Nonlinear Analysis 71 (2009) 6084–6092

Contents lists available at ScienceDirect

Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Stability of some waves in the dissipative Boussinesq systemClaudia VallsDepartamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

a r t i c l e i n f o

Article history:Received 2 April 2008Accepted 19 May 2009

MSC:primary 35Qxx37Kxx

Keywords:Boussinesq systemNormal formsHamiltonian formalism

a b s t r a c t

In this paper we study analytically a class of waves in the variant of the classical dissipativeBoussinesq system given by

∂tu = −∂xv − α∂xxxv + β∂xxtu− ε∂x(uv),∂tv = −∂xu+ c∂xxxu+ β∂xxtv − εv∂xv,

where β, c > 0, ε is a small parameter and α ∈ (0, 1). This equation is ill-posed and mostinitial conditions do not lead to solutions. Nevertheless, we show that, for almost every β ,c and almost every α ≤ 1, it contains solutions that are defined for large values of time andthey are very close (of order O(ε)) to a linear torus for long times (of order O(ε−1)). Theproof uses the fact that the equation leaves invariant a smooth center manifold and for therestriction of the system to the center manifold, uses arguments of classical perturbationtheory by considering the Hamiltonian formulation of the problem, the Birkhoff normalform and Neckhoroshev-type estimates.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we consider the variant of the classical Boussinesq system given by

∂tu = −∂xv − α∂xxxv + β∂xxtu− ε∂x(uv),∂tv = −∂xu+ c∂xxxu+ β∂xxtv − εv∂xv,

(1)

where α, β, c > 0 and ε is a small parameter. It is proved in [1–3] that this system has the capacity to capture the maincharacteristics of the flow in an ideal fluid. More concretely, it provides a realistic study of two-directional waves on thesurface of an inviscid fluid in a flat channel without excluding the effects of wave interactions and/or wave reflections. Thismodel has been extensively studied by several authors. In particular, it was studied the well-posedness of the initial-valueproblem (locally in time) for the linear part, the integrability of the system, the existence of solitary-wave solutions andperiodic solutions as well as the existence of accurate and efficient numerical schemes for approximating solutions of theinitial boundary-value problems. See for instance [1,2,4–9] for details and further references.In the previouswork [10], the author proved the existence of periodic and quasiperiodic solutions (with two frequencies)

for system (1) that generalizes the linear oscillations of the normal flow to the complete system. The present paper dealswithdescribing the phase space near those solutions, more concretely, it concernswith the stability of suchmotions emphasizingtheir spatial features and the transfer of energy between the various modes.These solutions are obtained inside a certain invariant center manifold for the dynamics, which turns out to be finite-

dimensional. The approach of reducing to a center manifold can be traced back to the work of Kirchgässner [11] and issometimes called Kirchgässner reduction. However, the application of this approach strongly depends on the desired results

E-mail address: cvalls@math.ist.utl.pt.

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.05.073

C. Valls / Nonlinear Analysis 71 (2009) 6084–6092 6085

and appropriate additional techniques may need to be developed in each particular case, particularly due to the infinite-dimensional nature of the problems. For later development and applications in the context of elliptic problems on cylindricaldomains we refer the reader to the works [12–17] and to the references therein.Once we have proven the existence of a center manifold, we will study the dynamics of the restriction to this center

manifold. To do it, we proceed as follows: first we observe that system (1) can be written as a Hamiltonian in infinitelymany coordinates, which turns out to be real analytic near the origin. The theoretical tool will be the use of a Birkhoffnormal form calculation to construct a suitable normal form up to third order of the Hamiltonian restricted to the centermanifold. Then, we will prove the existence of quasiperiodic solutions for this transformed Hamiltonian truncated at orderthree andwewill show that they have a strong stability property with respect to the complete dynamics: they evolve slowly(O(ε)) for long times (up to O(1/ε)) (see Theorem 3.1 and (12)). These stability estimates will be obtained by bounding theremainder of this normal form.A technical difficulty when one computes these normal forms is the existence of small divisors at each step of the normal

form procedure. More concretely, if one computes more steps in the normal form process in order to kill more terms inthe part of the remainder that obstructs the existence, in the center manifold, of the corresponding invariant torus for thecomplete Hamiltonian system and thus obtain better estimates for the diffusion time around that torus, we need to haveuniformly bounded away from zero the combinations of intrinsic frequencies and normal eigenvalues that appear in thedivisions of the series appearing in the normal form process. This implies so tedious calculations that is the main reasonwhy in this paper we deal only with the normal form up to the third degree for which we can control very well those smalldivisors, and also we restrict the values of the parameter α to the interval α ∈ (1/9, 1/4) and the values of the parametersc and β to the set {c, β > 0 : β ∈ (0, (1+

√2√3+ 2c)/(2c))}.

The restriction ofαwill imply (see Section 3 for further explanations) thatwewill study only the stability properties of el-liptic toriwith two frequencies. Again one could dealwith a larger interval of values ofα and obtain the stability properties ofelliptic toriwithmore than two frequencies but thiswill again imply very tedious calculations andwedonot compute it here.We recall that there is an extensive literature concerning with the problem of finding periodic solutions, quasiperiodic

solutions and studying their stability/dynamical properties for different partial differential equations, we refer the readerto the works [18–31] and the references therein. However, due to the infinite-dimensional nature of the problems, thetechniques applied can be of very different nature since they depend strongly on each particular case.The paper has been organized in the following way: Section 2 summarizes the main known results concerning with this

model. Section 3 deals with the details concerning with the normal form and the bounds on the ‘‘diffusion’’ time, and finallywe have included an Appendix that contains some basic lemmas used along Section 3.

2. Summary

Here we have included a technical description of the problem and some known results from [10] that will be used alongthe paper. We have omitted the proofs since they are all contained in [10].

2.1. Notation and existence of center manifolds

We consider the system (1) on the finite x-interval [0, 2π ], with the periodic boundary condition u(t, 0) = u(t, 2π),v(t, 0) = v(t, 2π) for 0 ≤ t < ∞ and zero mean,

∫ 2π0 u(t, x)dx = 0,

∫ 2π0 v(t, x)dx = 0. Furthermore, we rewrite it as a

Hamiltonian system. Namely,

H(u, v) = −12

∫ 2π

0

[u2 + v2 − α(∂xv)2 + c(∂xu)2 + εuv2

]dx, (2)

the system (1) can be written in the form

∂t(u, v) = J∇(u,v)H(u, v) with J(uv

)=

((1− β∂2x )

−1∂xv

(1− β∂2x )−1∂xu

). (3)

We will study system (3) on the space Hm−10 ([0, 2π ]) × Hm−10 ([0, 2π ]), where Hm0 ([0, 2π ]) is the Sobolev space offunctions on [0, 2π ]with periodic boundary conditions and zero mean. Moreover,m ≥ 1. We will work with the variablesu(t, x) and v(t, x) decomposed in Fourier series with respect to the x variables. We will focus our attention to the variablesof the form

u(t, x) =∑k≥1

λk(t) cos(kx), v(t, x) =∑k≥1

bk(t) sin(kx). (4)

The variables u(t, x) and v(t, x) correspond to an invariant subspace of (formal) solutions of system (1). We note that thefact that we restrict to this subset guarantees that the Birkhoff normal form computed below is not resonant.Note that the linear variational equation of (1) is

(1− β∂2x )∂tu = −∂xv − α∂xxxv, (1− β∂2x )∂tv = −∂xu+ c∂xxxu.

6086 C. Valls / Nonlinear Analysis 71 (2009) 6084–6092

with the frequencies

νk(α, β, c) = ±iνk(α, β, c), νk(α, β, c) =k

1+ βk2√1− αk2

√1+ ck2, k ≥ 1.

We remark that there is a finite number of coefficients for which νk(α) is purely imaginary: the values of k such thatk ∈ Λ(α), where

k(α) := [1/√α], Λ(α) := {k : 1 ≤ k ≤ k(α)},

with [·] indicating the integer part of a real number. Then, we have elliptic behavior on tori of dimension k(α). Restrictingthe values of α to the full Lebesgue measure set S = {α > 0, νk(α) 6= 0, ∀k ≥ 1}, in [10] applying the work of Mielkein [16, Chapter 2], we can prove for each r ≥ 1 the existence of a C r center manifold around the origin,Wc , for Eq. (1). Welist here some important properties of the center manifold:

(1) The space T0Wc is spanned by the eigenvectors of the linear part of (1) with purely imaginary eigenvalues. Thus, therestriction of the dynamics to the center manifold is finite-dimensional. We note that the center manifold is preciselywhere one can expect to have invariant tori persisting, and thus, some kind of stability. Since in our case the dynamicsrestricted to the center manifold is finite-dimensional this will allow us to use (finite-dimensional) perturbativetechniques to establish this stability.

(2) The dynamics on the centermanifold isHamiltonianwith aHamiltonian that coincideswith the restriction of the originalone in (2) to the center manifold (see [16] for details).

2.2. The Hamiltonian formalism

We consider the Hamiltonian for Eq. (1). We will work with the variables u(t, x) and v(t, x) as in (4). From (2) and byParseval’s identity, we obtain that the Hamiltonian H(u, v) can be written as

−12

∫ 2π

0

[u2 + v2 − α

(∂v

∂x

)2+ c

(∂u∂x

)2+ εuv2

]dx = H2(λ, b)+ εH3(λ, b),

with λ = (λk)k≥1, b = (bk)k≥1 and H2, H3 given by

H2 = −π

2

∑k≥1

(λ2k(1+ ck

2)+ b2k(1− αk2)),

H3 =π

4

∑k,l≥1

∑σ∈{0,1}

(−1)σ+1bkblλklσ ,

where we have introduced the notation klσ = |k+ (−1)σ l|, with σ ∈ {0, 1}. As pointed out in [10], it is very convenient to

simplify H2, such that it has a diagonal form. To do it, we set

w1,k(c) = 1+ ck2, w2,k(α) = 1− αk2, Tk(β) = 1+ βk2

and we introduce the variables

qk =

√k

2Tk(β)

(bkw

1/42,k (α)

w1/41,k (c)

− iλkw

1/41,k (c)

w1/42,k (α)

),

pk =

√k

2Tk(β)

(λkw

1/41,k (c)

w1/42,k (α)

− ibkw

1/42,k (α)

w1/41,k (c)

).

With respect to those variables, we can define the Poisson bracket of two functionals S(p, q) and R(p, q) as

{S, R} =∑n≥1

{∂S∂pn

∂R∂qn−∂S∂qn

∂R∂pn

}and in the variables p, q,

H2 = −iπ∑k≥1

µk(α, β, c)pkqk, with µk(α, β, c) =

√w2,k(α)w1,k(c)Tk(β)

k. (5)

Furthermore, setting N1 = {0, 1} × N2, and taking the notation

T (β, k, l, klσ ) =

√Tk(β)Tl(β)Tklσ (β)

klklσ,

C. Valls / Nonlinear Analysis 71 (2009) 6084–6092 6087

then in the variables p, q

H3 =π

8√2

∑(σ ,k,l)∈N1

(−1)σw1/41,k (c)w

1/41,l (c)w

1/42,klσ(α)

w1/42,k (α)w

1/42,l (α)w

1/41,klσ(c)T (β, k, l, klσ )

×

{pkplpklσ + pkqlqklσ + qkplqklσ − qkqlpklσ + i

[pkplqklσ − qkqlqklσ − qkplpklσ − pkqlpklσ

]}. (6)

2.3. The process of normal form: A formal description

In this subsection we compute the normal form up to third order of the reduction to the center manifold for theHamiltonian H = H2 + εH3 (introduced in (5) and (6)). This means that we want to compute a normal form such thaton the center manifold H = H2 + ε2R1 where R1 contain the terms with degree greater than or equal to four.The process of reducing to the centermanifold is based on removing somemonomials in the expansion of theHamiltonian

in order to produce an invariant manifold tangent to the elliptic directions of H2, that is, the modes with indices k ∈ Λ(α).To do it, and since we already have proven the existence of a center manifold, we just need to cancel the monomials in H3with at most one hyperbolic direction (i.e., the monomials with k > k(α)) and that in this direction have degree one (thisensures that when restricted to the elliptic directions (i.e., the monomials with k ≤ k(α)), the Hamiltonian H = H2 + εH3exhibits a decoupling in these two directions and approximates the dynamics in the center manifold up to third degree.Furthermore, the monomials in H3 with all elliptic directions are in normal form). Indeed, considering that H = H2 + εH3with H2 given in (5) and the monomials pk, qk with k > k(α) in H3 have degree greater than or equal to two, then setting inH , pk = qk = 0 for k > k(α), we get that the equations of motion coming from H satisfy pk = qk = 0 for k > k(α). Thus,the set

C = {pk = qk = 0 for k > k(α)}

is invariant under the Hamiltonian flow and H restricted to C only contain monomials pj, qj with j ≤ k(α) and thus,represents the dynamics inside the center manifold up to third degree. Since we already have proved the existence of thecenter manifold, and in this section we just want to compute it approximately up to third degree, together with the fact thatto compute a term in the expansion of the center manifold we only need to work with a finite number of Fourier modes(which is an algebraic expression of the coefficients of the Hamiltonian), we can proceed formally ignoring questions ofdomains of the operators and convergence of series.In view of the explanation given above, to produce a HamiltonianH3 of order threewith atmost one hyperbolic direction,

that in this direction has degree one and that all the elliptic directions inH3 are in normal form,we need to cancel, if possible,in H3 all the monomials such that for any σ ∈ {0, 1}2, have indices (k, l, klσ ) ∈ Γ3(α) where Γ3(α) =

⋃4j=1 Γ3,j(α) with

Γ3,1(α) = {(k, l, klσ ) ∈ Λ3(α)} and

Γ3,2(α) = {(k, l, klσ ) ∈ N3 : (k, l) ∈ Λ2(α)} \ Γ3,1(α)

Γ3,3(α) = {(k, l, klσ ) ∈ N3 : (k, klσ ) ∈ Λ2(α)} \ Γ3,1(α)

Γ3,4(α) = {(k, l, klσ ) ∈ N3 : (l, klσ ) ∈ Λ2(α)} \ Γ3,1(α).

We take Γ1 = X tG3|t=1 the time 1-map of the flow of the Hamiltonian vector field XG3 given by the Hamiltonian G3 := εG3

defined as:

G3 =i

8√2

∑σ=0,1

∑(k,l,klσ )∈Γ3(α)

(−1)σ+1w1/41,k (c)w

1/41,l (c)w

1/42,klσ(α)

w1/42,k (α)w

1/42,l (α)w

1/41,klσ(c)T (β, k, l, klσ )

×

{C−1,−1,−1k,l,klσ

pkplpklσ + C−1,1,1k,l,klσ

pkqlqklσ + C1,−1,1k,l,klσ

qkplqklσ − C1,1,−1k,l,klσ

qkqlpklσ

+ i[C−1,−1,1k,l,klσ

pkplqklσ − C1,1,1k,l,klσ

qkqlqklσ − C1,−1,−1k,l,klσ

qkplpklσ − C−1,1,−1k,l,klσ

pkqlpklσ

]}(7)

where, for a fixed (k1, k2, k3) ∈ N3, in the notation given in (5),

Cn,m,lk1,k2,k3=

1nµk1(α, β, c)+mµk2(α, β, c)+ lµk3(α, β, c)

.

Now we introduce some notation: we say that h ∈ M(Γ3(α)) if hk,l,klσ = 0 when (k, l, klσ ) 6∈ Γ3(α). With this notation,

we clearly have G3 ∈ M(Γ3(α)). Furthermore, we can write H2 = H2,1 + H2,2 and H3 = H3,1 + H3,2, where for j = 2, 3,

6088 C. Valls / Nonlinear Analysis 71 (2009) 6084–6092

Hj,1 ∈ M(Γ3(α)), and Hj,2 ∈ Mc(Γ3(α)) (here Mc(Γ3(α)) denotes the complementary of M(Γ3(α))). Then it is provedin [10] that with the definition of G3 given in (7), we get

H∗ = H ◦ X G31 = H2,1 + H2,2 + εH3,1 + εH3,2 + ε{H2,1,G3} + ε{H2,2,G3}

+ ε2{H3,G3} + ε2∫ 1

0(1− t){{H2 + εH3,G3},G3} ◦ X

G3t dt

= H2,1 + H2,2 + εH3,2 + ε{H2,2,G3} + ε2{H3,G3} + ε2∫ 1

0(1− t){{H2 + εH3,G3},G3} ◦ X

G3t dt

:= H2,1 + H2,2 + εH3,2 + ε{H2,2,G3} + ε2({H3,G3} + R3). (8)

Now,wehave all the ingredients to state the result proved in [10].Wedenote byΠWc (·) the restriction to the centermanifold.

Theorem 2.1 ([10]). For almost every α > 0, the formal change of variables Γ1 takes the reduction of H to the center manifold,ΠWc (H), into its normal form up to third order,ΠWc (H

∗), i.e.,

ΠWc (H∗) = ΠWc (H2)+ ε

2ΠWc ({H3,G3} + R3) := ΠWc (H2)+ ε2ΠWc (R3).

We first note that the restriction of H2 to the center manifold corresponds to set in (5), pj = qj = 0 for j ≥ 3. FurthermoreH3,2 and {H2,2,G3} do not belong to the center manifold, and since this center manifold is invariant under the dynamics,from (8),ΠWc (R3) is equal to

ΠWc

({ΠWc (H3),G3} +

∫ 1

0(1− t){{ΠWc (H),G3},G3} ◦ X

G3t dt

).

3. Normal form and stability

In this sectionwe include the technical details of the normal formprocesswith rigorous bounds on the remainder, aswellas bounds on the diffusion time around any elliptic torus inside the center manifold. As pointed out in the introduction anddue to technicalities (see Lemma3.2), we restrict the values ofα to the set [1/9+δ1, 1/4−δ2]with δ1, δ2 > 0, δ1+δ2 < 5/36and the values of the parameters c, β to the set

Dc,β =

{c, β > 0 : β ∈

(0,1+√2√3+ 2c

2c

)}. (9)

The restriction of α implies that, in this case, k(α) = 2 and thus Λ(α) = {k = 1, 2}. This means that in this paper we willbound the diffusion time around a torus with two frequencies and that the center manifold is four dimensional. As pointedout in the introduction, some results may also be true for almost every α > 0 for which k(α) > 2 obtaining bounds onthe diffusion time around torus with more than two frequencies. However the process imply very tedious calculations andtherefore we do not compute them here.We note that since k(α) = 2, then in the definition of Γ3(α)we get

Γ3,1(α) = {(1, 2, 1), (1, 1, 2), (2, 1, 1)}, Γ3,2(α) = {(1, 2, 3), (2, 1, 3), (2, 2, 4)},Γ3,3(α) = {(1, 3, 2), (2, 3, 1), (2, 4, 2)}, Γ3,4(α) = {(3, 1, 2), (3, 2, 1), (4, 2, 2)}.

(10)

3.1. Notation

We first consider some notation. Let p = (p1, . . . , p4), q = (q1, . . . , q4) and consider functions f = f (p, q) defined onD(R) for some real number 0 < R < 1, where

D(R) = {(p, q) ∈ C4 × C4 : |p| ≤ R, |q| ≤ R},

and | · | denotes the infinity norm of a complex vector. We introduce the multi-index notation. If f is analytic, we denote

f =∑

(l,j)∈N4×N4fl,jplqj, and ‖f ‖R =

∑(l,j)∈N4×N4

|fl,j|R|l|1+|j|1

with |l|1 =∑4k=1 lk, and |j|1 =

∑4k=1 jk. Some basic properties of this norm are given in the Appendix. We just note here

that when they converge they are bounds of the supremum norms of f on D(R).If f ∈ C j with j ≥ 1, then we will denote by ‖f ‖C j(R) its norm on the domain D(R), that is,

‖f ‖C j(R) = sup(p,q)∈D(R)

j∑k=0

|Dkf (p, q)|.

C. Valls / Nonlinear Analysis 71 (2009) 6084–6092 6089

We also introduce the action angle variables I = (I1, . . . , I4), θ = (θ1, . . . , θ4)with (Ik, θk)k=1,...,k defined by

pk =√Ikjeikθk , qk = −i

√Ikje−ikθk , k = 1, . . . , 4. (11)

3.2. Main result and ideas

The main result of this section is the following that ensures the stability of elliptic torus inside the center manifold.

Theorem 3.1. We consider the Hamiltonian H as in (5)–(6) restricted to the center manifold defined on D(R0) for some 0 <R0 < 1. Then, there exist constants C1 > 0, C2 > 0 such that for ε sufficiently small any solution of the Hamiltonian equations ofmotion given byΠWc (H) that at t = 0 starts in D(Re

−δ), with δ > 0, satisfies

‖I(t)− I(0)‖C3(Re−6δ) ≤ C2ε for any 0 ≤ t ≤ T0 :=C3ε

(12)

with C3 = e3δ/(2C1R‖ΠWc (R3)‖C3(Re−3δ)).

We observe that the smallness condition of ε is explicitly given in Theorem 3.3 (see (14)).Theorem 3.1 ensures the long time stability of any real trajectory close to an elliptic torus inside the center manifold. The

main idea of the proof is first to estimate the domain of the canonical transformation provided by Theorem 2.1 and also tobound its remainder. This will provide the distance between the original and the transformed variables. Then, we will findtimes for which the trajectories of the equations provided by the Hamiltonian ΠWc (H

∗) (i.e., already in normal form) staywithin the domain of the transformation and we will estimate the drift of the actions.

3.3. Bounding the remainder of the normal form

To show that the construction of the normal form given in Theorem 2.1 is not formal, we are going to prove the well-defined character of the transformation Γ1 = X1G3 and bound the remainder ε

2ΠWc (R3). This is done in Theorem 3.3.For this purpose, we first need to bound the small divisors that appear in (7). The key tool is the observation in the

following lemma that all the relevant divisors in (7) are independent of α and uniformly bounded away from zero. Fortechnical reasonswewill have to restrict the values of the parameterα to [1/9+δ1, 1/4−δ2]with δ1, δ2 > 0, δ1+δ2 < 5/36,and the parameters c, β to Dc,β (see (9)).

Lemma 3.2. For each δ1, δ2 > 0, δ1 + δ2 < 5/36, any (k, l, klσ ) ∈ Γ3(α) with α ∈ [1/9 + δ1, 1/4 − δ2], c, β ∈ Dc,β andj1, j2, j3 ∈ {0, 1}3 satisfy∣∣∣C (−1)j1 ,(−1)j2 ,(−1)j3

k,l,klσ

∣∣∣ ≤ max{ 1√δ1,1√δ2,

6

(√7−√5)√1+ 4c(1+ 4β)

}.

Proof. We consider three different cases.Case 1: (k, l, klσ ) ∈ Γ3,2(α). In this case, since µk ∈ R, µl ∈ R and µklσ ∈ iR it holds that for any (k, l, k

lσ ) as in (10) and

j1, j2, j3 ∈ {0, 1}3 we get∣∣∣C (−1)j1 ,(−1)j2 ,(−1)j3k,l,klσ

∣∣∣ ≤ 1|µklσ |

=1√

α − 1(klσ )2(α)

√1+ c(klσ )2(α)(1+ β(klσ )2(α))

≤1√

α − 1(klσ )2(α)

≤1

√α − 1/9

=1√δ1.

Case 2: (k, l, klσ ) ∈ Γ3,j(α)with j = 3, 4. In this case, the proof of the lemma follows analogously as in Case 1 (see also (10)).

Case 3: (k, l, klσ ) ∈ Γ3,1(α). In this case, µk, µl, µklσ ∈ R and to bound C (−1)j1 ,(−1)j2 ,(−1)j3

k,l,klσfor any (k, l, klσ ) as in (10) and

(j1, j2, j3) ∈ {0, 1}3 it is enough to bound the three quantities C1,1,11,1,2 , C

1,−1,11,1,2 , C

1,1,−11,1,2 . Clearly,∣∣∣C1,1,11,1,2

∣∣∣ = 1

2√1− α

√1+ c(1+ β)+

√1− 4α

√1+ 4c(1+ 4β)/2

≤1

2√1− α +

√1− 4α/2

≤1

2√3/4+ δ2 +

√δ2≤

1√δ2.

6090 C. Valls / Nonlinear Analysis 71 (2009) 6084–6092

Furthermore,∣∣∣C1,−1,11,1,2

∣∣∣ = 2√1− 4α

√1+ 4c(1+ 4β)

≤2

√1− 4α

≤2

√1− 4(1/4− δ2)

≤1√δ2.

Finally∣∣∣C1,1,−11,1,2

∣∣∣ = 1

2√1− α

√1+ c(1+ β)−

√1− 4α

√1+ 4c(1+ 4β)/2

. (13)

To bound (13) we consider the function fc,β(x) = 2√1− x√1+ c(1 + β) − 1

2

√1− 4x

√1+ 4c(1 + 4β). For x ∈ R, with

x < 1/4, and c, β ∈ Dc,β (see (9)), we obtain that ∂xf (x) = 0 if and only if x = 0 and that x = 0 is a minimum. Furthermore,if c, β ∈ Dc,β it yields

2√1+ c(1+ β) ≥

12

√1+ 4c(1+ 4β)

and thus, for x ∈ (1/9+ δ1, 1/4− δ2) and c, β ∈ Dc,β we get that

fc,β(x) ≥ fc,β(1/9+ δ1) ≥ 2

√1−

19− δ1√1+ c(1+ β)−

12

√1−

49− 4δ1

√1+ 4c(1+ 4β)

≥12

√1+ 4c(1+ 4β)

[√89− δ1 −

√59− 4δ1

].

Now, using that since δ1, δ2 > 0, δ1+δ2 < 5/36, then 0 < δ1 < 5/36, we obtain that |C1,1,−11,1,2 | ≤ 6/(

√7−√5)√1+ 4c(1+

4β). Thus, the lemma is proved. �

The following result describes the precise bounds of the normal form procedure given in Theorem 2.1.

Theorem 3.3. We assume that ΠWc (H) is defined on D(R) for some 0 < R < 1 and we introduce Rj = Re−jδ , with δ > 0 andj ≥ 0. Then, it holds:

(1) the Hamiltonian G3 is defined on D(R) and

‖G3‖R ≤ ‖H3‖C3(R)max{1√δ1,1√δ2,

6

(√7−√5)√1+ 4c(1+ 4β)

};

(2) if we denote by X G3t the flow at time t of the Hamiltonian G3 = εG3 defined in (7), then if ε is sufficiently small so that

ε‖G3‖R ≤ R2e−δ(1− e−δ), (14)

holds, we have that X G31 , XG3−1:D(R2)→ D(R1);

(3) if we use the notation X G3t − I = (Z1, . . . , Z4,W1, . . . ,W4) := (Z,W ) then

‖Z‖R1 ≤ε‖G3‖RR(1− e−δ)

, ‖W‖R1 ≤ε‖G3‖RR(1− e−δ)

;

(4) there exists K > 0 such that ‖ΠWc (R3)‖C3(R3) is bounded by

‖{ΠWc (H3),G3}‖C3(R3) + ‖{{ΠWc (H),G3},G3}‖C3(R3)

≤K‖G3‖R

R3(1− e−δ)3

(‖ΠWc (H3)‖C3(R) +

‖ΠWc (H)‖C3(R)‖G3‖RR3(1− e−δ)3

).

Proof. The first statement of the theorem is straightforward by construction and in view of Lemma 3.2 (see (6) and (7)).Now, using Lemma A.1 we have∥∥∥∥∥∂G3∂z

∥∥∥∥∥R1

≤ε‖G3‖RR(1− e−δ)

,

∥∥∥∥∥∂G3∂w∥∥∥∥∥R1

≤ε‖G3‖RR(1− e−δ)

.

Since (14) holds, we get that∥∥∥∥∥∂G3∂z∥∥∥∥∥R1

≤ R(1− e−δ),

∥∥∥∥∥∂G3∂w∥∥∥∥∥R1

≤ R(1− e−δ),

C. Valls / Nonlinear Analysis 71 (2009) 6084–6092 6091

and thus, if (p, q) ∈ D(R2) then the biggest time t for the trajectories of the Hamilton equations for G3 to be outside D(R1)satisfies that is bigger than one. Thus, the second statement of the theorem is proved.To prove the third statement of the theorem we note that

‖Z‖R1 ≤

∥∥∥∥∥∂G3∂W

∥∥∥∥∥R1

≤ε‖G3‖RR(1− e−δ)

, ‖W‖R1 ≤

∥∥∥∥∥∂G3∂Z∥∥∥∥∥R1

≤ε‖G3‖RR(1− e−δ)

.

Finally, the fourth statement of the theorem follows directly using Lemmas A.1 and A.2 and the fact that

‖G3‖C3(R3) ≤4‖G3‖R

R3(1− e−δ)3. � (15)

3.4. Stability estimates: Proof of Theorem 3.1

An immediate consequence of Theorem 3.3 is that we can bound the diffusion speed around a linear stable torus of aHamiltonian system with the two frequencies µ1(α), µ2(α) (see (5)).Let (p(0), q(0)) ∈ D(R2). Then, by the second and third statements of Theorem 3.3 we have that if we denote

(P(0),Q (0)) = X G31 (p(0), q(0)), then (P(0),Q (0)) ∈ D(R1) and

‖p(0)− P(0)‖R2 ≤ε‖G3‖RR(1− e−δ)

, ‖q(0)− Q (0)‖R2 ≤ε‖G3‖RR(1− e−δ)

.

We introduce the variables I = (I1, . . . , I4)with Ij = iPjQj, for j = 1, . . . , 4. Then, (see also (11) for the definition of I)

‖I(0)− I(0)‖R2 ≤ ‖p(0)− P(0)‖R2‖q(0)‖R2 + ‖q(0)− Q (0)‖R2‖P(0)‖R2

≤2ε‖G3‖Re−2δ

(1− e−δ). (16)

Furthermore, since

I(t) = (P(t)Q (t)+ Q (t)P(t)) = ε2(∂ΠWc (R3)∂Q

Q +∂ΠWc (R3)

∂PP),

if we denote by T0 the maximum time for which (P(t),Q (t)) ∈ D(R3), we have

‖I(t)− I(0)‖C3(R3) ≤ T0‖I(t)‖C3(R3) ≤ 2ε2T0e−3δR‖ΠWc (R3)‖C3(R3),

with ‖R3‖C3(R3) given in the fourth statement of Theorem 3.3. Then, there exists a constant C1 > 0 such that

‖I(t)− I(0)‖C3(R3) ≤ C1ε (17)

for 0 ≤ t ≤ T0 with T0 as in (12). Now, proceeding in a similar manner as in the second and third statements of Theorem 3.3we have that since (P(t),Q (t)) ∈ D(R3) then (p(t), q(t)) ∈ D(R2) and thus, for any 0 ≤ t ≤ T0, we have

‖I(t)− I(t)‖R3 ≤2ε‖G3‖Re−3δ

(1− e−δ). (18)

Therefore, from (16)–(18), for any 0 ≤ t ≤ T0 with T0 as in (12), we obtain (see also (15))

‖I(t)− I(0)‖C3(R6) ≤ ‖I(t)− I(t)‖C3(R6) + ‖I(t)− I(0)‖C3(R6) + ‖I(0)− I(0)‖C3(R6)

≤4‖I(t)− I(t)‖R3R3(1− e−δ)3

+ ‖I(t)− I(0)‖C3(R3) +4‖I(t)− I(0)‖R2R3(1− e−δ)3

≤ C2ε,

for some constant C2. Thus, the theorem is proved.

Acknowledgements

The author wishes to thank Luis Barreira for his useful comments and remarks.Partially supported by FCT through CAMGSD, Lisbon.

6092 C. Valls / Nonlinear Analysis 71 (2009) 6084–6092

Appendix

We include two technical lemmas used in Section 3. Since the lemmas are variants of the well-known facts, the proofswill be just indicated. We continue to use the notation in Section 3.

Lemma A.1. Let f (x, y) be analytic functions on D(R). Then, for every 0 < χ < 1, we have that∥∥∥∥∂ f∂x∥∥∥∥Rχ≤

|f |R(1− χ)R

,

∥∥∥∥∂ f∂y∥∥∥∥Rχ≤

|f |R(1− χ)R

.

Proof. The proof of the lemma is done by applying Cauchy estimates to the function∑l,j∈N4×N4 |fl,j|x

jyj. �

Lemma A.2. Let us take 0 < R0 < R and let us consider analytic functions Z,W with values in C4, all defined for (z, w) ∈ D(R0).We assume that |Z |R0 and |W |R0 are both bounded by R. Let f (z

∗, w∗)z∗ = (z∗1 , . . . , z∗

4 ),w∗= (w∗1, . . . , w

∗

4) be a given analyticfunction on D(R). If we introduce

F(z1, . . . , z4, w1, . . . , w4) = f (Z1, . . . , Z4,W1, . . . ,W4),

then |F |R0 ≤ |f |R.

Proof. The proof of the lemma can be directly checked by expanding f in Taylor series and using that if f , g ∈ D(R), then‖fg‖R ≤ ‖f ‖R‖g‖R. �

References

[1] J. Bona, M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D 15 (1998) 191–224.[2] J. Bona, M. Chen, J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation andlinear theory, J. Nonlinear Sci. 12 (2002) 283–318.

[3] J. Bona, G. Pritchard, L. Scott, An evaluation of a model equation for water waves, Philos. Trans. R. Soc. Lond. Ser. A 302 (1981) 457–510.[4] R. Conte, M. Mussette, A. Pickering, Factorization of the classical Boussinesq system, J. Phys. A 27 (1994) 2831–2836.[5] W. Craig, An existence theory for water waves, and Boussinesq and Korteweg-de Vries scaling limits, Comm. PDE 10 (1985) 787–1003.[6] W. Craig, M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion 19 (1994) 367–389.[7] G. El, R. Grimshaw, M. Pavlov, Integrable shallow-water equations and undular bores, Stud. Appl. Math. 106 (2001) 157–186.[8] A. Kamchatnov, R. Kraenkel, B. Umarov, Asymptotic soliton train solutions of Kaup–Boussinesq equations, Wave Motion 38 (2003) 355–365.[9] V. Matveev, M. Yavor, Solutions presque périodiques et à N-solitons de l’équation hydrodynamique non linéaire de Kaup, Ann. Inst. H. Poincaré Sect.A (N.S.) 31 (1979) 25–41.

[10] C. Valls, Quasiperiodic solutions for dissipative Boussinesq systems, Comm. Math. Phys. 265 (2) (2006) 305–331.[11] K. Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations 45 (1982) 113–127.[12] M. Groves, J. Toland, On variational formulations for steady water waves, Arch. Ration. Mech. Anal. 137 (1997) 203–226.[13] G. Iooss, K. Kirchgässner, Water waves for small surface tension: An approach via normal form, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992) 267–299.[14] G. Iooss, A. Mielke, Bifurcating time-periodic solutions of Navier–Stokes equations in infinite cylinders, J. Nonlinear Sci. 1 (1991) 107–146.[15] A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988) 51–66.[16] A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, in: Lect. Notes in Math., vol. 1489, Springer, 1991.[17] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations 110 (1994) 322–355.[18] D. Bambusi, Lyapunov center theorem for some nonlinear PDE’s: A simple proof, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 29 (2000) 823–837.[19] D. Bambusi, S. Paleari, Normal form and exponential stability for some nonlinear string equations, Z. Angew. Math. Phys. 52 (2001) 1033–1052.[20] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 243 (2003) 315–328.[21] M. Berti, P. Bolle, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal. 56 (2004) 1011–1046.[22] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995) 629–639.[23] C. Chierchia, J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000) 497–525.[24] W. Craig, C. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993) 1409–1498.[25] A.I. Dyachenko, V.E. Zakharov, Is free surface hydrodynamic an integrable system? Phys. Lett. A 190 (1994) 114–118.[26] F. Fasso, M. Guzzo, G. Benettin, Neckhoroshev-stability of elliptic equilibria of Hamiltonian systems, Preprint.[27] G. Gentile, V. Mastropietro, Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series

method, J. Math. Pures Appl. (9) 83 (2004) 1019–1065.[28] S. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl. 21 (1987) 192–205.[29] S. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2) 143 (1996)

149–179.[30] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996) 119–148.[31] P. Panayotaros, Near-monochromatic water waves on the sphere, Preprint.