Luc Molinet

Sharp ill-posedness results for the KdV and mKdV equations on the torus

Keywords: AMS Subject Classification : 35Q53, 35A01, 35D30

We establish a new a priori bound for L 2 -bounded sequences of solutions to the mKdV equations on the torus. This first enables us to construct weak solutions in L 2 for this equation and to check that the "solutions" constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the KdV equation are discontinuous for the H s (T) topology for respectively s < 0 and s < -1. These last results are sharp.

Introduction

In this paper we study different properties of the Cauchy problems posed on the flat torus T := R/2πZ associated with the Korteweg-de Vries (KdV) equation w t + w xxx -6ww x = 0 (1.1)

and the modified Korteweg-de Vries (mKdV) equation

v t + v xxx ∓ 6v 2 v x = 0 (1.2)
Here, w and v are real-valued functions on T. For some results we will have to distinguish between two mKdV equations depending on the sign in front of the nonlinear term. (1.2) is called the defocussing mKdV equation when there is a minus sign in front of the nonlinear term and the focussing mKdV 1 equation when it is a plus sign. The Cauchy problem associated with these equations in space of rough functions on the torus has been extensively studied these last two decades. In a seminal paper [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, I. Schrödinger equations II. The KdV equation[END_REF], Bourgain proved that the Cauchy problem associated with the KdV equation is globally wellposed in H s (T), s ≥ 0, whereas the one associated with the mKdV equation is globally well-posed in H s (T), s ≥ 1, and locally well-posed in H s (T) for s ≥ 1/2. The local well-posedness of the KdV equation was pushed down to H s (T), s ≥ -1/2 by Kenig, Ponce and Vega [START_REF] Kenig | A bilinear estimate with applications to the KdV equation[END_REF] (see [START_REF] Colliander | Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T[END_REF] for the globalwellposedness of the KdV and the mKdV equations in H s (T) for respectively s ≥ -1/2 and s ≥ 1/2.) The local well-posednesss results proved in these papers mean the following : for any initial data u 0 ∈ H s (T) there exists a time T = T ( u 0 H s ) > 0 only depending on u 0 H s and a solution u that satisfies the equation at least in some weak sense and is unique in some function space X ֒→ C([0, T ]; H s (T)). Moreover, for any R > 0, the flowmap u 0 → u is continuous from the ball centered at the origin with radius R of H s (T) into C([0, T (R)]; H s (T)). Note that in all these works, by a change of variables, the study of the KdV equation is actually restricted to initial data with mean value zero and the mKdV equation is substituting by the following "renormalized" equation :

u t + u xxx ∓ 6(u 2 --u 2 )u x = 0 (1.3)
where -u 2 denotes the mean value of u 2 . The best results quoted above are known to be sharp if one requires moreover the smoothness of the flowmap (cf. [START_REF] Bourgain | Periodic Korteweg de Vries equation with measures as initial data[END_REF]) or the uniform continuity on bounded sets of the solutionmap (cf. [START_REF] Christ | Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations[END_REF]) associated respectively with the KdV equation on space of functions with mean value zero and with (1.3). On the other hand, they have been improved if one only requires the continuity of the flow-map. In this direction, in [START_REF] Kappeler | Global wellposedness of KdV in H -1 (T, R)[END_REF]- [START_REF] Kappeler | Global well-posedness of mKdV in L 2 (T, R)[END_REF], Kappeler and Topalov introduced the following notion of solutions which a priori does not always corresponds to the solution in the sense of distribution : A continuous curve γ : (a, b) → H β (T) with 0 ∈ (a, b) and γ(0) = u 0 is called a solution of KdV equation (resp. mKdV equation) in H β (T) with initial data u 0 iff for any C ∞ -sequence of initial data {u 0,n } converging to u 0 in H β (T) and for any t ∈]a, b[, the sequence of emanating solutions {u n } of the KdV equation (resp. mKdV equation) satisfies : u n (t) → γ(t) in H β (T) Note that a solution in the sense of this definition is necessarily unique. With this notion of solution they proved the global well-posedness of the KdV and the defocusing mKV equations in H s (T) for respectively s ≥ -1 and s ≥ 0, with a solution-map which is continuous from H -1 (T) (resp.

L 2 (T)) into C(R; H -1 (T)) (resp. C(R; L 2 (T))). Their proof is based on the inverse scattering method and thus depends in a crucial way of the complete integrability of these equations. It is worth noticing that, by Sobolev embedding theorem, their solutions of the defocussing mKdV equation satisfy the equation in the distributional sense as soon as s ≥ 1/6. Independently, Takaoka and Tsutsumi ( [START_REF] Takaoka | Well-posedness of the Cauchy problem for the modied KdV equation with periodic boundary condition[END_REF]) extended the local well-posedness of the mKdV equation (with the classical notion of solutions) to H s (T) for s > 3/8 by modifying in a suitable way the Bourgain's space used as resolution space. This approach has been very recently improved by Nakanashi, Takaoka and Tsutsumi [START_REF] Nakanishi | Local well-posedness in low regularity of the mKdV equation with periodic boundary condition[END_REF] and local well-posedness has been pushed to H s (T) for s > 1/3 (local existence of solutions is shown in H s (T) for s > 1/4)

In this paper we first establish an a priori estimate for L 2 -bounded sequences of solutions to the mKdV equation. To this aim we slightly modify the spaces introduced by Ionescu-Kenig and Tataru in [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF]. Recall that these spaces are constructed by localizing in time the Bourgain spaces with a localization in time that depends inversely on the space frequencies of the functions (see [START_REF] Koch | A priori bounds for the 1D cubic NLS in negative Sobolev spaces[END_REF] and [START_REF] Christ | A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order[END_REF] for previous works in this direction). Note that, to some extent, this approach is a version for the Bourgain's spaces of the approach developped by Koch and Tzvetkov [START_REF] Koch | Tzvetkov Local well-posedness of the Benjamin-Ono equation in H s (R)[END_REF] in Strichartz spaces. Once our a priori estimate is established we translate it in the Bourgain's type spaces introduced in [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF] . This enables us to pass to the limit on the nonlinear term by separating resonant and non resonant parts. Following some ideas developped in [START_REF] Molinet | On ill-posedness for the periodic cubic Schrödinger equation[END_REF], we then derive a non continuity result for the mKdV equation in H s (T) for s < 0. On the other hand, we obtain the existence of weak L 2 -solutions of (1.3) and prove that the L 2 -solutions constructed in [START_REF] Kappeler | Global wellposedness of KdV in H -1 (T, R)[END_REF] of the defocusing mKdV equation satisfy the equation in some weak sense. Finally, we follow some ideas of [START_REF] Molinet | A note on ill-posedness for the KdV equation[END_REF] and use properties of the Riccati map proved in [START_REF] Kappeler | Riccati map of L 2 0 (T) and its applications[END_REF] to derive a non continuity result for the KdV equation in H s (T) for s < -1.

Statement of the results

Our results can be summarized in the two following theorems. The first one deals with the discontinuity of the solution-map associated with the KdV and mKdV equations.

Theorem 1.1. The Cauchy problems associated with the KdV equation and the mKdV equation are ill-posed in H s (T) for respectively s < -1 and s < 0. More precisely, i) for any T > 0 and any s < -1, the solution-map u 0 → u associated with the KdV equation is discontinous at any u 0 ∈ H ∞ 0 (T) from H ∞ 0 (T), endowed with the topology inducted by H s (T), into D ′ (]0, T [×T).

ii) for any T > 0 and any s < 0, the solution-map u 0 → u associated with the mKdV equation is discontinous at any non constant function u 0 ∈ H ∞ (T) from H ∞ (T), endowed with the topology inducted by

H s (T), into D ′ (]0, T [×T).
Remark 1.1. Actually we prove the following assertions :

i') For any T > 0, the solution-map u 0 → u associated with the KdV equation is discontinous, at any u 0 ∈ H ∞ 0 (T), from H ∞ 0 (T) endowed with the weak topology of

H -1 (T) into D ′ (]0, T [×T).
ii') For any T > 0, the solution-map u 0 → u associated with the mKdV equation is discontinous, at any non constant function u 0 ∈ H ∞ (T), from H ∞ (T) endowed with the weak topology of L 2 (T) into D ′ (]0, T [×T).

ii") Let u 0 ∈ H ∞ (T) be a non constant function. There exists no T > 0 such that for all t ∈]0, T [ the flow-map u 0 → u(t) associated with the mKdV equation is continuous, at u 0 , from H ∞ (T) endowed with the weak topology of L 2 (T) into D ′ (T) .

Remark 1.2. Our proof of the ill-posedness of the KdV equation below H -1 (T) is heavily related to the algebraic structure of the equation via the Miura map. However, it is interesting to notice that, in [START_REF] Molinet | Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case[END_REF], a similar result is proved for the KdV-Burgers equation with a completely different method.

The second one deals with the existence of weak L 2 -solutions to the mKdV equation.

Theorem 1.2. i) For any u 0 ∈ L 2 (T) there exists a weak solution u ∈ C w (R; L 2 (T)) ∩ (∪ s<0 F s,1/2 ) of mKdV such that u(0) = u 0 . Moreover, u(t) → u 0 in L 2 (T) as t → 0.
ii) The C(R; L 2 (T))-functions determined by the unique continuous extension to C(R; L 2 (T)) of the C(R; H ∞ (T)) solution-map of the defocusing mKdV equation, constructed in [START_REF] Kappeler | Global well-posedness of mKdV in L 2 (T, R)[END_REF], are weak solutions of the mKdV equation and belong to ∪ s<0 F s,1/2 .

Remark 1.3. The function spaces F s,1/2 are defined by (2.4) when substituting the linear KdV group U (•) by the linear group V (•) defined in (5.13).

Our notion of weak solution to mKdV is described in Definition 5.2 (see also (5.14) for assertion ii)).

Remark 1.4. Once, the second assertion of Theorem 1.2 is established, the first assertion seems to have no more interest for the defocusing mKdV equation (note that the first assertion is, up to our knowledge, the only available existence result of global weak L 2 (T)-solutions for the focusing mKdV equation). However, the proof of assertion 2 uses the complete integrability of the equation which is not a priori conserved by perturbations. On the other hand, the proof of assertion 1 seems to be widely more tractable and for instance certainly works for a wide class of perturbations of the defocusing mKdV equation. For instance, Damped mKdV :

u t + u xxx + νu ∓ u 2 u x = 0, ν > 0 KdV-mKdV : u t + u xxx ∓ u 2 u x ∓ uu x = 0, Remark 1.5.
We also construct in Proposition 5.2 global weak solutions for the "renormalized" mKdV equation (1.3).

This paper is organized as follows: In the next section we introduce the notations and the functions spaces we will work with. We also give some useful estimates for time-localized functions. In Section 3 we recall general linear estimates in such functions spaces and some linear and bilinear estimates relating to the KdV group. Section 4 is devoted to the proof of the uniform bound for L 2 (T)-bounded sequence of solutions to mKdV. We prove Theorem 1.2 in Section 5 and Theorem 1.1 in Section 6. Finally, in the appendix we first give a simplified proof of the continuous embedding in L 4 (R × T) of some Bourgain's space related to the KdV group. Then, for sake of completeness, we prove some needed bilinear estimates and sketch the proof of some properties of the Riccati map u → u ′ + u 2 --u 2 .

Notations and functional spaces 2.1 Notations

For x, y ∈ R * + , x ∼ y means that there exist C 1 , C 2 > 0 such that C 1 x ≤ y ≤ C 2 x. x
y and x y mean that there exists C 2 > 0 such that respectively x ≤ C 2 y and x ≥ C 2 y. For a Banach space X, we denote by • X the norm in X.

We will use the same notations as in [START_REF] Colliander | Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T[END_REF] and [START_REF] Colliander | Multilinear estimates for periodic KdV equations, and applications[END_REF] to deal with Fourier transform of space periodic functions with a large period λ. (dξ) λ will be the renormalized counting measure on λ -1 Z :

a(ξ) (dξ) λ = 1 λ ξ∈λ -1 Z a(ξ) .
As written in [START_REF] Colliander | Multilinear estimates for periodic KdV equations, and applications[END_REF], (dξ) λ is the counting measure on the integers when λ = 1 and converges weakly to the Lebesgue measure when λ → ∞. In all the text, all the Lebesgue norms in ξ will be with respect to the measure (dξ) λ .

For a (2πλ)-periodic function ϕ, we define its space Fourier transform on λ -1 Z by

φ(ξ) = λT e -iξx f (x) dx, ∀ξ ∈ λ -1 Z .
We denote by U (•) the free group associated with the linearized Korteweg-de Vries equation,

U (t)ϕ(ξ) = e ip(ξ)t φ(ξ), ξ ∈ λ -1 Z , p(ξ) = ξ 3 .
The Lebesgue spaces L q λ , 1 ≤ q ≤ ∞, for (2πλ)-periodic functions, will be defined as usually by

ϕ L q λ = λT |ϕ(x)| q dx 1/q
with the obvious modification for q = ∞. We define the Sobolev spaces H s λ for (2πλ)-periodic functions by

ϕ H s λ = ξ s ϕ(ξ) L 2 ξ = J s x ϕ L 2 λ , where • = (1 + | • | 2 )
1/2 and J s x ϕ(ξ) = ξ s ϕ(ξ). Note that the closed subspace of zero mean value functions of H s λ will be denoted by H s 0,λ (it is equipped with the H s λ -norm). In the same way, for a function u(t, x) on R × λT, we define its space-time Fourier transform by

û(τ, ξ) = F t,x (u)(τ, ξ) = R λT e -i(τ t+ξx) u(t, x) dxdt, ∀(τ, ξ) ∈ R×λ -1 Z .
L p t L q λ and L p T L q λ will denote respectively the Lebesgue spaces

u L p t L q λ = R u(t, •) p L q λ dt 1/p and u L p T L q λ = T 0 u(t, •) p L q λ dt 1/p
with the obvious modification for p = ∞. For any (s, b) ∈ R 2 , we define the Bourgain space X s,b λ , of (2πλ)-periodic (in x) functions as the completion of S(λT × R) for the norm

u X s,b λ = τ -p(ξ) b ξ s û L 2 τ,ξ = τ b ξ s F t,x (U (-t)u) L 2 τ,ξ , (2.1)
For T > 0 and a function space B λ , we denote by B T,λ the corresponding restriction in time space endowed with the norm

u B T,λ = inf w∈B λ { w B λ , w(•) ≡ u(•) on ] -T, T [ } .
Finally, for all function spaces of (2πλ)-periodic functions, we will drop the index λ when λ = 1.

Bourgain's spaces on frequency dependent time intervals

We will need a Littlewood-Paley analysis. Let ψ ∈ C ∞ 0 (R) be an even function such that ψ ≥ 0, supp ψ ⊂ [-3/2, 3/2], ψ ≡ 1 on [-5/4, 5/4]. We set η 0 := ψ and for all k

∈ N * , η k (ξ) := ψ(2 -k ξ) -ψ(2 -k+1 ξ) and η ≤k := ψ(2 -k •) = k j=0 η k . We also set η0 := ψ(2 -1 •) and for all k ∈ N * , ηk (ξ) := ψ(2 -k-1 ξ) -ψ(2 -k+2 ξ)
The Fourier multiplicator operators by η j , ηj and η ≤j will be denoted respectively by ∆ j , ∆j and S j , and the projection on the constant Fourier mode will be denoted by P 0 , i.e. for any u ∈ L 2 (λT) ∆ j u := η j u, ∆j u := ηj u, S j u := η ≤j û and P 0 (u) = 1 2πλ λT u(x) dx.

By a slight abuse of notations, we will also define the operator ∆ j , ∆j and P 0 on L 2 (λT × R)-functions by the same formula. Finally, for any l ∈ N we define the functions ν l on λ -1 Z × R by

ν l (ξ, τ ) := η(τ -p(ξ)) . (2.2) Let 0 ≤ b < 1, k ∈ N, t 0 ∈ R and f ∈ C ∞ (λT×]t 0 -2 -k , t 0 + 2 -k [), we define f F b λ,k,t 0 := inf f ∈X 0,b,1 λ f X 0,b,1 λ , f = f on ]t 0 -2 -k , t 0 + 2 -k [
where X 0,b,1 λ is the Bourgain's type space defined by

X 0,b,1 λ =      f ∈ S ′ (λT × R), f X 0,b,1 λ := ∞ j=0 2 jb ν j (ξ, τ ) f L 2 ξ,τ < ∞      .
Our a priori estimate will take place in the normed space G λ defined as the completion of C ∞ (λT × R) for the norm

f 2 G λ := sup t∈R k≥0 ∆ k f 2 F 1/2 λ,k,t . (2.3)
Once our a priori estimate will be established we will make use of the spaces F s,b λ , introduced in [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF], that are endowed with the norm

f 2 F s,b λ := k≥0 sup t∈R 2 ks ∆ k f F b λ,k,t 2
.

(2.4)

To handle the nonlinear term, for k ∈ N, we will also need to introduce the function space Z b λ,k defined as the completion of L 2 (λT×R) for the following norm :

f Z b λ,k = j≥0 2 bj ν j (ξ, τ ) τ -p(ξ) + i2 k -1 f L 2 . (2.5) Finally, for t 0 ∈ R and f ∈ L 2 (T×]t 0 -2 -k , t 0 + 2 -k [), we define f Z b λ,k,t 0 := inf f ∈Z b λ,k f Z b λ,k , f = f on ]t 0 -2 -k , t 0 + 2 -k [ .

Some useful estimates for localized in time functions

Following [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF] , we state the lemma below that will be useful in the linear estimates and also in the nonlinear estimates when we will localize the functions on time interval of length of order 2 -l .

Lemma 2.1. Let be given b ∈ [0, 1[, λ ≥ 1 and f ∈ X 0,b,1 λ

. Then for all l ∈ N it holds

2 bl η ≤l (τ -p(ξ)) R | f (ξ, τ ′ )| (2 -l (1+2 -l |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ (1∨2 (b-1/2)l ) f X 0,b,1 λ (2.6) and j≥l+1 2 bj η j (τ -p(ξ)) R | f (ξ, τ ′ )| 2 -l (1 + 2 -l |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ (1 ∨ 2 (b-1/2)l ) f X 0,b,1 λ .
(2.7)

Proof. For b ∈ [0, 1/2] and l ∈ N, we get by Cauchy-Schwarz in τ ′ ,

I b,l := 2 bl η ≤l (τ -p(ξ)) R | f (ξ, τ ′ )| 2 -l (1 + 2 -l |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ 2 bl η ≤l (τ -p(ξ)) ∞ q=0 R ν q (ξ, τ ′ )| f (ξ, τ ′ )| 2 -l (1 + 2 -l |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ 2 (b-1)l η ≤l (τ -p(ξ)) ∞ q=0 I b,l q ν q (ξ, τ ′ )| f (ξ, τ ′ )| τ ′ -p(ξ) b L 2 τ ′ L 2 ξ,τ
where

I b,l q := sup |τ -p(ξ)|≤2 l+1 R ηq (τ ′ -p(ξ))(1 + 2 -l |τ -τ ′ |) -8 τ ′ -p(ξ) -2b dτ ′ 1/2
But, for q ≤ l + 2,

I b,l q R ηq (τ ′ -p(ξ)) τ ′ -p(ξ) -2b dτ ′ 1/2 (2 q 2 -2bq ) 1/2 2 ( 1 2 -b)l
and for q ≥ l + 3, noticing that |τ -τ ′ | ∼ 2 q >> 2 l in the region where the integrand is not vanishing, we obtain

I b,l q (2 q 2 -2bq 2 8(l-q) ) 1/2 2 ( 1 2 -b)l .
Gathering the above estimates, we eventually get

I b,l 2 (b-1)l 2 l/2 2 ( 1 2 -b)l f X 0,b,1 λ = f X 0,b,1 λ , which proves (2.6) for b ∈ [0, 1/2].
The case b > 1/2 follows immediately by using the result for b = 1/2 and controlling the X 0,1/2,1 λ -norm by the X 0,b,1 λ -norm. Now to prove (2.7), we first notice that by the mean-value theorem,

|η j (τ -p(ξ)) -η j (τ ′ -p(ξ))| 2 -j |τ -τ ′ |
and it thus would be sufficient to estimate

j≥l+1 2 bj R η j (τ ′ -p(ξ))| f (ξ, τ ′ )| 2 -l (1 + 2 -l |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ + j≥l+1 2 (b-1)j R ηj (τ ′ -p(ξ))| f (ξ, τ ′ )| 2 -l |τ -τ ′ | (1 + 2 -l |τ -τ ′ |) 4 dτ ′ L 2 ξ,τ
.

By identifying a convolution term and applying generalized Young's inequality, we can bound the first term for any b ≥ 0 by

j≥l+1 2 bj ν j f L 2 R 2 -l (1 + 2 -l |τ ′ |) -4 dτ ′ j≥l+1 2 bj ν j f L 2
We call by J b,l the second term. For b ∈ [0, 1/2] we proceed as for (2.6) to get

J b,l 2 -l j≥l+1 2 (b-1)j ηj (τ -p(ξ)) ∞ q=0 J b,l q,j ν q (ξ, τ ′ )| f (ξ, τ ′ )| τ ′ -p(ξ) b L 2 τ ′ L 2 τ,ξ with J b,l q,j := sup τ -p(ξ) ∈[2 j-2 ,2 j+2 ] R ηq (τ ′ -p(ξ))|τ -τ ′ | 2 (1 + 2 -l |τ -τ ′ |) -8 τ ′ -p(ξ) -2b dτ ′ 1/2
For q ≤ j -5, we use that |τ -τ ′ | ∼ 2 j ≥ 2 l in the region where the integrand is not vanishing to get

J b,l q,j 2 j 2 4(l-j) R ηq (τ ′ -p(ξ)) τ ′ -p(ξ) -2b dτ ′ 1/2 2 j 2 4(l-j) 2 ( 1 2 -b)q 2 4l 2 -( 5 2 +b)j
whereas for q ≥ j -4, we easily get

J b,l q,j 2 -bj |τ | (1 + 2 -l |τ |) 4 L 2 τ 2 -bj 2 3l 2 .
Gathering these estimates we conclude that

J b,l f X 0,b,1 λ 2 -l 2 3l 2 ∞ j=l+1 2 (b-1)j 2 j/2 2 -bj f X 0,b,1 λ 2 l/2 ∞ j=l+1 2 -j/2 f X 0,b,1 λ . This proves (2.7) for b ∈ [0, 1/2]. Finally, in the case b ∈]1/2, 1] we observe that J b,l 2 -l j≥l+1 2 (b-1 2 )j 2 -j/2 ηj (τ -p(ξ)) ∞ q=0 J 1/2,l q,j ν q (ξ, τ ′ )| f (ξ, τ ′ )| τ ′ -p(ξ) 1/2 L 2 τ ′ L 2 τ,ξ
and the above bounds on J

1/2,l q,j lead to J b,l f X 0,1/2,1 λ 2 l/2 ∞ j=l+1 2 (b-1 2 )j 2 -j/2 2 (b-1 2 )l f X 0,b,1 λ . Remark 2.1. A very useful corollary of the preceding lemma is the follow- ing: Let b ∈ [0, 1/2], f ∈ X 0,b,1 λ and γ ∈ C ∞ c (R) with support in ] -2, 2[. Then for all k ∈ N, it holds γ(2 k t)f X 0,b,1 λ f X 0,b,1 λ (2.8) and η ≤k (τ -p(ξ))F xt (γ(2 k t)f ) L 2 ξ,τ 2 -bk f X 0,b,1 λ . (2.9) Remark 2.2. It is easy to check that G λ is continuously embedded in L ∞ (R; L 2 λ (T)). Indeed, for any u ∈ G λ , t 0 ∈ R and k ∈ N, taking a function ũ ∈ X 0,1/2,1 λ such that ũ ≡ u on ]t 0 -2 -k , t 0 + 2 -k [ and ũ X 0,1/2,1 λ ≤ 2 u F 1/2 λ,k,t 0 , it holds F x (∆ k u(t 0 ))(ξ) = R F(∆ k ũ(ξ, τ ))e it 0 τ dτ According to the obvious estimate R | f (ξ, τ ′ )| dτ ′ L 2 ξ f X 0,1/2,1 λ , ∀f ∈ X 0,1/2,1 λ , (2.10)
this leads to

∆ k u(t 0 ) L 2 λ ∆ k ũ X 0,1/2,1 λ ∆ k u F 1/2 λ,k,t 0 .
Squaring and summing in k one obtains that

u L ∞ t L 2 (λT)) u G λ . (2.11)
On the other hand, it seems pretty clear that G λ is not included in C(R; L 2 λ (T)).

3 Some linear and bilinear estimates

General linear estimates

We first derive linear estimates that do not depend on the dispersive linear group associated with our functional space. We mainly follow [START_REF] Koch | A priori bounds for the 1D cubic NLS in negative Sobolev spaces[END_REF]- [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF].

Lemma 3.1. Let be given b ∈ [1/2, 1[. Then ∀ϕ ∈ L 2 λ and all k ∈ N, it holds U (t)ϕ F b λ,k,0 2 (b-1 2 )k ϕ L 2 λ . (3.1)
Proof. Clearly, it suffices to prove that for any k ∈ N,

η 0 (2 k t)U (•)ϕ X 0,b,1 λ 2 (b-1 2 )k ϕ L 2 λ .
Notice that the left-hand side member of the above inequality is bounded by

∞ j=0 2 bj η j (τ )2 -k η0 (2 -k τ ) L 2 τ ϕ L 2 λ . Since η 0 ∈ C ∞ c (R), η 0 decays at least as (1 + |y|) -4 and thus η j (τ )2 -k η0 (2 -k τ ) L 2 τ η j (τ )2 -k (1+2 -k |τ |) -4 ) L 2 τ 2 -k 2 j/2 min(1, 2 4(k-j) ) . (3.2) Hence, k+2 j=0 2 bj η j (τ )2 -k η0 (2 -k τ ) L 2 τ 2 (b-1 2 )k and ∞ j=k+2 2 bj η j (τ )2 -k η0 (2 -k τ ) L 2 ∞ j=k+2 2 j(b-7 2 ) 2 3k 2 (b-1 2 )k . Lemma 3.2. Let be given b ∈ [1/2, 1[. Then for any k ∈ N and any f ∈ Z b λ,k,0 it holds t 0 U (t -t ′ )f (t ′ ) dt ′ F b λ,k,0 2 (b-1 2 )k f Z b λ,k,0 . (3.3) Proof. Let f ∈ Z b λ,k be and extension of f such that f Z b λ,k ≤ 2 f Z b λ,k,0 . We set v := η 0 (2 k t) t 0 U (t -t ′ ) f (t ′ ) dt ′ . Then F tx (v)(ξ, τ ) = F t η 0 (2 k t) R e itτ -e itp(ξ) i(τ -p(ξ)) F t,x ( f ) dτ (ξ, τ ) = F t (η 0 (2 k t)) * F t,x ( f ) i(τ -p(ξ)) -F t (e itp(ξ) η 0 (2 k t)) R F t,x ( f ) i(τ ′ -p(ξ)) dτ ′ = R F t,x ( f )(τ ′ , ξ) 2 -k η0 (2 -k (τ -τ ′ )) -2 -k η0 (2 -k (τ -p(ξ))) i(τ ′ -p(ξ)) dτ ′
Now we claim that

I = 2 -k η0 (2 -k (τ -τ ′ )) -2 -k η0 (2 -k (τ -p(ξ))) |τ ′ -p(ξ)| |τ ′ -p(ξ) + i2 k | 2 -k (1 + 2 -k |τ -τ ′ |) -4 + 2 -k (1 + 2 -k |τ -p(ξ)|) -4 (3.4)
Assuming (3.4) for a while, it follows that

v X 0,b,1 ∞ j=0 2 bj η j (τ -p(ξ)) R f (ξ, τ ′ ) τ ′ -p(ξ) + i2 k 2 -k (1 + 2 -k |τ -τ ′ |) -4 dτ ′ L 2 ξ,τ + ∞ j=0 2 bj η j (τ -p(ξ))2 -k (1 + 2 -k |τ -p(ξ)|) -4 R f (ξ, τ ′ ) τ ′ -p(ξ) + i2 k dτ ′ L 2 ξ,τ
The desired bound on the first term of the above right-hand side member follows directly from (2.6)-(2.7). To obtain the desired bound on the second term, we combine (2.10) and (3.2). It thus remains to prove (3.4). For this we first notice that, since η 0 (y) = η 0 (|y|), by the mean-value theorem there exists

θ ∈]|τ -τ ′ |, |τ -p(ξ)|[ such that 2 -k η0 (2 -k (τ -τ ′ ))-2 -k η0 (2 -k (τ -p(ξ))) 2 -2k η′ 0 (2 -k θ)|τ ′ -p(ξ)| . (3.5) Furthermore, since η 0 ∈ S(R), |η 0 (y)| + |η ′ 0 (y)| (1 + |y|) -10 .
Let us now separate three cases :

• |τ -p(ξ)| ≤ 2 k . Then (3.4) is obvious whenever |τ ′ -p(ξ)| ≥ 2 k and follows directly from (3.5) whenever |τ ′ -p(ξ)| ≤ 2 k . • |τ -p(ξ)| ≥ 2 k and |τ -τ ′ | ∼ |τ -p(ξ)|. Then we must have |θ| ∼ |τ -p(ξ)| and |τ ′ -p(ξ) + i2 k | |τ -p(ξ)|.Therefore (3.5) leads to I 2 -2k (1 + 2 -k |τ -p(ξ)|) -5 |τ -p(ξ)| 2 -k (1 + 2 -k |τ -p(ξ)|) -4 • |τ -p(ξ)| ≥ 2 k and |τ -τ ′ | ∼ |τ -p(ξ)| . Then |τ ′ -p(ξ)| ∼ (|τ -p(ξ)|∨ |τ ′ -τ |)
2 k and (3.4) follows directly from the decay of η0 .

Specific linear and bilinear estimates

We will also need estimates that are specific for Bourgain's spaces associated with the KdV linear group. We first recall the following Strichartz's type estimate proved in [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, I. Schrödinger equations II. The KdV equation[END_REF] (we give a simplified proof of this estimate in the appendix) :

Lemma 3.3. For any λ ≥ 1 and any u ∈ X 0,1/3 λ , it holds u L 4 t,λ u X 0,1/3 λ (3.6)
Finally we will make a frequent use of the following bilinear estimates that can be deduced for instance from [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF] (we give a proof of these estimates in the appendix since we need to quantify the dependence of these estimates with respect to the period λ): Lemma 3.4. Let λ ≥ 1 and let u 1 and u 2 be two real valued L 2 functions defined on R × (λ -1 Z) with the following support properties

(τ, ξ) ∈ supp u i ⇒ τ -ξ 3 L i , i = 1, 2.
Then for any N > 0 the following estimates holds:

u 1 ⋆ u 2 L 2 τ L 2 (|ξ|≥N ) (L 1 ∧ L 2 ) 1/2 (L 1 ∨ L 2 ) 1/4 N 1/4 + λ -1/2 u 1 L 2 u 2 L 2 ,
(3.7) and

Λ[N ](u 1 , u 2 ) L 2 τ,ξ (L 1 ∧ L 2 ) 1/2 (L 1 ∨ L 2 ) 1/2 N + λ -1/2 u 1 L 2 u 2 L 2 . (3.8) where Λ[N ] : (L 2 (R × λ -1 Z)) 2 → L ∞ (R × λ -1 Z) is defined by Λ[N ](u 1 , u 2 )(τ, ξ) = R |ξ 1 |-|ξ-ξ 1 | ≥ N 100 u 1 (τ 1 , ξ 1 )u 2 (τ -τ 1 , ξ -ξ 1 ) (dξ 1 ) λ dτ 1 4 A priori estimate for smooth solutions to (1.3)
As in previous works on mKdV on the torus (cf. [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, I. Schrödinger equations II. The KdV equation[END_REF], [START_REF] Colliander | Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T[END_REF]), we actually work with the "renormalized " mKdV equation (1.3) instead of the mKdV equation itself. This permits to cancel some resonant part in the nonlinear term. Recall that for v ∈ C(R; H ∞ λ ), a smooth solution to mKdV with initial data v 0 , the L 2 -norm of v is a constant of the motion and thus

u(t, x) := v(t, x ∓ 6t 2πλ v 0 2 L 2 λ ) satisfies (1.3).
Denoting by N (u) the nonlinear term of (1.3), it holds for any ξ ∈ λ -1 Z,

F x [N (u)](ξ) = - 6i λ 2 ξ 1 +ξ 2 +ξ 3 =ξ ξ 1 +ξ 2 =0 û(ξ 1 )û(ξ 2 )ξ 3 û(ξ 3 ) = - 2iξ λ 2 ξ 1 +ξ 2 +ξ 3 =ξ (ξ 1 +ξ 2 )(ξ 1 +ξ 3 )(ξ 2 +ξ 3 ) =0 û(ξ 1 )û(ξ 2 )û(ξ 3 ) + 6iξ λ 2 û(ξ)û(ξ)û(-ξ) := -iξ F x A(u, u, u) (ξ) + F x B(u, u, u) (ξ) i.e. 6(u 2 -P 0 (u 2 ))u x = ∂ x A(u, u, u) + B(u, u, u) . (4.1)
According to the resonance relation (4.15), A is non resonant whereas B is a resonant term. As pointing out in [START_REF] Takaoka | Well-posedness of the Cauchy problem for the modied KdV equation with periodic boundary condition[END_REF], A is a "good term" as far as one wants to solve the equation in H s (T) for s ≥ 1/4. On the other hand, B is a bad term as soon as one wants to solve the equation below H 1/2 (T), giving rise to rapid oscillations that breaks the uniform continuity on bounded set of the flow-map. Then,

u G λ u L ∞ (R;L 2 λ ) + u 3 G λ (4.2)
Proof. From the definition of the norm G λ , we have to bound sup

t 0 ∈R k≥0 ∆ k u 2 F 1/2 λ,k,t 0 .
We use that for any (t 0 , t) ∈ R 2 , it holds

u(t) = U (t -t 0 )u(t 0 ) + t t 0 U (t -t ′ )∂ x A(u(t ′ )) + B(u(t ′ ) dt ′ .
By translation in time we can always assume that t 0 = 0 and according to Lemmas 3.1-3.2,

U (t)∆ k u(0) F 1/2 λ,k,0 ∆ k u(0) L 2 λ and t 0 U (t-t ′ )∂ x ∆ k (A(u(t ′ ))+B(u(t ′ ))dt ′ F 1/2 λ,k,0 ∆ k ∂ x (A(u))+B(u)) Z 1/2 λ,k,0 . Since k≥0 ∆ k u(0) 2 L 2 λ ∼ u 2 L ∞ (R;L 2 λ ) , it remains to prove that ∞ k=0 2 2k ∆ k A(u) 2 Z 1/2 λ,k,0 + ∆ k B(u) 2 Z 1/2 λ,k,0 u 6 G λ .
This is the aim of the two following lemmas.

Lemma 4.1. Let be given

t 0 ∈ R, λ ≥ 1 and u i ∈ G λ for i = 1, 2, 3. Then it holds ∞ k=0 2 2k ∆ k B(u 1 , u 2 , u 3 ) 2 Z 1/2 λ,k,t 0 3 i=1 u i 2 G λ .
Proof. By translation in time, we can take t 0 = 0. For any fixed k ∈ N, we take a time extension ũ1 of

u 1 such that ∆ k ũ1 X 0,1/2,1 λ ≤ 2 ∆ k u 1 F 1/2 λ,k,0 .
Then, in view of the structure of B, it holds

2 k ∆ k B(u 1 , u 2 , u 3 ) Z 1/2 λ,k,0 l≥0 2 l/2 2 k η l (τ -ξ 3 ) τ -ξ 3 +i2 k -1 F xt ∆ k B(v 1 , v 2 , v 3 ) L 2 l≥0 2 l/2 2 k η l (τ -ξ 3 ) τ -ξ 3 + i2 k -1 F xt B(∆ k v 1 , ∆k v 2 , ∆k v 3 )] L 2
where v 1 = η 0 (2 k t)ũ 1 and v i = η 0 (2 k t)u i , i = 2, 3. By duality it suffices to prove that

I k := 2 k F xt B(∆ k v 1 , ∆k v 2 , ∆k v 3 ) , τ -ξ 3 + i2 k -1 w L 2 sup l (2 -l/2 ν l w L 2 ) ∆ k v 1 X 0,1/2,1 λ 3 i=2 S k+1 v i X 0,1/2,1 λ . (4.3)
Indeed, first, according to (2.8) for any k ∈ N * and any

u ∈ C(R × λT), η 0 (2 k t)S k-1 u 2 X 0,1/2,1 λ ∼ k-1 j=0 η 0 (2 k t)∆ j u 2 X 0,1/2,1 λ ≤ k-1 j=0 ∆ j u 2 F 1/2 λ,j,0 ≤ u 2 G λ . (4.4) Second, taking a C ∞ -function γ : R → [0, 1] with compact support in [-1, 1] satisfying γ ≡ 1 on [-1/2, 1/2] and m∈Z γ(t -m) ≡ 1 on R, we get for any j ≥ k ∈ N , η 0 (2 k t)∆ j u X 0,1/2,1 λ ≤ |m|≤2 j-k γ(2 j t -m)η 0 (2 k t)∆ j u X 0,1/2,1 λ ≤ |m|≤2 j-k ∆ j u F 1/2 λ,j,2 -j m 2 j-k u G λ . (4.5)
Therefore, (4.3) will lead to

2 k ∆ k B(u 1 , u 2 , u 3 ) Z 1/2 λ,k,0 ∆ k u 1 X λ,k,0 3 i=2 u i G λ
which will gives the result by squaring and summing in k.

Since the norms in the right-hand side of (4.3) only see the size of the modulus of the Fourier transform of the functions we can assume that all the functions have non negative Fourier transforms. In view of the structure of B, using Cauchy-Schwarz, we get

I k 2 k η k v 1 * ηk v 2 L 2 ( ξ ∼2 k ) τ -ξ 3 + i2 k -1 ηk w * ηk ˇ v 3 L 2 ( ξ ∼2 k )
where ˇ v(τ, ξ) = v(-τ, -ξ) for all (τ, ξ) ∈ R × λ -1 Z. For k = 0, 1, 2 this yields directly the result by using the Strichartz inequality (3.6). For k ≥ 3 we introduce the following notation : we set νl (τ, ξ) := η l (τ -ξ 3 ) for l > k and νl (τ, ξ) := k j=0 η j (τ -ξ 3 ) for l = k .

I k 2 k min(l 1 ,l 2 ,l 3 )≥k l≥0 2 (l 1 ∧l 2 )/2 2 (l 1 ∨l 2 )/4 2 -k/4 + 1 νl 1 η k v 1 L 2 νl 2 ηk v 2 L 2 2 (l∧l 3 )/2 2 (l∨l 3 )/4 2 -k/4 + 1 2 -(l∨k) νl 3 ηk v 3 L 2 ν l ηk w L 2 2 k l≥0 2 3l 8 2 -(l∨k) ν l ηk w L 2 min(l 1 ,l 2 ,l 3 )≥k νl 1 η k v 1 L 2 νl 2 ηk v 2 L 2 νl 3 ηk v 3 L 2 max 2 3 8 (l 1 +l 2 +l 3 ) 2 -k/2 , 2 (l 1 +l 2 ) 4 2 3l 3 8 2 -k/4 , 2 3 8 (l 1 +l 2 ) 2 -k/4 2 l 3 8 , 2 (l 1 +l 2 ) 4 2 l 3 8 sup l (2 -l/2 ν l ηk w L 2 ) sup l≥k (2 l/2 νl η k v 1 L 2 ) 3 i=2 sup l≥k 2 l/2 νl ηk v i L 2 sup l (2 -l/2 ν l ηk w L 2 ) ∆ k v 1 X 0,1/2,1 λ 3 i=2 S k+1 v i X 0,1/2,1 λ (4.7)
where we used (2.9) in the last step.

Lemma 4.2. Let be given

t 0 ∈ R, λ ≥ 1 and u i ∈ G λ for i = 1, 2, 3. Then it holds J := ∞ k=0 2 2k ∆ k A(u 1 , u 2 , u 3 ) 2 Z 1/2 λ,k,t 0 3 i=1 u i 2 G λ . (4.8)
Proof. Again by translation in time, we can take t 0 = 0. Denoting by ξ i the Fourier modes of u i , we can always assume by symmetry that

|ξ 1 | ≥ |ξ 2 | ≥ |ξ 3 |. We divide A in different terms corresponding to regions of (λ -1 Z) 3 . 1.|ξ 1 | ≤ 2 4 .
Then it holds |ξ 1 + ξ 2 + ξ 3 | ≤ 2 6 . By Sobolev inequalities and (3.6),

J 7 k=0 3 i=1 η 0 (2 k t)S 6 u i X 0,1/2,1 λ 2 ,
which is acceptable thanks to (4.4)-(4.5).

2.|ξ

1 | ≥ 2 4 and |ξ 1 | ≤ 4|ξ|. In this region, it holds |ξ| ∼ |ξ 1 |. We rewrite η k (ξ) as η k (ξ 1 ) + η k (ξ) -η k (ξ 1 )
and notice that by the mean-value theorem,

|η k (ξ) -η k (ξ 1 )| min 1, 2 -k |ξ| -|ξ 1 | . (4.9) Therefore, J ∞ k=0 (J 2 1,k + J 2 2,k ) with J 1,k := 2 k ηk (ξ) τ -ξ 3 + i2 k -1 F tx A 1 (∆ k u 1 , u 2 , u 3 ) F 1/2 λ,k,0 and J 2,k := 2 k ηk (ξ) τ -ξ 3 + i2 k -1 F tx A 2 (u 1 , u 2 , u 3 ) F 1/2 λ,k,0
, where

F x A 1 (v 1 , v 2 , v 3 ) (ξ) := 1 λ 2 (ξ 1 ,ξ 2 ,ξ 3 )∈Θ(ξ) 2 4 ≤|ξ 1 |≤4|ξ| v1 (ξ 1 )v 2 (ξ 2 )v 3 (ξ 3 ) , (4.10)
F x A 2 (v 1 , v 2 , v 3 ) (ξ) := 1 λ 2 (ξ 1 ,ξ 2 ,ξ 3 )∈Θ(ξ) 2 4 ≤|ξ 1 |≤4|ξ| [η k (ξ) -η k (ξ 1 )] v1 (ξ 1 )v 2 (ξ 2 )v 3 (ξ 3 ) , and
Θ(ξ) := (ξ 1 , ξ 2 , ξ 3 ) ∈ λ -1 Z, 3 i=1 ξ i = ξ, 3 i,j=1 i =j (ξ i +ξ j ) = 0 and |ξ 3 | ≤ |ξ 2 | ≤ |ξ 1 | . • Estimate on J 1,k
For any fixed k ∈ N, we take a time extension ũ1 of

u 1 such that ∆ k ũ1 X 0,1/2,1 λ ≤ 2 ∆ k u 1 F 1/2 λ,k,0
. We set v 1 = η 0 (2 k t)ũ 1 and v i = η 0 (2 k t)u i , i = 2, 3. By duality it suffices to prove that

H 1,k := 2 k F xt A 1 (∆ k v 1 , v 2 , v 3 ) , τ -ξ 3 + i2 k -1 ∆w L 2 sup l (2 -l/2 ν l w L 2 ) ∆ k v 1 X 0,1/2,1 λ 3 i=2 S k+3 v i X 0,1/2,1 λ (4.11)
which is acceptable thanks to (4.4)-(4.5). Again, since the norms in the right-hand side of (4.11) only see the size of the modulus of the Fourier transform of the functions we can assume that all the functions have non negative Fourier transforms. We separate four cases :

A. ξξ 1 ≤ 0. Then |ξ 2 + ξ 3 | = |ξ -ξ 1 | ∼ |ξ|.
Proceeding as in (4.7), we get Therefore we can proceed exactly as in the previous case.

H 1,k 2 k τ -ξ 3 + i2 k -1 ηk w * η k ˇ v 1 L 2 (|ξ|∼2 k ) η ≤k+3 v 2 * η ≤k+3 v 3 L 2 (|ξ|∼2 k ) sup l (2 -l/2 ν l ∆k w L 2 ) ∆ k v 1 X 0,1/2,1 λ 3 i=2 S k+3 v i X 0,1/2,1 λ . ( 4
• Estimate on J 2,k For any fixed k ∈ N, we set v i := η 0 (2 k t)u i , i = 1, 2, 3. By duality and (4.4)-(4.5), it suffices to prove that

H 2,k := 2 k F xt A 2 (v 1 , v 2 , v 3 ) , τ -ξ 3 + i2 k -1 ∆k w L 2 2 -k/8 sup l (2 -l/2 ν l w L 2 ) 3 i=1 S k+3 v i X 0,1/2,1 λ (4.13)
Since the norms in the right-hand side of (4.13) only see the size of the modulus of the Fourier transform of the functions we can assume that all the functions have non negative Fourier transforms. In view of (4.9), we thus infer that

H 2,k 2 k F xt Ã2 (v 1 , v 2 , v 3 ) , τ -ξ 3 + i2 k -1 ∆k w L 2
where

F x Ã2 (v 1 , v 2 , v 3 ) (ξ) := 1 λ 2 (ξ 1 ,ξ 2 ,ξ 3 )∈Θ(ξ) 2 4 ≤|ξ 1 |≤4|ξ| 1∧(2 -k |ξ|-|ξ 1 | v1 (ξ 1 )v 2 (ξ 2 )v 3 (ξ 3 ) .
First to estimate the contribution of the region |ξ|-|ξ 1 | ≤ |ξ| 3/4 , we proceed exactly as for J 1,k by separating the four cases A, B, C and D to obtain

H 2,k 2 -k/4 sup l (2 -l/2 ν l ∆k w L 2 ) 3 i=1 S k+3 v i X 0,1/2,1 λ which is acceptable. Now in the region |ξ| -|ξ 1 | ≥ |ξ| 3/4 , we notice that |ξ 2 + ξ 3 | = |ξ -ξ 1 | ≥ |ξ| 3/4 . Therefore, setting σ := σ(τ, ξ) = τ -ξ 3 and σ i := σ(τ i , ξ i ), i = 1, 2, 3, (4.14)
we claim that the well-known resonance relation

σ -σ 1 -σ 2 -σ 3 = 3(ξ 1 + ξ 2 )(ξ 1 + ξ 3 )(ξ 2 + ξ 3 ) (4.
H 2,k 2 k η ≤k+2 v 2 * η ≤k+3 v 3 L 2 (|ξ| 2 3k/4 ) Λ[2 3k/4 ] τ -ξ 3 + i2 k -1 ηk w , η ≤k+3 ˇ v 1 L 2 min(l 1 ,l 2 ,l 3 )≥k,l≥0 max(2 l ,2 l i )≥λ -1 2 7k/4 2 k 2 (l 2 ∧l 3 )/2 2 (l 2 ∨l 3 )/4 2 -3 16 k + λ -1/2 νl 2 η ≤k+3 v 2 L 2 νl 3 η ≤k+3 v 3 L 2 2 (l 1 ∧l)/2 2 (l 1 ∨l)/4 2 -3 16 k + λ -1/2 1/4 2 (l 1 ∨l)/2 2 -3k/4 + λ -1/2 3/4 2 -(l∨k) νl 1 η ≤k+3 v 1 L 2 ν l ηk w L 2 2 -k 32 sup l (2 -l/2 ν l ηk w L 2 ) 3 i=1 sup l≥k 2 l/2 νl S k+3 v i L 2 λ 2 -k 32 sup l (2 -l/2 ν l ∆k w L 2 ) 3 i=1 S k+3 v i X 0,1/2,1 λ
where in the last step we used (2.8)-(2.9).

3.|ξ

1 | ≥ 2 4 and |ξ 1 | ≥ 4|ξ|.
In this region it holds

|ξ 2 + ξ 3 | ≥ 3 4 |ξ 1 |, |ξ 2 | ≥ 3 8 |ξ 1 | and |ξ 3 | ≤ 5 8 |ξ 1 | (4.17)
The two first above inequalities are clear. To prove the third one, we notice that in this region ξ 1 ξ 2 ≤ 0 and we proceed by contradiction by assuming that |ξ 3 | > 5 8 |ξ 1 |. Then we first notice that if ξ 1 ξ 3 ≥ 0 then we must have

|ξ| ≥ |ξ 3 | > 5 8 |ξ 1 | which contradicts |ξ 1 | ≥ 4|ξ|. Second, if ξ 1 ξ 3 ≤ 0 then we have |ξ| = |ξ 2 | + |ξ 3 | -|ξ 1 | > 5 4 |ξ 1 | -|ξ 1 | = 1 4 |ξ 1 | which again contradicts |ξ 1 | ≥ 4|ξ| . Therefore the resonance relation yields max(|σ|, |σ i |) λ -1 |ξ 1 | 2 . (4.18)
In this region J k∈N J 2 3,k where, for any fixed k ∈ N, J 3,k is defined by

J 3,k := k 1 ≥k l≥0 2 l/2 2 k η l (τ -ξ 3 ) τ -ξ 3 +i2 k -1 ηk (ξ)F tx A 3 (∆ k 1 v 1 , v 2 , v 3 ) L 2 (4.

19) with

F x A 3 (v 1 , v 2 , v 3 ) (ξ) := 1 λ 2 (ξ 1 ,ξ 2 ,ξ 3 )∈Λ(ξ) |ξ 1 |≥2 4 , |ξ 1 |≥4|ξ| v1 (ξ 1 )v 2 (ξ 2 )v 3 (ξ 3 ) (4.20)
and

v i := η 0 (2 k t)u i , i = 1, 2, 3. Let γ be a C ∞ -function γ : R → [0, 1] with compact support in [-1, 1] satisfying γ ≡ 1 on [-1/2, 1/2] and m∈Z γ(t -m) 3 ≡ 1 on R. (4.21)
We set

v k 1 ,m i := η 0 (2 k t)γ(2 k 1 t -m)u i = γ(2 k 1 t -m)v i , i = 1, 2, 3, m ∈ Z . Clearly, A 3 (v 1 , v 2 , v 3 ) ≡ |m| 2 k 1 -k A 3 (v k 1 ,m 1 , v k 1 ,m 2 , v k 1 ,m 3
). Therefore, by duality it suffices to prove that

I k := k 1 ≥k |m| 2 k 1 -k l≥0 2 k F xt A 3 (∆ k 1 v k 1 ,m 1 , v k 1 ,m 2 , v k 1 ,m 3 ) , τ -ξ 3 + i2 k -1 η l (τ -ξ 3 )η k w L 2 2 -k 16 sup l (2 -l/2 ν j ηk w L 2 ) sup m∈Z 3 i=1 S k 1 +1 v k 1 ,m i X 0,1/2,1 λ . (4.22)
Indeed, this will be acceptable, since by proceeding as in (4.4)-(4.5), it is easily checked that

S k 1 +1 v k 1 ,m i 2 X 0,1/2,1 λ ≤ k 1 -1 j=0 ∆ j u i 2 F 1/2 λ,j,m2 -k 1 + 1 q=-1 ∆ k 1 u i 2 F 1/2 λ,k 1 ,m2 -k 1 +q2 -(k 1 +1) + 2 q=-2 ∆ k 1 +1 u i 2 F 1/2 λ,k 1 +1,m2 -k 1 +q2 -(k 1 +2) u i 2 G λ .
Since the norms in the right-hand side of (4.22) only see the size of the modulus of the Fourier transform of the functions we can assume that all the functions have non negative Fourier transforms. We will use the following notations :

νl (τ, ξ) := η l (τ -ξ 3 ) for l > k 1 and νl (τ, ξ) := η ≤k 1 (τ -ξ 3 ) for l = k 1 . (4.23) In view of (4.17

I k k 1 ≥k |m| 2 k 1 -k l≥0 2 k Λ[2 k 1 ] η k 1 v k 1 ,m 1 , η ≤k 1 +1 v k 1 ,m 3 L 2 τ -ξ 3 + i2 k -1 η k w * η ≤k 1 +1 ˇ v k 1 ,m 2 L 2 (|ξ| 2 k 1 ) k 1 ≥k |m| 2 k 1 -k min(l 1 ,l 2 ,l 3 )≥k 1 ,l≥0 max(2 l ,2 l i )≥λ -1 2 2k 1 2 k 2 (l 1 ∧l 3 )/2 (2 (l 1 ∨l 3 )/2 2 -k 1 + λ -1/2 ) 3/4 (2 (l 1 ∨l 3 )/4 2 -k 1 /4 + λ -1/2 ) 1/4 2 (l∧l 2 )/2 (2 (l∨l 2 )/4 2 -k 1 /4 + λ -1/2 )2 -(l∨k) νl 2 η ≤k+1 v k 1 ,m 2
L 2 νl 3 η ≤k 1 +1 v k 1 ,m 3 L 2 νl 1 η k 1 v k 1 ,m 1 L 2 ν l η k w L 2 k 1 ≥k |m| 2 k 1 -k 2 k 2 -17 16 k 1 sup l 2 -17 32 l ν l η k w L 2 3 i=1 sup l≥k 1 2 15 32 l νl η ≤k 1 +1 v k 1 ,m i L 2 k 1 ≥k 2 -k 1 16 sup l 2 -17 32 l ν l η k w L 2 sup |m| 2 k 1 -k 3 i=1 S k 1 +1 v k 1 ,m i X 0,15/32,1 λ (4.24)
which yields (4.22) by summing in k 1 and concludes the proof of the lemma.

Corollary 4.1. There exists ε 0 > 0 such that any λ ≥ 1 and any solution

u ∈ C ∞ (R; H ∞ (λT)) to (1.3) satisfying u 0 L 2 λ ≤ ε 0 , it holds u G λ u 0 L 2 λ . (4.25)
Proof. We are going to implement a continuity argument on the space period. Recall that if u(t, x) is a smooth global 2λπ-periodic solution of (1.3) with initial data

u 0 then u β (t, x) = β -1 u(β -3 t, β -1 x) is a (2πλβ)-periodic solution of (1.3) emanating from u 0,β = β -1 u 0 (β -1 x). Moreover, u 0,β L 2 λβ = β -1/2 u 0 L 2 λ and u 0,β H 1 λβ ≤ β -1/2 u 0 H 1 λ .
From the conservation of the L 2 -norm and of the Energy,

E(u) := 1 2 T u 2 x ∓ 1 6 T u 4 ,
and Sobolev inequalities (in the focusing case) , we get

u β L ∞ (R;L 2 λβ ) = u 0,β L 2 λβ and u β L ∞ (R;H 1 λβ ) u 0,β H 1 λβ (1 + u 0,β 2 L 2 λβ )
In particular, it follows from (4.2) that for β ≥ 1,

u β G λβ ≤ C u 0 L 2 λ + u β 3 G λβ , (4.26)
for some constant C > 0. Now, from classical linear estimates in Bourgain's spaces and the Duhamel formula, it holds

u β G λβ sup t 0 ∈R η 0 (t -t 0 )u β X 0,1/2,1 λβ u β (t 0 ) L 2 λβ + (u 2 β -P 0 (u 2 β ))∂ x u β L 2 (]t 0 -2,t 0 +2[;L 2 λβ ) u β L ∞ (R;L 2 λβ ) + u β 3 L ∞ (R;H 1 λβ ) β -1/2 u 0 L 2 λ + β -3/2 u 0 3 H 1 λ . Therefore for β ≥ 1 large enough (depending on u 0 H 1 ), u β G λβ ≤ C u 0 L 2 λ
. Recalling that thanks to the infinite number of conservation laws, u actually belongs to

C ∞ b (R; H ∞ (λT)), β → u β G λβ is continuous on R * +
and the result follows from (4.26) and a classical continuity argument.

Proof of Theorem 1.2

In this section we follow the process proposed in [START_REF] Molinet | On ill-posedness for the periodic cubic Schrödinger equation[END_REF] to identify the limit of a L 2 -bounded sequence of solutions to mKdV. To pass to the limit on the nonlinear term in (1.3), we will make use of the space F s,b λ,T introduced in [START_REF] Ionescu | Global well-posedness of the initial value problem for the KP-I equation in the energy space[END_REF]. We will need the following lemma which states that for any s < 0, b < 1/2 and T > 0, G λ is compactly embedded in F s,b λ,T .

Lemma 5.1. Let λ ≥ 1 and {u n } n∈N be a bounded sequence of G λ . Then for any T > 0, s < 0 and b < 1/2, {u n } n∈N is relatively compact in F s,b λ,T .

Proof. First we observe that for any s < 0 and any u ∈ G λ ,

u 2 F s,1/2 λ = k∈N 2 2ks sup t∈R ∆ k u 2 F 1/2 λ,k,t ( k∈N 2 2ks ) sup k∈N sup t∈R ∆ k u 2 F 1/2 λ,k,t ≤ C(s) u G λ 2 .
(5.1)

Hence G λ ֒→ F s,1/2 λ
. Second, proceeding as in Remark 2.2, it is easy to check that there exists C 0 > 0 such that for all s < 0 and v ∈ F

s,1/2 λ , v L ∞ (R;H s λ ) ≤ C 0 v F s,1/2 λ . (5.2)
Let us now prove the desired result. We will use a diagonal extraction argument. Let us fix s ′ < 0, b ′ < 1/2 and T > 0. We notice that for any fixed k 0 ∈ N, there exists C k 0 > 0 such that

S k 0 v F s ′ ,b ′ λ,T S k 0 v X s ′ ,b ′ λ,T ≤ C k 0 S k 0 v F s ′ ,b ′ λ,T
Indeed, the first inequality is obvious and the second one follows from the following chain of inequalities for any v ∈ F s ′ ,b ′ λ,T ,

S k 0 v 2 X s ′ ,b ′ λ,T ≤ k 0 +1 k=0 2 2ks ′ η 0 ( t 2T )∆ k S k 0 ṽ 2 X 0,b ′ λ k 0 +1 k=0 2 2k T 2 sup t∈R 2 2ks ′ ∆ k S k 0 ṽ 2 F b ′ λ,k,t C k 0 S k 0 v F s ′ ,b ′ λ,T 2
where ṽ is an extension of v such that

S k 0 v F s ′ ,b ′ λ ≤ 2 S k 0 v F s ′ ,b ′ λ,T
. Since, according to (5.1), {u n } n≥0 is bounded in

F s ′ ,1/2 λ , it follows that {S k 0 u n } n≥0 is bounded in X s ′ ,b ′ λ for any k 0 ≥ 0. Now, using that for s < s ′ and b < b ′ , X s ′ ,b ′ λ,T +1 is compactly embedded in X s,b λ,T +1
, we deduce that there exists a subsequence {u nq } of {u n } and a sequence {w k } ⊂ F s,b λ,T +1 such that for any k ∈ N,

S k u nq → w k in F s,b λ,T +1 . (5.3) We define w ∈ S ′ (] -T -1, T + 1[×T) by ∆ k w = ∆ k w k+1 for all k ∈ N.
Clearly, for all k 0 ≥ 0,

k 0 k=0 2 2ks sup t∈-]T,T [ ∆ k w 2 F b λ,k,t ∞ k=0 2 2ks sup n∈N sup t∈R ∆ k u n 2 F b λ,k,t sup n∈N u n 2 G λ .
which ensures that w ∈ F s,b λ,T . Moreover, proceeding as in (5.1), it easy to check that lim

k 0 →∞ sup n∈N ∞ k=k 0 ∆ k u n F s,b λ,T = 0 . It thus follows from (5.3) that u nq -w F s,b λ,T → 0 as q → ∞.
The following proposition states that our sequence of solution {u n } is uniformly equi-continuous in H s (T) for any s < 0. Proposition 5.1. Let λ ≥ 1 and {u n } n∈N be a sequence of smooth solutions to (1.3) that is bounded in G λ . Then, for any ε > 0 and s < 0, there exists

δ ε,s > 0 such that ∀(t 1 , t 2 ) ∈ R 2 , |t 1 -t 2 | < δ ε,s ⇒ u n (t 1 ) -u n (t 2 ) H s λ < ε, ∀n ∈ N .
Proof. First we claim that if we prove that {u n } is bounded in F s,b λ for some b > 1/2 and s < 0 then we are done. To prove this claim, we fix s := s 0 < 0 and we set

M := sup n∈N ( u n G λ + u n F s 0 ,b λ ) .
For any given ε > 0, we denote by k ε the smaller integer such that

∞ k=kε 2 2ks 0 1/2 < ε 4C 0 C(s 0 )M . (5.4)
From (5.1)-( 5.2) and (5.4) we infer that

∞ k=kε+1 ∆ k u n (t 1 )- ∞ k=kε+1 ∆ k u n (t 2 ) H s 0 λ ≤ 2C 0 ∞ k=kε+1 ∆ k u n F s 0 ,1/2 λ ≤ ε/2 . Now, let γ : R → [0, 1] be a C ∞ -function with compact support in [-1, 1] satisfying γ ≡ 1 on [-1/2,
∆ k u n (t 1 ) -∆ k u n (t 2 ) 2 H s 0 λ = kε k=0 γ(2 k (t 1 -t 1 ))∆ k u n (t 1 ) -γ(2 k (t 2 -t 1 ))∆ k u n (t 2 ) 2 H s 0 λ ∼ kε,s k=0 2 2ks 0 R F t,x γ(2 k (t -t 1 ))∆ k u n (ξ, τ )(e it 1 τ -e it 2 τ ) dτ 2 L 2 ξ C(s 0 ) sup 0≤k≤kε R 2 |t 1 -t 2 | 2 R -R F t,x ∆ k γ(2 k (t -t 1 ))u n (ξ, τ ) dτ 2 L 2 ξ + |τ |>R F t,x ∆ k γ(2 k (t -t 1 ))u n (ξ, τ ) dτ 2 L 2 ξ C(s 0 )R 2 |t 1 -t 2 | 2 sup 0≤k≤kε ∆ k γ(2 k (t -t 1 ))u n 2 X 0,1/2,1 λ +R 1-2b ∆ k γ(2 k (t -t 1 ))u n 2 X 0,b,1 λ C(s 0 )M |t 1 -t 2 | 2 R 2 + R 1-2b 2 (2b-1)kε , (5.5)
where in the last step we use that, according to (2.6)-(2.7) , for all b ∈ [1/2, 1[ and all k ∈ N,

γ(2 k t)f X 0,b,1 λ 2 (b-1 2 )k f X 0,b,1 λ , ∀f ∈ X 0,b,1 λ . Taking R := |t 1 -t 2 | -1 b+1/2 , (5.5) leads to kε k=0 ∆ k u n (t 1 ) -∆ k u n (t 2 ) H s 0 λ C(ε, M, b)|t 1 -t 2 | b-1/2 b+1/2
This gives the desired result for s = s 0 and δ ε,

s 0 := min[ ε 2M , ( ε 2C(ε,M,b) ) b+1/2 b-1/2
]. Finally, the result for any s ∈]s 0 , 0[ follows by interpolating with (2.11).

We are thus reduced to prove that {u n } is bounded in F -2,b λ for some b > 1/2. We proceed as in Proposition 4.1 and write that for any (t 0 , t) ∈ R 2 , it holds

u(t) = U (t -t 0 )u(t 0 ) + t t 0 U (t -t ′ )∂ x A(u(t ′ )) + B(u(t ′ ) dt ′ .
By translation in time we can always assume that t 0 = 0 and according to Lemmas 3.1-3.2, for any b ∈]1/2, 1[,

U (t)∆ k u(0) F b λ,k,0 2 (b-1/2)k ∆ k u(0) L 2 λ 2 (b-1/2)k ∆ k u L ∞ t L 2 λ and t 0 U (t-t ′ )∂ x ∆ k (A(u(t ′ ))+B(u(t ′ ))dt ′ F b λ,k,0 2 (b-1 2 )k 2 k ∆ k (A(u))+B(u)) Z b λ,k,0 . Therefore, for any b ∈]1/2, 1[, U (t)u 2 F -2,b λ = k∈N 2 -4k sup t∈R U (t)∆ k u 2 F b λ,k,t k∈N 2 -4k 2 (2b-1)k ∆ k u 2 L ∞ t L 2 λ C(b) u 2 L ∞ t L 2 λ C(b) u 2 G λ , (5.6)
where we used (2.11) in the last step. In the remaining we assume that

b ∈]1/2, 1/2 + 1 32 [. It remains to prove that ∞ k=0 sup t∈R ∆ k A(u) 2 Z b λ,k,t + ∆ k B(u) 2 Z b λ,k,t u 6 G λ .
To control the term involving B, we proceed as in Lemma 4.1. By duality it suffices to prove that

Ĩk := F xt B(∆ k v 1 , ∆k v 2 , ∆k v 3 ) , τ -ξ 3 + i2 k -1 w L 2 2 -k/2 sup l (2 -bl ν l w L 2 ) 3 i=1 S k+1 v i X 0,1/2,1 λ .
where v i := η 0 (2 k t)u i . For k = 0, 1, 2, this follows directly from (3.6), whereas for k ≥ 3 we deduce from (4.7) that

Ĩk 2 -k 2 (b-1 2 )k sup l (2 -bl ν l w L 2 ) 3 i=1 S k+1 v i X 0,1/2,1 λ
which is acceptable. Now to estimate the term involving A we denote by ξ i the Fourier modes of u i and we assume by symmetry that 4 . This region can be treated as in Lemma 4.2 by using (3.6). 2.|ξ 1 | ≥ 2 4 and |ξ 1 | ≤ 4|ξ|. By duality it suffices to prove that

|ξ 1 | ≥ |ξ 2 | ≥ |ξ 3 |. We divide A in different terms corresponding to regions of (λZ) 3 . 1.|ξ 1 | ≤ 2
Hk := 2 k F xt A 1 (v 1 , v 2 , v 3 ) , τ -ξ 3 + i2 k -1 ∆w L 2 2 -k/2 sup l (2 -bl ν l w L 2 ) 3 i=1 S k+3 v i X 0,1/2,1 λ
where A 1 is defined in (4.10) and v i := η 0 (2 k t)u i . This can be done by proceeding as above for B, separating as in Lemma 4.2, the 4 cases : ξξ 1 ≤ 0, ξξ 2 ≤ 0, ξξ 3 ≤ 0 and all the ξ i 's of the same sign. 3.|ξ 1 | ≥ 2 4 and |ξ 1 | ≥ 4|ξ|. In this last region it suffices to prove that Jk :=

k 1 ≥k l≥0 2 bl 2 k η l (τ -ξ 3 ) τ -ξ 3 + i2 k -1 ηk (ξ)F tx A 3 (∆ k 1 v 1 , v 2 , v 3 ) L 2 2 -k 2 3 i=1 u i G λ
where A 3 is defined in (4.20). But this follows directly from (4.24) for any b ∈] 1 2 , 1 2 + 1 32 [. This completes the proof of the lemma. As noticed in [START_REF] Molinet | On ill-posedness for the periodic cubic Schrödinger equation[END_REF], B has got a nice structure for passing to the limit in the sense of distributions. More precisely, we have the following lemma: Lemma 5.2. For any λ ≥ 1 and T > 0, the operator

∂ x B is continuous from (F -1/3,1/3 λ,T ) 3 into X -4,-1/3 λ,T . Proof. Taking w ∈ X 4,1/3 λ supported in time in ] -2T, 2T [ and extensions ũi ∈ F -1/3,1/3 λ of u i ∈ F -1/3,1/3 λ,T such that ũi F -1/3,1/3 λ ≤ 2 u i F -1/3,1/3 λ,T , it holds I := w, ∂ x B(u 1 , u 2 , u 3 ) L 2 (R×λT) ∞ k=0 ∆ k w x , B( ∆k ũ1 , ∆k ũ2 , ∆k ũ3 ) L 2 (R×λT) ∞ k=0 |m| T 2 k ∆ k w x , B( ∆k v k,m 1 1 , ∆k v k,m 2 2 , ∆k v k,m 3 3 ) L 2 (R×λT)
where for any k ∈ N, i ∈ {1, 2, 3} and m ∈ Z, we set v k,m i := γ(2 k t -m)ũ i with γ defined as in (4.21). (3.6) and Bernstein inequality then ensure that

I ∞ k=0 |m| T 2 k 2 2k F -1 (| ∆ k w|) L 4 λ 3 i=1 2 -k/3 F -1 (| ∆k v k,m i |) L 4 λ T ∞ k=0 2 3k ∆ k w X 0,1/3 λ 3 i=1 sup m∈Z ∆k v k,m i X -1/3,1/3 λ (5.7)
But on account of (2.8) (with obvious modification for k = 0),

∆k v k,m i X -1/3,1/3 λ 1 j=-1 ∆ k+j γ(2 k t -m)ũ i X -1/3,1/3 λ 1 j=-1 2 -(k+j)/3 sup t∈R ∆ k+j ũi F 1/3 λ,k+j,t ũi F -1/3,1/3 λ u i F -1/3,1/3 λ,T
.

Therefore, leads to

I T w X 4,1/3 λ 3 i=1 u i F -1/3,1/3 λ,T (5.8)
which concludes the proof of the lemma.

Let us now prove a continuity result for the non resonant part A.

Lemma 5.3. For any λ ≥ 1 and T > 0, the operator

∂ x A is continuous from (F -2 -6 ,15/32 λ,T ) 3 into X -4,-1/2 λ,T . Proof. Taking w ∈ X 4,1/2 λ supported in time in ] -2T, 2T [ and exten- sions ũi ∈ F -2 -6 ,15/32 λ of u i ∈ F -2 -6 ,15/32 λ,T such that ũi F -2 -6 ,15/32 λ ≤ 2 u i F -2 -6 ,15/32 λ,T
, it holds

J := w, ∂ x A(u 1 , u 2 , u 3 ) L 2 (R×λT) (k,k 1 ,k 2 ,k 3 )∈N 4 ∆ k w x , A(∆ k 1 ũ1 , ∆ k 2 ũ2 , ∆ k 3 ũ3 ) L 2 (R×λT)
By symmetry we may assume that k 1 ≥ k 2 ≥ k 3 . Now, the sum in the reion k 1 ≤ 6k + 10 can clearly be treated exactly in the same way as we treat B in the preceding lemma. It thus remains to consider the sum over the region k 1 > 6k + 10. We follow the proof of Lemma 4.2. First we notice that (4.17) -(4.18) hold in this region. Then setting,

v k 1 ,m i := γ(2 k 1 t -m)ũ i , i = 1, 2, 3, m ∈ Z,
with γ defined as in (4.21), we obtain in this region

J k≥0 k 1 ≥k 2 ≥k 3 k 1 ≥6k+10 |m| T 2 k 1 ∆ k w x , A(∆ k 1 v k 1 ,m 1 , ∆ k 2 v k 1 ,m 2 , ∆ k 3 v k 1 ,m 3 ) L 2 (R×λT)
Proceeding exactly as in (4.24), with (2.8) in hand, we get

J k≥0 k 1 ≥k 2 ≥k 3 k 1 ≥6k+10 2 -k 1 16 ∆ k w X 2,1/2 λ 3 i=1 sup m∈Z ∆ k i v k 1 ,m i X 0,15/32 λ T k≥0 k 1 ≥0 2 -k 1 16 k 2 1 ∆ k w X 2,1/2 λ sup m∈Z ( ∆ k 1 v k 1 ,m 1 X 0,15/32 λ ) 3 i=2 sup 0≤k 3 ≤k 2 ≤k 1 m∈Z ∆ k i v k 1 ,m i X 0,15/32 λ T w X 3,1/2 λ 3 i=1 sup t∈R,k∈N 2 -2 -6 k ∆ k ũi F 15/32 λ,k,t T w X 3,1/2 λ 3 i=1 u i F -2 -6 ,15/32 λ,T (5.9)
which completes the proof of the lemma .

For any T > 0, let us define the operator Υ T which to u associates

Υ T (u) := F -1 tx Γ T (u)
where . In particular, for any u ∈ F -2 -6 ,15/32 λ and any T > 0, this operator defined an element of X -4,-1/2 λ with a X -4,-1/2 λ -norm which is, according to (5.8) and (5.9), of order at most O(T ). This ensures that we can pass to the limit in S ′ on Γ T (u) as T → ∞. We can thus define the operator Γ from F tx (Γ(u)) ≡ 6(u 2 -P 0 (u 2 ))u x on λT × R for any u ∈ S(λT × R). Definition 5.1. We will say that a function u is a weak solution of (1.3) if it satisfies (1.3) in the sense of distributions, when (u 2 -P 0 (u 2 ))u x is interpreted as the inverse Fourier transform of Γ(u). Proposition 5.2. Let {u 0,n } ⊂ H ∞ (T) be such that u 0,n ⇀ u 0 in L 2 (T). Then there exist a weak solution u ∈ C w (R; L 2 (T)) ∩ s<0 F s,1/2 to (1.3), with u(0) = u 0 , and a subsequence of emanating solutions {u n k } to (1.3) such that for all T > 0 and φ ∈ L 2 (T), (u n k (t), φ) L 2 (T) → (u(t), φ) L 2 (T) in C([-T, T ]) .

Γ T (u)(ξ, τ ) := 6iξ λ 2 τ 1 +τ 2 +τ 3 =τ ψ T u(ξ, τ 1 ) ψ T u(ξ, τ 2 ) ψ T u(-ξ, τ 3 ) - 1 3 ξ 1 +ξ 2 +ξ 3 =ξ (ξ 1 +ξ 2 )(ξ 1 +ξ 3 )(ξ 2 +ξ 3 ) =0 ψ T u(ξ 1 , τ 1 ) ψ T u(ξ 2 , τ 2 ) ψ T u(ξ 3 , τ 3 ) dτ 1 dτ 2 dτ 3 ,( 5
(5.11)

Moreover, P 0 (u(t)) = P 0 (u 0 ) for all t ∈ R.

Proof. We proceed as in [START_REF] Molinet | On ill-posedness for the periodic cubic Schrödinger equation[END_REF]. By Banach-Steinhaus theorem, the sequence {u 0,n } is bounded in L 2 (T). We start by assuming that sup n∈N u 0,n L 2 ≤ ε 0 so that the conclusions of Corollary 4.1 hold. From the conservation of the L 2 -norm, the sequence of emanating solutions {u n } to (1.3) is bounded in L ∞ (R; L 2 (T)) and thus, up to a subsequence, {u n } converges weakly star in L ∞ (R; L 2 (T)) to some u ∈ L ∞ (R; L 2 (T)). In particular, {∂ t u n } and {∂ 3 x u n } converge in the distributional sense to respectively u t and u xxx . It remains to pass to the limit on the nonlinear term. By Corollary 4.1, {u n } is bounded in G. From Lemma 5.1 and lemmas 5.2-5.3 it follows that ∂ x (A(u n )+B(u n )) converges to F -1 tx (Γ(u)) in the distributional sense on ]-T, T [×T. Since this holds for all T > 0, the convergence holds actually in the distributional sense on R × T. Therefore, u is a weak solution to (1.3) in the sense of Definition 5.1. Note also that, in view of (5.1), u ∈ ∪ s<0 F s,1/2 . Moreover, according to Proposition 5.1, for any time-independent 2π-periodic smooth function φ, the family {t → (u n (t), φ) L 2 x } is bounded and uniformly equi-continuous on [-T, T ]. Ascoli's theorem then ensures that {t → (u n k , φ)} converges to t → (u, φ) in C([-T, T ]). Since {u n } is bounded in L ∞ (R; L 2 (T)), this convergence also holds for any φ ∈ L 2 (T). This proves (5.11) and that u ∈ C w (R; L 2 (T)). In particular, u(0) = u 0 and for all t ∈ R,

T u(t) dx = lim k→∞ T u n k (t) dx = lim k→∞ T u n k (0) dx = T u 0 dx .
This concludes the proof of the proposition whenever sup n∈N u 0,n L 2 ≤ ε 0 . Now, if sup n∈N u 0,n L 2 = M > ε 0 we use the dilation symmetry of (1.3). Recall that if u(t, x) is a smooth global 2π-periodic solution of (1.3) with initial data u 0 then u λ (t, x) = λ -1 u(λ -3 t, λ -1 x) is a 2πλ-periodic solution of (1.3) emanating from

u λ 0 = λ -1 u 0 (λ -1 x). Setting λ M = M 2 /ε 2 so that sup n∈N u λ M 0,n L 2 λ M
≤ ε 0 , it follows from above that the conclusions of the proposition hold if one replaces {u 0,n }, {u n } and u by respectively

{u λ M 0,n }, {u λ M n } and u λ M ∈ C w (R; L 2 (λ 0 T)) ∩ ∪ s<0 F s,1/2 λ M
. This ensures 1 that these conclusions also hold for {u 0,n }, {u n } and u with u(t, x) := λ M u λ M (λ 3 M t, λ M x) and completes the proof of the proposition.

Proof of Theorem 1.2 : Again we have to introduce a notion of weak solution for mKdV: Definition 5.2. We will say that a function v ∈ C w (R; L 2 (T)) is a weak solution of (1.2) with initial data v 0 if it satisfies the following equation in the sense of distributions,

v t + v xxx ∓ 6(v 2 -P 0 (v 2 ))v x ± 6P 0 (v 2 0 )v x = 0 , (5.12)
when (v 2 -P 0 (v 2 ))v x is interpreted as the inverse Fourier transform of Γ(v).

Of course, any smooth solution of the mKdV equation is a weak solution since the L 2 -norm is a constant of the motion for smooth solutions.

Let

v 0 ∈ L 2 (T). It is easy to construct a sequence {v 0,n } ⊂ H ∞ (T) such that v 0,n L 2 = v 0 L 2 for any n ≥ 0 and v 0,n → v 0 in L 2 (T). The emanating solutions v n satisfies v n,t + v n,xxx ± 6P 0 (v 2 0 )v x ∓ 6(v 2 n -P 0 (v 2 n ))v x = 0 .
Let u 0 ∈ H ∞ (T) be a non constant function and κ = 0 be a real number. We set u 0,n = u 0 + κ cos(nx) so that u 0,n ⇀ u 0 in L 2 (T) and u 0,n 2

L 2 → u 0 2 L 2 + κ 2 π.
According to Proposition 5.2 there exists a subsequence {u n k } of the emanating solutions {u n } to (1.3) and u ∈ C w (R; L 2 (T)) a weak solution of (1.3) , with u(0) = u 0 , satisfying

u n k (t) ⇀ u(t) in L 2 (T) for all t ∈ R. Let now {v n k := u n k (•, • -6t 2π u 0,n k 2 L 2
)} be the associated subsequence of solutions to mKdV emanating from {u 0,n k }. We proceed by contradiction. Assuming that the solution-map is continuous at u 0 from L 2 (T) equipped with its weak topology into D ′ (]0, T [×T), we obtain that

{v n k } converges in the sense of distributions in ]0, T [×T to the solution v ∈ C ∞ (R; H ∞ (T)) of mKdV emanating from u 0 . It follows that {u n k } converges in the same sense to v •, • + 6t 2π ( u 0 2 L 2 + κ 2 π) and thus u ≡ v •, • + 6t 2π ( u 0 2 L 2 + κ 2 π) on ]0, T [ . (6.1)
This ensures that u is actually a strong solution of (1.3) and satisfies this equation everywhere on ]0, T [×T. On the other hand, according to (6.1), u is also solution of

u t + u xxx + 6 u 2 -P 0 (u 2 ) -κ 2 /2 u x = 0 .
This forces u x to be identically vanishing on ]0, T [ which contradicts that u(0) = u 0 is not a constant and u ∈ C w (R; L 2 (T)). Note that in the contradiction process, we can replace the assumption on the continuity of the solution-map by the assumption that the flow-map u 0 → u(t) is continuous from L 2 (T), equipped with its weak topology, into D ′ (T) for all t ∈]0, T [. Since for each t ∈ R, u n k (t) ⇀ u(t) in L 2 (T) we also get a contradiction. This proves assertion ii") of Remark 1.1.

The KdV equation

First, recall that Miura discovered that the Miura map M (u) := u ′ +u 2 maps smooth solutions to the defocusing mKdV equation into smooth solutions to the KdV equation. Actually, it was observed in [START_REF] Colliander | Symplectic nonsqueezing of the Korteweg-de Vries flow[END_REF] that the Riccati map R(u) := u ′ + u2 -P 0 (u 2 ) maps smooth solutions to the defocusing version of equation (1.3) (i.e. with the + sign in front of the nonlinear term) into smooth solutions to the KdV equation (1.1), i.e. if u is a smooth solution of the defocusing (1.3) then R(u) is a smooth solution of (1.1). Moreover, according to [START_REF] Kappeler | Riccati map of L 2 0 (T) and its applications[END_REF], this map enjoys the following property : Theorem 6.1 ( [START_REF] Kappeler | Riccati map of L 2 0 (T) and its applications[END_REF]). The Riccati map R is an isomorphism from L 2 0 (T) into H -1 0 (T). Actually, we will only need the Riccati map to be a bijection from H ∞ 0 (T) into itself. For sake of completeness, we give in the appendix the outline of the proof of this property. Now, let w 0 ∈ H ∞ 0 (T) and κ ∈ R * be given. We set θ

0 := w 0 -κ 2 cos x, u 0 := R -1 (θ 0 ) ∈ H ∞ 0 (T) 2 and u 0,n := u 0 + κ[cos(nx) + cos((n + 1)x)] . Clearly, u 0,n ⇀ u 0 in L 2 (T), u 0,n 2
L 2 → u 0 2 L 2 + 2κ 2 π and R(u 0,n ) ⇀ θ 0 + κ 2 cos(x) = w 0 in H -1 (
T). According to Proposition 5.2 there exists a subsequence {u n k } of the emanating solutions {u n } to (1.3) and u ∈ C w (R; L 2 0 (T)) such that u n k (t) ⇀ u(t) in L 2 (T) for all t ∈ R and u is a weak solution to (1.3). To identify R(u) we will need the following lemma Lemma 6.1. The operator u → u

2 -P 0 (u 2 ) is continuous from F -1/16,7/16 T into D ′ (] -T, T [×T).
Proof. We set

C(u, u) := u 2 -P 0 (u 2 ) = ξ∈Z * (ξ 1 ,ξ 2 )∈Z 2 ξ 1 +ξ 2 =ξ û(ξ 1 )û(ξ 2 ) e iξx
Taking w ∈ D(]-T, T [×T) and extensions ũi ∈ F -1/16,7/16 of u i ∈ F -1/16,7/16 T such that ũi F -1/16,7/16 ≤ 2 u i F -1/16,7/16 T , it holds

J := w, C(u 1 , u 2 ) L 2 (R×T) (k,k 1 ,k 2 )∈N 3 ∆ k w, C(∆ k 1 ũ1 , ∆ k 2 ũ2 ) L 2 (R×T)
By symmetry we may assume that k 1 ≥ k 2 . We set

v k 1 ,m i := γ(2 k 1 t -m)ũ i , i = 1, 2, m ∈ Z, where γ : R → [0, 1] is a C ∞ -function with compact support in [-1, 1] satisfying γ ≡ 1 on [-1/2, 1/2] and m∈Z γ(t -m) 2 ≡ 1 on R. We then obtain J (k,k 2 ,k 3 )∈N 3 k 1 ≥k 2 |m| T 2 k 1 ∆ k w, C(∆ k 1 v k 1 ,m 1 , ∆ k 2 v k 1 ,m 2 ) L 2 (R×T)
.

We separate N 3 into two regions.

1. The region : k 1 ≤ 4k + 4. Then by (3.6) and (2.8), we can write

J (k,k 2 ,k 3 )∈N 3 k 1 ≥k 2 |m| T 2 k 1 ∆ k w L 2 ∆ k 1 v k 1 ,m 1
L 4 ∆ k 2 v k 1 ,m 2 L 4 T (k,k 2 ,k 3 )∈N 3 k 1 ≥k 2 2 k 1 2 2k 1 /3 ∆ k w L 2 2 i=1 sup m∈Z 2 -k 1 /3 ∆ k i v k 1 ,m i X 0,1/3 T (sup k∈N 2 5k/3 ∆ k w L 2 ) 2 i=1 sup k i ∈N sup m∈Z ∆ k i v k 1 ,m i X -1/3,1/3 T w L 2 (R;H 2 (T)) u 1 F -1/3,1/3 T u 2 F -1/3,1/3 T (6.2)
which is acceptable.

2. The region : k 1 > 4k + 4. Note that in this region,

k 2 ≥ k 1 -2 ≥ 4k + 2.
In this region we will need the well-known resonance relation

|σ -σ 1 -σ 2 | = |3ξξ 1 ξ 2 | 2 2k 1 , (6.3)
where ξ = ξ 1 + ξ 2 and σ, σ 1 , σ 2 are defined by (4.14). We subdivide this region into subregions with respect to the maximum of (|σ|, |σ i |).

• |σ| = max(|σ|, |σ 1 |, σ 2 |). In this subregion, making use of (6.3), (3.6) and (2.8), we get

J (k,k 2 ,k 3 )∈N 3 k 1 ≥k 2 , k 1 >4k+4 |m| T 2 k 1 ∆ k w L 2 ∆ k 1 v k 1 ,m 1 L 4 ∆ k 2 v k 1 ,m 2 L 4 T (k,k 2 ,k 3 )∈N 3 k 1 ≥k 2 , k 1 >4k+4 2 k 1 2 2k 1 /3 2 -2k 1 ∆ k w X 0,1 2 i=1 sup m∈Z 2 -k 1 /3 ∆ k i v k 1 ,m i X 0,1/3 T w X 0,1 u 1 F -1/3,1/3 T u 2 F -1/3,1/3 T . (6.4) • |σ 1 | = max(|σ|, |σ 1 |, σ 2 |).
In this subregion, defining νl , l ≥ k 1 as in (4.23) and making use of (6.3), (3.7) and (2.8)-(2.9), we get

J (k,k 1 ,k 2 )∈N 3 k 1 ≥k 2 , k 1 >4k+4 |m| T 2 k 1 l≥0,l 2 ≥k 1 ν l ∆ k w * νl 2 ˇ v k 1 ,m 2 L 2 (|ξ| 2 k 1 ) ∆ k 1 v k 1 ,m 1 L 2 T (k,k 1 ,k 2 )∈N 3 k 1 ≥k 2 , k 1 >4k+4 l≥0,l 2 ≥k 1 2 l/2 2 k 1 (2 l 2 /4 2 -k 1 /4 + 1) ∆ k w L 2 sup m∈Z νl 2 ∆ k 2 v k 1 ,m 2 L 2 2 -7k 1 /8 sup m∈Z ∆ k 1 v k 1 ,m 1 X 0,7/16 T (k,k 1 ,k 2 )∈N 3 k 1 ≥k 2 , k 1 >4k+4 l≥0 2 l/2 2 k 1 2 -21k 1 /16 ∆ k w L 2 sup m∈Z ∆ k 2 v k 1 ,m 2 X 0,7/16,1 sup m∈Z ∆ k 1 v k 1 ,m 1 X 0,7/16
T w X 0,1 . Gathering (6.2), (6.4) and (6.5), we obtain the desired continuity result. By Corollary 4.1, Lemma 5.1 and possibly dilation arguments as in the proof of Proposition 5.2, we know that the sequence {u n } is relatively compact in F -1/16,7/16 T for any T > 0. From the above lemma we thus infer that R(u n ) ⇀ R(u) weak star in L ∞ (R; H -1 (T)) .

We proceed now by contradiction. Let us assume that the solution-map is continuous at w 0 = θ 0 + κ 2 cos x from H -1 (T) equipped with its weak topology into D ′ (]0, T [×T). Since w 0 ∈ H ∞ 0 (T), we deduce from Lemma 6.1 that w 0 = R(Θ 0 ) for some Θ 0 ∈ H ∞ 0 (T) with Θ 0 = u 0 . By the continuity of the solution-map at w 0 , R(u) must be equal on ]0, T [ to the solution of KdV emanating from w 0 which is nothing else but R(Θ) where Θ is the smooth solution to (1.3) emanating from Θ 0 . This ensures that u(t) ∈ H ∞ 0 (T) for all t ∈]0, T [ and the injectivity of R on H ∞ 0 (T) then forces u = Θ on ]0, T [. This contradicts that u(0) = u 0 = Θ 0 = Θ(0) and both functions are weakly continous with values in L 2 (T). supported in {(τ, ξ) ∈ R × Z + }. Let us recall that these arguments are based on the fact that the operator j : L 2 (R × Z) → L 2 (R × Z), defined by j(u)(τ, ξ) = u(-τ, -ξ), is an isometry of L 2 (R × Z) satisfying, for any real-valued L 1 ∩ L 2 -functions u 1 and u 2 ,

u 1 ⋆ u 2 L 2 (R×Z) = u 1 ⋆ j(u 2 ) L 2 (R×Z) .
By Cauchy-Schwarz in (τ 1 , ξ 1 ) we infer that

(β l 1 u 1 ) ⋆ (β l 2 u 2 ) 2 L 2 = τ ξ∈Z τ 1 ξ 1 ∈Z (β l 1 u 1 )(τ 1 , ξ 1 ) (β l 2 u 2 )(τ -τ 1 , ξ -ξ 1 )
2 τ ξ∈Z α(τ, ξ)

τ 1 ξ 1 ∈Z (β l 1 u 1 )(τ 1 , ξ 1 ) (β l 2 u 2 )(τ -τ 1 , ξ -ξ 1 ) 2 sup τ ∈R,ξ∈Z + α(τ, ξ) β l 1 u 1 2 L 2 β l 2 u 2 2 L 2 ,
where α(τ, ξ) mes (τ 1 , ξ 1 ) ∈ R × Z + / ξ -ξ 1 ∈ Z + , τ 1 -ξ 3 1 ∼ 2 l 1 and τ -τ 1 -(ξ -ξ 1 ) 3 ∼ 2 l 2 (2 l 1 ∧ 2 l 2 ) #A(τ, ξ) , with A(τ, ξ) := {ξ 1 ≥ 0/ ξ -ξ 1 ≥ 0 and τ -ξ 3 1 -(ξ -ξ 1 ) 3 2 l 1 ∨ 2 l 2 } .

We separate two regions. In the region ξ 3 ≥ 2 l 1 ∨ 2 l 2 , we notice that ∂ 2 y [τy 3 -(ξ -y) 3 ] = -6ξ which leads to #A(τ, ξ)

2 l 1 ∨ 2 l 2 ξ 1/2 + 1 2 l 1 ∨ 2 l 2 1/3 .
In the region 0 ≤ ξ 3 ≤ 2 l 1 ∨ 2 l 2 , we use that 0 ≤ ξ 1 ≤ ξ to obtain that

#A(τ, ξ) ≤ #{ξ 1 , 0 ≤ ξ 3 1 ≤ 2 l 1 ∨ 2 l 2 } 2 l 1 ∨ 2 l 2 1/3 .
This completes the proof of (7.1). which leads to mes[C(τ, ξ)]

1 λ λ L 1 ∨ L 2 N 2 ∨ 1 .
7.3 Outline of the proof of the bijectivity of the Riccati map from H ∞ 0 (T) into itself.

We follow the arguments in [START_REF] Colliander | Symplectic nonsqueezing of the Korteweg-de Vries flow[END_REF]. For u ∈ H ∞ 0 (T) we denote by L u the Schrödinger operator with potential u, i.e. Since L u is a self adjoint operator with compact resolvent, it has a discrete spectrum λ 1 ≤ λ 2 ≤ ... with λ n → +∞. By the definition of E u one must have E u (φ) ≥ λ 1 T φ 2 for any φ ∈ H 1 (T) with equality if and only if φ is a λ 1 -eigenfunction. For u ≡ 0, noticing that E(1) = 0 and that 1 is not an eigenfunction, it follows that the first eigenvalue λ 1 is negative. Then, by standard arguments, one can check that λ 1 is a simple eigenvalue with an eigenfunction that is a non vanishing H ∞ (T)-function. On the other hand, if u ≡ 0 it is well-known that the first eigenvalue of L 0 is 0 and that the associated eigenspace is spanned by 1. In both cases, we normalized this eigenfunction by requiring it to be positive and L 2 -normalized and we call it φ 1 . Introducing the logarithmic derivative v of φ 1 , defined by

v := d dx ln(φ 1 ) = φ 1,x φ 1 ∈ H ∞ 0 (T)
we observe that

v x = φ 1,xx φ 1 - φ 1,x φ 1 2 = u -λ 1 -v 2
and thus u = v x + v 2 + λ 1 . Taking the means of both sides of this equality it leads to λ 1 = -P 0 (v 2 ) which ensures that u = R(v). This proves that R is surjective from H ∞ 0 (T) into itself. Now, let w ∈ H ∞ 0 (T) be such that R(w) = u. Setting

Proposition 4 . 1 .

 41 Let λ ≥ 1 and u ∈ C(R; H ∞ (λT)) be a solution to (1.3).

( 4 . 6 )

 46 According to Lemma 3.4 we obtain

. 12 )

 12 B. ξξ 2 ≤ 0. Then |ξ 1 + ξ 3 | = |ξ -ξ 2 | ∼ |ξ|. Therefore, exchanging the role of v 1 and v 2 , we can proceed exactly as in the previous case. C. ξξ 3 ≤ 0. Then |ξ 1 + ξ 2 | = |ξ -ξ 3 | ∼ |ξ|. Therefore, exchanging the role of v 1 and v 3 , we can proceed exactly as in the case A. D. ξ 1 , ξ 2 and ξ 3 are of the same sign. Then |ξ -ξ 3 | = |ξ 1 + ξ 2 | ∼ |ξ|.

  ), in this region it holds |ξ 1 | -|ξ 3 | |ξ 1 | and |ξ -ξ 2 | |ξ 1 |. Lemma 3.4, (4.18) and (2.8)-(2.9) thus lead to

. 10 )- 2 - 6 , 15

 102615 with ψ T (•) := ψ(•/T ) for ψ defined as in Section 2.2. In view of (4.1), for any smooth function u ∈ S(λT × R), Υ T (u) ≡ 6(u 2 -P 0 (u 2 ))u x on ] -T, T [. From the two above lemmas we infer that Υ T can be continuously extended in F

- 2 - 6

 26 ,15/32 λ into S ′ (λT × R) by setting Γ(u), φ S ′ ,S , := lim T →∞ Γ T (u), φ S ′ ,S , ∀φ ∈ S(λT × R) Obviously, F -1

• |σ 2 | 1 and v k 1 ,m 2

 212 = max(|σ|, |σ 1 |, σ 2 |).Then we can proceed exactly as in the preceding case by exchanging the role of v k 1 ,m

7. 2 8 ) 1 λ ξ 1 2 L 2 u 1 (τ 1 , ξ 1 ) 2 L 2 u 2 2 L 2 , 1 ∨ L 2 ) 1 / 2 N 1 /2 ∨ 1 .

 281122111222121211 Proof of (3.7)-(3.We take λ ≥ 1. Let A ⊂ λ -1 Z and let, for any ξ ∈ λ -1 Z, B(ξ) ⊂ λ -1 Z. To prove (3.7)-(3.8) we first notice that, by Cauchy-Schwarz,R ∈B(ξ) u 1 (τ 1 , ξ 1 ) u 2 (τ -τ 1 , ξ -ξ 1 ) dτ 1 u 2 (τ -τ 1 , ξ -ξ 1 )whereα(τ, ξ) mes (τ 1 , ξ 1 ) ∈ R×B(ξ) / (τ 1 , ξ 1 ) ∈ supp(u 1 ) and (τ -τ 1 , ξ-ξ 1 ) ∈ supp(u 2 ) . Therefore, assuming that supp(u i ) ⊂ {(τ, ξ)/ τ -ξ 3 L i }, we get α(τ, ξ) (L 1 ∧ L 2 ) mes[C(τ, ξ)] with C(τ, ξ) := {ξ 1 ∈ B(ξ)/ τ -ξ 3 1 -(ξ -ξ 1 ) 3 L 1 ∨ L 2 } .(3.7) then follows by noticing that ∂ 2 y [τ -y 3 -(ξ -y) 3 ] = -6ξ and thus, for any |ξ| ≥ N > 0, mes[C(τ, ξ)] Finally, (3.8) follows by noticing that ∂ y [τ -y 3 -(ξ -y) 3 ] = -3 y + (ξ -y) y -(ξ -y) and thus, on B(ξ) = {ξ 1 ∈ λ -1 Z / |ξ 1 | -|ξ -ξ 1 | ≥ N > 0}, it holds ξ 1 + (ξ -ξ 1 ) ξ 1 -(ξ -ξ 1 )N 2

L u := - d dx 2

 2 + u with domain H 2 (T). One can associate to L u the energy E u (•) defined on H 1 (T) byE u (φ) := L u φ, φ = T φ 2x + uφ 2 .

dx 2 +

 2 ds)it is easy to check that w = ρ ′ /ρ. Observing that ( R(w) + P 0 (w 2 ) = -d 2 dx 2 + u + P 0 (w 2 ) ,

  Indeed, either we are in the cases B ,C or D described above and then there exists i ∈ 2, 3 such that |ξ 1 + ξ i | ∼ |ξ| so that we are done or we are in the case A. In this last case, we first notice that |ξ 2 + ξ 3 | ∼ |ξ|. Second, since ξξ 1 ≤ 0 and |ξ| -|ξ 1 | ≥ |ξ| 3/4 we must have |ξ| ≤ |ξ 1 | -|ξ| 3/4 and |ξ| = |ξ 2 | + |ξ 3 | -|ξ 1 |. It follows that |ξ 3 | ≤ |ξ 1 | -1 2 |ξ| 3/4 and ensures that (4.16) holds also in this region. Using (3.7)-(3.8) and the notations (4.6), we get

		15)
	leads to (recall that |ξ| ∼ |ξ 1 |)
	max(|σ|, |σ i |) λ -1 |ξ| 7/4	(4.16)

  1/2]. By (2.10), for |t 1 -t 2 | < 2 -kε /4 and any R > 0, it holds

	kε
	k=0

It can be easily checked that the dilation symmetry u → λu(λ 3 t, λx) is an isomorphism from F s,1/2 into F s,1/2 λ for any λ ≥ 1 and s ∈ R.

It is easy to check that u0 ∈ L 2 (T) and R(u0) ∈ H ∞ 0 (T) ensure that u0 ∈ H ∞ 0 (T).

Acknowledgements: L.M. was partially supported by the ANR project "Equa-Disp".

We will consider 6P 0 (v 2 0 )v x as a part of the linear group of the equation. The v n satisfy the same Duhamel formula as the solution of (1.3) where we substitute the linear group of the KdV equation U (t) by the linear group V (t) defined by V (t)ϕ(ξ) = e iq(ξ)t φ(ξ), ξ ∈ λ -1 Z , q(ξ) = ξ 3 ∓ 6P 0 (v 2 0 )ξ .

(5.13)

It is direct to check that all linear estimates remain true when changing the functional spaces in consequence. Also the bilinear estimates in Lemma 3.4 remain true since the resonance relation remain unchanged (see the proof of this lemma in the appendix). Therefore, the results of Section 4 and the conclusions of Proposition 5.2 remain true when substituting the function spaces associated with U (•) by those associated with V (•) and (1.3) by (5.12).

We thus obtain a weak solution v ∈ C w (R; L 2 (T)) ∩ ∪ s<0 F s,1/2 to (5.12)

with initial data v 0 , where F s,1/2 is defined as F s,1/2 but for the group V (•). Moreover, by the weak convergence result (5.11), v(t) L 2 ≤ v 0 L 2 for all t ∈ R and thus the weak continuity of v ensures that v(t) → v(0) in L 2 (T) as t → 0. This completes the proof of assertion i). Now in the defocusing case, according to [START_REF] Kappeler | Global well-posedness of mKdV in L 2 (T, R)[END_REF], the sequence {v n } of solutions to mKdV emanating from {v 0,n } converges in C(R; L 2 (T)) to some function w such that w(t) L 2 = v 0 L 2 for all t ∈ R. By the uniqueness of the limit in D ′ (R × T), w ≡ v on R and thus w is a weak solution of the defocussing mKdV equation in the sense of Definition 5.2. Actually, using the conservation of the L 2 -norm for w, we also obtain that w satisfy the following equation in the sense of distributions,

when (w 2 -P 0 (w 2 ))w x is interpreted as the inverse Fourier transform of Γ(w). This concludes the proof of assertion ii).

6 Proof of Theorem 1.1

The mKdV equation

We will prove that the solution-map u 0 → u is not continuous at any u 0 ∈ H ∞ (T) from L 2 (T) equipped with its weak topology into D ′ (]0, T [×T). This obviously leads to the desired result since L 2 (T) is compactly embedded in H s (T) for any s < 0. Since the sign in front of the nonlinear term will not play any role in the proof, we choose to take the plus sign to simplify the notations.

Appendix

7.1 A simplified proof of (3.6)

We give below a very simple proof of (3.6). Recall that this estimate was first established in [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, I. Schrödinger equations II. The KdV equation[END_REF]. To simplify the notations we take λ = 1 (this corresponds to 2π-periodic in space functions) but it easy to check that the proof is exactly the same for any λ ≥ 1 (see for instance Section 7.2 for the slight modifications in the case λ ≥ 1). By the triangle inequality, we write

where the β l , l ∈ N, are defined in (2.2). The proof of (3.6) will then follow from the following lemma.

Lemma 7.1. Let u 1 and u 2 be L 2 (Z × R)-real valued functions then for any

Indeed, with this lemma in hand, rewritting l 1 as l 1 = l 2 + l with l ∈ N, we get the following chain of inequalities

It thus remains to prove Lemma 7.1. Following the arguments given in [START_REF] Bourgain | On the Cauchy problem for the Kadomtsev-Petviashvili equation[END_REF] (see also [[20], page 460]for more details) we can assume that u and w are easy calculations then lead to E u (φ) = T (φ ′ -wφ) 2 -P 0 (w 2 ) T φ 2 , ∀φ ∈ H 1 (T) .

It follows that E(ρ) = -P 0 (w 2 ) T ρ 2 and E(φ 1 ) ≥ -P 0 (w 2 ) T φ 2 1 . This ensures that -P 0 (w 2 ) = λ 1 . Therefore, ρ/ ρ 2 = φ 1 and thus w = φ ′ 1 /φ 1 = v. This proves the injectivity of R in H ∞ 0 (T).