Joseph Thirouin
email: joseph.thirouin@math.u-psud.fr

On the growth of high Sobolev norms for certain one-dimensional Hamiltonian PDEs

Keywords: 37K40, 35B45 Hamiltonian systems, fractional nonlinear Schrödinger equation, nonlinear wave equation, dispersive properties

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus :

where α is a real parameter. We show that, apart from the case α = 1, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.

MSC 2010

Introduction

In the study of Hamiltonian partial differential equations, understanding the large time dynamics of solutions is an important issue. In usual cases, the conservation of the Hamiltonian along trajectories enables to control one Sobolev norm of the solution (in the so-called energy space), but when solutions are globally defined and regular, higher norms could grow despite the conservation laws, reflecting an energy transfer to high frequencies. Even for notorious equations, such as the nonlinear Schrödinger equation on manifolds, it is an old problem to know whether such instability occurs [START_REF] Bourgain | Problems in Hamiltonian PDE's[END_REF], and often still an open question.

We start from the particular case of the defocusing Schrödinger equation on the torus of dimension one, with a cubic nonlinearity :

i∂ t u = -∂ 2 x u + |u| 2 u, u(0, •) = u 0 . (1)
Here, u is a function of time t ∈ R and of space variable x ∈ T, and u 0 ∈ H 1 (T). Equation ( 1) is Hamiltonian, and because of the energy conservation, its trajectories are bounded is H 1 (T). But it is also well-known that (1) is integrable (see [START_REF] Grébert | The defocusing NLS equation and its normal form[END_REF], [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF]), with conservation laws ensuring that if u 0 belongs to H s (T) for some s ∈ N\{0}, then the solution u remains bounded in H s (this is even true for any real s ≥ 1 [START_REF] Kappeler | Scattering-like phenomena of the periodic defocusing NLS equation[END_REF]).

1

In order to track down large time instability for Hamiltonian systems, Majda, McLaughlin and Tabak [START_REF] Majda | A one-dimensional model for dispersive wave turbulence[END_REF] suggested to replace the Laplacian in (1) by a whole family of pseudo-differential operators : the operators |D| ρ (sometimes written as ( -∂ 2

x ) ρ ) for real ρ. Recall that if w = k∈Z w k e ikx is a function on the torus, then

|D| ρ w = k∈Z |k| ρ w k e ikx .
So we consider the following fractional Schrödinger equation :

i∂ t u = |D| α u + |u| 2 u, u(0, •) = u 0 , (2)
where α is any positive number. If α = 2, we recognize the classical Schrödinger equation [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF]. In the case α = 1, (2) is a non-dispersive equation, called the (defocusing) "half-wave" equation :

i∂ t u = |D|u + |u| 2 u, u(0, •) = u 0 . (3)
This half-wave equation has been studied by Gérard and Grellier in [START_REF] Gérard | Effective integrable dynamics for a certain nonlinear wave equation[END_REF]. In particular, they showed that the dynamics of ( 3) is related to the behaviour of the solutions of a toy model equation, called the cubic Szegő equation :

i∂ t u = Π + (|u| 2 u), u(0, •) = u 0 , (4)
where Π + := 1 D≥0 is the projection onto nonnegative Fourier modes. In a more precise way, equation (4) appears to be the completely resonant system of (3). All the equations (2) derive from the Hamiltonian H α (u) := 1 2 (|D| α u, u) + 1 4 u 4 L 4 for the symplectic structure endowed by the form ω(u, v) = ℑm(u, v), where (u, v) := T uv denotes the standard inner product on L 2 (T). The functional H α is therefore conserved along trajectories. Gauge invariance as well as translation invariance also imply the existence of two other conservation laws for equation [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF] :

Q(u) := 1 2 u 2 L 2
M (u) := (Du, u), where D := -i∂ x , i.e. the mass and the momentum respectively. Starting from these observations, it has been proved that for α = 1, equation (3) admits a globally defined flow in H s with s ≥ 1 2 (see [START_REF] Gérard | Effective integrable dynamics for a certain nonlinear wave equation[END_REF]). In the case of the half-wave equation, the Brezis-Gallouët inequality [START_REF] Brezis | Nonlinear Schrödinger evolution equations[END_REF] also ensures that H s -norms of solutions grow at most like e exp B|t| , for some constant B > 0 depending on s and on the initial data.

The question of the large time instability of global solutions of (2) thus naturally arises : is it possible to find smooth initial data whose corresponding orbits are not bounded in some space H s , or at least not polynomially bounded 1 ?

The cubic Szegő equation discloses this kind of instability, as recently shown in [START_REF] Gérard | On the growth of Sobolev norms for the cubic Szegő equation[END_REF], [START_REF] Gérard | The cubic Szegő equation and Hankel operators[END_REF] : for generic smooth initial data, the corresponding solution of the Szegő equation in H s is polynomially unbounded, for any s > 1 2 . Therefore it is reasonable to think that the same statement should hold for the half-wave equation [START_REF] Bourgain | Problems in Hamiltonian PDE's[END_REF], though such a result seems far beyond our reach at this point. Nevertheless the theorem we prove in this paper gives an a priori bound for all solutions of the half-wave equation : Theorem 1. Let u 0 ∈ C ∞ (T), and t → u(t) the solution of the half-wave equation (3) such that u(0) = u 0 . Given any integer n ≥ 0, there exist real constants B, C > 0 such that ∀t ∈ R,

u(t) H 1+n ≤ Ce B|t| 2 .
(

) 5
Here, C depends on n and u 0 H 1+n , whereas B can be chosen equal to B n u 0 8 H 1/2 , with B n > 0 depending only on n.

The bound appearing in [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF] is an improvement the "double exponential bound" mentioned above. But as a matter of fact, finding any explicit non-trivial solution of [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF] is still an open problem, and nothing is known about the optimality of [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF]. Solution with rapidly growing H s -norms could perfectly well exist. H. Xu [START_REF] Xu | Large time blowup for a perturbation of the cubic Szegő equation[END_REF] typically proved the existence of exponentially growing solutions for a perturbation of the Szegő equation. See also the result of Hani-Pausader-Tzvetkov-Visciglia [START_REF] Hani | Modified scattering for the cubic Schrödinger equation on product spaces and applications[END_REF], in the context of the Schrödinger equation, as well as its recent counterpart in [START_REF] Xu | Unbounded sobolev trajectories and modified scattering theory for a wave guide nonlinear schrödinger equation[END_REF].

Notice that in the case of the Szegő equation, the best bound quantifying the growth of Sobolev norms of solutions is e B|t| : it is obtained by Gérard and Grellier in [8, section 3]. Hence ( 5) is likely to be improved, but recall that, as far as we know, the only way of proving the simple exponential bound for Szegő solutions makes use of the Lax pair structure associated with the equation. Elementary methods would only give an e B|t| 2 bound (see Appendix A). Unfortunately, such a Lax pair structure apparently does not exist as regards the half-wave equation.

Even so, the proof of (5) in theorem 1 suggests that we could get a simple exponential bound, instead of e B|t| 2 , if we could deal with a quadratic nonlinearity. Simply putting an L 3 -norm in the energy H 1 , instead of the L 4 one, would give rise to a nonlinearity of the form |u|u, and the singularity at the origin may lead to solutions less regular than their initial data. It is possible to avoid this phenomenon, by consider a system of two equations rather than a single scalar equation :

   i∂ t u 1 = |D|u 1 + u 2 u 1 , i∂ t u 2 = |D|u 2 + u 2 1 2 , (6)
with

(u 1 , u 2 )| t=0 = (u 0 1 , u 0 2 )
. System (6) only involves (analytic) quadratic nonlinearities. It happens that Schrödinger systems of that kind frequently appear in physics : they are closely linked with the SHG (Second-Harmonic Generation) theory in optics, and the study of propagation of solitons in so-called χ (2) (or quadratic) media or materials (for a review, see e.g. [18, section 4]). Quadratic systems are also relevant in fluid mechanics, to describe the interaction between long nonlinear waves in fluid flows [START_REF] Gottwald | Stable two-dimensional parametric solitons in fluid systems[END_REF]. From a mathematical point of view, interest on quadratic systems is more recent [START_REF] Ozawa | On a system of nonlinear schrödinger equations with quadratic interaction[END_REF].

For the case of system (6), we prove the following theorem :

Theorem 2. Let (u 0 1 , u 0 2 ) ∈ C ∞ (T) × C ∞ (T), and t → (u 1 (t), u 2 (t)) the solution of (6) such that (u 1 (0), u 2 (0)) = (u 0 1 , u 0 2 ). Given any integer n ≥ 0, there exist real constants B ′ , C > 0 such that ∀t ∈ R, u 1 (t) H 1+n , u 2 (t) H 1+n ≤ Ce B ′ |t| . (7)
Here, C depends on n and on the sum u 0 1 H 1+n + u 0 2 H 1+n , whereas B ′ can be chosen equal to

B ′ n ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 ), with B ′ n > 0 depending only on n.
Let us now return to equation [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF]. When α = 1, (2) has dispersive properties. Using them for α > 1 and proving some Strichartz estimate for the operator e -it|D| α , Demirbas, Erdoğan and Tzirakis show in [START_REF] Demirbas | Existence and Uniqueness theory for the fractional Schrödinger equation on the torus[END_REF] that ( 2) is globally well-posed in the energy space H α 2 (and even below). Their method rely on Bourgain's high-low frequency decomposition, but it does not say anything about the possible growth of solutions.

Still for α > 1, a naive calculation leads to an exponential bound for H s -norms of solutions, i.e. a bound of the form e A|t| , but results such as Bourgain's [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF] or Staffilani's [START_REF] Staffilani | On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations[END_REF] suggest that because of dispersion, solutions should be polynomially bounded. On the other hand, the polynomial growth of solutions of (2) for α > 1 is announced to be true in the work by Demirbas-Erdoğan-Tzirakis. Indeed we establish a theorem also involving (part of) the case α < 1 :

Theorem 3. Let α ∈ ( 2 3 , 1) ∪ (1, 2)
, and u 0 ∈ C ∞ (T). There exist a unique u ∈ C ∞ (R, C ∞ (T)) solution of (2) with u(0) = u 0 . Furthermore, given any integer n ≥ 0, there exist real constants A, C > 0, and a constant C ′ > 0 depending only on u 0 H α+n , such that ∀t ∈ R,

u(t) H α+n ≤ C ′ (1 + C|t|) A . (8)
When α ∈ (1, 2), we can choose

A = 2n + α α -1 , C = C α,n u 0 4+ α-1 2n+α H α/2
, where C α,n > 0 is a constant which only depends on α and n.

When α ∈ ( 2 3 , 1), a possible choice is

A = 2n(23α -2) (2α -1)(3α -2) + 10α 3α -2 .
Here again, it is not known whether (8) is optimal or not. When α > 1, Demirbas [START_REF] Demirbas | Almost sure global well-posedness for fractional cubic Schrödinger equation on torus[END_REF] found, by a probabilistic way inspired by the works of Bourgain [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF], solutions growing at most like a power of log(1 + |t|). But in any case, proving that some solutions of (2) do blow up for large time, even at a very low rate, would be a big step forward.

Combining theorem 1 and 3 thus indicate that α = 1 is an isolate point in the family of equations [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF]. Notice that, when α < 1, theorem 3 includes the existence of a flow, which had not been proved so far. As for the condition α > 2 3 , it appears to be convenient in the proof for technical reasons ; but since the heart of our work is to prove that the case α = 1 is more likely to disclose weak turbulence phenomena than other cases, we postpone discussions and comments concerning the relevance of the value 2 3 until Appendix B.

The proof of theorems 1, 2 and 3 is based on an idea developped by Ozawa and Visciglia in [START_REF] Ozawa | An improvement on the Brezis-Gallouët technique for 2D NLS and 1D half-wave equation[END_REF] : the authors introduce a modified energy method, in order to sharpen H 1 -estimates and thus prove well-posedness for the half-wave equation with quartic nonlinearity. To put it shortly, their idea is to introduce a nonlinear energy which is in fact a perturbation of the norm they wish to bound. The perturbation does not modify the size of the norm, but induces simplifications while differentiating, so that time-differentiation behaves like a self-adjoint operator.

In the sequel of this paper, we begin by proving theorems 1, 2, and the first part of theorem 3, with elementary tools. Then we address the case of α < 1, using Bourgain spaces as in [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF], which we fully develop for the convenience of the reader.

We were about to finish this paper when we were informed of a work by Planchon and Visciglia also applying the modified energy method to solutions of nonlinear Schrödinger equations on certain Riemaniann manifolds, and for every power nonlinearity.

The author would like to express his gratitude towards P. Gérard for his deep insight and generous advice. He also thanks J.-C. Saut for the references concerning quadratic medias.

The case α ≥ 1

Throughout this section, we suppose that α ∈ [1, 2). We fix u 0 ∈ C ∞ (T), and we study t → u(t), the associated solution of (2) (or (3), depending on the value of α).

The modified energy method

Fix n ∈ N. The following lemma gathers some standard inequalities of which we will make an extensive use.

Lemma 2.1.

There exist an absolute constant C > 0, a constant C α depending on α and a constant C n depending on n such that, for every t ∈ R,

(i) u(t) H α/2 ≤ C u 0 2 H α/2 , (ii) if α > 1, u(t) L ∞ ≤ C α u 0 2 H α/2 , (iii) if α = 1, u(t) L ∞ ≤ C n u 0 2 H 1/2 log 1 + u(t) 2 H 1+n C 2 u 0 4 H 1/2 .
We justify briefly these inequalities : (i) derives from the conservation of Q + H α together with the Sobolev embeddings in dimension one, and (ii) is a consequence of (i) and the injection H α/2 ֒→ L ∞ . As for (iii), it follows from the classical Brezis-Gallouët inequality : for s > 1 2 , and w ∈ H s (T),

w L ∞ ≤ C s w H 1/2 log 1 + w H s w H 1/2 1/2 (9)
(see e.g. [START_REF] Gérard | The cubic Szegő equation[END_REF]). Here, to infer (iii), we begin by squaring the ratio of the two norms, and we then take into account the fact that the function x → x log(1 + 1 x 2 ) is increasing.

To prove the estimates we have in mind, we are going to establish an inequality between the H α+n -norm of the solution and its derivative, and apply a Gronwall lemma. As announced, we define for this purpose a well-chosen nonlinear functional2 :

E α,n (u) := u 2 L 2 + |D| α+n u 2 L 2 + 2ℜe(|D| α+n u, |D| n (|u| 2 u)) -1 2 |D| α 2 +n (|u| 2 ) 2 L 2 . ( 10
)
Roughly speaking, E α,n is a perturbation of the square of the H α+n -norm of u (the first two terms) by the means of two corrective quantities. First of all, let us show that the latter do not substantially modify the size of u 2 H α+n . To turn this into a rigorous statement, we begin by restricting ourselves to intervals of time on which u H α+n is larger than a certain constant M depending on u 0 H α/2 , and we show that on such intervals,

1 2 u 2 H α+n ≤ E α,n (u) ≤ 2 u 2 H α+n , (11)
for a suitable choice of M which we precise later. Set J 1 (u) := 2ℜe(|D| α+n u, |D| n (|u| 2 u)). We can write3

|J 1 (u)| |D| α+n u L 2 |D| n (|u| 2 u) L 2 u H α+n u 2 L ∞ u H n
, where we used the tame estimates for products in H s , for s ≥ 0. Then interpolate H n between H α+n and H α/2 (or just bound the L 2 -norm by a constant if n = 0), and using lemma 2.1, get

|J 1 (u)| u 2-εα,n H α+n u 0 2εα,n H α/2 u 2 L ∞ , where ε α,n := min 1, 2α 2n + α .
On the other side, introducing J 2 (u) := -|D| α 2 +n (|u| 2 ) 2 L 2 , we similarly obtain :

|J 2 (u)| u 2 L ∞ u 2 H α 2 +n u 2-εα,n H α+n u 0 2εα,n H α/2 u 2 L ∞ .
In sight of (ii) and (iii), all these estimates show that J 1 (u) and J 2 (u) are of lower order than u 2 H α+n . More precisely, if we now set, for real x,

g α (x) :=          x εα,n/2 if α > 1, x ε 1,n /2 log 1 + x 2 C 2 u 0 4 H 1/2 if α = 1,
we see that it suffices to request for instance that g α ( u H α+n ) ≫ u 0 4+2εα,n H α/2 , which holds true whenever u H α+n is greater than a certain M. As a conclusion, ( 11) is proved on intervals of the form [T * , T ] which satisfy

u(t) H α+n ≥ M := max{M, 2 u 0 H α+n }, ∀t ∈ [T * , T ], and u(T * ) H α+n = M. Now we study the evolution of E α,n (u) on [T * , T ]. As the L 2 -norm of u is conserved, we denote by J 0 (u) := |D| α+n u 2 L 2
, and compute at once :

d dt J 0 (u) = 2ℜe(|D| α+n u, |D| α+n u),
where the dot refers to the time-derivative, and commutes with |D| ρ for all ρ. According to equation ( 2),

|D| α u = i u -|u| 2 u, so d dt J 0 (u) = 2ℑm(|D| α+n u, |D| n u) -2ℜe(|D| α+n u, |D| n (|u| 2 u)).
Because of the imaginary part, the first term of this sum is zero. As for the second one, it combines with the time-derivative of J 1 (u), and we thus have

d dt [J 0 + J 1 ](u) = 2ℜe(|D| α+n u, |D| n (|u| 2 u)˙) = 2ℜe ∂ n x |D| α u, ∂ n x ( u|u| 2 + u[|u| 2 ]˙) .
Applying Leibniz formula, we get three terms (or only two when n = 0) :

2ℜe(∂ n x |D| α u, (∂ n x u)|u| 2 ) + 2ℜe ∂ n x |D| α u, n k=1 n k [(∂ n-k x u)∂ k x (|u| 2 ) + (∂ k x u)∂ n-k x (|u| 2 )˙ + 2ℜe(∂ n x |D| α u, u∂ n x (|u| 2 )˙).
Each of these terms has to be estimated. The first one and the third one are more tricky, since all the time-and space-derivative are concentrated on the same function. First term : A simplification fortunately occurs. Rewrite

∂ n x |D| α u = i∂ n x u -∂ n x (|u| 2 u),
and observe that ℜe(i∂

n x u, (∂ n x u)|u| 2 ) = 0. The first term then equals -2ℜe(∂ n x (|u| 2 u), (∂ n x u)|u| 2 ). Let Q 1 be this new quantity. Assuming that n ≥ 1, we can bound |Q 1 | |u| 2 u H n u 2 L ∞ u H n u 4 L ∞ u H n ( u H α+n + u 2 L ∞ u H n ).
Indeed, because of the equation, we have u

H s u H α+s + u 2 L ∞ u H s for any s ≥ 0, so that u H n u H α+n (using again the property of the interval [T * , T ]). Hence |Q 1 | u 2-εα,n H α+n u 0 2εα,n H α/2 u 4
L ∞ with the same ε α,n as above (notice that it is true even if n = 0). Second term : As announced, we suppose here that n ≥ 1, and fix a

k ∈ {1, • • • , n}. We must estimate Q (k) 2 := 2ℜe(∂ n x |D| α u, (∂ n-k x u)∂ k x (|u| 2 )).
Using the Sobolev embedding H 1/4 ֒→ L 4 as well as tame estimates again, write

|Q (k) 2 | ∂ n x |D| α u L 2 ∂ n-k x u L 4 ∂ k x (|u| 2 ) L 4 u H α+n u H α+n-k+ 1 4 u L ∞ u H k+ 1 4 .
Interpolate the H s -norms between H α/2 et H α+n , and get finally

|Q (k) 2 | u 2-θ(α,n) H α+n u 0 2+2θ(α,n) H α/2 u L ∞ ,
where θ(α, n) := α-1 2n+α . In the same way, Q

′(k) 2 := 2ℜe(∂ n x |D| α u, (∂ k x u)∂ n-k x (|u| 2 )˙
) can be proven to be controlled by the same quantity (with the same exponents).

Third term : This term is the most delicate. We have

2ℜe(∂ n x |D| α u, u∂ n x (|u| 2 )˙) = 2ℜe ū∂ n x |D| α u, ∂ n x (|u| 2 )˙ ≃ 2ℜe ∂ n x (ū|D| α u), ∂ n x (|u| 2 )˙ ,
where the ≃ sign means that the equality is true up to terms of order u

2-θ(α,n) H α+n u 0 2+2θ(α,n) H α/2
u L ∞ (which we control in the same way as for the second term). Thus we would like to estimate

Q 3 := 2ℜe(|D| n (ū|D| α u), |D| n (|u| 2 )˙), but as |u| 2 is real, it appears, expanding the real part, that Q 3 = (|D| n (ū|D| α u + u|D| α ū), |D| n (|u| 2 )˙
), which combines with the time-derivative of J 2 (u), and finally leads to the following expression :

|D| n [ū|D| α u + u|D| α ū -|D| α (ūu)] , |D| n (|u| 2 )˙ . ( 12
)
Now we have a very simple Leibniz lemma on the operator |D| α , which we will prove in section 2.2 :

Lemma 2.2. Let α ∈ [1, 2).
For any integer n ∈ N, there is a constant C n > 0 depending only on n, such that for all function u ∈ H α+n (T),

ū|D| α u + u|D| α ū -|D| α (ūu) H n ≤ C n u 1+ α-1 2n+α H α/2 u 1-α-1 2n+α H α+n .
Such a result is better than the crude L ∞ -H α+n estimate, because of the exponent of the H α+n -norm (which is strictly less than 1 as soon as α > 1).

Consequently, expression ( 12) is controlled by u

2-θ(α,n) H α+n u 0 2+2θ(α,n) H α/2
u L ∞ as well. To sum up, only the second and the third term really matter, whence

d dt E α,n (u) u 2-θ(α,n) H α+n u 0 2+2θ(α,n) H α/2 u L ∞ .
First, assume α > 1. In this situation, we can incorporate the

L ∞ -norm into the H α/2 one (see (ii) of lemma 2.1), so for t ∈ [T * , T ], t T * d dτ E α,n (u)dτ ≤ t T * d dτ E α,n (u) dτ u 0 4+2θ(α,n) H α/2 t T * u(τ ) 2-θ(α,n) H α+n dτ.
Furthermore, remembering our estimates (11

) on E α,n (u), t T * d dτ E(u)dτ = |E(u)(t) -E(u)(T * )| ≥ 1 2 u(t) 2 H α+n -2 u(T * ) 2 H α+n .
Let f (t) := u(t) 2 H α+n . The above calculation ensures that for some C α,n > 0 depending on α and n,

f (t) ≤ 4f (T * ) + C α,n u 0 4+2θ(α,n) H α/2 t T * f (τ ) 1-1 2 θ(α,n) dτ. Now θ(α, n) is positive.
A "Gronwall's lemma" argument (which is also known as "Osgood's lemma") thus proves that

f (t) ≤ 4f (T * )(1 + C|t -T * |) A for t ∈ [T * , T ].
Notice that the value of f (T * ), i.e. of M, only depends on the value of u 0 H α+n . Moreover, this inequality remains true even for t outside any interval of type [T * , T ], so it globally holds and the first part of theorem 3 is proved. At last, the constant A can be set to 2 θ(α,n) , i.e. 4n+2α α-1 , which implies the statement.

It remains to consider the case α = 1. This time, θ(1, n) = 0 for any n, and in addition, the L ∞ -norm of u is not bounded by a constant anymore. Using part (iii) of lemma 2.1, and going on as in the previous case with an auxiliary function

g(t) = u(t) 2 H 1+n /C 2 u 0 4 H 1/2 , we find, g(t) ≤ 4g(T * ) + C n u 0 4 H 1/2 t T * g(τ ) log(1 + g(τ ))dτ, for all t ∈ [T * , T ].
Osgood's lemma then yields g(t) ≤ Cg(T * )e B|t| 2 , and the proof of theorem 1 is complete.

A Leibniz lemma

We eventually turn to the Proof of lemma 2.2. Let u = k∈Z u k e ikx ∈ H α+n (T). We intend to control the H n -norm of F α (u) := ū|D| α u + u|D| α ū -|D| α (ūu). A straightforward computation yields

F α (u) = +∞ k=-∞ e ikx +∞ l=-∞ (|l| α + |k -l| α -|k| α ) u l u l-k .
The key idea is to replace |l| α + |k -l| α -|k| α by a more symmetric coefficient, and then to recognize a convolution product. More precisely, define a continuous function ϕ of the real variable

x ∈ R \ {0, 1} with ϕ(x) := |x| α + |1 -x| α -1 |x| α 2 |1 -x| α 2
, and ϕ(0) = ϕ(1) = 0. For every l, k ∈ Z with k = 0, we then have

|l| α + |k -l| α -|k| α = ϕ( l k )|l| α 2 |k -l| α 2
, which is true even if k = 0, once we have assigned ϕ(±∞) = 2, by convention. Now, to show that ϕ is bounded on R, it suffices to check that it is bounded near 0 and -∞, and then to invoke the symmetry of ϕ around x = 1 2 . And actually, since α ≤ 2, lim x→0 ϕ(x) = 0 and lim x→-∞ ϕ(x) = 2. Studying the variations of ϕ even show that ϕ L ∞ (R) = 2, and hence is independent of α.

As a consequence, +∞ k=-∞ |k| 2n +∞ l=-∞ (|l| α + |k -l| α -|k| α )u l u l-k 2 ≤ 2 +∞ k=-∞ |k| 2n +∞ l=-∞ |l| α 2 |k -l| α 2 |u l ||u l-k | 2 ≤ 2 2n +∞ k=-∞   +∞ l=-∞ |l| α 2 +n |k -l| α 2 |u l ||u l-k | 2 + +∞ l=-∞ |l| α 2 |k -l| α 2 +n |u l ||u l-k | 2   , because of the inequality |k| n ≤ 2 n-1 (|l| n + |k -l| n ), satisfied for any n ≥ 1, any k and l.
When n = 0, the result is rather immediate, since

+∞ k=-∞ +∞ l=-∞ |l| α 2 |k -l| α 2 |u l ||u l-k | 2 = |D| α 2 ũ 2 2 L 2 = |D| α 2 ũ 4 L 4 ,
where ũ := k∈Z |u k |e ikx . Using as always the embedding L 4 ֒→ H 1/4 , and interpolating between α/2 and α, we get the result.

From here on, we suppose n = 0. We shall deal with the first part of the above sum (the last one follows identically). We consider two sequences v := (|l|

α 2 +n |u l |) l∈Z and w := (|l| α 2 |u -l |) l∈Z . With these notations, +∞ k=-∞ +∞ l=-∞ |l| α 2 +n |k -l| α 2 |u l ||u l-k | 2 = v ⋆ w 2 ℓ 2 . By Schur's lemma, v ⋆ w ℓ 2 ≤ v ℓ 2 w ℓ 1 . But v ℓ 2 ≤ u H α 2 +n . As for w ℓ 1 , write +∞ k=-∞ |w k | = |k|≤N |w k | + |k|>N |w k | ≤   |k|≤N |w k | 2   1 2 √ 2N + 1 +   |k|>N |w k | 2 (1 + |k| 2 )   1 2   |k|>N 1 1 + |k| 2   1 2 ≤ √ 2N + 1 w ℓ 2 + 2 N w h 1 .
Taking the infimum over N ∈ N, we finally get

w ℓ 1 ≤ w ℓ 2 w h 1 . In our case, w ℓ 2 ≤ u H α 2 ,

and similarly

w h 1 ≤ u H α 2 +1
. Now we can conclude :

F α (u) 2 H n u 2 H α 2 +n u H α 2 u H α 2 +1 u 2(1+ α-1 2n+α ) H α 2 u 2(1-α-1 2n+α )
H α+n , which corresponds to the statement. Remark 1. Lemma 2.2 probably takes the best advantage of the form of [START_REF] Ginibre | Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain)[END_REF]. There is still a kind of "virial" identity which holds for any value of α :

d dt |u| 2 = i (u|D|ū -ū|D|u) , but the inequality u|D|ū -ū|D|u L 2 u H 1/2 u H 1 , for instance, is false. As a counter- exemple, choose u N (x) = 1 √ log N N n=1 e inx n ,
and let N → +∞.

Quadratic half-wave equations

We come to the system of equations ( 6) and the proof of [START_REF] Demirbas | Existence and Uniqueness theory for the fractional Schrödinger equation on the torus[END_REF]. To see the Hamiltonian structure of ( 6), choose L 2 (T) × L 2 (T) as a phase space, endowed with the inner product

(u 1 , u 2 ), (v 1 , v 2 ) := (u 1 , v 1 ) + (u 2 , v 2 ).
Taking the imaginary part of •, • as our symplectic form, we infer from a simple calculation that the Hamiltonian

H(u 1 , u 2 ) := 1 2 [(|D|u 1 , u 1 ) + (|D|u 2 , u 2 ) + T ℜe(u 2 1 u 2 )
] is associated to the system [START_REF] Demirbas | Almost sure global well-posedness for fractional cubic Schrödinger equation on torus[END_REF]. Notice that the functional H is invariant under the flow (u 1 , u 2 ) → (e iθ u 1 , e 2iθ u 2 ), with θ varying in R. It follows then from the Noether theorem that

Q(u 1 , u 2 ) := u 1 2 L 2 + 2 u 2 2
L 2 is a conservation law for the system [START_REF] Demirbas | Almost sure global well-posedness for fractional cubic Schrödinger equation on torus[END_REF]. As a consequence, the L 2 -norms of u 1 , u 2 stay bounded along the flow lines. In addition, the conservation of H as well as Q claims the uniform boundedness of u 1 H 1/2 and u 2 H 1/2 with respect to time.

Immediately, we get, for each n ≥ 0,

u 1 H 1+n , u 2 H 1+n e B|t| 2 ,
where B > 0 is independent of time : this follows from a straightforward application of inequality (iii) in lemma 2.1. But now, set F (t) := ( u 1 2

H 1+n + u 2 2 H 1+n ).
Here we won't repeat the details of section 2.1, but we suppose from the beginning that F is "big enough", and we compute

d dt |D| 1+n u 1 2 L 2 + |D| 1+n u 2 2 L 2 = 2ℜe (|D| 1+n u1 , |D| 1+n u 1 ) + (|D| 1+n u2 , |D| 1+n u 2 ) = -2ℜe (|D| 1+n u1 , |D| n (u 2 u 1 )) + |D| 1+n u2 , |D| n u 2 1 2 .
Then, correcting the initial quantity with terms of lower order than F (t), we rather estimate

A := 2ℜe (|D| 1+n u 1 , |D| n ( u2 u 1 + u 2 u1 )) + |D| 1+n u 2 , |D| n ( u1 u 1 )
.

Now apply the Leibniz formula :

A = 2ℜe [(∂ n x |D|u 1 , (∂ n x u2 )u 1 ) + (∂ n x |D|u 2 , (∂ n x u1 )u 1 )] + 2ℜe ∂ n x |D|u 1 , n-1 k=0 n k [(∂ k x u2 )∂ n-k x u 1 + (∂ n-k x u 2 )∂ k x u1 + 2ℜe ∂ n x |D|u 2 , n-1 k=0 n k (∂ k x u1 )∂ n-k x u 1 + 2ℜe(∂ n x |D|u 1 , u 2 ∂ n x u1 ). (13)
We aim at showing that each of these terms is controlled by F (t).

Exactly as before, we have a cancellation occuring in the first line of ( 13) : replace u1 by i|D|u 1 , and u2 by i|D|u 2 (here again the nonlinearities can be neglected), and observe that

2ℑm [(∂ n x |D|u 1 , (∂ n x |D|u 2 )u 1 ) + (∂ n x |D|u 2 , (∂ n x |D|u 1 )u 1 )] = 0.
Concerning the second and the third line of ( 13), straightforward Sobolev estimates and interpolation inequalities are enough to conclude.

As for the fourth line of ( 13), where all time-and space-derivatives concentrate on the same function, we need an equivalent of lemma 2.2 for the operator |D| 1/2 , namely : Lemma 2.3 (Kenig-Ponce-Vega, see [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF]). For f , g : T → C, we have

f |D| s g + g|D| s f -|D| s (f g) L p |D| s 1 f L p 1 |D| s 2 g L p 2 ,
provided that 0 < s < 1, s = s 1 + s 2 and s 1 , s 2 ≥ 0, and on the other side,

1 p = 1 p 1 + 1 p 2 , with p, p 1 , p 2 ∈ (1, +∞).
With this lemma, we can write

2ℜe(∂ n x |D|u 1 , u 2 ∂ n x u1 ) = 2ℜe(|D| 1/2 (∂ n x u 1 ), (|D| 1/2 u 2 )∂ n x u1 ) (14)
+ 2ℜe(|D| 1/2 (∂ n x u 1 ), u 2 |D| 1/2 (∂ n x u1 )) (15)
+ 2ℜe(|D| 1/2 (∂ n x u 1 ), |D| 1/2 u 2 ∂ n x u1 -(|D| 1/2 u 2 )∂ n x u1 -u 2 |D| 1/2 (∂ n x u1 )). (16)
We estimate separately

|(14)| |D| 1/2 ∂ n x u 1 L 4 |D| 1/2 u 2 L 4 ∂ n x u1 L 2 u 1 H 3 4 +n u 2 H 3 4 u 1 H 1+n ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 )F (t).
On the other hand, (15) can be rewritten as 2ℜe

(|D| 1/2 (∂ n x u1 )|D| 1/2 (∂ n x u 1 ), u 2 ), i.e. d dt ℜe (∂ n x |D| 1/2 u 1 ) 2 , u 2 -ℜe (∂ n x |D| 1/2 u 1 ) 2 , u2 .
Thus, perturbing the initial quantity by a term of lower order than F (t), it is enough to control

ℜe (∂ n x |D| 1/2 u 1 ) 2 , |D|u 2 ≤ u 1 2 H 3 4 +n u 2 H 1 ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 )F (t).
Eventually, using L 4 -L 4 3 duality, and lemma 2.3 with s = 1 2 , s 1 = 1 2 , s 2 = 0, and p = 4 3 , p 1 = 4, p 2 = 2, we infer that

|(16)| |D| 1/2 ∂ n x u 1 L 4 |D| 1/2 u 2 L 4 ∂ n x u1 L 2 ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 )F (t),
as above.

To sum up, there exist a constant C n depending only on n, such that, for all times t ∈ R,

F (t) ≤ 4F (0) + C n ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 )
t 0 F (s)ds, which means, by Gronwall's lemma, that

F (t) ≤ 4F (0)e Cn ( u 0 1 2 H 1/2 + u 0 2 2 H 1/2 )|t| .
This tells us that u 1 H 1+n and u 2 H 1+n grow at most exponentially, and theorem 2 is proved.

Dispersion estimates and Bourgain spaces

We come back to equation ( 2) and to the end of the proof of theorem 3. We have to deal now with the case when α < 1. In this case, the boundedness of the H α/2 -norm of the solutions is not enough to get a pointwise control of their L ∞ -norm. In other terms, we need to prove a Strichartz estimate for solutions of (2).

The Strichartz estimate

From now on, α is fixed, with 2 3 < α < 1. For u 0 ∈ D ′ (T) and t ∈ R, denote by

S(t)u 0 := e -it|D| α u 0 = +∞ k=-∞ u 0 (k)e i(kx-|k| α t) ,
the solution of the homogeneous equation i∂ t u = |D| α u, with value u 0 at time t = 0. We are also going to use the Littlewood-Paley decomposition. For this purpose, let ψ be a nonnegative C ∞ function on R + , such that ψ > 0 on 1 2 + 1 10 , 2 -1 10 and ψ ≡ 0 outside 1 2 , 2 . Without loss of generality, we can assume that +∞ j=1 ψ(2 -j x) ≡ 1 on [2, +∞). Let then u ∈ D ′ (T), and N = 2 j for some integer j ≥ 1. We define

∆ N u := ψ |D| N u = |k|∈[ N 2 ,2N ] ψ |k| N û(k)e ikx ,
and ∆ 1 u := u -j≥1 ∆ 2 j u. In the sequel, capital letters will always refer to dyadic integers, and we will use the simplified notation N for a sum over all dyadic integers, starting from 1.

Let us recall a few facts about the Littlewood-Paley decomposition :

u = N ∆ N u, (17)
u 2 H s ≃ N N 2s ∆ N u 2 L 2 , (18)
u 2 L p ≃ N ∆ N u 2 L p . (19)
By the ≃ sign, we just mean that the two quantities, as norms, are equivalent.

We can now state our Strichartz lemma :

Lemma 3.

(Strichartz inequality).

There exist a constant C α > 0, depending on α, such that, for every u ∈ L 2 (T) and every N = 2 j ,

e -it|D| α ∆ N u L 4 ((0,1)t,L ∞ (T)) ≤ C α u L 2 N 1 2 -α 4 . (20)
This lemma also has a non-localized version :

Corollary 3.2. For every γ > 1 2 -α 4 and every u ∈ H γ (T), we have

e -it|D| α u L 4 ((0,1)t,L ∞ (T)) ≤ C α,γ u H γ .
Proof of the corollary. Using the triangle inequality, [START_REF] Ozawa | On a system of nonlinear schrödinger equations with quadratic interaction[END_REF], and the Cauchy-Schwarz inequality, we can write that

e -it|D| α u L 4 L ∞ = N e -it|D| α ∆ N u L 4 L ∞ ≤ N e -it|D| α ∆ N u L 4 L ∞ ≤ C N N 1 2 -α 4 ∆ N u L 2 ≤ C N N 2γ ∆ N u 2 L 2 1 2 N N -2(γ-( 1 2 -α 4 )) 1 2 ≤ C α,γ u H γ ,
where the last bound comes from [START_REF] Yu | Bright and dark spatial solitons in non-kerr media[END_REF].

Proof of lemma 3.1. To proove the Strichartz inequality, we proceed in a quite usual manner : we begin by showing a dispersion estimate, and to conclude, we apply a T T * -argument, combined with the Hardy-Littlewood-Sobolev inequality.

Let N = 2 j , j ≥ 1, be a dyadic integer 4 . For t ∈ R, we have where κ N stands for the following kernel :

e -it|D| α ∆ N u(x) = k∈Z ∆ N u(k)e i(kx-|k| α t) = k∈Z T ψ |k| N u(y)e i(k(x-y)-|k| α t) dy =: (u * x κ N )(x, t),
κ N (x, t) := k∈Z ψ |k| N e i(kx-|k| α t) .
Our first step will be to estimate κ N (•, t) L ∞ (T) for fixed t ∈ (-1, 1), t = 0. Applying the Poisson summation formula to the function F x,t (y) := ψ(|y|/N )e i(yx-|y| α t) , which is C ∞ and compactly supported, we have

κ N (x, t) = k∈Z F x,t (k) = n∈Z F x,t (2πn) = n∈Z R N ψ(|ξ|)e i[N ξ(x-2πn)-N α t|ξ| α ] dξ,
and to study κ N L ∞ , we naturally restrict ourselves to x ∈ (-π, π].

The integrals above will be estimated by a stationnary phase result called the Van der Corput lemma (see [START_REF] Stein | Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals[END_REF]) : Lemma 3.3 (Van der Corput). Let ϕ, Ψ : R → C be two smooth functions, with Ψ compactly supported on R. Suppose in addition that there exist

A > 0 such that |ϕ ′′ | ≥ A on supp(Ψ). Then R e iϕ(x) Ψ(x)dx ≤ C √ A R |Ψ ′ (x)|dx,
where C > 0 is an absolute constant.

In our case, the phase reads N ξ(x -2πn) -N α t|ξ| α , and we denote it by

φ n (ξ). Compute φ ′ n (ξ) = N (x -2πn) -αN α tξ|ξ| α-2 . In particular, φ ′ n (ξ) = 0 if and only if sgn(ξ) (N |ξ|) 1-α = αt x -2πn . ( 21
)
Because of ψ cutting all frequences below 1 2 , and because of the condition 1 -α > 0, we have (N |ξ|) 1-α ≥ 1 on supp(ψ), whereas |tα| ≤ 1. Furthermore, when n = 0, |x -2πn| -1 ≤ π -1 : in that case, (21) cannot hold.

So suppose first n = 0. Then φ ′ n L ∞ (supp(ψ)) ≥ N |x -2πn| -N α 2 1-α , and integrating by parts, it is easy to estimate the integral

I n := R N ψ(|ξ|)e iφn (ξ) dξ = R N d dξ 1 iφ ′ n (ξ) d dξ ψ(|ξ|) iφ ′ n (ξ)
e iφn(ξ) dξ.

Thus, we have

|I n | ≤ CN φ ′ n - 2
L ∞ ψ H 2 ,
where C is proportionnal to the size of supp(ψ). Finally, we sum on n :

n =0 I n ≤ C N n =0 |x -2πn| - 2 N 1-α -2 ≤ C N n =0 (2π|n| -π -1) -2 ≤ C N , (22)
with C just depending on ψ.

The only difficult part, then, is I 0 , because (21) could be satisfied. At this point, we apply lemma 3.3, and calculate φ ′′ 0

(ξ) = α(1-α)N α t|ξ| α-2 . It is clear that |φ ′′ 0 (ξ)| ≥ α(1-α)N α |t|2 2-α on supp(ψ), so we have |I 0 | ≤ C N 1-α 2 |t| , (23)
where C > 0 is a constant depending on α and ψ. So far, ( 22) and [START_REF] Stein | Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals[END_REF] show that there exist a constant C > 0, depending only on α and ψ, such that κ N (•, t) L ∞ ≤ C|t| -1/2 N 1-α/2 for all t ∈ (-1, 1), t = 0. In particular, for fixed t, considering S(t)∆ N as an operator mapping L 1 (T) to L ∞ (T), we have

S(t)∆ N L 1 →L ∞ ≤ C N 1-α 2 |t| . ( 24
)
Now comes the T T * -argument. Define a linear operator T : u → S(t)∆ N u. We want to prove that T maps L 2 (T) into L 4 ((0, 1) t , L ∞ (T)), as well as to bound its norm. To this end, we rather study the operator T T * , where T * : L 4/3 ((0, 1) t , L 1 (T)) → L 2 (T) is (a restriction of) the adjoint of T . We can find T * explicitely. Let g ∈ L 4/3 L 1 : for u ∈ L 2 (T),

(0,1)×T [S(t)∆ N u(x)]g(t, x)dxdt = u, 1 0 S(-s)∆ N g(s, x) L 2 (T) . Thus, T T * (g)(t, x) = 1 0 ∆ N S(t -s)∆ N g(s,
x)ds, and by [START_REF] Xu | Large time blowup for a perturbation of the cubic Szegő equation[END_REF], for all t ∈ (0, 1),

T T * (g)(t, •) L ∞ (T) ≤ C 1 0 N 1-α 2 |t -s| g(s, •) L 1 (T) ds.
In the integral of the left hand side, we recognize a convolution product between t → g(t, •) L 1 (T) and the function ω : t → |t| -1/2 . The Hardy-Littlewood-Sobolev inequalities guarantee that the convolution with ω maps L 4/3 ((0, 1) t ) to L 4 ((0, 1) t ). In other terms, T T * (g) L 4 L ∞ ≤ CN 1-α/2 g L 4/3 L 1 , which implies that the operator norm of T is bounded :

T L 2 →L 4 ((0,1)t,L ∞ (T)) ≤ CN 1 2 -α 4 .
This finishes the proof of lemma 3.1.

Remark 2. Notice that the results of lemma 20 and corollary 3.2 remain true, with the same constants, when replacing (0, 1) t by any time interval of length 1. This follows from the fact that S(t) is an isometry in any H s (T), s ≥ 0.

Bourgain spaces and embedding results

The Strichartz estimate of corollary 3.2 will enable us to prove the local well-posedness of equation ( 2) in a certain Hilbert space, usually called a Bourgain space, which we are now going to define.

Definition. Let u : R × T → C, and s, b ∈ R.

• We say that u ∈ H s,b if for all t ∈ R, u(t) ∈ H s (T), and if in addition, the function

t → u(t) H s (T) belongs to H b (R). Then, the norm u H s,b is just the H b -norm of t → u(t) H s (T) .
• We say that u ∈ X s,b α if the function v : (t, x) → S(-t)u(t, x) belongs to H s,b . Then, we define u X s,b α

:= v H s,b = S(-t)u(t, x) H s,b . The space X s,b
α is called a Bourgain space. Explicitely,

u 2 X s,b α = 1 2π k∈Z R (1 + |k| 2 ) s (1 + |τ + |k| α | 2 ) b |Fu(τ, k)| 2 dτ, (25)
where Fu(•, k) stands for the Fourier transform of u(•, k) with respect to time, i.e. the Fourier transform in both time-and space-variables. Bourgain spaces are very convenient for several reasons. Playing on the two exponents s and b, we begin by showing two embedding results :

Lemma 3.4. For any b > 1 4 , we have u L 4 (Rt,L 2 (T)) u X 0,b α . Proof. Assuming u ∈ X 0,2b α , write Fu(τ, k) = R u(t, k)e -itτ
dt, so by the inverse Fourier transform and the Cauchy-Schwarz inequality (observing that 4b > 1), [START_REF] Xu | Unbounded sobolev trajectories and modified scattering theory for a wave guide nonlinear schrödinger equation[END_REF] and the Parseval formula. Interpolating between these two statements gives the result.

| u(t, k)| 2 = 1 2π R Fu(τ, k)e itτ dτ 2 ≤ C R |Fu(τ, k)| 2 (1 + |τ + |k| α | 2 ) 2b dτ R dτ (1 + |τ + |k| α | 2 ) 2b . Summing over k ∈ Z, we find u(t) 2 L 2 (T) ≤ C b u 2 X 0,2b α , or equivalently u L ∞ (Rt,L 2 (T)) ≤ C u X 0,2b α . But the equality u L 2 (Rt,L 2 (T)) = u X 0,0 α also follows from
The following lemma is a consequence of the Strichartz inequality.

Lemma 3.5. For any b > 1 2 and γ > 1 2 -α 4 , we have u L 4 (Rt,L ∞ (T)) u X γ,b α . Proof. Let u ∈ X γ,b
α , and v := S(-t)u. Suppose at first that t → u(t, •) is supported on an interval I t of length 1. Thus it is possible to apply corollary 3.2 directly : indeed,

u L 4 (Rt,L ∞ ) = S(t)v L 4 (It,L ∞ ) = S(t) R v(τ )e itτ dτ L 4 (It,L ∞ ) ≤ R S(t)v(τ ) L 4 (It,L ∞ ) dτ ≤ C R v(τ ) H γ dτ ≤ C R v(τ ) 2 H γ (1 + |τ | 2 ) b dτ 1 2 R dτ (1 + |τ | 2 ) b 1 2 ≤ C b v H γ,b . Since v H γ,b = u X γ,b α
, this finishes the first part of the proof. Now, we remove the special assumption on u. By simple construction, it is possible to find a function ϑ ∈ C ∞ 0 ((0, 1)), such that 0 ≤ ϑ ≤ 1 on R, and n∈Z ϑ(t -n/2) = 1, for all t ∈ R. We have

u 4 L 4 (Rt,L ∞ (T)) = R dt n∈Z u(t, •)ϑ t - n 2 4 L ∞ (T) ≤ R dt n∈Z ϑ t - n 2 u(t, •) L ∞ (T) 4 = R dt n 1 ∈Z n 2 ,n 3 ,n 4 ∈{n 1 -1,n 1 ,n 1 +1} ϑ t - n 1 2 ϑ t - n 2 2 ϑ t - n 3 2 ϑ t - n 4 2 u(t, •) 4 L ∞ (T) ≤ C n 1 ∈Z ϑ t - n 1 2 u(t, •) 4 L 4 (Rt,L ∞ (T))
, thanks to the elementary inequality : abcd ≤ 1 4 (a 4 + b 4 + c 4 + d 4 ). To each term of the sum, we apply the first part of the proof, and we find, using the embedding ℓ 4 (N) ֒→ ℓ 2 (N) :

u L 4 (Rt,L ∞ (T)) n∈Z ϑ t - n 2 S(-t)u(t, •) 4 H γ,b 1 4 n∈Z ϑ t - n 2 S(-t)u(t, •) 2 H γ,b 1 2 S(-t)u(t, •) H γ,b .
The very last bound comes from the following classical lemma : Lemma 3.6. For any b ∈ [0, 1], any function w ∈ H b (R), any smooth ϑ as above, the following norms are equivalent :

w H b (R) ≃ n∈Z ϑ(• -n/2)w(•) 2 H b (R) 1 2
.

The nonlinear estimate

To solve (2) locally in time, we need to introduce a restriction space. For T > 0, let X s,b α (T ) be the set of all functions u defined on [-T, T ] such that there exist a function ũ ∈ X s,b α with ũ| [-T,T ] ≡ u. Endowed with the norm

u X s,b α (T ) := inf ũ X s,b α ũ ∈ X s,b α , ũ ≡ u on [-T, T ] ,
X s,b α (T ) is a Banach space, so we can apply Picard's fixed-point theorem in X s,b α (T ). From here on, we strictly follow the scheme of the proof of [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF]Theorem 3].

Fix a function ϕ ∈ C ∞ 0 (R), such that ϕ(t) = 1 when |t| ≤ 1. Then for T ∈ (0, 1], a solution of (2) on the interval [-T, T ] is a fixed point of the application

K : u -→ ϕ(t)S(t)u 0 -iϕ t T t 0 S(t -s) |u(s)| 2 u(s) ds. (26)
Thanks to the definition of Bourgain spaces, estimating the first part of K is elementary :

ϕ(t)S(t)u 0 X s,b α = ϕ H b (Rt) u 0 H s (T)
. All the difficulty lies in the nonlinear part, and we now turn to our central proposition.

Proposition 3.7. Let γ > 1 2 -α 4 , b > 1 2 , b ′ > 1 4 such that b + b ′ < 1.
Then

u 1 u 2 u 3 X γ,-b ′ α ≤ C u 1 X γ,b α u 2 X γ,b α u 3 X γ,b α .
Proof. We prove this proposition thanks to a duality argument. Since

(X σ,β α ) ′ = X -σ,-β α for any σ, β ∈ R, it is sufficent to show that, for a given u 0 ∈ X -γ,b ′ α , we have R×T u 1 u 2 u 3 u 0 ≤ u 0 X -γ,b ′ α 3 j=1 u j X γ,b α . (27)
In fact, we restrict the proof of (27) to the case of smooth functions u j , u 0 , with compact support in time -and then conclude by density of such functions. To start with, we introduce new functions w 0 , w 1 , w 2 , w 3 by

Fw j (τ, k) := (1 + |k| 2 ) γ 2 (1 + |τ + |k| α | 2 ) b 2 Fu j (τ, k), for j ∈ {1, 2, 3}, Fw 0 (τ, k) := (1 + |k| 2 ) -γ 2 (1 + |τ + |k| α | 2 ) b ′ 2 Fu 0 (τ, k),
and we can replace the right hand side of (27) by w 0 L 2 (R×T) 3 j=1 w j L 2 (R×T) . To estalish (27), we perform a time-and space-localization. By L 0 , L j , N 0 , N j , we refer to dyadic integers, and L := (L 0 , L 1 , L 2 , L 3 ) will concern time, whereas N := (N 0 , N 1 , N 2 , N 3 ) will hint at space. Define, for j ∈ {1, 2, 3},

u L j N j j (t, x) := 1 2π |n|∈[N j ,2N j [ e inx L j ≤|τ +|n| α |<2L j (1 + |n| 2 ) -γ 2 (1 + |τ + |n| α | 2 ) -b 2 Fw j (τ, n)e itτ dτ, u L 0 N 0 0 (t, x) := 1 2π |n|∈[N 0 ,2N 0 [ e inx L 0 ≤|τ +|n| α |<2L 0 (1 + |n| 2 ) γ 2 (1 + |τ + |n| α | 2 ) -b ′ 2 Fw 0 (τ, n)e itτ dτ.
A simple calculus shows that, for j ∈ {1, 2, 3} first,

u L j N j j 2 X σ,β α = C |n|≃N j |τ +|n| α |≃L j (1 + |n| 2 ) σ-γ (1 + |τ + |n| α | 2 ) β-b |Fw j (τ, n)| 2 dτ L 2(β-b) j N 2(σ-γ) j |n|≃N j |τ +|n| α |≃L j |Fw j (τ, n)| 2 dτ =: c j (L j ,N j ) 2
, where c j (L j , N j ) satisfies L j N j c j (L j , N j ) 2 ≤ w j 2 L 2 (R×T) . (Here, and in all the sequel, the summation over L j or N j means the summation over all dyadic integers.) We have a similar result for u L 0 N 0 0 , so finally, for any σ, β ∈ R,

u L j N j j X σ,β α L β-b j N σ-γ j c j (L j , N j ), (28)
u L 0 N 0 0 X σ,β α L β-b ′ 0 N σ+γ 0 c 0 (L 0 , N 0 ). (29)
Now we are going to estimate

I(L, N ) := R×T u L 1 N 1 1 u L 2 N 2 2 u L 3 N 3 3 u L 0 N 0 0 .
Notice that integrating on T implies that I(L, N ) = 0 unless N 0 ≤ 2(N 1 +N 2 +N 3 ). From now on, we suppose that this condition is fulfilled. Moreover, the proof below does not take into account the precise role of the u j 's, neither the conjugate bar, so we can assume that

N 1 ≥ N 2 ≥ N 3 .
Using the Hölder inequalities, lemmas 3.4 and 3.5, and finally (28) and ( 29), choosing any

β ∈ ( 1 2 , b), β ′ ∈ ( 1 4 , b ′ ), γ ′ ∈ ( 1 2 -α 4 , γ), we bound I(L, N ) : I(L, N ) ≤ u L 1 N 1 1 L 4 (Rt,L 2 (T)) • u L 2 N 2 2 L 4 (Rt,L ∞ (T)) • u L 3 N 3 3 L 4 (Rt,L ∞ (T)) • u L 0 N 0 0 L 4 (Rt,L 2 (T)) ≤ u L 1 N 1 1 X 0,β ′ α • u L 2 N 2 2 X γ ′ ,β α • u L 3 N 3 3 X γ ′ ,β α • u L 0 N 0 0 X 0,β ′ α ≤ (L 0 L 1 ) β ′ (L 2 L 3 ) β L b ′ 0 (L 1 L 2 L 3 ) b (N 2 N 3 ) γ ′ N γ 0 (N 1 N 2 N 3 ) γ 3 j=0 c j (L j , N j ) ≤ L -ε 0 0 L -ε 1 1 (L 2 L 3 ) -ε ′ (N 2 N 3 ) -η ′ N 0 N 1 γ 3 j=0 c j (L j , N j ),
where ε 0 , ε 1 , ε ′ , η ′ are some positive constants. Consequently, we can sum on L 2 , N 2 , L 3 , N 3 , making use of the bound c j (L j , N j ) ≤ w j L 2 (R×T) for j ∈ {2, 3}. Then, by Cauchy-Schwarz,

L 0 ,L 1 L 2 ,N 2 ,L 3 ,N 3 I(L, N ) N 0 N 1 γ w 2 L 2 w 3 L 2   L 0 c 0 (L 0 , N 0 )L -ε 0 0     L 1 c 1 (L 1 , N 1 )L -ε 1 1   N 0 N 1 γ K 0 (N 0 ) 1 2 K 1 (N 1 ) 1 2 w 2 L 2 w 3 L 2 ,
introducing K j (N j ) := L j c j (L j , N j ) 2 for j ∈ {0, 1}. It remains to sum on N 0 , N 1 , remembering that N 0 ≤ 6N 1 , and using Cauchy-Schwarz again :

L,N I(L, N ) +∞ ℓ=-3 N 0 N 0 2 ℓ N 0 γ K 0 (N 0 ) 1 2 K 1 (2 ℓ N 0 ) 1 2 w 2 L 2 w 3 L 2 +∞ ℓ=-3 2 -γℓ   N 0 K 0 (N 0 )   1 2   N 0 K 1 (2 ℓ N 0 )   1 2 w 2 L 2 w 3 L 2 w 0 L 2 w 1 L 2 w 2 L 2 w 3 L 2 .
Hence ( 27) is proven, and so is proposition 3.7.

Remark 3. The condition b + b ′ < 1 has not been used in the proof, except for the fact that b ′ < b ; but it will be crucial in the next proposition.

Local and global well-posedness

We are now ready to state our Proposition 3.8. Let γ > 1 2 -α 4 and 1 2 < b < 1. If u 0 ∈ H γ (T), there exist T 0 > 0, depending only on u 0 H γ , such that the problem (2) admits a unique solution u ∈ X γ,b α (T ) for all T ≤ T 0 . This solution satisfies u X γ,b α (T ) ≤ C u 0 H γ (T) , where C > 0 is an absolute constant. If in addition u 0 belongs to H s (T) for some s > γ, then u(t) ∈ H s (T) for all t ∈ [-T 0 , T 0 ].

Proof. We intend to show that the functional K, defined in [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF], is a contraction in some ball of the space X γ,b α (T ), for well-chosen T . Since the first part of K has been previously bounded, we turn to

M 2 (u) := ϕ t T t 0 S(t -s) |u(s)| 2 u(s) ds X γ,b α (T ) = ϕ t T t 0 S(-s) |u(s)| 2 u(s) ds H γ,b
.

Here, we take advantage of the regularizing property of time-integration. Let U ∈ C ∞ 0 (R × T), and b, b ′ as in proposition 3.7. Set G(t, x) := ϕ t T t 0 U (s, x)ds. For fixed k ∈ Z, a lemma of Ginibre [12, lemma (3.11)] guarantees that

Ĝ(t, k) H b (Rt) = ϕ t T t 0 Û (s, k)ds H b (Rt) T 1-b-b ′ Û (t, k) H -b ′ (Rt) ,
with an implicit constant which only depends on ϕ, b, b ′ . Squaring this identity, we find

R (1 + |τ | 2 ) b |F(G)(τ, k)| 2 dτ T 1-b-b ′ R (1 + |τ | 2 ) -b ′ |F(U )(τ, k)| 2 dτ.
Eventually, multiply by (1 + |k| 2 ) γ and sum over k to get G H γ,b T 1-b-b ′ U H γ,-b ′ , which remains true, by approximation, for less regular functions.

As a consequence,

M 2 (u) T 1-b-b ′ S(-t) |u(t)| 2 u(t) H γ,-b ′ = T 1-b-b ′ |u| 2 u X γ,-b ′ α T 1-b-b ′ u 3 X γ,b α
, by proposition 3.7. This calculation is valid for any ũ ∈ X γ,b α such that ũ = u on [-T, T ]. Thus, we proved that

K(u) X γ,b α (T ) ≤ ϕ H b (Rt) u 0 H γ (T) + CT 1-b-b ′ u 3 X γ,b α (T ) . (30)
So K stabilizes the ball B centered at the origin, of radius C u 0 H γ (for some arbitrary C > ϕ H b (Rt) ), provided that T ≤ T 0 with

T 0 := C -ϕ H b (Rt) C3 C u 0 2 H γ 1 1-b-b ′ .
Reasoning in the same way, by means of the identity

|u| 2 u -|v| 2 v = u(u -v)u + (u -v)v(u + v),
we show that for T ≤ T 0 5 , there exist a positive constant c < 1 such that

K(u)-K(v) X γ,b α (T ) ≤ c u -v X γ,b
α (T ) for all u, v ∈ B. Hence K : B → B is a contraction, and has a fixed point, also called u. Since u ∈ B, we have u X γ,b α (T ) ≤ C u 0 H γ (T) . To prove the uniqueness of u, notice first, in view of (30), that any other fixed point of K in X γ,b α (T ), as a function of some X γ,b α ( T ) for some smaller T ≤ T , lies in the ball centered at 0 and of radius C u 0 H γ , so equals u in that space, by unicity of the fixed point. Now let v ∈ X γ,b α (T ) be another solution of [START_REF] Bourgain | On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE[END_REF]. Observe that, because b > 1 2 , both u and v are continuous functions from [-T, T ] to H γ (T). Define T 1 := sup{t ∈ [0, T ] | u(t) = v(t)}, and suppose T 1 < T . Then, translating time, and restarting the equation with u(T 1 ) = v(T 1 ) as an initial data, we get a contradiction, by the previous remark.

Finally, if u 0 ∈ H s (T) for s > γ, and if u is the associated solution in X γ,b α (T 0 ), let us show that u(t) ∈ H s (T) for all |t| ≤ T 0 . It is crucial to see, modifying slightly the proof of proposition 3.7, that whenever s > γ > 1 2 -α 4 , and b, b ′ as above,

u 1 u 2 u 3 X s,-b ′ α ≤ C 3 j=1   u j X s,b α k =j u k X γ,b α   .
Given ũ in the intersection of the ball of radius C u 0 H s in the space X s,b α (T ), and of the ball of radius C ′ u 0 H γ in the space X γ,b α (T 0 ), (30) becomes, for T ≤ T 0 :

K(ũ) X s,b α (T ) ≤ ϕ H b (Rt) u 0 H s (T) + CT 1-b-b ′ ũ 2 X γ,b α (T ) ũ X s,b α (T ) ≤ ϕ H b (Rt) + C CC ′2 T 1-b-b ′ u 0 2 H γ (T) u 0 H s (T) .
This shows that if T is chosen small enough, regardless of the size of u 0 H s , K stabilizes the set we described above. The same can be done while estimating K(u) -K(v) X s,b α (T ) , and K therefore has a fixed point ũ in some X s,b α (T ). Obviously, since X s,b α (T ) ֒→ X γ,b α (T ), we have ũ ≡ u on [-T, T ]. Repeating this argument after translating time and restarting the equation from ũ(T ) = u(T ), the claim is proved.

The next corollary follows as an immediate consequence, and uses explicitly the condition α > 2 3 :

Corollary 3.9. Let u 0 ∈ C ∞ (T). Then (2) admits a unique global solution u ∈ C(R, C ∞ (T)). Besides, let b ∈ ( 1 2 , 1). For any γ ∈ ( 1 2 -α 4 , α 2 ], there exist T (γ) 0 , C γ > 0, such that for any t ∈ R, u(• -t) X γ,b α (T (γ) 0 ) ≤ C γ u 0 H γ (T) . (31)
5. Or possibly a fixed fraction of T0.

Proof. It suffices to notice that α > 2 3 ⇐⇒ 1 2 -α 4 < α 2 , so there exist T 0 > 0, only depending on u 0 H α/2 , such that (2) can be solved locally in X α/2,b α (T 0 ). But the energy H α and the mass Q are conserved along the trajectory, so u(t) H α/2 remains bounded by 2(H α + Q)(u 0 ). This proves that the solution is global, and we have u(t) ∈ C ∞ (T) for all t ∈ R because of the second part of the previous proposition. (31) also follows.

End of the proof of theorem 3

All the needed results are gathered : now we can study the growth of the Sobolev norms of the solutions of (2) for α ∈ ( 2 3 , 1). We begin with the H α -norm, introducing as in [START_REF] Gérard | On the growth of Sobolev norms for the cubic Szegő equation[END_REF] the modified energy :

E α (u) := u 2 L 2 + |D| α u 2 L 2 + 2ℜe(|D| α u, |u| 2 u) =:J 1 (u) -1 2 (|D| α (|u| 2 ), |u| 2 ) =:J 2 (u)
, when u is a solution of (2). J 1 and J 2 are of lower order than u 2 H α , so that when u H α is big enough, arguing as in section 2.1, we have

1 2 u 2 H α ≤ E α (u) ≤ 2 u 2 H α .
Let us study the evolution of E α (u). The L 2 -norm is conserved, so we directly pass on to

d dt |D| α u 2 L 2 = 2ℜe(|D| α u, |D| α u) = -2ℜe(|D| α u, |u| 2 u).
This combines with the derivative of J 1 (u), and gives rise to two terms :

d dt [ |D| α u 2 L 2 + J 1 (u)] = 2ℜe(|D| α u, u|u| 2 ) + 2ℜe(|D| α u, u(|u| 2 )˙) =: Q 1 (u) + Q 2 (u).
Thanks to the equation, a simplification occurs : Q 1 (u) = -2ℑm(|D| α u, |u| 4 u). The Sobolev embedding H 2/5 ֒→ L 10 and interpolation between H α/2 and H α then allows to bound

|Q 1 (u)| u 2-ε H α , where ε := 6 α α -2 3 > 0.
On the other hand, Q 2 (u) combines with the derivative of J 2 (u) :

Q 2 (u) + d dt J 2 (u) = (ū|D| α u + u|D| α ū -|D| α (ūu), (|u| 2 )˙). Since (|u| 2 )˙= i(u|D| α ū -ū|D| α u), we bound (|u| 2 )˙ L 2 u L ∞ u H α .
As for the other side of the scalar product, we appeal to lemma 2.3, since α < 1 :

ū|D| α u + u|D| α ū -|D| α (ūu) L 2 |D| α 2 u 2 L 4 .
At this point, we have a Gagliardo-Nirenberg inequality : Lemma 3.10. For any p > 2, s > 0, there exist C > 0 such that

|D| s f L p ≤ C f L ∞ + |D| s p 2 f 2 p L 2 f 1-2 p L ∞ .
for every function f : T → R.

Choose a real γ ∈ (

= γ. Thus |D| α 2 u 2 L 4 |D| α 2 +γ u L 2 |D| α 2 -γ u L ∞
-the other terms can be neglected. All these calculations lead to the following fact : there exist a small θ > 0 (which can be chosen to be 3α-2 8α ), and constants C 1 , C 2 > 0 such that for all t ∈ R,

u(t) 2 H α ≤ C 1 u 0 2 H α + C 2 t 0 u(τ ) 2-2θ H α u(τ ) L ∞ |D| α 2 -γ u(τ ) L ∞ dτ.
Denoting by f (t) the right hand side of this inequality, and assuming that t ≥ 0 without loss of generality, this implies that

f (t) θ -f (0) θ = t 0 f ′ (τ )dτ f (τ ) 1-θ ≤ C 2 t 0 u(τ ) L ∞ |D| α 2 -γ u(τ ) L ∞ dτ ≤ C 2 √ t • u L 4 ([0,t],L ∞ ) |D| α 2 -γ u L 4 ([0,t],L ∞ ) ≤ C 2 √ t     ⌈ t T (γ) 0 ⌉ k=0 u(• -kT (γ) 0 ) X γ,b α (T (γ) 0 )         ⌈ t T (α/2) 0 ⌉ k=0 u(• -kT (α/2) 0 ) X α 2 ,b α (T (α/2) 0 )     ≤ C 2 C γ C α 2 u 0 H γ u 0 H α 2 √ t t T (γ) 0 + 1 • t T (α/2) 0 + 1 ,
where we fixed a real b ∈ ( 1 2 , 1), and used the localized version of lemma 3.5, as well as (31). This achieves to show that the H α -norm of the solution of (2) grows at most polynomially, with the power of t being less than 5 4θ , hence than 10α 3α-2 .

The end of the proof crucially relies on this first step. Indeed, to estimate the evolution of the H α+n -norm of u, with n ≥ 1, we follow exactly the same scheme as for the proof of theorems 1 and 3 in section 2.1. Each time the L ∞ -norm of u appears, we bound it by u H α (recall that α > 1 2 ). Besides, we do not interpolate the H s -norms between H α/2 and H α+n anymore, but between H α and H α+n .

The only difference is that we need a new (and, to some extent, rougher) version of lemma 2.2 : Lemma 3.11. Let 1 2 < α < 1. For any integer n ≥ 1, there is a constant C n > 0 (independent of α) such that for all function u ∈ H α+n (T), All of this proves that there exist a small θ ′ > 0, and constants C 1 , C 2 > 0 such that for all t ∈ R,

ū|D| α u + u|D| α ū -|D| α (ūu) H n ≤ C u 1+ 1 n (α-1 2 ) H α u 1-1 n (α-1 2 ) H α+n
u(t) 2 H α+n ≤ C 1 u 0 2 H α+n + C 2 t 0 u(τ ) 2-2θ ′ H α+n u(τ ) 2+2θ ′ H α dτ.
This holds with θ ′ = 1 2n (α -1 2 ). On the other hand, we know that for some A > 0, u(τ ) H α (1 + |τ |) A for all τ ∈ R. By Osgood's lemma, theorem 3 is then fully established.

Appendices A Growth of Sobolev norms for the Szegő equation : an elementary bound

Let u 0 ∈ C ∞ (T) with only nonnegative frequencies (which we denote by u 0 ∈ C ∞ + (T)), and consider t → u(t, x) the solution of the cubic Szegő equation (4) starting from u 0 at time t = 0 : u satisfies i∂ t u = Π + (|u| 2 u), and for all t ∈ R, u(t) also belongs 6 to C ∞ + (T). The purpose of this section is to give an elementary proof of the following estimate, which is the equivalent of theorem 1 : Proposition A.1. For all n ∈ N, there exist positive constants C and B such that

u(t) H 1+n ≤ Ce B|t| 2 . ( 32
)
C is a constant depending on n and u 0 H 1+n , whereas B can be chosen equal to

B n u 0 8 H 1/2
(here, B n depends only on n, not on the considered solution).

We recall here that, though (32) is not the best bound available, it is the best one we can prove without resorting to the Lax pair formalism. The proof below only involves the boundedness of trajectories in the space H 1/2 (T) (which is due to the conservation of mass and momentum). It also uses a standard fact about Hankel operators.

6. All the claims in this section can be found in [START_REF] Gérard | The cubic Szegő equation[END_REF] and are proven there.

Definition. Let v ∈ H 1/2 + (T) (i.e. v has vanishing negative frequencies). The Hankel operator of symbol v is the following C-antilinear operator : A simple application of the Cauchy-Schwarz inequality gives

H v : L 2 + (T) -→ L 2 + (T) h -→ Π + (v h).
Π + (v h) 2 L 2 ≤ k≥0   l≥0 |v l+k | 2     l≥0 |h l | 2   = h 2 L 2 k≥0
(1 + k)|v k| 2 , which is the yielded result.

Since the embedding H 1/2 (T) ֒→ L ∞ (T) fails, proposition A.2 is an improvement of the standard L ∞ -L 2 estimate of a product in L 2 , similarly to lemma 2. The last equality comes from the equation, and the fact that the frequencies of u are nonnegative (so we get rid of Π + here). Then, using Leibniz rule, we expand ∂ 1+n

d dt u 2 H 1+n u 2 H 1+n u H 1/2 u L ∞ u 2 H 1+n u 2 H 1/2 log(1 + u 2 H 1+n ),
and the proof is complete, once we have recalled that u H 1/2 ≤ C 0 u 0 2 H 1/2 .

4. For N = 1 ,

 1 [START_REF] Ozawa | On a system of nonlinear schrödinger equations with quadratic interaction[END_REF] holds trivially.

.Proof.( 2 +∞ 2 +

 22 Denote by L the left hand side of the inequality we intend to prove. Writing u = k∈Z u k e ikx , we clearly have|l| α + |k -l| α -|k| α )u l u l-k |l| n |k -l| α |u l ||u l-k | +∞ l=-∞ |l| α |k -l| n |u l ||u l-k | 2   ,where we used the elementary inequality |k| n ≤ 2 n-1 (|l| n + |k -l| n ) and the triangle inequality associated to the concave function x → x α on R + . Now, define ũ := k∈Z |u k |e ikx , so thatL |D| n ũ • |D| α ũ L 2 u H n+1/4 u H α+1/4. Interpolating these norms between H α and H α+n leads to the result.

Proposition A. 2 .

 2 For h ∈ L 2 + (T), we have H v (h) L 2 ≤ v H 1/2 h L 2 .Proof. Expand h = k≥0 h k e ikx and v = k≥0 v k e ikx . With these notations,

2 .

 2 It is the key of theProof of proposition A.1. We compute, for n ≥ 0,

x (|u| 2 u) = |u| 2 + u∂ 1+n x |u| 2 .

 22 The first term cancels because of the imaginary part, and the "crossed terms" are easily estimated thanks to Sobolev injections and interpolation inequalities, as in the proof of theorem 1. As for the last term, we have(u∂ 1+n x |u| 2 , ∂ 1+n x u) = (Π + (u∂ 1+n x |u| 2 ), ∂ 1+n x u) = (H u (∂ 1+n x |u| 2 ), ∂ 1+n x u),because |u| 2 is real. So, by proposition A.2, and then inequality (9),

From now on, the time-dependence of the terms will always be implicit. In addition, we will always restrict ourselves to nonnegative times t ≥ 0, since it is possible to reverse the evolution of (2) via the transformation u(t) ↔ ū(-t).

The symbol is understood as refering to constants depending only on n, or absolute constants, whose explicit form is not particularly meaningful.

B Some comments on the threshold α = 2 3 In this section, we discuss the bound α = 2 3 that naturally appears in the proof of theorem 3. The crucial point, in the proof above, is to know which Strichartz estimate we are able to establish. In particular, following the strategy of [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces[END_REF], we would like to know for which values of the parameter γ the inequality e -it|D| α u(x) L 4 ((0,1)t,L 4 (Tx)) ≤ C u H γ (33) holds, whenever u ∈ H γ (T). If true, (33) would imply that equation ( 2) is well-posed in H α/2 , provided that 2γ < α 2 , and we could then adapt the arguments we developped in section 3.5 to prove that solutions are polynomially bounded.

Looking back at [START_REF] Ozawa | On a system of nonlinear schrödinger equations with quadratic interaction[END_REF], and interpolating the L 4 ((0, 1) t , L ∞ (T x )) estimate we obtained with the trivial L ∞ ((0, 1) t , L 2 (T x )) one, we find a bound for e -it|D| α u(x) L 8 ((0,1)t,L 4 (Tx)) , so (33) is proved with γ = 1 4 -α 8 . The condition 2γ < α 2 exactly means that α > 2 3 . However, a scaling heuristic suggests that the natural value of γ should rather be γ 0 := 1 4 -α 4 . Here, the condition 2γ 0 < α 2 would enable us to extend the conclusions of theorem 3 until α = 1 2 . In [START_REF] Demirbas | Existence and Uniqueness theory for the fractional Schrödinger equation on the torus[END_REF], by a different method, Demirbas, Erdoğan and Tzirakis also prove a Strichartz estimate in the case α > 1, but their result corresponds to ours (notice that in their work, they use other notations : what they call α is in fact half ours).

So far, we don't know if (33) can be proved with γ = γ 0 . Usual counter-examples (such as localized functions) only confirm that the scaling exponent γ 0 is the best we can hope. Besides, the difficulty is not due to the particular framework of the torus, since when α < 1 the speed of propagation of the waves (i.e. the group velocity) is finite.


Mahdi Bennara
email: mahdi.bennara@liris.cnrs.fr

Youssef Amghar
email: youssef.amghar@liris.cnrs.fr

Michael Mrissa
email: michael.mrissa@liris.cnrs.fr

Managing Web Resource Compositions

Keywords: RESTful Web services, linked services, semantic Web, composition

Nowadays, the use of RESTful Web services promotes stateless service interaction and decentralized hypermedia-driven discovery and composition. However, there is a need for models and tools to drive user interaction as well as description, discovery and composition of RESTful services. In this paper, we provide a solution to help users manage, share and discover workflows of RESTful Web services. We annotate RESTful Web services with semantic information, and introduce the notion of composition directory as a Web resource that assists a user in sharing, managing and discovering workflows. Users' composition directories form a decentralized repository of service workflows connected by hypermedia links. We illustrate the benefits of our approach with a typical scenario and show through a set of experiments that the breadth-first search algorithm combined with the exploitation of semantic annotations efficiently answers users' goals by crawling through composition directories.

I. INTRODUCTION

The Web has moved from a Web of documents to a distributed application platform where applications are exposed as Web resources, as witnesses the growing number of available APIs 1 . Leading research topics are related to discovery, composition and invocation of Web resources via their API. In addition, the emergence of semantic Web technologies gives the opportunity to improve the use of APIs with semantic annotation over Web resources. Semantic annotation helps to drive the interaction with APIs by providing explicit description of domain-specific information about resources.

Another key concept that drives today's Web is distributed affordance. Affordance is the ability for a user to use a Web resource. The idea is to dynamically create affordance based on the information already present in a resource representation, with knowledge from distributed sources [START_REF] Verborgh | Distributed affordance: an open-world assumption for hypermedia[END_REF]. Distributed affordance combines the information on resources and the knowledge on service providers, as well as user profiles in order to generate possibilities for manipulating Web resources. It should allow client-side software to dynamically drive the interaction with Web resources, therefore service providers do not have to anticipate user interaction and avoid deploying static business processes that constrain users.

In order to enable distributed affordance, Web resources must be semantically described, and user agents needs to be able to exploit such descriptions. In this paper, we build on previous work to semantically annotate Web resources [START_REF] Bennara | An approach for composing restful linked services on the web[END_REF] and facilitate resource discovery and browsing. We introduce the concept of composition directory to help users manage and share compositions, and show that the breadth-first search algorithm can be used in this context to crawl and discover resources according to a composition workflow the user provides.

Our paper is organized as follows: Section II introduces the challenges we want to overcome while trying to solve the composition problem. Section III presents related work and highlights the advantages our solution offers. Section IV details the main aspects of our contribution and shows its innovation. Section V shows a scenario example and details how our prototype operates in the context of the scenario to demonstrate the applicability of our solution. Section VI discusses our results and gives some guidelines for future work.

II. CONTEXT AND CHALLENGES

In the context of our work, Web services are seen as Web resources that comply with the REST architectural style. The REST architectural style is based on the notion of resource as a conceptual entity that represents abstract or concrete things such as books, orders, payments, etc. Resources are identified by URIs, their state is passed to the client through representations using the adequate media type according to the principle of context negotiation. In this paper we consider a RESTful Web service as a set of resources that provide a coherent access to the state and functionality of the software it represents [START_REF] Pautasso | Restful web service composition with bpel for rest[END_REF].

Another principle that drives the REST architectural style is the HATEOAS 2 principle. Using HATEOAS requires hyperlinks to be established between Web resources that form an open and very large graph. HATEOAS means that the discovery process is realized progressively, user agents should be able to discover other Web resources accessible from any given resource in the graph.

In this context, we identified several challenges to address, which can be summarized as follows :

• Web resource description and interlinking: resources need to be appropriate described with semantic annotations and also linked to each other with hyperlinks to enable client-side discovery (how to interact with the resource) and crawling (how to discover other resource from a given one according to the HATEOAS principle). • Web Resource discovery: as a follow-up to the first challenge, user agents should be able to implement an efficient algorithm to crawl between resources and exploit their annotations to realize users' objectives. In order to answer these challenges, we build on previous work to annotate resources. We introduce the notion of composition directory to manage and share composition workflows and we show that the breadth-first search algorithm can be used to crawl through and and efficiently discover Web resources.

III. RELATED WORK

In this section, we overview existing work about Web resource management and discovery algorithms.

A. Resource Discovery

RESTdesc [START_REF] Verborgh | Description and Interaction of RESTful Services for Automatic Discovery and Execution[END_REF] is based on the Notation3 RDF syntax. It involves all the operational semantics of Notation3 which allows for a versatile discovery methods. We can take advantage of all the advances in the reasoning domain in the Notation3 syntax in order to determine whether a resource satisfies a set of conditions defined for the discovery. More advanced reasoning is performed in order to achieve service matching. Authors estimate that this is an important prerequisite for services in order for them to contribute to the future Web of clients, because new functionality can only be obtained by on-demand compositions tailored to a specific problem [START_REF] Verborgh | Description and Interaction of RESTful Services for Automatic Discovery and Execution[END_REF].

RESTdoc [START_REF] John | RESTDoc: Describe, Discover and Compose RESTful Semantic Web Services using Annotated Documentations[END_REF] is a format that combines multiple microformats in order to semantically describe RESTful resources. RESTdoc offers a discovery mechanism that distinguishes two different aspects of REST services discovery problem:

(1)the discovery as a client, or discovery as you browse, concerns the client-side browsers. This discovery uses on HTML Link element on a Web site in order to point to other resource descriptions. and (2) the discovery as a service, also called automated discovery, is the ability for a service to access and link to other related resources in the same application domain. The solution provided by RESTdoc describes a fully peer to peer discovery mechanism. The main idea is to construct a graph by running through links and identifying resources. This graph can be subject to later extending in order to explore new related resources.

LinkedWS [START_REF] Maamar | Linkedws: A novel web services discovery model based on the metaphor of "social networks[END_REF] is a Web service discovery model based on social networks. The idea behind LinkedWS is to construct a social network for every service on the Web in order to allow social-based discovery process. The social network is built the moment the service participates a first time in a composition. Generally speaking, a social network of a Web service consists of nodes and edges. The nodes represent an object or an entity (book, person, organization, etc.) and the edges represent the relationships between nodes (distance between two cities, relationship between two persons, etc.) Each edge has its own weight, that is used by the search or ranking algorithms that navigate through the network in order to find Web services that suit specific purposes. Every Web service is the entry point of his own social network. LinkedWS allow the discovery of additional Web services in a specific composition. The discovery itself triggers the reevaluation of the weight of the edge that led to this discovery.

Our proposition aims to enhance the discovery process by embedding the related resources directly in the resource description. It allows also the discovery of resources using existing composition work-flows.

B. Resource Composition

Many researchers are interested in the problem of describing the semantics of the sequences in the executions flows, and have proposed many solutions.

One of the most important works in this domain is the BPMN3 [START_REF] Wong | A process semantics for bpmn[END_REF] specification. BPMN specifies a set of flow control sequences that allows us to describe the progressing of a process. The main interest of BPMN for us is that it can be used in order to construct and store dynamic service composition processes which can be reused afterward by another user that wants to do similar service composition. The use of BPMN relies on a Process-oriented approach rather than being Resource-oriented. This may conflict with the principles of the REST architectural style, nevertheless some concepts can be used naturally on resource oriented architectures.

Linked USDL4 [START_REF] Pedrinaci | Linked usdl: A vocabulary for web-scale service trading[END_REF], is another work in this perspective. Unlike, BPMN, Linked USDL vocabulary has been designed especially for the service-oriented domain, making it easier to adapt for our solution. Some of the important concepts introduced by this vocabulary include: Service, ServiceOffering, InteractionPoint as well as services roles including: Producer, Provider, Intermediary, etc. which constitute the main semantic concepts of workflow control in serviceoriented architectures. Linked USDL is being used in many projects, and it proved its efficient for the service community.

BPEL for REST [START_REF] Pautasso | Restful web service composition with bpel for rest[END_REF] is a work that proposes to reuse the BPEL language principles and apply it on the REST architectural style. BPEL for REST either uses the WSDL 2.0 description language without changing the current BPEL or extends BPEL in order to support HTTP operations on the resource API. The main drawback of BPEL comes with its centralized approach that relies on a static composition engine, which does not fit with the HATEOAS principle.

Our approach aims to use work-flows in order to enable reuse of popular compositions. This is also a flexible way of composing resources as the work-flows can be duplicated and edited.

C. Graph discovery algorithms

Exploring very large graphs such as the Web requires efficient algorithms in order to have acceptable response times. As we are discovering resources on the Web, the efficiency of the exploration algorithm is one of the most important elements of our research work. Practically the classic algorithms are not used as such because they may result in important response times due to the Web size. Instead, variants of these algorithms are used with specific parameters (often limiters) in order to yield reasonable response times and acceptable results. The most known examples of algorithms are Breadth First Search and Depth First Search algorithms. Other algorithms include variants of these two with limiting parameters, for example limiting the depth of the search (number of consecutive edges counting from the root) also known as depth-limited search or limiting the total number of nodes accessed during the whole process [START_REF] Russell | A modern approach[END_REF] [START_REF] Förster | Directed graph exploration[END_REF].

According to [START_REF] Najork | Breadth-first crawling yields high-quality pages[END_REF] the Breadth-First Search graph traversal algorithm yields high-quality pages early on in a crawl. In other words, the most relevant pages/resources to the search are discovered early on in the process. In our work, this means Breadth-First Search finds the most relevant resources to answer a user's request by finding multiple (or single) resources that can perform the tasks needed in order to answer the request. In addition to that, Breadth-First Search is a very natural search strategy in the context of Web. Also, compared to other efficient search algorithms, it has a relatively low computational cost for a large scale graph such as the Web.

IV. CONTRIBUTION

Based on the related work presented above, we have built our solution that promotes the concept of composition directory and uses the breadth-first search algorithm to manage, share and crawl Web resources. Our solution must respect the following requirements in order to facilitate resource discovery and composition:

• Scalability: the increasing number of today's Web APIs makes the scalability of solutions important. • Responsiveness: the increasing number of users generates an important load of requests on servers. We want server responses to be as fast as possible in order to handle all the requests in a reasonable time.

• Diversity: We want our resource descriptions to propose rich and diverse links to other resources in order to give them a chance for being used. In other words, the users which make a request should have different propositions rather than only popular services in a given field and thus giving the chance to less popular services to emerge if the users are interested in the services they offer.

• Dynamism: the results of the resource discovery process should not be static, in other words it should be different from one request to another, because on one hand the availability of resources involved in the request as well as the context of the request may have changed in the meantime, and on the other hand, the user context may also have changed, which implies that users might not get same results because they browsed new resources which may impact the response. • Serendipity: the serendipity concept allows APIs to be used in a non-specific process. In other words we do not want the clients to use APIs in a deterministic way where every next API to use is already known in advance Today, the main advances in Web resource composition are centered around the description of the resources. The focus of these advances is how to describe a resource in order to give as much information as possible to identify the nature of the resource, its activity and what type of data it exchanges. Too few efforts focus on how it links to other resources and how to follow these links, as well as how to manage and share composition workflows. The latter aspects are presented in the following in order to enable value-added resource discovery and composition.

A. Describing and Discovering Resources

In order to semantically describe resources, we rely on the notion of resource descriptor discussed in [START_REF] Bennara | An approach for composing restful linked services on the web[END_REF]. The resource descriptor has been slightly modified and the data model we use to implement the descriptor concept is the Hydra core vocabulary [START_REF] Lanthaler | Hydra Core Vocabulary[END_REF]. The main reason for this choice is that Hydra defines explicit semantics of its Operation and Link elements, the major two elements present in the descriptor.

The resource representation contains the business-level information about the given resource, while the descriptor contains semantically annotated information on how to use this resource (the available HTTP Operations on the current resource, not on other resources) plus information about the related resources (links to other related resources). Hence, resource descriptors separate resource representations from their descriptions, to promote separation of concerns between resource interaction and management (discovery and composition). We deem appropriate to make sure that every resource must have a descriptor to enable machineto-machine interaction. As every resource must have a descriptor, and descriptors are also considered as resources, descriptors also must have their own descriptors. To address this issue, we define the Universal Descriptor5 , which describes all the resources including itself.

With the help of resource descriptors, a generic-client is able to interact with resources and to crawl from one to another in order to compose resources. From a resource URI, a client can get its descriptor by executing a GET/HEAD operation on the given URI, and checking the LINK header element in the HTTP response. Another GET operation on the retrieved link returns a HTTP response with the descriptor containing all the necessary meta-data that describes not only the interaction model with the current resource, but also annotated links to internal or external resources that can be composed together with the given resource. In order to document the semantics of sets of links as well as operations that are present in a given resource descriptor, we rely on ontology concepts. On the operation side, we want to know the exact concept that is accomplished by the operation being described. The concept is part of a larger global composition process that implies several other operations on different resources. Describing operations requires ontologies of operational semantics to describe how to realize a complex task with a combination of simpler, specific tasks.

B. Discovery Process

The discovery process we introduce in this work relies on semantic annotations of operations in the descriptors. When a user enters a request, it is processed by a reasoner in order to know what actions should be realized in order to prepare a response, as presented in previous work [START_REF] Mrissa | Semantic discovery and invocation of functionalities for the web of things[END_REF]. These actions are represented by an ordered list of ontology concepts. The discovery process will take as input this list of concepts as well as an entry point (the URL of a Web resource). Starting from this entry point, the process tries to find resources on the Web that provide the required operations to respond to the request, on the basis of the concept list taken as input. We consider the Web as a big oriented cyclic graph, where nodes are resources and links are Web hyperlinks. We adopt the Breadth-First Search (BFS) algorithm in order to traverse the Web in search of resources that provide the operations corresponding to the concepts of the request.

C. Managing and Sharing Compositions

In order to enable users to record, reuse, manage and share their composition workflows, we propose a specific resource called composition directory. The Composition Directory resource contains information about its owner, and a subresource called repository to store as sub-resources the composition workflows the user creates. The Composition Directory of a user links to other connected users Composition Directories. Note that this is completely compatible with the descriptor concept because the links to other Composition Directories, the Repository and the created compositions scenarios represent the external part of our descriptor.

We define the following API in order to qualify possible interactions with Composition Directories 1) GET on the base URI of a Composition Directory should send back the information about this Composition Directory and its owner. Composition attributes and its accessibility should be indicated by the use beforehand in the representation. 5) POST from the user on his own composition directory in order to add a new Composition Directory URI of another user that exposes interesting compositions for him. This offers many advantages, first it is a scalable and decentralized solution as every user stores a part of compositions on the web, it also respects the serendipity concept as a given client may find part of the solution to the user's problem in another user's composition set. Our model includes access control features relying on HTTP authentication. We define public compositions that everyone on the Web can access from its URI, and private compositions that are not disclosed and require authentication.

V. TESTS AND EVALUATION

In this section, we illustrate our contribution with different scenarios, we detail our implementation setup, and discuss the results obtained.

A. Illustrative Scenarios and Experiment Setup

In order to illustrate our approach, we consider three scenarios. The first scenario involves three different Web resources and illustrates how resources and descriptors are disposed, and how resources link to each other using descriptors. It includes :

• A book selling service: users can select books, read abstract and place an order. • A shipping service: its task is to deliver goods that users buy online. • An online payment service: the task of this service is to debit money from users' bank accounts to the benefit of online stores for the goods they buy. The second scenario involves two users to illustrate the fact that users do not use Web resources in the same way.

Each user orchestrates services in a different manner and shares it with the other user or with everyone on the Web using the Composition Directory mechanisms. The Web resources involved are similar to the first scenario with an additional computer accessories store, where users can select accessories, read description and place an order. The first user's objective is to buy a book online, pay it online and receive it by mail. The second user wants to buy some computer accessories, pay them online and receive them by mail. The third scenario involves 25 resources with their descriptors. These resources can perform multiple actions, which are annotated in the descriptors. The objective of this scenario is to illustrate the discovery process with the descriptor mechanism. The prototype illustrating our work is available online 6 .

In our work, we rely on Java TM language and the Jersey framework 7 in order to implement our services and JavaScript as a client-side scripting language in our Web pages. We use the Google Gson module8 in order to manipulate Json objects in Java. We use Apache Tomcat 9 as an application server in order to accommodate our different Web resources. Our descriptors rely on the Hydra core vocabulary [START_REF] Lanthaler | Hydra: A Vocabulary for Hypermedia-Driven Web APIs[END_REF] to describe resources. The machine used for the experiments has an Intel TM Core i5-3340M CPU and 8GB RAM. The Web browser used is Mozilla TM Firefox. The tests were performed on the local university network.

B. Evaluation and Discussion

In this section, we discuss the choices of our implementation and the impact of these choices on the challenges and the properties we want to achieve. As we are using the REST architectural style to build our solution, the respect of Web constraints is ensured by design.

1) Resource Description: We evaluate and discuss our contribution for the resource description challenge on the basis of the first scenario. The use of descriptors allows to benefit from separation of concerns between businesslevel information and description-level information (metadata contained in the descriptor). The descriptor also allows access to relations with other Web resources, which enables interactions according to HATEOAS. The first scenario illustrates well this separation.

In our example, the book selling resource only contains business layer information such as the name of the service as well as a brief description. Its descriptor, on the other hand, contains links to internal and external related resources (the shipping and payment services for instance) as well as the functioning of the resource itself with the help of semantic annotations. The order resource in the book selling service gives another example about resource description: a GET operation returns the representation of the repository with all the placed orders and their statuses (with per user authentication), a POST operation takes an order as request body and adds it to the repository as an unpaid order, and PUT operation changes the order state and a DELETE operation (only available when the order has been fully completed) deletes an order from the repository.

2) Resource Composition: We evaluate and discuss our contribution for the composition challenge with the second scenario. Composition directories allow users to create, store and share compositions of Web resources. Compositions can be thereafter entirely of partially reused by the user himself or other authorized users. This solution allows for dynamic creation of new compositions rather than follow inflexible server-side compositions. It allows also a large scale sharing of popular compositions that users find useful and offers flexible ways to reuse and adapt compositions to user's needs. We do not rely on a central repository to store the compositions.

The second scenario illustrates these statements: the first user can create a composition for the process of buying a book involving the shipping and online payment services. He can share this composition with the second user to buy things online with goods delivery and payment services. The second user can reuse the same composition if he wants to buy a book, or only a part of this composition if he wants to buy computer accessories. The creation of a new composition as well as its reuse depend on the discovery process discussed below.

3) Resource Discovery: We evaluate and discuss the application of the Breadth-First Search algorithm for discovery of resources in our work.

One important thing to note while applying the algorithm is that nodes are not simple in the context of descriptors, but are double. A node of our graph is the resource and its descriptor while the edges are simply the links in the descriptor of a given resource. In order to avoid unnecessary use of bandwidth, we use HEAD requests on resources instead of GET, in order to retrieve the descriptor link, and then we use a GET request in order to retrieve the descriptor contents.

We illustrate the application of the algorithm on our resources with the third scenario. The algorithm takes a set of concepts and an entry point as input data, it crawls through the graph formed by resources and their descriptors in order to give back the results. The timings of our experiment are shown in Table I and discussed below.

The number of nodes indicates the total number of nodes in the graph that have been traversed in order to give a full response to the request. The response time indicates the time in milliseconds taken in order to respond to the request. The response time per node is simply the response time divided by the number of nodes. The response times per node are under 20 millisecond if the research involves a small number of nodes, but as the number of nodes grows, it becomes lower due to the caching mechanism of the Web browser and the factoring of descriptors for resources that share the same descriptor contents. In other words, additional GET requests on the same descriptor that is shared by multiple resources in the graph are processed faster, as the results are stored in the local cache. This experiment demonstrates that response times are below a linear progression as the global response times indicate. Hence, our approach scales quite well as the composition directories as well as the descriptions are decentralized and each resource stores a part of the global graph. Our approach also ensures dynamic results, as the content of descriptors may be subject to change especially the links part. The diversity is also ensured as the algorithm may find multiple resources that implement a specific operation needed in the process, this number can be limited before the algorithm starts and is given as input with the concepts. The serendipity has also its share in our approach, as the order in which the links of a descriptor are crawled may influence the final result, giving chance to more or less popular resources to be used within the composition process.

VI. CONCLUSION

In this paper, we have introduced the notion of composition directory as a REST resource that allows users to record, manage and share composition workflows. We have proposed a solution to link users' composition directories over the Web to each other to form a distributed directory of composition workflows that can be crawled with graph traversal algorithms. We demonstrate the adequacy of our solution with a set of experiments that rely on the well-known breadth-first search algorithm to discover Web resources according to a set of concepts that represent a user's goal. The obtained results show the scalability of our proposal. As future work, we envision extending our solution with quality models combined with user-side reasoning to enhance the discovery algorithm efficiency. We aim to enable automatic reuse of compositions by reasoning about their semantic annotations in order to respond to a user's request.

2 )

 2 GET on the Repository of the Composition Directory should send back the set of links to every composition on the Repository. 3) GET on a specific composition URI should send back the representation of the composition. This may require an authentication and may send a 401 code (Unauthorized) in case the authentication fails. 4) POST from the user on the Repository of his Composition Directory should create a new composition.

http://www.programmableweb.com

Hypermedia As The Engine Of Application State

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Unified Service Description Language

http://soc.univ-lyon1.fr/universal.md

https://liris.cnrs.fr/ ∼ mbennara/doku.php?id=wetice2015

https://jersey.java.net/

https://code.google.com/p/google-gson/

http://tomcat.apache.org/