Resonant decomposition and the $I$-method for the two-dimensional Zakharov system

The initial value problem of the Zakharov system on two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some 'resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).

(1.1) is described as a Hamiltonian PDE with the Hamiltonian given by Local well-posedness in the energy space H 1 × L 2 × |∇|L 2 was obtained in [4] for Z = R 2 and in [11] for Z = T 2 γ . In particular, using conservation of mass and the Hamiltonian and the sharp Gagliardo-Nirenberg inequality (the last term in the right hand side is required only in the periodic case; see [15,6]), we have the a priori control of the energy norm of solutions in the energy class if u 0 L 2 < Q L 2 (R 2 ) , where Q is the ground state of the cubic NLS on R 2 . More 1 precisely, if η := 1 − u 0 2 L 2 / Q 2 L 2 (R 2 ) > 0, then we have L 2 . Therefore, we have the following a priori estimate as long as the solution (u(t), n(t)) exists in the energy class. Consequently, (1.1) is globally well-posed for initial data in the energy space with u 0 L 2 < Q L 2 (R 2 ) . In fact, the solution also exists globally for initial data in H 1 × L 2 × H −1 with u 0 L 2 ≤ Q L 2 (R 2 ) (see [10] for Z = R 2 and [12] for Z = T 2 γ ). The present article addresses the global well-posedness of (1.1) for some initial data without finite energy. The proof will rely on the I-method, which was originally introduced by Colliander, Keel, Staffilani, Takaoka, and Tao to deal with the nonlinear Schrödinger equations and has been applied to a wide variety of nonlinear dispersive equations. For the details of the I-method, we refer to [7,14,8] and references therein.
The I-method for the Zakharov system was initiated by Fang, Pecher, and Zhong [9] for the R 2 case, who established the global well-posedness in H s × L 2 × |∇|L 2 with 1 > s > 3 4 . Their estimate of the modified energy was mainly based on the Strichartz estimate for the Schrödinger equation and its bilinear refinement, as well as some crude estimates with the Hölder inequality and the Sobolev embedding. It is worth noting that they did not use the scaling argument in the I-method; thus it was quite important for global well-posedness under the minimal regularity assumptions to obtain the best estimate for the lower bound of local existence time in terms of the size of initial data.
Our principal aim is to apply the I-method in the periodic case Z = T 2 γ , where the local well-posedness of (1.1) below the energy space is known for 1 2 ≤ s ≤ 1, r = 0 ( [11]). However, it turns out not to be trivial at all to adjust their argument to the periodic setting. In fact, since the dispersive effect is limited on torus, the same estimate as for R 2 cannot be expected in general. For example, the L 4 Strichartz estimate for the Schrödinger equation on T 2 γ cannot hold without some loss of derivative (see [2,5]). To obtain the best decay in the almost conservation law, we will use the sharp trilinear estimates established in [11] which control various interactions between two Schrödinger solutions and a wave solution.
We remark that, in [9], the trilinear terms have the biggest contribution in the increment of the modified energy and force them to assume s > 3 4 . To improve further, we shall introduce a new modified energy based on the concept of 'resonant decomposition' (see [8], for instance). The trilinear terms then become harmless; in fact, we find that these terms are acceptable for wider regularity range s > 1 2 . However, some portion of the quartilinear terms in the modified energy increment still has a large contribution, which will require the regularity s > 2 3 even for the case of R 2 if we estimate it in the same manner as [9]. To control these quartilinear terms, we make more refined analysis with the Strichartz estimate for the wave equation. At the end, we will push down the threshold to s > 9 14 .
Theorem 1.1. Let 1 > s > 9 14 and r = 0. Then, for any spatial period γ, (1.1) on T 2 γ is globally well-posed for initial data with u 0 L 2 (T 2 γ ) < Q L 2 (R 2 ) . Moreover, the global solutions satisfy for any T > 0, where the constant C > 0 depends on s, the implicit constant in the exponent, and the size of initial data.
The period γ has nothing to do with our results, as in the local theory [11].
(ii) In contrast to the nonperiodic problem, we know ( [11]) that the data-tosolution map for (1.1) on T 2 γ cannot be smooth (nor C 2 ) for r < 0. That is why we restrict our attention to the case r = 0 in the above theorem. Compare this to Theorem 1.3 below.
Of course, these approaches are also effective for the R 2 case. Recently, Pecher [13] extended the previous result [9] on R 2 to a wider regularity range, in H s ×H r ×|∇|H r with r ≤ 0, s < r + 1, s(r + 3 2 ) > 1. The new ingredient was the global well-posedness with regularity for the wave data below L 2 . Note that even local well-posedness was not known in these regularities before. He first established the local well-posedness of (1.1) with the operator I, and then applied the argument in [9] to obtain an almost conservation law of the modified energy. Even for the case r = 0 he could improve the previous threshold s > 3 4 to s > 2 3 by refining the analysis of the worst trilinear terms in the increment of the modified energy. However, since he used the same modified energy as [9], the trilinear terms still require the regularity s > 2 3 . Therefore, it is strongly expected that his result, combined with our approaches, can be improved further. We carry out this and obtain the following result. Theorem 1.3. Let s < 1, r ≤ 0 be such that r ≥ s − 1 and s > 9+3r 14+8r . Then, (1.1) on R 2 is globally well-posed for initial data with u 0 L 2 (R 2 ) < Q L 2 (R 2 ) . Moreover, for any T > 0, where the constant C > 0 depends on s, r, the implicit constant in the exponent, and the size of initial data. Remark 1.4. (i) If we consider the particular case r = 0, then the above result shows the global well-posedness for 1 > s > 9 14 just as the periodic case. (ii) See Figure 1 for the range of regularity in the theorem. The previous result of Pecher [13] is indicated by , and the optimal corner is A = ( 1 4 ( √ 17−1), 1 4 ( √ 17− 5)) ≈ (0.781, −0.219). We extend it to the range , and the optimal corner is The plan of this article is as follows. In Section 2, we recall some definitions and estimates given in the previous results. In Section 3, we construct our modified energy. A proof of the almost conservation law for the periodic case and Theorem 1.1 will be given in Section 4. We indicate in Section 5 how to apply our ideas to the nonperiodic case, obtaining Theorem 1.3. In Appendix we give an elementary proof of the Strichartz estimate for the periodic wave equation, which is used in Section 4.

Function spaces and preliminary lemmas
We will use the same notations as used in [11]. Definition 2.1 (Littlewood-Paley decomposition). Let η ∈ C ∞ (R) be an even function with the properties Define a partition of unity on R, η N for dyadic N ≥ 1, by Define the frequency localization operator P N on functions f : Z → C by We also use the notation P N to denote the operator on functions in (t, x), Also, define the operators Q S L , Q W ± L on spacetime functions by for dyadic numbers L ≥ 1. We will write P S N, for brevity. Finally, we define several dyadic frequency regions: In what follows, capital letters N and L are always used to denote dyadic numbers ≥ 1. We will often use these capital letters with various subscripts and the notation The following will be used for the specific indices; Definition 2.2 (Function spaces X s,b,p ). For s, b ∈ R and 1 ≤ p < ∞, define the spaces X s,b,p S and X s,b,p W ± by the completion of Schwartz functions on R × Z, Z = R 2 or T 2 γ , with respect to the following norm For T > 0, define the restricted space X s,b,p * (T ) ( * = S or W ± ) by the restrictions of distributions in X s,b,p Define the Duhamel operators Lemmas 2.4-2.11 below are stated for spatially periodic functions (or the Fourier transform of them) but equally hold for functions on the whole space (in this case, however, some of them are rougher than known estimates). (i) Suppose that u 1 , u 2 ∈ L 2 (R × T 2 γ ) satisfy We also assume N 0 ≥ 2. Then we have Then we have Lemma 2.5 ( [11], Proposition 3.1). Let N j , L j ≥ 1 be dyadic numbers and f, g 1 , g 2 ∈ L 2 ζ (R × Z 2 γ ) be real-valued nonnegative functions with the support properties Assume L max N 2 max . Then, we have Lemma 2.6 ([11], Proposition 3.2). Let f, g 1 , g 2 ∈ L 2 ζ (R × Z 2 γ ) be functions as in Lemma 2.5, and assume N 1 ≫ N 2 or N 2 ≫ N 1 . Then, we have Lemma 2.7 ( [11], Corollary 3.4). Let f, g 1 , g 2 ∈ L 2 ζ (R × Z 2 γ ) be functions as in Lemma 2.5, and assume that N 0 1. Then, we have Assume that 1 ≪ N 1 ∼ N 2 and L max ≪ N 1 . Then, we have (2.4) . (2.5) Next, we give a Strichartz-type estimate for the periodic (reduced) wave equation. It seems that the Strichartz estimates in periodic setting do not follow immediately from that on the whole space, because the finite speed of propagation does not hold for the reduced wave linear propagator e ∓it|∇| . An elementary proof of it will be given in Appendix.
Finally, we collect some estimates valid only for the nonperiodic case. The next one is a refinement of Lemma 2.4 (ii) above.
Then we have The last estimate is one of the main consequences in [1]. . (2.6)

Modified energy and resonant decomposition
In this section we introduce our almost conservation quantity and prepare some basic lemmas in the I-method, treating Z = R 2 and Z = T 2 γ at once. With n + := n + i|∇| −1 ∂ t n and n +0 : where n − := n + , which conserves (formally) the L 2 norm of u(t) and although H(u, n + ) cannot be in general defined for (u(t), n + (t)) ∈ H s × H r with s < 1 or r < 0. We can recover (1.1) from (3.1) by putting n := ℜn + since n is real valued.
For s < 1, r ≤ 0, and N ≫ 1, we define the operator I S s,N for the Schrödinger equation and the operator I W + r,N for the reduced wave equation as Note that I S s,N ∈ B(H s , H 1 ), I W + r,N ∈ B(H r , L 2 ), and I W + 0,N is the identity operator.
Define the modified energy of (u, n + ) by The operators I S s,N and I W + r,N only act u orū and n ± , respectively, so in what follows we abbreviate as For an integer p ≥ 2, we write Σp to denote for the case Z = T 2 γ . Also, we use the notations ξ ij := ξ i + ξ j , m q,j := m q,N (ξ j ). Note that Hence, we need an almost conservation law for the modified energy, as well as the local well-posedness with the existence time written in terms of Iu 0 H 1 + In +0 L 2 . For better decay of the increment of the modified energy, we introduce another quantity where the multipliers σ ± will be defined soon. A direct calculation using the equation shows that d dt H(u, n + ) An initial guess for σ ± would be which kills all the trilinear terms. Under this definition, however, σ ± have singularities and we will fail to estimate the quartilinear terms. Here arises an essential difficulty in applying the I-method to the Zakharov system. In [9,13], they used so that the worst terms including two derivatives would be cancelled with σ ± in the trilinear terms. It is easy to check that σ S is bounded. However, the remaining trilinear terms are still much more massive than the quartilinear terms. In fact, it was exactly these terms that determined the regularity threshold for global wellposedness, both in [9] (s > 3 4 ) and in [13] (s > 2 3 ). We will use both (3.2) and (3.3) to obtain a slightly better estimate. It turns out that the biggest contribution in the remaining trilinear terms comes from the frequency region for high-low interactions (|ξ 1 | ∼ |ξ 2 |), which has no intersection with the region |ξ 1 | 2 − |ξ 2 | 2 ∼ |ξ 12 |, where σ Z ± become unbounded. Motivated by this fact, we shall employ the following definition.
We next show that the new quantity H(u, n + ), which is our almost conserved quantity, is always close to the (first generation) modified energy H(Iu, In + ).
Then, for any t ∈ R, we have Proof. From the definition and boundedness of multipliers, we have We may assume that all of u, ū, n ± are real-valued and non-negative. Symmetry allows us to assume |ξ 1 | ≥ |ξ 2 |. Also, it suffices to consider the case of n + . Then the above is bounded by Applying the Cauchy-Schwarz inequality to each summation we reach the claim.

Global solutions for the periodic case
In this section we consider the periodic case and prove Theorem 1.1. Since we always assume the wave data to be in L 2 , the operator I is operated only to the Schrödinger equation, so we use the notation m(k) to denote m 1−s,N (k) for simplicity.
Estimate of (4.2). Motivated by the argument in [9], we add to (4.2) and consider the estimate of It is then sufficient to estimate with an arbitrary choice of ±. However, since the choice of n ± plays no role in the following, we consider the case n + only, and write for simplicity. We thus need to estimate First, we state an estimate which will be frequently used later.
Proof. From Lemma 2.4, we have On the other hand, an application of the Hölder inequality shows that The required estimate is obtained from an interpolation between them.
Let us begin to estimate (4.6). First of all, we remark that the multiplier σ ± (k 13 , k 2 )− m 2 13 vanishes if N 2 , N 4 ≪ N. We consider some cases separately. Case 1. N 2 N 4 . In this case we can assume N 2 N and bound the multiplier by 1. Also, we see that either N 1 or N 2 has to be comparable to the biggest one among N j 's.
(i) Consider the case N 1 N. We use (4.7) twice to have (4.6) Since s > 1 2 , there remains N 0− 1 N 0− 2 if we choose ε > 0 sufficiently small. Summing over N j 's and then applying (2.2) and (4.5), we obtain a bound of (ii) Consider the case N 1 ≪ N, where we may assume N 2 ≫ N 1 and N 2 is comparable to the max. We further decompose the integral as Observe that if ζ 1 + · · · + ζ 4 = 0, then We begin with the case L 34 = L 1234 . Without loss of generality we assume L 3 is the biggest one. We apply the Hölder inequality and Lemma 2.4 (ii) to obtain that At the last inequality we have used L 0+ 24 ≤ L 0+ 3 . We perform the summation in N j 's and use (2.2) and (4.5), concluding We next treat L 12 = L 1234 ≫ L 34 , which is actually the worst case. (When L 1 is the max, however, we can have some better bound than obtained below.) If L 2 is the max, (4.8) is bounded by . Now, we use the L 4 Strichartz estimate for wave (Lemma 2.11) to bound this by If L 1 is the max, we first apply the Hölder inequality as t,x and then make a similar argument, concluding the same bound.
Without loss of generality we assume N 1 ≥ N 2 , which implies N 1 N; otherwise the multiplier vanishes. We may also assume that at least two of N j 's are N Case 1. Two of N j 's ≪ N. It will be sufficient to consider the particular case N 1 , N 2 N ≫ N 3 , N 4 , where N 1 ∼ N 2 is the max. From a Hölder argument, (4.9) Case 2. More than two of N j 's N. Prepare the following lemma.
for some dyadic N 1 , N 2 ≥ 1. Then, for any 0 < ε ≪ 1, we have Proof. Making dyadic decompositions, we have We use Lemma 2.4 (i) for N 0 ≥ 2 and Lemma 2.7 for N 0 = 1, On the other hand, we apply the Hölder inequality to obtain (4.11) The required estimate is obtained from an interpolation between them.
We go back to the estimate of (4.9). Define the biggest, the second biggest and the smallest one among N 2 , N 3 , N 4 as N a , N b and N c , respectively. Then, we may assume that N a N 1 , N b , N c . From (4.10), we obtain (4.9) 1 2 , and ε > 0 sufficiently small. Consequently, we obtain a bound of (N −2+ + N − 3 2 + δ Proof. The second estimate immediately follows from (2.5), since u such that the following estimate holds: In particular, we have sup −δ≤t≤δ

4.3.
Proof of Theorem 1.1. Let (u 0 , n 0+ ) ∈ H s × L 2 be an initial datum with u 0 L 2 < Q L 2 (R 2 ) . The datum then satisfies and its modified energy obeys Since H(Iu, n + )(t) and the (a priori bounded) L 2 norm of Iu(t) control Iu(t) H 1 + n + (t) L 2 , we see from Proposition 4.5 that the solution to the initial value problem on [0, t 0 ] can be extended up to t = t 0 + δ with a uniform time δ ∼ N −2(1−s)− and satisfies as long as If we could iterate the local theory M times, then Propositions 3.2 and 4.1 imply that the increment of the modified energy would be bounded by which means that we can repeat O(N min{1, Going back to the original Zakharov system (1.1), we obtain the a priori estimate concluding the proof of Theorem 1.1.

Global solutions for the nonperiodic case
In this section we treat the R 2 case and also put the operator I on the wave equation.

5.1.
Almost conservation law. An adaptation of the argument for periodic problem easily implies the following almost conservation law.
Proposition 5.1 (Almost conservation law). Let 1 > s > 1 2 , 0 ≥ r ≥ s − 1 be such that r > 1 − 2s and r > − 1 2 s. Let 0 < δ ≤ 1 and (u, n + ) be a smooth solution to Proof. We follow the proof of Proposition 4.1 and only indicate the difference from it. We have to consider the following three terms: Estimate of (5.1). We bound the multiplier by 1 as in the periodic case. We should consider (4.4). This is bounded by Estimate of (5.2). We can obtain simpler estimate instead of (4.7) by using Lemma 2.12 instead of Lemma 2.4.
(i) N 1 N. In this case we need to consider the quantity Considering the worst case N N 1 ≪ N 3 ∼ N 4 ∼ N 2 , we can bound the above by (ii) N 1 ≪ N. Make the same decomposition as (4.8). When L 34 = L 1234 , we use Lemma 2.12 instead of Lemma 2.4 to obtain the following bound, Even the worst case N 2 ∼ N 3 ∼ N 4 N can be estimated with decay factor N −2+ whenever 1 − s − 2r < 2. When L 12 = L 1234 ≫ L 34 , we follow the argument for periodic case precisely to encounter the quantity This can be treated appropriately if 1−s−2r < 5 4 . The decay With a modification of the argument for periodic case similar to Case 1 (i), we estimate We obtain the decay N −2+ in this case.
(ii) N 1 ≪ N. If N 2 2 N 4 , then we have , we can employ the same argument as Case 1 (ii) and obtain the decay N − 5 4 + δ 1 4 − . Estimate of (5.3). This is identical with the periodic case, because (5.3) includes no n + . We have the bound (N −2+ + N −1+ δ 1− ) Iu 4 (We can choose 1 + r− > 1 2 because r > − 1 2 . Note that L max N 2 1 is required for nonzero contribution under this assumption.) To apply this, we have to decompose I S (n ± u) as If L 0 = L max (similar for the case L 2 = L max ), we use the above estimate and Lemma 2.3 to obtain where at the last inequality we have used the assumption 1 + r − s ≥ 0. Squaring and summing up the above in N 1 we obtain (5.6) (note that N 0 ∼ N 1 ). In the case L 1 = L max , a similar argument yields We can carry out the sum in L 1 using the fact L 1 ∼ max{L 02 , N 2 1 }, and have the same bound as the previous case. This completes the proof of (5.6).
In the remaining case, |ξ 2 | ≪ |ξ 1 | and |ξ 2 | ≪ N, we have |ξ 0 | ∼ |ξ 1 | and then By a standard argument, we can deduce from Lemma 5.2 the following local well-posedness. Proposition 5.3. Let 1 > s > 1 2 , 0 ≥ r ≥ s−1. Then, for any (u 0 , n +0 ) ∈ H s ×H r , there exists a unique solution to (3.1) on R 2 , (u, n + ) ∈ X s, 1 2 ,1 S (δ) × X r, 1 2 ,1 W + (δ), with the existence time such that the following estimate holds: In particular, we have We remark that our local existence time δ ∼ data − 2 1+r − is longer than that obtained in [13], which was δ ∼ data − 2 1+2r − . Compare the bilinear estimate (5.4) with Lemma 2.1 in [13]. In fact, a longer local existence time will lead to the global well-posedness for a lower regularity. Let (u 0 , n 0+ ) ∈ H s × H r be an initial datum with u 0 L 2 < Q L 2 (R 2 ) . The modified energy H(Iu, In + )(t), satisfying the initial bound controls Iu(t) H 1 + In + (t) L 2 . Proposition 5.3 shows that the solution on [0, t 0 ] can be extended up to t = t 0 + δ with a uniform time δ ∼ N − 2(1−s) 1+r − and satisfies as long as H(Iu, n + )(t 0 ) ≤ 2C 0 N 2(1−s) . If we could iterate the local theory M times, then from Propositions 3.2 and 5.1, We obtain the same a priori estimate for solutions to the original equation (1.1), concluding the proof of Theorem 1.3.
Appendix A. Proof of Lemma 2.11 Here we shall give a proof of the following bilinear estimate.
Lemma 2.11 then follows by letting v = u. The standard argument reduces the problem to the following; for details, see e.g. the proof of Lemma 2.5 in [11].
Proposition A.2. Let N, L ≥ 1. Then, for any k ∈ R 2 and A ≥ |k|, the set is covered with at most O(N We begin with preparing the following lemma.
Then, there exists no unit square in R 2 intersecting with both E < and E > . The same holds for Proof. We only prove the first half of the claim. The second half will be shown by a similar argument. Assume for contradiction that there existed such a square of side length 1. Then, it would hold for some (x, y) ∈ E < and (x ′ , y ′ ) ∈ E > that (x − x ′ ) 2 + (y − y ′ ) 2 ≤ 2, (A.1) x 2 a 2 + y 2 b 2 ≤ x ′2 (a + 100 a b ) 2 + y ′2 (b + 100) 2 . (A.2) Note that x ′2 (a + 100 a b ) 2 − From these estimates and the fact (x ′ , y ′ ) ∈ E > , which is, from (A.1) and (x, y) ∈ E < , This contradicts (A.2).
Proof of Proposition A.1. We may assume |k| ≤ 2N, otherwise the set is empty. Treat several cases separately. (i) L N. In this case, we use the condition |k ′ | ≤ N to estimate the number of squares by N 2 N 3 2 L 1 2 .
(ii) L ≪ N, |k| 1. In this case we have |k ′ | ≤ N and A − C ≤ 2|k ′ | ≤ A + L + C. It is easy to see that such a region, which is a disk of radius L or the intersection of a disk of radius N and an annulus of width L, can be covered with NL unit squares. L N implies the claim.
Therefore, we can cover this region with N × √ NL unit squares.
We remark that k ′ is confined to the region between two ellipses with common foci 0, k, longer axis A and A + L, respectively. (iv) L ≪ N, A ≥ 10N. In this case the region is close to an annulus. In fact, with 2a, 2a ′ (resp. 2b, 2b ′ ) the length of the long (resp. short) axes of inner and outer ellipses. We first change the scale in the direction of short axis to make the inner ellipse a circle. Then, the new region R ′ is included in an annulus of width max{a ′ − a, a b (b ′ − b)}. We see a ′ − a = L and Hence, the intersection of any ball of radius 2N and R ′ is covered with NL unit squares, which shows that the intersection of any ball of radius N and the original R is also covered with the same number of unit squares.
From Lemma A.3, we see that the smallest (axis-aligned) lattice polygon including the inside of outer boundary of R is included in the inside of an ellipse with long axis 2(a ′ + 100 a ′ b ′ ) and short axis 2(b ′ + 100). In the same manner, the biggest (axisaligned) lattice polygon included in the inside of inner boundary of R includes an ellipse with long axis 2(a − 100 a b ) and short axis 2(b − 100). Therefore, the number of needed unit squares is estimated by We find b ′ + 100 N, |a − 100 a b | N, a ′ − a L, and We also see a ′ /b ′ N/L in the same manner. Finally, With all of them together, we reach the bound N