A remark on norm inflation for nonlinear Schr\"odinger equations

We consider semilinear Schr\"odinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.


Introduction
We consider the initial value problem for semilinear Schrödinger equations: where the spatial domain Z is of the form Z = R d 1 × T d 2 , d 1 + d 2 = d, and F (u,ū) is a polynomial in u,ū without constant and linear terms, explicitly given by F (u,ū) = n j=1 ν j u q jū p j −q j with mutually different indices (p 1 , q 1 ), . . . , (p n , q n ) satisfying p j ≥ 2, 0 ≤ q j ≤ p j and non-zero complex constants ν 1 , . . . , ν n . The aim of this article is to prove norm inflation for the initial value problem (1.1) in some negative Sobolev spaces. We say norm inflation in H s (Z) ("NI s " for short) occurs if for any δ > 0 there exist φ ∈ H ∞ and T > 0 satisfying φ H s < δ, 0 < T < δ such that the corresponding smooth solution u to (1.1) exists on [0, T ] and Clearly, NI s implies the discontinuity of the solution map φ → u (which is uniquely defined for smooth φ locally in time) at the origin in the H s topology, and hence the ill-posedness of (1.1) in H s . However, NI s is a stronger instability property of the flow than the discontinuity, which only requires 0 < T 1 and u(T ) H s 1. Let us begin with the case of single-term nonlinearity: i∂ t u + ∆u = νu qūp−q , (t, x) ∈ [0, T ] × Z, u(0, x) = φ(x), (1.2) where p ≥ 2 and 0 ≤ q ≤ p are integers, ν ∈ C \ {0} is a constant. The equation is invariant under the scaling transformation u(t, x) → λ 2 p−1 u(λ 2 t, λx) (λ > 0), and the critical Sobolev index s for which λ 2 p−1 φ(λ·) Ḣs = φ Ḣs is given by s = s c (d, p) := d 2 − 2 p−1 . The scaling heuristics suggests that the flow becomes unstable in H s for s < s c (d, p). In addition, we will demonstrate norm inflation phenomena by tracking the transfer of energy from high to low frequencies (that is called "high-to-low frequency cascade"), which naturally restrict us to negative Sobolev spaces. In fact, we will show NI s with any s < min{s c (d, p), 0} for any Z and (p, q), as well as with some negative but scale-subcritical regularities for specific nonlinearities. Precisely, our result reads as follows: Theorem 1.1. Let Z be a spatial domain of the form R d 1 ×T d 2 with d 1 +d 2 = d ≥ 1, and let p ≥ 2, 0 ≤ q ≤ p be integers. Then, the initial value problem (1.2) exhibits NI s in the following cases: (i) Z and (p, q) are arbitrary, s < min{s c (d, p), 0}.
There is an extensive literature on the ill-posedness of nonlinear Schrödinger equations, and a part of the above theorem has been proved in previous works.
Concerning ill-posedness in the sense of norm inflation, Christ, Colliander, and Tao [10] treated the case of gauge-invariant power-type nonlinearities ±|u| p−1 u on R d and proved NI s when 0 < s < s c (d, p) or s ≤ − d 2 (with some additional restriction on s if p is not an odd integer). For the remaining range of regularities − d 2 < s < 0 (when s c ≥ 0) they proved the failure of uniform continuity of the solution map. Note that this milder form of ill-posedness is not necessarily incompatible with wellposedness in the sense of Hadamard, for which continuity of the solution map is required. Moreover, since their argument is based on scaling consideration and some ODE analysis, it does not apply in any obvious way to the cases of periodic domains, 1 non gauge-invariant nonlinearities, and complex coefficients. Later, Carles, Dumas, and Sparber [6] and Carles and Kappeler [7] studied norm inflation in Sobolev spaces of negative indices for the problem with smooth nonlinearities (i.e., ±|u| p−1 u with an odd integer p ≥ 3) in R d and in T d , respectively. They used a geometric optics approach to obtain NI s for d ≥ 2 and s < − 1 p in the R d case 2 and for s < 0 in the T d case with the exception of (d, p) = (1, 3) for which s < − 2 3 was assumed. (See [5,1] for related ill-posedness results.) In fact, they showed stronger instability property than NI s for these cases; that is, norm inflation with infinite loss of regularity (see 1 One can still adapt their idea to the periodic setting with additional care. Moreover, although their original argument did not apply to the 1d cubic case with the scaling critical regularity s = − 1 2 , one can modify the argument to cover that case. See [35] for details. 2 In [6] they also proved norm inflation for generalized nonlinear Schrödinger equations and the Davey-Stewartson system including non-elliptic Laplacian.
where for R d with d = 1, 2, 3 they proved norm inflation in Besov spaces B −1/4 2,σ of regularity − 1 4 with 4 < σ ≤ ∞. 4 It turns out that the method of Iwabuchi and Ogawa [20] proving norm inflation has a wide applicability. The purpose of the present article is to apply this method to NLS with general nonlinearities. In the last few years the method has been used to a wide range of equations; see for instance [30,31,19,8,37]. 5 In [33,37], norm inflation based at general initial data was proved for NLS and some other equations. 6 We make some additional remarks on Theorem 1.1.
Remark 1.2. (i) Concerning one-dimensional periodic cubic NLS below L 2 , the renormalized (or Wick ordered) equation is known to behave better than the original one (1.2) with nonlinearity ±|u| 2 u; see [34] for a detailed discussion. We note that our proof can be also applied to the renormalized cubic NLS. In fact, the solutions constructed in Theorem 1.1 is smooth and its L 2 norm is conserved. Then, a suitable gauge transformation, which does not change the H s norm at any time, gives smooth solutions to the renormalized equation that exhibit norm inflation.
(ii) In the periodic setting, our proof does not rely on any number theoretic consideration. Hence, it can be easily adapted to the problem on general anisotropic tori, whether rational or irrational; that is, (iii) When Z = R and (p, q) = (4, 2), the example in [15,Example 5.3] suggests that a high-to-low frequency cascade leads to instability of the solution map when s < − 1 8 . However, our argument does not imply NI s for − 1 6 ≤ s < − 1 8 so far. There are far less results on ill-posedness for multi-term nonlinearities than for (1.2). However, such nonlinear terms naturally appear in application. For instance, the nonlinearity 6u 5 − 4u 3 appears in a model related to shape-memory alloys [13], and (u + 2ū + uū)u is relevant in the study of asymptotic behavior for the Gross-Pitaevskii equation (see e.g. [18]). Note that norm inflation for a multi-term nonlinearity does not immediately follow from that for each nonlinear term. Our next result concerns the equation (1.1) of full generality: 4 Essentially, they also proved NI s for s < − 1 4 , i.e., the case (iv) of our Theorem 1.1.

5
In the first version of this article, we only considered gauge-invariant smooth nonlinearities ν|u| 2k u, k ∈ Z >0 and linear combinations of them. Note, however, that the method of Iwabuchi and Ogawa [20] had been applied before only to quadratic nonlinearities and it was the first result dealing with nonlinearities of general degrees in a unified manner. The authors of [8,33] informed us that their proofs of norm inflation results followed the argument in the first version of this article. We also remark that an estimate proved in the first version (Lemma 3.6 below) was employed later in [31,19,37]. 6 In [37] non gauge-invariant nonlinearities were first treated in a general setting. In fact Theorem 1.1 follows as a corollary of [37, Proposition 2.5 and Corollary 2.10]. However, we decide to include the non gauge-invariant cases in the present version in order to state Theorem 1.3 (for multi-term nonlinearities) with more generality. Theorem 1.3. The initial value problem (1.1) exhibits NI s whenever s satisfies the condition in Theorem 1.1 for at least one term u q jū p j −q j in F (u,ū), except for the case where Z = T and F (u,ū) contains uū.
When Z = T and F (u,ū) contains uū, NI s occurs in the following cases: has a quintic or higher term, or one of u 3ū , u 2ū2 , uū 3 .
has a cubic term but no quartic or higher terms. (iv) s < 0 if F (u,ū) has no cubic or higher terms.
In the above theorem, the range of regularities is restricted when Z = T and F (u,ū) has uū; note that the nonlinear term uū by itself leads to NI s for s < 0 as shown in Theorem 1.1. This restriction seems unnatural and an artifact of our argument.
The rest of this article is organized as follows. In the next section, we recall the idea of [2], [20] and discuss some common features and differences between them. Section 3 is devoted to the proof of Theorem 1.1 for the single-term nonlinearities. Then, in Section 4 we see how to treat the multi-term nonlinearities, proving Theorem 1.3. In Appendices, we consider norm inflation with infinite loss of regularity in Section A and inflation of various norms with the critical regularity for the one-dimensional cubic problem in Section B.

Strategy for proof
We will use the power series expansion of the solutions to prove norm inflation. To see the idea, let us consider the simplest case of quadratic nonlinearity u 2 in (1.2). This amounts to considering the integral equation (2.1) We first recall the argument of Bejenaru and Tao [2]. By Picard's iteration, the power series formally gives a solution to (2.1). To justify this, we basically need the linear and bilinear estimates for the space of initial data D and some space S ⊂ C([0, T ]; D) in which we construct a solution. In fact, they showed (roughly speaking) the following: Assume that (2.2) holds with the Banach space D of initial data and some Banach space S. Then, (i) for any k ≥ 1 the operators U k : D → S are well-defined and satisfies U k [φ] S ≤ (C φ D ) k , and (ii) there exists ε 0 > 0 (depending on the constants in (2.2)) such that the solution map φ → u[φ] := ∞ k=1 U k [φ] is well-defined on B D (ε 0 ) := φ ∈ D φ D ≤ ε 0 and gives a solution to (2.1). Next, consider some coarser topologies on D and S induced by the norms D ′ and S ′ weaker than D and S , respectively. They claimed the following: Assume further that the solution map φ → u[φ] given above is continuous from (B D (ε 0 ), D ′ ) (i.e., B D (ε 0 ) equipped with the D ′ topology) to (S, S ′ ). Then, for each k the operator U k is continuous from (B D (ε 0 ), D ′ ) to (S, S ′ ). To show the continuity of U k in coarser topologies, by its homogeneity one can restrict to sufficiently small initial data. Then, by the estimates (2.2), contribution of higher order terms k ′ >k U k ′ [φ] can be made arbitrarily small compared to U k [φ]. Combining this fact with the hypothesis that k≥1 U k [φ] is continuous, one can show the claim by an induction argument on k. Now, this claim gives a way to prove ill-posedness in coarse topologies. Namely, one can show the discontinuity of the solution map φ → ∞ k=1 U k [φ] in coarse topologies by simply establishing the discontinuity of the (more explicit) map φ → U k [φ] for at least one k. 7 We notice that this proof of ill-posedness includes evaluating higher terms by using (2.2), that is, estimates (or well-posedness) in stronger topology.
Here, we observe two facts on this method. First, it cannot yield norm inflation in coarse topologies. This is because the image of the continuous solution map with domain B D (ε 0 ) is bounded in S, and hence it must be bounded in weaker norms.
Secondly, the 'well-posedness' estimates (2.2) in D, S and discontinuity of some U k in D ′ , S ′ would imply the discontinuity of U k in any 'intermediate' norms D ′′ , S ′′ satisfying In particular, if we work in Sobolev spaces: then ill-posedness in H s 1 as a consequence of the argument in [2] should actually yield ill-posedness in any H s , s 1 ≤ s < s 0 , while we have (2.2), i.e., well-posedness in H s 0 . Therefore, the regularity s 0 in which we invoke (2.2) must be automatically the 7 It is worth noticing that the continuity of U k from (B D (ε 0 ), D ′ ) to (S, S ′ ) does not imply its continuity from (D, D ′ ) to (S, S ′ ) in general, even though U k can be defined for all functions in D. By the k-linearity of U k , the latter continuity is equivalent to the boundedness: Hence, only disproving the boundedness of U k in coarse topologies (which may imply that the solution map is not k times differentiable) is not sufficient to conclude the discontinuity of the solution map. threshold regularity for well-/ill-posedness. This explains why the same argument cannot be applied to the two-dimensional quadratic NLS with nonlinearity u 2 . In fact, as mentioned in Introduction, (2.2) are obtained in D = H s when s > −1 (with a suitable S) but fails if s ≤ −1 (for any S continuously embedded into C([0, T ]; H s )), and hence well-posedness at the threshold regularity is not available in this case.
We next recall Iwabuchi and Ogawa's result [20], which settled the aforementioned two-dimensional case. Indeed, the argument in [20] is similar to that of [2] in that it exploits the power series expansion and shows that one term in the series exhibits instability and dominates all the other terms. Now, we notice that the existence time T > 0 is allowed to shrink for the purpose of establishing norm inflation, while in [2] it is fixed and uniform with respect to the initial data. The main difference of the argument in [20] from that of [2] is that they worked with the estimates like for the data space D, S T ⊂ C([0, T ]; D), and δ > 0, and consider the expansion up to different times T according to the initial data. In fact, this enables us to take a sequence of initial data which is unbounded in D (but converges to 0 in a weaker norm), and such a set of initial data actually yields unbounded sequence of solutions. Another feature of the argument in [20] is that higher-order terms were estimated directly in D ′ by using properties of specific initial data they chose; in [2] these terms were simply estimated in D by (2.2) that hold for general functions. 8 At a technical level, another novelty in [20] is the use of modulation space M 2,1 as D instead of Sobolev spaces. The bilinear estimate in (2.3) is then straightforward thanks to the algebra property of M 2,1 . Finally, we remark that the strategies of [2,20] work well in the case that the operator U k involves a significant high-to-low frequency cascade, as mentioned in [2]. However, the situation is different in the case of system of equations, as there are more than one regularity indices and one cannot simply order two pairs of regularity indices; see e.g. [30], where the argument of [20] was employed to derive norm inflation from nonlinear interactions of "high×low→high" type.

Proof of Theorem 1.1
Let us first consider the case of single-term nonlinearity and prove Theorem 1.1. The argument in this section basically follows that in [20]. Since the coefficient ν = 0 plays no role in our proof, we assume ν = 1 for simplicity. We write so that u qūp−q = µ p,q (u). 8 In fact, we do not need 'well-posedness in D', i.e., such estimates as (2.3) that hold for all functions in D and S. It is enough to estimate the terms U k [φ] just for particularly chosen initial data φ. In some problems this consideration becomes essential; see [37], Theorem 1.2 and its proof.
of a (unique) solution u to (1.2) will play a crucial role in the proof. To make sense of this representation, we use modulation spaces. The notion of modulation spaces was introduced by Feichtinger in the 1980s [14] and nowadays it has become one of the common tools in the study of nonlinear evolution PDEs; see e.g. the survey [38] and references therein.
We will only use the following properties of the space M A . The proof is elementary, and thus it is omitted.
Since the space M A is a Banach algebra and the linear propagator e it∆ is unitary in M A , we can easily show the following multilinear estimates.
for any t ≥ 0 and k ≥ 1.
Proof. Let {a k } ∞ k=1 be the sequence defined by As observed in [2,Eq. (16)], one can show inductively that a k ≤ C k for some C > 0. To be more precise, we state it as the following lemma. The p = 2 case can be found in [31,Lemma 4.2] with a detailed proof.
for some C 1 > 0. This is trivial if k = 1. Let k ≥ 2, and assume the above estimate for U 1 , U 2 , . . . , U k−1 . Using Lemma 3.4, we have The estimate for U k follows by setting C 1 to be C and that Ψ is a contraction on a ball in by an argument similar to Lemma 3.5. By letting K → ∞, we obtain Ψ φ [u] = u.
Remark 3.8. (i) In M A we have unconditional local well-posedness. In particular, the embedding (Lemma 3.4 (i)) shows that the unique solution with initial data in some high-regularity Sobolev space exists on a time interval [0, T ] and coincides with the solution constructed in Corollary 3.7.
(ii) In the following proof of Theorem 1.1 we will take initial data that are localized in frequency on several cubes of side length O(A) located in {|ξ| ≫ max(1, A)}. For such initial data the L 2 norm is comparable with the M A norm, but much smaller than the Sobolev norms of positive indices. In the L 2 -supercritical cases (i.e., s c (d, p) > 0), no reasonable well-posedness is expected in L 2 , while the use of higher Sobolev space would verify the power series expansion only on a smaller time interval. In this regard, the space M A is suitable for our purpose.
Let N, A be dyadic numbers to be specified so that N ≫ 1 and 0 < A ≪ N (1 ≤ A ≪ N when Z has a periodic direction). In the proof of norm inflation, we will use initial data φ of the following form: We derive Sobolev bounds of U k [φ](t) with φ satisfying the above condition.
Lemma 3.9. There exists C > 0 such that for any φ satisfying (3.2) and k ≥ 1, we have is determined by a spatial convolution of k copies ofφ orφ =φ(−·), it is easily seen that Since #S k ≤ 6 k , we have Lemma 3.10. Let φ satisfy (3.2). Assume that s < 0. Then, there exists C > 0 depending only on d, p, s such that the following holds.
. By Young's inequality, the above is bounded by where B D ⊂ R d is the ball centered at the origin with |D| = |B D |. This implies that ξ s L 2 (D) ≤ ξ s L 2 (B D ) . Moreover, it follows from Lemma 3.5 with M = CrN −s that Hence, we apply Lemma 3.9 to bound the above by which is the desired one.
We observe the following lower bounds on the H s norm of the first nonlinear term in the expansion of the solution.
). Then, for any 0 < T ≪ 1 we have Then, for any 0 < T ≪ 1 we have Proof. Note that where the sum is taken over the set which is non-empty for any (p, q). 9 Since |Φ| N 2 in the integral, for 0 < T ≪ N −2 we have and thus (ii) In this case we have and in the integral, for ξ = ξ 1 − ξ 2 ∈ Q N −1 , . 9 If p is even, we can choose η l to be N e d or −N e d so that q l=1 η l − p m=q+1 η m = 0. If p is odd, we choose η 1 = 2N e d and η 2 to be N e d or −N e d so that the output from these two frequencies is either N e d or −N e d . Then, the other η j can be chosen as for p even.
Hence, if 0 < T ≪ 1, we have for any s ∈ R.
Therefore, we have for any s ∈ R and T > 0.
We follow the argument in Case 1 again, but with Then, T ≪ N −2 , ρ ∼ (log N) − 1 2 ≪ 1 and Take φ as in Lemma 3.11 (iii) and choose r, T as r = (log N) −1 and T = N s , which implies From Lemmas 3.10 and 3.11, we have u(T ) H s ∼ U 2 [φ](T ) H s ∼ (log N) −2 N −s ≫ 1, and norm inflation occurs.
We use anisotropic modulation space M defined by the norm We have the product estimate in this space. Thus, we follow the proof of Lemma 3.5 to obtain for any k ≥ 1, which is used to justify the expansion of the solution in M up to time T such that ρ := rN − 1 2 −s T ≪ 1. Then, by the same argument as in the proofs of Lemmas 3.9 and 3.10, we see that In particular, U 2 [φ](T ) H s ∼ r ρN −s for 0 < T ≪ 1 by Lemma 3.11 (iii). Now, we take r = (log N) −1 ≪ 1, T = (log N) 3 N 2s+ 1 2 ≪ 1, so that ρ = (log N) 2 N s ≪ 1, r ρN −s = log N ≫ r. From the estimates above, we have u(T ) H s ∼ log N ≫ 1, which shows norm inflation.

Proof of Theorem 1.3
Here, we see how to use the estimates for single-term nonlinearities for the proof in the multi-term cases. We write p := max 1≤j≤n p j .
For the initial value problem (1.1), the k-th order term U k [φ] in the expansion of the solution is given by U 1 [φ] := e it∆ φ and The following lemmas are verified in the same manner as Lemmas 3.6, 3.5, and Corollary 3.7.

.1) exists and has the expansion
. The next lemma can be verified similarly to Lemma 3.10. ( We now begin to prove Theorem 1.3. Proof of Theorem 1.3. We divide the proof into two cases: (I) One of the terms of order p (highest order) is responsible for norm inflation, or (II) a lower order term determines the range of regularities for norm inflation. Note that (II) occurs only when Z = R, p = 3, F (u,ū) has the term uū and s ∈ (− 1 2 , − 1 4 ). (I): Rewrite the nonlinear terms as F (u,ū) = p q=0 ν p,q µ p,q (u) + (terms of order less than p).
We divide the series into four parts: Note that U low = 0 if p = 2.
The following lemma indicates how U low is dominated by U main , and how the contributions of the (p + 1) terms in U main can be 'separated'. Lemma 4.5. We have the following: (i) Let φ satisfy (3.2) and s < 0. Let 0 < T ≤ 1, and assume that ρ = rA (ii) Let q * ∈ {0, 1, . . . , p} be such that ν p,q * = 0. Then, for any T ≥ 0 there exists j ∈ {0, 1, . . . , p} such that Proof. (i) We notice that the nonlinear terms of highest order p have nothing to do with U low [φ]. Hence, we estimate by Lemma 4.4 (ii) with p replaced by p − 1 and have (ii) We observe that ζ p := e i π p+1 satisfies p j=0 ζ 2qj for an appropriate j. Hence, from Lemma 4.4 (i) and Lemma 4.5 (i), If we take the same choice for r, A, T as in Case 1 of the proof of Theorem 1.1; all the required conditions for norm inflation are satisfied when p = 2. Even for p ≥ 3, it suffices to check that This is equivalent to ρ p−2 ≫ T 1 p−2 − 1 p−1 , which we can easily show. Case 2-4-5-7: p = 2. We need to deal with the following situations: , which correspond to Cases 2, 4, 5, and 7 in the proof of Theorem 1.1, respectively. As seen in the preceding case, we do not have to care about U low and the proof is the same as the single-term cases, except that we need to pick up the appropriate one among u 2 , uū,ū 2 by using Lemma 4.5 (ii).
Case 3: d = 1, p = 3, s = − 1 2 . We take the initial data e i jπ 4 φ with φ as in (3.2) and parameters r, A, T as in Case 3 for Theorem 1.1. Following the argument in Case 1, it suffices to check the condition for U main H s ≫ U low H s ; Actually, we see that L.H.S. ∼ (log N) for 0 < T ≤ 1. For U 3 , observing that the Fourier support is in the region |ξ| ∼ N, we modify the estimate in Lemma 4.4 to obtain For U 2 the contribution from u 2 andū 2 has the Fourier support in high frequency, thus being dominated by the contribution from uū. By Lemma 3.11 (ii), we have if 0 < T ≪ 1. We set r = (log N) −1 and T = (log N) 3 N 2s+ 1 2 as before (Case 7 in the single-term case), then it holds that T ≪ 1, ρ = (log N) , which gives the claimed norm inflation. This concludes the proof of Theorem 1.3.

Appendix A. Norm inflation with infinite loss of regularity
In this section, we derive norm inflation with infinite loss of regularity for the problem with smooth gauge-invariant nonlinearities: where ν is a positive integer. The initial value problem (A.1) on R d is invariant under the scaling u(t, x) → λ This is also the key ingredient in the proof of the previous results [6,7], and hence the restriction on the range of s in Proposition A.1 is the same as that in [6,7].
A complete characterization of the resonant set (for k ∈ Z d given) is easily obtained in the ν = 1 case; see [7, Proposition 4.1] for instance. In Proposition A.3 below, we will provide a complete characterization of the set R 1,2 (0), which may be of interest in itself. Since (k m ) 5 m=1 ∈ R 1,2 (k) if and only if (k m − k) 5 m=1 ∈ R 1,2 (0), we have a characterization of R 1,2 (k) for any k ∈ Z as well. However, in the proof of Proposition A.1 we only need the fact that R d,ν (0) has an element consisting of non-zero vectors in Z d , except for (d, ν) = (1, 1).
Proof of Proposition A.1. We follow the proof of Theorem 1.1 but take different initial data to show infinite loss of regularity.
Let N ≫ 1 be a large positive integer and define φ ∈ H ∞ (Z) by where r = r(N) > 0 is a constant to be chosen later, For the first nonlinear term U 2ν+1 [φ], we observe that Now, we restrict ξ to the low-frequency region Q 1/2 . If d = ν = 1, then we have and Φ = O(N 2 ) in the integral. If d ≥ 2 and ν = 1, we have T 0 e itΦ dt, and the resonant property implies that in the integral. Therefore, in these cases we have the following lower bound: The quintic and higher cases are slightly different. On one hand, there are "almost resonant" interactions such as j=1,3 for which it holds in the integral. On the other hand, some non-resonant interactions such as also create low-frequency modes, with |Φ| ∼ N 2 in the integral. Hence, if we choose T > 0 as otherwise.
We see that, under the assumption on s, φ H s ∼ r ≪ 1, T ≪ 1, ρ ≪ 1, and We conclude the proof by letting N → ∞.
At the end of this section, we give a characterization of resonant interactions creating the zero mode in the one-dimensional quintic case.

Appendix B. Norm inflation for 1D cubic NLS at the critical regularity
In this section, we consider the particular equation (B.1) We will show the inflation of the Besov-type scale-critical Sobolev and Fourier-Lebesgue norms with an additional logarithmic factor: p,q -norm by p,q .
We also define the D s p,q -norm for s ∈ R by .
Remark B.2. (i) We see that D p,q is scale invariant for any p, q.
(iii) We will not consider the space D [α] p,q with p = ∞ here, since our argument seems valid only in the space of negative regularity. Proposition B.3. For the Cauchy problem (B.1), norm inflation occurs in the following cases: p,q for any 1 ≤ q ≤ ∞ and α < 1 p,q and D s p,q for any 1 ≤ q ≤ ∞, α ∈ R and s < − 2 3 , if 1 ≤ p < 3 2 . Remark B.4. (i) If 3 2 ≤ p < ∞ and 1 ≤ q < ∞, Proposition B.3 shows inflation of a "logarithmically subcritical" norm (i.e., D [α] p,q with α > 0). Moreover, if 1 ≤ p < 3 2 we show norm inflation in D s p,q for subcritical regularities − 2 3 > s > − 1 p . However, for q = ∞ and p ≥ 3 2 , inflation is not detected even in the critical norm D 2,2 and D [2] 2,∞ . Recently, Oh and Wang [36] proved global-in-time bound in F L 0,p for Z = T and 2 ≤ p < ∞. There are still some gaps between these results and ours. In fact, Proposition B.3 shows inflation of D 2,∞ norms, as well as in a norm only logarithmically stronger than F L − 1 p ,p for p ≥ 2. (iii) Guo [16] also studied (B.1) on R in "almost critical" spaces. It would be interesting to compare our result with [16,Theorem 1.8], where he showed wellposedness (and hence a priori bound) in some Orlicz-type generalized modulation spaces which are barely smaller than the critical one M 2,∞ . There is no conflict between these results, because the function spaces for which norm inflation is claimed in Proposition B.3 are not included in M 2,∞ due to negative regularity. Note also that the function spaces in [16, Theorem 1.8] admit the initial data φ of the form φ(ξ) = [log(2 + |ξ|)] −γ only for γ > 2 (see [16,Remark 1.9]), while it belongs to D [α] p,q if γ > α + 1 q . (iv) In contrast to the results in [24,36], complete integrability of the equation will play no role in our argument. In particular, Proposition B.3 still holds if we replace the nonlinearity in (B.1) with any of the other cubic terms u 3 ,ū 3 , uū 2 or any linear combination of them with complex coefficients.
Proof of Proposition B.3. We follow the argument in Section 3. For 1 ≤ ρ < ∞ and A > 0, let M ρ A be the rescaled modulation space defined by the norm It is easy to see that M ρ A is a Banach algebra with a product estimate: Mimicking the proof of Lemma 3.5, we see that the operators U k defined as in Definition 3.1 satisfy We also recall that from Corollary 3. p,q , we restrict the initial data φ to those of the form (3.2); for given N ≫ 1, we set where r > 0 and 1 ≪ A ≪ N will be specified later according to N. Then, since φ M 2 A ∼ rA Moreover, it holds that p,q ∼ r(log N) α , T ≥ 0, (B.5) and similarly to Lemma 3.11 (i), that For estimating U 2l+1 [φ], l ≥ 2 in D [α] p,q , we first observe that p,q U k [φ](T ) L ∞ . A simple computation yields that p,q for any measurable set Ω ⊂ R of finite measure. From Lemma 3.9, we have supp U k [φ](T ) ≤ C k A, T ≥ 0, k ≥ 1, and hence, p,q ≤ C k f α p,q (A).
Moreover, similarly to Lemma 3.10 (ii), we use Young's inequality, (B.2) and Lemma 3.6 to obtain In particular, when α > − 1 q this condition requires which shows the necessity of the restriction α < 1 2q in our argument. We now see the possibility of choosing r, A, T with the condition (B.8) in the following two cases separately: (a) If 1 ≤ q < ∞ and 0 ≤ α < 1 2q , we may take for instance r = (log N) −α (log log N) −1 , A = N(log log N) −1 , T = 1 100 N −2 .
Finally, we assume 1 ≤ p < 3 2 and prove norm inflation in D s p,q for s < − 2 3 . We use the initial data φ of the form Repeating the argument above we also verify that