Long-time stability of the quantum hydrodynamic system on irrational tori

We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a"non trivial"time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from $\varepsilon$-small initial conditions, remain bounded in $H^s$ for a time scale of order $O(\varepsilon^{-1-1/(d-1)+})$, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schr\"odinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising from three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes.This is due to the irrationality of the torus which prevent to have"good separation"properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable \emph{high/low} frequencies analysis, which gives an \emph{a priori} estimate on the solutions.


INTRODUCTION
We consider the quantum hydrodynamic system on an irrational torus of dimension 2 or 3    ∂ t ρ = −m∆φ − div(ρ∇φ) where m > 0, κ > 0, the function g belongs to C ∞ (R + ; R) and g(m) = 0. The function ρ(t, x) is such that ρ(t, x) + m > 0 and it has zero average in x. The space variable x belongs to the irrational torus T d ν := (R/2πν 1 Z) × · · · × (R/2πν d Z) , d = 2, 3 , (1.1) with ν = (ν 1 , . . . , ν d ) ∈ [1,2] d . We assume the strong ellipticity condition g ′ (m) > 0. (1.2) We shall consider an initial condition (ρ 0 , φ 0 ) having small size ε ≪ 1 in the standard Sobolev space H s (T d ν ) with s ≫ 1. Since the equation has a quadratic nonlinear term, the local existence Felice Iandoli has been supported by ERC grant ANADEL 757996. theory (which may be obtained in the spirit of [7,13]) implies that the solution of (QHD) remains of size ε for times of magnitude O(ε −1 ). The aim of this paper is to prove that, for generic irrational tori, the solution remains of size ε for longer times.
For φ ∈ H s (T d ν ) we define Our main result is the following.
Phase space and notation. In the paper we work with functions belonging to the Sobolev space of functions with zero average. Despite this fact we prefer to work with a couple of variable (ρ, φ) ∈ H s 0 (T d ν ) × H s (T d ν ) but at the end we control only the norm which in fact is the relevant quantity for (QHD). To lighten the notation we shall write · H s ν to denote · H s (T d ν ) . In the following we will use the notation A B to denote A ≤ CB where C is a positive constant depending on parameters fixed once for all, for instance d and s. We will emphasize by writing q when the constant C depends on some other parameter q.
Ideas of the proof. The general (EK) is a system of quasi-linear equations. The case (QHD), i.e. the system (EK) with the particular choice (1.6), reduces, for small solutions, to a semi-linear equation, more precisely to a nonlinear Schrödinger equation. This is a consequence of the fact that the Madelung transform (introduced for the first time in the seminal work by Madelung [18]) is well defined for small solutions. In other words one can introduce the new variable ψ := √ m + ρe iφ/ (see Section 2 for details), where = 2 √ k, obtaining the equation Since g(m) = 0, such an equation has an equilibrium point at ψ = √ m. The study of the stability of small solutions for (QHD) is equivalent to the study of the stability of the variable z = ψ − √ m. The equation for the variable z reads where f is a smooth function having a zero of order 2 at z = 0, i.e. |f (z)| |z| 2 , and |D| 2 ν is the Fourier multiplier with symbol The aim is to use a Birkhoff normal form/modified energy technique in order to reduce the size of the nonlinearity f (z). To do that, it is convenient to perform some preliminary reductions. First of all we want to eliminate the addendum −i mg ′ (m) z. In other words we want to diagonalize the matrix . (1.12) To achieve the diagonalization of this matrix it is necessary to rewrite the equation in a system of coordinates which does not involve the zero mode. We perform this reduction in Section 2.2: we use the gauge invariance of the equation as well as the L 2 norm preservation to eliminate the dynamics of the zero mode. This idea has been introduced for the first time in [11]. After the diagonalization of the matrix in (1.12) we end up with a diagonal, quadratic, semi-linear equation with dispersion law where j is a vector in Z d \ {0}. At this point we are ready to define a suitable modified energy. Our primary aim is to control the derivative of the H s -norm of the solution d dt wherez is the variable of the diagonalized system, for the longest time possible. Using the equation, such a quantity may be rewritten as the sum of trilinear expressions inz. We perturb the Sobolev energy by expressions homogeneous of degree at least 3 such that their time derivatives cancel out the main contribution (i.e. the one coming from cubic terms) in (1.13), up to remainders of higher order. In trying to do this small dividers appear, i.e. denominators of the form It is fundamental that the perturbations we define is bounded by some power of z H s , with the same s in (1.13), otherwise we obtain an estimate with loss of derivatives. Therefore we need to impose some lower bounds on the small dividers. Here it enters in the game the irrationality of the torus ν. We prove indeed that for almost any ν ∈ [1, 2] d , there exists γ > 0 such that if ±j 1 ± j 2 ± j 3 = 0, we denoted by M(d) a positive constant depending on the dimension d and µ i the i-st largest integer among |j 1 |, |j 2 | and |j 3 |. It is nowadays well known, see for instance [3,5], that the power of µ 3 is not dangerous if we work in H s with s big enough. Unfortunately we have also a power of the highest frequency µ 1 which represents, in principle, a loss of derivatives. However, this loss of derivatives may be transformed in a loss of length of the lifespan through partition of frequencies, as done for instance in [10,17,12,6].
Some comments. As already mentioned, an estimate on small divisors involving only powers of µ 3 is not dangerous. We may obtain such an estimate when the equation is considered on the squared torus T d , using as a parameter the mass m. In this case, indeed, one can obtain better estimates by following the proof in [11]. This is a consequence of the fact that the set of differences of eigenvalues is discrete. This is not the case of irrational tori with fixed mass, where the set of eigenvalues is not discrete. Having estimates involving only µ 3 one could actually prove an almost-global stability. More precisely one can prove, for instance, that there exists a zero Lebesgue measure set N ⊂ [1, +∞), such that if m is in [1, +∞) \ N , then for any N ≥ 1 if the initial condition is sufficiently regular (w.r.t. N) and of size ε sufficiently small (w.r.t. N) then the solution stays of size ε for a time of order ε −N . The proof follows the lines of classical papers such as [3,5,4] by using the Hamiltonian structure of the equation. More precisely, the system (QHD) can be written in the form where ∂ denotes the L 2 -gradient and H(ρ, φ) is the Hamiltonian function We do not know if the solution of (QHD) are globally defined. There are positive answers in the case that the equation is posed on the Euclidean space R d with d ≥ 3, see for instance [2]. Here the dispersive character of the equation is taken into account. For recent developments in this direction see [1] and reference therein. It is worth mentioning also the scattering result for the Gross-Pitaevsii equation [16]. Since we are considering the equation on a compact manifold, the dispersive estimates are not available. It would be interesting to obtain a long time stability result also for solutions of the general system (EK). In this case the equation may be not recasted as a semi-linear Schrödinger equation. Being a quasi-linear system, we expect that a para-differential approach, in the spirit of [17,12] should be applied. However, in this case, the quasi-linear term is quadratic, hence big. In [17,12] the quasi-linear term is smaller. Therefore new ideas have to be introduced in order to improve the local existence theorem. By using para-compositions (in the spirit of [9,14,15]), in the case d = 1, i.e. on the torus T 1 , it is possible to obtain stronger results. This is the argument of a future work of one of us with other collaborators [8].
2. FROM (QHD) TO NONLINEAR SCHRÖDINGER 2.1. 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 provide a well-posedness result for the Madelung transform.
The following holds.
(i) Let s > 1 and Proof. The bound (2.3) follows by (M) and classical estimates on composition operators on Sobolev spaces (see for instance [19]). Let us check the (2.4). By the first of (2.1), for any σ ∈ T, we have Therefore, by the arbitrariness of σ and using that ( Moreover we note that Then by the second in (2.1), (2.2), composition estimates on Sobolev spaces and the smallness condition We now rewrite equation (QHD) in the variable (ψ, ψ).
for some ε > 0 small enough. Then the function ψ defined in (M) solves We remark that the assumption of Lemma 2.2 can be verified in the same spirit of the local well-posedness results in [13] and [7].
Notice that the (2.8) is an Hamiltonian equation of the form Elimination of the zero mode. In the following it would be convenient to rescale the space variables x ∈ T d ν ν · x with x ∈ T d and work with functions belonging to the Sobolev space the Sobolev space in (1.10) with ν = (1, . . . , 1). By using the notation ψ = (2π) − d 2 j∈Z d ψ j e ij·x , we introduce the set of variables which are the polar coordinates for j = 0 and a phase translation for j = 0. Rewriting (2.14) in Fourier coordinates one has where H is defined in (2.14). We define also the zero mean variable By (2.16) and (2.18) one has ψ = (α + z)e iθ , (2.19) and it is easy to prove that the quantity is a constant of motion for (2.8). Using (2.16), one can completely recover the real variable α in terms of {z j } j∈Z d \{0} as (2.20) Note also that the (ρ, φ) variables in (2.1) do not depend on the angular variable θ defined above. This implies that system (QHD) is completely described by the complex variable z. On the other hand, using Taking the real part of the first equation in (2.21) we obtain We resume the above discussion in the following lemma.
There is C = C(s) > 1 such that, if C(s)δ ≤ 1, then the function z in (2.18) satisfies near the equilibrium z = 0. Note also that the natural phase-space for (2.28) is the complex Sobolev space H s 0 (T d ; C), s ∈ R, of complex Sobolev functions with zero mean. 2.3. Taylor expansion of the Hamiltonian. In order to study the stability of z = 0 for (2.28) it is useful to expand K m at z = 0. We have where for any r = 3, · · · , N − 1, K m (z, z) is an homogeneous multilinear Hamiltonian function of degree r of the form The vector field of the Hamiltonian in (2.29) has the form (recall (1.14)) (2.36) Therefore system (2.32) becomes (2.37)

SMALL DIVISORS
As explained in the introduction we shall study the long time behaviour of solutions of (2.37) by means of Birkhoff normal form approach. Therefore we have to provide suitable non resonance conditions among linear frequencies of oscillations ω(j) in (2.33). This is actually the aim of this section.
Throughout this section we assume, without loss of generality, |j 1 | a ≥ |j 2 | a ≥ |j 3 | a > 0, for any j i in Z d , moreover, in order to lighten the notation, we adopt the convention ω i := ω(j i ) for any i = 1, 2, 3. The main result is the following.
The proof of this proposition is divided in several steps and it is postponed to the end of the section. The main ingredient is the following standard proposition which follows the lines of [3,6]. Here we give weak lower bounds of the small divisors, these estimates will be improved later.
Proposition 3.2. Consider I and J two bounded intervals of R + \ {0}; r ≥ 2 and j 1 , . . . , j r ∈ Z d such that j i = ±j k if i = k, n 1 , . . . n r ∈ Z \ {0} and h : J d−1 → R measurable. Then for any γ > 0 we have Remark 3.3. We shall apply this general proposition only in the case r = 3, however we preferred to write it in general for possible future applications.

Proof of Prop. 3.2.
For simplicity in the proof we assume |j 1 | (1,b) > . . . > |j r | (1,b) . Since by assumption we have j i = j k for any i = k then one could easily prove that for any η > 0 (later it will be chosen in function of γ) we have We define P η = ∪ i =k P i,k η , and We have to estimate from above the measure of the last set. We define the function For any ℓ ≥ 1 we have Therefore we can write the system of equations    . . .
We denote by V the Vandermonde matrix above. We have that V is invertible since where in the penultimate passage we have used that b / ∈ P η and j 1 2 · · · j r 2 r,n η r j 1 2 · · · j r 2 .
At this point we are ready to use Lemma 7 in appendix A of the paper [20], we obtain Summarizing we obtained we may optimize by choosing η = γ 1 2r ( j 1 · · · j r ) 1 r and we obtain the thesis.
As a consequence of the preceding proposition we have the following.
Corollary 3.4. Let r ≥ 1, consider j 1 , . . . , j r ∈ Z d such that j k = j i if i = k and n 1 , . . . , n k ∈ Z \ {0}. For any γ > 0 we have where we have set The determinant of its inverse is bounded by a constant depending only on d. Therefore the result follows by applying Prop. 3.2 and the change of coordinates (a 1 , . . . , a d ) → ( 1 a 1 , b). Owing to the corollary above we may reduce in the following to the study of the small dividers when we have 2 frequencies much larger then the other. (3.4). If there exists i ∈ {1, . . . , d} such that

5)
then for anyγ > 0 we have Proof. We give a lower bound for the derivative of the functionΛ with respect to the parameter a i .
Therefore a i →Λ is a diffeomorphism and applying this change of variable we get the thesis.
Proposition 3.6. There exists a set of full Lebesgue measure A 3 ⊂ (1, 4) d such that for any a in A 3 there exists γ > 0 such that for any σ ∈ ±1, for any j 1 , j 2 , j 3 in Z d satisfying |j 1 | a > |j 2 | a ≥ |j 3 | a , the momentum condition σj 3 + j 2 − j 1 = 0 and
We are now in position to prove Prop. 3.1.
Proof of Prop. 3.1. The case σ 1 σ 2 = 1 is trivial, we give the proof if σ 1 σ 2 = −1. From Prop. 3.6 we know that there exists a full Lebesgue measure set A 3 and γ > 0 such that the statement is proven if |j 3 | ≤ J(j 1 , γ). Let us now assume |j 3 | > J(j 1 , γ). Let us define whereγ will be chosen in function of γ and M(d) big enough w.r.t. d. (3.6)) and Corollary 3.4 with r = 3, we have

Let us set
If the exponent M(d) (and hence p) is chosen large enough we get the summability in the r.h.s. of the inequality above. We now chooseγ 1/6 γ −p = γ m , we eventually obtain µ(B γ ) γ m . If one can reason similarly. The wanted set of full Lebesgue measure is therefore obtained by choosing A := A 3 ∩ (∪ γ>0 B c γ ).

ENERGY ESTIMATES
In this section we construct a modified energy for the Hamiltonian K m in (2.36). We first need some convenient notation.
We define G  Proof. Fix c 0 > 0. By (4.5) and Lemma 4.5, we deduce Proof of Lemma 4.8. We study how the equivalent energy norm E s (w) defined in (4.20) evolves along the flow of (4.17). Notice that ∂ t E s (w) = −{E s , H}(w) .
Proof of Theorem 1.1. In the same spirit of [13], [7] we have that for any initial condition (ρ 0 , φ 0 ) as in (1.4) there exists a solution of (QHD) satisfying for some T > 0 possibly small.