Norm Inflation for Nonlinear Schrödinger Equations in Fourier–Lebesgue and Modulation Spaces of Negative Regularity

Abstract

We consider nonlinear Schrödinger equations in Fourier–Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schrödinger equations in a general abstract functional setting.

1 Introduction

1.1 General Setting

We consider the nonlinear Schrödinger (NLS) equations of the form

$$\begin{aligned} i\partial _t \psi + \frac{1}{2} \Delta \psi = \mu |\psi |^{2\sigma } \psi , \quad x\in {\mathbb {R}}^d;\quad \psi (0, x)=\psi _0(x), \end{aligned}$$

(1.1)

where $\psi =\psi (t,x)\in \mathbb C$, $\sigma \in \mathbb N$, $\mu \in \{ 1, -1\}.$ We prove some ill-posedness results in Fourier–Lebesgue and modulation spaces, involving negative regularity in space. We recall the notion of well-posedness in the sense of Hadamard.

Definition 1.1

Let $X, Y \hookrightarrow \mathcal {S}'({\mathbb {R}}^d)$ be a Banach spaces. The Cauchy problem for (1.1) is well posed from X to Y if, for all bounded subsets $B\subset X$, there exist $T>0$ and a Banach space $X_{T} \hookrightarrow C([0,T], Y)$ such that:

• (i) For all $\phi \in X$, (1.1) has a unique solution $\psi \in X_T$ with $\psi _{|t=0}=\phi$.

• (ii) The mapping $\phi \mapsto \psi$ is continuous from $(B, \Vert \cdot \Vert _{X})$ to C([0, T], Y).

The negation of the above definition is called a lack of well-posedness or instability. In connection with the study of ill-posedness of (1.1) and nonlinear wave equations Christ, Colliander, and Tao introduced in [14] the notion of norm inflation with respect to a given (Sobolev) norm, saying that there exist a sequence of smooth initial data $(\psi _n(0))_{n\ge 1}$ and a sequence of times $(t_n)_{n\ge 1}$, both converging to 0, so that the corresponding smooth solution $\psi _n$, evaluated at $t_n$, is unbounded (in the same space).

The solutions to (1.1) is invariant under the scaling transformation

$$\begin{aligned} \psi (t,x) \mapsto \lambda ^{1/\sigma } \psi \left( \lambda ^2 t, \lambda x\right) , \quad \lambda >0. \end{aligned}$$

(1.2)

The homogeneous Sobolev space $\dot{H}^{s}({\mathbb {R}}^d)$ is invariant exactly for $s=s_c$, where

$$\begin{aligned} s_c= \frac{d}{2} -\frac{1}{\sigma }. \end{aligned}$$

Another important invariance of (1.1) is the Galilean invariance: if $\psi (t,x)$ solves (1.1), then so does

$$\begin{aligned} e^{iv\cdot x -i |v|^2 t/2}\psi (t,x-vt) \end{aligned}$$

(1.3)

for any $v \in {\mathbb {R}}^d$. This transform does not alter the $L^2({\mathbb {R}}^d)$ norm of the function. From these two invariances, well-posedness is not expected to hold in $H^s({\mathbb {R}}^d)$ as soon as $s<\max (0,s_c)$. In this paper, we consider the case of negative regularity, $s<0$, in Fourier–Lebesgue and modulation spaces, instead of Sobolev spaces.

Kenig, Ponce and Vega [26] established instability for the cubic NLS in $H^{s}({\mathbb {R}})$ for $s<0$. Christ, Colliander and Tao [14] generalized this result in $H^{s}({\mathbb {R}}^d)$ for $s<0$ and $d\ge 1$. In the periodic case $x\in {\mathbb {T}}^d$, instability in $H^{s}({\mathbb {T}}^d)$ for $s<0$ was established in [15] ($d=1$) and [10] ($d\ge 1$). Stronger results for the cubic NLS on the circle were proven by Molinet [29]. In [31, Theorem 1.1], Oh established norm-inflation for (1.1) in the cubic case $\sigma =1$, in $H^{s}({\mathbb {R}})$ for $s\le -1/2$ and in $H^s({\mathbb {R}}^d)$ for $s<0$ if $d\ge 2.$ He actually proved that the flow map fails to be continuous at any function in $H^s$, for s as above. Norm inflation in the case of mixed geometries, $x\in {\mathbb {R}}^d\times {\mathbb {T}}^n$, for sharp negative Sobolev regularity in (1.1), is due to Kishimoto [27], who also considers nonlinearities which are not gauge invariant.

The general picture to prove ill-posedness results is typically as following, as explained in e.g. [14]: at negative regularity, one relies on a transfer from high frequencies to low frequencies, while to prove ill-posedness at positive regularity, one uses a transfer from low frequencies to high frequencies. In particular, the proofs are different whether a negative or a positive regularity is considered.

Stronger phenomena than norm inflation have also been proved, showing that the flow map fails to be continuous at the origin from $H^s$ to $H^k$ even for (some) $k<s$, and so a loss of regularity is present. This was proven initially for $0<s<s_c$ by Lebeau [28] in the case of the wave equation, then in [8] (cubic nonlinearity) and [1, 36] for NLS. In the case of negative regularity, an infinite loss of regularity was established in [11] for (1.1) in $H^s({\mathbb {R}}^d)$ ($d\ge 2$ and $s<-1/(2\sigma +1)$), and in the periodic case $x\in {\mathbb {T}}^d$ in [12], in Fourier–Lebesgue spaces. Typically, the NLS flow map fails to be continuous at the origin from $H^s({\mathbb {R}}^d)$ to $H^k({\mathbb {R}}^d)$, for any $k\in {\mathbb {R}}$.

1.2 Fourier–Lebesgue Spaces

The Fourier–Lebesgue space ${\mathcal {F}}L_s^p({\mathbb {R}}^d)$ is defined by

$$\begin{aligned} {\mathcal {F}}L_s^p({\mathbb {R}}^d)= \left\{ f\in \mathcal {S}'({\mathbb {R}}^d): \Vert f\Vert _{{\mathcal {F}}L_s^{p}}:= \Vert \hat{f} \langle \cdot \rangle ^s \Vert _{L^{p}}< \infty \right\} , \end{aligned}$$

where the Fourier transform is defined as

$$\begin{aligned} \hat{f}(\xi )= {\mathcal {F}} f(\xi ) = \frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d}e^{-ix\cdot \xi }f(x)dx,\quad f\in {\mathcal {S}}({\mathbb {R}}^d), \end{aligned}$$

and where $1\le p \le \infty$, $s\in {\mathbb {R}}$, and $\langle \xi \rangle ^{s} = (1+ |\xi |^2)^{s/2}$ ($\xi \in {\mathbb {R}}^d$). For $p=2$, ${\mathcal {F}}L_s^2=H^s$ the usual Sobolev space. For $s=0$, we write ${\mathcal {F}}L^p_0({\mathbb {R}}^d) = {\mathcal {F}}L^p({\mathbb {R}}^d).$

The scaling (1.2) leaves the homogeneous ${\mathcal {F}}\dot{L}_s^p({\mathbb {R}}^d)$-norm (replace the Japanese bracket $\left\langle \cdot \right\rangle ^s$ with the length $|\cdot |^s$ in the definition of ${\mathcal {F}}L_s^p({\mathbb {R}}^d)$) invariant for $s=s_c(p)$, where

$$\begin{aligned} s_c(p):= d \left( 1-\frac{1}{p} \right) -\frac{1}{\sigma }. \end{aligned}$$

Of course when $p=2$, we recover the previous value $s_c$. On the other hand, the Galilean transform (1.3) does not alter the ${\mathcal {F}} L^p({\mathbb {R}}^d)$ norm of $\psi$, and so well-posedness is not expected to hold in ${\mathcal {F}} L^p_s$ for $s<\max (0,s_c(p))$. Note however that the recent results from [22] show that this heuristical argument is not always correct: in the case $p=2$, $d=1=\sigma$, well-posedness may hold for $\min (0,s_c(2))=-\tfrac{1}{2}<s<\max (0,s_c(2))=0$. Therefore, $s<\min (0,s_c(p))$ is a safer assumption to obtain ill-posedness results. In this paper, we consider cases where $s<0$.

In [24, Theorem 1], Hyakuna-Tsutsumi established local well-posedness for the cubic NLS in ${\mathcal {F}}L^p({\mathbb {R}}^d)$ for $p \in (4/3, 4)\setminus \{2\}.$ Later this result is generalized in [23, Theorem 1.1] for $p\in [1,2].$

Our first results concern norm inflation of the type discussed above:

Theorem 1.2

Assume that $1\le p \le \infty , d, \sigma \in \mathbb N$ and $s< \min \left( 0, s_c(p)\right)$. For any $\delta >0,$ there exists $\psi _0 \in {\mathcal {F}}L^p_s({\mathbb {R}}^d)$ and $T>0$ satisfying

$$\begin{aligned} \Vert \psi _0 \Vert _{{\mathcal {F}}L^p_s}< \delta \quad \text {and} \quad 0<T< \delta , \end{aligned}$$

such that the corresponding solution $\psi$ to (1.1) exists on [0, T] and

$$\begin{aligned} \Vert \psi (T)\Vert _{{\mathcal {F}}L^p_s}> \delta ^{-1}. \end{aligned}$$

As discussed above, in the case $s_c(p)>0$, norm inflation is expected in ${\mathcal {F}}L^p_s({\mathbb {R}}^d)$ for $0<s<s_c(p)$, but with different arguments. The proof of Theorem 1.2 is inspired by the two-scale analysis of Kishimoto [27]. We also prove norm inflation with an infinite loss of regularity: the initial regularity must be sufficiently small, and we leave out the cubic one-dimensional nonlinearity.

Theorem 1.3

Let $\sigma \in {\mathbb {N}}$, $s< -\frac{1}{2 \sigma +1}$ and assume $d\sigma \ge 2.$ There exist a sequence of initial data $\left( \psi _n(0)\right) _{n\ge 1}$ in ${\mathcal {S}}({\mathbb {R}}^d)$ such that

$$\begin{aligned} \Vert \psi _n(0)\Vert _{ {\mathcal {F}}L^p_{s}} \mathop {\longrightarrow }\limits _{n\rightarrow \infty }0, \quad \forall p\in [1,\infty ], \end{aligned}$$

and a sequence of times $t_n \rightarrow 0$ such that the corresponding solutions $\psi _n$ to (1.1) satisfies

$$\begin{aligned} \Vert \psi _n(t_n)\Vert _{{\mathcal {F}}L^p_{k}} \mathop {\longrightarrow }\limits _{n\rightarrow \infty }\infty , \quad \forall k\in {\mathbb {R}},\ \forall p\in [1,\infty ] . \end{aligned}$$

Remark 1.4

There is no general comparison between the assumptions on s in Theorems 1.2 and 1.3: for $p=1$, $\min (0,s_c(1))=-1/\sigma <-1/(2\sigma +1)$, while if $s_c(p)\ge 0$, we obviously have $\min (0,s_c(p))=0>-1/(2\sigma +1)$.

1.3 Modulation Spaces

We now turn our attention to the theory of modulation spaces. The idea of modulation spaces is to consider the decaying properties of space variable and its Fourier transform simultaneously. Specifically, we consider the short-time Fourier transform (STFT) (sliding-window transform/wave packet transform) of f with respect to Schwartz class function $g\in \mathcal {S}({\mathbb {R}}^d)$:

$$\begin{aligned} V_{g}f(x,\xi )= \int _{{\mathbb {R}}^{d}} f(t) \overline{g(t-x)} e^{- i \xi \cdot t}dt, \quad (x, \xi ) \in {\mathbb {R}}^{2d}, \end{aligned}$$

whenever the integral exists. Then the modulation spaces $M^{p,q}_s({\mathbb {R}}^d)$ ($1 \le p, q \le \infty$, $s\in {\mathbb {R}})$ is defined as the collection of tempered distributions $f\in \mathcal {S}'({\mathbb {R}}^d)$ such that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_s}= \left\| \Vert V_gf\Vert _{L^p_x} (1+|\xi |^2)^{s/2}\right\| _{L^q_{\xi }} < \infty , \end{aligned}$$

with natural modification if a Lebesgue index is infinite. For $s=0,$ we write $M^{p,q}_0({\mathbb {R}}^d)= M^{p,q}({\mathbb {R}}^d).$ When $p=q=2,$ modulation spaces coincide with usual Sobolev spaces $H^{s}({\mathbb {R}}^d).$ For the last two decades, these spaces have made their own place in PDEs and there is a tremendous ongoing interest to use these spaces as a low regularity Cauchy data class for nonlinear dispersive equations; see e.g. [2, 5, 6, 32, 33, 37, 38]. Using the algebra property and boundedness of Schrödinger propagator on $M^{p,q}_s({\mathbb {R}}^d)$, (1.1) is proved to be locally well-posed in $M^{p,1}_s({\mathbb {R}}^d)$ for $1\le p \le \infty$, $s\ge 0$, and in $M^{p,q}_{s}({\mathbb {R}}^d)$ for $1\le p, q \le \infty$ and $s> d(1-1/q)$, via fixed point argument; see [2, 5, 7]. Using uniform-decomposition techniques, Wang and Hudzik [37] established global well-posedness for (1.1) with small initial data in $M^{2,1}({\mathbb {R}}^d).$ Guo [21] proved local well-posed for the cubic NLS in $M^{2,q}({\mathbb {R}})$ $(2\le q \le \infty )$, and later Oh and Wang [32], established global existence for this result. In [13], Chaichenets et al. established global well-posedness for the cubic NLS in $M^{p, p'}({\mathbb {R}})$ for p sufficiently close to 2. The well-posedness problems for some other PDEs in $M^{p,q}_s({\mathbb {R}}^d)$ are widely studied by many authors, see for instance the excellent survey [33] and references therein. We complement the existing literature on well-posedness theory for (1.1) with Cauchy data in modulation spaces. First, observe that, in view of Proposition 2.6 below,

$$\begin{aligned} \left\| \psi \left( \lambda \, \cdot \right) )\right\| _{M_s^{2,q}} \lesssim {\left\{ \begin{array}{ll} \lambda ^{- \frac{d}{2}} \max \left( 1,\lambda ^s\right) \Vert \psi \Vert _{M_s^{2,q}}, &{}\quad \text {if} \quad 1\le q \le 2,\\ \lambda ^{-d \left( 1-\frac{1}{q} \right) } \max \left( 1,\lambda ^s\right) \Vert \psi \Vert _{M_s^{2,q}} ,&{}\quad \text {if} \quad 2\le q \le \infty , \end{array}\right. } \end{aligned}$$

for all $\lambda \le 1$ and $s\in {\mathbb {R}}$. Invoking the general belief that ill-posedness at positive regularity is due to the transfer from low frequencies ($0<\lambda \ll 1$) to high frequencies, the scaling (1.2) suggests that ill-posedness occurs in $M_s^{2,q}({\mathbb {R}}^d)$ if

$$\begin{aligned} s< {\left\{ \begin{array}{ll} s_c= \frac{d}{2}-\frac{1}{\sigma } &{}\quad \text {if}\quad 1\le q\le 2,\\ d\left( 1-\frac{1}{q}\right) &{}\quad \text {if}\quad 2\le q\le \infty . \end{array}\right. } \end{aligned}$$

The following analogue of Theorem 1.2 then appears rather natural.

Theorem 1.5

Let $d, \sigma \in \mathbb N$ and assume that

• $s< \min \left( \frac{d}{2}-\frac{1}{\sigma }, 0 \right)$ when $1\le q \le 2,$ and

• $s< \min \left( d\left( 1- \frac{1}{q} \right) -\frac{1}{\sigma }, 0\right)$ when $2\le q \le \infty .$

For any $\delta >0,$ there exists $\psi _0 \in M^{2,q}_s({\mathbb {R}}^d)$ and $T>0$ satisfying

$$\begin{aligned} \Vert \psi _{0} \Vert _{M^{2,q}_s}< \delta \quad \text {and} \quad 0<T< \delta \end{aligned}$$

such that the corresponding solution $\psi$ to (1.1) exists on [0, T] and

$$\begin{aligned} \Vert \psi (T)\Vert _{M^{2,q}_s}>\delta ^{-1}. \end{aligned}$$

We also have some infinite loss of regularity of the flow map (1.1) at the level of modulation spaces with negative regularity. We no longer assume $p=2$, and show a stronger result, provided that the negative regularity s is sufficiently small, and (again) that we discard the one-dimensional cubic case.

Theorem 1.6

Let $\sigma \in {\mathbb {N}}$, $s< -\frac{1}{2 \sigma +1}$ and assume $d\sigma \ge 2.$ There exists a sequence of initial data $\left( \psi _n(0)\right) _{n\ge 1}$ in ${\mathcal {S}}({\mathbb {R}}^d)$ such that

$$\begin{aligned} \Vert \psi _n(0)\Vert _{M_s^{p,q}} \mathop {\longrightarrow }\limits _{n\rightarrow \infty }0, \quad \forall p,q\in [1,\infty ], \end{aligned}$$

and a sequence of times $t_n \rightarrow 0$ such that the corresponding solutions $\psi _n$ to (1.1) satisfies

$$\begin{aligned} \Vert \psi _n(t_n)\Vert _{M_{k }^{p,q}} \mathop {\longrightarrow }\limits _{n\rightarrow \infty }\infty , \quad \forall k \in {\mathbb {R}},\ \forall p,q\in [1,\infty ]. \end{aligned}$$

Remark 1.7

Contrary to the Fourier–Lebesgue case, the assumption regarding s is always weaker in Theorem 1.5 than in Theorem 1.6 (recall that the cubic one-dimensional case is ruled out in Theorem 1.6).

1.4 Comments and Outline of the Paper

As pointed out before, the numerology regarding the norm inflation phenomenon (Theorems 1.2 and 1.5) is probably sharp, up to the fact that the minimum should be replaced by a maximum in the assumption on s, and that at positive regularity, different arguments are required. On the other hand, we believe that the restriction $s<-\frac{1}{2\sigma +1}$ in Theorems 1.3 and 1.6 is due to our approach, and we expect that the result is true under the mere assumption $s<0$ if $d\sigma \ge 2$, and for $s<-1/2$ if $d=\sigma =1$.

The analogue of our results remains true if we replace $\Delta$ by the generalized dispersion of the form $\Delta _{\eta } = \sum _{j=1}^{d} \eta _{j}\partial ^2_{x_j}, \eta _j=\pm 1.$ The (1.1) associated $\Delta _{\eta }$ (with the non uniform signs of $\eta _j$) arises in the description of surface gravity waves on deep water, see e.g. [35].

In [34], Sugimoto-Wang-Zhang established some local well-posedness results for Davey–Stewartson equation in some weighted modulation spaces. We note that our method of proof can be applied to get norm-inflation results for Davey–Stewartson equation, and infinite loss of regularity in the spirit of [11], in some negative modulation and Fourier–Lebesgue spaces.

Theorems 1.3 and 1.6 cover any smooth power nonlinearity in multidimension, and power nonlinearities which are at least quintic in the one-dimensional case. Our method of proof seems too limited to prove loss of regularity in the case of the cubic nonlinearity on the line. It turns out that the method followed to treat the cubic nonlinearity on the circle in [12] seems helpless in the case of the line. On the other hand, Theorems 1.2 and 1.5 include the cubic one-dimensional Schrödinger equation.

The rest of this paper is organized as follows, In Sect. 2, we recall various properties associated to modulation spaces. In Sect. 3, we prove Theorem 1.2, and we adapt the argument in Sect. 4 to prove Theorem 1.5. In Sect. 5, we show how the theory of weakly nonlinear geometric optics makes it possible to prove loss of regularity at negative regularity for (1.1). A general framework where multiphase weakly nonlinear geometric optics is justified is presented in Sect. 6, and it is applied in Sect. 7 to prove Theorems 1.3 and 1.6.

Notations The notation $A \lesssim B$ means $A \le cB$ for a some constant $c > 0$, Let $(\Lambda ^{\varepsilon })_{0<\varepsilon \le 1}$ and $(\Upsilon ^{\varepsilon })_{0<\varepsilon \le 1}$ be two families of positive real numbers.

• We write $\Lambda ^{\varepsilon }\ll \Upsilon ^{\varepsilon }$ if $\limsup _{\varepsilon \rightarrow 0} \Lambda ^{\varepsilon }/\Upsilon ^{\varepsilon } =0.$

• We write $\Lambda ^{\varepsilon }\lesssim \Upsilon ^{\varepsilon }$ if $\limsup _{\varepsilon \rightarrow 0} \Lambda ^{\varepsilon }/\Upsilon ^{\varepsilon } <\infty .$

• We write $\Lambda ^{\varepsilon }\approx \Upsilon ^{\varepsilon }$ if $\Lambda ^{\varepsilon }\lesssim \Upsilon ^{\varepsilon }$ and $\Upsilon ^{\varepsilon }\lesssim \Lambda ^{\varepsilon }$.

2 Preliminary: Modulation Spaces

Feichtinger [18] introduced a class of Banach spaces, the so-called modulation spaces, which allow a measurement of space variable and Fourier transform variable of a function, or distribution, on ${\mathbb {R}}^d$ simultaneously, using the short-time Fourier transform (STFT). The STFT of a function f with respect to a window function $g \in {\mathcal S}({\mathbb {R}}^d)$ is defined by

$$\begin{aligned} V_{g}f(x,y)= \int _{{\mathbb {R}}^{d}} f(t) \overline{g(t-x)} e^{- i y\cdot t}dt, \ (x, y) \in {\mathbb {R}}^{2d}, \end{aligned}$$

(2.1)

whenever the integral exists. For $x, y \in {\mathbb {R}}^d$, the translation operator $T_x$, and the modulation operator $M_y$, are defined by $T_{x}f(t)= f(t-x)$ and $M_{y}f(t)= e^{ i y\cdot t} f(t).$ In terms of these operators the STFT may be expressed as

$$\begin{aligned} V_{g}f(x,y)=\langle f, M_{y}T_{x}g\rangle = e^{- i x \cdot y} \left( f*M_y g^*\right) (x) , \end{aligned}$$

(2.2)

where $\langle f, g\rangle$ denotes the inner product for $L^2$ functions, or the action of the tempered distribution f on the Schwartz class function g, and $g^*(y) = \overline{g(-y)}.$ Thus $V: (f,g) \mapsto V_g(f)$ extends to a bilinear form on $\mathcal {S}'({\mathbb {R}}^d) \times \mathcal {S}({\mathbb {R}}^d)$, and $V_g(f)$ defines a uniformly continuous function on ${\mathbb {R}}^{d} \times {\mathbb {R}}^d$ whenever $f \in \mathcal {S}'({\mathbb {R}}^d)$ and $g \in \mathcal {S}({\mathbb {R}}^d)$.

Definition 2.1

(Modulation spaces) Let $1 \le p,q \le \infty$, $s \in {\mathbb {R}}$ and $0\ne g \in {\mathcal S}({\mathbb {R}}^d)$. The weighted modulation space $M_s^{p,q}({\mathbb {R}}^d)$ is defined to be the space of all tempered distributions f for which the following norm is finite:

$$\begin{aligned} \Vert f\Vert _{M_s^{p,q}}= \left( \int _{{\mathbb {R}}^d}\left( \int _{{\mathbb {R}}^d} |V_{g}f(x,y)|^{p} dx\right) ^{q/p} (1+|y|^2)^{sq/2} \, dy\right) ^{1/q}, \end{aligned}$$

for $1 \le p,q <\infty$. If p or q is infinite, $\Vert f\Vert _{M_s^{p,q}}$ is defined by replacing the corresponding integral by the essential supremum.

Remark 2.2

The definition of the modulation space given above, is independent of the choice of the particular window function. See [20, Proposition 11.3.2(c)].

We recall an alternative definition of modulation spaces via the frequency-uniform localization techniques, providing another characterization which will be useful to prove Theorem 1.5. Let $Q_n$ be the unit cube with the center at n, so $\left( Q_{n}\right) _{n \in {\mathbb {Z}}^d}$ constitutes a decomposition of ${\mathbb {R}}^d,$ that is, ${\mathbb {R}}^d = \cup _{n\in {\mathbb {Z}}^{d}} Q_{n}.$ Let $\rho \in \mathcal {S}({\mathbb {R}}^d),$ $\rho : {\mathbb {R}}^d \rightarrow [0,1]$ be a smooth function satisfying $\rho (\xi )= 1 \ \text {if} \ \ |\xi |_{\infty }\le \frac{1}{2}$ and $\rho (\xi )= 0 \ \text {if} \ \ |\xi |_{\infty }\ge 1.$ Let $\rho _n$ be a translation of $\rho ,$ that is,

$$\begin{aligned} \rho _n(\xi )= \rho (\xi -n),\quad n \in {\mathbb {Z}}^d. \end{aligned}$$

Denote

$$\begin{aligned} \sigma _{n}(\xi )= \frac{\rho _{n}(\xi )}{\sum _{\ell \in {\mathbb {Z}}^{d}}\rho _{\ell }(\xi )}, \quad n \in {\mathbb {Z}}^d. \end{aligned}$$

Then $\left( \sigma _n(\xi )\right) _{n\in {\mathbb {Z}}^d}$ satisfies the following properties:

$$\begin{aligned} \left\{ \begin{aligned}&|\sigma _{n}(\xi )|\ge c, \forall \xi \in Q_{n},\\&{\text {supp}} \sigma _{n} \subset \{\xi : |\xi -n|_{\infty }\le 1 \},\\&\sum _{n\in {\mathbb {Z}}^{d}} \sigma _{n}(\xi )\equiv 1, \forall \xi \in {\mathbb {R}}^d,\\&|D^{\alpha }\sigma _{n}(\xi )|\le C_{|\alpha |}, \forall \xi \in {\mathbb {R}}^d, \alpha \in ({\mathbb {N}}\cup \{0\})^{d}. \end{aligned} \right. \end{aligned}$$

(2.3)

The frequency-uniform decomposition operators can be exactly defined by

$$\begin{aligned} \square _n = {\mathcal {F}}^{-1} \sigma _n {\mathcal {F}}. \end{aligned}$$

For $1\le p, q \le \infty , s\in {\mathbb {R}}$, it is known [18] that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_s}\asymp \left( \sum _{n\in \mathbb Z^d} \left\Vert \square _n(f)\right\Vert ^q_{L^p} (1+|n|)^{sq} \right) ^{1/q}, \end{aligned}$$

with natural modifications for $p, q= \infty .$

Lemma 2.3

([20, 33, 38]) Let $p,q, p_j, q_j\in [1, \infty ]$, $s, s_j \in {\mathbb {R}}$ ($j=1,2$). Then

• (1) $M^{p_{1}, q_{1}}_{s_1}({\mathbb {R}}^{d}) \hookrightarrow M^{p_{2}, q_{2}}_{s_2}({\mathbb {R}}^{d})$ whenever $p_{1}\le p_{2}$ and $q_{1}\le q_{2}$ and $s_2\le s_1.$ In particular, $H^{s}({\mathbb {R}}^d) \hookrightarrow M_s^{p,q}({\mathbb {R}}^d)$ for $2\le p,q \le \infty$ and $s\in {\mathbb {R}}.$

• (2) $M^{p_1, q_1}_{s_1} ({\mathbb {R}}^d) \hookrightarrow M^{p_2, q_2}_{s_2} ({\mathbb {R}}^d)$ for $q_1> q_2, s_1>s_2$ and $s_1-s_2> d/q_2-d/q_1.$

• (3) $M^{p,q_{1}}({\mathbb {R}}^{d}) \hookrightarrow L^{p}({\mathbb {R}}^{d}) \hookrightarrow M^{p,q_{2}}({\mathbb {R}}^{d})$ holds for $q_{1}\le \text {min} \{p, p'\}$ and $q_{2}\ge \text {max} \{p, p'\}$ with $\frac{1}{p}+\frac{1}{p'}=1.$

• (4) $M^{\min \{p', 2\}, p}({\mathbb {R}}^d) \hookrightarrow {\mathcal {F}} L^{p}({\mathbb {R}}^d)\hookrightarrow M^{\max \{p',2\},p}({\mathbb {R}}^d), \frac{1}{p}+\frac{1}{p'}=1.$

• (5) $\mathcal {S}({\mathbb {R}}^{d})$ is dense in $M^{p,q}({\mathbb {R}}^{d})$ if p and q are finite.

• (6) $M^{p,p}({\mathbb {R}}^d) \hookrightarrow L^p({\mathbb {R}}^d) \hookrightarrow M^{p,p'}({\mathbb {R}}^d)$ for $1\le p \le 2$ and $M^{p,p'}({\mathbb {R}}^d) \hookrightarrow L^p({\mathbb {R}}^d) \hookrightarrow M^{p,p}({\mathbb {R}}^d)$ for $2 \le p \le \infty .$

• (7) The Fourier transform ${\mathcal {F}}:M_s^{p,p}({\mathbb {R}}^{d})\rightarrow M_s^{p,p}({\mathbb {R}}^{d})$ is an isomorphism.

• (8) The space $M_s^{p,q}({\mathbb {R}}^{d})$ is a Banach space.

• (9) The space $M_s^{p,q}({\mathbb {R}}^{d})$ is invariant under complex conjugation.

Theorem 2.4

(Algebra property) Let $p,q, p_{i}, q_{i}\in [1, \infty ]$ $(i=0,1,2).$ If $\frac{1}{p_1}+ \frac{1}{p_2}= \frac{1}{p_0}$ and $\frac{1}{q_1}+\frac{1}{q_2}=1+\frac{1}{q_0},$ then

$$\begin{aligned} M^{p_1, q_1}({\mathbb {R}}^{d}) \cdot M^{p_{2}, q_{2}}({\mathbb {R}}^{d}) \hookrightarrow M^{p_0, q_0}({\mathbb {R}}^{d}); \end{aligned}$$

(2.4)

with norm inequality $\Vert f g\Vert _{M^{p_0, q_0}}\lesssim \Vert f\Vert _{M^{p_1, q_1}} \Vert g\Vert _{M^{p_2,q_2}}.$ In particular, the space $M^{p,q}({\mathbb {R}}^{d})$ is a pointwise ${\mathcal {F}}L^{1}({\mathbb {R}}^{d})$-module, that is, we have

$$\begin{aligned} \Vert fg\Vert _{M^{p,q}} \lesssim \Vert f\Vert _{{\mathcal {F}}L^{1}} \Vert g\Vert _{M^{p,q}}. \end{aligned}$$

Proof

The product relation (2.4) between modulation spaces is well known and we refer the interested reader to [5] and since ${\mathcal {F}}L^{1}({\mathbb {R}}^{d}) \hookrightarrow M^{\infty , 1}({\mathbb {R}}^{d})$, the desired inequality (2.4) follows. $\square$

For $f\in \mathcal {S}({\mathbb {R}}^{d}),$ the Schrödinger propagator $e^{i\frac{t}{2}\Delta }$ is given by

$$\begin{aligned} e^{i\frac{t}{2}\Delta }f(x)= \frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d} e^{ix\cdot \xi }e^{-i\frac{t}{2}|\xi |^2}\hat{f}(\xi ) d\xi . \end{aligned}$$

The first point in the following statement was established in [4], and the second, in [37, Proposition 4.1].

Proposition 2.5

([4, 37])

• (1) Let $t\in {\mathbb {R}}, \ p, q\in [1, \infty ].$ Then

  $$\begin{aligned} \Vert e^{i\frac{t}{2} \Delta } f\Vert _{M^{p,q}} \le C(t^{2}+1)^{d/4}\Vert f\Vert _{M^{p,q}} \end{aligned}$$

  where C is some constant depending on d.

• (2) Let $2 \le p \le \infty , 1\le q \le \infty .$ Then

  $$\begin{aligned} \Vert e^{i\frac{t}{2}\Delta } f \Vert _{M^{p,q}}\le (1+|t|)^{- d\left( \frac{1}{p}-\frac{1}{2} \right) } \Vert f\Vert _{M^{p',q}}. \end{aligned}$$

For $(1/p, 1/q)\in [0,1] \times [0,1],$ we define the subsets

$$\begin{aligned}&I_1=\{(p,q);\ \max (1/p,1/p')\le 1/q\},\quad I_1^{*}=\{(p,q);\ \min (1/p,1/p')\ge 1/q\},\\&I_2=\{(p,q);\ \max (1/q,1/2)\le 1/p'\},\quad I_2^{*}=\{(p,q);\ \min (1/q,1/2)\ge 1/p'\},\\&I_3=\{(p,q);\ \max (1/q,1/2)\le 1/p\}, \quad I_3^{*}=\{(p,q);\ \min (1/q,1/2)\ge 1/p\}. \end{aligned}$$

We now define the indices:

$$\begin{aligned} \mu _1(p,q)={\left\{ \begin{array}{ll} -1/p &{} \text { if } (1/p,1/q)\in I_1^{*},\\ 1/q-1 &{} \text { if } (1/p,1/q)\in I_2^{*},\\ -2/p+1/q &{} \text { if } (1/p,1/q)\in I_3^{*}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \mu _2(p,q)={\left\{ \begin{array}{ll} -1/p &{} \text { if } (1/p,1/q)\in I_1,\\ 1/q-1 &{} \text { if } (1/p,1/q)\in I_2,\\ -2/p+1/q &{} \text { if } (1/p,1/q)\in I_3. \end{array}\right. } \end{aligned}$$

The dilation operator $f_{\lambda }$ is given by

$$\begin{aligned} f_{\lambda }(x)= f(\lambda x), \quad \lambda >0. \end{aligned}$$

Proposition 2.6

(See Theorem 3.2 in [17]) Let $1\le p, q \le \infty$, $s\in {\mathbb {R}}.$ There exists a constant $C>0$ such that for all $f\in M^{p,q}_{s}({\mathbb {R}}^d), 0< \lambda \le 1,$ we have

$$\begin{aligned} C^{-1} \lambda ^{d \mu _1(p,q)} \min \{1, \lambda ^s \} \Vert f\Vert _{M^{p,q}_s} \le \Vert f_{\lambda } \Vert _{M^{p,q}_s} \le C \lambda ^{d \mu _2(p,q)} \max \{1, \lambda ^s\} \Vert f\Vert _{M^{p,q}_s}. \end{aligned}$$

3 Norm Inflation in Fourier–Lebesgue Spaces

Define

$$\begin{aligned} \mu _{\sigma } (z_1,\dots ,z_{2\sigma +1})= \prod _{\ell =1}^{\sigma +1} z_{\ell } \prod _{m=\sigma +2}^{2\sigma +1} \bar{z}_m. \end{aligned}$$

Definition 3.1

For $\psi _0\in L^2({\mathbb {R}}^d),$ define $U_1[\psi _0](t)= e^{i \frac{t}{2} \Delta } \psi _0,$

$$\begin{aligned} U_k[\psi _0](t) = -i \sum _{\begin{array}{c} {k_1, \dots , k_{2\sigma +1} \ge 1}\\ { k_1+\dots +k_{2\sigma +1} =k} \end{array}}\int _0^t e^{i \frac{(t-\tau )}{2} \Delta }\mu _{\sigma }\left( U_{k_1}[\psi _0],...,U_{k_{2\sigma +1}}[\psi _0] \right) (\tau ) d\tau , \quad k\ge 2. \end{aligned}$$

It is known that the solution $\psi$ of (1.1) can be written as a power series expansion $\psi = \sum _{k=1}^{\infty } U_k[\psi _0]$, see [3], and [25, 27] for later refinements of the method.

Definition 3.2

Let $A>0$ be a dyadic number. Define the space $M_{A}$ as the completion of $C_0^{\infty }({\mathbb {R}}^d)$ with respect to the norm

$$\begin{aligned} \Vert f\Vert _{M_{A}}= \sum _{\xi \in A{\mathbb {Z}}^d} \Vert \hat{f}\Vert _{L^2 (\xi + Q_A)},\quad Q_A= [-A/2, A/2)^d. \end{aligned}$$

Lemma 3.3

([3, 27]) Let $A>0$ be a dyadic number.

• (1) $M_A \sim _A M_1$, and for all $\epsilon >0$, $H^{\frac{d}{2}+ \epsilon } \hookrightarrow M_1 \hookrightarrow L^2$.

• (2) $M_A$ is a Banach algebra under pointwise multiplication, and

  $$\begin{aligned} \Vert fg\Vert _{M_A} \le C(d) A^{d/2} \Vert f\Vert _{M_A} \Vert g\Vert _{M_A} \quad \forall f, g \in M_A. \end{aligned}$$

• (3) Let $A \ge 1$ be a dyadic number and $\phi \in M_A$ with $\Vert \psi _0\Vert _{M_A} \le M.$ Then, there exists $C>0$ independent of A and M such that

  $$\begin{aligned} \Vert U_k[\psi _0] (t) \Vert _{M_A} \le t^{\frac{k-1}{2\sigma }} (C A^{d/2} M)^{k-1} M , \end{aligned}$$

  for any $t\ge 0$ and $k \ge 1.$

• (4) Let $\left( b_k\right) _{k=1}^{\infty }$ be a sequence of nonnegative real numbers such that

  $$\begin{aligned} b_k \le C \sum _{\begin{array}{c} {k_1,\dots , k_{2\sigma +1} \ge 1}\\ {k_1+\dots + k_{2\sigma +1} =k} \end{array}} b_{k_1}\cdots b_{k_{2\sigma +1}} \quad \forall k \ge 2. \end{aligned}$$

  Then we have

  $$\begin{aligned} b_k \le b_1 C_0^{k-1}, \quad \forall k \ge 1,\text { where } C_0= \frac{\pi ^2}{6} \left( C (2\sigma +1)^2\right) ^{1/(2\sigma )} b_1. \end{aligned}$$

Corollary 3.4

(See Corollary 1 in [27]) Let $A\ge 1$ be dyadic and $M>0$. If $0<T\ll (A^{d/2} M)^{-2\sigma },$ then for any $\psi _0 \in M_A$ with $\Vert \psi _0\Vert _{M_A} \le M$:

• (i) A unique solution $\psi$ to the integral equation associated with (1.1),

  $$\begin{aligned} \psi (t) = e^{i \frac{t}{2} \Delta } \psi _0 - i \int _0^t e^{i\frac{(t-\tau )}{2} \Delta } \mu _{\sigma } (\psi (\tau )) d\tau \end{aligned}$$

  exists in $C([0, T], M_A).$

• (ii) The solution $\psi$ given in (i) has the expression

  $$\begin{aligned} \psi = \sum _{k=1}^{\infty } U_k[\psi _0]= \sum _{\ell =0}^{\infty } U_{2\sigma \ell +1}[\psi _0] \end{aligned}$$

  (3.1)

  which converges absolutely in $C([0, T], M_A).$

Remark 3.5

By Definition 3.1, we obtain $U_k[\psi _0](t)=0$ unless $k\equiv 1$ mod $2\sigma$. For instance, $U_k[\psi _0](t) \equiv 0$ for all $k \in 2\sigma {\mathbb {N}}.$ To see this, fix $\sigma \in {\mathbb {N}}.$ Then clearly $U_{2\sigma }[\psi _0]\equiv 0$ because there does not exist $k_j\ge 1$ such that $k_1+\cdots + k_{2\sigma +1}=2\sigma .$ Now since $U_{2\sigma }[\psi _0] \equiv 0,$ it follows that $U_{4\sigma }[\psi _0] \equiv 0$ and so on. Thus, $U_k[\psi _0](t) \equiv 0$ for all $k \in 2\sigma {\mathbb {N}}.$

The general idea from [3] to prove instability is to show that one term in the sum (3.1) dominates the sum of the other terms, and rules out the continuity of the flow map. Usually, the first Picard iterate accounting for nonlinear effects, that is, $U_{2\sigma +1}[\psi _0]$ in our case, does the job. The proof of Theorems 1.2 and 1.5 indeed relies on this idea, for a suitable $\psi _0$ as in [27].

Let N, A be dyadic numbers to be specified so that $N\gg 1$ and $0<A\ll N$. We choose initial data of the following form

$$\begin{aligned} \widehat{\psi _0} = R A^{-d/p} N^{-s} \chi _{\Omega }, \end{aligned}$$

(3.2)

for a positive constant R and a set $\Omega$ satisfying

$$\begin{aligned} \Omega = \bigcup _{\eta \in \sum } (\eta + Q_A) , \end{aligned}$$

for some $\sum \subset \{ \xi \in {\mathbb {R}}^d: |\xi | \sim N\}$ such that $\# \sum \le 3.$ Then we have

$$\begin{aligned} \Vert \psi _0\Vert _{{\mathcal {F}}L^p_s} \sim R, \quad \Vert \psi _0\Vert _{M_A} \sim R A^{d (\frac{1}{2}-\frac{1}{p})} N^{-s}. \end{aligned}$$

In fact, we have

$$\begin{aligned} \Vert \psi _0\Vert _{{\mathcal {F}}L^p_s}^p&= R^p A^{-d} N^{-sp} \int _{\Omega } (1+|\xi |^2)^{ps/2} d\xi \\&= R^p A^{-d} N^{-sp} \sum _{\eta } \int _{\eta + Q_A} (1+|\xi |^2)^{ps/2} d\xi . \end{aligned}$$

Since $A<N$ and $|\eta | \sim N,$ we have $N^{2} \lesssim (1+|\xi |^2) \lesssim N^2$ for $\xi \in \eta +Q_A$ and so $N^{ps} \lesssim (1+|\xi |^2)^{ps/2} \lesssim N^{ps}$. As $\# \sum \le 3$ and $|\eta + Q_A| \sim A^{d},$ we infer that $\Vert \psi _0\Vert _{{\mathcal {F}}L^p_s}^p \sim R^p.$

Lemma 3.6

(See Lemma 3.6 in [27]) There exists $C>0$ such that for any $\psi _0$ satisfying (3.2) and $k\ge 1$, we have

$$\begin{aligned} \left| {\text {supp}}\widehat{ U_k[\psi _0]} (t)\right| \le C^k A^d ,\quad \forall t\ge 0. \end{aligned}$$

The next result is the analogue of [27, Lemma 3.7].

Lemma 3.7

Let $\psi _0$ given by (3.2), $s<0$ and $1\le p \le \infty .$ Then there exists $C>0$ depending only on $d, \sigma$ and s such that following holds.

$$\begin{aligned}&\Vert U_1[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} \le C R ,\quad \forall T\ge 0, \end{aligned}$$

(3.3)

$$\begin{aligned}&\Vert U_k[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} \lesssim \rho _1^{k-1} C^{k} A^{-d/p} RN^{-s} \Vert \langle \cdot \rangle ^s\Vert _{L^p (Q_A)}, \end{aligned}$$

(3.4)

where $\rho _1= R N^{-s} A^{d \left( 1- \frac{1}{p} \right) } T^{\frac{1}{2\sigma }}$.

Proof

The Schrödinger group is a Fourier multiplier,

$$\begin{aligned} \Vert U_1[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} = \left\| (e^{i\frac{T}{2^{}}|\cdot |^2} \widehat{\psi _0} )\langle \cdot \rangle ^s \right\| _{L^p} = \Vert \psi _0\Vert _{{\mathcal {F}}L^p_s} \le CR, \end{aligned}$$

hence (3.3). We note that

$$\begin{aligned} I&:= \Vert U_k[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} \\&\le \Vert \langle \cdot \rangle ^s \Vert _{L^p( {\text {supp}}\widehat{U_k}[\psi _0](t))} \sup _{\xi \in {\mathbb {R}}^d} \left| \widehat{U_k[\psi _0]} (t, \xi )\right| \\&\le \Vert \langle \cdot \rangle ^s \Vert _{L^p( {\text {supp}} \widehat{U_k}[\psi _0](t))} \sum _{k_1+\dots + k_{2\sigma +1} =k} \int _0^t \left\| |v_{k_1} (\tau )|*\dots *|v_{k_{2\sigma +1}}(\tau )| \right\| _{L^{\infty }} d\tau , \end{aligned}$$

where $v_{k_{\ell }}$ is either $\widehat{U_{k_{\ell }}[\psi _0]}$ or $\widehat{\overline{ U_{k_{\ell }}[\psi _0]}}$. By Young and Cauchy-Schwarz inequalities,

$$\begin{aligned} \Vert v_{k_1}*\cdots *v_{k_{2\sigma +1}}\Vert _{L^{\infty }}&\le \Vert v_{k_1}*v_{k_2}\Vert _{L^{\infty }}\Vert v_{k_3} *\cdots *v_{k_{2\sigma +1}}\Vert _{L^1}\\&\le \Vert v_{k_1}^{\vee } v_{k_2}^{\vee }\Vert _{L^1} \prod _{\ell =3}^{2\sigma +1} \Vert v_{k_{\ell }}\Vert _{L^1}\\&\le \Vert v_{k_1} \Vert _{L^2} \Vert v_{k_2}\Vert _{L^2}\prod _{\ell =3}^{2\sigma +1} \Vert v_{k_{\ell }}\Vert _{L^1}\\&\le \prod _{\ell =3}^{2\sigma +1} \left| {\text {supp}} \widehat{ U_{k_{\ell }}[\psi _0]} \right| ^{1/2} \prod _{\ell =1}^{2\sigma +1} \Vert \widehat{ U_{k_{\ell }}[\psi _0]} \Vert _{L^2}. \end{aligned}$$

Thus, we have

$$\begin{aligned} I \le \Vert \langle \cdot \rangle ^s \Vert _{L^p( {\text {supp}}\widehat{U_k}[\psi _0](t))} I_1, \end{aligned}$$

where

$$\begin{aligned} I_1:=\sum _{k_1+\cdots + k_{2\sigma +1} =k} \int _0^t \prod _{\ell =3}^{2\sigma +1} \left| {\text {supp}} \widehat{ U_{k_{\ell }}[\psi _0]} (\tau ) \right| ^{1/2} \prod _{\ell =1}^{2\sigma +1} \Vert \widehat{ U_{k_{\ell }}[\psi _0]}(\tau ) \Vert _{L^2} d\tau . \end{aligned}$$

By Lemma 3.3 (3) (with $M= CR N^{-s} A^{\frac{d}{2}-\frac{d}{p}}$), we have, for all $k\ge 1$,

$$\begin{aligned} \Vert U_{k}[\psi _0](t)\Vert _{L^2} \le \Vert U_{k}[\psi _0](t)\Vert _{M_A} \le Ct^{\frac{k-1}{2\sigma }} \left( C^2R A^{d/2} N^{-s} A^{\frac{d}{2}-\frac{d}{p}}\right) ^{k-1} R N^{-s} A^{\frac{d}{2}- \frac{d}{p}}. \end{aligned}$$

Note that, by Lemma 3.6,

$$\begin{aligned} I_1&\lesssim \sum _{k_1+\dots + k_{2\sigma +1} =k} \int _0^t \prod _{\ell =3}^{2\sigma +1} A^{d/2} \prod _{\ell =1}^{2\sigma +1}\left[ \tau ^{\frac{k_{\ell }-1}{2\sigma }} \left( R A^{d \left( 1-\frac{1}{p} \right) } N^{-s} \right) ^{k_{\ell }-1} R N^{-s} A^{\frac{d}{2}-\frac{d}{p}}\right] d\tau \\&\lesssim (RN^{-s})^{k} A^{ \frac{d(2\sigma -1)}{2}} A^{d \left( 1- \frac{1}{p} \right) (k-2\sigma -1)} A^{\left( \frac{d}{2}- \frac{d}{p}\right) (2\sigma +1)} \int _0^t \tau ^{\frac{k-2\sigma -1}{2\sigma }} d\tau \\&\lesssim A^{d \left( 1-\frac{1}{p} \right) (k-1)} A^{-d/p}(RN^{-s})^{k} t^{\frac{k-1}{2\sigma }}. \end{aligned}$$

Since $s<0$, for any bounded set $D\subset {\mathbb {R}}^d$, we have

$$\begin{aligned} \left| \{ \langle \xi \rangle ^{s}> \lambda \} \cup D \right| \le \left| \{ \langle \xi \rangle ^{s}> \lambda \} \cup B_D \right| , \quad \forall \lambda >0, \end{aligned}$$

where $B_D\subset {\mathbb {R}}^d$ is the ball centered at origin with $|D|=|B_D|.$ This implies that $\Vert \langle \xi \rangle ^s\Vert _{L^p(D)} \le \Vert \langle \xi \rangle ^s\Vert _{L^p(B_D)}.$ In view of this and performing simple change of variables ($\xi = C^{k/d} \xi '$), we obtain

$$\begin{aligned} \Vert \langle \cdot \rangle ^s \Vert _{L^p( {\text {supp}} \widehat{U_k}[\psi _0](t))}\le \Vert \langle \cdot \rangle ^s \Vert _{L^p ( \{ |\xi | \le C^{k/d} A \})} \lesssim C^{k} \Vert \langle \cdot \rangle ^s \Vert _{L^p(\{ |\xi | \le A\})}, \end{aligned}$$

and the lemma follows. $\square$

In the next lemma we establish a crucial lower bound on $U_{2\sigma +1}[\psi _0].$

Lemma 3.8

Let $1\le p \le \infty ,$ $1\le A \ll N$ and $\sum =\{ Ne_d, - Ne_d, 2Ne_d\}$ where $e_d= (0,\dots ,0, 1) \in {\mathbb {R}}^d$. If $0<T\ll N^{-2},$ then we have

$$\begin{aligned} \Vert U_{2\sigma +1} [\psi _0] (T)\Vert _{{\mathcal {F}}L_s^p} > rsim R A^{-\frac{d}{p}} N^{-s} \rho _1^{2\sigma }\Vert \langle \cdot \rangle ^s\Vert _{L^p (Q_A)}, \end{aligned}$$

where $\rho _1= R N^{-s} A^{d- \frac{d}{p}} T^{\frac{1}{2\sigma }}$.

Proof

Note that

$$\begin{aligned} \widehat{U_{2\sigma +1}[\psi _0]}(T, \xi )= c e^{-i \frac{T}{2} |\xi |^2} \int _{\Gamma } \prod _{\ell =1}^{\sigma +1} \widehat{\psi _0} (\xi _{\ell }) \prod _{m=\sigma +2}^{ 2\sigma +1} \overline{ \widehat{\psi _0} (\xi _m)} \int _0^{T} e^{i\frac{t}{2} \Phi } dt d\xi _1...d\xi _{2\sigma }\!, \end{aligned}$$

where

$$\begin{aligned}&\Gamma = \left\{ (\xi _1,\dots , \xi _{2\sigma +1})\in {\mathbb {R}}^{(2\sigma +1)d}: \sum _{\ell =1}^{\sigma +1} \xi _{\ell } - \sum _{m= \sigma +2}^{2\sigma +1} \xi _m = \xi \right\} , \\&\Phi = |\xi |^2- \sum _{\ell =1}^{\sigma +1}|\xi _{\ell }|^2 + \sum _{m= \sigma +2}^{2\sigma +1} |\xi _m|^2. \end{aligned}$$

By the choice of initial data (3.2), we have

$$\begin{aligned} \int _{\Gamma } \prod _{\ell =1}^{\sigma +1} \widehat{\psi _0} (\xi _{\ell }) \prod _{m=\sigma +2}^{ 2\sigma +1} \overline{ \widehat{\psi _0} (\xi _m)}&= \int _{\Gamma } \prod _{\ell =1}^{\sigma +1} R A^{-d/p} N^{-s} \chi _{\Omega } (\xi _{\ell }) \prod _{m=\sigma +2}^{ 2\sigma +1} R A^{-d/p} N^{-s} \chi _{\Omega } (\xi _{m}) \\&= \left( R A^{-d/p} N^{-s} \right) ^{2\sigma +1} \int _{\Gamma } \prod _{\ell =1}^{2\sigma +1} \chi _{\Omega } (\xi _{\ell }) d\xi _1 \dots d\xi _{2\sigma }\\&= \left( R A^{-d/p} N^{-s} \right) ^{2\sigma +1} \sum _{\mathcal C} \int _{\Gamma } \prod _{\ell =1}^{2\sigma +1} \chi _{\eta _{\ell } + Q_{A}} (\xi _{\ell }) d\xi _1 \dots d\xi _{2\sigma }\!, \end{aligned}$$

where the sum is taken over the non-empty set

$$\begin{aligned} \mathcal C =\left\{ (\eta _1, \dots , \eta _{2\sigma +1}) \in \{ \pm Ne_d, 2 Ne_d\}^{2\sigma +1}: \sum _{\ell =1}^{\sigma +1} \eta _{\ell } - \sum _{m= \sigma +2}^{2\sigma +1} \eta _{m} =0 \right\} . \end{aligned}$$

For $\xi \in Q_{A},$ we have $|\xi _{i}|^2\le |\xi |^2 \le A^2 \ll N^2$ and so $\left| \Phi \right| \lesssim N^2.$ Then $| \frac{t}{2}\Phi (\xi )|\ll 1$ for $0<T \ll N^{-2}.$ In view of this, together with the fact that the cosine function decreasing on $[0, \pi /4],$ we obtain

$$\begin{aligned} \left| \int _0^{T} e^{i\frac{t}{2} \Phi (\xi )} dt\right| \ge Re\int _0^{T} e^{i\frac{t}{2} \Phi (\xi )} dt \ge \frac{1}{2} T. \end{aligned}$$

Taking the above inequalities into account, we infer

$$\begin{aligned} \left| \widehat{U_{2\sigma +1}[\psi _0]}(T, \xi )\right| > rsim \left( R A^{-d/p} N^{-s} \right) ^{2\sigma +1} (A^d)^{2\sigma } T \chi _{(2\sigma +1)^{-1} Q_{A}} (\xi ). \end{aligned}$$

(3.5)

Hence, we have

$$\begin{aligned} \Vert U_{2\sigma +1}[\psi _0] (T) \Vert _{{\mathcal {F}}L^p_s}& > rsim \left( R A^{-d/p} N^{-s} \right) ^{2\sigma +1} (A^d)^{2\sigma } T \Vert \langle \cdot \rangle ^{s}\Vert _{L^p ( (2\sigma +1)^{-1} Q_A)}\\& > rsim R A^{-\frac{d}{p}} N^{-s} \rho _1^{2\sigma } \Vert \langle \cdot \rangle ^s\Vert _{L^p (Q_A)}, \end{aligned}$$

where $\rho _1= R N^{-s} A^{d- \frac{d}{p}} T^{\frac{1}{2\sigma }}$. $\square$

For the convenience of reader, we compute the $L^p$-norm of weight $\langle \cdot \rangle ^s$ on the cube $Q_A.$

Lemma 3.9

Let $A\gg 1$, $d\ge 1$, $s<0$ and $1\le p < \infty$. We define

$$\begin{aligned} f^p_s(A)= {\left\{ \begin{array}{ll} 1 \quad \text {if} \ s< -\frac{d}{p},\\ \left( \log A \right) ^{1/p} \quad \text {if} \ s= -\frac{d}{p},\\ A^{d/p+s} \quad \text {if} \ s> -\frac{d}{p}. \end{array}\right. } \end{aligned}$$

Then we have $f^p_s(A) \lesssim \left\| \langle \cdot \rangle ^s \right\| _{L^p(Q_A)} \lesssim f^p_s(A)$ and $f_s^{\infty }(A)= \left\| \langle \cdot \rangle ^s \right\| _{L^{\infty }(Q_A)} \sim 1.$ In particular, $f^{p}_s(A) > rsim A^{\frac{d}{p}+s}$ for any $s<0.$

Proof

We first compute the $\Vert \cdot \Vert _{L^{p}}$-norm on ball of radius $R_1$ in ${\mathbb {R}}^d$, say $B_{R_1}(0).$ Since $\langle \cdot \rangle ^s$ is radial, we have

$$\begin{aligned} I(R_1):= \int _{B_{R_1}(0)} \frac{1}{(1+|\xi |^2)^{-sp/2}} d\xi = \frac{2 \pi ^{d/2}}{\Gamma (d/2)} \int _0^{R_1} \frac{r^{d-1}}{(1+r^2)^{-sp/2}} dr. \end{aligned}$$

Notice that $(1+r^2)^{-sp/2} \ge \max \{1,r^{-sp}\}$, and assuming that $R_1 \gg 1$, we obtain:

$$\begin{aligned} I(R_1)&\lesssim \int _0^1 \frac{r^{d-1}}{\max \{1,r^{-sp}\}} \ \text {d} t + \int _1^{R_1} \frac{r^{d-1}}{\max \{1,r^{-sp}\}} dr \\&= \int _0^1 r^{d-1} dr + \int _1^{R_1} \frac{1}{r^{-sp-d+1}} dr. \end{aligned}$$

Using conditions on s,  we have $I(R_1) \lesssim \left( f_s^p(R_1)\right) ^{p}.$ Notice that $Q_{A} \subset B_{\sqrt{d}A/2}(0),$ we have $\Vert \langle \cdot \rangle ^s \Vert _{L^p (Q_A)} \le \left( I (\sqrt{d} A/2) \right) ^{1/p} \lesssim f_s^p(A).$ On the other hand, we notice that $1+r^2 \le 2$ if $0<r<1$ and $1+r^2 \le 2r^2$ if $1<r< R_2$ for some appropriate $R_2$. Using this together with the above ideas, we obtain $f_s^p(A) \lesssim \Vert \langle \cdot \rangle ^s \Vert _{L^p (Q_A)}.$ This completes the proof. $\square$

Proof of Theorem 1.2

By Corollary 3.4, we have the existence of a unique solution to (1.1) in $M_{A}$ up to time T whenever $\rho _1=RN^{-s} A^{d \left( 1- \frac{1}{p}\right) } T^{1/(2\sigma )} \ll 1$. In view of Lemma 3.7 and since $\rho _1<1$, $\sum _{\ell =2}^{\infty } \Vert U_{2\sigma \ell +1}[\psi _0] (T)\Vert _{{\mathcal {F}}L^{p}_s}$ can be dominated by the sum of the geometric series. Specifically, we have

$$\begin{aligned} \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0] (T) \right\| _{{\mathcal {F}}L^p_s}&\lesssim A^{-d/2} R N^{-s} f_s^p(A) \sum _{\ell =2}^{\infty } \rho _1^{2\sigma \ell } \nonumber \\&\lesssim A^{-d/2} R N^{-s} f_s^p(A)\rho _1^{4\sigma }. \end{aligned}$$

(3.6)

By Corollary 3.4 and the triangle inequality, we obtain

$$\begin{aligned} \Vert \psi (T)\Vert _{{\mathcal {F}}L^p_s}&= \left\| \sum _{\ell =0}^{\infty } U_{2\sigma \ell +1}[\psi _0] \right\| _{{\mathcal {F}}L^p_s} \\&\ge \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} - \Vert U_1[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s}- \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0](T) \right\| _{{\mathcal {F}}L^p_s}. \end{aligned}$$

In order to ensure

$$\begin{aligned} \Vert \psi (T)\Vert _{{\mathcal {F}}L^p_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s}, \end{aligned}$$

we rely on the conditions

$$\begin{aligned}&\Vert U_{2\sigma +1}[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} \gg \Vert U_1[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s}, \end{aligned}$$

(3.7)

$$\begin{aligned}&\Vert U_{2\sigma +1}[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} \gg \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0](T) \right\| _{{\mathcal {F}}L^p_s}. \end{aligned}$$

(3.8)

To use Lemma 3.8, we require

• (i) $T\ll N^{-2}.$

In view of Lemma 3.7, to prove (3.7) it is sufficient to prove

• (ii) $R \rho _1^{2\sigma } A^{-\frac{d}{p}} N^{-s} f^p_s(A) \gg R$, with $\rho _1= R N^{-s} A^{d- \frac{d}{p}} T^{\frac{1}{2\sigma }}$.

Finally, in view of Lemmas 3.7, 3.8 and 3.9, and (3.6), to prove (3.8) it is sufficient to prove:

• (iii) $\rho _1\ll 1$,

• (iv) $R \rho _1^{2\sigma } A^{-\frac{d}{p}} N^{-s} f^p_s(A) \gg R \rho _1^{4\sigma } A^{-\frac{d}{p}} N^{-s} f_s^p(A).$

We now choose A, R and T so that conditions (i)- (iv) are satisfied. To this end, we set

$$\begin{aligned} R= (\log N)^{-1}, \quad A \sim (\log N)^{-\frac{2\sigma +2}{|s|}} N, \quad T= (A^{d(\frac{1}{p}-1)} N^{s})^{2\sigma }. \end{aligned}$$

Then we have

$$\begin{aligned} \rho _1= R N^{-s} A^{d- \frac{d}{p}} T^{\frac{1}{2\sigma }}=(\log N)^{-1}\ll 1. \end{aligned}$$

Hence, condition (iii) is satisfied and so condition (iv). Note that

$$\begin{aligned} T= (\log N)^{-\frac{2\sigma +2}{|s|} d (\frac{1}{p} -1) 2\sigma } N^{d (\frac{1}{p} -1) 2\sigma + 2\sigma s}. \end{aligned}$$

Since $s< d \left( 1-\frac{1}{p} \right) - \frac{1}{\sigma }$ and $\log N = \mathcal {O}(N^{\epsilon })$ for any $\epsilon >0,$ we have

$$\begin{aligned} T \ll N^{-2}, \end{aligned}$$

and hence (i) is satisfied. By Lemma 3.9, we have $f^{p}_s(A) > rsim A^{\frac{d}{p}+s}$ for any $s<0$ and $A\ge 1$ and so

$$\begin{aligned} R \rho _1^{2\sigma } A^{-\frac{d}{p}} N^{-s} f^p_s(A) > rsim \log N \gg (\log N)^{-1}=R \end{aligned}$$

and hence (ii) is satisfied. Thus, we have $\Vert \psi (T)\Vert _{{\mathcal {F}}L^p_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{{\mathcal {F}}L^p_s} > rsim \log N.$ Since $\Vert \psi _0\Vert _{{\mathcal {F}}L^p_s} \sim R = (\log N)^{-1}$ and $T\ll N^{-2},$ we get norm inflation by letting $N\rightarrow \infty .$ $\square$

4 Norm Inflation in Modulation Spaces

The proof of Theorem 1.5 follows the same general lines as the proof of Theorem 1.2 from the previous section. Let N, A be dyadic numbers to be specified so that $N\gg 1$ and $0<A\ll N$. We choose initial data of the following form

$$\begin{aligned} \widehat{\psi _0} = {\left\{ \begin{array}{ll} R A^{-d/2} N^{-s} \chi _{\Omega },\quad \text {if}\quad 1\le q\le 2,\\ R A^{-d/q} N^{-s} \chi _{\Omega },\quad \text {if}\quad 2\le q\le \infty , \end{array}\right. } \end{aligned}$$

(4.1)

where

$$\begin{aligned} \Omega = \bigcup _{\eta \in \sum } (\eta + Q_A) ,\quad Q_A=[-A/2, A/2), \end{aligned}$$

for some $\sum \subset \{ \xi \in {\mathbb {R}}^d: |\xi | \sim N\}$ such that $\# \sum \le 3$.

4.1 A Priori Estimates: $1\le q\le 2$

Then we have, for any $s\in {\mathbb {R}}$,

$$\begin{aligned} \Vert \psi _0\Vert _{H^s} \sim R, \quad \Vert \psi _0\Vert _{M_A} \sim RN^{-s}. \end{aligned}$$

Lemma 4.1

Let $q\in [1,2]$, $\psi _0$ given by (4.1), $s<0.$ Then there exists $C>0$ depending only on $d, \sigma$ and s such that following holds.

$$\begin{aligned}&\Vert U_1[\psi _0](T)\Vert _{M^{2,q}_s} \le C R ,\quad \forall T\ge 0, \end{aligned}$$

(4.2)

$$\begin{aligned}&\Vert U_k[\psi _0](T)\Vert _{M^{2,q}_s} \lesssim \rho ^{k-1} C^{k} A^{-d/2} RN^{-s} \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le A \right) }, \end{aligned}$$

(4.3)

where $\rho = R N^{-s} A^{d/2} T^{\frac{1}{2\sigma }}$.

Proof

By Lemma 2.3 and Proposition 2.5, we have

$$\begin{aligned} \Vert U_{1}[\psi _0](T)\Vert _{M^{2,q}_s} \lesssim \Vert \psi _0(T)\Vert _{M^{2,q}_s} \lesssim \Vert \psi _0(T)\Vert _{M^{2,1}_s} \lesssim R, \end{aligned}$$

hence (4.2). By Plancherel theorem and (2.3), for $s<0,$ we have

$$\begin{aligned} \Vert U_k[\psi _0](T)\Vert _{M^{2,q}_s}&= \left\| (1+|n|)^s \Vert \sigma _n \widehat{U_k[\psi _0]}(T)\Vert _{L^2} \right\| _{\ell ^q}\\&\le \sup _{\xi \in {\mathbb {R}}^d} \left| \widehat{U_k[\psi _0]} (t, \xi )\right| \left\| (1+|n|)^s \left\| \sigma _n \right\| _{L^2 \left( Q_n \cap \ {\text {supp}} \widehat{U_k[\psi _0]}(t) \right) } \right\| _{\ell ^q}\\&\le \sup _{\xi \in {\mathbb {R}}^d} \left| \widehat{U_k[\psi _0]} (t, \xi )\right| \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le C A \right) }. \end{aligned}$$

This yields the desired inequality in (4.3). $\square$

Lemma 4.2

Let $s<0$, $q\in [1,2]$, $2\le A \ll N$ and $\sum =\{ Ne_d, - Ne_d, 2Ne_d\}$ where $e_d= (0,\dots ,0, 1) \in {\mathbb {R}}^d.$ If $0<T\ll N^{-2},$ then we have

$$\begin{aligned} \Vert U_{2\sigma +1} [\psi _0] (T)\Vert _{M^{2,q}_s} > rsim R A^{-\frac{d}{2}} N^{-s} \rho ^{2\sigma } \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le A \right) }, \end{aligned}$$

where $\rho = R N^{-s} A^{d/2} T^{\frac{1}{2\sigma }}.$

Proof

By (2.3), we note that

$$\begin{aligned} \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s}^q&= \sum _{n\in {\mathbb {Z}}^d} \Vert \square _n(U_{2\sigma +1}[\psi _0](T))\Vert ^q_{L^2} (1+|n|)^{sq} \\&= \sum _{n\in {\mathbb {Z}}^d} \Vert \sigma _n \widehat{ U_{2\sigma +1}[\psi _0]}(T)\Vert ^q_{L^2} (1+|n|)^{sq}\\& > rsim \sum _{n\in {\mathbb {Z}}^d} \frac{1}{(1+|n|)^{-sq}} \left( \int _{Q_n} | \widehat{ U_{2\sigma +1}[\psi _0]}(\xi , T)|^{2} d\xi \right) ^{q/2}, \end{aligned}$$

where $Q_n$ is a unit cube centered at $n\in {\mathbb {Z}}^d.$ Arguing as before in the proof of Lemma 3.8 (specifically, by (3.5)), for $\xi \in Q_A= [-A/2, A/2)^d$, we have

$$\begin{aligned} \left| \widehat{U_{2\sigma +1}[\psi _0]}(T, \xi )\right| > rsim \left( R A^{-d/2} N^{-s} \right) ^{2\sigma +1} (A^d)^{2\sigma } T \chi _{(2\sigma +1)^{-1} Q_{A}} (\xi ). \end{aligned}$$

It follows that

$$\begin{aligned} \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s}& > rsim \left( R A^{-d/2} N^{-s} \right) ^{2\sigma +1} (A^d)^{2\sigma } T \left( \sum _{|n|= \lfloor -A/2 \rfloor }^{\lfloor A/2 \rfloor } \frac{1}{(1+|n|)^{-sq}} \right) ^{1/q}\\& > rsim R A^{-d/2} N^{-s} \rho ^{2\sigma } \left( \sum _{|n|= \lfloor -A/2 \rfloor }^{\lfloor A/2 \rfloor } \frac{1}{(1+|n|)^{-sq}} \right) ^{1/q}, \end{aligned}$$

where the floor function is ${\displaystyle \lfloor x\rfloor =\max \left( m\in {\mathbb {Z}}\mid m\le x\right) }$ and $\rho = R N^{-s} A^{d/2} T^{\frac{1}{2\sigma }}$. $\square$

4.2 A Priori Estimates: $2\le q\le \infty$

Then we have, for any $s\in {\mathbb {R}},$

$$\begin{aligned} \Vert \psi _0\Vert _{{\mathcal {F}}L^q_s} \sim R, \quad \Vert \psi _0\Vert _{M_A} \sim R A^{d (\frac{1}{2}-\frac{1}{q})} N^{-s}. \end{aligned}$$

Lemma 4.3

Let $s<0$ and $2 \le q \le \infty .$ Then there exists $C>0$ depending only on $d, \sigma$ and s such that following holds.

$$\begin{aligned}&\Vert U_1[\psi _0](T)\Vert _{M^{2,q}_s} \le C R ,\quad \forall T\ge 0, \end{aligned}$$

(4.4)

$$\begin{aligned}&\Vert U_k[\psi _0](T)\Vert _{M^{2,q}_s} \lesssim \rho _2^{k-1} C^{k} A^{-d/q} RN^{-s} \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le A \right) }, \end{aligned}$$

(4.5)

where $\rho _2= R N^{-s} A^{d \left( 1- \frac{1}{q} \right) } T^{\frac{1}{2\sigma }}.$

Proof

By Lemma 2.3, we have

$$\begin{aligned} \Vert U_1[\psi _0](T)\Vert _{M_s^{2,q}} \lesssim \Vert U_1[\psi _0](T)\Vert _{{\mathcal {F}}L_s^q} \le C R. \end{aligned}$$

The proof of (4.5) is similar to Lemmas 4.1, (4.3) and 3.7, (3.4), we omit the details. $\square$

The next lemma is the analogue of Lemmas 3.8 and 4.2, so we leave out its proof.

Lemma 4.4

Let $s<0$, $2 \le q \le \infty ,$ $1\le A \ll N$ and $\sum =\{ Ne_d, - Ne_d, 2Ne_d\}$ where $e_d= (0,\dots ,0, 1) \in {\mathbb {R}}^d$. If $0<T\ll N^{-2},$ then we have

$$\begin{aligned} \Vert U_{2\sigma +1} [\psi _0] (T)\Vert _{M^{2,q}_s} > rsim R A^{-\frac{d}{q}} N^{-s} \rho _2^{2\sigma } \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le A \right) }, \end{aligned}$$

where $\rho _2= R N^{-s} A^{d- \frac{d}{q}} T^{\frac{1}{2\sigma }}$.

4.3 Proof of Theorem 1.5

Lemma 4.5

Let $s<0$. In the limit $A\rightarrow \infty$, we have, for $1\le q<\infty$,

$$\begin{aligned} \left\| (1+|n|)^s \right\| _{\ell ^q\left( 0\le |n| \le A \right) } \mathop {\sim }\limits _{A\rightarrow \infty }g_s^q(A):= {\left\{ \begin{array}{ll} 1 \quad \text {if} \ -sq>d,\\ \left( \log A\right) ^{1/q} \quad \text {if} \ -sq=d,\\ A^{1/q+s} \quad \text {if} \ -sq<d, \end{array}\right. } \end{aligned}$$

and $\displaystyle \left\| (1+|n|)^s\right\| _{\ell ^{\infty }\left( 0\le |n| \le A \right) } \mathop {\sim }\limits _{A\rightarrow \infty }1$.

Proof

Since $(1+|\xi |)^{sq}$ is a decreasing function in $|\xi |$, in view of the integral test and Lemma 3.9, we have, for $1\le q < \infty$,

$$\begin{aligned} \left\| (1+|n|)^s\right\| _{\ell ^q\left( 0\le |n|\le A \right) }&= \left( \sum _{0\le |n|\le A} \frac{1}{(1+|n|)^{-sq}} \right) ^{1/q} \mathop {\sim }\limits _{A\rightarrow \infty }\left( \int _{|\xi |\le A} \frac{d\xi }{(1+|\xi |)^{-sq}}\right) ^{1/q} \\&\mathop {\sim }\limits _{A\rightarrow \infty }\left( \int _0^A \frac{r^{d-1}dr}{(1+r)^{-sq}}\right) ^{1/q} , \end{aligned}$$

hence the result for q finite. The case $q=\infty$ is straightforward. $\square$

To prove Theorem 1.5, we distinguish two cases.

First case: $1\le q \le 2.$ By Corollary 3.4, we have the existence of solution to (1.1) in $M_{A}$ up to time T whenever $\rho =RN^{-s} A^{d/2} T^{1/(2\sigma )} \ll 1.$ In view of Lemma 3.7 and since $\rho <1,$ $\sum _{\ell =2}^{\infty } \Vert U_{2\sigma \ell +1}[\psi _0] (T)\Vert _{M^{2,q}_s}$ can be dominated by the sum of a geometric series. Specifically, we have

$$\begin{aligned} \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0] (T) \right\| _{M^{2,q}_s}&\lesssim A^{-d/2} R N^{-s} g_s^q(A) \sum _{\ell =2}^{\infty } \rho ^{2\sigma \ell } \nonumber \\&\lesssim A^{-d/2} R N^{-s} g_s^q(A)\rho ^{4\sigma }. \end{aligned}$$

(4.6)

By Corollary 3.4 and the triangle inequality, we obtain

$$\begin{aligned} \Vert \psi (T)\Vert _{M^{2,q}_s}&= \left\| \sum _{\ell =0}^{\infty } U_{2\sigma \ell +1}[\psi _0] \right\| _{M^{2,q}_s} \\&\ge \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} - \Vert U_1[\psi _0](T)\Vert _{M^{2, q}_s}- \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0](T) \right\| _{M^{2,q}_s}. \end{aligned}$$

In order to ensure

$$\begin{aligned} \Vert \psi (T)\Vert _{M^{2,q}_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s}, \end{aligned}$$

we rely on the conditions

$$\begin{aligned}&\Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} \gg \Vert U_1[\psi _0](T)\Vert _{M^{2,q}_s}, \end{aligned}$$

(4.7)

$$\begin{aligned}&\Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} \gg \left\| \sum _{\ell =2}^{\infty } U_{2\sigma \ell +1}[\psi _0](T) \right\| _{M^{2,q}_s}. \end{aligned}$$

(4.8)

In view of Lemmas 4.1 and 4.2, (4.7) amount to the condition

$$\begin{aligned} R \rho ^{2\sigma } A^{-\frac{d}{2}} N^{-s} g^q_s(A) \gg R \end{aligned}$$

(4.9)

In view of Lemmas 4.1 and 4.2, and (4.6), (4.8) amount to the condition

$$\begin{aligned} T\ll N^{-2}, \quad \rho \ll 1, \quad R \rho ^{2\sigma } A^{-\frac{d}{2}} N^{-s} g^q_s(A) \gg R \rho ^{4\sigma } A^{-d/2} N^{-s} g_s^q(A). \end{aligned}$$

(4.10)

We now choose A, R and T so that conditions (4.9) and (4.10) are satisfied. To this end, we set

$$\begin{aligned} R= (\log N)^{-1}, \quad A \sim (\log N)^{-\frac{2\sigma +2}{|s|}} N, \quad T= (A^{-d/2} N^{s})^{2\sigma }. \end{aligned}$$

Then we have

$$\begin{aligned} \rho = R N^{-s} A^{d/2} T^{\frac{1}{2\sigma }}=(\log N)^{-1}\ll 1. \end{aligned}$$

Note that

$$\begin{aligned} T= (\log N)^{-\frac{2\sigma +2}{|s|} d (\frac{1}{2} -1) 2\sigma } N^{-d \sigma + 2\sigma s}. \end{aligned}$$

Since $s< \frac{d}{2}- \frac{1}{\sigma }$ and $\log N = \mathcal {O}(N^{\epsilon })$ for any $\epsilon >0,$ we have

$$\begin{aligned} T \ll N^{-2}. \end{aligned}$$

We have $g^{q}_s(A) > rsim A^{\frac{d}{q}+s}$ for any $s<0$ and $A\ge 1.$ Thus, for $1\le q \le 2,$ we have

$$\begin{aligned} R \rho ^{2\sigma } A^{-\frac{d}{2}} N^{-s} g^q_s(A)& > rsim (\log N)^{- (2\sigma +1)} (\log N)^{2\sigma +2} A^{d \left( \frac{1}{q}- \frac{1}{2} \right) } > rsim \log N \\&\gg (\log N)^{-1}=R, \end{aligned}$$

and (4.9) is satisfied. Thus, we have $\Vert \psi (T)\Vert _{M^{2,q}_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} > rsim \log N.$ Since $\Vert \psi _0\Vert _{M^{2,q}_s} \lesssim R = (\log N)^{-1}$ and $T\ll N^{-2},$ we get norm inflation by letting $N\rightarrow \infty .$ This completes the proof for $q\in [1,2]$.

Second case: $2\le q \le \infty$. By Corollary 3.4, we have the existence of solution to (1.1) in $M_{A}$ up to time T whenever $\rho _2=RN^{-s} A^{d \left( 1- \frac{1}{q}\right) } T^{1/(2\sigma )} \ll 1.$ By Lemmas 4.3 and 4.4, the conditions

$$\begin{aligned} T\ll N^{-2}, \quad \rho _2\ll 1 \quad \text {and} \quad R \rho _2^{2\sigma } A^{-\frac{d}{q}} N^{-s} g^q_s(A) \gg R \end{aligned}$$

(4.11)

ensures that

$$\begin{aligned} \Vert \psi (T)\Vert _{M^{2,q}_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} \sim R \rho _1^{2\sigma } A^{-\frac{d}{q}} N^{-s} g^q_s(A). \end{aligned}$$

We now choose A, R and T so that conditions (4.11) satisfied. To this end, we set

$$\begin{aligned} R= (\log N)^{-1}, \quad A \sim (\log N)^{-\frac{2\sigma +2}{|s|}} N, \quad T= (A^{d(\frac{1}{q}-1)} N^{s})^{2\sigma }. \end{aligned}$$

Then we have

$$\begin{aligned} \rho _2= R N^{-s} A^{d- \frac{d}{q}} T^{\frac{1}{2\sigma }}=(\log N)^{-1}\ll 1. \end{aligned}$$

Note that

$$\begin{aligned} T= (\log N)^{-\frac{2\sigma +2}{|s|} d (\frac{1}{q} -1) 2\sigma } N^{d (\frac{1}{q} -1) 2\sigma + 2\sigma s}. \end{aligned}$$

Since $s< d \left( 1-\frac{1}{q} \right) - \frac{1}{\sigma }$ and $\log N = \mathcal {O}(N^{\epsilon })$ for any $\epsilon >0,$ we have

$$\begin{aligned} T \ll N^{-2}. \end{aligned}$$

Note that $g^{q}_s(A) > rsim A^{\frac{d}{q}+s}$ for any $s<0$ and $A\ge 1$ and so

$$\begin{aligned} R \rho _2^{2\sigma } A^{-\frac{d}{q}} N^{-s} g^q_s(A) > rsim \log N \gg (\log N)^{-1}=R. \end{aligned}$$

Thus, we have $\Vert \psi (T)\Vert _{M^{2,q}_s} > rsim \Vert U_{2\sigma +1}[\psi _0](T)\Vert _{M^{2,q}_s} > rsim \log N.$ Since $\Vert \psi _0\Vert _{M^{2,q}_s} \lesssim R = (\log N)^{-1}$ and $T\ll N^{-2},$ we get norm inflation by letting $N\rightarrow \infty .$ This completes the proof of Theorem 1.5.

5 Norm Inflation as a By-product of Geometric Optics

The proof of Theorems 1.3 and 1.6 follows the same strategy as in [11]: through a suitable rescaling, we turn the ill-posedness result into an asymptotic result, which can be expressed in the framework of weakly nonlinear geometric optics. More precisely, we change the unknown function $\psi$ to u, via

$$\begin{aligned} u^{\varepsilon } (t,x) = \varepsilon ^{\frac{2-J}{2\sigma }} \psi (\varepsilon t, x), \end{aligned}$$

(5.1)

where the parameter $\varepsilon$ will tend to zero. For $\psi$ solution to (1.1), $u^{\varepsilon }$ solves

$$\begin{aligned} i\varepsilon \partial _t u^{\varepsilon } + \frac{\varepsilon ^2}{2}\Delta u^{\varepsilon } = \mu \varepsilon ^{J}|u^{\varepsilon }|^{2s}u^{\varepsilon }. \end{aligned}$$

(5.2)

The case $J=1$ corresponds to weakly nonlinear geometric optics (WNLGO), as defined in [9]. As noticed in [11] in the framework of Sobolev spaces, a phenomenon of infinite loss of regularity can be proved via this WNLGO setting, under the assumption $s<-1/(2\sigma )$ in the analogue of Theorems 1.3 and 1.6. Like in that paper, in order to weaken the assumption on s to $s<-1/(2\sigma +1)$, we will have to consider some value $J>1$, and perform some “asymptotic sin”, in the sense that we change the hierarchy in an asymptotic expansion involving the limit $\varepsilon \rightarrow 0$.

The heuristic idea is the same as in [15]: when negative regularity is involved, the zero Fourier mode plays a stronger role than (large) non-zero modes, which come with a small factor. With the scaling (5.1) in mind, our goal is to show that we may consider initial data (for u) of the form

$$\begin{aligned} u(0,x) = \sum _{j\not =0}e^{ij\cdot x/\varepsilon }\alpha _j(x), \end{aligned}$$

that is containing only rapidly oscillatory terms, and such that the evolution under (5.2) creates a non-trivial non-oscillatory term.

To be more specific, recall the strategy of multiphase nonlinear geometric optics (see [10] for more details): we plug an ansatz of the form

$$\begin{aligned} u(t,x) = \sum _j e^{i\phi _j(t,x)/\varepsilon } a_j(t,x) \end{aligned}$$

into (5.2), and order the powers of $\varepsilon$. The most singular term is of order $\varepsilon ^0$, it is the eikonal equation:

$$\begin{aligned} {\partial }_t \phi _j +\frac{1}{2}|\nabla \phi _j|^2=0. \end{aligned}$$

In the case of an initial phase $\phi _j(0,x) = j\cdot x$, no caustic appears, and the global solution is given by

$$\begin{aligned} \phi _j(t,x) = j\cdot x-\frac{|j|^2}{2}t. \end{aligned}$$

(5.3)

In the sequel, we consider such phases, for $j\in {\mathbb {Z}}^d$. The next term in the hierarchy is of order $\varepsilon ^1$, but as evoked above, we “cheat”, and incorporate some nonlinear effects even if $J>1$ (and $J<2$),

$$\begin{aligned} {\partial }_t a_j +j\cdot \nabla _x a_j = -i\mu \varepsilon ^{J-1}\sum _{\phi _{k_1}-\phi _{k_2}+\dots +\phi _{k_{2\sigma +1}} = \phi _j}a_{k_1}\bar{a}_{k_2}\dots a_{k_{2\sigma +1}}, \end{aligned}$$

(5.4)

where we have used $\nabla \phi _j(t,x)=j$. Again in view of the specific form of the phase (5.3), the condition on the sum involves a resonant condition, $(k_1,k_2,\dots ,k_{2\sigma +1})\in {{\mathcal {R}}}_j$, where

$$\begin{aligned} { {\mathcal {R}}}_j&= \left\{ (k_\ell )_{1\le \ell \le 2\sigma +1},\ \sum _{\ell =1}^{2\sigma +1} (-1)^{\ell +1} k_\ell =j,\ \sum _{\ell =1}^{2\sigma +1} (-1)^{\ell +1} |k_\ell |^2=|j|^2\right\} . \end{aligned}$$

In the cubic case $\sigma =1$, those sets are described exactly:

Lemma 5.1

(See [10, 16]) Suppose $\sigma =1$.

• If $d=1$, then ${{\mathcal {R}}}_j=\{ (j,\ell ,\ell ),\ (\ell ,\ell ,j)\ ;\ \ell \in {\mathbb {Z}}\}$.

• If $d\ge 2$, then $(k_1,k_2,k_3)\in {{\mathcal {R}}}_j$ precisely when the endpoints of the vectors $k_1,k_2,k_3,j$ for four corners of a non-degenerate rectangle with $k_2$ and j opposing each other, or when this quadruplet corresponds to one of the following two degenerate cases: $(k_1=j,k_2=k_3)$ or $(k_1=k_2, k_3=j)$.

The above lemma explains why our approach distinguishes the one-dimensional case and the multi-dimensional case, and in particular why the cubic one-dimensional case is left out in Theorems 1.3 and 1.6.

5.1 Multi-dimensional Case

The leading idea in [11] is to start from three non-trivial modes only, in the case $d\ge 2$, and create at least one new mode (possibly more if $\sigma \ge 2$), corresponding to $j=0$.

Lemma 5.2

Let $d\ge 2$ and $\sigma \in {\mathbb {N}}^*$. Define $k_1,k_2,k_3\in {\mathbb {Z}}^d$ as

$$\begin{aligned} k_1=(1,0,\dots ,0), \quad k_2=(1,1,0,\dots ,0),\quad k_3=(0,1,0,\dots ,0). \end{aligned}$$

For initial data of the form

$$\begin{aligned} u(0,x) = \alpha (x)\sum _{j=1}^3 e^{ik_j\cdot x/\varepsilon },\quad \alpha \in {\mathcal {S}}({\mathbb {R}}^d)\setminus \{0\}, \end{aligned}$$

we have $a_{0\mid t=0}=0$ and ${\partial }_t a_{0\mid t=0}=\varepsilon ^{J-1}c_0\alpha (x)$, with $c_0=\sharp {\mathcal {R}}_0\ge 1$.

This lemma is straightforward, in view of (5.4), and since $(k_1,k_2,k_3)\in {\mathcal {R}}_0$ if $\sigma =1$, $(k_1,k_2,k_3,k_1,k_1,\dots ,k_1)\in {\mathcal {R}}_0$ if $\sigma \ge 2$.

5.2 One-Dimensional Case

In the one-dimensional case, we have a similar result, provided that the nonlinearity is at least quintic, in view of [12, Lemma 4.2] (and Example 4.3 there):

Lemma 5.3

Let $d=1$ and $\sigma \ge 2$. Define $k_1,k_2,k_3,k_4,k_5\in {\mathbb {Z}}$ as

$$\begin{aligned} (k_1,k_2,k_3,k_4,k_5)= (2,-1,-2,4,3). \end{aligned}$$

For initial data of the form

$$\begin{aligned} u(0,x) = \alpha (x)\sum _{j=1}^5 e^{ik_j x/\varepsilon },\quad \alpha \in {\mathcal {S}}({\mathbb {R}})\setminus \{0\}, \end{aligned}$$

we have $a_{0\mid t=0}=0$ and ${\partial }_t a_{0\mid t=0}=\varepsilon ^{J-1}c_0\alpha (x)$, with $c_0=\sharp {\mathcal {R}}_0\ge 1$.

5.3 How to Conclude

Supposing that we can prove that the geometric expansion recalled above provides an approximation $u_\mathrm{app}$ for the solution u to (5.2), suitable in the sense that the error is measured in a sufficiently strong norm, the idea is that both $u_{\mid t=0}$ and $u-u_\mathrm{app}$ are small in spaces involving negative regularity in x, while Lemma 5.2 or 5.3 implies that $u_\mathrm{app}$ is large in many spaces.

6 (Very) Weakly Nonlinear Geometric Optics

6.1 A Convenient Functional Framework

Throughout this section, we denote by X a Banach algebra in the space variable, that is

$$\begin{aligned} \exists C>0,\quad \Vert fg\Vert _X\le C\Vert f\Vert _X\Vert g\Vert _X,\quad \forall f,g\in X. \end{aligned}$$

(6.1)

We suppose that X is translation invariant and, denoting by $\tau _kf(x) = f(x-k)$,

$$\begin{aligned} \Vert \tau _k f \Vert _X = \Vert f\Vert _{X},\quad \forall k\in {\mathbb {R}}^d,\ \forall f\in X. \end{aligned}$$

(6.2)

We assume in addition that the multiplication by plane wave oscillations leaves the X-norm invariant,

$$\begin{aligned} \Vert f e_k\Vert _X= \Vert f\Vert _X,\quad \forall k\in {\mathbb {R}}^d,\text { where } e_k(x)= e^{ik\cdot x}. \end{aligned}$$

(6.3)

Note that this assumption rules out Sobolev spaces $H^s({\mathbb {R}}^d)$, unless $s=0$. Finally, we assume that the Schrödinger group acts on X, at least locally in time:

Assumption 6.1

There exists $T_0$ such that $e^{i\frac{t}{2}\Delta }$ maps X to X for $t\in [0,T_0]$, and

$$\begin{aligned} \exists C>0,\quad \Vert e^{i\frac{t}{2}\Delta }\Vert _{\mathcal L(X,X)}\le C,\quad \forall t\in [0,T_0]. \end{aligned}$$

In [10, 19], the case $X= {\mathcal {F}} L^1(M)$ was considered, with $M={\mathbb {T}}^d$ (a choice resumed in [12]) or ${\mathbb {R}}^d$. In [11, 30], the choice $X= {\mathcal {F}} L^1\cap L^2({\mathbb {R}}^d)$ was motivated by the presence of more singular nonlocal nonlinearities. We shall consider later two sorts of X spaces: ${\mathcal {F}} L^1\cap {\mathcal {F}} L^p$ or ${\mathcal {F}} L^1\cap M^{1,1}$.

We denote by

$$\begin{aligned} Y=\left\{ (a_j)_{j\in {\mathbb {Z}}^d},\ \sum _{j\in {\mathbb {Z}}^d} \Vert a_j\Vert _X<\infty \right\} =\ell ^1(X), \end{aligned}$$

and

$$\begin{aligned} Y_2=\left\{ (a_j)_{j\in {\mathbb {Z}}^d}\in Y,\ \sum _{j\in {\mathbb {Z}}^d} \left( \left\langle j\right\rangle ^2\Vert a_j\Vert _X+\left\langle j\right\rangle \Vert \nabla a_j\Vert _X+\Vert \Delta a_j\Vert _X\right) <\infty \right\} . \end{aligned}$$

We suppose that u solves (5.2) with initial data

$$\begin{aligned} u(0,x) = \sum _{j\in {\mathbb {Z}}^d}\alpha _j(x) e^{ij\cdot x/\varepsilon }. \end{aligned}$$

6.2 Construction of the Approximate Solution

The approximate solution is given by

$$\begin{aligned} u_\mathrm{app}(t,x) =\sum _{j\in {\mathbb {Z}}^d} a_j(t,x) e^{i\phi _j(t,x)/\varepsilon }, \end{aligned}$$

where $\phi _j$ is given by (5.3) and the $a_j$’s solve (5.4), with initial data $\alpha _j$.

Lemma 6.2

Let $d\ge 1$, $\sigma \in {\mathbb {N}}^*$ and $J\ge 1$.

• If $(\alpha _j)_{j\in {\mathbb {Z}}^d}\in Y$, then there exists $T>0$ independent of $\varepsilon \in [0,1]$ and a unique solution $(a_j)_{j\in {\mathbb {Z}}^d}\in C([0,T];Y)$ to the system (5.4), such that $a_{j\mid t=0}=\alpha _j$ for all $j\in {\mathbb {Z}}^d$.

• If in addition $(\alpha _j)_{j\in {\mathbb {Z}}^d}\in Y_2$, then $(a_j)_{j\in {\mathbb {Z}}^d}\in C([0,T];Y_2)$.

Proof

In view of Duhamel’s formula for (5.4),

$$\begin{aligned} a_j(t,x) = a_j(0,x-jt) -i\lambda \sum _{(k_1,k_2,\dots ,k_{2\sigma +1})\in {{\mathcal {R}}}_j} \int _0^t \left( a_{k_1}\bar{a}_{k_2}\dots a_{k_{2\sigma +1}}\right) (s,x - j(t-s))ds, \end{aligned}$$

the first point of the lemma is straightforward, as an easy consequence of (6.2) and (6.1), and a fixed point argument. The second point requires a little bit more care: it was proven in [10, Proposition 5.12] in the case $X= {\mathcal {F}} L^1$ (Wiener algebra), and the proof relies only on the properties of X required at the beginning of this section. $\square$

From now on, we assume $(\alpha _j)_{j\in {\mathbb {Z}}^d}\in Y_2$.

6.3 Error Estimate

First, in view of the assumptions made in this section, a standard fixed point argument yields, in view of (6.1) and Assumption 6.1:

Lemma 6.3

Let $d\ge 1$, $\sigma \in {\mathbb {N}}^*$ and $J\ge 1$. If $u_0\in X$, then there exists $T^\varepsilon >0$ and a unique solution $u\in C([0,T^\varepsilon ];X)$ to (5.2) such that $u_{\mid t=0}=u_0$.

To construct the approximate solution $u_\mathrm{app}$, we have discarded two families of terms:

• Non-resonant terms, involving the source term

  $$\begin{aligned} r_1:= \mu \varepsilon ^J\sum _j\sum _{(k_1,k_2,\dots ,k_{2\sigma +1})\not \in {\mathcal {R}}_j}a_{k_1}\bar{a}_{k_2}\dots a_{k_{2\sigma +1}} e^{i(\phi _{k_1}-\phi _{k_2}+\dots +\phi _{2\sigma +1})/\varepsilon }. \end{aligned}$$

• Higher order terms, involving

  $$\begin{aligned} r_2:= \frac{\varepsilon ^2}{2}\sum _j \Delta a_j e^{i\phi _j/\varepsilon }. \end{aligned}$$

Indeed, $u_\mathrm{app}$ solves

$$\begin{aligned} i\varepsilon {\partial }_t u_\mathrm{app} + \frac{\varepsilon ^2}{2}\Delta u_\mathrm{app} =\mu \varepsilon ^J|u_\mathrm{app} |^{2\sigma }u_\mathrm{app} +r_1+r_2. \end{aligned}$$

(6.4)

Duhamel’s formula for $u-u_\mathrm{app}=:w$ reads

$$\begin{aligned} w(t)&= -i\mu \varepsilon ^{J-1}\int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta }\left( |u|^{2\sigma }u -|u_\mathrm{app} |^{2\sigma }u_\mathrm{app} \right) (\tau )d\tau \\&\quad -i\mu \varepsilon ^{J-1}\sum _{j=1,2}\int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta }r_j(\tau )d\tau . \end{aligned}$$

In view of our assumptions on X, we readily have, thanks to Minkowski inequality,

$$\begin{aligned} \Vert w(t)\Vert _X&\le C \int _0^t \left( \Vert u_\mathrm{app}(\tau )\Vert _X^{2\sigma } + \Vert w(\tau )\Vert _X^{2\sigma } \right) \Vert w(\tau )\Vert _Xd\tau \\&\quad + C\varepsilon ^{-1}\sum _{j=1,2}\left\| \int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta }r_j(\tau )d\tau \right\| _X, \end{aligned}$$

for some C independent of $\varepsilon \in [0,1]$ and $t\in [0,T_0]$. In view of the second point of Lemma 6.2, we readily have

$$\begin{aligned} \left\| \int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta }r_2(\tau )d\tau \right\| _X\lesssim \varepsilon ^2. \end{aligned}$$

By construction, $r_1$ is the sum of terms of the form $g(t,x) e^{ik\cdot x/\varepsilon -\omega t/(2\varepsilon )}$, with $k\in {\mathbb {Z}}^d$, $\omega \in {\mathbb {Z}}$, and the non-resonance property reads exactly $|k|^2\not =\omega$.

Lemma 6.4

Let $k\in {\mathbb {R}}^d$, $\omega \in {\mathbb {R}}$, with $|k|^2\not =\omega$. Define

$$\begin{aligned} D^\varepsilon (t,x) =\int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta } \left( g(\tau ,x) e^{ik\cdot x/\varepsilon -i\omega \tau /(2\varepsilon )}\right) d\tau . \end{aligned}$$

Then we have

$$\begin{aligned} D^\varepsilon (t,x)&= \frac{-2i\varepsilon }{|k|^2-\omega } e^{i\varepsilon \frac{t-\tau }{2}\Delta } \left( g(\tau ,x)e^{ik\cdot x/\varepsilon -i\omega \tau /(2\varepsilon )}\right) \Big |_0^t\\&+\,\frac{2i\varepsilon }{|k|^2-\omega } \int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta } \left( e^{ik\cdot x/\varepsilon -i\omega \tau /(2\varepsilon )}\left( \frac{i}{2}\left( \varepsilon \Delta g +2k\cdot \nabla g\right) +{\partial }_t g\right) (\tau ,x)\right) d\tau . \end{aligned}$$

In particular, for $t\in [0,T_0]$,

$$\begin{aligned} \Vert D^\varepsilon (t)\Vert _X\lesssim \frac{\varepsilon }{\left||k|^2-\omega \right|}&\big (\Vert g\Vert _{L^\infty ([0,t];X)} + \Vert \Delta g\Vert _{L^\infty ([0,t];X)} + |k| \Vert \nabla g\Vert _{L^\infty ([0,t];X)}\\&\quad +\Vert {\partial }_t g\Vert _{L^\infty ([0,t];X)}\big ). \end{aligned}$$

Proof

The last estimate follows directly from the identity of the lemma, (6.3) and Assumption 6.1, so we only address the identity, which is essentially established in [10, Lemma 5.7] (up to the typos there). Setting $\eta = \xi -k/\varepsilon$, the (spatial) Fourier transform of D is given by

$$\begin{aligned} \widehat{D}^\varepsilon (t,\xi )&= e^{-i\varepsilon t|\eta +k/\varepsilon |^2/2}\int _0^t e^{i\varepsilon \tau |\eta +k/\varepsilon |^2/2} \, \hat{b}\left( \tau ,\eta \right) e^{-i\omega \tau /(2\varepsilon )}d\tau \\&=e^{-i\varepsilon t|\eta +k/\varepsilon |^2/2}\int _0^t e^{i\tau \theta /2} \, \hat{b}\left( \tau ,\eta \right) d\tau \\&=e^{-i\varepsilon t|\eta +k/\varepsilon |^2/2}\int _0^t e^{i\tau \theta _2/2} \,e^{i\tau \theta _1/2}\, \hat{b}\left( \tau ,\eta \right) d\tau , \end{aligned}$$

where we have denoted

$$\begin{aligned} \theta = \varepsilon \left|\eta +\frac{k}{\varepsilon }\right|^2 -\frac{\omega }{\varepsilon } = \underbrace{\varepsilon |\eta |^2 +2 k\cdot \eta }_{\theta _1} +\underbrace{\frac{|k|^2-\omega }{\varepsilon }}_{\theta _2}. \end{aligned}$$

Integrate by parts, by first integrating $e^{i\tau \theta _2/2}$:

$$\begin{aligned} e^{i\varepsilon \frac{t}{2}|\xi |^2} \widehat{D}^\varepsilon (t,\xi ) = -\frac{2i}{\theta _2} e^{i\tau \theta /2} \hat{b}\left( \tau ,\eta \right) \Big |_0^t +\frac{2i}{\theta _2}\int _0^t e^{i\tau \theta /2} \left( i\frac{\theta _1}{2}\widehat{b}\left( \tau ,\eta \right) + \widehat{{\partial }_t b}\left( \tau ,\eta \right) \right) d\tau . \end{aligned}$$

The identity follows by inverting the Fourier transform. $\square$

We infer:

Proposition 6.5

Let $d\ge 1$, $\sigma \in {\mathbb {N}}^*$, $J\ge 1$, and $(\alpha _j)_{j\in {\mathbb {Z}}^d}\in Y_2$. Then for T as in Lemma 6.2,

$$\begin{aligned} \Vert u-u_\mathrm{app}\Vert _{L^\infty ([0,T];X)}\lesssim \varepsilon . \end{aligned}$$

Proof

First, Lemma 6.2 and (5.4) imply that we also have $({\partial }_t a_j)_{j\in {\mathbb {Z}}^d}\in C([0,T];Y)$. Then, in view of these properties and Lemma 6.4, we have

$$\begin{aligned} \left\| \int _0^t e^{i\varepsilon \frac{t-\tau }{2}\Delta }r_1(\tau )d\tau \right\| _X\lesssim \varepsilon ^{J+1}, \end{aligned}$$

where we have used the fact that in the application of Lemma 6.4, $\left||k|^2-\omega \right|\ge 1$, since now $k\in {\mathbb {Z}}^d$ and $\omega \in {\mathbb {Z}}$. We infer

$$\begin{aligned} \Vert w(t)\Vert _X\le C \int _0^t \left( \Vert u_\mathrm{app}(\tau )\Vert _X^{2\sigma } + \Vert w(\tau )\Vert _X^{2\sigma } \right) \Vert w(\tau )\Vert _Xd\tau + C\varepsilon ^J + C\varepsilon , \end{aligned}$$

where C is independent of $\varepsilon \in [0,1]$ and $t\in [0,T]$. Lemmas 6.2 and 6.3 yield $w\in C([0,\min (T,T^\varepsilon )];X)$. Since $w_{\mid t=0}=0$, the above inequality and a standard continuity argument imply that $u\in C([0,T];X)$ provided that $\varepsilon >0$ is sufficiently small, along with the announced error estimate. $\square$

7 Norm Inflation with Infinite Loss of Regularity

7.1 Proof of Theorem 1.3

For $1<J<2$ to be fixed later, let $u^\varepsilon$ defined by (5.1), and consider the initial data given by Lemma 5.2 (if $d\ge 2$) or Lemma 5.3 (if $d=1$ and $\sigma \ge 2$). We apply the analysis from Sect. 6 with $X= {\mathcal {F}} L^1\cap {\mathcal {F}} L^\infty$. This is obviously a Banach algebra, (6.1) holds, thanks to Young inequality, the X-norm is invariant by translation, and by multiplication by plane wave oscillations as in (6.3). Assumption 6.1 is satisfied with $C=1$ for any $T_0>0$, since the Schrödinger group is a Fourier multiplier of modulus one. We can therefore invoke the conclusions of Lemma 5.2 (if $d\ge 2$), Lemma 5.3 (if $d=1$ and $\sigma \ge 2$), and Proposition 6.5 (in all cases). In order to translate these properties involving u solving (5.2) in terms of $\psi$ solving (1.1), we use the following lemma:

Lemma 7.1

Let $d\ge 1$. For $f\in \mathcal {S}'({\mathbb {R}}^d)$ and $j \in {\mathbb {R}}^d,$ denote

$$\begin{aligned} I^{\varepsilon }(f, j)(x) = f(x) e^{ij \cdot x/\varepsilon }. \end{aligned}$$

• For all $s \in {\mathbb {R}}$, $p\in [1,\infty ]$, and $f\in {\mathcal {F}}L^{p}_{s}({\mathbb {R}}^d)$, $\Vert I^{\varepsilon }(f, 0)\Vert _{{\mathcal {F}}L^{p}_{s}} = \Vert f\Vert _{{\mathcal {F}}L^p_{s}}$.

• Let $j \in {\mathbb {R}}^d\setminus \{ 0\}$. For all $s \le 0$, there exists $C= C(j)$ independent of $p\in [1,\infty ]$ such that for all $f\in \mathcal {S}({\mathbb {R}}^d)$,

  $$\begin{aligned} \Vert I^{\varepsilon } (f, j) \Vert _{{\mathcal {F}}L^{p}_{s}}\le C \varepsilon ^{ |s| } \Vert f\Vert _{{\mathcal {F}}L^{p}_{|s|}}. \end{aligned}$$

Proof

We have obviously

$$\begin{aligned} \widehat{ I^{\varepsilon }(f, j)} (\xi )= \hat{f} \left( \xi - \frac{ j}{\varepsilon }\right) . \end{aligned}$$

The first point is thus trivial. For the second one, if p is finite,

$$\begin{aligned} \Vert I^{\varepsilon }(f, j)\Vert _{ {\mathcal {F}}L^p_{s}}^p =\int \langle \xi \rangle ^{ps} \left| \hat{f} \left( \xi - \frac{ j}{\varepsilon }\right) \right| ^p d\xi . \end{aligned}$$

Note that, for $s \le 0$,

$$\begin{aligned} \Vert I^{\varepsilon }(f, j)\Vert _{ {\mathcal {F}}L^p_{s}}^p&= \int \left\langle \xi \right\rangle ^{ps}\left\langle \xi -\frac{j}{\varepsilon }\right\rangle ^{ps} \left\langle \xi -\frac{j}{\varepsilon }\right\rangle ^{p|s|} \left| \hat{f} \left( \xi - \frac{ j}{\varepsilon }\right) \right| ^p d\xi \\&\le \sup _{\xi \in {\mathbb {R}}^d} \left( \left\langle \xi \right\rangle ^{-1}\left\langle \xi -\frac{j}{\varepsilon }\right\rangle ^{-1} \right) ^{p|s|} \Vert f\Vert _{ {\mathcal {F}}L^p_{|s|}}^p \end{aligned}$$

For $j\not =0$,

$$\begin{aligned} \inf _{\xi \in {\mathbb {R}}^d} \left\langle \xi \right\rangle \left\langle \xi -\frac{j}{\varepsilon }\right\rangle > rsim \frac{1}{\varepsilon }, \end{aligned}$$

hence the second point of the lemma in the case p finite. The case $p=\infty$ follows from the same estimate, controlling the supremum of a product by the product of the suprema. $\square$

With $u_{\mid t=0}$ as in Lemmas 5.2 or  5.3, the above lemma yields, in view of (5.1), and for $s<0$,

$$\begin{aligned} \Vert \psi (0)\Vert _{ {\mathcal {F}} L^p_s}\lesssim \varepsilon ^{\frac{J-2}{2\sigma }+|s|}. \end{aligned}$$

This sequence of initial data is small (in ${\mathcal {F}} L^p_s$ for all p) provided that

$$\begin{aligned} |s|>\frac{2-J}{2\sigma }. \end{aligned}$$

(7.1)

Lemmas 5.2 and 5.3 show that there exists $\tau >0$ independent of $\varepsilon$ such that

$$\begin{aligned} \Vert a_0(\tau )\Vert _{ {\mathcal {F}} L^p_k} > rsim \varepsilon ^{J-1}, \quad \forall p\in [1,\infty ],\ \forall k\in {\mathbb {R}}. \end{aligned}$$

By construction,

$$\begin{aligned} \hat{u}_\mathrm{app}(t,\xi ) = \sum _{j\in {\mathbb {Z}}^d}e^{-it\frac{|j|^2}{2\varepsilon }}\hat{a}_j\left( t,\xi -\frac{j}{\varepsilon }\right) , \end{aligned}$$

so we infer, at least for $\varepsilon$ sufficiently small ($(a_j(\tau ))_{j\in {\mathbb {Z}}^d}\in Y$ from Lemma 6.2),

$$\begin{aligned} \Vert u_\mathrm{app}(\tau )\Vert _{ {\mathcal {F}} L^p_k} > rsim \varepsilon ^{J-1}, \quad \forall p\in [1,\infty ],\ \forall k\in {\mathbb {R}}. \end{aligned}$$

On the other hand, since $X= {\mathcal {F}} L^1\cap {\mathcal {F}} L^\infty \subset {\mathcal {F}} L^p$, Proposition 6.5 yields

$$\begin{aligned} \Vert u(\tau )-u_\mathrm{app}(\tau )\Vert _{ {\mathcal {F}} L^p_k} \lesssim \Vert u(\tau )-u_\mathrm{app}(\tau )\Vert _{X}\lesssim \varepsilon ,\quad \forall k\le 0, \end{aligned}$$

hence, if $J<2$,

$$\begin{aligned} \Vert u(\tau )\Vert _{ {\mathcal {F}} L^p_k} > rsim \varepsilon ^{J-1}, \quad \forall p\in [1,\infty ],\ \forall k\le 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \psi (\varepsilon \tau )\Vert _{ {\mathcal {F}} L^p_k} > rsim \varepsilon ^{\frac{J-2}{2\sigma }}\times \varepsilon ^{J-1}, \quad \forall p\in [1,\infty ],\ \forall k\in {\mathbb {R}}. \end{aligned}$$

The right hand side is unbounded as $\varepsilon \rightarrow 0$ provided that

$$\begin{aligned} \frac{J-2}{2\sigma }+J-1<0,\quad \text {that is},\quad J<\frac{2\sigma +2}{2\sigma +1}. \end{aligned}$$

Then given $s<-1/(2\sigma +1)$, we can always find a $J\in ]1,2[$ satisfying (7.1) and the above constraint. Theorem 1.3 follows in the case $k\le 0$, by taking for instance $\varepsilon _n=1/n$ and $t_n= \varepsilon _n\tau$. In the case $k>0$, we just recall the obvious estimate

$$\begin{aligned} \Vert \psi (\varepsilon \tau )\Vert _{ {\mathcal {F}} L^p_k}\ge \Vert \psi (\varepsilon \tau )\Vert _{ {\mathcal {F}} L^p}, \end{aligned}$$

and the proof of Theorem 1.3 is complete.

7.2 Proof of Theorem 1.6

In the case of modulation spaces, the proof goes along the same lines as above, up to adapting the space X and Lemma 7.1.

We choose $X= {\mathcal {F}} L^1\cap M^{1,1}$. Theorem 2.4 shows that the Banach algebra property (6.1) is satisfied. ${\mathcal {F}} L^1$ is translation invariant, and for $M^{1,1}$,

$$\begin{aligned} V_g\left( \tau _k f\right) (x,y)&= \int _{{\mathbb {R}}^d} f(t-k) \overline{g(t-x)}e^{-iy\cdot t}dt = e^{-iy\cdot k}\int _{{\mathbb {R}}^d} f(t) \overline{g(t+k-x)}e^{-iy\cdot t}dt\\&=e^{-iy\cdot k} V_g(f)(x-k,y), \end{aligned}$$

and thus

$$\begin{aligned} \Vert \tau _k f \Vert _{M^{1,1}} = \Vert V_g\left( \tau _k f\right) \Vert _{L^1_{x,y}} = \Vert f\Vert _{M^{1,1}} . \end{aligned}$$

We have used already the fact that (6.3) is satisfied on ${\mathcal {F}} L^1$. On $M^{1,1}$, this is the case too, since for $k\in {\mathbb {R}}^d$,

$$\begin{aligned} V_g\left( f e_k\right) (x,y) = \int _{{\mathbb {R}}^d} f(t) e^{ik\cdot t} \overline{g(t-x)}e^{-i y\cdot t}dt = V_g\left( f \right) (x,y-k), \end{aligned}$$

and so

$$\begin{aligned} \Vert fe_k\Vert _{M^{1,1}} = \Vert V_g\left( fe_k\right) \Vert _{L^1_{x,y}} = \Vert V_g\left( f\right) \Vert _{L^1_{x,y}} = \Vert f\Vert _{M^{1,1}} . \end{aligned}$$

Finally, Assumption 6.1 is satisfied thanks to Proposition 2.5, and we can again invoke Lemmas 5.2, 5.3, and Proposition 6.5.

Like before, Lemmas 5.2 and 5.3 show that there exists $\tau >0$ independent of $\varepsilon$ such that

$$\begin{aligned} \Vert a_0(\tau )\Vert _{M^{p,q}_k} > rsim \varepsilon ^{J-1}, \quad \forall p,q\in [1,\infty ],\ \forall k\in {\mathbb {R}}. \end{aligned}$$

By the same asymptotic decoupling phenomenon as in the case of ${\mathcal {F}} L^p$ spaces, we infer

$$\begin{aligned} \Vert u_\mathrm{app}(\tau )\Vert _{M^{p,q}_k} > rsim \varepsilon ^{J-1}, \quad \forall p,q\in [1,\infty ],\ \forall k\in {\mathbb {R}}. \end{aligned}$$

In view of Lemma 2.3, $X \hookrightarrow M^{p,q}_k$ for all $p,q\ge 1$ and all $k\le 0$, and so

$$\begin{aligned} \Vert u(\tau )-u_\mathrm{app}(\tau )\Vert _{M^{p,q}_k}\lesssim \Vert u(\tau )-u_\mathrm{app}(\tau )\Vert _X\lesssim \varepsilon , \quad \forall p,q\in [1,\infty ],\ \forall k\le 0. \end{aligned}$$

The analogue of Lemma 7.1 is the following:

Lemma 7.2

Let $d\ge 1$. For $f\in \mathcal {S}'({\mathbb {R}}^d)$ and $j \in {\mathbb {R}}^d,$ denote

$$\begin{aligned} I^{\varepsilon }(f, j)(x) = f(x) e^{ij \cdot x/\varepsilon }. \end{aligned}$$

• For all $s \in {\mathbb {R}}$, $p,q\in [1,\infty ]$, and $f\in M^{p,q}_s({\mathbb {R}}^d)$, $\Vert I^{\varepsilon }(f, 0)\Vert _{M^{p,q}_s} = \Vert f\Vert _{M^{p,q}_s}$.

• Let $j \in {\mathbb {R}}^d\setminus \{ 0\}$. For all $s \le 0$, there exists $C= C(j)$ independent of $p,q\in [1,\infty ]$ such that for all $f\in \mathcal {S}({\mathbb {R}}^d)$,

  $$\begin{aligned} \Vert I^{\varepsilon } (f, j) \Vert _{M^{p,q}_s}\le C \varepsilon ^{ |s| } \Vert f\Vert _{M^{p,q}_s}. \end{aligned}$$

Proof

The first point is proven like (6.3) above. For the second point, write

$$\begin{aligned} \Vert I^{\varepsilon } (f, j) \Vert _ {M^{p,q}_s}&= \left\| \left\| V_{g}f\left( x,y-\frac{j}{\varepsilon }\right) \right\| _{L^p_x} \left\langle y\right\rangle ^s\right\| _{L_y^q} = \left\| \Vert V_{g}f(x,y)\Vert _{L^p_x} \left\langle y+ \frac{j}{ \varepsilon }\right\rangle ^s\right\| _{L_y^q}\\&\le \left\langle \frac{j}{\varepsilon }\right\rangle ^s \left\| \Vert V_{g}f(x,y)\Vert _{L^p_x} \left\langle y\right\rangle ^{|s|}\right\| _{L_y^q} , \end{aligned}$$

where we have used Peetre inequality $\left\langle a+b\right\rangle ^s\le \left\langle a\right\rangle ^s\left\langle b\right\rangle ^{|s|}$. The lemma follows. $\square$

At this stage, we can repeat the same arguments as in the previous subsection, and Theorem 1.6 follows.