## Abstract

We study analytically a class of solutions for the elliptic equation
where *α*>0 and *ε* is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every *α*>0, it contains solutions that are defined for large values of time and they are very close (of order *O*(*ε*)) to a linear torus for long times (of order *O*(*ε*^{−1})). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

## 1. Introduction

Elliptic problems in cylindrical domains occur in many situations, such as in the study of the deformation of beams or of inviscid channel flows. As such, they are clearly scientifically relevant. These problems lead to elliptic equations in which the axial variable formally takes the role of time. This often allows one to use ideas and techniques from the theory of dynamical systems and, in turn, to reduce the study of these systems to that of certain (perhaps multi-valued) evolution operators associated with the equation. Elliptic problems in cylindrical domains may be ill-posed (in the sense that some initial conditions may not lead to solutions, or that the solutions are not sufficiently regular), but they may still have many smooth solutions. This is quite relevant, even if the smooth solutions only exist for a relatively small region on the phase space. As such, it is also clearly scientifically relevant to study ill-posed elliptic problems, and particularly whether they have ‘many’ smooth solutions and also what type of solutions. Here, we consider a particular relevant case of such a system and use ideas and techniques from the theory of dynamical systems to show that there are solutions that are defined for large values of time and are very close (of order *O*(*ε*)) to a linear torus for long times (of order *O*(*ε*^{−1})). The solutions are found inside a central manifold of infinite-dimensional dynamics. More precisely, we obtain quasi-periodic solutions on tori. We note that it is natural to look for recurrent dynamics inside an invariant submanifold that is neither stable nor unstable. The study of the existence of quasi-periodic solutions for nonlinear partial differential equations was started somewhat recently by Kuksin (1987) and Wayne (1996).

The system that we consider can be regarded as a nonlinear perturbation with a small parameter of the ill-posed version of the Klein–Gordon equation. We recall that the latter is the equation of motion of a quantum scalar field, and can be regarded as the relativistic version of the Schrödinger equation. Although we consider only a particular model, it should be emphasized that it is also a first step towards the full comprehension of ill-posed equations. Thus, it is quite reasonable to start by considering ill-posed versions of somewhat simple well-posed equations that are completely understood (even though this does not mean that the ill-posed model remains simple), instead of trying any general approach that seems completely out of reach with the present state-of-the-art.

In this paper, we consider the variant of an elliptic equation on a cylindrical domain
1.1
where *α*>0, *ε* is a small parameter and
This equation is ill-posed in the sense that ‘most’ initial conditions do not lead to a solution. In a previous work (Valls 2006), the author proved the existence of periodic and quasi-periodic solutions (with two frequencies) for system (1.1) that generalize the linear oscillations of the normal flow to the complete system. The present paper deals with the description of the phase space near those solutions; more specifically, it is concerned with the stability of such motions, emphasizing their spatial features and the transfer of energy between the various modes.

These solutions are obtained inside a certain invariant centre manifold for the dynamics, which turns out to be finite-dimensional. The approach of reducing to a centre manifold can be traced back to the work of Kirchgässner (1982) and is sometimes called the Kirchgässner reduction. However, the application of this approach strongly depends upon the results desired, and appropriate additional techniques may need to be developed in each particular case due to the infinite-dimensional nature of the problems. For later development and applications in the context of elliptic problems on cylindrical domains, we refer the reader to the works of Mielke (1988, 1991, 1994), Iooss & Mielke (1991), Iooss & Kirchgassner (1992), Groves & Toland (1997) and references therein.

Once we have proven the existence of a centre manifold, we study the dynamics of the restriction to this centre manifold. To do this, we proceed as follows. Firstly we observe that system (1.1) can be written as a Hamiltonian in infinitely many coordinates, which turns out to be real analytic near the origin. The theoretical tool will be the calculation of a Birkhoff normal form to construct a suitable normal form up to third order of the Hamiltonian restricted to the centre manifold. Then, we prove the existence of quasi-periodic solutions for this transformed Hamiltonian truncated at order three, and we 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 equation (3.4)). These stability estimates are obtained by bounding the remainder 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 substantially, if one computes more steps in the normal-form process in order to kill more terms in the part of the remainder that obstructs the existence, in the centre manifold, of the corresponding invariant torus for the complete Hamiltonian system, and thus obtain better estimates for the diffusion time around that torus, we need to have, uniformly bounded away from zero, the combinations of intrinsic frequencies and normal eigenvalues that appear in the divisions of the series appearing in the normal-form process. This implies calculations so tedious that it is the main reason why, in this paper, we deal only with the normal form up to the third degree, for which we can control very well those small divisors. We study only the stability properties of elliptic tori with two frequencies. Again, one could obtain the stability properties of elliptic tori with more than two frequencies, but this implies very tedious calculations and we do not compute it here.

We recall that there is extensive literature concerning the problem of finding periodic and quasi-periodic solutions and studying their stability/dynamical properties for different partial differential equations. We refer the reader to the works of Kuksin (1987), Craig & Wayne (1993), Dyachenko & Zakharov (1994), Bourgain (1995), Kuksin & Pöschel (1996), Pöschel (1996), Panayotaros (1998), Fasso *et al.* (1999), Bambusi (2000), Chierchia & You (2000), Bambusi & Paleari (2001), Berti & Bolle (2003, 2004), Gentile & Mastropietro (2004) and references therein. However, due to the infinite-dimensional nature of the problems, the techniques 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 this model. Section 3 deals with the details concerning the normal form and the bounds on the ‘diffusion’ time. Finally, we have included an appendix that contains some basic lemmas used in §3.

## 2. Summary

Here, we have included a technical description of the problem and some known results from Valls (2006) that are used in the paper. We have omitted the proofs since they are all contained in Valls (2006).

### (a) Notation and existence of centre manifolds

We consider the system (1.1) on the finite *x*-interval [0,1] with the periodic boundary condition *u*(*t*,0)=*u*(*t*,1), (∂*u*/∂*t*)(*t*,0)=(∂*u*/∂*t*)(*t*,1) for and zero mean, i.e. , . Furthermore, we rewrite it as a Hamiltonian system, namely, introducing the variables *w*=(*q*,*p*), with *q*(*t*,*x*)=*u*(*t*,*x*) and *p*(*t*,*x*)=(∂*u*/∂*t*)(*t*,*x*), and the Hamiltonian
2.1
The system (1.1) can then be written in the form
2.2

We study system (2.2) on the space , where is the Sobolev space of functions on [0,1] with periodic boundary conditions and zero mean. Moreover, *m*≥2. We work with the variables *q*(*t*,*x*)=*u*(*t*,*x*) and *p*(*t*,*x*)=∂*u*/∂*t*(*t*,*x*) decomposed in Fourier series with respect to the *x* variable. We recall that we are interested in solutions that are not travelling waves. In order to force that our solutions are not travelling waves, we require that *q*(*t*,*x*) be odd in *x*. This leads us to restrict our attention to variables of the form
2.3
where *β*_{k}(*t*)=*λ*′_{k}(*t*). The variables *q*(*t*,*x*) and *p*(*t*,*x*) correspond to an invariant subspace of (formal) solutions. We note that the fact that we restrict ourselves to this subset guarantees that the Birkhoff normal form computed below is not resonant.

Note that the linear variational equation of (1.1) is
The functions are the basic modes of the linear variational equation and the numbers
are the corresponding frequencies. Each solution of the linear variational equation of (1.1) is a superposition of the basic modes
in which *a*_{k} and *b*_{k} are determined by the initial data. We denote by [.] the integer part of a real number and set
Then, we have elliptic behaviour on tori of dimension *k*(*α*). We can assume that the values of *α* are such that *k*(*α*)=2, that is *α*∈[16*π*^{2},36*π*^{2}). We recall that this is not a restriction on the values of *α* since if we have system (1.1) with 0≠*α*_{1}∉[16*π*^{2},36*π*^{2}) making, for example, the rescaling
for *δ*_{2}>0 sufficiently small, we have that system (1.1) becomes
which has *k*(*α*)=2 (with *α*=36*π*^{2}−*δ*_{2}), sufficiently small and

Restricting the values of *α* to the full Lebesgue measure set §={*α*>0, ν_{k}(*α*)≠0, ∀*k*≥1}, in Valls (2006) applying the work of Mielke (1991, ch. 2), we can prove, for each *r*≥1, the existence of a *C*^{r} centre manifold around the origin, *W*_{c}, for equation (1.1). We list here some important properties of the centre manifold.

The space

*T*_{0}*W*_{c}is spanned by the eigenvectors of the linear part of (1.1) with purely imaginary eigenvalues: the values of*k*such that*k*∈*λ*(*α*) and, on those values, μ_{k}(*α*)=±*i**w*_{k}(*α*). These correspond to elliptic behaviour to a torus of dimension*k*(*α*), with*k*(*α*) being finite. Thus, the restriction of the dynamics to the centre manifold is finite-dimensional. We note that the centre manifold is precisely where one can expect to have invariant tori persisting, and thus, some kind of stability. Since, in our case, the dynamics restricted to the centre manifold is finite-dimensional, this allows us to use (finite-dimensional) perturbative techniques to establish the stability.The dynamics on the centre manifold is Hamiltonian with a Hamiltonian that coincides with the restriction of the original one in (2.1) to the centre manifold (see Mielke 1991 for details).

### (b) The Hamiltonian formalism

We consider the Hamiltonian for equation (1.1). We work with the variables *u*(*t*,*x*) and *v*(*t*,*x*) as in equation (2.3). From equation (2.1) and by Parseval’s identity, we obtain that the Hamiltonian *H*(*w*) can be written as
where
where *R*_{1} comprises the terms of order greater than or equal to five. Introducing the notation
we obtain
As was pointed out in Valls (2006), it is very convenient to simplify *H*_{2} so that it has a diagonal form. To do it, we set
Introducing the variables
we obtain
2.4
and
2.5

### (c) 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 centre manifold for the Hamiltonian *H*=*H*_{2}+*ε**H*_{4} (introduced in equations (2.4) and (2.5)). This means that we want to compute a normal form such that, on the centre manifold, *H*=*H*_{2}+*ε*^{2}*R*_{1}, where *R*_{1} contains the terms with degrees greater than or equal to five.

The process of reducing to the centre manifold is based on removing some monomials in the expansion of the Hamiltonian in order to produce an invariant manifold tangent to the elliptic directions of *H*_{2}, that is the modes with indices *k*∈*λ*(*α*). To do this and, since we already have proven the existence of a centre manifold, we just need to cancel the monomials in *H*_{4} with at most one hyperbolic direction (i.e. the monomials with *k*>*k*(*α*)) and that, in this direction, have degree one. This ensures that, when restricted to the elliptic directions (i.e. the monomials with *k*≤*k*(*α*)), the Hamiltonian *H*=*H*_{2}+*ε**H*_{4} exhibits a decoupling in these two directions and approximates the dynamics in the centre manifold up to the fourth degree. Furthermore, the monomials in *H*_{4} with all elliptic directions are in normal form. Indeed, considering that *H*=*H*_{2}+*ε**H*_{4} with *H*_{2} given in equation (2.4) and the monomials *x*_{k},*y*_{k} with *k*>*k*(*α*) in *H*_{4} have degree greater than or equal to two, setting in *H*, *x*_{k}=*y*_{k}=0 for *k*>*k*(*α*), we get that the equations of motion coming from *H* satisfy for *k*>*k*(*α*). Thus, the set
is invariant under the Hamiltonian flow and *H* restricted to *C* only contains monomials *x*_{j}, *y*_{j} with *j*≤*k*(*α*) and thus represents the dynamics inside the centre manifold up to the fourth degree. Since we have already proved the existence of the centre manifold and, in this section, we just want to compute it approximately up to the fourth degree, together with the fact that to compute a term in the expansion of the centre 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 of domains of the operators and convergence of series.

In view of the explanation given above, to produce a Hamiltonian *H*_{4} of order four with at most one hyperbolic direction that, in this direction, has degree one and that all the elliptic directions in *H*_{4} are in normal form, we need to cancel, if possible, in *H*_{4} all the monomials such that, for any (*σ*,*ρ*)∈{0,1}^{2} have indices , where with and
We take the time 1-map of the flow of the Hamiltonian vector field given by the Hamiltonian , which is defined as
2.6
where, for a fixed , in the notation given in equation (2.4),
We note that
2.7

Now, we introduce some notation: we say that if when . With this notation, we clearly have . Furthermore, we can write *H*_{2}=*H*_{2,1}+*H*_{2,2} and *H*_{4}=*H*_{4,1}+*H*_{4,2}, where for *j*=2 and *j*=4, and () denote the complementary of ). Then, it is proved in Valls (2006) that, with the definition of *G*_{4} given in equation (2.6), we get
2.8
Now we have all the ingredients to state the result proved in Valls (2006). We denote by *π*_{Wc}(.) the restriction to the centre manifold.

### Theorem 2.1 Valls 2006

*For almost every α>0, the formal change of variables Γ*

_{1}

*takes the reduction of H to the centre manifold*,

*(*

*π*W_{c}*H*),

*into its normal form up to third order,*

*π*_{Wc}(

*H**),

*i.e.*

We firstly note that the restriction of *H*_{2} to the centre manifold corresponds to setting, in equation (2.4), *x*_{j}=*y*_{j}=0 for *j*≥3. Furthermore, *H*_{3,2} and {*H*_{2,2},*G*_{4}} do not belong to the centre manifold and, since this centre manifold is invariant under the dynamics, from equation (2.8) is equal to

## 3. Normal form and stability

In this section, we include the technical details of the normal-form process with rigorous bounds on the remainder, as well as bounds on the diffusion time around any elliptic torus inside the centre manifold. Due to technicalities (see lemma 3.2), we consider that the values of *α* belong to the set [16*π*^{2}+*δ*_{1},36*π*^{2}−*δ*_{2}] with *δ*_{1},*δ*_{2}>0. This implies that, in this case, *k*(*α*)=2 and thus *λ*(*α*)={*k*=1,2}. As pointed out in §2*a*, this is not a restriction on the values of *α*. This means that, in this paper, we bound the diffusion time around a torus with two frequencies and that the centre manifold is four-dimensional. As was pointed out in §1, some results may also be true for almost every *α*>0 for which *k*(*α*)>2 obtains bounds on the diffusion time around the torus with more than two frequencies. However, the process implies very tedious calculations and therefore we do not compute them here.

We note that, since *k*(*α*)=2, in the definition of *Γ*_{4}(*α*) we get
3.1
3.2

### (a) Notation

We firstly consider some notation. Let *p*=(*p*_{1},…,*p*_{4}), *q*=(*q*_{1},…,*q*_{4}) and consider functions *f*=*f*(*p*,*q*) defined on *D*(*S*) for some real number 0<*S*<1, where
and |.| denotes the infinity norm of a complex vector. We introduce the multi-index notation. If *f* is analytic, we denote
with and . 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*(*S*).

If *f*∈*C*^{j}, with *j*≥1, we denote by ∥*f*∥_{Cj(S)} its norm on the domain *D*(*S*),
We also introduce the action angle variables *I*=(*I*_{1},…,*I*_{4}) and θ=(θ_{1},…,θ_{4}), with (*I*_{k},θ_{k})_{k=1,…,k} defined by
3.3

### (b) Main result and ideas

The main result of this section is the following, which ensures the stability of the elliptic torus inside the centre manifold.

### Theorem 3.1

*We consider the Hamiltonian H as in equations* (*2.4*) *and* (*2.5*) *restricted to the centre manifold defined on D(S _{0}) for some* 0<

*S*

_{0}<1.

*Then, there exists constants*

*C*

_{1}>0

*and*

*C*

_{2}>0

*such that, for*

*ε*sufficiently small, any solution of the Hamiltonian equations of motion given by*π*_{Wc}(

*H*)

*that at t*=0

*starts in*

*D*(

*S*e

^{−δ})

*with*

*δ*>0

*satisfies*3.4

*with*.

We observe that the smallness condition of *ε* is explicitly given in theorem 3.3 (see equation (3.7)).

Theorem 3.1 ensures the long time stability of any real trajectory close to an elliptic torus inside the centre manifold. The main idea of the proof is firstly to estimate the domain of the canonical transformation provided by theorem 2.1 and also to bound its remainder. This provides the distance between the original and the transformed variables. Then, we find times for which the trajectories of the equations provided by the Hamiltonian *π*_{Wc}(*H**) (i.e. already in normal form) stay within the domain of the transformation and we estimate the drift of the actions.

### (c) 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 and bound the remainder . This is done in theorem 3.3.

For this purpose, we firstly need to bound the small divisors that appear in equation (2.6). The key tool is the observation in the following lemma that all the relevant divisors in equation (2.6) are independent of *α* and uniformly bounded away from zero. For technical reasons, we have to consider the values of the parameter *α* that belong to [16*π*^{2}+*δ*_{1},36*π*^{2}−*δ*_{2}] with *δ*_{1},*δ*_{2}>0.

### Lemma 3.2

For each *δ*_{1},*δ*_{2}>0, any with *α*∈[16*π*^{2}+*δ*_{1},36*π*^{2}−*δ*_{2}] and *j*_{1},*j*_{2},*j*_{3},*j*_{4}∈ {0,1}^{4} satisfies

### Proof.

We consider three different cases.

*Case 1*: . In this case, since , , and , it holds that, for any as in equation (3.1) and *j*_{1},*j*_{2},*j*_{3},*j*_{4}∈{0,1}^{4}, we get

*Case 2*: with *j*=2,3,4. In this case, the proof of the lemma follows analogously as in case 1 (see also equation (3.2)).

Case 3: . In this case, and to bound for any ( *j*_{1},*j*_{2},*j*_{3},*j*_{4})∈{0,1}^{4} and any , as in equation (3.1), in view of equation (2.7) it is enough to bound the quantities in the following groups:

(G.1) and . Note that the case , with 3.5 does not contribute to

*G*_{4}.(G.2) and . Note again that the case , with (

*j*_{1},*j*_{2},*j*_{3},*j*_{4}) as in equation (3.5), does not contribute to*G*_{4}.(G.3) , , , , and . Note that the case , with does not contribute to

*G*_{4}.

Clearly, in the case (G.1), we have

In the case (G.2), we obtain

Finally, in the case (G.3), we obtain
and
Since, on the interval *α*∈[16*π*^{2}+*δ*_{1},36*π*^{2}−*δ*_{2}], we have the inequality , we get
3.6
and hence
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 πW_{c}(H) is defined on D(S) for some* 0<

*S*<1

*and we introduce S*e

_{j}=S^{−jδ},

*with*

*δ*>0

*and j*≥0.

*Then, it holds that*

*the Hamiltonian G*_{4}*is defined on D(S) and**if we denote by**the flow at time t of the Hamiltonian**defined in equation (2.6), then, if*3.7*ε*is sufficiently small so that*holds, we have that*,*if we use the notation*,*then**there exists K*>0*such that**is bounded by*

### Proof.

The first statement of the theorem is straightforward by construction and in view of lemma 3.2 (see equations (2.5) and (2.6)).

Now using lemma A.1, we have
Since equation (3.7) holds, we get that
3.8
and thus if is the flow time *t* of the Hamiltonian system given by , we can readily deduce, using the Hamilton equations together with equation (3.8), that the transformations and act as we describe in the second statement of the theorem.

To prove the third statement of the theorem, we note that Finally, the fourth statement of the theorem follows directly using lemmas A.1 and A.2 and the fact that 3.9 ▪

### (d) 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 a Hamiltonian system with the two frequencies τ_{1}(*α*) and τ_{2}(*α*) (see equation (2.4)).

Let (*p*(0),*q*(0))∈*D*(*S*_{2}). Then, by the second and third statements of theorem 3.3, we have that, if we denote , then (*P*(0),*Q*(0))∈*D*(*S*_{1}),
We introduce the variables , with , for *j*=1,…,4. Then, (see also equation (3.3) for the definition of *I*)
3.10
Furthermore, since
if we denote by *T*_{0} the maximum time for which (*P*(*t*),*Q*(*t*))∈*D*(*S*_{3}), we have
with given in the fourth statement of theorem 3.3. Then, there exists a constant *C*_{1}>0, such that
3.11
for 0≤*t*≤*T*_{0}, with *T*_{0} as in equation (3.4).

Now proceeding in a similar manner as in the second and third statements of theorem 3.3, we have that, since (*P*(*t*),*Q*(*t*))∈*D*(*S*_{3}), then (*p*(*t*),*q*(*t*))∈*D*(*S*_{2}) and thus for any 0≤*t*≤*T*_{0}, we have
3.12
Therefore, from equations (3.10)–(3.12) for any 0≤*t*≤*T*_{0}, with *T*_{0} as in equation (3.4), we obtain (see also equation (3.9))
for some constant *C*_{2}. Thus, the theorem is proved.

## Acknowledgements

Partially supported by FCT through CAMGSD, Lisbon.

## Appendix A

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

## Lemma A.1.

*Let f(x,y) be analytic functions on D(S). Then, for every*0<*χ*<1, *we have that*

## Proof.

The proof of the lemma is carried out by applying Cauchy estimates to the function . ▪

## Lemma A.2.

*Take* 0<*S*_{0}<*S* *and consider analytic functions Z and W with values in C*^{4}, *all defined for (z,w)∈D(S*_{0}). *We assume that |Z|S*_{0} *and* |*W*|_{S0} *are both bounded by R. Let f(z*,w*), where* *and* , *be a given analytic function on D(S). If we introduce*
*then* |*F*|_{S0}≤|*f*|_{S}.

## Proof.

The proof of the lemma can be directly checked by expanding *f* in Taylor series and using that, if *f*,*g*∈*D*(*S*), then ∥*f**g*∥_{S}≤∥*f*∥_{S}∥*g*∥_{S}. ▪

## Footnotes

- Received February 26, 2009.
- Accepted May 11, 2009.

- © 2009 The Royal Society