Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain’s Lemma which provides a partition of the “resonant sites” of the Laplace operator on irrational tori.


Introduction
The problem of giving an upper bound on the growth of Sobolev norms in Hamiltonian nonlinear PDEs, has been widely investigated. The results typically obtained are known as "almost global existence": they ensure that solutions corresponding to smooth and small initial data remain smooth and small for times of order ǫ −r with arbitrary r; here ǫ is the norm of the initial datum.
We recall that there exist quite satisfactory results for semilinear equations in one space dimension [15,1,5], which have also been extended to some semilinear PDEs with unbounded perturbations [29] and to some quasilinear wave equations [17], gravity capillary water waves [11] (see also [12]), quasi-linear Schrödinger [23] and pure gravity water waves [13] still in dimension one. On the contrary for the case of higher dimensional manifolds only particular examples are known [20,4,19] and for PDEs in higher space dimension with unbounded perturbations only partial results have been obtained [27,22,25]. A slightly different point of view is the one developed in [28], [26], in which the authors analyze the phenomenon of energy transfer to high modes, for initial data Fourier supported in a box for the cubic NLS on the irrational square torus in dimension two.
To discuss the main difficulty met in order to obtain almost global existence in more than one space dimension, we recall that all the known results deal with perturbations of linear systems whose eigenvalues are of the form ±iω j with ω j real numbers playing the role of frequencies. The main point is that in all known results, the frequencies are assumed to verify a non-resonance condition of the form |ω j 1 ± ω j 2 ± .... ± ω jr | ≥ (max with max 3 {|j 1 |, ..., |j r |} the third largest number among |j 1 |, ..., |j r | and γ, τ positive numbers; condition (1.1) is a kind of second Melnikov condition since it requires to control linear combinations involving two frequencies with index arbitrarily large. This is a Diophantine type condition which is typically violated in more than one space dimensions.
In the present paper we prove an abstract result of almost global existence (see Theorem 2.9) for some Hamiltonian PDEs in which the linear frequencies are assumed to fulfill the much weaker condition for all possible choices of indexes j 1 , ..., j r . This is a condition typically fulfilled in any space dimension. The key point is that we also require the frequencies ω j and the indexes j to fulfill a structural property ensured by a Lemma by Bourgain on the "localization of resonant sites" in T d . This allows to prove a theorem ensuring that the Hamiltonian of the PDE can be put in a suitable block-normal form which can be used to control the growth of Sobolev norms. More precisely, following [5], we decompose the variables in variables of large index (high modes) and variables of small index (low modes); the normal form we construct is the standard Birkhoff normal form for low modes, while it is a normal form in which the equations of the high modes are linear time dependent equations. Furthermore the equations for the high modes have a block diagonal structure with dyadic blocks and this allows to control the growth in time of the Sobolev norms.
We emphasize that one of the points of interest of our paper is that it shows the impact of results of the kind of [16,18] dealing with linear time dependent systems on nonlinear systems, thus, in view of the generalizations [6,7,8], it opens the way to the possibility of proving almost global existence in more general systems, e.g. on some manifolds with integrable geodesic flow.
In the present paper, after proving the abstract result, we apply it to a few concrete equations for which almost global existence was out of reach with previous methods. Precisely we prove almost global existence of small amplitude solutions (1) for nonlinear Schrödinger equations with convolution potential, (2) for nonlinear beam equations and (3) for a quantum hydrodinamical model (QHD) (see for instance [24]). We also prove Sobolev stability of plane waves for the Schrödinger equation (following [21]).
To present in a more precise way the result, we recall that an arbitrary torus can be easily identified with the standard torus endowed by a flat metric. This is the point of view we will take. For the Schrödinger equation we show that, without any restrictions on the metric of the torus, one has that if the potential belongs to a set of full measure then one has almost global existence. For the case of the beam equation, we use the metric in order to tune the frequencies and to fulfill the nonresonance condition, thus we prove that if the metric of the torus is chosen in a set of full measure then almost global existence holds. Examples of tori fulfilling our property are rectangular tori with diophantine sides, but also more general tori are allowed.
The result for the QHD model is very similar to that of the beam equation: if the metric is chosen in a set of full measure, then almost global existence holds. Also the result of Sobolev stability of plane waves in the Schrödinger equation is of the same kind: if the metric belongs to a set of full measure, one has stability of the plane waves over times longer than any inverse power of ǫ.
We recall that results of this kind were only known for square tori in which the frequencies have a structure identical to that of typical 1 dimensional systems. For irrational tori the only result ensuring at least a quadratic lifespan of nonlinear Schrödinger equations with unbounded, quadratic nonlinearities has been proved in [25] (see also [24] for the Euler-Kortweg system and [10] for the Beam equation).
Finally we recall the result [9] in which the authors consider a nonlinear wave equation on T d and prove that if the initial datum is small enough in some Sobolev norm then the solution remains small in a weaker Sobolev norm for times of order ǫ −r with arbitrary r. The main difference is that this result involves a loss of smoothness of the solution which is not present in our result; however, we emphasize that at present our method does no apply to the wave equation since no generalizations of Bourgain's Lemma to systems of first order are known.
We define In the following we will simply write ℓ 2 s for ℓ 2 s (Z d ; C) and ℓ 2 for ℓ 2 0 . We denote by B s (R) the open ball of radius R and center 0 in ℓ 2 s . Furthermore in the following U s ⊂ ℓ 2 s will always denote an open set containing the origin. We endow ℓ 2 by the symplectic form i j∈Z d u (j,+) ∧ u (j,−) , which, when restricted to ℓ 2 s (s > 0), is a weakly symplectic form. Correspondingly, given a function H ∈ C 1 (U s ), for some s, its Hamilton equations are given bẏ .
We will also denote by the corresponding (formal) Hamiltonian vector field. In the following we will work on the space ℓ 2 s with s large. More precisely, all the properties we will ask will be required to hold for all s large enough.

The class of functions (and perturbations)
Given an index J ≡ (j, σ) ∈ Z d we define the involution Given a multindex J ≡ (J 1 , ..., J r ), with J l ∈ Z d , l = 1, ..., r, we definē J := (J 1 , ...,J r ). On the contrary, for a complex number the bar will simply denote the complex conjugate.
Definition 2.1. On ℓ 2 s we define the involution I by The sequences such that Iu = u will be called real sequences.
Given a multi-index J ≡ (J 1 , ..., J r ), we also define its momentum by In particular in the following we will deal almost only with multi indexes with zero momentum, so we define Given a homogeneous polynomial P of degree r, namely P : ℓ 2 s → C for some s, it is well known that it can be written in a unique way in the form with P J 1 ,...,Jr ∈ C symmetric with respect to any permutation of the indexes.
We are now ready to specify the class of functions we will consider.
For R > 0 we endow the space P r with the family of norms Given r 2 ≥ r 1 ≥ 1 we denote by P r 1 ,r 2 := r 2 l=r 1 P l the space of polynomials P (u) that may be written as endowed with the natural norm Remark 2.3. By the reality condition (P.2) in Definition 2.2, one can note that if P ∈ P r then • P (u) ∈ R for all real sequence u (see Def. 2.1).
• Fix J 1 , J 2 ∈ Z d and define Then, for all real sequence u, one has this "formal selfadjointness" will play a fundamental role in the following.
Definition 2.4. (Functions). We say that a function P ∈ C ∞ (U s ; C) belongs to class P, and we write P ∈ P, if • all the terms of its Taylor expansion at u = 0 are of class P r for some r; • the vector field X P (recall (2.5)) belongs to C ∞ (U s ; ℓ 2 s ) for all s > d/2.
The Hamiltonian systems that we will study are of the form with P ∈ P and H 0 of the form and ω j ∈ R a sequence on which we are going to make some assumptions in the next subsection.

Statement of the main result
We need the following assumption.
Hypothesis 2.5. The frequency vector ω = (ω j ) j∈Z d satisfies the following.
F.1 There exist constants C 1 > 0 and β > 1 such that, ∀j large enough one F.2 For any r ≥ 3 there exist γ r > 0 and τ r such that the following condition holds for all N large enough with the following properties: F.3.1 * either Ω α is finite dimensional and centered at the origin, namely there exists C 1 such that j ∈ Ω α , =⇒ |j| ≤ C 1 , * or it is dyadic, namely there exists a constant C 2 independent of α such that sup j∈Ωα |j| ≤ C 2 inf j∈Ωα |j| .
(2.18) F.3.2 There exist δ > 0 and C 3 = C 3 (δ) such that, if j ∈ Ω α and i ∈ Ω β with α = β, then Finally, we need a separation property of the resonances, namely that the resonances do not couple very low modes with very high modes. To state this precisely, we first define an equivalence relation on Z d Definition 2.6. For i, j ∈ Z d , we say that i ∼ j if ω i = ω j . We denote by [i] the equivalence classes with respect to such an equivalence relation.
Hypothesis 2.7. The frequency vector ω = (ω j ) j∈Z d satisfies the following.
(NR.1) The equivalence classes are dyadic, namely there exists C > 0 such that (2.20) (NR.2) Non-resonance: ∀l ∈ N and for any choice of j 1 , ..., j l such that Remark 2.8. We point out that the Hypothesis 2.7 is only used in Section 4 in order to prove energy estimates for the system in normal form, see Lemma 4.1.
Our main abstract theorem pertains the Cauchy problem Theorem 2.9. Consider the Cauchy problem (2.22) where H has the form (2.14) with H 0 as in (2.15) and P ∈ P vanishing at order at least 3 at u = 0. Assume that the frequencies ω j fulfill Hypotheses 2.5, 2.7. For any integer r there exists s r ∈ N such that for any s ≥ s r there exists ǫ 0 > 0 and c > 0 with the following property: if the initial datum u 0 ∈ ℓ 2 s is real and small, namely if then the Cauchy problem (2.22) has a unique solution The main step for the proof of Theorem 2.9 consists in proving a suitable normal form lemma which is given in the next section.

Normal form
In the following we will use the notation a b to mean there exists a constant C, independent of all the relevant parameters, such that a ≤ Cb. If we want to emphasize the fact that the constant C depends on some parameters, say r, s, we will write a s,r b. We will also write a ≃ b if a b and b a.
We need the following definition.
2.2 there exist α such that j 1 , j 2 ∈ Ω α , namely both the large indexes belong to the same cluster 1 Ω α .
We now state the main result of this section.
Theorem 3.2. Fix any N ≫ 1, s 0 > d/2 and consider the Hamiltonian (2.14) with ω j fulfilling Hypothesis 2.5 and P ∈ P. For anyr ≥ 3 there is τ > 0 such that for any s > s 0 there exist R s,r , C s,r > 0 such that for any R < R s,r the following holds. If then there exists an invertible canonical transformation • Z (r) ∈ P 3,r is in N-block normal form and fulfills The rest of the section is devoted to the proof of this theorem and is split in a few subsections.

Properties of the class of functions P
First we give the following lemma. Lemma 3.3. (Estimates on the vector field). Fix r ≥ 3, R > 0. Then for any s > s 0 > d/2 there exists a constant C r,s > 0 such that, ∀P ∈ P r , the following inequality holds: Proof. Let P ∈ P r . Then (recalling (2.5)) one has X P = ((X P ) J ) J∈Z d with (X P ) (j,+) = i∂ u (j,−) P = ir and similarly for (X P ) (j,−) . Remark that the r.h.s. of (3.7) defines a unique symmetric (r − 1)-linear form ( X P ) (j,+) (u (1) , ..., u (r−1) ) := ir In order to apply Lemma A.1 we decompose Substituting in the previous expression we have Now each of the addenda of (3.10) fulfills the assumptions of Lemma A.1. Therefore, since u s 0 ≤ u s one has Taking all the u (l) equal to u ∈ B s (R) (i.e. u s < R) and recalling the norm in (2.12) one gets the thesis for (X P ) + . Similarly one gets the thesis for (X P ) − and this concludes the proof of the lemma.
As usual given two functions f 1 , f 2 ∈ C ∞ (ℓ 2 s ; C) we define their Poisson Brackets by which could be ill defined (but will turn out to be well defined in the cases we will consider). We recall that if both f 1 and f 2 have smooth vector field then with [· ; ·] denoting the commutator of vector fields.
We now fix some large N > 0, but will track the dependence of all the constants on N. Corresponding to N we define a decomposition of u in low and high modes. Precisely, we define the projectors Π ≤ u := (u J ) |J|≤N , Π ⊥ u := (u J ) |J|>N (3.13) and denote so that u = u ≤ + u ⊥ . As in [1,5], a particular role is played by the polynomials P ∈ P r which are quadratic or cubic in u ⊥ . We are now going to give a precise meaning to this formal statement. First, given f ∈ C ∞ (U s ; C), we denote by the l-th differential of f evaluated at u and applied to the increments h 1 , ..., h l . Definition 3.5. We say that P ∈ P r has a zero of order at least k in u ⊥ if We say that it is homogeneous of degree k in u ⊥ if it has a zero of order at least k, but not of order at least k + 1.
Remark 3.6. By the very definition of normal form, one can decompose  (i) if P ∈ P r has a zero of order at least 2 in u ⊥ , then (ii) if P ∈ P r has a zero of order at least 3 in u ⊥ , then Proof. Consider first the case (i) and remark that, using the notation (3.9), we have Π ≤ X P (u) ± = ±i∇ u ≤ ± P , so that Π ≤ X P (u) has a zero of order 2 in u ⊥ . It follows that both in the case (i) and in the case (ii) we have to estimate a polynomial function X(u) of the form (3.7) with a zero of second order in u ⊥ . To exploit this fact consider first the + component and consider again the multilinear form ( X) + as in (3.8): we have but, since X + (u) has a zero of at least second order in u ⊥ , one has .
Consider the first addendum (which is the one giving rise to worst estimates): proceeding as in the proof of Lemma 3.3 one can apply Lemma A.1 and get the estimate and thus The other cases can be treated similarly.

Lie Tranfsorm
Given G ∈ P r,r , consider its Hamilton equationsu = X G (u), which, by Lemma 3.3, are locally well posed in a neighborhood of the origin. Denote by Φ t G the corresponding flow, then we have the following Lemma whose proof is equal to the finite dimensional case.
Lemma 3.8. Considerr ≥ r 1 ≥ r ≥ 3 and s > s 0 > d/2. There exists C r,s > 0 such that for any G ∈ P r,r 1 and any R > 0 satisfying the following holds. For any |t| ≤ 1 one has Φ t G (B s (R)) ⊂ B s (2R) and the estimate is called the Lie transform generated by G.
In order to describe how a function is transformed under Lie transform we define the operator and its k-th power Ad k G f := {Ad k−1 G f ; G} for k ≥ 1. Also the following Lemma has a standard proof equal to that of the finite dimensional case. Lemma 3.10. Letr ≥ r ≥ 3 and s > s 0 > d/2 and consider G ∈ P r,r . There exists C r,s > 0 such that for any R > 0 satisfying (3.15) the following holds. For any f ∈ C ∞ (B s (2R); C) and any n ∈ N one has ∀u ∈ B s (R) and any t with |t| ≤ 1.
From Lemma 3.4 one has the following corollary.
Corollary 3.11. Let G ∈ P r 1 ,r 2 , F ∈ P r 3 ,r 4 , with r 1 , r 2 , r 3 , r 4 ≤r and 3 ≤ r 1 ≤ r 2 . Letn ∈ N be the smallest integer such that (n + 1)(r 1 − 2) + r 2 >r. Then there exists Cr > 0 such that for any k ≤n, one has A further standard Lemma we need is the following.
Lemma 3.12. Let G ∈ P r 1 ,r 2 , 3 ≤ r 1 , r 2 ≤r and let Φ G be the Lie transform it generates. Let R s by the largest value of R such that (3.15) holds. Then there exists C > 0 such that for any one has sup From Lemma 3.10, Corollary 3.11 and Lemma 3.12, one has the following Corollary which is the one relevant for the perturbative construction leading to the normal form lemma.
Corollary 3.13. There exists µ 0 > 0 such that for any G ∈ P r,r , 3 ≤ r ≤r, the following holds. If with Cr the constant of Corollary 3.11, then, for any F ∈ P r 1 ,r , r 1 ≤r, one has which fulfill the following estimates withn as in Corollary 3.11 and C as in Lemma 3.12.

Homological equation
In order to construct the transformation T (r) of Theorem 3.2, we will use the Lie transform generated by auxiliary Hamiltonian functions G 3 , ..., Gr, with G ℓ ∈ P ℓ,r , which in turn will be constructed by solving the homological equation with F ∈ P ℓ,r a given polynomial of order 2 in u ⊥ and Z to be determined, but in N-block normal form. In order to solve the homological equation we need a nonresonance condition seemingly stronger than (2.16), but which actually follows from F.1, F.2, F.3 of Hypothesis 2.5.
To state the non-resonance condition we need the following definition. (I.2) if there exist two indexes, say J 1 and J 2 with |J 1 | > N and |J 2 | > N, with j 1 and j 2 belonging to the same cluster 2 Ω α then σ 1 σ 2 = 1, Multi-indexes J ∈ J N l are called non resonant multi-indexes. Remark 3.15. By Definitions 3.1 and 3.14 we notice that an Hamiltonian Z ∈ P r , r ≥ 3, of the form (2.11) but supported only on multi-indexes J ∈ I r \ J N r is in N-block normal form. Lemma 3.16. Assume Hypothesis 2.5 and let r ∈ N. Then there exist τ ′ r and γ ′ r > 0, such that for any 3 ≤ p ≤ r and any multi-index J ∈ J N p one has Proof. For the case where all the indexes j l are smaller than N there is nothing to prove. Consider now the case where there is only one index, say J 1 , larger than N and the length of the multi-index is n + 1 ≤ r. The quantity to be estimated is now n l=2 σ l ω j l + σ 1 ω j 1 . (3.19) which implies the bound (3.18) by choosing γ ′ r ≤ γ r 2 τr (rC 2 1 ) τr/β . Consider now the case where there are two indexes larger than N, say J 1 and J 2 . The case σ 1 σ 2 = 1 is dealt with similarly to the previous case.
We discuss now to the case σ 1 σ 2 = −1. By condition (I.2) in Definition 3.14 there exist α = β such that j 1 ∈ Ω α and j 2 ∈ Ω β . It follows that either for some C > 0. Assume for concreteness that |j 1 | ≥ |j 2 | and σ 1 = 1, Consider first the case where (3.20) holds. The quantity to be estimated is Notice that (3.20) implies |ω j 1 − ω j 2 | ≥ C|j 1 | δ and that we also have Then it follows that (3.18) is automatic if Hence the bound on (3.22) is nontrivial only if all the indexes are smaller than N 2 . In this case we can apply (2.16) with N 2 in place of N, getting which is the wanted estimate, in particular with τ ′ r ≥ β δ τ r . It remains to bound (3.22) from below with indexes fulfilling (3.21). By the zero momentum condition we have It follows that in our set there are no indexes with C|j 1 | δ > rN (otherwise the zero momentum condition cannot be fulfilled), so all the indexes must be smaller than N 3 := (rN/C) 1/δ , and again we can estimate (3.22) using (2.16) with N substituted by N 3 , thus getting the thesis.  .15) and ω j satisfying Hypotheses 2.5 and where F ∈ P r,r is a polynomial having a zero of order 2 in u ⊥ . Then equation (3.17) has solutions Z ∈ P r,r and G ∈ P r,r where Z is in N-block normal form, N ≫ 1 and moreover Proof. Notice that, denoting u J := u J 1 ...u Jr and recalling (3.11), one has It follows that, writing P = one can solve the Homological equation (3.17) by defining (recall Def. 3.14) By Remark 3.15 we have that Z is in N-block normal form. The estimates (3.23)-(3.24) immediately follow using Lemma 3.16.

Proof of the normal form Lemma
Theorem 3.2 is an immediate consequence of the forthcoming Lemma 3.18.
To introduce it, we first split withP ∈ P 3,r and R T,0 having a zero of order at leastr + 1 at the origin. A relevant role will be played by the quantity P R . In order to simplify the notation, we remark that, for R sufficiently small there exists K s,r such that Lemma 3.18. (Iterative lemma). Assume Hypothesis 2.5 and fixr ≥ 3.
There exists µr > 0 such that for any 3 ≤ k ≤r and any s > s 0 > d/2 there exist R s,k > 0, C s,k , τ > 0 such that for any R < R s,k and any N ≫ 1 the following holds. If one has then there exists an invertible canonical transformation such that (3.32) The proof occupies the rest of the section and is split in a few Lemmas. We reason inductively. First, we consider the Taylor expansion of P k in u ⊥ and we write with P k,ef f containing only terms of degree 0, 1 and 2 in u ⊥ , while R k,⊥ has a zero of order at least 3 in u ⊥ . Then we determine G k+1 and Z k+1 by solving the homological equation so that, by Lemma 3.17 and the inductive assumption (3.30), we get Consider the Lie transform Φ G k+1 (recall Def. 3.9) generated by G k+1 . By the estimate (3.35) and the condition (3.25) we have that there is R s,k+1 > 0 such that (3.15) is fulfilled for R < R s,k+1 . Hence Lemma 3.8 applies and so we deduce that the map Φ G k+1 is well-posed.
We study now H (k) • Φ G k+1 . To start with we prove the following Lemma.
Lemma 3.19. Let G k+1 be the solution of (3.34), then one has

37)
withH 0 ∈ P k+1,r , and, provided R < R 0 k+1 , for some R 0 k+1 , one has H 0 R µ k−2 P R . (3.38) Furthermore, there exists C 0 > 0 such that one has Proof. Letn be such that (n + 1)(k − 2) + k >r; using the expansion (3.16) one gets where we can rewrite explicitly the remainder term as Since G k+1 fulfills the Homological equation one has Hence, definingH 0 to be the sum in Eq. (3.40), one has provided R is small enough. Analogously one gets and, since k +n ≥r the thesis follows.
In an analogous way one proves the following simpler Lemma whose proof is omitted.
Lemma 3.20. Let G k+1 ∈ P k,r fulfills the estimate (3.35), then we have

41)
and the following estimates hold End of the proof of Lemma 3.18. We consider the Lie transform Φ G k+1 generated by G k+1 determined by the equation (3.34) and we define By estimate (3.35), condition (3.25), taking R small enough, we have that Lemma 3.8 applied to G k+1 and the inductive hypothesis on T (k) imply that T (k+1) satisfies (3.26)-(3.27) with k k + 1 and some constant C s,k+1 . Recalling (3.37), (3.41) we define Then the iterative estimates follow from the estimates of Lemmas 3.19 and 3.20. This concludes the proof.
An important consequence of Theorem 3.2 is the following.
Corollary 3.21. Consider the Hamiltonian (2.14) with ω j fulfilling Hypotheses 2.5 and P ∈ P (see Def. 2.4). For any r ≥ 3 there exists N r > 0, τ > 0 and s r > d/2 and a canonical transformation T r such that for any s ≥ s r there exists R s > 0 and C s > 0 such that the following holds for any R < R s :

43)
where • Z r ∈ P 3,r is in N r -block normal form according to Def. 3.1; according to the splitting (3.13)-(3.14) with N replaced by N r and we set Z r = Z 0 + Z 2 (see Remark 3.6) where Z 0 is the part independent of z ⊥ and Z 2 is the part homogeneous of order 2 in z ⊥ . Then we have Proof. Let us fixr consider τ = τ r given by Lemma 3.16 and fix We now take N r = N such that With this choices the assumption (3.1) holds taking R < R s with R s small enough. Then Theorem 3.2 applies with s ≥ s r , N = N r and τ = τ r chosen above. First of all notice that

Dynamics and proof of the main result
In this section we conclude the proof of Theorem 2.9. Consider the Cauchy problem (2.22) (with Hamiltonian H as in (2.14)) with an initial datum u 0 satisfying (2.23) and fix any r ≥ 3. Recalling Hypotheses 2.5, 2.7, setting ǫ ≃ R , (4.1) then for s ≫ 1 large enough and ǫ small enough (depending on r), we have that the assumption of Corollary 3.21 are fulfilled. Therefore we set and we consider the Cauchy probleṁ with H r given in (3.43). By (3.42) we have that the bound (2.24) on the solution u(t) of (2.22) follows provided that we show where z(t) is the solution of the problem (4.2) and where we denoted the (possibly infinite) escape time of the solution from the ball of radius R.
The rest of the section is devoted to the proof of the claim (4.3). To do this we now analyze the dynamics of the system (4.2) obtained from the normal form procedure. To this end we write the Hamilton equations in the form of a system for the two variables (z ≤ , z ⊥ ) and also split the normal form Z r = Z 0 + Z 2 as in item (ii) in Corollary 3.21. We geṫ where Λ is the linear operator such that Λz = X H 0 (z). The key points to analyze the dynamics are the following: (i) Z 0 is in standard Birkhoff normal form; (ii) by item (i) of Lemma 3.7 one has that Π ≤ X Z 2 (z ≤ , z ⊥ ) is a remainder term (see item (ii) in Corollary 3.21); (iii) Π ⊥ X Z 2 (z ≤ , z ⊥ ) is linear in z ⊥ . Furthermore, for any given trial solution z ≤ (t) it is a time dependent family of linear operators, which by the property (2.13) are selfadjoint and thus conserve the L 2 norm; (iv) since Z 2 is in normal form it leaves invariant the dyadic decomposition Ω α on which the ℓ 2 norm is equivalent to all the ℓ 2 s norms. Formally we split the analysis in a few lemmas. The first is completely standard and provides a priori estimates on the low frequency part z ≤ of the solution of (4.5).

7)
where T R is given in (4.4).
Proof. For i ∈ Z d , define the "superaction" Then by (3.44)-(3.45) the last quantity is estimated by a constant times R r+2 . From this, denoting by K 0 the constant in the above inequality, one gets from which, writing K 1 := K 0 C one gets the estimate (4.7). We now provide a priori estimates on the high frequencies z ⊥ which evolve according to (4.6).

8)
where T R is given by (4.4).
Proof. First, we denote by Z(z ≤ ) : Π ⊥ ℓ 2 → Π ⊥ ℓ 2 the family of linear operator s.t. X Z 2 (z ≤ , z ⊥ ) = Z(z ≤ )z ⊥ ; We also write Z(t) := Z(z ≤ (t)), with z ≤ (t) the projection on low modes of the considered solution. We now introduce some further notations. For any z ∈ Π ⊥ ℓ 2 , we introduce the projector Π α associated to the block Ω α of the partition. More precisely, for any α, we define Then any sequence z ∈ Π ⊥ ℓ 2 can be written as z = α z α , z α := Π α u . (4.10) By the property 2.2 of Definition 3.1, the normal form operator Z(t) has a block-diagonal structure, namely it can be written as (4.12) by (4.9), (4.10), (4.11), it is block diagonal, namely it is equivalent to the decoupled system Since Z α is self-adjoint, one immediately has that s , so that, denoting by U(t, τ ) the flow map of (4.12), one has Consider now (4.6). Using Duhamel formula one gets from which (using also (3.44)) one deduces and the estimate (4.8).

Applications
Let e 1 , ..., e d be a basis of R d and let 2πn j e j , n j ∈ Z (5.1) be a maximal dimensional lattice. We denote T d Γ := R d /Γ. To fit our scheme it is convenient to introduce in T d Γ the basis given by e 1 , ..., e d , so that the functions turn out to be defined on the standard torus T d := R d /(2πZ) d , but endowed by the metric g ij := e j · e j . In particular the Laplacian turns out to be where g ln is the inverse of the matrix g ij . The positive definite symmetric quadratic form of equation (2.1) is then defined by The coefficients g ln , l, n = 1, . . . , d, of the metric g above can be seen as parameters that will be chosen in the set we now introduce. We also assume the symmetry g ij = g ji for any i, j = 1, . . . , d, hence we identify the metric g with (g ij ) i≤j , namely we identify the space of symmetric metrics with R d(d+1) 2 . We denote by g 2 2 := i,j |g ij | 2 Definition 5.1. Consider the open set We then define the set of admissible metrics as follows.
), implying that G has full measure in G 0 (we denote by | · | the Lebesgue measure). We also point out that in Section 5.1, we only take the metric g ∈ G 0 and we shall use the convolution potential in order to impose the non-resonance conditions. For the other applications, namely in sections 5.2, 5.3, 5.4 we shall use that the metric g is of the form g = βḡ, withḡ in the set of the admissible metrics G. We then use the parameter β, in order to verify the non-resonance conditions required.

Schrödinger equations with convolutions potentials
We consider Schrödinger equations of the form where ∆ g is in (5.2) with g ∈ G 0 (see Def. 5.1), V is a potential, * denotes the convolution and the nonlinearity f is of class C ∞ (R, R) in a neighborhood of the origin and f (0) = 0.

Equation (5.3) is Hamiltonian with Hamiltonian
where F is a primitive of f and ϕ is a variable conjugated to ψ. To get equation (5.3) one has to restrict to the invariant manifold ϕ = ψ.
Fix n ≥ 0 and R > 0, then the potential V is chosen in the space V given by which we endow with the product probability measure. Here and below |T d | g is the measure of the torus induced by the metric g.
Theorem 5.3. There exists a set V (res) ⊂ V with zero measure such that for any V ∈ V \ V (res) the following holds. For any r ∈ N, there exists s r > d/2 such that for any s > s r there is ǫ s > 0 and C > 0 such that if the initial datum for (5.3) belongs to H s and fulfills ǫ := |ψ| s < ǫ s then We are now going to prove this theorem. To fit our scheme simply introduce the Fourier coefficients In these variables the equation ( and P obtained by substituting in the F dependent term of the Hamiltonian (5.4). It is easy to see that the perturbation is of class P of Definition 2.4. In order to apply our abstract Birkhoff normal form theorem, we only need to verify the Hypotheses 2.5, 2.7. The hypothesis (F.1) in Hyp. 2.5 holds trivially with β = 2 using (5.6).
The hypothesis (F.3) follows by the generalization of the Bourgain's Lemma proved in [14]. Precisely we now prove the following lemma. Proof. Let Ω α be the partition of Z d constructed in Theorem 2.1 of [14]. It satisfies the properties for some C 0 > 0 and δ ∈ (0, 1). Clearly, one has that if j ∈ Ω α , j ′ ∈ Ω β with α = β, one has that and remark that its cardinality #K r N ≤ N dr . For k ∈ Z Z d N , consider V N k (γ) := {V ∈ V : |ω · k| < γ} .
with n the number in the definition of V in (5.5).
Proof. If V N k (γ) is empty there is nothing to prove. Assume thatṼ ∈ V N k (γ). Since k = 0, there existsj such that kj = 0 and thus kj ≥ 1; so we have

It means that if V N k (γ) is not empty it is contained in the layer
whose measure is γ j n ≤ 2γN n . This implies (5.7).
Proof. From Lemma 5.5 it follows that the measure of the set is estimated by a constant times γ. It follows that the set has zero measure and with this definition the lemma is proved.

Beam equation
In this section we study the beam equation in a neighborhood of the origin and having a zero of order 2 at the origin. Introducing the variable ϕ =ψ ≡ ψ t , it is well known that (5.8) can be seen as an Hamiltonian system in the variables (ψ, ϕ) with Hamiltonian function In order to fulfill the diophantine non-resonance conditions on the frequencies we need to make some restrictions on the metric g whereas, we only require that the mass m > 0 is strictly positive. More precisely, we considerḡ be a metric in the set of the admissible metrics G given in the definition 5.1. We consider a metric g of the form we shall use the parameter β in order to tune the resonances and to impose the non-resonance conditions required in order to apply Theorem 2.9. The precise statement of the main theorem of this section is the following one.
Theorem 5.7. Let g ∈ G, There exists a set of zero measure B (res) ⊂ B such that if β ∈ B \ B (res) then for all r ∈ N there exist s r > d/2 such that the following holds. For any s > s r there exist ǫ rs , c, C such that if the initial datum for (5.8) fulfills then the corresponding solution satisfies We actually state also a corollary which state that there exists a full measure set of metrics (not only constrained to a given direction g) for which the statements of Theorem 5.7 hold. Let 0 < β 1 < β 2 and define where G 0 is given in the definition 5.1.

2
. For any β 1 ≤ β ≤ β 2 , we denote by σ β the surface n − 1 dimensional measure on the sphere ∂B β := { g 2 = β}. We now prove the following two claims • Claim 1. One has that the surface measure of all diophantine metrics G in G 0 with norm equal 1 has full surface measure in G 0 ∩ ∂B 1 , namely • Claim 2. Let g ∈ G ∩ ∂B 1 and let B g ⊂ (β 1 , β 2 ) the full measure set provided in Theorem 5.7. We shall prove that G (nr) has full measure in G 0 (β 1 , β 2 ).
Proof of claim 2. By Fubini, the Lebesgue measure |G (nr) The claimed statement has then been proved.
To prove Theorem 5.7 we first show how to fit our scheme and then we prove that the Hypotheses of Theorem 2.9 are verified.
To fit our scheme we first introduce new variables and consider their Fourier series, namely, for σ = ±1 In these variables the beam equation ( and P obtained by substituting (5.17)-(5.18) in the F dependent term of the Hamiltonian (5.9). Thanks to the regularity assumption on F , it is easy to see that the perturbation P is of class P.
The verification of (F.3) in Hyp. 2.5 goes exactly as in the case of the Schrödinger equation, since the asymptotic of ω j in (5.19) is ω j = |j| 2 g + O(1). The asymptotic condition (F.1) is also trivially fulfilled with β = 2. The main point is to verify the non-resonance conditions (F.2) and the conditions (NR.1), (NR.2) in Hyp. 2.7. This will occupy the rest of this subsection.
First of all we remark that the equivalence classes of Definition 2.6 are simply defined by [j] ≡ i ∈ Z d : |i| g = |j| g .
Now, recall that g = βg with g ∈ G and β ∈ B = [β 1 , β 2 ]. One can easily verify that |j| g = β|j| g (5.20) implying that |j| g = |k| g if and only if |j| g = |k| g . Hence the equivalence We are going to prove the following Lemma We first need a lower bound on the distance between points with different modulus in Z d . The following lemma holds Lemma 5.10. Fix any N > 1 and let g ∈ G, β ∈ B = (β 1 , β 2 ), g = βg. One has that if j, k ∈ Z d such that |ℓ|, |k| ≤ N, |j| g = |k| g , then there exists γ > 0 and a constant C(β 1 ) such that where τ * is given in the definition 5.1.
Proof. By recalling the Definition 5.1 of the admissible set G, since g ∈ G, one has that there exists γ > 0 such that By (5.20), since g = βg, one has that Since |ℓ|, |k| ≤ N, one has the following chain of inequalities: The latter inequality, together with (5.21) imply that there exists a constant C(β 1 ) such that β 2 ). The claimed statement has then been proved.
Using the property (5.20), one can easily verify that the frequencies ω j ≡ ω j (β) assume the form is an analytic diffeomorphism, we can introduce ζ = m/β 4 as parameter in order to tune the resonances. Hence we verify non resonance conditions on the frequencies Ω j (ζ) = |j| 4 g + ζ, j ∈ Z d . (5.24) Lemma 5.11. Letḡ ∈ G. For any K ≤ N, consider K indexes j 1 , ..., j K with |j 1 | g < ... |j K | g ≤ N; and consider the determinant D ≥ C N ηK 2 , for some constant η ≡ η d > 0 depending only on the dimension d.
The proof was given in [2]. For sake of completeness we insert it.
Proof. For any i = 1, . . . , K, for any n = 0, . . . , K − 1, one computes for some constant C n = 0. This implies that where the matrix A is defined as This is a Van der Monde determinant. Thus we have which implies the thesis.
Exploiting this Lemma, and following step by step the proof of Lemma 12 of [2] one gets Lemma 5.12. Let g ∈ G. Then and for any r there exists τ ≡ τ r with the following property: for any positive γ small enough there exists a set I γ ⊂ (ζ 1 , ζ 2 ) such that ∀ζ ∈ I γ one has that for any N ≥ 1 and any multiindex J 1 , ..., J r with |J l | ≤ N ∀l, one has Moreover, End of the proof of Lemma 5.9 Let γ > 0. By recalling the diffeomorphism (5.23), one has that the set By the above result, one has that, if then (NR.2) holds and furthermore γ>0 I γ has full measure. Hence the claimed statement follows by defining B (res) := B \ γ>0 I γ .

The Quantum hydrodinamical system
We consider the following quantum hydrodynamic system on an irrational where m > 0, κ > 0, the function p belongs to C ∞ (R + ; R) and p(m) = 0. The function ρ(t, x) is such that ρ(t, x) + m > 0 and it has zero average in x. The variable x is on the irrational torus T d (as in the previous two applications). We assume the conditions p ′ (m) > 0 . (5.25) We shall use Theorem 2.9 in order to prove the following almost global existence result. In order to give a precise statement of the main result, we shall introduce the following notation. Given a function u : T d → C, we define Letḡ be a metric in the set of the admissible metrics G given in the definition 5.1. Exactly as in the case of the Beam equation, we consider a metric g of the form we shall use the parameter β in order to tune the resonances and to impose the non-resonance conditions required in order to apply Theorem 2.9. The precise statement of the long time existence for the QHD system is the following.
The key tool in order to prove the latter almost global existence result 5.13 is to use a change of coordinates (the so called Madelung transformation) which allows to reduce the system (QHD) to a semilinear Schrödinger type equation. We shall implement this in the next sections.

Madelung transform
For λ ∈ R + , we define the change of variable (Madelung transform) Notice that the inverse map has the form In the following lemma we state a well-posedness result for the Madelung transform.

(5.34)
Taking the real part of the first equation in (5.34) we obtain By (5.35), (5.34) and using that ∂ψ j H(α + z, α +z) = ∂z jH (α, z,z) , one obtains We resume the above discussion in the following lemma.
Lemma 5.17. The following holds.
(i) Let s > d 2 and There is C = C(s) > 1 such that, if C(s)δ ≤ 1, then the function z in (5.32) satisfies and where ω(D) is the Fourier multiplier with symbol Notice that, by using (5.25), the matrix C is bounded, invertible and symplectic, with estimates Consider the change of variables then the Hamiltonian (5.39) reads From the latter properties, one deduces that the perturbation is in the class P of Def. 2.4. The verification of (F.3) goes exactly as in the case of the Schrödinger equation, since also in this case ω j = |j| 2 g + O(1). The asymptotic condition (F.1) is also trivially fulfilled with β = 2. The main point is to verify the nonresonance conditions (F.2) and (NR.1), (NR.2). This will be done in the next subsection.

Non-resonance conditions for (QHD)
According to the section 5.2 on the Beam equation, we fix the metricḡ ∈ G and we consider g = βḡ, β 1 ≤ β ≤ β 2 . We shall verify the non-resonance conditions on the frequencies ω j in (5.40). By the property (5.20), one can verify non resonance conditions on Ω j . Since the map is an analytic diffeomorphism, we can introduce ζ = δ/β 2 as parameter in order to tune the resonances. Hence we verify non resonance conditions on the frequencies Lemma 5.20. Assume that the metricḡ ∈ G. For any K ≤ N, consider K indexes j 1 , ..., j K with |j 1 | g < . . . < |j K | g ≤ N; and consider the determinant Proof. The dispersion relation is slightly different w.r. to the one of the Beam equation, hence in this proof we just highlight the small differences w.r. to Lemma 5.11. For any i = 1, . . . , K, for any n = 0, . . . , K − 1, one computes ∂ n ζ Ω j i (ζ) = C n |j i |ḡ(|j i | 2 g + ζ) 1 2 −n for some constant C n = 0. This implies that where the matrix A is defined as This is a Van der Monde determinant. Thus we have which implies the thesis.
Exploiting this Lemma, and following step by step the proof of Lemma 12 of [2] one gets Lemma 5.21. Letḡ ∈ G. Then for any r there exists τ r with the following property: for any positive γ small enough there exists a set I γ ⊂ (ζ 1 , ζ 2 ) such that ∀ζ ∈ I γ one has that for any N ≥ 1 and any set J 1 , ..., J r with |J l | ≤ N ∀l, one has Moreover one has |[ζ 1 , ζ 2 ] \ I γ | ≤ Cγ 1/r .
By recalling the diffeomorphism (5.42), one has that the set Now, if we take β ∈ I γ and if r i=1 σ i ω j i = 0, one has that (recall (5.41)) By the above result, one has that, if then (NR.2) holds and furthermore γ>0 I γ has full measure. Hence the claimed statement follows by defining B (res) := B \ γ>0 I γ .

Stability of plane waves in NLS
Consider the NLS iψ t = −∆ g ψ + f (|ψ| 2 )ψ , with ν = |m| 2 g + f (a 2 ) and a > 0. In order to state the next stability theorem, we need that a suitable condition between f ′ (a 2 ) and the metric g is satisfied. For this reason, we slightly modify the definition of G 0 in 5.1. We then re-define G 0 in the following way: fix K > 0, we define The definition of the admissible set G is then the same in which one replace this new set G 0 with its hold definition. The main theorem of this section is the following.
We start by reducing the problem to a problem of stability of the origin of a system of the form (2.14).
First it is easy to see that introducing the new variables ϕ by ϕ(x, t) = e −im·x e −it|m| 2 ψ(x + 2mt, t) , then ϕ still fulfills (5.44), but ψ * ,m (x, t) is changed to ae −iνt with ν = f (a 2 ). The idea of [21] is to exploit that ϕ(x) = a appears as an elliptic equilibrium of the reduced Hamiltonian system obtained applying Marsden Weinstein procedure to (5.44) in order to reduce the Gauge symmetry. We recall that according to Marsden Weinstein procedure (following [21]), when one has a system invariant under a one parameter symmetry group, then there exists an integral of motion (the L 2 norm in this case), and the effective dynamics occurs in the quotient of the level surface of the integral of motion with respect to the group action. This is the same procedure exploited in section 5.3 for the QHD system. The effective system has a Hamiltonian which is obtained by restricting the Hamiltonian to the level surface. Such a Hamiltonian is invariant under the symmetry group associated to the integral of motion.
More precisely, consider the zero mean variable z(x) := 1 z j e ij·x , and the substitution ϕ(x) = e iθ ( a 2 − |T d | g z 2 L 2 + z(x)) (5.49) where θ ∈ T is a parameter along the orbit of the Gauge group, Notice that ϕ belongs to the level surface ϕ L 2 = a |T d | g and z(x) is the new free variable. In this case it also turns out that this is a canonical variable(as it can be verified by the theory of [3]). Thus the Hamiltonian for the reduced system turns out to be with ϕ given by (5.49). The explicit form of the Hamiltonian and its expansion were computed in [21] who showed that all the terms of the Taylor expansion of H a have zero momentum and that all the nonlinear terms are bounded, so, with our language, the nonlinear part is of class P. Considering the quadratic part, [21] showed that there exists a linear transformation preserving H s norms and the zero momentum condition, such that the quadratic part takes the form (2.15) with ω j = |j| 4 g − f ′ (a 2 )|j| 2 g . (5.50) The system is now suitable for the application of Theorem 2.9. We do not give the details, since the verification of the nonresonance and structural assumptions are done exactly in the same way as in the previous cases. Indeed one can prove the nonresonance conditions on the frequencies (5.50) reasoning as done in section 5.3.4.
Then, for any s > s 0 > d/2 there exists a constant C s,r > 0 such that one has X(u (1) , ..., u (r) ) s ≤ C s,r sup j,j 1 ,...,jr∈Z d |X j,j 1 ,...,jr | × then the thesis immediately follows. To deal with the case of different signs every time one has σ l = −1 one simply substitutes u (l) to u (l) . This concludes the proof.