Global Well-Posedness for a periodic nonlinear Schr\"odinger equation in 1D and 2D

The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold. The main ingredient in the proof is a"refinement"of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $\Bbb T^{d}_{\lambda} = \Bbb R^{d}/{\lambda \Bbb Z^{d}}$, $d=1,2$.


Introduction
In this paper we study the L 2 critical Cauchy problem u(x, 0) = u 0 (x) ∈ H s (T d ), (1.2) where d = 1, 2 and T d = R d /Z d is the d-dimensional torus.
We say that a Cauchy problem is locally well-posed in H s if for any choice of initial data u 0 ∈ H s , there exists a positive time T = T ( u 0 H s ) depending only on the norm of the initial data, such that a solution to the initial value problem exists on the time interval [0, T ], is unique and the solution map from H s x to C 0 t H s x depends continuously on the initial data on the time interval [0, T ]. If T = ∞ we say that a Cauchy problem is globally well-posed.
In the case when x ∈ R d local well-posedness for (1.1)-(1.2) in H s (R d ) has been studied extensively (see, for example, [8,16,20]). In particular if one solves the equivalent integral equation by Picard's fixed point method and controls the nonlinearity in the iteration process by using Strichartz's type inequalities, then the problem can be shown to be locally well-posed for all s > 0.
Bourgain [1] adjusted this approach to the periodic case, where there are certain difficulties due mainly to a "lack of dispersion". In [1] number theoretic methods were used to show that (1.1)-(1.2) is locally well-posed in H s (T d ), d = 1, 2 for any s > 0.
Assuming local existence there are many issues to be addressed about the behavior of the solution as t → ∞, • global well-posedness/blow-up behavior • asymptotic stability • behavior of higher order Sobolev norms of smooth solutions.
Date: February 16, 2006. In this note we address the question of global well-posedness for (1.1)-(1.2). We recall that solutions of (1.1) satisfy mass conservation u(t) L 2 = u 0 L 2 and smooth solutions also satisfy energy conservation For initial data in H s (T d ) , s ≥ 1, the energy conservation together with local well-posedness imply global well-posedness for s ≥ 1. However it is a more subtle problem to extend the global theory to infinite energy initial data. Bourgain [3] established global well posedness for (1.1) in H s (T) for any s > 1/2−, by combining a "normal form" reduction method (see also [5]), the "I-method", and a refined trilinear Strichartz type inequality. The normal form reduction is achieved by symplectic transformations that, in some sense, reduce the nonlinear part of the equation to its "essential part". The I-method, introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao in [10,13,14], is based on the almost conservation of certain modified Hamiltonians. These two methods together yield global well-posedness in H s (T) with s ≥ 1/2. The refined trilinear Strichartz inequality established in [3] is a qualitative estimate needed to achieve global well-posedness in H s (T), s * < s < 1/2 for some s * .
In this paper we continue to fill in the gap between what is known locally and what is known globally in H s (T d ) for s > 0, when d = 1, 2. Our approach is based on an implementation of the I-method itself adjusted to the periodic setting via elementary number theoretic techniques.
In order to present our method, we briefly review global well-posedness results on R d . Using an approximation of the modified energy in the I-method, Tzirakis [25] showed that the Cauchy problem (1.1)-(1.2) is globally well posed in H s (R) for any s > 4/9. For u 0 ∈ H s (R 2 ) the best known global well posedness result for (1.1)-(1.2) is in [14] where the authors proved global well posedness for any s > 4/7. In both results, apart from the application of the I-method, a key element in the proof is the existence of a bilinear refined Strichartz's estimate due to Bourgain, [4] (see also [9] [10]). In general dimensions d ≥ 2 this estimate reads as follows. Let f and g be any two Schwartz functions whose Fourier transforms are supported in |k| ∼ N 1 and |k| ∼ N 2 respectively. Then we have where U t denotes the solution operator associated to the linear Schrödinger equation. In one dimension the estimate fails if the two frequencies are comparable but continues to hold if the frequencies are separated (N 1 >> N 2 ). Such an estimate is very useful when f is in high frequency and g is in low frequency since we can move derivatives freely to the low frequency factor. The main difficulty in obtaining global well-posedness results in the periodic context is exactly the absence of a quantitative refinement of the bilinear Strichartz's estimate. The reader can consult the paper of Kenig, Ponce and Vega, [21], where the difference between the real and the periodic case is clearly exposed when one tries to prove bilinear estimates in different functional spaces.
In order to overcome the non-availability of a quantitative refined Strichartz's estimate, we develop a different strategy, closer in spirit to the approach in [13]. By exploiting the scaling symmetry of the equation, we analyze (1.1) on T d λ ×R, d = 1, 2, where T d λ = R d /λZ d . The main novelty in this analysis consists in establishing a bilinear Strichartz's inequality for λ-periodic functions which are well separated in frequency space. The constant in the inequality is quantified in terms of λ. As λ → ∞, our estimate reduces to the refined bilinear Strichartz's inequality 1 on R d , d = 1, 2, see, for example, [4,10,23]. Such an estimate allows us to use the I-method machinery in an efficient way.
More precisely, when d = 1, beside rescaling, we follow the argument in [25] (see also [12]), where one applies the I-operator to the equation on T λ , and defines a modified second energy functional as the energy corresponding to the new "I-system". We prove that such modified second energy is "almost conserved", that is the time derivative of this new energy decays with respect to a very large parameter N . Roughly speaking, N denotes the stage at which the I-operator stops behaving like the identity and starts smoothing out the solution. We prove a local existence result for the "I-system", under a smallness assumption for the initial data, which is guaranteed by choosing λ in terms of N . The decay of the modified energy enables us to iterate the local existence preserving the same bound for u λ H s during the iteration process. By undoing the scaling we obtain polynomial in time bounds for u in T × R + , and this immediately implies global well posedness.
The precise statement of our 1D result reads as follows.
. For the two dimensional case Bourgain already announced in [3] that the I-method based only on the first energy would give global well-posedness for s > 2/3. While we were explicitly writing up the calculations to recover this claim, we noticed that in one particular case 2 of the estimate of the first energy, a better Strichartz inequality was needed to successfully conclude the argument. We then proceeded by determining a qualitative "ǫ-refined" Strichartz type estimate, see Proposition 4.6, which allowed us to implement the I-method described above to obtain indeed the following result: . We remark that, as we mentioned above, in 1D we introduce an approximation to the modified energy, by adding correction terms, in the spirit of [25] and [12]. In the 2D problem we did not use correction terms, since the Fourier multipliers corresponding to the approximated modified energy would be singular. This singularity is not only caused by the presence of zero frequencies, but also by orthogonality issues. This is a whole new ground that we are exploring in a different paper.
Organization of the paper. In section 2 we review the notation and some known Strichartz estimates for the periodic Schrodinger equations in 1D and 2D. In section 3 we present the proof of Theorem 1.1, while in section 4 we give a proof of Theorem 1.2.
Sciences Research Institute (MSRI) at Berkeley. The authors would like to thank these institutions for their hospitality.

Notation
In what follows we use A B to denote an estimate of the form A ≤ CB for some constant C. If A B and B A we say that A ∼ B. We write A ≪ B to denote an estimate of the form A ≤ cB for some small constant c > 0. In addition a := 1 + |a| and a± := a ± ǫ with 0 < ǫ << 1.
We recall that the equation (1.1) is L 2 invariant under the following scaling ( Since in our argument we exploit the scaling symmetry let us recall some properties of λ−periodic functions. Define (dk) λ to be the normalized counting measure on ( 1 λ Z) d : We define the Fourier transform of For an appropriate class of functions the following Fourier inversion formula holds: Moreover we know that the following identities are true: We define the Sobolev space H s = H s ([0, λ] d ) as the space equipped with the norm We write U λ (t) for the solution operator to the linear Schrödinger equation We denote by X s,b = X s,b (T d λ ×R) the completion of S(T d λ ×R) with respect to the following norm, see, for example, [15] Furthermore for a given time interval J, we define Often we will drop the subscript J.
In what follows we will need the following known estimates. Since where in the last inequality we applied the definition of the X s,b spaces, the basic estimate u L ∞ ≤ û L 1 , and the Cauchy-Schwartz inequality.
1D estimates. If u is on T 1 then the estimate (2.1) combined with the Sobolev embedding theorem implies that Also the following linear Strichartz's estimates was obtained in [1] in the case of the torus: We note that (2.2) remains true for the λ-periodic problem. This is the case also for (2.3). The proof is essentially in [18]. In fact, it is enough to show, by the standard X s,b method, that 1 λ sup This is done in [18], the only difference being that there are O(λ) numbers k ∈ 1 λ Z, such that |k + x 0 | < 1 and |k − x 0 | < 1, where x 2 0 = |2τ + ξ 2 |. But then summing the above series, using Cauchy-Schwartz inequality and the fact that b > 3/8, one gets that the left hand side of the inequality is 1 λ (c 1 λ + c 2 ) C for any λ > 1. So from now on we will use (2.3) for the λ-periodic solutions without any further comment. On the other hand, using scaling we can see that for the λ-periodic solutions, (2.4) takes the form If we interpolate equations (2.2) and (2.5) we get 2D estimates. On T 2 Bourgain proved, [1], Again using scaling we can prove that for the λ-periodic solutions, (2.7) takes the form The estimate (2.1) together with Sobolev embedding gives: Hence, by interpolation, we get with α 2 (p) = (1 − 4 p )+ and 4 ≤ p ≤ ∞. Remark 2.1. (Decomposition remark) Our approach to prove Theorem 1.1 and Theorem 1.2 is based on obtaining certain multilinear estimates in appropriate functional spaces which are L 2 -based. Hence, whenever we perform a Littlewood-Paley decomposition of a function we shall assume that the Fourier transforms of the Littlewood-Paley pieces are positive. Moreover, we will ignore the presence of conjugates.
3. The I-method and the proof of Theorem 1.1 In this section we present the proof of Theorem 1.1. We start by recalling the definition of the operator I introduced by Colliander et al, see, for example, [10,14]. For s < 1 and a parameter N >> 1 let m(k) be the restriction to Z of the following smooth monotone multiplier: We define the multiplier operator I : H s → H 1 by The operator I is smoothing of order 1 − s and we have that: We refer to E 1 (u) as the first modified energy. As observed by Colliander et al a hierarchy of modified energies can be formally considered for different nonlinear dispersive equations. The goal of the I-method is to prove that the modified energies are "almost conserved" i.e. they decay in time with respect to N . Since in 1D we base our approach on the analysis of a second modified energy, it is appropriate to collect some facts concerning the calculus of multilinear forms used to define the hierarchy, see, for example [25]. If n ≥ 2 is an even integer we define a spatial multiplier of order n to be the function M n (k 1 , k 2 , . . . , k n ) on Γ n = {(k 1 , k 2 , . . . , k n ) ∈ 1 λ Z n : k 1 + k 2 + . . . + k n = 0} which we endow with the standard measure δ(k 1 + k 2 + . . . + k n ). If M n is a multiplier of order n, 1 ≤ j ≤ n is an index and l ≥ 1 is an even integer we define the elongation X l j (M n ) of M n to be the multiplier of order n + l given by . . , k n+l ). In addition if M n is a multiplier of order n and f 1 , f 2 , ..., f n are functions on T λ we define where we adopt the notation Λ n (M n ; f ) = Λ n (M n ; f,f , ..., f,f ). Observe that Λ n (M n ; f ) is invariant under permutations of the even k j indices, or of the odd k j indices.
If f is a solution of (1.1) the following differentiation law holds for the multilinear forms Λ n (M n ; f ): Note that in this notation the first modified energy (3.2) reads as: where m j = m(k j ). We define the second modified energy where M 6 (k 1 , k 2 , ..., k 6 ) is the following multiplier: Notice that the zero set of the denominator corresponds to the resonant set for a six-waves interaction. We remark that M 6 contains more "cancellations" than the multiplier m 1 ...m 6 that appears in E 1 . The differentiation rule (3.3) together with the fundamental theorem of calculus implies the following Lemma, which will be used to prove that E 2 is almost conserved.
Lemma 3.1. Let u be an H 1 solution to (1.1). Then for any T ∈ R and δ > 0 we have As it was observed in [25] where the equation (1.1) was considered on R one has For the proof see [25].
Now we proceed to present the steps leading to the proof of Theorem 1.1.
Step 1: Local well-posedness for the I-system. The first step towards the proof of Theorem 1.1 is to apply the I-operator to (1.1) and prove a local well-posedness result for the I-initial value problem In order to obtain the local well-posedness we need the following technical lemma: and equals 1 on (2) The following inequality is true and its proof can be found in [15], [21], But then (2) follows if we apply the above inequality for fixed ξ, multiplying by < ξ > 2s and taking the L 2 norm in ξ.

Remark. It is shown in
where the constants in the inequality above are independent of N . From now on we use this fact and refer to it as the "invariant lemma". For details see Lemma 12.1 in [11]. Now we present the local well-posedness result for (3.6). Because of the loss in the periodic Strichartz estimates (2.5) represented by the λ 0+ factor, the local well-posedness is not as straightforward as in the real case. (There one can essentially use the Leibnitz rule for the fractional derivatives and the Cauchy-Schwartz inequality along with interpolated Strichartz's estimates to bound the nonlinearity). In 1D we can still follow the same approach thanks to (2.3), which is valid for the λ-periodic problem. In 2D for the λ-periodic problem we need a more complicated argument that is due to Bourgain, [2], and uses a local variant of the periodic Strichartz inequality that is proved there. For details see [2]. As a final comment, note that our local well-posedness results are not optimal but they suffice for the purposes of the global theory that we establish. Thus without aiming at sharpness, we prove local well-posedness for s > 5/16 in 1D (see Proposition 3.4 and Corollary 3.5) and s > 7/20 in 2D (see Proposition 4.1 and Corollary 4.2).
We shall prove that . By the invariant lemma we know that to prove such an estimate it suffices to prove where J s is the Bessel potential of order s. Using (2.6) with λ = 1 we obtain for s > 5 16 . Note that we also use (2.3) in order to bound J s u L 4 t L 4 x . Thus gives . Now by standard nonlinear techniques, see, for example, [12] we have that for δ ∼ 1 where a is a fixed positive integer (a ∼ 20 will do it).
Proof. The proof is line by line the same as in the previous proposition. The important step is that in the iteration process where x ∈ [0, λ], δ has a fixed power and this fact gives us a local well-posedness result on the time interval [0, 1 λ aǫ ], where 1 λ aǫ ∼ 1 λ 0+ for all practical purposes.
Step 2: E 2 (u) is a small perturbation of E 1 (u). Now we shall prove that the second energy E 2 (u) is a small perturbation of the first energy E 1 (u). This is the content of the following proposition.
Proposition 3.6. Assume that u solves (3.6) with s ≥ 2/5. Then Proof. By the definition of first and second modified energy we have Therefore it suffices to prove the following pointwise in time estimate H 1 x .
Combining the Decomposition Remark 2.1 with the fact that M 6 is bounded (by Proposition 3.2) and that m is bounded (by its definition) it is enough to show that H 1 x .
Towards this aim, we break all the functions into a sum of dyadic constituents u j , each with frequency support k ∼ 2 j , j = 0, .... We will be able to sum over all frequency pieces u j of u since our estimate will decay geometrically in these frequencies. Hence we need to show the following: the decreasing rearrangement of N 1 , ..., N 6 , and by u ⋆ j the function u j supported in Fourier space around N ⋆ j , j = 1, ..., 6. Since we are integrating over Γ 6 , we have that N ⋆ 1 ∼ N ⋆ 2 . Moreover, we may assume that Step 3: Decay of E 2 (u). In order to estimate the decay of E 2 (u) in time we need to bound the right hand side of (3.5). Towards this aim we prove a bilinear estimate using an elementary number theory argument. Let η ∈ C ∞ 0 be a bump function that is supported on [−2, 2] and equals 1 on [−1, 1].
Proposition 3.7. Let φ 1 and φ 2 be λ−periodic functions whose Fourier transforms are supported on {k : |k| ∼ N 1 } and {k : |k| ∼ N 2 } respectively with N 1 >> N 2 . Then Remark. We observe that as λ → ∞ we recover the refined bilinear estimate in R with constant C(N 1 ) = ( 1 N 1 ) 1/2 , see, for example, [4,10,23]. Proof. Let ψ be a positive even Schwartz function such that ψ =η. Then we have where to obtain (3.12) we used Cauchy-Schwartz and the following definition of ψ ∈ S τ =τ 1 +τ 2 An application of Hölder gives us the following upper bound on (3.12) Now by integration in τ followed by Fubini in k 1 , k 2 and two applications of Plancharel we have which combined with (3.12), (3.13) gives We find an upper bound on M as follows: and #S denotes the number of elements of S. When N 1 ≤ 1 then #S O(λ), which implies C(λ, N 1 ) 1. When N 1 > 1 rename k 1 = z. Then Let z 0 be an element of S i.e.
. In order to obtain an upper bound on #S, we shall count the number ofz's ∈ 1 λ Z, such that z 0 +z ∈ S. Thus However by (3.17) we can rewrite the left hand side of (3.19) as follows Hence it suffices to countz's ∈ 1 λ Z satisfying (3.18) and such that On the other hand, by (3.16) we have that Sincez ∈ 1 λ Z this implies that the number ofz's of size 1/N 1 is λ/N 1 . Hence which combined with (3.15) gives , and therefore Now we are ready to prove desired decay of E 2 (u) which follows from the following proposition: Proof. We perform a dyadic decomposition as in Proposition 3.6, and we borrow the same notation. From now on we do not keep track of all the different λ kǫ , k ∈ R, and we just write λ 0+ We observe that M 10 is bounded as an elongation of the bounded multiplier M 6 . Since the multiplier M 10 vanishes on Γ 10 when all frequencies are smaller than N , we can assume that there are N * 1 , N * 2 N . Now we divide the proof in two cases.
Hence using the Cauchy-Schwartz inequality we get: where to obtain (3.26) we use Proposition 3.7, to obtain (3.27) we use Hölder's inequality and the facts that λ ≥ N and N * 1 N , to obtain (3.28) we use (2.6), and in the last line we use that s > 11/28. Now we are ready to prove our main theorem on T 1 .
We choose λ ∼ N 1−s s , which for s < 1/2 implies λ N . Then we have that E 1 (u λ 0 ) 1. Therefore which allows us to apply Propositions 3.4, 3.6. The main idea of the proof is that if in each step of the iteration we have the same bound for Iu λ 0 H 1 then we can iterate again with the same timestep. Because we iterate the u λ solutions, the interval of local existence is of order 1 λ aǫ . But this creates no additional problem since λ is given in terms of N . Moreover N is a fixed number and thus the interval of local existence does not shrink. This is the reason why we don't write down the λ 0+ dependence in the following string of inequalities. Now, by Lemma 3.1, Proposition 3.6, Proposition 3.8 and (3.29) we have that for N large enough. Combining the inequality above with Proposition 3.6 we have,

Thus
goes to infinity as N → ∞ since s > 4/9.

The I-method and the proof of Theorem 1.2.
In this section we present the proof of Theorem 1.2. We will focus on the analysis of the λ-periodic problem: for 0 < s < 1. As for the one dimensional case, we will use the I-method. The definitions of the multiplier m and of the operator I are the same as in the 1D context. Precisely, given a parameter N >> 1 to be chosen later, we define m(k) to be the following multiplier: Then I : H s → H 1 will be defined as the following multiplier operator: This operator is smoothing of order 1 − s, that is: The first modified energy of a function u ∈ H s is defined by where we recall that Our main objective will be to prove that the modified energy of a solution to the λperiodic problem (4.1)-(4.2) is almost conserved in time.
We now proceed to present the steps leading to the proof of Theorem 1.2.
Step 1: Local well-posedness for the I-system. The first step towards the proof of Theorem 1.2 is to apply the I-operator to (1.1) and prove a local well-posedness result for the I-initial value problem This is the content of the next proposition.
By the "invariant lemma" in [11] we know that the estimate Iu 3 is implied by u 3 X s,1/2+ . Since the rest of the proof is identical to Proposition 3.4 we briefly recall how (4.5) can be obtained. Note that this estimate was first proved by Bourgain [2] and his proof is based upon a local variant of the well known periodic Strichartz estimate Here we omit the proof and refer the reader to [2] for details. Proof. The proof is identical to the argument of Bourgain in [2], the only difference being the fact that (4.6) holds true for the λ-periodic solutions, but now with a factor of order λ 0+ on the right hand side. Again the fixed power of δ in the local theory, enables us to prove the local well-posedness result for the λ-periodic problem in the interval [0, 1 λ aǫ ].
Step 2: Decay of the first energy. In order to prove that the first energy is almost conserved, we will use the bilinear estimate for λ-periodic functions stated in Proposition 4.6. Its proof is based on some number theoretic facts that we recall in the following three lemmas; see also related estimates in the work of Bourgain [6].
The following lemma is known as Pick's Lemma [24]:  Proof. We prove the lemma by contradiction. Assume that there are 3 lattice points P 1 , P 2 and P 3 on an arc γ = AB of C, and denote by T (P 1 , P 2 , P 3 ) the triangle with vertices P 1 , P 2 and P 3 . Then, by Lemma 4.3 we have We shall prove that under the assumption that |γ| < 3 4 R 1/3 , then hence γ must contain at most two lattice points. We observe that (see Figure 1) Area of the sector ABO = R 2 θ, Area of the triangle ABO = R 2 sin θ cos θ.
(a) Then for any ǫ > 0. Hence Remark. Note that as λ → ∞ we recover the improved bilinear Strichartz with constant (a) Let ψ be a positive even Schwartz function such that ψ =η. Then we have where to obtain (4.12) we used Cauchy-Schwartz and the following definition of ψ ∈ S τ =τ 1 +τ 2 An application of Hölder gives us the following upper bound on (4.12) Now by integration in τ followed by Fubini in k 1 , k 2 and two applications of Plancharel we have which combined with (4.12) and (4.13) gives (4.14) We find an upper bound on M as follows: and #A denotes the number of lattice points of a set A.
For notational purposes, let us rename k 1 = z, that is i.e.
Hence, it suffices to count the l ′ s ∈ Z 2 satisfying (4.20) and (4.21) where z 0 is such that (4.22) holds. Let w = (a, b) denote the vector z 0 − k 2 . Thus we need to count the number of points in the set A λ (4.23) and for any integer n, let C λ n be the circle C λ n : (x + λa) 2 + (y + λb) 2 = n + λ 2 (a 2 + b 2 ). Finally, let D λ denote the disk

Figure 2. Circular sector
We need to count the number of lattice points inside D λ that are on arcs of circles C λ n , with −cλ 2 ≤ n ≤ cλ 2 .
Precisely, the total number of lattice point in A λ can be bounded from above by (4.25) 2cλ 2 × #(C λ n ∩ D λ ). Denote by γ n the arc of circle C λ n which is contained in D λ . Notice that (see Figure 2) where R M = cλ 2 + k 1 λ 2 N 2 1 for some constant k 1 > 0, and θ M is the angle between the line segment CB and CD, which lie along the tangent lines from C = (−λa, −λb) to the circle x 2 + y 2 = (k 2 λN 2 ) 2 . Hence, for some constant k > 0. Since N 1 >> N 2 , we can assume that sin θ M > 1 2 θ M . Hence, In order to count efficiently the number of lattice points on each γ n , we distinguish two cases based on the application of Lemma 4.4.
M . In this case (4.26)-(4.27) guarantee that the hypothesis of Lemma 4.4 is satisfied by each arc of circle γ n . Hence, on each γ n there are at most two lattice points.
In the rest of the proof for arbitrary two points P 1 , P 2 on a circle with center C we denote by S(C, P 1 , P 2 ) the solid sector contained between the lines CP 1 ,CP 2 and the circle arc between P 1 and P 2 . Figure 3. Circular sectors when λ = 1

PSfrag replacements
We observe that (see Figure 3 ) where CB and CD lie along the tangent lines from C = (−a, −b) to the circle x 2 + y 2 = (k 2 N 2 ) 2 . Hence, setting S 1 = S(C, B, D), S 2 = S(C, A, E), and H = S 1 \ S 2 we have where to obtain (4.31) we used Lemma 4.5 twice. In order to compute the area of H notice that since a 2 + b 2 ∼ N 2 1 , we have (see Figure 3) Hence, by the assumption N 1 >> N 2 , we get θ ∼ N 2 N 1 . Therefore This combined with (4.32) implies that that together with (4.15) gives . Now we are ready to prove the desired decay of the first modified energy, which is the content of the following proposition: Proposition 4.7. Let s > 1/2, λ N, t > 0, and u 0 ∈ H s (T 2 λ ) be given. If u is a solution to (4.1) − (4.2) then Proof. The definition of E together with equation (4.1) give that: Therefore, integrating in time, and using the Parseval's formula we get: Using the equation (4.1), we then get: where Tr 1 = (4.34) We wish to prove that: (4.36) |Tr 1 | + |Tr 2 | 1 N 1− , for a constant C = C( Iu X 1,1/2+ ). According to the Decomposition Remark in our estimates we may ignore the presence of the complex conjugates. In order to obtain the estimate (4.36), we break u into a sum of dyadic pieces u j , each with frequency support k ∼ 2 j , j = 0, .... We will then be able to sum over all the frequency pieces u j of u, since our estimates will decay geometrically in these frequencies.
We start by analyzing Tr 1 . First notice that, ∆(Iu) X −1,1/2+ ≤ Iu X 1,1/2+ , therefore, the estimate on Tr 1 will follow, if we prove that: for any λ-periodic function φ i , i = 1, ..., 4 with positive spatial Fourier transforms supported on By the symmetry of the multiplier, we can assume that Moreover, since we are integrating over Γ 4 , we have that N 1 N 2 . Therefore, we only need to get the factor N 0− 2 in the estimate (4.37), in order to be able to sum over all dyadic pieces. Now we consider three different cases, by comparing N to the size of the N i 's.
Case I. N >> N 2 . In this case, the symbol in Tr 1 is identically zero, and the desired bound follows trivially.
Case II. N 2 N >> N 3 ≥ N 4 . Since we are on Γ 4 , we also have that N 1 ∼ N 2 . By the mean value theorem, we have Using the pointwise bound above, the Cauchy-Schwartz inequality, and Plancharel's theorem we get: x By two applications of Hölder's inequality, followed by four applications of the Strichartz estimate (2.8) we obtain: Since N 1 ∼ N 2 N , and λ N , it follows immediately that in this case Case III. N 2 ≥ N 3 N. We will use the following crude bound on the multiplier: which follows from the fact that we are integrating on Γ 4 . We also need the following estimate: for α ≥ 1/2−, |k| N . We distinguish two subcases: Using Plancharel's, followed by Holder's inequality, and then by four applications of the Strichartz estimate (2.8), we get: where to obtain (4.43) we use that N 1 ≤ N 2 together with (4.42) with α = 1− (N 3 N ). Now if N 4 N , we conclude with an application of the monotonicity property (4.42). If N 4 << N , then we use that m(k 4 ) = 1 and N 4 ≥ 1. Hence, since λ N , we get IIIb)N 2 >> N 3 N . Since we are integrating on Γ 4 , we get N 1 ∼ N 2 . Let us emphasize that in the case IIIb) we could not use the same strategy as in the case IIIa). More precisely, the approach of IIIa) consisting of Plancharel's, followed by Holder's inequality, and then by four applications of the Strichartz estimate (2.8) would produce a bound of the type m(k 1 ) m(k 2 )m(k 3 )m(k 4 ) λ 0+ (N 1 ) 1+ (N 2 N 3 N 4 ) 1− × φ 1 X −1,1/2+ φ 2 X 1,1/2+ φ 3 X 1,1/2+ φ 4 X 1,1/2+ 1 N 1− λ 0+ N 0+ 1 φ 1 X −1,1/2+ φ 2 X 1,1/2+ φ 3 X 1,1/2+ φ 4 X 1,1/2+ which we cannot sum in N 1 (here we cannot cancel N 0+ 1 by N 0+ 3 since N 3 << N 1 ). Thus to handle the case IIIb) we do need an improved Strichartz estimate given in Proposition 4.6.
Step 3: Proof of Theorem 1.2. We are now ready to exhibit the proof of Theorem 1.2. As we explained in Theorem 1.1 we do not keep track of the λ 0+ factors.
Proof. Let u 0 be an initial data, and let N >> 1 be given. Recall that, for any λ ∈ R, we set u λ 0 (x, t) = 1 λ u 0 ( x λ , t λ 2 ). Now, let us choose a rescaling parameter λ so that E 1 (u λ 0 ) 1, that is λ ∼ N 1−s s . Then, by Corollary 4.2 and Proposition 4.7, there exists a δ, and a solution u λ to (4.1), with initial condition u λ 0 , such that Therefore, we can continue our solution u λ until the size of E 1 (u λ )(t) reaches 1, that is at least C · N 1− times. Hence E 1 (u λ )(C · N 1− δ) ∼ 1. Given T >> 1, we choose our parameter N >> 1, so that We observe that the exponent of N is positive as long as s > 2/3, hence N is well-defined for all times T. Now, undoing the scaling, we get that: