Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 20, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu REMARKS ON PERIODIC ZAKHAROV SYSTEMS NOBU KISHIMOTO Abstract. In this article, we consider the Cauchy problem associated with the Zakharov system on the torus. We obtain unconditional uniqueness of solutions in low regularity Sobolev spaces including the energy space in one and two dimensions. We also prove convergence of solutions in the energy space, as the ion sound speed tends to infinity, to the solution of a cubic nonlinear Schrödinger equation, for dimensions one and two. Our proof of unconditional uniqueness is based on the method of infinite iteration of the normal form reduction; actually, we simply show a certain set of multilinear estimates, which was proposed as a criterion for unconditional uniqueness in [11]. The convergence result is obtained by a similar argument to the non- periodic case [13], which uses conservation laws and unconditional uniqueness for the limit equation. 1. Introduction We consider the Cauchy problem associated with the Zakharov system under a periodic boundary condition i∂tu+ ∆u = nu, 1 α2 ∂2 t n−∆n = ∆(|u|2), t ∈ R, x ∈ Tdλ, (u, n, ∂tn) ∣∣ t=0 = (u0, n0, n1) ∈ Hs,l(Tdλ), (1.1) where α > 0 is a constant, λ ∈ (0,∞)d, and Tdλ := Rd/(2πλ1Z) × · · · × (2πλdZ) is the torus with period 2πλ = (2πλ1, . . . , 2πλd). We treat the torus of arbitrary period and (by rescaling) normalize the coefficient of the Laplace operator: ∆ := ∂2 x1 + · · · + ∂2 xd . Write Zdλ to denote the lattice 1 λ1 Z × · · · × 1 λd Z corresponding to Tdλ. The unknown functions u, n are C- and R-valued, respectively, and Hs,l(Tdλ) := Hs(Tdλ;C)×H l(Tdλ;R)×H l−1(Tdλ;R) for s, l ∈ R. For an interval I ⊂ R, we denote by C(I;Hs,l(Tdλ)) the space of all functions (u, n) such that u ∈ C(I;Hs(Tdλ;C)), n ∈ C(I;H l(Tdλ;R)) ∩ C1(I;H l−1(Tdλ;R)). If I = [0, T ], we further abbreviate it as CTHs,l(Tdλ). The (vector-valued) Zakharov system was derived as a model for propagation of Langmuir waves in a plasma; see [17] for more details. There is a wealth of literature on local and global well-posedness, as well as asymptotic behavior of global solutions, of the Cauchy problem (1.1) on Rd and on Td; we refer to the 2020 Mathematics Subject Classification. 35Q55, 35A02. Key words and phrases. Zakharov system; periodic boundary condition; unconditional uniqueness; subsonic limit. ©2022. This work is licensed under a CC BY 4.0 license. Submitted July 5, 2020. Published March 18, 2022. 1 2 N. KISHIMOTO EJDE-2022/20 recent article [5] and references therein. The aim of this note is to give two results on the property of the solutions to the periodic Cauchy problem (1.1): unconditional uniqueness and convergence to a cubic nonlinear Schrödinger equation as α → ∞ (the subsonic limit). These properties have also been studied in the non-periodic case, while there seems no result in the periodic setting. Let us recall the following result on local well-posedness of the periodic Cauchy problem (1.1) in Sobolev spaces, which was given by Takaoka [18] for d = 1 and the author [9] for d ≥ 2 (see also an earlier work of Bourgain [3]). Theorem 1.1 ([9, 18]). The Cauchy problem (1.1) is locally well-posed in Hs,l(Tdλ) in the following cases: • d = 1, αλ 6∈ Z, − 1 2 ≤ l ≤ 2s− 1 2 , 0 ≤ s− l ≤ 1; • d = 1, αλ ∈ Z, 0 ≤ l ≤ 2s− 1, 0 ≤ s− l ≤ 1; • d = 2, α, λ are arbitrary, 0 ≤ l ≤ 2s− 1, 0 ≤ s− l ≤ 1; • d ≥ 3, α, λ are arbitrary, d−2 2 < l ≤ 2s− d 2 , 0 ≤ s− l ≤ 1. The above result was obtained by an iteration argument using the Fourier restric- tion norm (Bourgain norm), and thus uniqueness is ensured only for those solutions with such an auxiliary norm being finite. In very low regularities (e.g., the case d = 1, αλ 6∈ Z, and (s, l) = (0,− 1 2 ) in the theorem), one has to impose some additional requirement on solutions (not only to be in CTHs,l) to ensure that both of the nonlinear terms nu, ∆(|u|2) are well-defined in a certain sense. However, at least when s+ l ≥ 0 and s ≥ 0, these nonlinear terms make sense in the framework of distributions for any (u, n) ∈ Hs,l, so that one can ask uniqueness within the class of all (distributional) solutions in CTHs,l, which we refer to as unconditional uniqueness. Our result on unconditional uniqueness reads as follows. Theorem 1.2. Let T > 0. For any (u0, n0, n1) ∈ Hs,l(Tdλ), there is at most one solution (in the sense of distributions) to the Cauchy problem (1.1) in CTHs,l(Tdλ) in the following cases: • d = 1, αλ 6∈ Z, s > 1 6 , l > − 1 2 and s+ l ≥ 0; • ([11, Theorem 6.1]) d = 1, αλ ∈ Z, s ≥ 1 2 and l ≥ 0; • d = 2, α, λ are arbitrary, s ≥ 1/2 and l ≥ 0; • d ≥ 3, α, λ are arbitrary, s > d−1 2 and l > d−2 2 . A result on unconditional uniqueness for the non-periodic problem was obtained in [14] by means of various estimates in Strichartz- and Bourgain-type norms. We prove the theorem by a different approach: infinite iteration of the Poincaré-Dulac normal form reduction. In [11], the author developed this methodology for uncon- ditional uniqueness, which had been introduced in the work of Guo, Kwon, and Oh [6] for the cubic nonlinear Schrödinger equation on T, in an abstract setting and proved that the overall argument can be reduced to a certain set of multilinear estimates associated with the nonlinearity of the equation. In this note, we rely on the abstract theory in [11] and simply show these multilinear estimates. The case d = 1, α = λ = 1 of Theorem 1.2 was treated in [11] for a demonstration of the method, and it is easy to see that the same proof works in the case αλ ∈ Z. Note that, in the above theorem, we only consider (s, l) satisfying s ≥ 0 and s + l ≥ 0, so that the nonlinear terms make sense in the framework of distributions. Combining Theorems 1.1 and 1.2, we obtain unconditional well-posedness of (1.1). In particular, when d = 1, 2, the energy space (s, l) = (1, 0) is included for arbitrary α, λ. EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 3 Corollary 1.3. The Cauchy problem (1.1) is unconditionally locally well-posed in Hs,l(Tdλ) if: • d = 1, αλ 6∈ Z, −s ≤ l ≤ 2s− 1 2 , 0 ≤ s− l ≤ 1 and (s, l) 6= ( 1 6 ,− 1 6 ), ( 1 2 ,− 1 2 ); • d = 1, αλ ∈ Z, 0 ≤ l ≤ 2s− 1, 0 ≤ s− l ≤ 1; • d = 2, α, λ are arbitrary, 0 ≤ l ≤ 2s− 1, 0 ≤ s− l ≤ 1; • d ≥ 3, α, λ are arbitrary, d−2 2 < l ≤ 2s− d 2 , 0 ≤ s− l ≤ 1. Next, we study convergence of the solutions (uα, nα) of the periodic Zakharov system i∂tu α + ∆uα = nαuα, 1 α2 ∂2 t n α −∆nα = ∆(|uα|2), t ∈ R, x ∈ Tdλ, (uα, nα, ∂tn α) ∣∣ t=0 = (uα0 , n α 0 , n α 1 ) (1.2) as α → ∞. This problem has also been well studied in the Rd case. In principle, the Schrödinger part uα of the solution converges to the unique solution u of the focusing cubic nonlinear Schrödinger equation i∂tu+ ∆u = −|u|2u, t ∈ R, x ∈ Rd (1.3) with initial condition u(0) = lim α→∞ uα0 , while the wave part nα converges to −|u|2. In the non-compatible case nα0 + |uα0 |2 6→ 0, the strong convergence of the wave part is verified after correction by a fast oscillating linear wave solution; this is called the initial layer. The strong convergence in Sobolev spaces was first proved in [16] for compatible data, and then the initial layer phenomenon and the rate of convergence were investigated in subsequent works [1, 15, 8]. While a certain amount of regularity (H5, for instance) had been assumed in the above results, Masmoudi and Nakanishi [13] proved the strong convergence in the energy class H1 × L2 × Ḣ−1(Rd). Their proof is substantially simpler than the previous ones, only using local well-posedness (conservation laws) of (1.2), (1.3) and unconditional uniqueness for the limit equation (1.3) in the energy class, though the rate of convergence is difficult to obtain by this approach. We aim here to give an analogous result of [13] in the periodic setting. We focus on one and two dimensions, because local well-posedness for (1.2) in the energy class has been shown only in one and two dimensions. In the limit α → ∞, we formally obtain ∆(nα + |uα|2) ∼ 0, namely, P6=c(n α + |uα|2) ∼ 0, where Pc and P6=c denote the orthogonal projections onto zero and non-zero frequency modes, respectively. In contrast to the non-periodic (spatially decaying) case, one cannot determine the asymptotic behavior of the zero mode (spatial mean) of nα from the relation ∆(nα + |uα|2) ∼ 0. In the periodic case, however, the zero mode of the wave part of the system (1.2) can be decoupled and explicitly solved as ∂2 t Pcn α = 0,( Pcn α, ∂tPcn α )∣∣ t=0 = (Pcn α 0 , Pcn α 1 ) implies Pcn α(t) = Pcn α 0 + tPcn α 1 (t ∈ R). This suggests that nα(t, x) = P6=cn α(t, x) + Pcn α(t) ∼ −P6=c(|uα|2)(t, x) + Pcn α 0 + tPcn α 1 4 N. KISHIMOTO EJDE-2022/20 as α→∞, and that the Schrödinger part uα converges to the solution of a “shifted” cubic NLS: i∂tu+ ∆u = − ( |u|2 − Pc(|u|2)− lim α→∞ [ Pcn α 0 + tPcn α 1 ]) u. Note that, even in the case of mean-zero wave initial data Pcn α 0 = Pcn α 1 ≡ 0, the expected limit equation in the periodic setting differs by Pc(|u|2) from the usual focusing cubic NLS (1.3). This is also different from the renormalized (or Wick- ordered) cubic NLS, where 2Pc(|u|2) is subtracted. We also remark that, if the initial data (uα0 , n α 0 ) do not satisfy the condition P6=c(n α 0 + |uα0 |2) = 0 in the limit α→∞ (i.e., non-compatible), the initial layer should appear as α→∞. We denote by P≤R, P>R the projections in spatial frequency onto {|k| ≤ R} and {|k| > R}, respectively. Here is our theorem on convergence. Theorem 1.4. Let d = 1, 2 and λ ∈ (0,∞)d be arbitrary. Let {uα0 , nα0 , nα1 }α ⊂ H1,0(Tdλ) be a family of initial data satisfying ∃ u∞0 := lim α→∞ uα0 in H1(Tdλ), sup α ∥∥(P6=cn α 0 , |α∇|−1P6=cn α 1 ) ∥∥ L2×L2 <∞, lim R→∞ lim sup α→∞ ∥∥(P>Rn α 0 , |α∇|−1P>Rn α 1 ) ∥∥ L2×L2 = 0, and ∃ (ν0, ν1) := lim α→∞ ( Pcn α 0 , Pcn α 1 ) . (1.4) Let (uα, nα) ∈ C([0, Tα);H1,0(Tdλ)) be the (forward-in-time) maximal-lifespan solu- tion of (1.2). The maximal-lifespan solution is uniquely defined in the energy class CtH1,0 x by the existence result given in [18, 9] and the uniqueness result established in Theorem 1.2. Let u ∈ C([0, T∞);H1) be the (forward-in-time) maximal-lifespan solution of i∂tu+ ∆u = − ( |u|2 − Pc(|u|2)− ν0 − ν1t ) u, t ∈ (0, T∞), x ∈ Tdλ, u ∣∣ t=0 = u∞0 . (1.5) This is also uniquely defined in CtH 1 x. See Remark 3.4 (i) below. Then, we have T∞ ≤ lim infα→∞ Tα, and for any T ∈ (0, T∞), uα → u in C([0, T ];H1), P6=cn α − nαil → −P 6=c(|u|2) in C([0, T ];L2), |α∇|−1∂t(P 6=cn α − nαil)→ 0 in C([0, T ];L2), Pcn α → ν0 + ν1t in C1([0, T ]) as α→∞, where the initial layer nαil is given by nαil(t) := cos ( t|α∇| ) P 6=c(n α 0 + |uα0 |2) + sin ( t|α∇| ) |α∇| P6=cn α 1 . Remark 1.5. The assumptions in the above theorem trivially hold if the initial data are independent of α; (uα0 , n α 0 , n α 1 ) ≡ (u0, n0, n1) ∈ H1,0. In this case, one can simply take nαil = cos ( t|α∇| ) P6=c(n α 0 + |uα0 |2) as the initial layer, since the remaining part is of O(α−1). On the other hand, (non-zero modes of) the initial data nα1 ∈ H−1 are allowed to diverge with growth order at most O(α) as α→∞. For instance, the data nα1 = αP6=cn1 + Pcn1 for a fixed n1 ∈ H−1 also satisfy the EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 5 assumptions. In this case, one needs to modify the initial layer depending on nα1 as in the theorem. Remark 1.6. The first three assumptions on initial data in the theorem are the same as those in the Rd case [13]. The last one (1.4), which was not assumed in [13], is necessary for the convergence of uα in the periodic case. To see this, we first note that, in the periodic case, for any solution (uα, nα) of (1.2) in the energy class, the transformation (uα, nα) 7→ (uαei(c0t+ 1 2 c1t 2), nα − c0 − c1t), c0, c1 ∈ R gives another energy-class solution of (1.2). Then, consider three families of solu- tions (uα, nα), (uαeit sinα, nα − sinα), (uαeit 2 sinα, nα − 2t sinα). We observe that the first three assumptions are equivalent for all of them. However, the claimed convergence cannot hold for any two of them at the same time, unless u ≡ 0. The rest of this note is devoted to the proofs of Theorems 1.2 and 1.4, which will be given in Sections 2 and 3, respectively. Throughout this note, we often use the notation X ∼ Y, X . Y, X � Y as abbreviations for C−1Y ≤ X ≤ CY, X ≤ CY, X > CY with a suitably large positive constant C. 2. Proof of unconditional uniqueness 2.1. Reduction to the fundamental bilinear estimates. For p ∈ [1,∞] and s ∈ R, let `ps = `ps(Zdλ) be the weighted `p space on Zdλ with the norm ‖fk‖`ps := ‖〈k〉sfk‖`p , where 〈k〉 := 1 + |k|. We employ the infinite normal form reduction machinery. As discussed in [11, Sections 1 and 6], unconditional uniqueness of solutions to (1.1) in Hs,l(Tdλ) is established once we have the following bilinear estimates with some ε > 0:∥∥∥ ∑ k1=k0+k2 fk0hk2 〈µ±〉1/2 ∥∥∥ `2s((Zdλ)k1 ) . ‖f‖`2l ‖h‖`2s ,∥∥∥|k0| ∑ k0=k1−k2 gk1hk2 〈µ±〉1/2 ∥∥∥ `2l ((Z d λ)k0 ) . ‖g‖`2s‖h‖`2s ,∥∥∥ ∑ k1=k0+k2 〈k0〉+ 〈k2〉 〈k1〉 fk0hk2 〈µ±〉1−ε ∥∥∥ `2s((Zdλ)k1 ) . ‖f‖`2l ‖h‖`2s ,∥∥∥|k0| ∑ k0=k1−k2 〈k1〉+ 〈k2〉 〈k0〉 gk1hk2 〈µ±〉1−ε ∥∥∥ `2l ((Z d λ)k0 ) . ‖g‖`2s‖h‖`2s , ‖f ∗ h‖`2s−1 . ‖f‖`2l ‖h‖`2s , ‖g ∗ h‖`2l . ‖g‖`2s‖h‖`2s for any non-negative sequences f ∈ `2l (Zdλ), g, h ∈ `2s(Zdλ), where µ± := |k1|2 − |k2|2 ± α|k0| 6 N. KISHIMOTO EJDE-2022/20 and ∗ denotes the convolution. In [11], µ̃± = |k1|2 − |k2|2 ± 〈αk0〉 was used instead of µ± (and α was taken to be 1). Since 〈µ±〉 ∼ 〈µ̃±〉, there is no difference in the above estimates. We see that the first four estimates are equivalent by duality to the trilinear estimates∑ k0,k1,k2∈Zdλ k0=k1−k2 Wj(k0, k1, k2)fk0gk1hk2 . ‖f‖`2‖g‖`2‖h‖`2 , j = 1, . . . , 4 (2.1) for non-negative sequences f, g, h ∈ `2(Zdλ), where W1 = 〈k1〉s 〈µ±〉1/2〈k0〉l〈k2〉s , W2 = 〈k0〉l|k0| 〈µ±〉1/2〈k1〉s〈k2〉s , W3 = 〈k1〉s−1(〈k0〉+ 〈k2〉) 〈µ±〉1−ε〈k0〉l〈k2〉s , W4 = 〈k0〉l−1|k0|(〈k1〉+ 〈k2〉) 〈µ±〉1−ε〈k1〉s〈k2〉s . The next proposition is the main ingredient of the proof of Theorem 1.2. Proposition 2.1. Estimate (2.1) holds with some ε > 0 in the following cases: (i) d = 1, αλ 6∈ N, 1/6 < s < 1/2, and l = −s; (ii) d = 2, (s, l) = (1/2, 0); (iii) d ≥ 3, s > d−1 2 , l = s− 1 2 . We observe that the last two estimates on the convolution, which are equivalent to the Sobolev estimates on the product, hold if and only if A1 := min{1− s+ l, s+ l} ≥ 0, B1 := 1 + l − d 2 ≥ 0 with (A1, B1) 6= (0, 0) and A2 := min{s− l, 2s} ≥ 0, B2 := 2s− l − d 2 ≥ 0 with (A2, B2) 6= (0, 0). These conditions are satisfied in each of the cases (i)–(iii) in Proposition 2.1. Finally, note that uniqueness of solution in CTHs,l implies that in CTHs ′,l′ for any s′ ≥ s and l′ ≥ l. Therefore, to establish Theorem 1.2 it suffices to show Proposition 2.1. 2.2. One dimensional case. In this subsection, we prove Proposition 2.1 (i). It is easy to check Wj . 1 when k0 = 0. This implies that the estimate (2.1) holds if the sum is restricted to {k0 = 0}. We therefore assume k0 6= 0, then it holds 〈µ±〉 = 〈k0(k0 + 2k2 ± α sgn(k0))〉 under the relation k0 = k1 − k2. If αλ 6∈ N, we have |k0 + 2k2 ± α sgn(k0)| ≥ dist( 1 λZ, α) > 0, and in particular, 〈µ±〉 ∼ 〈k0〉〈k0 + 2k2 ± α sgn(k0)〉 ∼ 〈k0〉〈k0 + 2k2〉. (2.2) Let l = −s. Using this factorization, for W1 and W2, we see that W1 ∼ 〈k0 + k2〉s 〈k0〉 1 2−s〈k0 + 2k2〉1/2〈k2〉s . 1|k0+k2|�|k2| 〈k0〉 1 2−s〈k0 + 2k2〉 1 2−s〈k2〉s + 1|k0+k2|.|k2| 〈k0〉 1 2−s〈k0 + 2k2〉1/2 , W2 ∼ 〈k0〉 1 2−s 〈k0 + 2k2〉1/2〈k0 + k2〉s〈k2〉s . 1|k0+2k2|&|k0| 〈k0 + 2k2〉s〈k0 + k2〉s〈k2〉s + 1|k0+2k2|�|k0| 〈k0 + 2k2〉1/2〈k2〉3s− 1 2 , EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 7 where 1A denotes the characteristic function of the set A or the set of variables satisfying the condition A. For W3, we take ε = 1 2 to have W3 ∼ 〈k0〉+ 〈k2〉 〈k0〉 1 2−s〈k0 + 2k2〉1/2〈k0 + k2〉1−s〈k2〉s . 1|k0+k2|�|k2| 〈k0〉1−2s〈k2〉s + 1|k0+k2|∼|k2| 〈k0〉 1 2−s〈k0 + 2k2〉1/2 + 1|k0+k2|�|k2| 〈k0 + k2〉1−s , and for W4 we take ε = 1/3, so that W4 ∼ 〈k0 + k2〉+ 〈k2〉 〈k0〉 2 3 +s〈k0 + 2k2〉2/3〈k0 + k2〉s〈k2〉s . 1|k0+2k2|�|k0| 〈k0〉 2 3 +s〈k2〉2s− 1 3 + 1|k0+2k2|∼|k0| 〈k0〉 1 3 +s〈k0 + k2〉s〈k2〉s + 1|k0+2k2|�|k0| 〈k0 + 2k2〉2/3〈k2〉3s− 1 3 . If 1/6 < s < 1/2, we deduce from these estimates that Wj . 1 〈k0〉 1 2 +δ + 1 〈k0 + 2k2〉 1 2 +δ + 1 〈k0 + k2〉 1 2 +δ + 1 〈k2〉 1 2 +δ , j = 1, . . . , 4 for some δ > 0. We then apply the Hölder inequality to obtain (2.1). 2.3. Two and higher dimensional cases. In this subsection, we shall prove Proposition 2.1 (ii), (iii). The main difficulty comes from the fact that we do not have a factorization like (2.2). We divide the analysis into three cases according to the size of |µ±|. Let kmax and kmin be the largest and the smallest quantities among {|k0|, |k1|, |k2|}, respectively. High modulation interactions. We begin with the case |µ±| & k2 max and prove (2.1) with ε = 1 2 . Under the condition l = s− 1 2 , it holds that Wj . 1 〈kmax〉1/2〈kmin〉s , j = 1, . . . , 4. This and the Sobolev inequality imply (2.1); in fact, the desired estimate∑ k0,k1,k2∈Zdλ k0=k1−k2 fk0gk1hk2 〈kmax〉1/2〈kmin〉s . ‖f‖`2‖g‖`2‖h‖`2 is the dual of the product estimate ‖uv‖L2(Tdα) . ‖u‖H1/2(Tdα)‖v‖Hs(Tdα), which holds if d ≥ 2 and s ≥ d−1 2 . Middle modulation interactions. Hereafter, we assume |µ±| � k2 max. This in par- ticular implies |k0| . |k1| ∼ |k2|. Taking s = l + 1 2 (≥ 1 2 ) and ε = 1 2 , we see that Wj . 1 〈µ±〉1/2〈k0〉l , j = 1, . . . , 4. 8 N. KISHIMOTO EJDE-2022/20 If |k0| . 1, then the left-hand side of (2.1) is bounded by ‖1|n0|.1f‖`1‖g‖`2‖h‖`2 , which is sufficient. It then suffices to prove∑ k0=k1−k2 1�|k0|.|k1|∼|k2| |µ±|�k2max fk0gk1hk2 〈µ±〉1/2〈k0〉l . ‖f‖`2‖g‖`2‖h‖`2 (2.3) for l = 0 if d = 2 and l > d−2 2 if d ≥ 3. Here, we consider the middle-modulation case kmax . |µ±| � k2 max, following the idea in [9, Section 3.2] for the corresponding bilinear estimates in Bourgain spaces. First, restrict k0, k1, k2 to 〈kj〉 ∼ Nj for dyadic numbers N0, N1, N2 with N1 ∼ N2 & N0 � 1, and then restrict µ± to 〈µ±〉 ∼M for a dyadic N1 .M � N2 1 , so that L.H.S. of (2.3) . ∑ N1∼N2 ∑ 1�N0.N1 N1.M�N2 1 1 M1/2N l 0 ∑ k0=k1−k2 〈kj〉∼Nj 〈µ±〉∼M fk0gk1hk2 . Since in the last sum we have∣∣|k1| − |k2| ∣∣ = ∣∣µ± ∓ α|k0| ∣∣ |k1|+ |k2| = O (M N1 ) , the following decomposition into annuli: gk1hk2 = ∑ m1,m2 (1Am1 g)k1(1Am2 h)k2 , Am := { k ∈ Zdλ : m M N1 ≤ |k| ≤ (m+ 1) M N1 } , m ∈ Z, m ∼ N2 1 M exhibits almost orthogonality. If N0 � N1, we make further decomposition into cubes: gk1hk2 = ∑ n1,n2 (1Qn1 g)k1(1Qn2 h)k2 , Qn := { k ∈ Zdλ ∣∣∣ |k − n| ∈ [0, N0]d } , n ∈ (N0Z)d, |n| ∼ N1 and make use of its almost orthogonality. Hence, L.H.S. of (2.3) . ∗∑ N1,N2 ∑ 1�N0.N1 N1.M�N2 1 1 M1/2N l 0 ∗∑ m1,m2 ∗∑ n1,n2 ∑ k0=k1−k2 〈k0〉∼N0, 〈µ±〉∼M k1∈Am1 ∩Qn1 k2∈Am2 ∩Qn2 fk0gk1hk2 , where ∑∗ stand for almost orthogonal sums (i.e., one index determines the other up to O(1) ambiguity). Now, we recall another identity k0 |k0| k1 = 1 2|k0| ( |k0|2 ∓ α|k0|+ µ± ) = 1 2|k0| ( |k0|2 ∓ α|k0| ) +O (M N0 ) , which restricts k0 |k0| -component of k1 into an interval of length O( MN0 ) for each k0 fixed. Therefore, for fixed k0, k1 is confined to the intersection of a cube, an annulus, EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 9 and a plate. An elementary computation (see [9, Lemma 2.9 (i)]) evaluates the number of frequencies k1 ∈ Zdλ in such a region by C min { Nd 0 , M N1 Nd−1 0 , M N0 M1/2Nd−2 0 } .MNd−2 0 (N0 N1 )1/2 min {N2 0 M , M1/2 N0 }1/2 . By the Cauchy-Schwarz inequality in k1, we have (for d ≥ 2 and l ≥ d−2 2 ) L.H.S. of (2.3) . ∗∑ N1,N2 ∑ N0,M 1 M1/2N l 0 ∗∑ m1,m2 ∗∑ n1,n2 [ MNd−2 0 (N0 N1 )1/2 min {N2 0 M , M1/2 N0 }1/2]1/2 × ∑ k0 fk0 (∑ k1 (1Am1 ∩Qn1 g)2 k1(1Am2 ∩Qn2 h)2 k1−k0 )1/2 . ‖f‖`2 ∗∑ N1,N2 ‖1〈k1〉∼N1 g‖`2‖1〈k2〉∼N2 h‖`2 ∑ N0.N1 (N0 N1 )1/4∑ M min {N2 0 M , M1/2 N0 }1/4 . ‖f‖`2‖g‖`2‖h‖`2 . Low modulation interactions. The remaining case |µ±| � kmax can also be treated by mimicking the proof of the corresponding bilinear estimates in [9, Section 3.3]. Note that we need more delicate analysis including decomposition with respect to the angles between frequencies. Here, we take a different approach. It was mentioned in [11, Remark 1.2] that some of the multilinear estimates required for the normal form reduction argument have close relationship with the standard multilinear estimates in Bourgain spaces, which are used to prove conditional well-posedness. In our setting, the desired estimate (2.3) corresponds to the bilinear estimate ∥∥∥ 1 〈τ1 + |k1|2〉b1 ∫ R ∑ k0∈Zdλ 1�|k0|.|k1|∼|k1−k0| |µ±|�kmax w̃(τ0, k0)ũ(τ1 − τ0, k1 − k0) dτ0 ∥∥∥ L2 τ1,k1 . ∥∥〈k0〉l〈τ0 ∓ α|k0|〉b0w̃(τ0, k0) ∥∥ L2 τ0,k0 ∥∥〈τ2 + |k2|2〉b2 ũ(τ2, k2) ∥∥ L2 τ2,k2 (2.4) with b0 = b1 = b2 = 1/2. It is not clear whether the equivalence of these estimates holds in a general setting. Nevertheless, we will see that (2.4) implies (2.3) if b0 + b1 + b2 < 1: Lemma 2.2. Let s1, s2, l ∈ R, γ ≥ 0, and Ω be a subset of {(k0, k1, k2) ∈ (Zdλ)3 : k0 = k1 − k2}. Assume that there exist b0, b ′ 0, b1, b2, b ′ 2 ≥ 0 with max{b0 + b1 + 10 N. KISHIMOTO EJDE-2022/20 b2, b ′ 0 + b1 + b′2} < 1 2 + γ such that∥∥∥ 〈k1〉s1 〈ρ1〉b1 ∫ τ1=τ0+τ2 ∑ k0,k2∈Zdλ 1Ω(k0, k1, k2)1〈ρ0〉≤〈ρ2〉.〈ρ1〉∼〈µ±〉 × w̃(τ0, k0)ũ(τ2, k2) dτ0 ∥∥∥ L2 τ1,k1 . ∥∥〈k0〉l〈ρ0〉b0w̃(τ0, k0) ∥∥ L2 τ0,k0 ∥∥〈k2〉s2〈ρ2〉b2 ũ(τ2, k2) ∥∥ L2 τ2,k2 ,∥∥∥ 〈k1〉s1 〈ρ1〉b1 ∫ τ1=τ0+τ2 ∑ k0,k2∈Zdλ 1Ω(k0, k1, k2)1〈ρ2〉≤〈ρ0〉.〈ρ1〉∼〈µ±〉 × w̃(τ0, k0)ũ(τ2, k2) dτ0 ∥∥∥ L2 τ1,k1 . ∥∥〈k0〉l〈ρ0〉b ′ 0w̃(τ0, k0) ∥∥ L2 τ0,k0 ∥∥〈k2〉s2〈ρ2〉b ′ 2 ũ(τ2, k2) ∥∥ L2 τ2,k2 , (2.5) where ρ0 := τ0 ∓ α|k0|, ρ1 := τ1 + |k1|2, and ρ2 := τ2 + |k2|2. Then, we have∥∥∥ ∑ k0,k2∈Zdλ 1Ω(k0, k1, k2)fk0hk2 〈µ±〉γ ∥∥∥ (`2s1 )k1 . ‖f‖`2l ‖h‖`2s2 . Proof. Let I := { ( 101 100 )n : n ∈ Z, n ≥ 0 } , ΩL,σ := { (k0, k1, k2) ∈ Ω : 1 + |µ±| ∈ [L, 101 100L), σµ± ≥ 0 } , L ∈ I, σ ∈ {±1}. Take arbitrary non-negative sequences f ∈ `2l , h ∈ `2s2 , and define w̃L(τ, k) := 1[− L 10 , L 10 ](τ ∓ α|k|)fk, ũL(τ, k) := 1[− L 10 , L 10 ](τ + |k|2)hk, L ∈ I. We observe that, for (k0, k1, k2) ∈ ΩL,σ and τ1 ∈ R,∫ R 1[− L 10 , L 10 ](τ0 ∓ α|k0|)1[− L 10 , L 10 ](τ1 − τ0 + |k2|2) dτ0 ≥ L 10 1[− L 10 , L 10 ](τ1 + |k1|2 − µ±) ≥ L 10 1[− L 20 , L 20 ](τ1 + |k1|2 − σ(L− 1)), and that ∫ R 1[− L 10 , L 10 ](τ0 ∓ α|k0|)1[− L 10 , L 10 ](τ1 − τ0 + |k2|2) dτ0 6= 0 implies 〈ρ1〉 ∼ 〈µ±〉 ∼ L. Hence, for each L ∈ I and σ ∈ {±1}, we have∥∥∥ ∑ k0,k2∈Zdλ 1ΩL,σ (k0, k1, k2)fk0hk2 〈µ±〉γ ∥∥∥ (`2s1 )k1 ∼ L−1/2 ∥∥∥〈k1〉s1 ∥∥1[− L 20 , L 20 ](ρ1 − σ(L− 1)) ∥∥ L2 τ1 ∑ k0,k2∈Zdλ 1ΩL,σfk0hk2 〈µ±〉γ ∥∥∥ (`2)k1 . L− 3 2 ∥∥∥〈k1〉s1 ∫ R ∑ k0,k2∈Zdλ 1ΩL,σ 〈µ±〉γ w̃L(τ0, k0)ũL(τ1 − τ0, k2) dτ0 ∥∥∥ L2 τ1,k1 . L− 3 2−γ+b1 ∥∥∥ 〈k1〉s1 〈ρ1〉b1 ∫ τ1=τ0+τ2 ∑ k0,k2∈Zdλ 1ΩL,σ1〈ρ0〉,〈ρ2〉.〈ρ1〉∼〈µ±〉 EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 11 × w̃L(τ0, k0)ũL(τ2, k2) dτ0 ∥∥∥ L2 τ1,k1 , and then, using (2.5), . L− 3 2−γ+b1 (∥∥〈k0〉l〈ρ0〉b0w̃L(τ0, k0) ∥∥ L2 τ0,k0 ∥∥〈k2〉s2〈ρ2〉b2 ũL(τ2, k2) ∥∥ L2 τ2,k2 + ∥∥〈k0〉l〈ρ0〉b ′ 0w̃L(τ0, k0) ∥∥ L2 τ0,k0 ∥∥〈k2〉s2〈ρ2〉b ′ 2 ũL(τ2, k2) ∥∥ L2 τ2,k2 ) . ‖f‖`2l ‖h‖`2s2 ( L− 1 2−γ+b0+b1+b2 + L− 1 2−γ+b′0+b1+b′2 ) . From the assumption on b0, b ′ 0, b1, b2, b ′ 2, we have∥∥∥ ∑ k0,k2∈Zdλ 1Ωfk0hk2 〈µ±〉γ ∥∥∥ (`2s1 )k1 ≤ ∑ L∈I, σ∈{±1} ∥∥∥ ∑ k0,k2∈Zdλ 1ΩL,σfk0hk2 〈µ±〉γ ∥∥∥ (`2s1 )k1 . ‖f‖`2l ‖h‖`2s2 , as desired. � From [9, Propositions 3.9, 3.6], we can easily deduce the bilinear estimates (2.5) for s1 = s2 = 0, l = 0 if d = 2 and l > d−2 2 if d ≥ 3, and Ω = {(k0, k1, k2) | k0 = k1 − k2, |µ±| � kmax, |k0| � 1}, under the condition that b1 = b2 = b′0 > 3 8 , b0 = b′2 > 0. In view of Lemma 2.2, the desired estimate (2.3) is obtained. This completes the proof of Proposition 2.1. 3. Proof of convergence as α→∞ Before the proof, we first reduce the problem to the case of mean-zero wave part. As mentioned in Section 1, any solution (uα, nα) ∈ CTH1,0 to (1.2) (in the sense of distributions) is also a solution to i∂tu α + ∆uα = ( P 6=cn α + Pcn α 0 + tPcn α 1 ) uα t ∈ (0, T ), x ∈ Tdλ, 1 α2 ∂2 t P6=cn α −∆P 6=cn α = ∆(|uα|2), t ∈ (0, T ), x ∈ Tdλ,( uα, P6=cn α, ∂tP6=cn α )∣∣ t=0 = (uα0 , P6=cn α 0 , P6=cn α 1 ). We introduce (ũα, ñα)(t) := ( uα(t)ei(tPcn α 0 + t2 2 Pcn α 1 ), P6=cn α(t) ) , which solves i∂tũ α + ∆ũα = ñαũα t ∈ (0, T ), x ∈ Tdλ, 1 α2 ∂2 t ñ α −∆ñα = ∆(|ũα|2), t ∈ (0, T ), x ∈ Tdλ,( ũα, ñα, ∂tñ α )∣∣ t=0 = (uα0 , ñ α 0 , ñ α 1 ) := (uα0 , P6=cn α 0 , P6=cn α 1 ) ∈ H1,0 0 (Tdλ), (3.1) where H l 0(Tdλ) := P6=cH l(Tdλ), H1,0 0 (Tdλ) := H1(Tdλ;C)× L2 0(Tdλ;R)×H−1 0 (Tdλ;R). Conversely, for any (uα0 , P6=cn α 0 , P6=cn α 1 ) ∈ H1,0 0 the maximal-lifespan solution of (3.1) exists uniquely in C([0, Tα);H1,0 0 ), and (with Pcn α 0 , Pcn α 1 ∈ R given) the 12 N. KISHIMOTO EJDE-2022/20 maximal-lifespan solution of the original equation (1.2) (with the same maximal existence time) is given by (uα, nα)(t) = ( ũα(t)e−i(tPcn α 0 + t2 2 Pcn α 1 ), ñα(t, x) + Pcn α 0 + tPcn α 1 ) . Clearly, Theorem 1.4 follows once we prove the following proposition. Proposition 3.1. Let {uα0 , ñα0 , ñα1 }α ⊂ H 1,0 0 (Tdλ) be a family of initial data such that ∃ u∞0 := lim α→∞ uα0 in H1(Tdλ), (3.2) sup α ∥∥(ñα0 , |α∇|−1ñα1 ) ∥∥ L2×L2 <∞, (3.3) lim R→∞ lim sup α→∞ ∥∥(P>Rñ α 0 , |α∇|−1P>Rñ α 1 ) ∥∥ L2×L2 = 0. (3.4) Let (ũα, ñα) ∈ C([0, Tα);H1,0 0 (Tdλ)) be the (unique) maximal-lifespan solution of (3.1), and let ũ ∈ C([0, T∞);H1) be the (unique) maximal-lifespan solution of the Cauchy problem i∂tũ+ ∆ũ = −P6=c(|ũ|2)ũ, t ∈ (0, T∞), x ∈ Tdλ, ũ ∣∣ t=0 = u∞0 . (3.5) Then, we have T∞ ≤ lim inf α→∞ Tα, (3.6) and for any T ∈ (0, T∞),( ũα, ñα − ñαil, |α∇|−1∂t(ñ α − ñαil) ) → ( ũ,−P6=c(|ũ|2), 0 ) in C([0, T ];H1 × L2 0 × L2 0) (3.7) as α→∞, where ñαil ∈ C(R;L2 0(Tdλ;R))∩C1(R;H−1 0 (Tdλ;R)) is the solution of the linear wave equation: 1 α2 ∂2 t ñ α il −∆ñαil = 0, t ∈ R, x ∈ Tdλ,( ñαil, ∂tñ α il )∣∣ t=0 = ( ñα0 + P6=c(|uα0 |2), ñα1 ) . For the solution of (3.1) in CTH1,0 0 , with the property Pc∂tñ α(t) ≡ 0, the mass and the energy M(ũα(t)) := ‖ũα(t)‖2L2 , Eα(ũα(t), ñα(t)) := ‖∇ũα(t)‖2L2 + 1 2 ‖ñα(t)‖2L2 + 1 2 ∥∥|α∇|−1∂tñ α(t) ∥∥2 L2 + ∫ Tdλ ñα(t)|ũα(t)|2 are well-defined and formally conserved. The solution of (3.5) (as well as that of the standard NLS (1.3)) in the energy class ũ ∈ H1 also (formally) conserves the mass M(ũ(t)) and the energy E(ũ(t)) := ‖∇ũ(t)‖2L2 − 1 2 ‖ũ(t)‖4L4 . It is worth noticing that the energy functionals for (3.1) and (3.5) satisfy the relation Eα(ũα, ñα) = E(ũα) + 1 2 ∥∥ñα + |ũα|2 − i|α∇|−1∂tñ α ∥∥2 L2 . EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 13 We recall the result on local well-posedness of these Cauchy problems in the energy space including (rigorous) conservation laws, which is a crucial tool to prove Proposition 3.1. Lemma 3.2 (Local well-posedness; [18, 9, 2, 4]). Let d = 1, 2 for (3.1) and d = 1, 2, 3 for (3.5), λ ∈ (0,∞)d be arbitrary. Then, the initial value problems for (3.1) (with any α > 0) and (3.5) on Tdλ are locally well-posed in the energy space H = H1,0 0 (Tdλ) and H1(Tdλ), respectively. In particular, for any initial data in H, there exists a local-in-time solution in C([0, T ];H), with existence time T > 0 depending only on the size of the initial data in H (and also on α in the case of (3.1)), which depends continuously on the initial data. Moreover, the mass and the energy are conserved for these solutions. These conservation laws can be deduced from the local well-posedness result in the energy space by a standard approximation argument based on persistence of regularity and continuous dependence of solutions upon initial data. Another important ingredient of the proof is the following lemma. Lemma 3.3 (Unconditional uniqueness; [7]). Let d = 1, 2, 3, λ ∈ (0,∞)d be ar- bitrary, and T > 0. For any u0 ∈ H1(Tdλ), there are at most one solution (in the sense of distributions) of (3.5) in L∞(0, T ;H1(Tdλ)) satisfying u(0) = u0. Any distributional solution u(t) in L∞(0, T ;H1) belongs to W 1,∞(0, T ;H−1) by the equation, and thus has limits in H−1 at endpoints t→ 0, T and is extended to a function in C([0, T ];H−1). The initial condition then makes sense in H−1. Remark 3.4. (i) The known results [2, 4, 7] on local well-posedness and uncondi- tional uniqueness in the energy space for the cubic NLS (1.3) on Tdλ are transformed into the same results for shifted NLS (3.5) and (1.5) by the following changes of the unknown function u(t, x) 7→ u(t, x) exp { i ∫ t 0 1 |Tdλ| ‖u(t′)‖2L2(Tdλ) dt ′ } for (3.5), u(t, x) 7→ u(t, x) exp { i ∫ t 0 ( 1 |Tdλ| ‖u(t′)‖2L2(Tdλ) + ν0 + ν1t ′ ) dt′ } for (1.5). As easily seen, these maps are homeomorphisms on L∞(0, T ;H1(Tdλ)) or on C([0, T ]; H1(Tdλ)) for any T > 0 and transform a solution (in the sense of distributions) of (3.5) and (1.5), respectively, to a solution of (1.3). (ii) In [7], uniqueness of solutions to (1.3) on Tdλ, d = 2, 3, was shown in the class of mild Hs-solutions (see [7, Definition 1.1]) for some s < 1. First, we see that any distributional solution in C([0, T ];Hs) turns out to be a mild Hs-solution if d = 2, 3 and s is close to 1; see [10, Remark 1.3] for details. Then, any distributional solution in L∞(0, T ;H1) belongs to W 1,∞(0, T ;H−1) ⊂ C([0, T ];H−1) by the equation and hence to C([0, T ];Hs) for any s < 1 by interpolation. Consequently, we can deduce uniqueness in L∞(0, T ;H1) from the result in [7]. In the one-dimensional case, uniqueness holds in C([0, T ];Hs) for s > 1/2 by the Sobolev inequality, which implies uniqueness in L∞(0, T ;H1) as above. (iii) To prove Proposition 3.1 we need uniqueness of the solution to (3.5) in L∞(0, T ;H1); in fact, uniqueness in C([0, T ];H1) is not sufficient. For the Za- kharov system (1.2), we have proved uniqueness in C([0, T ];Hs,l) as “unconditional uniqueness” in Theorem 1.2. Concerning the energy-space regularity, uniqueness 14 N. KISHIMOTO EJDE-2022/20 in a wider class L∞(0, T ;H1,0) follows from Theorem 1.2 in the case d = 1 and αλ 6∈ Z by the same argument as above, whereas it does not follow if αλ ∈ Z or in the two-dimensional case, since we do not have uniqueness in C([0, T ],Hs,l) with l < 0. Note, however, that uniqueness in L∞(0, T ;H1,0) for (1.2) will not be required in our proof of Proposition 3.1. Proof of Proposition 3.1. We follow closely the argument for the non-periodic case given in [13, Section 6]. We focus on the two-dimensional case; note that the one- dimensional case can be treated by the same argument with some modifications on exponents related to the Sobolev embedding. We proceed in several steps. Step 1. We shall show uniform-in-α a priori bound on the energy norm of (ũα, ñα): there exists T0 > 0 and C > 0 independent of α such that Xα,T0 := max 0≤t≤T0 ( ‖ũα(t)‖2H1 + 1 2 ‖ñα(t)‖2L2 + 1 2 ∥∥|α∇|−1∂tñ α(t) ∥∥2 L2 ) ≤ C. (3.8) In particular, by Lemma 3.2, it holds that Tα > T0 for any α. By the conservation laws and (3.2), (3.3), together with the Hölder inequality and the Sobolev embedding, the conserved quantities M(ũα(t)) and Eα(ũα(t), ñα(t)) are bounded uniformly in α as long as the solution exists. Since Xα,T = max 0≤t≤T ( M(ũα(t)) + Eα(ũα(t), ñα(t))− ∫ ñα(t)|ũα(t)|2 ) , it suffices to control the cubic term ∫ ñα|ũα|2. By the Hölder inequality, the Sobolev embedding, interpolation and the Duhamel formula, we see that, for t ∈ [0, T ],∣∣ ∫ ñα(t)|ũα(t)|2 ∣∣ . ‖ñα(t)‖L2 ( ‖eit∆uα0 ‖2H1/2 + ‖ũα(t)− eit∆uα0 ‖2H1/2 ) . X1/2 α,T ( ‖uα0 ‖2H1/2 + ‖ũα(t)− eit∆uα0 ‖ 4/3 H1 ‖ñαũα‖2/3L1(0,T ;H−1/2) ) , which is, by Sobolev and interpolation again as well as the mass conservation law, bounded by X 1/2 α,T ( ‖uα0 ‖2H1/2 + ( X 2/3 α,T + ‖uα0 ‖ 4/3 H1 ) T 2/3‖ñα‖2/3L∞(0,T ;L2)‖ũ α‖2/3 L∞(0,T ;H1/2) ) . X1/2 α,T ( ‖uα0 ‖2H1/2 + ( X 2/3 α,T + ‖uα0 ‖ 4/3 H1 ) T 2/3X 1/3 α,TX 1/6 α,T ‖u α 0 ‖ 1/3 L2 ) . ‖uα0 ‖2H1/2X 1/2 α,T + T 2/3‖uα0 ‖ 5/3 H1Xα,T + T 2/3‖uα0 ‖ 1/3 L2 X 5/3 α,T . Using (3.2) again, we have Xα,T ≤ C0(1 + T 2/3) + C1T 2/3X 5/3 α,T for some constants C0, C1 > 0 independent of α. Since Xα,T is continuous in T , a bootstrap argument shows Xα,T ≤ 2C0 if T is sufficiently small depending on C0, C1, which yields (3.8). Step 2. Let T0 be as in Step 1. We shall show that for any sequence αk → ∞ there exist a subsequence αkl and ũ∞ ∈ L∞(0, T0;H1) ∩C([0, T0];H1/2) such that ũαkl → ũ∞ in C([0, T0]; w-H1 ∩H1/2), ñαkl + |ũαkl |2 ⇀ Pc(|ũ∞|2) weakly in L2((0, T0)× T2 λ). Here, convergence in C([0, T0]; w-H1) means that sup 0≤t≤T0 ∣∣〈ũαkl (t)− ũ∞(t), ψ(t) 〉 H1 ∣∣→ 0, ψ ∈ C([0, T0];H1). EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 15 In particular, by the Sobolev embedding, ũαkl → ũ∞ strongly in C([0, T0];L4). Let us first establish the convergence of ũα. By Step 1, {(ũα, ñα)}α is bounded in C([0, T0];H1×L2), so that {∂tũα = i(∆ũα−ñαũα)}α is bounded in C([0, T0];H−1). This implies that {ũα}α is equicontinuous in H−1 at any t ∈ [0, T0], and thus in Hs for any s < 1 by interpolation. Since {ũα(t)}α is relatively compact in Hs for s < 1 by the compact embedding H1 ↪→ Hs, Ascoli’s theorem (cf. [12, Chapter III, Theorem 3.1]) shows that {ũα}α is relatively compact in C([0, T0];Hs) for s < 1. The case of s = 1 2 implies, for any {αk}k, existence of a subsequence {ũαkl }l converging to some ũ∞ strongly in C([0, T0];H1/2). Moreover, since for each t ∈ [0, T0] (any subsequence of) the bounded sequence {ũαkl (t)}l ⊂ H1 has a weakly convergent subsequence, we see the sequence itself converges to ũ∞(t) weakly in H1. The weak lower semi-continuity of the norm and the bound from Step 1 then show that ũ∞ ∈ L∞(0, T0;H1). Finally, for any ψ ∈ C([0, T0];H1), we use strong convergence in C([0, T0];H1/2) and boundedness of ũ∞(t) in H1 obtained so far and notice lim R→∞ ‖P>Rψ‖L∞(0,T0;H1) = 0 to have lim sup l→∞ sup 0≤t≤T0 ∣∣〈ũαkl (t)− ũ∞(t), ψ(t) 〉 H1 ∣∣ ≤ lim l→∞ ‖ũαkl − ũ∞‖L∞(0,T0;H1/2)‖P≤Rψ‖L∞(0,T0;H3/2) + ( sup l ‖ũαkl ‖L∞(0,T0;H1) + ‖ũ∞‖L∞(0,T0;H1) ) ‖P>Rψ‖L∞(0,T0;H1) → 0 (as R→∞), which shows convergence in C([0, T0]; w-H1). Next, we obtain weak convergence of ñα + |ũα|2. We see that ∆(ñαkl + |ũαkl |2) = α−2 kl ∂2 t ñ αkl → 0 in D′((0, T0)×T2 λ) by the uniform bound on ñα from Step 1. This particularly im- plies that ñαkl +P6=c(|ũαkl |2)→ 0 in D′((0, T0)×T2 λ). Moreover, strong convergence of {ũαkl }l obtained above shows Pc(|ũαkl |2) → Pc(|ũ∞|2) in C([0, T0]). Conse- quently, we have ñαkl + |ũαkl |2 → Pc(|ũ∞|2) in D′((0, T0)×T2 λ). On the other hand, (any subsequence of) {ñαkl + |ũαkl |2}l is bounded in L2((0, T0)×T2 λ) and therefore has a weakly convergent subsequence. Hence, the sequence {ñαkl + |ũαkl |2}l itself converges to Pc(|ũ∞|2) weakly in L2((0, T0)× T2 λ). In the non-periodic case [13], ∆(nαkl + |uαkl |2)→ 0 in D′((0, T0)×Rd) and weak convergence of a subsequence in L2((0, T0) × Rd) imply that nαkl + |uαkl |2 ⇀ 0 weakly in L2((0, T0)×Rd). That is why uαkl converges to a solution of the standard NLS (1.3). Step 3. We shall show that T∞ > T0 and ũα → ũ in C([0, T0]; w-H1 ∩ H1/2) as α→∞. We first prove that ũ∞ given in Step 2 is a solution of (3.5) on (0, T0)×T2 λ in the sense of distributions. The initial condition is easily verified from strong convergence in Step 2 and (3.2), so it suffices to show that ñαkl ũαkl → −P6=c(|ũ∞|2)ũ∞ in D′((0, T0)× T2 λ). For any ψ ∈ C∞0 ((0, T0)× T2 λ), we see that∣∣∣ ∫ T0 0 ∫ T2 λ ( ñαkl ũαkl + P6=c(|ũ∞|2)ũ∞ ) ψ dx dt ∣∣∣ 16 N. KISHIMOTO EJDE-2022/20 ≤ ∣∣∣ ∫ T0 0 ∫ T2 λ ñαkl ( ũαkl − ũ∞ ) ψ dx dt ∣∣∣ + ∣∣∣ ∫ T0 0 ∫ T2 λ ( ñαkl + |ũαkl |2 − Pc(|ũ∞|2) ) ũ∞ψ dx dt ∣∣∣ + ∣∣∣ ∫ T0 0 ∫ T2 λ ( |ũ∞|2 − |ũαkl |2 ) ũ∞ψ dx dt ∣∣∣ ≤ ‖ñαkl ‖L∞(0,T0;L2)‖ũαkl − ũ∞‖L∞(0,T0;L2)‖ψ‖L1(0,T0;L∞) + ∣∣∣〈ñαkl + |ũαkl |2 − Pc(|ũ∞|2), ũ∞ψ 〉 L2((0,T0)×T2 λ) ∣∣∣ + ‖ũ∞ − ũαkl ‖L∞(0,T0;L4) ( ‖ũ∞‖L∞(0,T0;L4) + ‖ũαkl ‖L∞(0,T0;L4) ) × ‖ũ∞‖L∞(0,T0;L2)‖ψ‖L1(0,T0;L∞). By the uniform bound given in Step 1 and the convergence results proved in Step 2, the right-hand side vanishes as l→∞. Hence, ũ∞ satisfies (3.5). Now, we invoke Lemma 3.3 to conclude that ũ∞ = ũ ∈ C([0, T0];H1). In particular, ũαkl → ũ in C([0, T0]; w-H1 ∩ H1/2) as l → ∞. This is true for any sequence αk →∞, so that {ũα}α itself converges to ũ as α→∞. Step 4. We shall show (3.7) with T = T0. Let Nα := ñα − i|α∇|−1∂tñ α and Nα il := ñαil − i|α∇|−1∂tñ α il. Note that PcN α(t) = PcN α il (t) ≡ 0. Nα and Nα il solve the following inhomogeneous and homogeneous linear Cauchy problems: ∂tN α = i|α∇|Nα + i|α∇|(|ũα|2), Nα ∣∣ t=0 = ñα0 − i|α∇|−1ñα1 , and ∂tN α il = i|α∇|Nα il , Nα il ∣∣ t=0 = ñα0 − i|α∇|−1ñα1 + P6=c(|uα0 |2). In particular, we have ‖Nα il (t)‖L2 ≡ ‖Nα il (0)‖L2 . To prove the claim, it suffices to show that sup 0≤t≤T0 ( ‖∇ ( ũα(t)− ũ(t) ) ‖2L2 + 1 2 ∥∥Nα(t)−Nα il (t) + P6=c ( |ũ(t)|2 )∥∥2 L2 ) → 0 as α→∞. By a direct calculation, we have ‖∇(ũα − ũ)‖2L2 + 1 2 ∥∥Nα −Nα il + P6=c(|ũ|2) ∥∥2 L2 = Eα(ũα, ñα)− E(ũ)− 1 2 ‖Nα il‖2L2 − 1 2 ‖Pc(|ũ|2)‖2L2 (3.9) + < 〈 Nα, |ũ|2 − |ũα|2 〉 L2 + 2< 〈 ∇(ũ− ũα),∇ũ 〉 L2 (3.10) −< 〈 Nα −Nα il + P6=c(|ũ|2), Nα il 〉 L2 . (3.11) The first line (3.9) consists of conserved quantities, and hence for any t, (3.9) = E(uα0 ) + 1 2 ‖Nα(0) + |uα0 |2‖2L2 − E(u∞0 )− 1 2 ‖Nα il (0)‖2L2 − 1 2 ‖Pc(|u∞0 |2)‖2L2 = ( E(uα0 )− E(u∞0 ) ) + 1 2 ( ‖Pc(|uα0 |2)‖2L2 − ‖Pc(|u∞0 |2)‖2L2 ) , EJDE-2022/20 REMARKS ON PERIODIC ZAKHAROV SYSTEMS 17 which vanishes as α → ∞ by (3.2). The second line (3.10) vanishes uniformly in t by the uniform-in-α bound from Step 1 and the convergence result from Step 3. Therefore, we only have to show that the last line (3.11) vanishes uniformly in t. By (3.4) and the Sobolev inequality ‖P>R(|uα0 |2)‖L2 . R−1/2 ∥∥|uα0 |2∥∥H1/2 . R −1/2‖uα0 ‖2H1 , we see lim supα→∞ ‖P>RNα il (t)‖L2 = lim supα→∞ ‖P>RNα il (0)‖L2 → 0 as R → ∞. Hence, the uniform-in-α bound from Step 1 implies that for any ε there exist R > 0 and α0 > 0 such that for any α ≥ α0 sup 0≤t≤T0 ∣∣∣〈Nα(t)−Nα il (t) + P6=c(|ũ(t)|2), P>RN α il (t) 〉 L2 ∣∣∣ < ε. We fix such an R > 0 and estimate the low-frequency part. Noticing sup 0≤t≤T0 ∣∣∣〈Nα(t)−Nα il (t) + P6=c(|ũ(t)|2), P≤RN α il (t) 〉 L2 ∣∣∣ . ‖Nα −Nα il + P6=c(|ũ|2)‖L∞(0,T0;H−5/2)R 5 2 ‖Nα il (0)‖L2 , we shall estimate the H−5/2 norm of Nα −Nα il + P 6=c(|ũ|2). By the Duhamel formula and an integration by parts in t, we have Nα(t)−Nα il (t) + P 6=c(|ũ(t)|2) = P6=c(|ũ(t)|2)− eit|α∇|P 6=c(|uα0 |2)− ∫ t 0 ei(t−s)|α∇|(−i)|α∇|(|ũα(s)|2) ds = P 6=c(|ũ(t)|2)− eit|α∇|P 6=c(|uα0 |2)− |ũα(t)|2 + eit|α∇|(|uα0 |2) + ∫ t 0 ei(t−s)|α∇|∂s(|ũα(s)|2) ds = P 6=c(|ũ(t)|2)− P6=c(|ũα(t)|2) + ∫ t 0 ei(t−s)|α∇|∂s(|ũα(s)|2) ds, where we have used the L2 conservation for ũα at the last equality. The Sobolev embedding gives a bound for the first two terms as∥∥|ũ|2 − |ũα|2∥∥ L∞(0,T0;H−5/2) . ( ‖ũ‖L∞(0,T0;L2) + ‖ũα‖L∞(0,T0;L2) ) ‖ũ− ũα‖L∞(0,T0;L2). On the other hand, by the equation for ũα we have ∂t(|ũα|2) = 2∇ · < ( iũα∇ũα ) . We shall apply integration by parts once more to deal with this term. In the non-periodic case [13], the integral term was dealt with by the Strichartz estimate for the reduced wave equation, which yields some negative power of α. Although the same argument may be valid in the periodic case as well, we take a different approach here. Since (in the two-dimensional case) the high-frequency components will be dif- ficult to control after integration by parts, we first remove them and then perform integration by parts, as follows. For t ∈ [0, T0] and R̃ > 0, we use the 2D Sobolev estimate ‖fg‖H−3/2 . ‖f‖H1/2‖g‖H−1/2 18 N. KISHIMOTO EJDE-2022/20 to have∥∥∥2∇ · ∫ t 0 ei(t−s)|α∇|< [ iũα(s)∇ũα(s)− iP≤R̃ũα(s)∇P≤R̃ũ α(s) ] ds ∥∥∥ L∞(0,T0;H−5/2) ≤ 2T0 ∥∥ũα∇ũα − P≤R̃ũα∇P≤R̃ũα∥∥L∞(0,T0;H−3/2) . ‖ũα‖L∞(0,T0;H1/2)‖P>R̃ũ α‖L∞(0,T0;H1/2) . R̃−1/2‖ũα‖2L∞(0,T0;H1). On the other hand, using the equation for ũα again we have ∂t ( iP≤R̃ũ α∇P≤R̃ũ α ) = ∆P≤R̃ũ α∇P≤R̃ũ α − P≤R̃ũα∇∆P≤R̃ũ α − P≤R̃ ( ñαũα ) ∇P≤R̃ũ α + P≤R̃ũ α∇P≤R̃ ( ñαũα ) , so the Sobolev inequality yields∥∥∂t(iP≤R̃ũα∇P≤R̃ũα)∥∥H−5/2 . R̃ 3‖ũα‖2L2 + R̃‖ñα‖L2‖ũα‖2H1/2 . Then, integration by parts implies∥∥∥2∇ · ∫ t 0 ei(t−s)|α∇|< [ iP≤R̃ũ α(s)∇P≤R̃ũ α(s) ] ds ∥∥∥ L∞(0,T0;H−5/2) ≤ 2 ∥∥∥|α∇|−1∇ · ( < [ iP≤R̃ũ α(t)∇P≤R̃ũ α(t) ] − eit|α∇|< [ iP≤R̃u α 0∇P≤R̃u α 0 ])∥∥∥ L∞(0,T0;H−5/2) + 2 ∥∥∥|α∇|−1∇ · ∫ t 0 ei(t−s)|α∇|<∂s [ iP≤R̃ũ α(s)∇P≤R̃ũ α(s) ] ds ∥∥∥ L∞(0,T0;H−5/2) . α−1 ( R̃‖ũα‖2L∞(0,T0;L2) + T0R̃ 3‖ũα‖2L∞(0,T0;L2) + T0R̃‖ñα‖L∞(0,T0;L2)‖ũα‖2L∞(0,T0;H1/2) ) . Using the above estimates and the uniform-in-α bound from Step 1, we obtain ‖Nα −Nα il + P6=c(|ũ|2)‖L∞(0,T0;H−5/2) . ‖ũα − ũ‖L∞(0,T0;L2) + R̃−1/2 + R̃3α−1 for any R̃ > 1, with the implicit constant independent of R̃, α. We set R̃ largely enough depending on ε > 0 and R > 0 fixed above, and recall strong convergence of ũα shown in Step 3, to verify sup 0≤t≤T0 ∣∣∣〈Nα(t)−Nα il (t) + P6=c(|ũ(t)|2), Nα il (t) 〉 L2 ∣∣∣ ≤ 2ε for all sufficiently large α, as desired. Step 5. We shall show (3.6) and (3.7) for any T ∈ (0, T∞), concluding the proof. This follows once we can show the following. Let T ∈ [T0,min{T∞, lim inf Tα}) be such that (3.7) holds on the time interval [0, T ]. Then, there exists T1 = T1(‖ũ(T )‖H1) > 0 such that min{T∞, lim inf Tα} > T + T1 and (3.7) holds on [0, T + T1]. Note that the hypothesis is true for T = T0 by the previous steps. If (3.7) holds for some T ∈ [T0,min{T∞, lim inf Tα}), then Tα > T for suffi- ciently large α and ũα(T ) → ũ(T ) in H1. 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