# Norm inflation with infinite loss of regularity for the generalized improved
Boussinesq equation

Pierre de Roubin

###### Abstract.

In this paper, we study the ill-posedness issue for the generalized improved
Boussinesq equation. In particular we prove there is norm inflation with
infinite loss of regularity at general initial data in
$\langle\nabla\rangle^{-s}\big{(}L^{2}\cap L^{\infty}\big{)}(\mathbb{R})$ for
any $s<0$. This result is sharp in the $L^{2}$-based Sobolev scale in view of
the well-posedness in $L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. We also
show that the same result applies to the multi-dimensional generalized
improved Boussinesq equation. Finally, we extend our norm inflation result to
Fourier-Lebesgue, modulation and Wiener amalgam spaces.

##  1\. Introduction

We consider the generalized improved Boussinesq equation (gIBq):

$\begin{cases}\partial_{t}^{2}u-\partial_{x}^{2}u-\partial_{t}^{2}\partial_{x}^{2}u=\partial_{x}^{2}\left(f(u)\right)\\\
(u,\partial_{t}u)|_{t=0}=(u_{0},u_{1}),\end{cases}\qquad(t,x)\in\mathbb{R}\times\mathcal{M},$
(1.1)

where $\mathcal{M}=\mathbb{R}$ or $\mathbb{T}$ with
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $f(u)=u^{k}$ for $k\geq 2$ an integer.
The equation (1.1) appears in a wide variety of physical problems. Makhankov
[21] derived the improved Boussinesq equation, namely with $f(u)=u^{2}$, in
order to study ion-sound wave propagation and mentioned the case of the
improved modified Boussinesq equation, with $f(u)=u^{3}$, as modeling
nonlinear Alfvén waves. He also mentioned the possibility to use the linear
term of (1.1) to describe waves propagating at right angles to the magnetic
field. Besides, Clarkson-LeVeque-Saxton [14] considered the cases
$f(u)=u^{p}$, with $p=3$ or $5$, as a model for the propagation of
longitudinal deformation waves in an elastic rod. See also [27] for further
discussion on the physical aspect of (1.1).

This equation has also attracted attention from a mathematical point of view.
In particular, the well-posedness of this problem has been studied extensively
and Constantin-Molinet [15] proved local well-posedness for (1.1) in
$L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ (and global well-posedness
under additional conditions). See also [11, 17, 33]. On the other hand, it is
known that (1.1) possesses some undesirable behaviour in negative Sobolev
spaces. Following the work of Bourgain [6], it was showed in [17] that the
solution map $\Phi\colon H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})\to
C([-T,T],H^{s}(\mathbb{R}))$ of (1.1), for nonlinearity $f(u)=u^{k}$, fails to
be $C^{k}$ for $s<0$. This result in particular implies that we can not study
the well-posedness of (1.1) via a contraction argument. Note however that
their result does not imply failure of continuity for the data-to-soluton map
in negative Sobolev spaces. Our main goal in this article is to complete the
well-posedness issue of this problem by proving the discontinuity of the data-
to-solution map of (1.1) in $\langle\nabla\rangle^{-s}\big{(}L^{2}\cap
L^{\infty}\big{)}(\mathcal{M})$ for any $s<0$. For more clarity, let us denote
$\mathcal{W}^{s,2,\infty}(\mathcal{M})\coloneqq
W^{s,2,\infty}(\mathcal{M})\times W^{s,2,\infty}(\mathcal{M})$ with

$W^{s,2,\infty}(\mathcal{M})=\langle\nabla\rangle^{-s}\big{(}L^{2}\cap
L^{\infty}\big{)}(\mathcal{M})=\\{f,\langle\nabla\rangle^{s}f\in
L^{2}(\mathcal{M})\cap L^{\infty}(\mathcal{M})\\}.$

We exhibit the following norm inflation with infinite loss of regularity
behaviour:

######  Theorem 1.1.

Let $k\geq 2$, $\sigma\in\mathbb{R}$ and $s<0$. Fix
$(u_{0},u_{1})\in\mathcal{W}^{s,2,\infty}(\mathcal{M})$. Then, given any
$\varepsilon>0$, there exists a solution $u_{\varepsilon}$ to (1.1) with
nonlinearity $f(u)=u^{k}$ on $\mathcal{M}$ and
$t_{\varepsilon}\in(0,\varepsilon)$ such that

$\|(u_{\varepsilon}(0),\partial_{t}u_{\varepsilon}(0))-(u_{0},u_{1})\|_{\mathcal{W}^{s,2,\infty}}<\varepsilon\quad\text{
and
}\quad\|u_{\varepsilon}(t_{\varepsilon})\|_{W^{\sigma,2,\infty}}>\varepsilon^{-1}.$
(1.2)

Note that, when $\sigma=s$ and $(u_{0},u_{1})=(0,0)$, (1.2) is already called
norm inflation. Since we have (1.2) for any arbitrary initial data
$(u_{0},u_{1})\in\mathcal{W}^{s,2,\infty}(\mathcal{M})$, we say that we have
norm inflation at general initial data. Besides, Theorem 1.3 also yields (1.2)
for any arbitrary $\sigma<s$, leading to the so-called infinite loss of
regularity. Observe as well that, as a corollary, Theorem 1.1 gives the
following discontinuity of the solution map.

######  Corollary 1.2.

Let $s<0$. Then, for any $T>0$, the solution map
$\Phi\colon(u_{0},u_{1})\in\mathcal{W}^{s,2,\infty}(\mathcal{M})\to(u,\partial_{t}u)\in
C([-T,T],W^{s,2,\infty}(\mathcal{M}))\times
C([-T,T],W^{s,2,\infty}(\mathcal{M}))$ to the generalized improved Boussinesq
equation (1.1) is discontinuous everywhere in
$\mathcal{W}^{s,2,\infty}(\mathcal{M})$.

Throughout the rest of this paper, we prove in fact the following result of
norm inflation at general initial data with infinite loss of regularity in
negative Sobolev spaces:

######  Theorem 1.3.

Let $k\geq 2$, $\sigma\in\mathbb{R}$ and $s<0$. Fix
$(u_{0},u_{1})\in\mathcal{H}^{s}(\mathcal{M})\coloneqq
H^{s}(\mathcal{M})\times H^{s}(\mathcal{M})$. Then, given any $\varepsilon>0$,
there exists a solution $u_{\varepsilon}$ to (1.1) with nonlinearity
$f(u)=u^{k}$ on $\mathcal{M}$ and $t_{\varepsilon}\in(0,\varepsilon)$ such
that

$\|(u_{\varepsilon}(0),\partial_{t}u_{\varepsilon}(0))-(u_{0},u_{1})\|_{\mathcal{H}^{s}}<\varepsilon\quad\text{
and }\quad\|u_{\varepsilon}(t_{\varepsilon})\|_{H^{\sigma}}>\varepsilon^{-1}.$
(1.3)

Once Theorem 1.3 is proved, Theorem 1.1 follows as a corollary. See Remark
3.10. In fact, we first prove the usual norm inflation at general initial
data, namely we prove (1.3) with $\sigma=s$. Then, the loss of regularity
follows with a slight modification. See Remark 3.7. Note also that, in a
similar manner, Theorem 1.3 implies the discontinuity of the solution map in
negative Sobolev spaces.

There are several ways to prove norm inflation results. Christ-Colliander-Tao
[13] first introduced norm inflation with a method based on a dispersionless
ODE approach. See also [7, 32]. On the other hand, Carles with his
collaborators in [1, 3, 8] proved norm inflation with infinite loss of
regularity results for nonlinear Schrödinger equations (NLS) by using
geometric optics.

Our strategy is to follow a Fourier analytic argument that originated from the
abstract work of Bejenaru-Tao [2] on quadratic nonlinear Schrödinger equation.
Their idea was to expand the solution into a power series and to exploit the
high-to-low energy transfer in one of the terms to prove discontinuity of the
solution map. This approach was refined later on by Iwabuchi-Ogawa [19] who
used it to extend the ill-posedness result in [2] into a norm inflation
result. Kishimoto [20] then further developed these methods to prove norm
inflation for the nonlinear Schrödinger equation. Meanwhile, Oh [23] refined
the argument of [19] in the context of cubic NLS by introducing a way to index
the power series by trees, and to estimate each term separately. Forlano-
Okamoto [18] proved afterwards norm inflation with an approach inspired by
[23] for nonlinear wave equations (NLW) in Sobolev spaces of negative
regularities, and we use the same reasoning in our proof. For other papers
with similar argument, the interested reader might turn to [5, 10, 12, 24, 26,
31]. See also [9] for an implementation of this method in probabilistic
settings.

One of the key ingredients to our proof is the use of the Wiener algebra
$\mathcal{F}L^{1}(\mathcal{M})$, which we define now. Given
$\mathcal{M}=\mathbb{R}$ or $\mathbb{T}$, let $\widehat{\mathcal{M}}$ denote
the Pontryagin dual of $\mathcal{M}$, i.e.

$\widehat{\mathcal{M}}=\begin{cases}\mathbb{R}&\text{if}\quad\mathcal{M}=\mathbb{R},\\\
\mathbb{Z}&\text{if}\quad\mathcal{M}=\mathbb{T}.\end{cases}$ (1.4)

Note that, when $\widehat{\mathcal{M}}=\mathbb{Z}$, we endow it with the
counting measure. We can then define the following Fourier-Lebesgue spaces:

######  Definition 1.4 (Fourier-Lebesgue spaces).

For $s\in\mathbb{R}$ and $p\geq 1$, we define the Fourier-Lebesgue space
$\mathcal{F}L^{s,p}(\mathcal{M})$ as the completion of the Schwartz class of
functions $\mathcal{S}(\mathcal{M})$ with respect to the norm

$\|f\|_{\mathcal{F}L^{s,p}(\mathcal{M})}=\|\langle\xi\rangle^{s}\widehat{f}(\xi)\|_{L^{p}_{\xi}(\widehat{\mathcal{M}})},$

where $\langle\xi\rangle\coloneqq(1+\lvert\xi\rvert^{2})^{\frac{1}{2}}$.

In particular, $\mathcal{F}L^{0,1}(\mathcal{M})$ corresponds to the Wiener
algebra, and its algebra property allows us to prove easily that (1.1) is
analytically locally well-posed in $\mathcal{F}L^{0,1}(\mathcal{M})$.

Another major point of our proof is the following power series expansion of a
solution $u$ to (1.1) with $(u,\partial_{t}u)|_{t=0}=\vec{u}_{0}$:

$u=\sum^{\infty}_{j=0}\Xi_{j}(\vec{u}_{0}),$

where $\Xi_{j}(\vec{u}_{0})$ denotes a multilinear term in $\vec{u}_{0}$ of
degree $(k-1)j+1$ (for nonlinearity $f(u)=u^{k}$). More precisely, these
multilinear terms are exactly the successive terms of a Picard iteration
expansion. See Section 2. Then, by explicit computation we show that
$\Xi_{1}(\vec{u}_{0})$ grows rapidly in a short time, achieving the desired
growth, while we control the other terms. See Section 3.

######  Remark 1.5.

In [29, 30], Wang-Chen studied the multi-dimensional generalized improved
Boussinesq equation

$\begin{cases}\partial_{t}^{2}u-\Delta u-\partial_{t}^{2}\Delta
u=\Delta\left(f(u)\right)\\\
(u,\partial_{t}u)|_{t=0}=(u_{0},u_{1}),\end{cases}\qquad(t,x)\in[0,+\infty)\times\mathbb{R}^{d},$
(1.5)

which corresponds essentially to the $d$-dimensional form of (1.1), for $d\geq
1$. More precisely, they proved this problem is locally well-posed in
$W^{2,p}\cap L^{\infty}$, for any $1\leq p\leq\infty$, and in $H^{s}$ for
$s\geq\frac{d}{2}$. They also showed global well-posedness under additional
conditions. We claim that norm inflation with infinite loss of regularity also
applies for this problem, for any dimension $d\geq 1$ and $s<0$. See Remark
3.8 for more details.

######  Remark 1.6.

[4] used this method to extend the result of Forlano-Okamoto [18] and prove
infinite loss of regularity for the nonlinear wave equation in some Fourier-
Lebesgue, modulation and Wiener amalgam spaces111However, we point out that,
while Forlano-Okamoto did not state it, their argument implies infinite loss
of regularity.. We claim that we can prove the same result for equation (1.1).
See Appendix A for more details.

##  2\. Power series extension

In this section, we prove (1.1) with nonlinearity $f(u)=u^{k}$ is well-posed
in the Wiener algebra, and it can be expanded into a power series. First, let
us fix some notations. We define

$\overrightarrow{\mathcal{F}L}^{s,p}(\mathcal{M})\coloneqq\mathcal{F}L^{s,p}(\mathcal{M})\times\mathcal{F}L^{s,p}(\mathcal{M})$

and, for more clarity, we denote
$\mathcal{F}L^{p}(\mathcal{M})\coloneqq\mathcal{F}L^{0,p}(\mathcal{M})$ and
$\overrightarrow{\mathcal{F}L}^{p}(\mathcal{M})\coloneqq\overrightarrow{\mathcal{F}L}^{0,p}(\mathcal{M})$.

We denote $S(t)$ the linear propagator:

$S(t)(u,v)=\cos\left(tP(D)\right)u+\frac{\sin\left(tP(D)\right)}{P(D)}v,$
(2.1)

with $P(D)\coloneqq\frac{\lvert\nabla\rvert}{\langle\nabla\rangle}$. Namely,
for any $\xi\in\mathcal{M}$,

$\mathcal{F}\left[P(D)f\right](\xi)=\frac{\lvert\xi\rvert}{(1+\xi^{2})^{1/2}}\widehat{f}(\xi)\eqqcolon\lambda(\xi)\widehat{f}(\xi).$

We also denote $\mathcal{I}_{k}$ the multilinear Duhamel operator.

$\mathcal{I}_{k}(u_{1},\dots,u_{k})(t)=\int^{t}_{0}\sin\left((t-t^{\prime})P(D)\right)P(D)\prod^{k}_{j=1}u_{j}(t^{\prime})\mathrm{d}t^{\prime}.$
(2.2)

This gives the following Duhamel formulation for (1.1)

$u(t)=S(t)(u_{0},u_{1})+\mathcal{I}_{k}(u).$ (2.3)

Note that, in the aforementioned formula, we used the short-hand notation
$\mathcal{I}_{k}(u)\coloneqq\mathcal{I}_{k}(u,\dots,u)$.

To index the power series we intend to create, we need the following tree
structure:

######  Definition 2.1 ($k$-ary trees).

  1. (1)

Given a set $\mathcal{T}$ with partial order $\leq$, we say that
$b\in\mathcal{T}$ with $b\leq a$ and $b\neq a$ is a child of
$a\in\mathcal{T}$, if $b\leq c\leq a$ implies either $c=a$ or $c=b$. If the
latter condition holds, we also say that $a$ is the parent of $b$.

  2. (2)

A tree $\mathcal{T}$ is a finite partially ordered set, satisfying the
following properties222We do not identify two trees even if there is an order-
preserving bijection between them.:

     * •

Let $a_{1},a_{2},a_{3},a_{4}\in\mathcal{T}$. If $a_{4}\leq a_{2}\leq a_{1}$
and $a_{4}\leq a_{3}\leq a_{1}$, then we have $a_{2}\leq a_{3}$ or $a_{3}\leq
a_{2}$,

     * •

A node $a\in\mathcal{T}$ is called terminal, if it has no child. A non-
terminal node $a\in\mathcal{T}$, for $\mathcal{T}$ a $k$-ary tree, is a node
with exactly $k$ children,

     * •

There exists a maximal element $r\in\mathcal{T}$, called the root node, such
that $a\leq r$ for all $a\in\mathcal{T}$,

     * •

$\mathcal{T}$ consists of the disjoint union of $\mathcal{T}^{0}$ and
$\mathcal{T}^{\infty}$, where $\mathcal{T}^{0}$ and $\mathcal{T}^{\infty}$
denote the collections of non-terminal nodes and terminal nodes, respectively.

  3. (3)

Let ${\bf T}(j)$ denote the set of all trees of $j$-th generation, namely
trees with $j$ non-terminal nodes.

Note that a tree of $j$-th generation $\mathcal{T}\in{\bf T}(j)$ has $kj+1$
nodes. Indeed, it has $j$ non-terminal nodes by definition and an induction
argument shows it has $(k-1)j+1$ terminal nodes. Besides, we have the
following bound on the number of trees of $j$-th generation:

######  Lemma 2.2.

There exists a constant $C_{0}>0$, depending only on $k$, such that, for any
$j\in\mathbb{N}$, we have

$\lvert{\bf T}(j)\rvert\leq\frac{C^{j}_{0}}{(1+j)^{2}}\leq C^{j}_{0}$ (2.4)

The following proof is an adaptation of the one in [23] for ternary trees, we
include it for completeness.

###### Proof.

We prove (2.4) by induction. Note that the right inequality is immediate, so
we only have to prove

$\lvert{\bf T}(j)\rvert\leq\frac{C^{j}_{0}}{(1+j)^{2}}$

for any $j\geq 0$. Observe first that

$\lvert{\bf T}(0)\rvert=\lvert{\bf T}(1)\rvert=1.$

Then, fix $j\geq 2$. Assume equation (2.4) holds for any $0\leq m\leq j-1$ and
take $\mathcal{T}\in{\bf T}(j)$. By Definition 2.1, there exist $k$ trees
$\mathcal{T}_{1}\in{\bf T}(j_{1}),\dots,\mathcal{T}_{k}\in{\bf T}(j_{k})$,
with $j_{1}+\cdots+j_{k}=j-1$, such that $\mathcal{T}$ is the tree consisting
of a root node whose children are $\mathcal{T}_{1},\dots,\mathcal{T}_{k}$.
Thus, applying the induction hypothesis we get

$\displaystyle\lvert{\bf T}(j)\rvert$
$\displaystyle=\sum_{\begin{subarray}{c}j_{1}+\cdots+j_{k}=j-1,\\\
j_{1},\cdots,j_{k}\geq 0\end{subarray}}\lvert{\bf
T}(j_{1})\rvert\times\cdots\times\lvert{\bf T}(j_{k})\rvert$
$\displaystyle\leq\sum_{\begin{subarray}{c}j_{1}+\cdots+j_{k}=j-1,\\\
j_{1},\cdots,j_{k}\geq
0\end{subarray}}\frac{C^{j_{1}}_{0}}{(1+j_{1})^{2}}\times\cdots\times\frac{C^{j_{k}}_{0}}{(1+j_{k})^{2}}\times\frac{(1+j)^{2}}{(1+j)^{2}}$
$\displaystyle\leq k^{2}\left(\sum_{j_{2},\cdots j_{k}\geq
0}\frac{1}{(1+j_{2})^{2}\times\cdots\times(1+j_{k})^{2}}\right)\frac{C^{j-1}_{0}}{(1+j)^{2}}$

where we used $k\max\left(1+j_{1},\dots,1+j_{k}\right)\geq 1+j$ and rearranged
the sum so the maximum is reached for $j_{1}$. Then, choosing
$C_{0}=k^{2}\sum_{j_{2},\cdots j_{k}\geq
0}\frac{1}{(1+j_{2})^{2}\cdots(1+j_{k})^{2}}<\infty$ ends the induction.

∎

From now on, for any $\vec{\phi}\in\overrightarrow{\mathcal{F}L}^{1}$, we will
associate to any tree $\mathcal{T}\in{\bf T}(j)$, $j\geq 0$, a space-time
distribution
$\Psi(\mathcal{T})(\vec{\phi})\in\mathcal{D}^{\prime}\left((-T,T)\times\mathcal{M}\right)$
as follows:

  * •

replace a non-terminal node by Duhamel integral operator $\mathcal{I}_{k}$
with its $k$ arguments being the children of the node,

  * •

replace a terminal node by $S(t)\vec{\phi}$.

For any $j\geq 0$ and $\vec{\phi}\in\overrightarrow{\mathcal{F}L}^{1}$, we
also define $\Xi_{j}$ as follows:

$\Xi_{j}(\vec{\phi})=\sum_{\mathcal{T}\in{\bf
T}(j)}\Psi(\mathcal{T})(\vec{\phi}).$ (2.5)

For instance, we have the two following terms:

$\Xi_{0}(\vec{\phi})=S(t)\vec{\phi},\quad\text{ and
}\quad\Xi_{1}(\vec{\phi})=\mathcal{I}_{k}(S(t)\vec{\phi},\cdots,S(t)\vec{\phi}).$

Let us now state some basic multilinear estimates that will be useful both for
norm inflation and for local well-posedness in the Wiener algebra
$\mathcal{F}L^{1}$.

######  Lemma 2.3.

There exists $C>0$ such that, for any
$\vec{\phi}\in\overrightarrow{\mathcal{F}L}^{1}$, $j\in\mathbb{N}$ and
$0<T\leq 1$, we have

$\big{\|}\Xi_{j}(\vec{\phi})(T)\big{\|}_{\mathcal{F}L^{1}}\leq
C^{j}T^{2j}\|\vec{\phi}\|^{(k-1)j+1}_{\overrightarrow{\mathcal{F}L}^{1}}$
(2.6)

Moreover, if $j\geq 1$ and
$\vec{\psi}\in\overrightarrow{\mathcal{F}L}^{1}\cap\mathcal{H}^{0}\left(\mathbb{R}\right)$,

$\left\|\Xi_{j}(\vec{\psi})(T)\right\|_{\mathcal{F}L^{\infty}}\leq
C^{j}T^{2j}\|\vec{\psi}\|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^{1}}\|\vec{\psi}\|^{2}_{\mathcal{H}^{0}}.$
(2.7)

###### Proof.

Let $\mathcal{T}\in{\bf T}(j)$. Using the same tree structure argument as for
Lemma 2.2, there exist $k$ trees $\mathcal{T}_{1}\in{\bf
T}(j_{1}),\dots,\mathcal{T}_{k}\in{\bf T}(j_{k})$, with
$j_{1}+\cdots+j_{k}=j-1$, such that the root nodes of
$\mathcal{T}_{1},\dots,\mathcal{T}_{1}$ are the children of the root node of
$\mathcal{T}$. Thus, we can write

$\Psi(\mathcal{T})(\vec{\phi})=\mathcal{I}_{k}(\Psi(\mathcal{T}_{1})(\vec{\phi}),\cdots,\Psi(\mathcal{T}_{k})(\vec{\phi})),$

and $\Psi(\mathcal{T})(\vec{\phi})$ consists essentially of
$j=\lvert\mathcal{T}^{0}\rvert$ iterations of the Duhamel operator
$\mathcal{I}_{k}$ with $(k-1)j+1$ times the term $S(t)\vec{\phi}$ as
arguments.

Meanwhile, we deduce from (2.1) that $S(t)$ is unitary in $\mathcal{F}L^{1}$
for any $0<T\leq 1$, and, since $\lvert\sin y\rvert\leq\lvert y\rvert$ for any
$y\in\mathbb{R}$ and $\lambda(\xi)\leq 1$ for any $\xi\in\mathbb{R}$, the
algebra property of $\mathcal{F}L^{1}$ gives

$\|\mathcal{I}_{k}[u_{1},\dots,u_{k}]\|_{C_{T}\mathcal{F}L^{1}}\leq\int^{T}_{0}\lvert
T-t^{\prime}\rvert\lvert\lambda(\xi)\rvert^{2}\|u_{1}\cdots
u_{k}\|_{C_{T}\mathcal{F}L^{1}}\mathrm{d}t^{\prime}\leq\frac{1}{2}T^{2}\prod^{k}_{j=1}\|u_{j}\|_{C_{T}\mathcal{F}L^{1}}$
(2.8)

Hence, (2.6) follows from an induction argument and Lemma 2.2. Besides,
Young’s inequality and a similar argument give (2.7).

∎

Let us now use Lemma 2.3 to prove local well-posedness of (1.1) in the Wiener
algebra and to justify the power series expansion.

######  Lemma 2.4.

Let $M>0$. Then, for any time $T$ such that $0<T\ll\min(M^{-\frac{k-1}{2}},1)$
and $\vec{u}_{0}\in\overrightarrow{\mathcal{F}L}^{1}$ with
$\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{1}}\leq M$,

  1. (1)

there exists a unique solution
$u\in\mathbb{C}([0,T];\mathcal{F}L^{1}(\mathbb{R}))$ to (1.1) satisfying
$(u,\partial_{t}u)=\vec{u}_{0}$.

  2. (2)

Moreover, u can be expressed as

$u=\sum_{j=0}^{\infty}\Xi_{j}(\vec{u}_{0})=\sum_{j=0}^{\infty}\sum_{\mathcal{T}\in{\bf
T}(j)}\Psi(\mathcal{T})(\vec{u}_{0})$ (2.9)

###### Proof.

Let us first prove our problem is locally well-posed in $\mathcal{F}L^{1}$. We
define the functional $\Gamma$ by

$\Gamma[u](t)\coloneqq S(t)\vec{u}_{0}+\mathcal{I}_{k}(u)(t)$

for any $t\in[0,T]$. Then, using the unitarity of $S(t)$ and (2.8), we have

$\left\|\Gamma[u]\right\|_{C_{T}\mathcal{F}L^{1}}\leq\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{1}}+\frac{1}{2}T^{2}\|u\|_{\mathcal{F}L^{1}}^{k}$

Using the multilinearity of $\mathcal{I}_{k}$, we ensure, for $0<T\leq 1$ such
that $\frac{1}{2}T^{2}M^{k-1}\ll 1$, that $\Gamma$ is a strict contraction on
the ball

$B_{2M}\coloneqq\\{v\in
C([0,T],\mathcal{F}L^{1}(\mathcal{M})),\|v\|_{C_{T}\mathcal{F}L^{1}}\leq
2M\\}.$

Then, the contraction mapping theorem and an a posteriori continuity argument
proves the local well-posedness. Let us move onto the power series expansion.

Fix $\varepsilon>0$. We choose $0<T\leq 1$ such that
$\frac{1}{2}T^{2}M^{k-1}\ll 1$. Then, from (2.6), the sum in (2.9) converges
absolutely in $C([0,T],\mathcal{F}L^{1}(\mathcal{M}))$. Let us denote

$U_{J}=\sum_{j=0}^{J}\Xi_{j}(\vec{u}_{0})\qquad\text{ and }\qquad
U=\sum_{j=0}^{\infty}\Xi_{j}(\vec{u}_{0}).$

There exists $J_{1}\geq 0$ such that

$\|U-U_{J}\|_{C_{T}\mathcal{F}L^{1}}<\frac{\varepsilon}{3}$ (2.10)

for any $J\geq J_{1}$. In particular, this implies that $U$ and $U_{J}$ belong
to the ball $B_{2M}$. Then, by continuity of $\Gamma$ as a map from $B_{2M}$
into itself, (2.10) implies there exists $J_{2}\leq 0$ such that, for any
$J\geq J_{2}$,

$\|\Gamma[U]-\Gamma[U_{J}]\|_{C_{T}\mathcal{F}L^{1}}<\frac{\varepsilon}{3}.$
(2.11)

All that is left now is to estimate $U_{J}-\Gamma[U_{J}]$. Fix an integer
$J\geq 1$. Then, from the tree structure argument we already used in the proof
of Lemma 2.3, we get

$\displaystyle U_{J}-\Gamma[U_{J}]$
$\displaystyle=\sum_{j=1}^{J}\Xi_{j}(\vec{u}_{0})-\sum_{0\leq
j_{1},\dots,j_{k}\leq
J}\mathcal{I}_{k}\left(\Xi_{j_{1}}(\vec{u}_{0}),\cdots,\Xi_{j_{k}}(\vec{u}_{0})\right)$
$\displaystyle=-\sum^{kJ}_{l=J}\sum_{\begin{subarray}{c}0\leq
j_{1},\dots,j_{k}\leq J,\\\
j_{1}+\cdots+j_{k}=l\end{subarray}}\mathcal{I}_{k}\left(\Xi_{j_{1}}(\vec{u}_{0}),\cdots,\Xi_{j_{k}}(\vec{u}_{0})\right).$

Now a crude estimation of the sums, along with (2.8) and (2.6), give

$\displaystyle\|U_{J}-\Gamma[U_{J}]\|_{C_{T}\mathcal{F}L^{1}}$
$\displaystyle\leq\frac{1}{2}T^{2}\sum^{kJ}_{l=J}\
\sum_{\begin{subarray}{c}0\leq j_{1},\dots,j_{k}\leq J,\\\
j_{1}+\cdots+j_{k}=l\end{subarray}}\
\prod^{k}_{m=1}\|\Xi_{j_{m}}(\vec{u}_{0})\|_{C_{T}\mathcal{F}L^{1}}$
$\displaystyle\leq\frac{1}{2}T^{2}\sum^{kJ}_{l=J}\
\sum_{\begin{subarray}{c}0\leq j_{1},\dots,j_{k}\leq J,\\\
j_{1}+\cdots+j_{k}=l\end{subarray}}\
\prod^{k}_{m=1}C^{j_{m}}T^{2j_{m}}\|\vec{u}_{0}\|^{(k-1)j_{m}+1}_{\overrightarrow{\mathcal{F}L}^{1}}$
$\displaystyle\leq\frac{1}{2}T^{2}J^{k}M^{k}\sum^{\infty}_{l=J}(CT^{2}M^{k-1})^{l}$

Since we assumed $0<T\ll\min(M^{-\frac{k-1}{2}},1)$, the sum converges and the
right-hand side is bounded by $\frac{1}{2}T^{2}J^{k}M^{k}(CT^{2}M^{k-1})^{J}$,
which tends to $0$ as $J$ tends to infinity. Thus, there exists $J_{3}\geq 1$
such that for every $J\geq J_{3}$,

$\|U_{J}-\Gamma[U_{J}]\|_{C_{T}\mathcal{F}L^{1}}<\frac{\varepsilon}{3}.$
(2.12)

Now, for any $J\geq\max(J_{1},J_{2},J_{3})$, (2.10), (2.11) and (2.12) imply

$\|U-\Gamma[U]\|_{C_{T}\mathcal{F}L^{1}}\leq\|U-U_{J}\|_{C_{T}\mathcal{F}L^{1}}+\|U_{J}-\Gamma[U_{J}]\|_{C_{T}\mathcal{F}L^{1}}+\|\Gamma[U]-\Gamma[U_{J}]\|_{C_{T}\mathcal{F}L^{1}}<\varepsilon.$

Therefore, $U$ is a fixed point of $\Gamma$ and this ends the proof by
uniqueness.

∎

##  3\. Norm inflation for IBq

In this section, we present first the proof of Theorem 1.3. Actually, our main
goal is to prove the following proposition:

######  Proposition 3.1.

Let $k\geq 2$ and $s<0$. Fix $(u_{0},u_{1})\in\mathcal{H}^{s}(\mathcal{M})$.
Then, given any $\varepsilon>0$, there exists a solution $u_{\varepsilon}$ to
(1.1) with nonlinearity $f(u)=u^{k}$ on $\mathcal{M}$ and
$t_{\varepsilon}\in(0,\varepsilon)$ such that

$\|(u_{\varepsilon}(0),\partial_{t}u_{\varepsilon}(0))-(u_{0},u_{1})\|_{\mathcal{H}^{s}}<\varepsilon\quad\text{
and }\quad\|u_{\varepsilon}(t_{\varepsilon})\|_{H^{s}}>\varepsilon^{-1}.$

Indeed, once Proposition 3.1 is proved, the proof of Theorem 1.3 follows in
the same way, with a slight modification that we treat separately for more
clarity. See Remark 3.7.

To do so, suppose first that we proved the following proposition:

######  Proposition 3.2.

Let $k\geq 2$ and $s<0$. Fix
$(u_{0},u_{1})\in\mathcal{S}(\mathcal{M})\times\mathcal{S}(\mathcal{M})$.
Then, for any $n\in\mathbb{N}$, there exists a solution $u_{n}$ to (1.1) with
nonlinearity $f(u)=u^{k}$ and $t_{n}\in(0,\frac{1}{n})$ such that

$\|(u_{n}(0),\partial_{t}u_{n}(0))-(u_{0},u_{1})\|_{\mathcal{H}^{s}}<\frac{1}{n}\quad\text{
and }\quad\|u_{n}(t_{n})\|_{H^{s}}>n.$ (3.1)

Then, let us fix $\varepsilon>0$ and choose $n\in\mathbb{N}$ such that
$n>\varepsilon^{-1}$. According to Proposition 3.2, there exists a solution
$u_{n}$ to (1.1) and a time $t_{n}\in(0,\frac{1}{n})\subset(0,\varepsilon)$
such that

$\|(u_{n}(0),\partial_{t}u_{n}(0))-(u_{0},u_{1})\|_{\mathcal{H}^{s}}<\frac{1}{n}<\varepsilon\quad\text{
and }\quad\|u_{n}(t_{n})\|_{H^{s}}>n>\varepsilon^{-1}.$

Therefore, Proposition 3.1 follows from Proposition 3.2 and the density of
$\mathcal{S}(\mathcal{M})$ in $H^{s}(\mathcal{M})$. Similarly, Theorem 1.3
follows from the following proposition:

######  Proposition 3.3.

Let $k\geq 2$, $\sigma\in\mathbb{R}$ and $s<0$. Fix
$(u_{0},u_{1})\in\mathcal{S}(\mathcal{M})\times\mathcal{S}(\mathcal{M})$.
Then, given any $n\in\mathbb{N}$, there exists a solution $u_{n}$ to (1.1)
with nonlinearity $f(u)=u^{k}$ on $\mathcal{M}$ and $t_{n}\in(0,\frac{1}{n})$
such that

$\|(u_{n}(0),\partial_{t}u_{n}(0))-(u_{0},u_{1})\|_{\mathcal{H}^{s}}<\frac{1}{n}\quad\text{
and }\quad\|u_{n}(t_{n})\|_{H^{\sigma}}>n.$

Consequently, the rest of this paper is devoted to the proofs of Proposition
3.2 and Proposition 3.3. In the following, we fix some
$\vec{u}_{0}\in\mathcal{S}(\mathcal{M})\times\mathcal{S}(\mathcal{M})$.

###  3.1. Multilinear estimates

In this subsection, we establish some multilinear estimates that are
essentials to our proof.

Given $n\in\mathbb{N}$, let us fix some $N=N(n)\gg 1$, $R=R(N)\gg 1$ and $1\ll
A=A(N)\ll N$ to be determined later. We define
$\vec{\phi}_{n}\coloneqq(\phi_{n},0)$ by setting

$\widehat{\phi_{n}}=R\chi_{\Omega}$ (3.2)

where

$\Omega=\bigcup_{\eta\in\Sigma}(\eta+Q_{A})$ (3.3)

with $Q_{A}=[-\frac{A}{2},\frac{A}{2}]$ and

$\Sigma=\\{-2N,-N,N,2N\\}.$ (3.4)

Note that the condition $A\ll N$ ensures the union (3.3) is disjoint. Besides,
observe that (3.2), (3.3) and (3.4) imply for any $s\in\mathbb{R}$

$\|\vec{\phi}_{n}\|_{\overrightarrow{\mathcal{F}L}^{1}}\sim RA\quad\text{ and
}\quad\|\vec{\phi}_{n}\|_{\mathcal{H}^{s}}\sim RN^{s}A^{1/2}.$ (3.5)

We define finally $\vec{u}_{0,n}\coloneqq\vec{u}_{0}+\vec{\phi}_{n}$. Suppose
$N$, $R$ and $A$ satisfy

$\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{1}}\ll RA.$ (3.6)

Therefore, for each $n\in\mathbb{N}$, provided

$0<T\ll\min((RA)^{-\frac{k-1}{2}},1),$ (3.7)

Lemma 2.4 implies there exists a unique solution $u_{n}\in
C([0,T],\mathcal{F}L^{1}(\mathcal{M}))$ to (1.1) with
$(u_{n},\partial_{t}u_{n})|_{t=0}=\vec{u}_{0,n}$ and admitting the power
series expansion:

$u_{n}=\sum_{j=0}^{\infty}\Xi_{j}(\vec{u}_{0,n})=\sum_{j=0}^{\infty}\Xi_{j}(\vec{u}_{0}+\vec{\phi}_{n}).$
(3.8)

The purpose of this subsection is then to estimate the terms of the power
series on the right-hand side of (3.8). But first, let us recall the following
lemma:

######  Lemma 3.4.

Let $a,b\in\mathbb{R}$ and $A>0$, then we have

$CA\chi_{a+b+Q_{A}}(\xi)\leq\chi_{a+Q_{A}}\ast\chi_{b+Q_{A}}(\xi)\leq\widetilde{C}A\chi_{a+b+Q_{2A}}(\xi)$

where $C,\widetilde{C}>0$ are constants.

The proof of the following lemma is essentially the same as in [18, Lemma 3.2]
and is included for completeness.

######  Lemma 3.5.

For any $s<0$, $t\in[0,T]$ and $j\in\mathbb{N}$, the following estimates hold:

$\displaystyle\|\vec{u}_{0,n}-\vec{u}_{0}\|_{\mathcal{H}^{s}}$
$\displaystyle\sim RN^{s}A^{1/2},$ (3.9)
$\displaystyle\|\Xi_{0}(\vec{u}_{0,n})(t)\|_{H^{s}}$ $\displaystyle\lesssim
1+RA^{1/2}N^{s},$ (3.10)
$\displaystyle\|\Xi_{1}(\vec{u}_{0,n})(t)-\Xi_{1}(\vec{\phi}_{n})(t)\|_{H^{s}}$
$\displaystyle\lesssim t^{2}R^{k-1}A^{k-1}\|\vec{u}_{0}\|_{\mathcal{H}^{0}},$
(3.11) $\displaystyle\|\Xi_{j}(\vec{u}_{0,n})(t)\|_{H^{s}}$
$\displaystyle\lesssim
C^{j}t^{2j}R^{(k-1)j}A^{(k-1)j}\left(\|\vec{u}_{0}\|_{\mathcal{H}^{0}}+Rg_{s}(A)\right),$
(3.12)

where $g_{s}(A)$ is defined by

$g_{s}(A)\coloneqq\begin{cases}1&\textup{if }s<-\frac{1}{2},\\\ \left(\log
A\right)^{\frac{1}{2}}&\textup{if }s=-\frac{1}{2},\\\
A^{\frac{1}{2}+s}&\textup{if }s>-\frac{1}{2}\end{cases}$ (3.13)

###### Proof.

The proofs of (3.9) and (3.10) follow directly from
$\vec{u}_{0,n}=\vec{u}_{0}+\vec{\phi}_{n}$, (3.5), the unitarity of $S(t)$ for
$t\leq 1$ and the fact that $\vec{u}_{0}$ is fixed, implying
$\|\vec{u}_{0}\|_{\mathcal{H}^{s}}\lesssim 1$. Besides, the definition (2.5)
of $\Xi_{j}$ and the multilinearity of $\mathcal{I}_{k}$ imply

$\Xi_{1}(\vec{u}_{0,n})(t)-\Xi_{1}(\vec{\phi}_{n})(t)=\sum\limits_{(v_{1},\dots,v_{k})\in
E}\mathcal{I}_{k}(v_{1},\cdots,v_{k})(t)$

where $E$ is the subset of $\\{S(t)\vec{u}_{0},S(t)\vec{\phi}_{n}\\}^{k}$ such
that at least one of the choice is $S(t)\vec{u}_{0}$. Thus, since $s<0$,
Young’s inequality, (2.8) and the unitarity of $S(t)$ imply

$\displaystyle\|\Xi_{1}(\vec{u}_{0,n})(t)-\Xi_{1}(\vec{\phi}_{n})(t)\|_{H^{s}}$
$\displaystyle\lesssim
t^{2}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}\left(\|\vec{u}_{0}\|^{k-1}_{\overrightarrow{\mathcal{F}L}^{1}}+\|\vec{\phi}_{n}\|^{k-1}_{\overrightarrow{\mathcal{F}L}^{1}}\right)$
$\displaystyle\lesssim
t^{2}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}\left(\|\vec{u}_{0}\|^{k-1}_{\overrightarrow{\mathcal{F}L}^{1}}+R^{k-1}A^{k-1}\right)$

which, combined with assumption (3.6), proves (3.11).

For the last inequality, we split the left-hand side into two terms:

$\Xi_{j}(\vec{u}_{0,n})(t)=\left(\Xi_{j}(\vec{u}_{0,n})(t)-\Xi_{j}(\vec{\phi}_{n})(t)\right)+\Xi_{j}(\vec{\phi}_{n})(t)$
(3.14)

which we will estimate separately.

Part 1: $\mathcal{F}\big{(}\Xi_{j}(\vec{\phi}_{n})(t)\big{)}$ essentially
consists of $(k-1)j+1$ convolutions of terms of the form
$\mathcal{F}(S(t)\vec{\phi}_{n})$. According to (2.1) and (3.2), the support
of each of these terms is contained within at most $4$ disjoint cubes of
volume approximately $A$. Thus, Lemma 3.4 and a countability argument show the
support of $\mathcal{F}(\Xi_{j}(\vec{\phi}_{n})(t))$ is contained within at
most $4^{(k-1)j+1}$ cubes of volume approximately $A$. Therefore, (2.5) and
Lemma 2.2 imply there exist $c,C>0$ such that

$|\operatorname*{supp}\mathcal{F}(\Xi_{j}(\vec{\phi}_{n})(t))|\leq C^{j}A\leq
c|C^{j}Q_{A}|.$

Since $s<0$, $\langle\xi\rangle^{s}$ is decreasing in $\lvert\xi\rvert$ and
Young’s inequality, (2.7) and the unitarity of $S(t)$ yield

$\displaystyle\|\Xi_{j}(\vec{\phi}_{n})(t)\|_{H^{s}}$
$\displaystyle\leq\|\langle\xi\rangle^{s}\|_{L^{2}(\operatorname*{supp}\mathcal{F}(\Xi_{j}(\vec{\phi}_{n})(t)))}\|\Xi_{j}(\vec{\phi}_{n})(t)\|_{\mathcal{F}L^{\infty}}$
$\displaystyle\lesssim\|\langle\xi\rangle^{s}\|_{L^{2}(cC^{j}Q_{A})}C^{j}t^{2j}\|\vec{\phi}_{n}\|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^{1}}\|\vec{\phi}_{n}\|^{2}_{\mathcal{H}^{0}}.$

Since $\|\langle\xi\rangle^{s}\|_{L^{2}(cC^{j}Q_{A})}\lesssim g_{s}(A)$ with
$g_{s}$ defined as in (3.13), we get from (3.5)

$\|\Xi_{j}(\vec{u}_{0,n})(t)\|_{H^{s}}\lesssim
C^{j}t^{2j}R^{(k-1)j}A^{(k-1)j}Rg_{s}(A).$ (3.15)

Part 2: On the other hand, since
$\mathcal{F}\big{(}\Xi_{j}(\vec{u}_{0})(t)-\Xi_{j}(\vec{\phi}_{n})(t)\big{)}$
is essentially a sum of terms made of $j$ integrals in time and $(k-1)j+1$
convolutions of terms of the form $\mathcal{F}(v)$, with
$v\in\\{S(t)\vec{u}_{0},S(t)\vec{\phi}_{n}\\}$, a similar argument as for
(3.11) along with $s<0$ yield

$\displaystyle\|\Xi_{j}(\vec{u}_{0,n})(t)-\Xi_{j}(\vec{\phi}_{n})(t)\|_{H^{s}}$
$\displaystyle\leq\|\Xi_{j}(\vec{u}_{0,n})(t)-\Xi_{j}(\vec{\phi}_{n})(t)\|_{L^{2}}$
$\displaystyle\lesssim
C^{j}t^{2j}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}\left(\|S(t)\vec{u}_{0}\|_{\mathcal{F}L^{1}}+\|S(t)\vec{\phi}_{n}\|_{\mathcal{F}L^{1}}\right)^{(k-1)j}.$

Thus, the unitarity of $S(t)$, (3.5) and (3.6) give

$\|\Xi_{j}(\vec{u}_{0,n})(t)-\Xi_{j}(\vec{\phi}_{n})(t)\|_{H^{s}}\lesssim
C^{j}t^{2j}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}(RA)^{(k-1)j}.$ (3.16)

Now, (3.12) follows from triangular inequality, (3.15) and (3.16).

∎

With the help of Lemma 3.4, we can also state the following crucial
proposition. This proposition exploits the high-to-low energy transfer
mentioned before to identify the first multilinear term in the Picard
expansion as the one responsible for the instability in Proposition 3.2 and in
Proposition 3.3.

######  Proposition 3.6.

Let $s<0$, $\vec{\phi}_{n}$ defined as in equation (3.2), $T\ll 1$ and $A\ll
N$. Then, there exists a constant $C>0$ such that

$\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{s}}\gtrsim
C^{k-1}R^{k}T^{2}A^{k-\frac{1}{2}+s}.$ (3.17)

###### Proof.

To simplify notation, we write

$\Lambda\coloneqq\left\\{(\xi_{1},\dots,\xi_{k})\in\widehat{\mathcal{M}}^{k}\colon\sum^{k}_{j=1}\xi_{j}=\xi\right\\}\quad\text{
and
}\quad\mathrm{d}\xi_{\Lambda}\coloneqq\mathrm{d}\xi_{1}\cdots\mathrm{d}\xi_{k-1}.$

Using (3.2) and a product-to-sum formula, we have

$\displaystyle\mathcal{F}[\Xi_{1}(\vec{\phi}_{n})](T,\xi)=R^{k}\sum\limits_{(\eta_{1},\dots,\eta_{k})\in\Sigma^{k}}$
$\displaystyle\int^{T}_{0}\sin((T-t^{\prime})\lambda(\xi))\lambda(\xi)$
$\displaystyle\times\int_{\Lambda}\prod^{k}_{j=1}\cos(t^{\prime}\lambda(\xi_{j}))\chi_{\eta_{j}+Q_{A}}(\xi_{j})\mathrm{d}\xi_{\Lambda}\mathrm{d}t^{\prime}.$

We split the set $\Sigma^{k}$, and hence the sum, into two parts $\Sigma_{1}$
and $\Sigma_{2}$ where

$\Sigma_{1}=\left\\{(\eta_{1},\dots,\eta_{k})\in\Sigma^{k}\colon\eta_{1}+\cdots+\eta_{k}=0\right\\}$

and $\Sigma_{2}=\Sigma^{k}\backslash\Sigma_{1}$. Note that, for any integer
$k\geq 2$, $\Sigma_{1}$ and $\Sigma_{2}$ are both non-empty. We denote then

$\mathcal{F}[\Xi_{1}(\vec{\phi}_{n})](T,\xi)=R^{k}\left(I_{1}(T,\xi)+I_{2}(T,\xi)\right)$
(3.18)

with, for $r=1,2$,

$I_{r}(T,\xi)=\sum\limits_{(\eta_{1},\dots,\eta_{k})\in\Sigma_{r}}\int^{T}_{0}\sin((T-t^{\prime})\lambda(\xi))\lambda(\xi)\int_{\Lambda}\prod^{k}_{j=1}\cos(t^{\prime}\lambda(\xi_{j}))\chi_{\eta_{j}+Q_{A}}(\xi_{j})\mathrm{d}\xi_{\Lambda}\mathrm{d}t^{\prime}.$

Since $T\ll 1$, $\lambda(\xi)\leq 1$, $\sin(x)\sim x$ for $\lvert x\rvert\leq
1$ and $\cos(x)\geq\frac{1}{2}$ for $\lvert x\rvert\ll 1$, we have

$\displaystyle I_{1}(T,\xi)$
$\displaystyle\gtrsim\sum\limits_{(\eta_{1},\dots,\eta_{k})\in\Sigma_{1}}\frac{1}{2}T^{2}\lambda(\xi)^{2}\int_{\Lambda}\frac{1}{2^{k}}\prod^{k}_{j=1}\chi_{\eta_{j}+Q_{A}}(\xi_{j})\mathrm{d}\xi_{\Lambda}$
$\displaystyle\gtrsim\frac{1}{2^{k+1}}T^{2}C^{k-1}_{1}A^{k-1}\lambda(\xi)^{2}\chi_{Q_{A}}(\xi),$

where the second inequality comes from $\lvert\Sigma_{1}\rvert>1$ and Lemma
3.4. Computing the $H^{s}$ norm, we get

$\|I_{1}(T)\|_{H^{s}}\gtrsim C^{k-1}_{1}T^{2}A^{k-\frac{1}{2}+s}.$ (3.19)

Meanwhile, since $\sin(x)\sim x$ for $\lvert x\rvert<1$,
$\lvert\cos(x)\rvert\leq 1$, and $\lambda(\xi)\leq 1$, we have

$\lvert
I_{2}(T,\xi)\rvert\lesssim\sum\limits_{(\eta_{1},\dots,\eta_{k})\in\Sigma_{2}}\frac{1}{2}T^{2}\int_{\Lambda}\prod^{k}_{j=1}\chi_{\eta_{j}+Q_{A}}(\xi_{j})\mathrm{d}\xi_{\Lambda}.$

Fix $(\eta_{1},\dots,\eta_{k})\in\Sigma_{2}$. Since
$\eta_{1}+\cdots+\eta_{k}\neq 0$, (3.4) implies there exists
$m\in\mathbb{Z}\backslash\\{0\\}$ such that

$\eta_{1}+\cdots+\eta_{k}=mN.$

Lemma 3.4 implies then

$\displaystyle\bigg{\|}\int_{\Lambda}\prod^{k}_{j=1}\chi_{\eta_{j}+Q_{A}}(\xi_{j})\mathrm{d}\xi_{\Lambda}\bigg{\|}_{H^{s}}$
$\displaystyle\lesssim C^{k-1}_{1}A^{k-1}\|\chi_{mN+Q_{kA}}(\xi)\|_{H^{s}}$
$\displaystyle\lesssim C^{k-1}_{1}A^{k-\frac{1}{2}}N^{s}.$

Since $\lvert\Sigma_{2}\rvert\leq\lvert\Sigma^{k}\rvert\leq 4^{k}$, we get

$\|I_{2}(T)\|_{H^{s}}\lesssim C^{k-1}_{1}T^{2}A^{k-\frac{1}{2}}N^{s}.$ (3.20)

Combining (3.19) and (3.20) with triangular inequality, we have

$\displaystyle\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{s}}$ $\displaystyle\geq
R^{k}\big{|}\|I_{1}(T)\|_{H^{s}}-\|I_{2}(T)\|_{H^{s}}\big{|}$
$\displaystyle\gtrsim
R^{k}C^{k-1}_{1}T^{2}A^{k-\frac{1}{2}}\big{(}A^{s}-N^{s}\big{)}$

which, with the assumption $A\ll N$, proves our result.

∎

###  3.2. Proof of Proposition 3.2

In this subsection we give the proof of Proposition 3.2. We claim it suffices
to show that, given $n\in\mathbb{N}$, the following inequalities hold:

(i) $\displaystyle\quad RA^{\frac{1}{2}}N^{s}<\frac{1}{n},$ (ii)
$\displaystyle\quad T^{2}R^{k-1}A^{k-1}\ll 1,$ (iii)
$\displaystyle\quad\|\vec{u}_{0}\|_{\mathcal{H}^{0}}\ll
RA^{\frac{1}{2}+s}\text{ and
}\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{1}}\ll RA,$ (iv)
$\displaystyle\quad T^{4}(RA)^{2(k-1)}Rg_{s}(A)\ll
T^{2}R^{k}A^{k-\frac{1}{2}+s},$ (v) $\displaystyle\quad
T^{2}R^{k}A^{k-\frac{1}{2}+s}\gg n,$ (vi) $\displaystyle\quad A\ll N$

for some particular $R$, $T$ and $N$, all depending on $n$.

Let us show conditions (i) through (vi) prove Proposition 3.2. First,
condition (i) along with (3.9) verifies the first estimate in (3.1). Besides,
conditions (ii) and the second of (iii), along with Lemma 2.4, prove existence
and uniqueness of a solution $u_{n}$ in $C([0,T],\mathcal{F}L^{1})$ with
$(u_{n},\partial_{t}u_{n})|_{t=0}=\vec{u}_{0,n}$, as well as the convergence
of the power series (3.8). Furthermore, these conditions, along with (3.12),
yield

$\displaystyle\sum^{\infty}_{j=2}\|\Xi_{j}(\vec{u}_{0,n})(T)\|_{H^{s}}$
$\displaystyle\lesssim
T^{4}(RA)^{2(k-1)}\left(\sum^{\infty}_{j=0}(CT^{2}R^{k-1}A^{k-1})^{j}\right)\left(\|\vec{u}_{0}\|_{\mathcal{H}^{0}}+Rg_{s}(A)\right)$
$\displaystyle\lesssim T^{4}(RA)^{2(k-1)}Rg_{s}(A).$

Then, using (3.8), Lemma 3.5 and Proposition 3.6 \- which is applicable by
condition (vi) - conditions (i) through (v) give:

$\displaystyle\|u_{n}(T)\|_{H^{s}}$
$\displaystyle\geq\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{s}}$
$\displaystyle\qquad-\bigg{\|}\Xi_{0}(\vec{u}_{0,n})(T)+\left(\Xi_{1}(\vec{u}_{0,n})(T)-\Xi_{1}(\vec{\phi}_{n})(T)\right)+\sum^{\infty}_{j=2}\Xi_{j}(\vec{u}_{0,n})(T)\bigg{\|}_{H^{s}}$
$\displaystyle\gtrsim
R^{k}T^{2}A^{k-\frac{1}{2}+s}-1-RA^{1/2}N^{s}-T^{2}R^{k-1}A^{k-1}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}-T^{4}(RA)^{2(k-1)}Rg_{s}(A)$
$\displaystyle\sim R^{k}T^{2}A^{k-\frac{1}{2}+s}\gg n.$

Thus, this verifies the second estimate in (3.1) at time $t_{n}\coloneqq T$.
Finally, a suitable choice of $T$ in terms of $N=N(n)$ ensures
$t_{n}\in(0,\frac{1}{n})$, for $N(n)$ sufficiently large. See details below.
So ends the proof of Proposition 3.2.

Consequently, it only remains to verify the conditions (i) through (vi). To do
so, we express $A$, $R$ and $T$ in terms of $N$. More precisely, let us choose

$A=10,\quad R=N^{-s-\delta}\quad\text{ and }\quad
T=N^{\frac{k-1}{2}(s+\frac{\delta}{2})}$

where $\delta$ is sufficiently small, namely
$0<\delta<\min\big{(}1,-\frac{2}{k+1}s\big{)}$. Since $A$ is a constant,
condition (vi) is trivially satisfied and, since $\vec{u}_{0}$ is fixed,
condition (iii) reduces to

$R\gg 1$

which is true from $-s-\delta>0$, for $N$ sufficiently large. Besides, for $N$
sufficiently large, we get

$\displaystyle RA^{\frac{1}{2}}N^{s}\sim N^{-\delta}\ll\frac{1}{n}$
$\displaystyle T^{2}R^{k-1}A^{k-1}\sim N^{-\frac{k-1}{2}\delta}\ll 1$
$\displaystyle T^{2}R^{k}A^{k-\frac{1}{2}+s}\sim N^{-s-\frac{k+1}{2}\delta}\gg
1$

which prove conditions (i), (ii) and (v). Lastly, condition (iv) is equivalent
to

$T^{2}R^{k-1}A^{k-2}g_{s}(A)\ll A^{-\frac{1}{2}+s}$

which, in our setting, is equivalent to condition (ii) since $A$ is constant.
Finally, since $\delta<-s$, observe that $T$ goes to $0$ as $N\to\infty$, so
$T\in(0,\frac{1}{n})$ for $N$ sufficiently large. Taking

$N=n^{\frac{2}{\delta}}$

completes the proof since $\delta<1$.

######  Remark 3.7.

In this remark, we use our previous arguments to prove infinite loss of
regularity, namely Proposition 3.3. The idea is to use the same construction
as before. Indeed, this construction allowed us to choose our parameter $A$ to
be constant. In the following, we observe that the change of regularity
between Proposition 3.2 and Proposition 3.3 is only expressed in powers of
$A$, so that this change has no major implications whatsoever.

Let $s<0$, $\sigma\in\mathbb{R}$ and
$\vec{u}_{0}=(u_{0},u_{1})\in\overrightarrow{\mathcal{F}L}^{1}$. Define
$\vec{\phi}_{n}$ and $\vec{u}_{0,n}$ as before. There exists, for any given
$n\in\mathbb{N}$, a unique solution $u_{n}$ to (1.1) in
$C([0,T],\mathcal{F}L^{1})$, with $T$ satisfying (3.7), such that
$(u_{n},\partial_{t}u_{n})|_{t=0}=\vec{u}_{0,n}$. Moreover, $u_{n}$ can be
expressed as the power series in (3.8). First, we claim that it suffices to
consider the case $\sigma<s$. Indeed, in the case $\sigma\geq s$ the estimate
on the initial data remains the same as for Proposition 3.2 and, by Sobolev
embedding, we have

$\|u_{n}(t_{n})\|_{H^{\sigma}}\geq\|u_{n}(t_{n})\|_{H^{s}}>n.$

Let us then assume $\sigma<s$. Lemma 3.5, (3.8), (3.7) and (3.6) yield

$\displaystyle\|u_{n}(T)-\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{\sigma}}$
$\displaystyle\leq\|u_{n}(T)-\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{s}}$
$\displaystyle\lesssim
1+RA^{1/2}N^{s}+T^{2}(RA)^{k-1}\|\vec{u}_{0}\|_{\mathcal{H}^{0}}+T^{4}(RA)^{2(k-1)}Rg_{s}(A)$

while Proposition 3.6 gives

$\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{H^{\sigma}}\gtrsim
R^{k}T^{2}A^{k-\frac{1}{2}+\sigma}.$

Therefore, the same arguments as before allow us to say that, to prove
Proposition 3.3, it suffices to verify the following hold:

(i) $\displaystyle\quad RA^{\frac{1}{2}}N^{s}<\frac{1}{n},$ (ii)
$\displaystyle\quad T^{2}R^{k-1}A^{k-1}\ll 1,$ (iii)
$\displaystyle\quad\|\vec{u}_{0}\|_{\mathcal{H}^{0}}\ll
RA^{\frac{1}{2}+\sigma}\text{ and
}\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{1}}\ll RA,$ (iv)
$\displaystyle\quad T^{4}(RA)^{2(k-1)}Rg_{s}(A)\ll
T^{2}R^{k}A^{k-\frac{1}{2}+\sigma},$ (v) $\displaystyle\quad
T^{2}R^{k}A^{k-\frac{1}{2}+\sigma}\gg n,$ (vi) $\displaystyle\quad A\ll N.$

Note that, compared to the conditions in the proof of Proposition 3.2, the
only changes are in conditions (iii), (iv) and (v), where the power of $A$
changes. Yet, if we choose $A$ to be constant again, the rest of the reasoning
stays the same. Hence, the choices

$A=10,\quad R=N^{-s-\delta},\quad
T=N^{\frac{k-1}{2}(s+\frac{\delta}{2})},\quad\text{ and }\quad
N=n^{\frac{2}{\delta}}$

prove also Proposition 3.3.

######  Remark 3.8.

The proof we gave applies directly to the study of the multi-dimensional
generalized improved Boussinesq equation (1.5). The only major change to make
is in the definition of the functions $\phi_{n}$. Indeed, let us denote
$e_{1}=(1,0,\cdots,0)\in\mathbb{R}^{d}$, we then define $\phi_{n}$ by

$\widehat{\phi}_{n}(\xi)=R\chi_{\Omega}=R\sum_{\eta\in\Sigma}\chi_{\\{\eta
e_{1}+Q_{A}\\}}$

with $\Sigma$ as in (3.4) and $Q_{A}=[-\frac{A}{2},\frac{A}{2}]^{d}$. Then,
the same arguments are still applicable. In particular, observe that Lemma 3.4
can be rephrased as:

######  Lemma 3.9.

Let $a,b\in\mathbb{R}^{d}$ and $A>0$, then we have

$C_{d}A^{d}\chi_{a+b+Q_{A}}(\xi)\leq\chi_{a+Q_{A}}\ast\chi_{b+Q_{A}}(\xi)\leq\widetilde{C}_{d}A^{d}\chi_{a+b+Q_{2A}}(\xi)$

where $C_{d},\widetilde{C}_{d}>0$ are constants depending only on the
dimension $d$.

Subsequently, the only changes caused from the change of dimension are
expressed in powers of $A$. However, since $A$ is chosen to be constant, it
does not change the argument, as for when we proved infinite loss of
regularity from “standard” norm inflation. Therefore, norm inflation with
infinite loss of regularity for initial data
$\vec{u}_{0}\in\mathcal{H}^{s}(\mathbb{R}^{d})$ with $s<0$ follows naturally
from the same choice of parameters $A$, $R$, $T$ and $N$.

Similarly, our result would still apply on the same multidimensional problem
on the torus $\mathbb{T}^{d}$, although such a problem has not been seen in
the litterature yet, at least not to our knowledge.

######  Remark 3.10.

In this remark, we show that our proof also implies Theorem 1.1. The idea is
to use exactly the same construction and to estimate the norm. Using the same
notations, observe that we have

$\|u_{n}(t_{n})\|_{W^{\sigma,2,\infty}}\geq\|u_{n}(t_{n})\|_{H^{\sigma}}>n.$

Therefore, we only need to show that

$\|\vec{u}_{0}-\vec{u}_{0,n}\|_{\mathcal{W}^{s,2,\infty}}<\frac{1}{n}$

which means that we want $\vec{u}_{0}$ and $\vec{u}_{0,n}$ to satisfy

$\|\vec{u}_{0}-\vec{u}_{0,n}\|_{\mathcal{H}^{s}}<\frac{1}{n}\quad\text{ and
}\quad\|\vec{u}_{0}-\vec{u}_{0,n}\|_{W^{s,\infty}\times
W^{s,\infty}}<\frac{1}{n}.$

We already proved that the first estimate is satisfied, so we are only
interested in the second one. Observe that
$\vec{u}_{0}-\vec{u}_{0,n}=\vec{\phi}_{n}=(\phi_{n},0)$ and $\phi_{n}$ is
frequently supported on $\Omega$, defined by (3.3). Then, we get by Cauchy-
Schwarz and inverse Fourier transform

$|\langle\nabla\rangle^{s}\phi_{n}|\leq\int_{\widehat{\mathcal{M}}}|\langle\xi\rangle^{s}\widehat{\phi}_{n}(\xi)|\mathrm{d}\xi=\int_{\Omega}\langle\xi\rangle^{s}|\widehat{\phi}_{n}(\xi)|\mathrm{d}\xi\leq
2A^{\frac{1}{2}}\|\phi_{n}\|_{H^{s}}$

Since $A=10$ in our proof, and
$\|\vec{\phi}_{n}\|_{\mathcal{W}^{s,2,\infty}}=\|\phi_{n}\|_{W^{s,2,\infty}}$,
this shows the desired result and Theorem 1.1. Furthermore, combining this
argument with Remark 3.8, this result still holds in higher dimension.

##  Appendix A Norm inflation for other spaces

In this appendix, we come back to Remark 1.6 and show how to prove norm
inflation with infinite loss of regularity at general initial data for
equation (1.1) in Fourier-Lebesgue, modulation and Wiener amalgam spaces.
However, we do not show the entire proof since it is mostly similar to the one
we gave for Sobolev spaces. Indeed, we use the same construction and
decomposition as before, so the first differences are in the estimates we find
in Lemma 3.5 and Proposition 3.6. Fortunately we find that the new estimates
are, under the condition $A=10$, equivalent to the ones we had for Sobolev
spaces, so the last part of our argument remains unchanged. Therefore, we just
show quickly how to get these estimates. Again, a whole proof would be quite
redundant with what was shown before, the arguments being essentially the
same. Thus, we just point out the key differences. Before that, we introduce
our spaces. Let us recall, at this point, that modulation and Wiener amalgam
spaces were introduced by Feichtinger [16].

###  A.1. New spaces and some preliminary results.

First, let us recall that Fourier-Lebesgue spaces are defined in Definition
1.4. Now, we introduce the modulation and Wiener amalgam spaces.

For any $n\in\mathbb{Z}^{d}$, we define
$Q_{n}=n+\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$ so that
$\mathbb{R}^{d}=\bigcup_{n\in\mathbb{Z}^{d}}Q_{n}$. Then, let
$\rho\in\mathcal{S}(\mathbb{R}^{d})$ such that
$\rho\colon\mathbb{R}^{d}\to[0,1]$ and

$\rho(\xi)=\begin{cases}1\quad\text{ if
}\quad\lvert\xi\rvert\leq\frac{1}{2}\\\ 0\quad\text{ if
}\quad\lvert\xi\rvert\geq 1\end{cases}$ (A.1)

We also denote $\rho_{n}(\xi)=\rho(\xi-n)$ for any $n\in\mathbb{Z}^{d}$. Let
us define

$\sigma_{n}=\frac{\rho_{n}}{\sum_{l\in\mathbb{Z}}\rho_{l}}$

and $P_{n}f=\mathcal{F}^{-1}(\sigma_{n}\mathcal{F}(f))$ for any suitable $f$.
Then, we define the modulation spaces in the following way:

######  Definition A.1 (Modulation spaces).

Let $s\in\mathbb{R}$ and $p,q\geq 1$. The Modulation space
$M^{p,q}_{s}(\mathbb{R}^{d})$ is the completion of the Schwartz class of
functions $\mathcal{S}(\mathbb{R}^{d})$ with respect to the norm

$\|f\|_{M^{p,q}_{s}(\mathbb{R}^{d})}=\big{\|}(1+\lvert
n\rvert^{s})\|P_{n}f\|_{L^{p}_{x}(\mathbb{R}^{d})}\big{\|}_{\ell^{q}_{n}(\mathbb{Z}^{d})}.$

We also define Wiener amalgam spaces in the following way:

######  Definition A.2 (Wiener amalgam spaces).

Let $s\in\mathbb{R}$ and $p,q\geq 1$. The Wiener amalgam space
$W^{p,q}_{s}(\mathbb{R}^{d})$ is the completion of the Schwartz class of
functions $\mathcal{S}(\mathbb{R}^{d})$ with respect to the norm

$\|f\|_{W^{p,q}_{s}(\mathbb{R}^{d})}=\big{\|}\|(1+\lvert
n\rvert^{s})P_{n}f\|_{\ell^{q}_{n}(\mathbb{Z}^{d})}\big{\|}_{L^{p}_{x}(\mathbb{R}^{d})}.$

We have the following relations between modulation and Wiener amalgam spaces:

######  Lemma A.3.

Let $1\leq p,q,p_{1},q_{1},p_{2},q_{2}\leq\infty$ and $d\geq 1$ be any finite
dimension. Then:

  1. (1)

For $p_{1}\leq p_{2}$ and $q_{1}\leq q_{2}$, we have

$M^{p_{1},q_{1}}_{s}(\mathbb{R}^{d})\hookrightarrow
M^{p_{2},q_{2}}_{s}(\mathbb{R}^{d}),$ (A.2)

and

$W^{p,q_{1}}_{s}(\mathbb{R}^{d})\hookrightarrow
W^{p,q_{2}}_{s}(\mathbb{R}^{d}),$ (A.3)

  2. (2)

for $q\leq p$, we have

$M^{p,q}_{s}(\mathbb{R}^{d})\hookrightarrow W^{p,q}_{s}(\mathbb{R}^{d}),$
(A.4)

and for $p\leq q$ we have

$W^{p,q}_{s}(\mathbb{R}^{d})\hookrightarrow M^{p,q}_{s}(\mathbb{R}^{d}),$
(A.5)

  3. (3)

and we finally have

$M^{p,\min(p,q)}_{s}(\mathbb{R}^{d})\hookrightarrow
W^{p,q}_{s}(\mathbb{R}^{d})\hookrightarrow
M^{p,\max(p,q)}_{s}(\mathbb{R}^{d}).$ (A.6)

The proof of this Lemma can be seen in [4], but split into different proofs.
For completeness and ease of reading, we include it here. However, we first
need the following lemma, which proof can be seen in [28, Lemma 6.1]:

######  Lemma A.4.

Let $\Omega$ be a compact subset of $\mathbb{R}^{d}$ with $d\geq 1$, such that
$\mathrm{diam}\ \Omega<2R$, with $R>0$, and $1\leq p\leq q\leq\infty$. Then,
there exists a constant $C>0$ depending only on $p$, $q$ and $R$ such that:

$\|f\|_{L^{q}(\mathbb{R}^{d})}\leq C\|f\|_{L^{p}(\mathbb{R}^{d})}$

for any function $f\in L^{p}(\mathbb{R}^{d})$ such that its Fourier transform
$\widehat{f}$ is compactly supported in $\Omega$.

###### Proof of Lemma A.3.

The embeddings (A.2) and (A.3) follow from Lemma A.4 and the fact that, for
any $p\leq q$,
$\ell^{p}(\mathbb{Z}^{d})\hookrightarrow\ell^{q}(\mathbb{Z}^{d})$.

Embeddings (A.4) and (A.5) follow from Minkowski’s inequality.

Finally, (A.6) is a combination of (A.2), (A.3), (A.4) and (A.5). ∎

Besides, we also have the following algebra property:

######  Lemma A.5.

For any $d\geq 1$, the space $M^{2,1}_{0}(\mathbb{R}^{d})$ is a Banach
algebra.

###### Proof.

We want to show that, for any $u,v\in M^{2,1}_{0}(\mathbb{R}^{d})$, we have

$\|uv\|_{M^{2,1}_{0}(\mathbb{R}^{d})}=\sum_{n\in\mathbb{Z}^{d}}\|P_{n}(uv)\|_{L^{2}(\mathbb{R}^{d})}\leq\|u\|_{M^{2,1}_{0}(\mathbb{R}^{d})}\|v\|_{M^{2,1}_{0}(\mathbb{R}^{d})}.$

Note that $u=\sum_{m\in\mathbb{Z}^{d}}P_{m}u$ and
$v=\sum_{k\in\mathbb{Z}^{d}}P_{k}v$. Therefore

$P_{n}(uv)=\sum_{m,k\in\mathbb{Z}^{d}}P_{n}(P_{m}uP_{k}v).$

The idea then is to study the Fourier support of each of the terms
$P_{n}(P_{m}uP_{k}v)$. Since, for any $\xi\in\mathbb{R}^{d}$,

$\mathcal{F}[P_{n}(P_{m}uP_{k}v)](\xi)=\sigma_{n}(\xi)\int_{\xi=\xi_{1}+\xi_{2}}\sigma_{m}(\xi_{1})\widehat{u}(\xi_{1})\sigma_{k}(\xi_{2})\widehat{v}(\xi_{2})\mathrm{d}\xi_{1}$

and, for any $N\in\mathbb{Z}^{d}$,
$\operatorname*{supp}\sigma_{N}\subset\\{\xi\in\mathbb{R}^{d},\lvert\xi-N\rvert\leq
1\\}$, we have then

$\operatorname*{supp}\big{[}(\sigma_{m}\widehat{u})\ast(\sigma_{k}\widehat{v})\big{]}\subset\\{\xi\in\mathbb{R}^{d},\lvert\xi-
m-k\rvert\leq 2\\}$

and

$\mathcal{F}[P_{n}(P_{m}uP_{k}v)]\equiv 0\quad\text{ if }\quad\lvert
n-m-k\rvert>3.$

Plancherel’s identity, Hölder’s inequality and (A.2) yield then:

$\displaystyle\|P_{n}(P_{m}uP_{k}v)\|_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\leq\|P_{m}uP_{k}v\|_{L^{2}(\mathbb{R}^{d})}\chi_{(\lvert
n-k-m\rvert\leq 3)},$
$\displaystyle\leq\|P_{m}u\|_{L^{\infty}(\mathbb{R}^{d})}\|P_{k}v\|_{L^{2}(\mathbb{R}^{d})}\chi_{(\lvert
n-k-m\rvert\leq 3)},$
$\displaystyle\lesssim\|P_{m}u\|_{L^{2}(\mathbb{R}^{d})}\|P_{k}v\|_{L^{2}(\mathbb{R}^{d})}\chi_{(\lvert
n-k-m\rvert\leq 3)}.$

Hence,

$\displaystyle\|uv\|_{M^{2,1}_{0}(\mathbb{R}^{d})}$
$\displaystyle\leq\sum_{n\in\mathbb{Z}^{d}}\sum_{m,k\in\mathbb{Z}^{d}}\|P_{n}(P_{m}uP_{k}v)\|_{L^{2}(\mathbb{R}^{d})},$
$\displaystyle\lesssim\sum_{n\in\mathbb{Z}^{d}}\sum_{m,k\in\mathbb{Z}^{d}}\|P_{m}u\|_{L^{2}(\mathbb{R}^{d})}\|P_{k}v\|_{L^{2}(\mathbb{R}^{d})}\chi_{(\lvert
n-k-m\rvert\leq 3)},$
$\displaystyle\lesssim\sum_{m,k\in\mathbb{Z}^{d}}\|P_{m}u\|_{L^{2}(\mathbb{R}^{d})}\|P_{k}v\|_{L^{2}(\mathbb{R}^{d})}$

which ends the proof.

∎

###  A.2. Proofs of norm inflation in other spaces

In this subsection, we claim that the following result is true:

######  Theorem A.6.

Assume that $s<0$ and $1\leq q\leq\infty$ and let

$Z_{s}^{q}\coloneqq\mathcal{F}L^{s,q}(\mathcal{M})\quad\text{ or }\quad
M_{s}^{2,q}(\mathbb{R})\quad\text{ or }\quad W_{s}^{2,q}(\mathbb{R}).$ (A.7)

Then, norm inflation with infinite loss of regularity occurs at the origin for
(1.1) in $Z_{s}^{q}$.

Namely, let $\theta\in\mathbb{R}$, $s<0$ and fix $\vec{u}_{0}\in
Z_{s}^{q}\times Z_{s}^{q}$. Given any $\varepsilon>0$, there exists a solution
$u_{\varepsilon}\in C([0,T],Z_{s}^{q})$ to (1.1) with $T>0$ and
$t_{\varepsilon}\in(0,\varepsilon)$ such that

$\|(u_{\varepsilon}(0),\partial_{t}u_{\varepsilon}(0))-\vec{u}_{0}\|_{Z_{s}^{q}\times
Z_{s}^{q}}<\varepsilon\qquad\text{ and
}\qquad\|u_{\varepsilon}(t_{\varepsilon})\|_{Z_{\theta}^{q}}>\varepsilon^{-1}.$

As explained before, the idea to prove our result is to use the same
construction as in Section 3. Actually, we show that, by keeping the same
notations and choosing $A$ to be constant, we have estimates that are
equivalent to the ones given in Lemma 3.5 and Proposition 3.6. Then, the proof
of Theorem A.6 follows with the same argument.

####  A.2.1. The case of Fourier-Lebesgue spaces

In the case of Fourier-Lebesgue spaces $\mathcal{F}L^{s,q}(\mathcal{M})$, we
get from the same computations as for the Sobolev spaces $H^{s}(\mathcal{M})$:

######  Proposition A.7.

Let $s<0$ and $1\leq q\leq\infty$. We have then

$\|\Xi_{0}[\vec{u}_{0,n}](T)\|_{\mathcal{F}L^{s,q}}\lesssim
1+RN^{s}A^{\frac{1}{q}},$ (A.8)

as well as

$\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{\mathcal{F}L^{s,q}}\lesssim
T^{2}(RA)^{(k-1)}\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{q}(\mathcal{M})},$
(A.9)

and, for any $j\geq 2$,

$\|\Xi_{j}[\vec{u}_{0,n}](T)\|_{\mathcal{F}L^{s,q}}\lesssim
T^{2j}(RA)^{(k-1)j}\big{(}\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{q}(\mathcal{M})}+Rf_{s,q}(A)\big{)}$
(A.10)

where

$f_{s,q}(A)=\|\langle\xi\rangle^{s}\|_{L^{q}_{\xi}(Q_{A})}.$ (A.11)

Also, if $A\ll N$, we have the following lower bound:

$\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{\mathcal{F}L^{s,q}}\gtrsim
R^{k}T^{2}A^{k-1+\frac{1}{q}+s}$ (A.12)

Under the condition $A=10$, we observe that (A.8), (A.9), (A.15) and (A.16)
are essentially equivalent respectively to (3.10), (3.11), (3.12) and (3.17),
so that the expected result follows from the same argument. There is one
condition to change though, which is
$\|\vec{u}_{0}\|_{\overrightarrow{\mathcal{F}L}^{q}(\mathcal{M})}\lesssim 1$
instead of $\|\vec{u}_{0}\|_{\mathcal{H}^{0}(\mathcal{M})}\lesssim 1$, but
this is quite reasonable since our initial data $\vec{u}_{0}$ is fixed.

####  A.2.2. The case of modulation spaces

The computations in this section follow the same ideas as in the previous
section, but the modulation spaces make them a bit trickier to apply. We first
state our results:

######  Proposition A.8.

Let $s<0$ and $1\leq q\leq\infty$. Keeping the same notations as before, and
asssuming

$\|\vec{u}_{0}\|_{M^{2,q}_{s}\times M^{2,q}_{s}}\lesssim 1\quad\text{ and
}\quad\|\vec{u}_{0}\|_{M^{2,1}_{0}\times M^{2,1}_{0}}\ll R\|(1+\lvert
n\rvert)^{s}\|_{\ell^{q}(1\leq\lvert n\rvert\leq A)},$

we have

$\|\Xi_{0}[\vec{u}_{0,n}](T)\|_{M_{s}^{2,q}}\lesssim 1+RN^{s}A^{\frac{1}{2}},$
(A.13)

as well as

$\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{M_{s}^{2,q}}\lesssim
T^{2}(RA)^{(k-1)}\|\vec{u}_{0}\|_{M^{2,1}_{0}(\mathcal{M})},$ (A.14)

and, for any $j\geq 2$

$\|\Xi_{j}[\vec{u}_{0,n}](T)\|_{M_{s}^{2,q}}\lesssim
T^{2j}(RA)^{(k-1)j}R\|(1+\lvert n\rvert)^{s}\|_{\ell^{q}(1\leq\lvert
n\rvert\leq A)}.$ (A.15)

Also, if $A\ll N$, we have the following upper bound:

$\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{M_{s}^{2,q}}\gtrsim
R^{k}T^{2}A^{k-1}A^{\frac{1}{q}+s}.$ (A.16)

Again, we see that if we choose $A$ to be constant, all these estimates are
equivalent to the estimates we had for the Sobolev spaces, and the norm
inflation result becomes straightforward. We include the proof of Proposition
A.8 both for completeness and to put an emphasis on the differences between
these spaces and Sobolev spaces.

###### Proof of Proposition A.8.

First, we have from $\lvert\cos x\rvert\leq 1$ and $\lvert\sin
x\rvert\lesssim\lvert x\rvert$,

$\|S(t)(u_{0},u_{1})\|_{M_{s}^{2,q}}\lesssim\|u_{0}\|_{M_{s}^{2,q}}+\lvert
t\rvert\|u_{1}\|_{M_{s}^{2,q}}\lesssim\|(u_{0},u_{1})\|_{M^{2,q}_{s}\times
M^{2,q}_{s}}$ (A.17)

for any $0\leq t\leq 1$ and $u_{0},u_{1}\in M^{2,q}_{s}$. Besides, we also
have

$\displaystyle\|\phi_{n}\|_{M_{s}^{2,q}}$ $\displaystyle=R\big{\|}(1+\lvert
n\rvert)^{s}\|P_{n}\mathcal{F}^{-1}(\sum_{\eta\in\Sigma}\chi_{\eta+Q_{A}})\|_{L^{2}}\big{\|}_{\ell^{q}_{n}}$
$\displaystyle=R\big{\|}(1+\lvert
n\rvert)^{s}\|\sigma_{n}\sum_{\eta\in\Sigma}\chi_{\eta+Q_{A}}\|_{L^{2}}\big{\|}_{\ell^{q}_{n}}$

but since the sets $\eta+Q_{A}$ are disjoint for any two
$\eta_{1}\neq\eta_{2}$ in $\Sigma$, we get

$\left\|\sigma_{n}\sum_{\eta\in\Sigma}\chi_{\eta+Q_{A}}\right\|^{2}_{L^{2}}\sim\chi_{N+Q_{A}}(n)\int_{\mathbb{R}}\sigma^{2}_{n}(\xi)\chi^{2}_{n+Q_{A}}(\xi)\mathrm{d}\xi\sim\chi_{N+Q_{A}}(n)$

and

$\|\phi_{n}\|_{M_{s}^{2,q}}\sim R\|(1+\lvert
n\rvert)^{s}\chi_{N+Q_{A}}(n)\|_{\ell^{q}_{n}}\sim RN^{s}A^{\frac{1}{q}}$

which, combined with (A.17), proves our first estimate. Besides, (A.2) and the
algebra property of $M^{2,1}_{0}$ yield, in a similar argument as for (3.11):

$\displaystyle\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{M_{s}^{2,q}}$
$\displaystyle\lesssim\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{M_{0}^{2,1}},$
$\displaystyle\lesssim T^{2}\|\vec{u}_{0}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}\big{(}\|\vec{u}_{0}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}+\|\vec{\phi}_{n}\|_{M^{2,1}_{0}\times M^{2,1}_{0}}\big{)}^{k-1},$
$\displaystyle\lesssim T^{2}\|(u_{0},u_{1})\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}(RA^{\frac{1}{2}})^{k-1}.$

Hence (A.14).

Now, let $j\geq 2$. Using the same argument on the support as in the proof for
(3.12), we have

$\displaystyle\|\sigma_{n}\mathcal{F}[\Xi_{j}(\vec{\phi}_{n})](T)\|_{L^{2}_{\xi}}$
$\displaystyle\leq\|\sigma_{n}\|_{L^{2}_{\xi}(\operatorname*{supp}\mathcal{F}[\Xi_{j}(\vec{\phi}_{n})](T))}\|\Xi_{j}(\vec{\phi}_{n})(T)\|_{\mathcal{F}L^{\infty}}$
$\displaystyle\lesssim\|\sigma_{n}\|_{L^{2}_{\xi}(Q_{A}\cap
Q_{n})}C^{j}T^{2j}\|\vec{\phi}_{n}\|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^{1}}\|\vec{\phi}_{n}\|^{2}_{\mathcal{H}^{0}}$
$\displaystyle\lesssim\mathds{1}_{\\{n\in Q_{A}\\}}T^{2j}(RA)^{(k-1)j}R$

so that

$\|\Xi_{j}[\vec{\phi}_{n}](T)\|_{M_{s}^{2,q}}\lesssim
T^{2j}(RA)^{(k-1)j}R\|(1+\lvert n\rvert)^{s}\mathds{1}_{\\{n\in
Q_{A}\\}}\|_{\ell^{q}_{n}}.$

Besides, a similar argument as for (A.14) yields

$\displaystyle\|\Xi_{j}[\vec{u}_{0,n}](T)-\Xi_{j}[\vec{\phi}_{n}](T)\|_{M_{s}^{2,q}}$
$\displaystyle\lesssim\|\Xi_{j}[\vec{u}_{0,n}](T)-\Xi_{j}[\vec{\phi}_{n}](T)\|_{M_{0}^{2,1}},$
$\displaystyle\lesssim T^{2j}\|\vec{u}_{0}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}\big{(}\|\vec{u}_{0}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}+\|\vec{\phi}_{n}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}\big{)}^{(k-1)j},$ $\displaystyle\lesssim
T^{2j}\|\vec{u}_{0}\|_{M^{2,1}_{0}\times
M^{2,1}_{0}}(RA^{\frac{1}{2}})^{(k-1)j},$

which gives our third estimate.

Finally, our fourth estimate follows from a straightforward adaptaton of the
proof of Proposition 3.6, the only difference being the norm used.

∎

####  A.2.3. The case of Wiener amalgam spaces

This section relies a lot on the interactions between modulation and Wiener
amalgam spaces. Indeed, using (A.6) and Proposition A.8, we get respectively

$\|\Xi_{0}[\vec{u}_{0,n}](T)\|_{W_{s}^{2,q}}\lesssim\|\Xi_{0}[\vec{u}_{0,n}](T)\|_{M_{s}^{2,\min(2,q)}}\lesssim
1+RN^{s}A^{\frac{1}{2}},$ (A.18)

as well as

$\displaystyle\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{W_{s}^{2,q}}$
$\displaystyle\lesssim\|\Xi_{1}[\vec{u}_{0,n}](T)-\Xi_{1}[\vec{\phi}_{n}](T)\|_{M_{s}^{2,\min(2,q)}}$
(A.19) $\displaystyle\lesssim
T^{2}(RA)^{(k-1)}\|\vec{u}_{0}\|_{M^{2,1}_{0}(\mathcal{M})},$ (A.20)

and, for any $j\geq 2$,

$\displaystyle\|\Xi_{j}[\vec{u}_{0,n}](T)\|_{W_{s}^{2,q}}$
$\displaystyle\lesssim\|\Xi_{j}[\vec{u}_{0,n}](T)\|_{M_{s}^{2,\min(2,q)}}$
(A.21) $\displaystyle\lesssim T^{2j}(RA)^{(k-1)j}R\|(1+\lvert
n\rvert)^{s}\|_{\ell^{\min(2,q)}(1\leq\lvert n\rvert\leq A)}.$ (A.22)

Besides, we also get the following lower bound:

$\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{W_{s}^{2,q}}\gtrsim\|\Xi_{1}(\vec{\phi}_{n})(T)\|_{W_{s}^{2,\max(2,q)}}\gtrsim
R^{k}T^{2}A^{k-1}A^{\frac{1}{q}+s}.$ (A.23)

Again, these estimates are equivalent to the ones we had for Sobolev spaces
under the condition $A=10$, so norm inflation with infinite loss of regularity
follows.

## Acknowledgments

The author would like to thank Tadahiro Oh for suggesting this problem and his
continuous support throughout this work. The author acknowledges support from
Tadahiro Oh’s ERC grant (no. 864138 “SingStochDispDyn”). The author would also
like to thank Younes Zine for several helpful discussions.

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