A Nekhoroshev Type Theorem for the Nonlinear Wave Equation in Gevrey Space?

Chunyong LIU Huayong LIU Rong ZHAO

Abstract In this paper,the authors prove a Nekhoroshev type theorem for the nonlinear wave equation

Keywords Gevrey space,Nonlinear wave equation,Normal form,Stability

1 Introduction

We consider the nonlinear wave(NLW for short)equation

on the finite x-interval[0,π]with Dirichlet boundary conditions

The parameter m is real and positive,and the nonlinearity f is assumed to be real analytic in u and of the form

(1.1)is a typical model of in finite-dimensional Hamiltonian system associated with the Hamiltonian function

where v=ut,,g=f(s)ds anddenotes the usual scalar product in L2.(1.1)is well studied by many authors as an in finite dimensional Hamiltonian system,such as the long time stability result(see[1–4,7–10,19])and the existence of invariant tori by Kolmogorov-Arnold-Moser(KAM for short)theory(see[5,11,16–18,21–26]).

In[3],Bambusi proved a Birkho ffnormal form theorem which is applied to(1.1)with Dirichlet boundary conditions and obtained the dynamical consequences on the long time behavior of the solutions with small initial Cauchy data in Sobolev spaces Hs.Afterwards,Bambusi and Gr′ebert[6]gave an abstract Birkho ffnormal form theorem for Hamiltonian partial differential equations,which is applied to semi-linear equations with nonlinearity satisfying tame modulus.They got that for s sufficiently large,the Sobolev norm of index s of the solution is bounded by 2? during a polynomial long time(of order ??rwith r arbitrary).Later,Cong-Liu-Yuan[14]and Cong-Gao-Liu[13]generalized the method in[6]and proved the KAM tori are stable in a polynomial long time for nonlinear Schr¨odinger equation and nonlinear wave equation.

Recently,in[15],Faou and Gr′ebert proved a Nekhoroshev type theorem for the nonlinear Schr¨odinger equation

in an analytic space.The authors proved that if the initial datum is analytic in a strip of width ρ >0 with a bound on this strip equal to ? then,if ? is small enough,the solution of the nonlinear Schr¨odinger equation above remains analytic in a strip of widthand bounded on this strip by C? during very long time of order ??α|ln?|δfor some constants C>0,α >0 and 0<δ<1.We should point out that there is no so-called tame property in analytic space compared to that in Sobolev space.Later,Mi-Liu-Shi-Zhao[20]generalized the method in[15]to prove the similar result for(1.1)with Dirichlet boundary.

As Bourgain[12]said,the topology is very important for studying the long time stability result.On the other hand,the Gevrey space is a phase space which is well studied for in finitedimensional Hamiltonian system(see[9–10]).It is natural to ask that whether such a stability result holds in Gevrey space.In this paper,we will prove that if the Gevrey norm of initial datum is small in a strip of width 2ρ >0 with a bound on this strip equal to ?,then,if ? is small enough,the Gevrey norm of the solution of the nonlinear wave equation above remains small in a strip of widthand bounded on this strip by C? during very long time of order ??σ|ln?|βfor some constants C>0,σ >0 and 0< β <.

Before stating the result,we firstly introduce the Gevrey space.Suppose a function u:[0,π]→ C that can be expressed as

For ρ>0,we denote

Note that(Aρ,|·|ρ)is a Banach space.Then our main result is as follows.

Theorem 1.1 There exist 0< β 0,and the following holds:There exist constants C>0 and ?0>0 such that if

then the solution of(1.1)with initial datum u0and v0exists infor timesand satisfies

The rest of the paper is organized as follow.In Section 2,(1.1)is turned into an in finite dimensional Hamiltonian system in complex coordinates.And we give an important lemma which will be used to prove the main result.In Section 3,some important definitions are given.Then,we obtain estimate of the nonlinearity,vector field and Passion bracket.In Section 4,we introduce recursive equation and give two lemmas which we will use to get the normal form result(i.e.,Theorem 4.1).In Section 5,two important lemmas are given.Theorem 1.1 is proved and the long time stability of solutions to NLW equation is obtained in Gevrey space.

2 Hamiltonian System

We study(1.1)as an in finite dimensional Hamiltonian system.As the phase space one may take,for example,the product of the usual Sobolev spacewith coordinates u and v=ut,i.e.,

then the Hamiltonian is

and the Hamiltonian equations(2.2)is changed into

Lρ(ρ >0)of all real valued sequences w=(w1,w2,···)with finite norm

Then the Hamiltonian(2.3)turns into

with equations of motions turned into

where

These are Hamiltonian equations of motion with respect to the standard symplectic structure.Since the nonlinearity f in(1.1)is real analytic in a neighborhood of zero and of the form(1.2),we have

According to(2.7)–(2.8),we get

where

Let Z:=Z1{0}.If the Hamiltonian is defined on the complex Banach space Lρ,bcollecting all the two-side sequences with norm

the corresponding symplectic structure is.Then for a function P of C1(Lρ,b,C),we define its Hamiltonian vector field by XP=J?P,where J is the symplectic operator on Lρ,b.For two functions P and Q,we define the Poisson bracket by

Definition 2.1 For a given ρ >0,we define the space of real Hamiltonian P by Hρsatisfying

Obviously,for P and Q in Hρ,the formula(2.11)is well defined.For a given Hamiltonian function H ∈Hρ,we associate the Hamiltonian system

which is equivalent to

Then,let us introduce the complex coordinates

To simplify calculation,we introduce another set of coordinates(···,w?2,w?1,w1,w2,···)in Lρ,bby setting

Therefore,the system(2.6)is turned into

with Hamiltonian(2.5)changed into

Notice that Fj1,···,j2k=F|j1|,···,|j2k|.

In the end of the section,we give a lemma that shows the relation between the space Aρand the space Lρ,b,which will be used in the proof of the main result.

Lemma 2.1 Let u,v be valued in finite differentiable functions on the closure of the xinterval[0,π],and let(wj)j∈Zbe the sequence of its coordinates defined by(2.4)and(2.13)–(2.14).Then for anyμ<ρ,we have

where cμ,ρis a constant depending on μ and ρ.

ProofDue to(2.13)–(2.14),we know

and

Due to(2.4),it is clear to know that

and

So,forμ< ρ and for any u,v∈ Aρ,we have the estimation

Therefore,(2.18)is also proved.

3 Space of Polynomial and Some Properties

Firstly,let us introduce some terminologies and notions about the polynomial on CZ.Let?≥ 2 and j=(j1,j2,···,j?)∈ Z?.We define

(1)the monomial associated with j,

(2)the divisor associated with j,

where for ji∈ Z,,i=1,2,···,?.

In addition,we also denote the set of indices with zero momentum by

Furthermore,if ? is even and j is of the form(j1,?j1,···,,?j2?)or some permutation of it,we say that j=(j1,j2,···,j?) ∈ Z?is resonant writing j∈ N?.Specially,if j is resonant,its associated divisor vanishes,i.e.,?(j)=0,and its associated monomials depend only on the actions

where for j≥1,Ij=wjw?jdenotes the action associated with the index j.

We note that if w is real,then Ij=|wj|2and if ? is odd,the resonant set N?is the empty set.So,we know that the orders of monomials in the nonlinearity F are all even.

Definition 3.1 If P is real,then for k≥2,a formal polynomial P(w)=Σajwj∈Pk,of degree k,has a zero of order at least 2 in w=0,and satisfies the following conditions:

(1)P contains only monomials having zero momentum(i.e.,such that j∈ I?for some ?,when aj≠0),and P denotes

(2)The coefficients ajare bounded,i.e.,for all ?=2,···,k.

We endow Pkwith the norm

Recall that we assume that the nonlinearity f in(1.1)is complex analytic in a neighbourhood of zero in C.Therefore,there exist two positive constants M and R0such that the Taylor expansion of its primary function(2.8)is uniformly convergent and bounded by M on the district of|u|≤ R0of C.So the formula(2.9)defines an analytic function on the ball kwkρ≤ R0of Lρ,band we have

where P2k∈P2kis a homogeneous polynomial of degree 2k.By Cauchy integral formula,(2.10)and(2.16),we have

Finally,we will give some useful estimates of the polynomial space which are used in the following section.And we note that the zero momentum will play an important role in the estimates.

Proposition 3.1 For k≥ 2 and ρ>0,we obtain Pk? Hρ.Moreover,for any homogeneous polynomial F∈Pk,of degree k,we get the following two estimates:

and

Proof Let

We obtain

where k·kl1denotes the l1-norm of vector.Thus(3.7)is proved.

To prove the second estimate,let us take ?∈ Z,by using the zero momentum condition,we have

So we have the estimate

Due to j1± j2± ···±jk?1= ±?,then we have the inequality

Therefore,after summing in j1,···,jk?1and ?,we have

So(3.8)is also proved.

Proposition 3.2 If F and G are homogeneous polynomials of degree k and ? respectively in Pkand P?,then{F,G} ∈ Pk+??2and we have the estimate

ProofNow we assume that F and G are homogeneous polynomials of degrees k and ?respectively and with coefficients ak,k ∈ Ikand bl,l∈ I?.It is clear that{F,G}is a homogeneous polynomial of degree k+??2 satisfying the zero momentum condition.Furthermore,we can write

where cjis expressed as a sum of coefficients akblfor which there exists a j∈Z such that

and such that if for instance j=k1and ?j= ?1,we necessarily have(k2,···,kk,?2,···,??)=j.Hence,for a given j,the zero momentum condition on k and on l determines the value of j which in turn determines 2min{k,?}?1possible values of j.

This proves(3.9)for monomials.The extension to polynomials follows from the definition of the norm(3.5).

The last assertion and the fact that the Poisson bracket of two real Hamiltonian is real follow immediately from the definitions.

4 Recursive Equation and Normal Form Results

In this section,let us firstly introduce the definition of the N-normal form.For r≥4 and j=(j1,···,jr)∈ Zr,we denote the third largest integer amongst|j1|,···,|jr|by μ(j),and set

Definition 4.1(N-Normal Form)Let N be an integer.We say that a polynomial W∈Pkis in N-normal form if it can be written as

Namely,W contains either monomials depending only on the actions or monomials whose indices j satisfyμ(j)>N,i.e.,monomials involving at least three modes with index greater than N.

4.1 Recursive equation

Firstly,we give a lemma which is an easy consequence of the nonresonance condition and the definition of the normal forms.

Lemma 4.1 Firstly,we suppose that the nonresonance condition(6.6)is satisfied,and let N be fixed.Also suppose thatis the integrable part of Hamiltonian(2.5)and Q is a homogenous polynomial of degree n.Then the homological equation

admits a polynomial solution(χ,W)homogenous of degree n such that W is in N-normal form,and such that

where ?(j)is given in(3.1).We then define

In view of(6.6),this leads to(4.2).

In the following section,we will introduce the recursive equation.The solutions of recursive equation can generate a canonical transformation Φ such that in the new variables,the Hamiltonian H0+F is in normal form modulo a small remainder term.To obtain the recursive equation,we consider the following problem.

Recall that for two Hamiltonian functions χ and K,we have for all k ≥ 0,

where adχK={χ,K}.Moreover,if K and L are homogeneous polynomials of degree respectively n and ?,then{K,L}is a homogeneous polynomial of degree n+ ?? 2.Hence,we obtain

by using the Taylor formula,where Orstands for any smooth function R satisfying ?αR(0)=0 for all α ∈ NZwith|α|≤ r.On the other hand,we know that for ζ∈ C,the following relation holds:

where Bkare the Bernoulli numbers defined by the expansion of the generating function.Hence,by defining the two differential operators

we get

where Cris a differential operator satisfying

Applying Brto the two sides of(4.4),we obtain

Plugging the decompositions in homogeneous polynomials of χ,W and F in the last equation and equating the terms of same degree,after a straightforward calculation,we obtain the recursive equations

where

In the last sum,?i≤ n?2k appears as a consequence of ?i≥ 4 and ?1+···+?k+1=n+2k.Once these recursive equations solved,we define the remainder term asBy construction,R is analytic on a neighborhood of the origin in Lρ,band R=Or.So,by the Taylor’s formula,

Lemma 4.2 Suppose that the nonresonance condition(6.6)is fulfilled for some constants γ,ν.Then there exists C>0 such that for all r,N,and for n=4,···,r,there exist two homogeneous polynomials χnand Wnof degree n,with Wnin N-normal form,which are solutions of the recursive equation(4.5)and satisfy

Proof We define χnand Wnby induction using Lemma 4.1.Note that(4.8)is clearly satisfied for n=4,provided C big enough.Estimate(4.2)yields

Using the definition(4.6)of the term Qnand the estimate on the Bernoulli numbers,|Bk|≤k!Ckfor some C>0,together with(3.9),which implies that for all ?≥ 4,kadχ?Rk ≤2min{n,?}?1n?kRkkχ?k for any polynomial R of degree less than n,we have for all n ≥ 4,

where

and C is a constant.It is easy to know C(n,k)≤4n.

We set βn=n(kχnk+kWnk).Equation(4.9)implies that

for some constant C independent of n.

where

and

where C depends on M and R0.It remains to prove by induction thatAssume that,j=4,···,n ? 1.Then for C>1,we have

so we get

provided C>2.

Using(4.14)again and the induction hypothesis,we obtain

for n≥5 and after adapting C if necessary.

4.2 Normal form result

For any R1>0,we set Bρ(R1)={w ∈ Lρ,b|kwkρ

Theorem 4.1 Suppose that F is analytic on a ball Bρ(R1)for some R1>0 and ρ >0.Also suppose that the nonresonance condition(6.6)is satisfied,and let β1 be fixed.Then there exists a constant ?0>0 such that for all ?< ?0,there exist a polynomial χ,a polynomial W in N-normal form,and a Hamiltonian R analytic on Bρ(M?),such that

Furthermore,for all w ∈ Bρ(M?),

Proof Using Lemma 4.2,for all N and r,we can construct polynomial Hamiltonians

with W in N-normal form,such that(4.15)holds with R=Or.Now for fixed ?>0,we choose

This choice is motivated by the necessity of balance between W and R in(4.15).The error induced by W is controlled as in Lemma 5.2,while the error induced by R is controlled by Lemma 4.2.By(4.8),we get

and thus

for ? small enough.

Similarly,we have for all k≤r,

and

for ? small enough.Similar bounds clearly hold for,which shows the first estimate in(4.16).

On the other hand,using adχ?kH0=W?k+Q?k(see(4.5))and then combining Lemma 4.2 with the definition of Qn(see(4.6)),we can obtain

where the last inequality proceeds as in(4.17).Therefore,due to(4.7),(4.17)and,we obtain by Proposition 3.1 that for w ∈ Bρ(M?),

Since r=|ln?|β>2(? small enough),we getfor w ∈ Bρ(M?).

5 Proof of the Main Result

Firstly,before giving the proof of the main theorem,we will introduce two important lemmas.

Lemma 5.1(see[15])Let f:R→R+be a continuous function and y:R→R+be a differentiable function satisfying the inequality

Then we have the estimate

Fix N>1.For all w ∈ Lρ,b,we set

Notice that if w ∈ Lρ+μ,b,then

Lemma 5.2 Let N∈N and k≥4.Suppose that W is a homogeneous polynomial of degree k in N-normal form.Let w(t)be a real solution of the flow generated by the Hamiltonian H0+W.Then we have

and

Proof Fix j∈Z and let Ij(t)=wj(t)w?j(t)be the actions associated with the solution of the Hamiltonian system generated by H0+W.Due to(3.9),we can obtain

Then using Lemma 5.1,we can get

Ordering the multi-indices in such a way that|j1|and|j2|are the largest,and making use of the fact that w(t)is real,we have,after summation in|j|>N,

Inequality(5.3)can be proved in the same way.

Now we are in position to prove the main theorem in Section 1 in which we will take advantage of the bootstrap argument.

Proof of Theorem 1.1 Let u0,v0∈ A2ρwith?,and denotes by w(0)the corresponding sequence of its Fourier coefficients which belongs towithby Lemma 2.1.Let w(t)be the local solution in Lρ,bof the Hamiltonian system associated with H=H0+F.

Let χ,W and R given by Theorem 4.1 with M=cρand let y(t)=(w(t)).We recall that sincethe transformationis close to the identity,and thus,for ? small enough,we haveSpecially,note that

Let T?be the maximum of time T such thatand ky(t)kρ≤cρ? for all|t|≤ T?.By construction,we have

So using(5.2)for the first vector field and(4.16)for the second one,we get for|t|≤ T?,

where in the last inequality we have used

Using Lemma 4.2,we then verify

and thus,for ? small enough,

Similarly,we obtain

In view of the definition of T?,(5.6)–(5.7)imply.Specially,kw(t)kρ≤ 2cρ? for.Using(2.18),we finally obtain(1.4).

6 Appendix

In this section,we will give some technical lemmas and nonresonance condition.This section can be also find in[20].

Lemma 6.1 For any K ≤ r,consider K indexes j1< ···

It holds

Proof By explicit computation,one has

Substituting(6.3)into the right hand site of(6.1),we get the determinant to be estimated.To obtain the estimate factorize from the j-th column term λj=(j2+m)12,and from the n-th row term.Forgetting the essential power of?1,we obtain that the determinant to be estimated is given by

where we denoted by xj=(j2+m)?1.The last determinant is a Vandermond determinant whose value is given by

Now we have

with a suitable C.So(6.4)is estimated by

from which,using the asymptotics of the frequencies,the thesis immediately follows.

Next we need the lemma from[8,Appendix B].

Lemma 6.2 (see[8])Let u(1),···,u(K)be K independent vectors with ku(i)kl1≤ 1.Let w ∈ RKbe an arbitrary vector.Then there exists i∈ [1,···,K],such that

where det(u(i))is the determinant of the matrix formed by the components of the vectors u(i).

Proof The proof can be found in the proposition of Appendix B in[8].

Combining Lemmas 6.1 and 6.2,we deduce the following lemma.

Lemma 6.3 Let w∈Z∞be a vector with K components different from zero,namely those with indices j1,···,jK.Assume that K ≤ r and j1< ···

where λ =(λj1,λj2,···,λjK)is the frequency vector.

From[24]we learn the following lemma.

Lemma 6.4 Suppose that g(m)is r times differentiable on an interval J?R.Let Jγ:={m ∈ J||g(m)|< γ},γ >0.If|g(r)(m)|≥ d>0 on J,then|Jγ|≤ Mγ1r,where M:=2(2+3+ ···+r+d?1).

Proof The proof can be found in[24,Lemma 2.1].

Nonresonance condition In order to control the divisors(3.1),we need to impose the nonresonance condition on the linear frequencies λj,j ∈ Z.

Recall that ?(j)=sgnj1λ|j1|+sgnj2λ|j2|+ ···+sgnjrλ|jr|.We define a set

and let k=(k?)?∈N,where

Recalling the definition ofμ(j)in Section 4,then we have the following proposition.

Proposition 6.1 For a given positive number N,there exists a set J satisfying Meas([m0,?]?J)→0 as N→+∞,such that for any m∈J,

where|k|≤ r+2,ε1,ε2∈ {?1,0,1},|j1|,|j2|>N and μ(j)

Proof For a given positive number N,we define the resonant setby

where|k|≤ r+2,ε1,ε2∈ {?1,0,1},|j1|,|j2|>N.

Case 1 ε1= ε2=0.

where the last inequality is based on N>r and C?1N<1 if N is large enough.Here|·|denotes the Lebesgue measure of set and C is a constant in Lemma 6.3.We set

Then we have

Case 2 ε1= ±1,ε2=0 or ε1=0,ε2= ±1 or ε1ε2=1.

In this case,we take ε1= ±1,ε2=0 without loss of generality.Denote the resonant.Due to,one has

when|j1|≥2(r+2)N+1.Then the resonant Rkj1is empty.So we only consider|j1|<.Settingin place ofandin place of N,then according to Case 1,we have

Case 3 ε1ε2= ?1.

In this case,we take ε1=1,ε2= ?1 without loss of generality.

Therefore,

with

and a=j1?j2.Then we get

Hence,

If j>j0,we get

Then it is sufficient to consider

and let

In view of(6.9)–(6.11),we obtain

Let J=[m0,?]?R.Then the proposition is proved.

AcknowledgementsThe authors would like to thank Hongzi Cong for his invaluable discussions and suggestions.The authors would also like to thank the anonymous referee for helpful comments and suggestions.