can be handled similarly.Then by applying Lemma 2.2,we can obtain

where

Hence we havewith

It is easy to verify that Pmful fills the hypotheses in Lemma 3.2,then we can find an operator Um(?,ω)de fined on× Πmful filling(4.34)and the corresponding coordinate transformation η=Um(ωt,ω)φ.Differentiating it with respect to t,we have=+Inserting it into(4.55),we obtain

where Qm+1is de fined by(3.33)–(3.34)and

Moreover,we can conclude thatsatis fies(4.33)in view of Remark 4.1 in this section.

Consider the system i˙φ?Λm+1φ+=0 de fined on× Πm+1,then applying Lemma 4.1,we can find a solution φm+1(?,ω)such that

with

Hence(4.64)becomes

Let ηm+1=Um+1φm+1,from(3.31)and(4.66)we have

Taking(3.31)and(3.33)–(3.34)into account,the homological equation(4.55)has the form:

and hm+1(?,ω)is real analytic in ? ∈with

Consequently,from(4.41),(4.54)and(4.69),we can obtain

and from(4.38)–(4.39),(4.68),(4.70)–(4.71)and Remarks 4.1 in this section,we get that

and all the assumptions in(T1)and(T2)hold with m replaced by m+1.By now we finish the proof.

Proof of Theorem 2.1It remains to find the first approximate solution η1to(2.11)and then to show that(T1)and(T2)in Lemma 4.2 are true for m=1.By using Taylor’s formula we hav

where F0(?)∈ C?(Tn;X0)and P0(?)∈ C??1(Tn;X1)satisfy

Then applying Lemma 2.2 to F0(?),we getwith

From(4.61)–(4.62),we get with the estimate

The homological equation we hope to investigate is

Fix k,i and j and setThen it is easy to check Meas(Π′Π0)lα0.Hence from Lemma 3.2 in[17]we know that for ω ∈ Π0,there exists an operatorand

such that by the coordinate transformation η=eB0φ,we can change the systeminto the new systemwhere Λ1apparently ful fills(H1)in Lemma 3.1 andThen by the same transformation,(4.76)can be changed into

withand

In addition,from(4.75)and(4.77)we obtain

Applying Lemma 4.1,we can find a real analytic solution φ1(?)such that

andSetting similarly to(4.69),we conclude

where

Hence

Thus by setting

which ful fills(T1)–(T2)in Lemma 4.2 with even better estimates.

Finally,lettingthen ζ∞(ωt,ω)is a quasi-periodic solution of frequency ω ∈ Π?for(2.11)andIn addition,We finally finish the proof of Theorem 2.1,and the proof of Theorem 1.1 immediately follows.

5 Appendix

Lemma 5.1De fine the Sobolev space

Then we have the interpolation estimate below

ProofThe proof can be found in[22],we omit it here.

Lemma 5.2Let F=(Fij)i,j≥1be a bounded operator on ?2.Assume that the matrixelements(Fij)are analytic functions of ?∈Let R=(Rij)i,j≥1be another operator withmatrix elements depending analytically onandRii=0.Then R is a bounded operator on ?2andMoreover,let f=(fj)j≥1be a vector in ?2with the vector elements fjbeing analytic function of ? ∈ Assume thatis another vector of which the vector elements depend analytically on ? ∈ andThen r ∈ ?2and the ?2-norm

ProofThe proof of this lemma can be found in[17].

Lemma 5.3Letbe an operator with the elementsful fillingbe another operator satisfying

Then the operator[P,B]=PB?BP satis fies(3.5)also.

ProofThe conclusion of this lemma can be obtained by a direct calculation.

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