# Illposedness for dispersive equations:
Degenerate dispersion and Takeuchi–Mizohata condition

In-Jee Jeong Department of Mathematical Sciences and Research Institute of
Mathematics, Seoul National University.
E-mail<EMAIL_ADDRESS>Sung-Jin Oh Department of Mathematics, UC Berkeley
and School of Mathematics, Korea Institute for Advanced Study. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We provide a unified viewpoint on two illposedness mechanisms for dispersive
equations in one spatial dimension, namely degenerate dispersion and (the
failure of) the Takeuchi–Mizohata condition. Our approach is based on a robust
energy- and duality-based method introduced in an earlier work of the authors
in the setting of Hall-magnetohydynamics. Concretely, the main results in this
paper concern strong illposedness of the Cauchy problem (e.g., non-existence
and unboundedness of the solution map) in high-regularity Sobolev spaces for
various quasilinear degenerate Schrödinger- and KdV-type equations, including
the Hunter–Smothers equation, $K(m,n)$ models of Rosenau–Hyman, and the
inviscid surface growth model. The mechanism behind these results may be
understood in terms of combination of two effects: degenerate dispersion –
which is a property of the principal term in the presence of degenerating
coefficients – and the evolution of the amplitude governed by the
Takeuchi–Mizohata condition – which concerns the subprincipal term. We also
demonstrate how the same techniques yield a more quantitative version of the
classical $L^{2}$-illposedness result by Mizohata for linear variable-
coefficient Schrödinger equations with failed Takeuchi–Mizohata condition.

##  1 Introduction

###  1.1 Quasilinear degenerate dispersive equations

In this paper, we study the issue of illposedness of the Cauchy problem for
various quasilinear dispersive equations in one spatial dimension in the
presence of _degenerate dispersion_. We consider both Schrödinger and KdV-type
equations. Examples of Schrödinger-type equations we treat include, for
instance, the Hunter–Smothers equation [15]

$\displaystyle i\partial_{t}\phi+\partial_{x}(|\phi|^{2}\partial_{x}\phi)=0,$
(1.1)

which was derived from the Majda–Rosales–Schonbeck equation [20] describing
the resonant reflection of sound waves off a sawtooth entropy wave, as well as
its Hamiltonian variant considered by Germain, Harrop-Griffith and Marzuola
[11, 14]

$i\partial_{t}\phi+\overline{\phi}\partial_{x}(\phi\partial_{x}\phi)-\mu_{0}|{\phi}|^{2}\phi=0\qquad(\mu_{0}=-1,0,1),$
(1.2)

where $\phi:\mathbb{R}_{+}\times\mathbb{T}\rightarrow\mathbb{C}$. In the KdV
case, our results cover the $K(m,n)$ equation of Rosenau–Hyman [30] with
$n=2$, i.e.,

$\partial_{t}u+(\tfrac{1}{m}u^{m})_{x}+(\tfrac{1}{2}u^{2})_{xxx}=0\quad(m\hbox{
is a nonnegative integer}),$ (1.3)

which has been studied extensively in connection with the remarkable nonlinear
phenomenon of the existence of _compactons_ (solitons with compact spatial
support) [27, 28, 29, 30, 35, 36] (see [31] for a recent review), as well as
the inviscid surface growth model (see [6, 7, 8, 26] for the full surface
growth model, with the dissipation $-\nu h_{xxxx}$ on the right-hand side)

$\partial_{t}h+((h_{x})^{2})_{xx}=0,$ (1.4)

where $u,h:\mathbb{R}_{+}\times\mathbb{T}\to\mathbb{R}$. Indeed, degenerate
KdV-type equations similar in form to (1.3) appear in various subjects
including sedimentation models ([37, 5]), shoreline problem in water waves
([19]) and magma dynamics ([32, 33]), to name a few. A more extensive list of
references on degenerate KdV equations can be found in [11, 10].

In each of these equations, observe that the the highest order term is
nonlinear – more specifically, quadratic or cubic – in the solution. Vanishing
of the solution, therefore, leads to some kind of “degeneracy” of the highest
order term, which in turn gives rise to delicate issues in the (local) well-
posedness of the associated Cauchy problem.

Indeed, for initial data that are uniformly bounded away from $0$ (a property
henceforth referred to as _nondegeneracy_), one expects local well-posedness
in high-regularity $L^{2}$-based Sobolev spaces $H^{s}(\mathbb{T})$. For
example, in the case of (1.1), for a sufficiently regular solution $\phi$, one
has the conservation of the $L^{2}$-norm:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\mathbb{T}}|\phi|^{2}\,\mathrm{d}x\right)=0.$

Obtaining higher regularity apriori estimates is a much more nontrivial task.
One can observe the following bound for $n\geq 1$ (in operator notation) at
each $t$:

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\mathbb{T}}|(\partial_{x}|\phi|^{2}\partial_{x})^{n}\phi|^{2}\,\mathrm{d}x\right)\lesssim_{n}\|{\phi}\|_{H^{2n}}^{4n+2}.\end{split}$

Furthermore, as long as the solution stays nondegenerate at $t$, in the sense
that $\inf_{x}|\phi(x,t)|^{2}>c$ for some $c>0$, a standard argument involving
the ellipticity of $(\partial_{x}|\phi(x,t)|^{2}\partial_{x})^{n}$ allows us
to bound $\|{\phi(\cdot,t)}\|_{H^{2n}}^{2}$ by the integral on the left-hand
side up to errors of the form $O(\|{\phi(\cdot,t)}\|_{H^{2n}}^{4n+2})$.
Putting these together, one can establish a short-time $H^{2n}$ apriori
estimate for the solution $\phi$ with nondegenerate initial data. However, in
the case of _degenerate_ initial data (i.e., those without a uniform bound
away from $0$), the above scheme for a short-time $H^{2n}$ apriori estimate
with $n\geq 1$ clearly breaks down.

In this paper, we show that this failure of proof of higher derivative apriori
estimates is, in fact, a manifestation of genuine illposedness in standard
function spaces. Despite the formal conservation of the $L^{2}$-norm, we
demonstrate that all of the equations above are rather strongly ill-posed – in
the sense of non-existence of solutions and unboundedness of the data-to-
solution map in suitable set-ups – in a neighborhood of degenerate initial
data (e.g., zero data) in high-regularity spaces ($C^{k-1,1}$, Sobolev or
Hölder spaces).

###  1.2 Main results for quasilinear degenerate dispersive equations

####  1.2.1 Results for Schrödinger-type equations

To treat the Hunter–Smothers equation (1.1) and the Hamiltonian equation (1.2)
simultaneously, we shall consider the general equation

$i\partial_{t}\phi+|{\phi}|^{2}\partial_{xx}\phi+\alpha_{1}\phi|{\partial_{x}\phi}|^{2}+\beta_{1}\overline{\phi}(\partial_{x}\phi)^{2}+\mu_{1}|{\phi}|^{2}\phi=0,$
(1.5)

where $\phi:\mathbb{R}_{t}\times\mathbb{R}_{x}\to\mathbb{C}$,
$\alpha_{1},\beta_{1}\in\mathbb{R}$ and $\mu_{1}\in\mathbb{C}$. Indeed, the
case $\alpha_{1}=\beta_{1}=1$ and $\mu_{1}=0$ corresponds to (1.1), while the
case $\alpha_{1}=0$ and $\beta_{1}=1$ corresponds to (1.2).

For the statement of the main results, we need to introduce the following
exponents. Given $\alpha_{1},\beta_{1}\in\mathbb{R}$, we introduce the
exponent

$\sigma_{c}=-(\frac{\alpha_{1}}{2}+\beta_{1}-1),$ (1.6)

and let $s_{c}$ be the smallest integer greater than $1$ and
$\sigma_{c}-\frac{1}{2}$, i.e.,

$s_{c}=\max\\{2,\lfloor\sigma_{c}-\tfrac{1}{2}\rfloor+1\\}.$ (1.7)

Note that $\sigma_{c}=-\frac{1}{2}$ and $s_{c}=2$ for (1.1), while
$\sigma_{c}=0$ and $s_{c}=2$ for (1.2). For the significance of $\sigma_{c}$
and $\lfloor\sigma_{c}-\tfrac{1}{2}\rfloor+1$, see Remark 1.3 and Section 1.3.
We note already that the lower bound $s_{c}\geq 2$ is a technical byproduct of
our proof, which we have not attempted to optimize.

Our first result is unboundedness of the solution map (i.e., norm inflation)
in $C^{s_{c}}$ near any solution with a linear degeneracy.

######  Theorem 1.1 (Unboundedness of the solution map near a linearly
degenerate solution).

Assume that there exists a solution $f\in
L^{\infty}([0,\delta];C^{s_{c}+1,1}(\mathbb{T}))$ to (1.5) with some
$\delta>0$ such that $f(t=0)=f_{0}$ is linearly degenerate; that is, there
exists $x_{0}\in\mathbb{T}$ with $f(x_{0})=0$ and $f_{0}^{\prime}(x_{0})\neq
0$.

Then, for any $\epsilon>0$, $s_{0}\geq s_{c}$, and
$0<\delta^{\prime}\leq\delta$, we can find $\widetilde{\phi}_{0}\in
C^{\infty}(\mathbb{T})$ such that
$\|{\widetilde{\phi}_{0}}\|_{C^{s_{0}}}\leq\epsilon$ and one of the following
holds:

  * •

there exists _no_ $L^{\infty}([0,\delta^{\prime}];C^{s_{c}}(\mathbb{T}))$
solution to (1.5) with initial data $f_{0}+\widetilde{\phi}_{0}$; or

  * •

_any_ $L^{\infty}([0,\delta^{\prime}];C^{s_{c}}(\mathbb{T}))$ solution $\phi$
with $\phi(t=0)=f_{0}+\widetilde{\phi}_{0}$ satisfies

$\begin{split}\sup_{0<t<\delta^{\prime}}\|{\phi(t,\cdot)-f(t,\cdot)}\|_{C^{s_{c}}(\mathbb{T})}>c_{0}(\delta^{\prime})^{-\frac{1}{2}},\end{split}$

with some $c_{0}>0$ depending only on $f$.

We remark that the norm inflation assertion immediately implies the inflation
of any norm that controls $C^{s_{c}}$, such as $H^{\sigma}$ with
$\sigma>s_{c}+\frac{1}{2}$. In fact, our proof readily extends to norm
inflation in $H^{\sigma}$ for any $\sigma>\sigma_{c}$ in the second
alternative, which is expected to be sharp according to Remark 1.3 and Section
1.3 below; see Remark 2.10 for further details. We also remark that the
statement of Theorem 1.1 should extend over to the case of solutions with
orders of degeneracy other than 1. For simplicity, however, we restrict
ourselves to the linearly degenerate case, which is “critical” in some sense;
see Section 1.3 below.

Our second result is the nonexistence of a regular local-in-time solution in
arbitrarily high-regularity $C^{s_{0}}$ spaces.

######  Theorem 1.2 (Nonexistence of regular local-in-time solution).

For any $\epsilon>0$ and $s_{0}\geq s_{c}+2$, there exists an initial data
$\phi_{0}\in C^{\infty}(\mathbb{T})$ satisfying
$\|{\phi_{0}}\|_{C^{s_{0}}}<\epsilon$ for which there is no corresponding
solution to (1.5) belonging to
$L^{\infty}([0,\delta];{C}^{s_{c}+2}(\mathbb{T}))$ with any $\delta>0$.

As an immediate corollary of the above, we have that (1.1) and (1.2) are ill-
posed in the strongest sense of Hadamard in function spaces which contains
$C^{\infty}$ and controls the $C^{4}$–norm (where $4=s_{c}+2$): there exists
$C^{\infty}$ initial data without a local solution in $L^{\infty}_{t}W^{s,p}$
with $s-\frac{1}{p}>4$ and $L^{\infty}_{t}C^{k,\alpha}$ with $k+\alpha\geq 4$.

######  Remark 1.3 (Exponents $\sigma_{c}$, $s_{c}$, Takeuchi–Mizohata
condition and degenerate dispersion).

Observe that the illposedness results, Theorems 1.1–1.2, hold for all possible
coefficients $\alpha_{1}$, $\beta_{1}$ in front of subprincipal terms,
although these possibly _alter_ the exponents $\sigma_{c}$ and $s_{c}$.
Heuristically, $\sigma_{c}$ is the expected critical $L^{2}$-Sobolev
regularity exponent above which the linearization of (1.5) around a regular
linearly degenerate solution is ill-posed. In fact, the negativity of
$\sigma_{c}$ already signals $L^{2}$-illposedness of the linearized equation
by the classical Takeuchi–Mizohata condition [25, Chapter VII]! Even if
$\sigma_{c}$ is positive, it turns out to be $L^{\infty}$-ill-posed after
taking $k$ many derivatives with $k>\sigma_{c}-\frac{1}{2}$. This
consideration motivates the exponent $s_{c}$ and our illposedness results. We
shall elaborate on this remark in Section 1.3.

####  1.2.2 Results for KdV-type equations

To unify our treatment of KdV-type equations, we consider the general equation

$\partial_{t}u+uu_{xxx}+\alpha_{1}u_{x}u_{xx}+\frac{\mu_{1}}{m}(u^{m})_{x}=0,$
(1.8)

where $u:\mathbb{R}_{t}\times\mathbb{R}_{x}\to\mathbb{R}$,
$\alpha_{1}\in\mathbb{R}$, $\mu_{1}\in\mathbb{R}$ and $m$ is an integer
greater than or equal to $2$. Note that there is no need to separately
consider the case $m=1$, as then this term can be easily removed by the change
of variables $(t,x)=(t^{\prime},x^{\prime}+\mu_{1}t^{\prime})$.

Note that $\alpha_{1}=3$ and $\mu_{1}=1$ corresponds to the $K(m,2)$ equation
(1.3). The inviscid surface growth model (1.4) reduces to the case
$\alpha_{1}=3$ and $\mu_{1}=0$ after making the change of variables
$u=\sqrt{2}h_{x}$.

In the present case, the role of linear degeneracy in the Schrödinger case is
played by cubic degeneracy, see Section 1.3, below. As before, we introduce
the constant

$\sigma_{c}=-(\alpha_{1}-\frac{3}{2}),$

and the regularity exponent

$s_{c}=\max\\{5,\lfloor\sigma_{c}-\tfrac{1}{2}\rfloor+1\\}.$

Here, $\sigma_{c}$ is the critical $L^{2}$-Sobolev regularity exponent above
which the linearization of (1.8) around a regular cubically degenerate
solution is ill-posed (see Section 1.3 below). The linearized equation is
$L^{\infty}$-ill-posed after taking $k$ many derivatives with
$k>\sigma_{c}-\frac{1}{2}$; this motivates the exponent $s_{c}$. The lower
bound $s_{c}\geq 5$ is again a nonoptimal technical byproduct of our proof;
see Proposition 3.2 for where it is used.

######  Theorem 1.4 (Unboundedness of the solution map near a cubically
degenerate solution).

Assume that there exists a solution $f\in L^{\infty}([0,\delta];C^{s-1,1}(I))$
of (1.8) with some $\delta>0$ and $I=[a,b]$, such that the initial data
$f_{0}$ is positive on $I\setminus\partial I$, vanishes cubically on $\partial
I$ and $f_{0}\in C^{s_{0}-1,1}$, where $s_{c}\leq s\leq s_{0}$. Then, for any
$\epsilon>0$, $s\leq m_{0}\leq s_{0}$, and $0<\delta^{\prime}\leq\delta$, we
can find $\phi_{0}\in C^{\infty}(\mathbb{T};\mathbb{R})$ such that
${\mathrm{supp}}\,\phi_{0}\subseteq I\setminus\partial I$,
$\|{\phi_{0}}\|_{C^{m_{0}}}\leq\epsilon$, and one of the following holds:

  * •

there exists no solution to (1.8) with initial data $f_{0}+\phi_{0}$ that
belongs to $L^{\infty}([0,\delta^{\prime}];C^{s-1,1}(I))$; or

  * •

any solution $u$ with $u(0)=f_{0}+\phi_{0}$ and belonging to
$L^{\infty}([0,\delta^{\prime}];C^{s-1,1}(I))$ satisfies, for every $s_{c}\leq
s^{\prime}\leq 2\lfloor\frac{s}{2}\rfloor$,

$\sup_{0<t<\delta^{\prime}}\|{u(t,\cdot)-f(t,\cdot)}\|_{C^{s^{\prime}}(I)}>(\delta^{\prime})^{-\frac{1}{2}}.$

That $s^{\prime}\leq 2\lfloor\frac{s}{2}\rceil$ is not essential and expected
be replaceable by $s^{\prime}\leq s$, but assumed here to simplify the proof
(see the proof of Theorem 1.4 below).

As in the Schrödinger case, the statement of Theorem 1.4 should extend over to
the case of solutions with orders of degeneracy other than 3, provided that
$s_{c}$ is modified suitably. We however focus on the cubic degeneracy case
for simplicity, which is “critical”; see Section 1.3 below.

The nonexistence result for (1.8) is as follows.

######  Theorem 1.5 (Nonexistence of regular local-in-time solution).

For any $\epsilon>0$ and $s_{c}\leq s\leq s_{0}$, where $s$ is an even
integer, there exist an initial data $u_{0}\in C^{\infty}(\mathbb{T})$
satisfying $\|{u_{0}}\|_{C^{s_{0}}}<\epsilon$ for which there is no
corresponding solution to (1.8) belonging to
$L^{\infty}([0,\delta];C^{s}(\mathbb{T}))$ for any $\delta>0$.

That $s$ is an even integer is not essential, but assumed here to simplify the
proof (see the proof of Theorem 1.5 below).

######  Remark 1.6.

We now give a few simple remarks regarding the above.

  * •

In all of the above, the physical domain could be taken to be $\mathbb{R}$
instead of $\mathbb{T}$.

  * •

Our proof easily extends to norm inflation in $H^{\sigma}$ for any
$\sigma>\sigma_{c}$ in the second alternative in Theorem 1.4. Moreover, in
contrast to the Schödinger case, we may also easily extend Theorem 1.5 to the
non-existence of solutions in $H^{\sigma}$ for any
$\sigma>\max\\{\sigma_{c},5+\frac{1}{2}\\}$ (see Remark 1.14 for why the
situations are different). These numerologies are expected to be sharp, as we
shall discuss in Section 1.3 below. We refer the reader to Remark 2.10 for
more details on this modification (which is for norm inflation in the
Schrödinger case, but the overall idea is the same).

  * •

As one can expect, the lower order term $(u^{m})_{x}$, for any $m\geq 2$, does
not play any essential role in the proof of illposedness of (1.8). Similarly,
one can treat versions of (1.1) which have some lower order terms.

  * •

We expect our results to generalize to $K(m,n)$ with $n>2$, as well as
$\mathcal{C}(m,a,b)$ equations [29] with $n:=a+b>2$, by considering
degeneracies of order $\frac{3}{n-1}$ (which are critical).

######  Remark 1.7 (Comparison with the work of Ambrose–Simpson–Wright–Yang
[3]).

In the pioneering paper [3], the illposedness of $u_{t}=uu_{xxx}$ in the
(fairly low regularity) Sobolev space $H^{2}$ has been proved based on the
construction of a compactly supported $H^{2}$ (but not smooth) self-similar
solution $A$. However, the existence of such a solution is specific to the
equation $u_{t}=uu_{xxx}$, and the proof does not extend to more general class
of equations (1.8), nor to higher regularity Sobolev spaces, as in our
results. Our approach is distinct from that of [3]: it does not involve self-
similar solutions, but is rather based on appropriate smooth wave packet-type
approximate solutions traveling towards the degeneracy; see Section 1.3 below.
While our results (Theorems 1.4 and 1.5) do not cover Sobolev regularities as
low as $H^{2}$ due to technical reasons, our heuristics suggest that our
illposedness mechanism should extend to $H^{\sigma}$ with
$\sigma>\sigma_{c}=\frac{3}{2}$.

Nevertheless, we point out that a key heuristic consideration of our approach,
namely, the combined effect of degenerate dispersion and sub-principal term,
can already be found in [3], albeit with a different viewpoint.

######  Remark 1.8 (Comparison with the works of Germain–Harrop-
Griffith–Marzuola [10] and Harrop-Griffith–Marzuola [14]).

For solutions to (1.1) and (1.2) with degenerate initial data (i.e., initial
data with a zero), our proof identifies and exploits, in a nonlinear fashion,
a mechanism by which energies in low frequencies are transferred to high
frequencies at arbitrarily fast rates, where the frequencies are defined with
respect to the original variable $x$. We emphasize, however, that it does
_not_ rule out the possibility of well-posedness in regularity classes adapted
to the degeneracies of the initial data, by working with a renormalized
variable and/or suitable weights. Indeed, such positive results have been
proved in the interesting works of Germain–Harrop-Griffith–Marzuola for a KdV-
type quasilinear dispersive equation [10], and Harrop-Griffith–Marzuola for
(1.2) [14], where Lagrangian-type coordinates adapted to the solution were
used to formulate the function spaces.

###  1.3 Key mechanism: degenerate dispersion and Takeuchi–Mizohata condition

The nonlinear illposedness results in this paper are firmly based on a
detailed and quantitative understanding of illposedness for the linearized
equation around a background solution $f$ whose initial data contains a
degeneracy. For simplicity, in this subsection we shall assume that the
linearization takes the form

$\begin{cases}\partial_{t}u-i\partial_{x}(a\partial_{x}u)-ib\partial_{x}u=(\hbox{lower
order})&\hbox{ in the Schr\"{o}dinger case},\\\
\partial_{t}u+\tfrac{1}{2}(\partial_{x}^{3}(au)+a\partial_{x}^{3}u)+b\partial_{x}^{2}u=(\hbox{lower
order})&\hbox{ in the KdV case},\end{cases}$ (1.9)

where $a=a(x)$ is real-valued in both cases, and $b=b(x)$ is also real-valued
in the KdV case.111For Schrödinger-type problems, we regard first-order terms
of the form $\widetilde{b}(x)\partial_{x}\overline{u}$ as (lower order), as it
can be removed by a suitable change of the dependent variable; see the
introduction of $\psi$ in §2.3.2 below.

######  Remark 1.9 (On time independence of the coefficients in (1.9)).

While we assumed that $a(x)$ and $b(x)$ are time independent, the actual
linearization of (1.5) and (1.8) on a dynamic background solution $f(t,x)$
would, of course, have time-dependent coefficients. Nevertheless, the time
scale of the illposedness mechanism is arbitrarily short, and hence we may
effectively approximate these coefficients by the initial values for the
purpose of our discussion.

It is conceptually useful to distinguish two intertwined mechanisms for
illposedness, _degenerate dispersion_ and _Takeuchi–Mizohata instability_ ,
which can be seen from the principal and sub-principal terms, respectively.
Both phenomena must be taken into account to obtain a comprehensive picture of
the illposedness of (1.5) and (1.8) in the presence of a degeneracy in the
initial data (and, more concretely, to explain the relevance of the exponents
$\sigma_{c}$ and $s_{c}$).

1\. Principal term: dynamics of bicharacteristics. The illposedness of (1.9)
from degenerate dispersion can be most easily described at the level of the
_bicharacteristic ODE system_ associated with the principal symbol $p$ of the
spatial part of (1.9), which is given by

$\left\\{\begin{aligned} \dot{X}&=\partial_{\xi}p(X,\Xi),\\\
\dot{\Xi}&=-\partial_{x}p(X,\Xi),\end{aligned}\right.$ (1.10)

where $p(x,\xi)=-a(x)\xi^{2}$ in the Schrödinger case, and $-a(x)\xi^{3}$ in
the KdV case. By geometric optics, the trajectory $(X(t),\Xi(t))$ describes
(at least on sufficiently short timescales) wave packets concentrated near
$X(t)$ in the physical space and $\Xi(t)$ in the frequency space; see (1.16)
and (1.19) below for further discussion. If (1.10) admits the growth of
$|\Xi|$ by a definite factor (e.g., $2$) in arbitrarily short timescales, we
would have a strong indication of illposedness of (1.9) in high-regularity
Sobolev spaces. In turn, such a growth may come from some degeneracy of $p$ in
$X$ – this phenomenon is what we shall refer to as _degenerate dispersion_.

To be concrete, let us assume that dynamics is given in
$(x,\xi)\in\mathbb{R}\times\mathbb{R}$ and the coefficient $a$ in $p$ is of
the degenerate form $a(x)\approx Ax^{n}$ $(n>0)$ for $|{x}|$ small, so that

$\begin{split}p(x,\xi)\approx-Ax^{n}\xi^{m}\quad\hbox{ for $|x|$ small and
$|\xi|$ large}.\end{split}$ (1.11)

Note that $m=2$ in the Schrödinger case and $3$ in the KdV case. The
associated bicharacteristic ODE system is

$\left\\{\begin{aligned} \dot{X}&\approx-AmX^{n}\Xi^{m-1}\\\ \dot{\Xi}&\approx
AnX^{n-1}\Xi^{m},\end{aligned}\right.$ (1.12)

In view of the fact that the group velocity $\dot{X}$ vanishes at the point
$x=0$ (since $n>0$), we shall say that $p$ is _degenerate_ at $x=0$.

We shall now describe the illposedness mechanism of degenerate dispersion in
this concrete case. (This analysis can be found in the introduction of [10] as
well.) With a change of the time variable, we may take $A=1$. Assume, for the
sake of this heuristic discussion, that $\approx$’s above are exact
equalities. Consider the solution to (1.12) with initial conditions
$(X_{0},\Xi_{0})$, where $0<X_{0}\ll 1$ and $\Xi_{0}\gg 1$. Then, appealing to
the fact that $X^{n}\Xi^{m}$ is conserved in time, we have

$\begin{split}\Xi(t)=\Xi_{0}\left(1+(n-m)X_{0}^{n-1}\Xi_{0}^{m-1}t\right)^{\frac{m}{m-n}}\end{split}$

for $m\neq n$. When $m=n$, which will be referred to as the _critical_ case,
we have instead

$\begin{split}\Xi(t)=\Xi_{0}\exp\left(mX_{0}^{m-1}\Xi_{0}^{m-1}t\right).\end{split}$
(1.13)

In all cases, the frequency magnitude doubles (i.e.,
$|{\Xi(\tau_{2})}|=2|{\Xi_{0}}|$) at time
$\tau_{2}\simeq|{\Xi_{0}}|^{1-m}|{X_{0}}|^{1-n}$. If the order of $p$ is
greater than $1$ (i.e., $m>1$), the doubling time $\tau_{2}$ may be taken to
be arbitrarily small by choosing $|{\Xi_{0}}|$ large, as we desired. Such an
arbitrarily fast growth of $\Xi$ suggests that high derivatives of the
solution following this bicharacteristic flow would also grow arbitrarily fast
– this is what we shall refer to as _illposedness via degenerate dispersion_.

Finally, let us connect the above model case to the equations considered in
this work. Recall that the principal coefficient $a$ in the linearized
operator is determined by the background solution $f$, where $a=|{f}|^{2}$ for
(1.5) and $a=f$ for (1.8). Since the relevant frequency doubling time scale
$\tau_{2}$ is arbitrarily small, it is reasonable to make the approximation
$f\approx f_{0}$. Assuming that $f_{0}$ is degenerate at $x=0$, in the sense
that $|{f_{0}}|$ vanishes to some finite order at $0$, we arrive at the ansatz
$a(x)\approx Ax^{n}$ for some $A\neq 0$ and $n>0$.

######  Remark 1.10 (Critical degeneracy).

In this work, for simplicity, we shall consider only background solutions with
critical degeneracy $n=m$. The heuristics suggest, however, that a similar
arbitrary fast growth of $|{\Xi}|$ is expected for any order of degeneracy
$n>0$. The techniques in this paper should be generalizable to these cases.

2\. Sub-principal term: evolution of wave packet amplitude and
Takeuchi–Mizohata condition. While sub-principal terms do not enter in the
dynamics of bicharacteristics, they need to be considered in order to fully
understand the well/illposedness issues for (1.9). In fact, the sub-principal
term may already cause illposedness in $L^{2}$ even when the principal term is
_nondegenerate_! This phenomenon is captured by the classical
_Takeuchi–Mizohata condition_ (after the works [34, 24] in the Schrödinger
case); see (1.22) below.

To understand this phenomenon, it is instructive to delve a little deeper into
the construction of wave packet (approximate) solutions for (1.9). Consider
the ansatz222In the KdV case, we can take the real or imaginary part of ${\bf
a}e^{i\boldsymbol{\Phi}}$ at the end to obtain a real-valued wave packet.
$u={\bf a}(t,x)e^{i\boldsymbol{\Phi}(t,x)}$ with the following properties: (i)
$\boldsymbol{\Phi}(t,x)$ is real-valued,
$\partial_{x}\boldsymbol{\Phi}(0,x)=\Xi_{0}$ on the support of ${\bf a}(0,x)$,
and (ii) ${\bf a}(t,x)$ is complex-valued, and ${\bf a}(0,x)$ is a smooth bump
function adapted to a small ball centered at $X_{0}$. With the expectation
that the $\partial_{x}\boldsymbol{\Phi}(t,x)$ stays large compared to the
characteristic frequencies of ${\bf a}$, $a$ and $b$, we may write

$\displaystyle
e^{-i\boldsymbol{\Phi}}(\partial_{t}-i\partial_{x}a(x)\partial_{x}-ib\partial_{x})({\bf
a}e^{i\boldsymbol{\Phi}})$
$\displaystyle=i(\partial_{t}\boldsymbol{\Phi}+a(\partial_{x}\boldsymbol{\Phi})^{2}){\bf
a}$ $\displaystyle\mathrel{\phantom{=}}+\partial_{t}{\bf
a}+2a\partial_{x}\boldsymbol{\Phi}\partial_{x}{\bf
a}+\left(\partial_{x}a+b+a\tfrac{\partial_{x}^{2}\boldsymbol{\Phi}}{\partial_{x}\boldsymbol{\Phi}}\right)\partial_{x}\boldsymbol{\Phi}{\bf
a}+\cdots,$

in the Schrödinger case (where we omitted terms that does not involve
$\partial_{x}\boldsymbol{\Phi}$), and

$\displaystyle
e^{-i\boldsymbol{\Phi}}(\partial_{t}+\tfrac{1}{2}(\partial_{x}^{3}a+a\partial_{x}^{3})+b\partial_{x}^{2})({\bf
a}e^{i\boldsymbol{\Phi}})$
$\displaystyle=i(\partial_{t}\boldsymbol{\Phi}-a(\partial_{x}\boldsymbol{\Phi})^{3}){\bf
a}$ $\displaystyle\mathrel{\phantom{=}}+\partial_{t}{\bf
a}-3a(\partial_{x}\boldsymbol{\Phi})^{2}\partial_{x}{\bf
a}-(b+\tfrac{3}{2}\partial_{x}a-3a\tfrac{\partial_{x}^{2}\boldsymbol{\Phi}}{\partial_{x}\boldsymbol{\Phi}})(\partial_{x}\boldsymbol{\Phi})^{2}{\bf
a}+\cdots$

in the KdV case (where we omitted terms of order $0$ and $1$ in
$\partial_{x}\boldsymbol{\Phi}$). To eliminate the main terms on the right-
hand sides, we are led to impose the following classical _Hamilton–Jacobi_ and
_transport equations_ for $\boldsymbol{\Phi}$ and ${\bf a}$:

$\displaystyle\left\\{\begin{array}[]{l}\partial_{t}\boldsymbol{\Phi}+a(\partial_{x}\boldsymbol{\Phi})^{2}=0\\\
\partial_{t}{\bf a}+2a\partial_{x}\boldsymbol{\Phi}\partial_{x}{\bf
a}+\partial_{x}(a\partial_{x}\boldsymbol{\Phi}){\bf
a}=-b\partial_{x}\boldsymbol{\Phi}{\bf a}\end{array}\right.$
$\displaystyle\hbox{ in the Schr\"{o}dinger case},$ (1.16)
$\displaystyle\left\\{\begin{array}[]{l}\partial_{t}\boldsymbol{\Phi}-a(\partial_{x}\boldsymbol{\Phi})^{3}=0\\\
\partial_{t}{\bf a}-3a(\partial_{x}\boldsymbol{\Phi})^{2}\partial_{x}{\bf
a}-\frac{3}{2}\partial_{x}(a(\partial_{x}\boldsymbol{\Phi})^{2}){\bf
a}=b(\partial_{x}\boldsymbol{\Phi})^{2}{\bf a}\end{array}\right.$
$\displaystyle\hbox{ in the KdV case}.$ (1.19)

Observe that $(X(t),\Xi(t))$ solving (1.10) are precisely the
bicharacteristics for the above equations in the method of characteristics [9,
Chapter 3], which explains the relevance of (1.10). Moreover, the transport
equations show how $b$ influences the evolution of the amplitude ${\bf a}$. In
fact, we may easily check that

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{{\bf
a}}\|_{L^{2}}^{2}=\begin{cases}-\langle{\operatorname{Re}b\,\partial_{x}\boldsymbol{\Phi}\,{\bf
a},{\bf a}}\rangle&\hbox{in the Schr\"{o}dinger case},\\\
\langle{b(\partial_{x}\boldsymbol{\Phi})^{2}{\bf a},{\bf a}}\rangle&\hbox{in
the KdV case},\end{cases}$ (1.20)

which clearly demonstrates how $b$ influence the evolution of the $L^{2}$-norm
(here $\langle{\cdot,\cdot}\rangle$ is the standard $L^{2}$-inner product).

We are now ready to give a heuristic derivation of the Takeuchi–Mizohata
conditions. By the method of characteristics, we expect, at least for a short
time, that $\partial_{x}\boldsymbol{\Phi}(t,X(t))=\Xi(t)$ and ${\bf a}$
remains a bump function adapted to a ball centered at $X(t)$. Hence, on
${\mathrm{supp}}\,{\bf a}$, we expect

$\displaystyle-\langle{\operatorname{Re}b\,\partial_{x}\boldsymbol{\Phi}\,{\bf
a},{\bf a}}\rangle\approx-\operatorname{Re}b(X(t))\Xi(t)\|{{\bf
a}}\|_{L^{2}}^{2}=-\frac{\operatorname{Re}b(X(t))}{2a(X(t))}\dot{X}(t)\|{{\bf
a}}\|_{L^{2}}^{2},$

in the Schrödinger case, and

$\displaystyle\langle{b(\partial_{x}\boldsymbol{\Phi})^{2}{\bf a},{\bf
a}}\rangle\approx b(X(t))\Xi(t)^{2}\|{{\bf
a}}\|_{L^{2}}^{2}=-\frac{b(X(t))}{3a(X(t))}\dot{X}(t)\|{{\bf
a}}\|_{L^{2}}^{2},$

in the KdV case, where we used (1.10) for the last equalities. Using (1.20)
and $\|{u}\|_{L^{2}}=\|{{\bf a}}\|_{L^{2}}$, we arrive at the expectations

$\|{u(t)}\|_{L^{2}}\simeq\begin{cases}\exp\left(-\int_{X_{0}}^{X(t)}\frac{\operatorname{Re}b}{2a}\,\mathrm{d}x\right)\|{u(t=0)}\|_{L^{2}}&\hbox{
in the Schr\"{o}dinger case},\\\
\exp\left(\int_{X(t)}^{X_{0}}\frac{b}{3a}\,\mathrm{d}x\right)\|{u(t=0)}\|_{L^{2}}&\hbox{
in the KdV case}.\end{cases}$ (1.21)

The _Takeuchi–Mizohata conditions_ (cf. [34, 24] in the Schrödinger case; see
[1, 4] and Remark 1.12 for the KdV case) are simply sufficient conditions for
the forward-in-time boundedness of $\|{{\bf a}}\|_{L^{2}}$ read off from
(1.21):

$\displaystyle\sup_{x_{0}<x_{1}}\left|{\int_{x_{0}}^{x_{1}}\frac{\operatorname{Re}b}{2a}\,\mathrm{d}x}\right|<+\infty$
$\displaystyle\quad\hbox{ in the Schr\"{o}dinger case},$ (1.22)
$\displaystyle\sup_{x_{0}<x_{1}}\int_{x_{0}}^{x_{1}}\frac{b}{3a}\,\mathrm{d}x<+\infty$
$\displaystyle\quad\hbox{ in the KdV case}.$ (1.23)

Conversely, the failure of the Takeuchi–Mizohata conditions (1.22), (1.23)
signals arbitrarily fast growth (i.e., norm inflation) of the $L^{2}$-norm of
$u$, since $X(t)$ may travel arbitrarily far from $X_{0}$ in any fixed
duration of time if $\Xi_{0}$ is large. In this paper, we shall refer to this
norm inflation (or illposedness) mechanism as the _Takeuchi–Mizohata
instability_. Below, we shall consider the interaction of degenerate
dispersion and the Takeuchi–Mizohata instability, which provides us with a
detailed heuristic understanding of the illposedness properties of the
linearization of (1.5) and (1.8) in the presence of a (critical) degeneracy in
the initial data.

######  Remark 1.11 (Rigorous results on Takeuchi–Mizohata-type conditions).

The necessity of (1.22) for the $L^{2}$-wellposedness of (1.9) in the
Schrödinger case has been known since in the early works [34, 24]; see also
[1] for the KdV case. On the other hand, whether such a condition alone is
sufficient for $L^{2}$ boundedness in general is less clear in general,
especially in higher dimensions. Nevertheless, some strengthened form of the
Takeuchi–Mizohata condition underlies many works on the wellposedness of the
Cauchy problem for linear and even nonlinear Schrödinger- and KdV-type
equations; see, e.g., [1, 2, 4, 12, 13, 17, 18, 21, 22, 23, 25].

######  Remark 1.12 (Role of ${\mathrm{sgn}}\,b$ in the KdV case).

Observe that the absolute value is needed in the Schrödinger case (1.22) since
$X(t)$ may travel in both directions, while it is not necessary in the KdV
case (1.23) since $X(t)$ is _always_ decreasing if $a$ is positive (resp.
increasing if $a$ is negative) according to (1.10). In particular, in the KdV
case, (1.23) is always satisfied if $b<0$, and even when $b$ has some positive
parts, it is possible that the Takeuchi–Mizohata condition is still satisfied
(e.g., when $b$ oscillates). This phenomenon has been explored by
Ambrose–Wright [4], who prove wellposedness of some variable coefficient
linear KdV-type equations in the periodic setting in the presence of the
positive part of $b$ (referred to as “anti-diffusion” in that paper).

######  Remark 1.13.

While our main focus is the interaction of Takeuchi–Mizohota instability with
degenerate dispersion, the method developed in this paper also provides a new
and effective way to rigorously establish the necessity of (1.22) and (1.23)
for the $L^{2}$-wellposedness of (1.9). We refer the reader to Section 1.5 and
Appendix A below for sample results in the Schrödinger case for $a=1$ (but in
arbitrarily dimensions).

3\. Combined effect of degenerate dispersion and Takeuchi–Mizohata
instability. We are now ready to discuss the combined effect of the principal
and sub-principal terms in (1.9) obtained by linearizing around a background
solution $f$ with a degeneracy. Keeping Remarks 1.9 and 1.10 in mind, we
consider the linearization of (1.5) and (1.8) around $f(t,x)=x$ and $x^{3}$
(for $|{x}|$ small), respectively. Then we arrive at (1.9) with

$\displaystyle a(x)=x^{2},\quad b(x)=3(\tfrac{\alpha_{1}}{2}+\beta_{1}-1)x$
$\displaystyle\hbox{ in the Schr\"{o}dinger case},$ $\displaystyle
a(x)=x^{3},\quad b(x)=3(\alpha_{1}-\tfrac{3}{2})x^{2}$ $\displaystyle\hbox{ in
the KdV case}.$

Recall from the above that we are considering bicharacteristics
$(X(t),\Xi(t))$ with $X_{0}>0$ and $X(t)$ traveling to the degeneracy $0$ in
both cases. Wave packets corresponding to such bicharacteristics shall be
called _degenerating wave packets_.

The relevant Takeuchi–Mizohata condition (see (1.24) below with $\sigma=0$)
for $\|{u}\|_{L^{2}}$ may or may not hold, meaning that degenerating wave
packets may or may not remain bounded in $L^{2}$. Nevertheless, it _always
fails for high enough derivatives_ , which is consistent with the heuristic
$\Xi(t)\to\infty$! Indeed, observe that commutation of (1.9) with
$\partial_{x}^{\sigma}$ leads to a similar equation for
$\partial_{x}^{\sigma}u$ but with the following coefficients:

$\displaystyle a(x)=x^{2},\quad
b(x)=3(\tfrac{\alpha_{1}}{2}+\beta_{1}-1+\sigma)x$ $\displaystyle\hbox{ in the
Schr\"{o}dinger case},$ $\displaystyle a(x)=x^{3},\quad
b(x)=3(\alpha_{1}-\tfrac{3}{2}+\sigma)x^{2}$ $\displaystyle\hbox{ in the KdV
case}.$

In view of $0<X(t)<X_{0}$, the Takeuchi–Mizohata condition for boundedness of
$\|{u}\|_{H^{\sigma}}$ is:

$\displaystyle\sup_{0<x_{0}<x_{1}\ll
1}\int_{x_{0}}^{x_{1}}(\tfrac{\alpha_{1}}{2}+\beta_{1}-1+\sigma)\frac{\mathrm{d}x}{x}<+\infty$
$\displaystyle\hbox{ in the Schr\"{o}dinger case},$ (1.24)
$\displaystyle\sup_{0<x_{0}<x_{1}\ll
1}\int_{x_{0}}^{x_{1}}(\alpha_{1}-\tfrac{3}{2}+\sigma)\frac{\mathrm{d}x}{x}<+\infty$
$\displaystyle\hbox{ in the KdV case},$

which fails exactly when $\sigma>\sigma_{c}$ in both cases. Moreover, the
preceding heuristic analysis suggests that the $H^{\sigma}$ norm of the
degenerating wave packet grows if $\sigma>\sigma_{c}$, stays constant if
$\sigma=\sigma_{c}$ and decays if $\sigma<\sigma_{c}$. This consideration
explains the relevance of the exponent $\sigma_{c}$.

Working directly with the transport equations for ${\bf a}$ in (1.16)–(1.19)
in place of (1.20), we may also see that for
$\widetilde{s}_{c}=\sigma_{c}+\frac{1}{2}$, the $W^{s,\infty}$ norm of the
wave packet grows if $s>\widetilde{s}_{c}$, stays constant if
$s=\widetilde{s}_{c}$ and decays if $s<\widetilde{s}_{c}$. This consideration
motivates the integer exponent $s_{c}$ in our results.

###  1.4 Discussion of the proof

Our discussion so far has been rather formal; deriving actual nonlinear
illposedness in standard function spaces requires more ideas. Our main
technical contribution in this work is developing a robust scheme for
establishing quantitative illposedness, which is not only able to deduce
strong illposedness in quasilinear cases but also yields much stronger
statements for linear equations. The scheme largely consists of three parts:
1. construction of degenerating wave packets for the linearized equation, 2.
duality testing argument and 3. incorporation of the nonlinearity.

1\. Degenerating wave packets. We first describe the ideas for construction of
a degenerating wave packet. Compared to the heuristic discussion above, the
actual construction of such an approximate solution $\widetilde{u}$ to (1.9)
has to (1) allow for time-dependent coefficients $a=a(t,x)$ and $b=b(t,x)$ (as
the background solution may depend on time, see Part 3 below), and (2) solve
the linearized equation up to an equation error
$\boldsymbol{\epsilon}_{\widetilde{u}}$ of size
$\mathcal{O}(\Xi_{0}^{m-1-\delta})$ (in a suitable norm) for some $\delta>0$
(here, $m=2$ for Schrödinger and $3$ for KdV). Property (2) is necessary to
justify the approximation on a longer time scale than $\Xi_{0}^{1-m}$, which
is the instability time scale; see (1.13).

Our idea is to make appropriate changes of the independent and dependent
variables from $(x,u)$ to $(y,v)$ to reduce the problem to the constant
coefficient case, for which the construction is standard. For time-dependent
coefficients $a=a(t,x)$ and $b=b(t,x)$, the transformation
$(x,u)\mapsto(y,\check{v})$ is of the form

$\mathrm{d}x=(a(t,x))^{\frac{1}{m}}\mathrm{d}y,\quad
u=(w\check{w})^{-1}\check{v},\quad\hbox{ where
}w^{-1}\partial_{x}w=\frac{\operatorname{Re}b}{ma},\,\check{w}^{-1}\partial_{x}\check{w}=\frac{\partial_{x}a}{2ma}.$

Roughly speaking, Takeuchi–Mizohata instability is renormalized by the
conjugation of the dependent variable by the weight $w$, in the sense that
$v(t,x):=wu(t,x)$ solves (1.9) with $b=0$ (with possibly different lower order
terms). Similarly, degenerate dispersion is renormalized by the change of
variables $x\mapsto y$ accompanied with the conjugation of the dependent
variable by the weight $\check{w}$, in the sense that the $\check{v}(t,y)$
solves the constant coefficient problem
$(\partial_{t}+i(i\partial_{y})^{m})\check{v}=(\hbox{lower order terms})$.
Now, starting from a standard wave packet for the constant coefficient problem
traveling towards the degeneracy and returning to original variables, we
obtain a degenerating wave packet $\widetilde{u}$ with the desired properties.

In order to make the above heuristic discussion precise, there are several
more factors to consider. For instance, we need to make sure that the
contribution of $\partial_{t}a$, $\partial_{t}b$ are indeed acceptable, which
ultimately relies on the estimates we have on the time derivative of the
background solution $f(t,x)$ in applications; see 3. below. In the Schrödinger
case, we need the following two additional ideas: (i) an extra change of
dependent variables to treat terms of the form
$\widetilde{b}\partial_{x}\overline{u}$, and (ii) an extra phase rotation
$e^{i\lambda S}$ for the wave packet $\check{v}$ to treat terms of the form
$-(\operatorname{Im}b+\partial_{t}y)\partial_{y}\check{v}$, both of which are
potentially problematic for achieving Property (2). For more details, see the
proofs of Propositions 2.7 and 3.3 for details.

######  Remark 1.14 (Numerologies in the Schrödinger vs. KdV cases).

In order to justify the properties of $\widetilde{u}$ needed for the proof of
the $H^{s}$\- or $C^{s}$-norm growth, (ii) above forces the technical
restriction that $f\in C^{s+1,1}$ with $s\geq 2$ in the Schrödinger case,
while $f\in C^{s-1,1}$ with $s\geq 4$ is sufficient in the KdV case; compare
the degeneration bounds in Propositions 2.7 and 3.3. This point explains the
different numerologies in Theorems 1.1 and 1.2 in the Schrödinger case versus
Theorems 1.4 and 1.5 in the KdV case.

2\. Modified energy estimate and duality testing argument. In order to upgrade
the norm growth for a degenerating wave packet $\widetilde{u}$ to an actual
solution $u$ to (1.9), we adapt the _energy estimate and duality method_
introduced in our previous work [16] on Hall–MHD. Here we shall briefly
explain the argument, in the simplest setup.

Given a degenerating wave packet $\widetilde{u}$ for (1.9), denote by $u$
any333Note that it is a-priori possible that uniqueness of the Cauchy problem
for (1.9) fails. Nevertheless, the method is still applicable and establishes
the norm growth of every solution $u$ with the same initial data satisfying
(1.25). solution to (1.9) with $u(t=0)=\widetilde{u}_{0}$, where
$\widetilde{u}_{0}:=\widetilde{u}(t=0)$. In view of the aforementioned fact
that $v=wu$ solves (1.9) with $b=0$, the following _modified energy estimates_
should hold (at least when $u$ is sufficiently regular):

$\begin{split}\|{wu}\|_{L^{\infty}_{t}([0,t_{0}];L^{2})}\lesssim\|{wu_{0}}\|_{L^{2}},\qquad\|{w\widetilde{u}}\|_{L^{\infty}_{t}([0,t_{0}];L^{2})}\lesssim\|{w\widetilde{u}_{0}}\|_{L^{2}},\end{split}$
(1.25)

where $0<t_{0}<1$. By the same token, the following _generalized (bilinear)
energy estimate_ should also hold:

$\displaystyle\left|\frac{\mathrm{d}}{\mathrm{d}t}\langle{wu,w\widetilde{u}}\rangle\right|\lesssim\|{w\boldsymbol{\epsilon}_{u}}\|_{L^{2}}\|{w\widetilde{u}}\|_{L^{2}}+\|{wu}\|_{L^{2}}\|{w\boldsymbol{\epsilon}_{\widetilde{u}}}\|_{L^{2}}.$

(We remark that $\partial_{t}w$ also arises, but in applications we shall have
$|{\partial_{t}w}|\lesssim w$.) Here, $\boldsymbol{\epsilon}_{u}$ and
$\boldsymbol{\epsilon}_{\widetilde{u}}$ are the errors associated with $u$ and
$\widetilde{u}$ viewed as approximate solutions to (1.9). Then, as long as the
error terms are bounded, we may deduce that
$\langle{wu,w\widetilde{u}}\rangle\simeq\langle{wu_{0},w\widetilde{u}_{0}}\rangle=\|{w\widetilde{u}_{0}}\|_{L^{2}}^{2}$,
which allows us to obtain behavior of $u$ in various norms by simply
estimating the degenerating wave packet $\widetilde{u}$ and using duality.

In actual applications, the errors often contain derivatives and hence
$\|{w\boldsymbol{\epsilon}_{\widetilde{u}}}\|_{L^{2}}$ (resp.
$\|{w\boldsymbol{\epsilon}_{u}}\|_{L^{2}}$) may diverge as
$|\Xi_{0}|\to\infty$. Nevertheless, in view of the fact the instability time-
scale is $\simeq|{\Xi_{0}}|^{1-m}$, for the above argument to work it suffices
to have
$\int_{0}^{t_{0}}\|{w\boldsymbol{\epsilon}_{\widetilde{u}}}\|_{L^{2}}\lesssim
1$ (resp. $\int_{0}^{t_{0}}\|{w\boldsymbol{\epsilon}_{u}}\|_{L^{2}}\lesssim
1$) for $t_{0}>|{\Xi_{0}}|^{1-m+\delta}$. For $\widetilde{u}$, this is
precisely Property (2) in the preceding discussion. For an actual solution $u$
to (1.9), this follows from the fact that $\boldsymbol{\epsilon}_{u}$ does not
contain principal nor sub-principal terms (except
$\widetilde{b}\partial_{x}\overline{u}$ in the Schrödinger case, which may be
integrated by parts away).

3\. Incorporation of the nonlinearity. The ideas discussed so far explain how
to prove the illposedness of (1.9) that arises from linearizing (1.5) and
(1.8) around a regular solution $f(t,x)$ whose initial data has a critical
degeneracy at $x=0$. As in [16], the nonlinear norminflation results (Theorems
1.1 and 1.4) is derived by assuming the existence of a nonlinear perturbation
$u$ around $f$ (i.e., $f+u$ solves the nonlinear equation) without the
instability behavior, then applying the above argument. Moreover, the
nonlinear non-existence results (Theorems 1.2 and 1.5) are proved by
superposition of infinitely many configurations exhibiting norm inflation
(with unbounded rates of growth), with disjoint supports in physical space. We
refer to [16, Section 1.6] for a more detailed summary of the ideas involved,
and to Sections 2.5, 2.6, 3.5 and 3.6 for details. A key new feature of the
present paper compared to [16], however, is that the background solution $f$
need not be stationary solutions, and are given as a part of the contradiction
hypothesis in the proof of the non-existence theorems.

###  1.5 Revisiting $L^{2}$-illposedness à la Takeuchi–Mizohata

In view of the extensive appearance of the Takeuchi–Mizohata instability in
this paper, it is perhaps not surprising that our techniques also apply to the
original setting considered by Takeuchi and Mizohata of $L^{2}$-illposedness
of linear nondegenerate Schrödinger-type equations. In Appendix A, we provide
a few results concerning the Takeuchi–Mizohata condition obtained through our
approach. In particular, we recover the following result of Mizohata [25,
§VII.2]:

######  Proposition 1.15.

Consider the following linear first-order perturbation of the Schrödinger
equation on $\mathbb{R}^{d}$:

$i\partial_{t}u+\Delta u+b^{j}(x)\partial_{j}u=0$ (1.26)

where $b\in C^{1,1}(\mathbb{R}^{d})$. Suppose that the Takeuchi–Mizohata
condition for (1.26) fails, i.e.,

$\sup\left\\{\int_{0}^{T}\operatorname{Re}b^{j}(x-2s\omega)\omega_{j}\,\mathrm{d}s:x\in\mathbb{R}^{d},\,\omega\in\mathbb{S}^{d-1},\,T>0\right\\}=+\infty.$
(1.27)

Then for any $\delta>0$, every solution map $L^{2}\to
L^{\infty}_{t}([0,\delta];L^{2})$ for (1.26), if it exists, is unbounded.

Note that Proposition 1.15 clearly implies the result proved in [25, §VII.2],
namely, the impossibility of having a solution map for the inhomogeneous
equation

$i\partial_{t}u+\Delta u+b^{j}(x)\partial_{j}u=f$ (1.28)

satisfying

$\|{u}\|_{L^{\infty}([0,\delta];L^{2})}\leq
C_{0}\left(\|{u_{0}}\|_{L^{2}}+\|{f}\|_{L^{1}([0,\delta];L^{2})}\right)$

for some $C_{0}<+\infty$. In fact, via Duhamel’s principle, this result is
equivalent to Proposition 1.15. Nonetheless, our techniques generalize easily
to other situations when such an equivalence is not obvious, e.g., when $b$
depends on time.

More interestingly, we also provide some new unconditional quantitative lower
bounds for (1.26), which are valid way past the trivial
$O(\tfrac{1}{\lambda})$ time scale (where $\lambda$ is the initial
characteristic frequency), up to a time when the $L^{2}$-norm may grow at a
_quantitative_ rate depending on $\lambda$; see Propositions A.1 and A.3
below. These results should be contrasted with the proofs of Proposition 1.15
and [25, §VII.2] rely on _qualitative_ contradiction arguments up to
$O(\frac{1}{\lambda})$ time scales.

###  1.6 Organization of the paper

The Schrödinger and KdV-type equations are treated respectively in Sections 2
and 3. In Appendix A, we prove Proposition 1.15 and related results for the
linear nondegenerate Schrödinger-type equation (1.26).

### Acknowledgments

We thank J. Hunter for bringing the equation (1.1) to our attention. We also
thank Benjamin Harrop-Griffiths and Jeremy Marzuola for several insightful
discussions on degenerate dispersive equations. I.-J. Jeong has been supported
by the National Research Foundation of Korea grant (No. 2022R1C1C1011051).
S.-J. Oh was partially supported the Sloan Research Fellowship and the
National Science Foundation CAREER Grant under NSF-DMS-1945615. I.-J. Jeong
acknowledges the support from the KIAS associate member program.

##  2 Schrödinger-type equations

This section is organized as follows. To motivate our approach, we analyze in
Section 2.1 a model problem (2.2) derived from (1.1). In Section 2.2, we study
the properties of linearly degenerate solutions – typically denoted by $f$ –
and in Section 2.3, we construct degenerating wave packets for the linearized
equation around $f$. In Section 2.4, we establish a modified and generalized
(bilinear) energy estimates for the perturbation (solving the nonlinear
difference equation) around $f$. Finally, in Sections 2.5 and 2.6, we prove
Theorems 1.1 and 1.2, respectively.

###  2.1 Degenerating wave packets for model linear equation

To motivate what is to follow, consider the case (1.1) (i.e., (1.5) with
$\alpha_{1}=\beta_{1}=1$ and $\mu_{1}=0$), which we recall here for
convenience:

$i\partial_{t}\phi+\partial_{x}(|\phi|^{2}\partial_{x}\phi)=0.$ (1.1)

We note that when the domain is taken to be $\mathbb{R}$, $f(t,x)=xe^{2it}$ is
formally a solution to (1.1).444In this section, we take the domain to be
$\mathbb{R}$ rather than $\mathbb{T}$. Indeed, for initial data $f_{0}(x)$
which is $x$ in some neighborhood of the origin, it is not difficult to show
that any hypothetical smooth solution to (1.1) with initial data $f_{0}(x)$
should approximate $xe^{2it}$ uniformly for small $|x|$ and $|t|$. The goal of
this section is to sketch a proof of the fact that the linearized equation
around the explicit solution $xe^{2it}$ is ill-posed. To derive the equation,
we write

$\begin{split}\phi(t,x)=xe^{2it}+\tilde{\phi}(t,x).\end{split}$ (2.1)

Plugging in (2.1) into (1.1) and dropping quadratic or higher terms in
$\tilde{\phi}$, we obtain that

$\begin{split}i\partial_{t}\tilde{\phi}+\mathcal{L}[\tilde{\phi}]=0,\quad\mathcal{L}[\cdot]=\partial_{x}(x^{2}\partial_{x}(\cdot))+2x\partial_{x}\operatorname{Re}(\cdot).\end{split}$
(2.2)

An important observation regarding the operator $\mathcal{L}$ is that for any
sufficiently regular $v$, we have the estimate

$\begin{split}\left|\langle{|x|^{\frac{1}{2}}\mathcal{L}[v],|x|^{\frac{1}{2}}v}\rangle\right|\leq
C\|{|x|^{\frac{1}{2}}v}\|_{L^{2}}^{2},\end{split}$ (2.3)

which suggests that the correct way to measure regularity for solutions of
(2.2) is to use $|x|^{\frac{1}{2}}$–weighted spaces555As we shall see below,
the exponents $2$ and $\frac{1}{2}$ in $xe^{2it}$ and $|{x}|^{\frac{1}{2}}$,
respectively, should be replaced by appropriate constants for (1.5) in
general.. To this end, we set
$\|{v}\|_{L^{2}_{w}}=\|{|x|^{\frac{1}{2}}v}\|_{L^{2}}$. The main result in
this section is the following.

######  Proposition 2.1.

The equation (2.2) is ill-posed in $L^{2}$. More specifically, for any profile
$g_{0}\in C^{\infty}(\frac{1}{2},1)$, _any_ $L^{\infty}_{t}L^{2}_{w}$-solution
$\tilde{\phi}_{(\lambda)}$ to (2.2) with initial data

$\begin{split}\tilde{\phi}_{(\lambda),0}(x)=g_{0}(x)\exp(i\lambda\ln|x|),\quad\lambda<0\end{split}$

satisfies the growth

$\begin{split}\|{\tilde{\phi}_{(\lambda)}}\|_{L^{2}}(t)\geq
c_{0}\exp(|\lambda|t),\quad\mbox{for any }0<t<T,\end{split}$

with constants $c_{0},T>0$ depending only on $g_{0}$.

######  Remark 2.2.

By an $L^{\infty}_{t}L^{2}_{w}$ solution to (2.2), we mean a weak solution
$\tilde{\phi}$ which satisfies the bound

$\begin{split}\|{\tilde{\phi}}\|_{L^{2}_{w}}(t)\leq\exp(Ct)\|{\tilde{\phi}_{0}}\|_{L^{2}_{w}}\end{split}$

where $C>0$ is the constant from (2.3) and attains the initial data in the
weak sense. Existence of an $L^{\infty}_{t}L^{2}_{w}$ solution given an
$L^{2}_{w}$ initial data follows from a standard argument involving
Aubin–Lions lemma (see [16, Appendix A] for instance). Note that
$\|{\tilde{\phi}_{(\lambda),0}}\|_{L^{2}},\|{\tilde{\phi}_{(\lambda),0}}\|_{L^{2}_{w}}\lesssim
1$ uniformly in $\lambda$. While we cannot rule out the possibility of non-
uniqueness, the above result applies to all $L^{\infty}_{t}L^{2}_{w}$
solutions.

In following the proof, the reader may already find Items 1 (Degenerating wave
packets) and 2 (Modified energy estimate and duality testing argument) in
Section 1.4 expanded in detail. See also the remarks following the proof,
which discuss additional ideas that go into the proof of Theorems 1.1 and 1.2.

###### Proof.

We demonstrate how to construct approximate solutions to (2.2), from which
Proposition 2.1 naturally follows. To begin with, we make a change of
variables $y=\ln x$ for $x\geq 0$. Then using $x\partial_{x}=\partial_{y}$,
(2.2) transforms into

$\begin{split}i\partial_{t}\tilde{\phi}+\partial_{yy}\tilde{\phi}+\partial_{y}\tilde{\phi}+2\partial_{y}\operatorname{Re}(\tilde{\phi})=0.\end{split}$

Defining $\varphi=e^{y}\tilde{\phi}$,

$i\partial_{t}\varphi+\partial_{yy}\varphi+\partial_{y}\overline{\varphi}-2\varphi-\overline{\varphi}=0.$

We then introduce

$\psi=\varphi+A\overline{\varphi},$ (2.4)

where $A=\frac{1}{2}\partial_{y}^{-1}$ is (formally) an operator of order
$-1$. Then

$\displaystyle i\partial_{t}\psi$
$\displaystyle=-\partial_{yy}\varphi+A\partial_{yy}\overline{\varphi}-\partial_{y}\overline{\varphi}+A\partial_{y}\varphi$
$\displaystyle\mathrel{\phantom{=}}-2\varphi-\overline{\varphi}+2A\overline{\varphi}+A\varphi$
$\displaystyle=-\partial_{yy}\psi+A\partial_{yy}\overline{\varphi}+\partial_{yy}A\overline{\varphi}-\partial_{y}\overline{\varphi}$
$\displaystyle\mathrel{\phantom{=}}+A\partial_{y}\varphi-2\varphi-\overline{\varphi}+2A\overline{\varphi}+A\varphi.$

Then we see that

$i\partial_{t}\psi+\partial_{yy}\psi=A\partial_{y}\varphi-2\varphi-\overline{\varphi}+2A\overline{\varphi}+A\varphi.$

Since the right hand side is of order zero, it suggests that a degenerating
wave packet $\varphi$ may be constructed by taking $\psi$ to be an approximate
wave packet solution to the one-dimensional Schrödinger equation, and then
going back to $\varphi$. More precisely, take

$\psi^{app}_{(\lambda)}(t,y)=e^{i\lambda y-i\lambda^{2}t}a_{0}(y-2\lambda t),$
(2.5)

where we fix $a_{0}$ to be $C^{\infty}$–smooth and supported in
$\\{-2<y<-1\\}$. We need to take $\lambda<0$, so that the support of
$\phi^{app}(t,\cdot)$ is confined to $\\{y<-1\\}$ for all $t\geq 0$. To invert
(2.4), we wish to take $\varphi\approx\psi-A\overline{\psi}$. Since
$A=\frac{1}{2}\partial_{y}^{-1}$ acts like $-\frac{1}{2i\lambda}$ on
$\overline{\psi}$, we are motivated to take

$\displaystyle\varphi^{app}(t,y)$
$\displaystyle=\psi^{app}_{(\lambda)}(t,y)+\frac{1}{2i\lambda}\overline{\psi^{app}_{(\lambda)}}(t,y)$
(2.6)

and then set

$\begin{split}\tilde{\phi}^{app}=e^{-y}\varphi^{app}=e^{-y}\left(e^{i\lambda(y-\lambda
t)}a_{0}(y-2\lambda t)+\frac{1}{2i\lambda}e^{-i\lambda(y-\lambda
t)}\overline{a_{0}}(y-2\lambda t)\right).\end{split}$

Returning to the $x$-coordinates and defining the error by
$\boldsymbol{\epsilon}_{\tilde{\phi}}=[i\partial_{t}+\mathcal{L}]\tilde{\phi}^{app}$,
we have

$\begin{split}\|{\boldsymbol{\epsilon}_{\tilde{\phi}}(t)}\|_{L^{2}_{w}}\lesssim\|{a_{0}}\|_{H^{2}_{x}},\qquad
t\geq 0,\end{split}$ (2.7)

uniformly in $\lambda$. In this sense, $\tilde{\phi}^{app}$ is an approximate
solution of (2.2). Moreover, $\tilde{\phi}^{app}$ itself satisfies the bound
$\|{{\tilde{\phi}^{app}}(t)}\|_{L^{2}_{w}}\lesssim\|{a_{0}}\|_{L^{2}_{x}}$.
The last key property is degeneration: with a weight higher than
$|x|^{\frac{1}{2}}$, $\tilde{\phi}^{app}(t)$ decays in the
$O(|\lambda|^{-1})$–timescale: for example, with the weight $|x|$, we have

$\begin{split}\|{|x|\tilde{\phi}^{app}(t,x)}\|_{L^{2}_{x}}\lesssim
e^{-|\lambda|t}\|{a_{0}}\|_{L^{2}_{x}}.\end{split}$ (2.8)

Interpolating (2.8) with $L^{2}_{w}$–estimate shows that
$\|{\tilde{\phi}^{app}(t,\cdot)}\|_{L^{2}}\gtrsim e^{|\lambda|t}$. Now, let
$\tilde{\phi}$ be an $L^{\infty}_{t}L^{2}_{w}$ solution of (2.2). Then, with a
direct computation, we have the generalized energy estimate for the weighted
$L^{2}$–estimate (see §2.4.2):

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle=\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\boldsymbol{\epsilon}_{\tilde{\phi}}}\rangle,\end{split}$

which gives, together with (2.7),

$\begin{split}\operatorname{Re}\,\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle(t)\geq\operatorname{Re}\,\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle(t=0)-Ct\|{\tilde{\phi}}\|_{L^{\infty}_{t}L^{2}_{w}}\|{a_{0}}\|_{H^{2}_{x}}.\end{split}$

At the initial time, by choosing $a_{0}$ in a way depending only on $g_{0}$,
we can guarantee that
$\operatorname{Re}\,\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle(t=0)\geq\frac{1}{2}\|{\tilde{\phi}_{0}}\|_{L^{2}_{w}}\|{\tilde{\phi}^{app}_{0}}\|_{L^{2}_{w}}$.
Then, for
$0<t<C\|{a_{0}}\|_{H^{2}_{x}}/(4\|{\tilde{\phi}_{0}}\|_{L^{2}_{w}})$, we
obtain with (2.8) that

$\begin{split}\frac{1}{C}e^{-|\lambda|t}\|{a_{0}}\|_{L^{2}}\|{\tilde{\phi}(t)}\|_{L^{2}}\geq\operatorname{Re}\,\langle{\tilde{\phi},|x|\tilde{\phi}^{app}}\rangle(t)=\operatorname{Re}\,\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle(t)\geq\frac{1}{4}\|{\tilde{\phi}_{0}}\|_{L^{2}_{w}}\|{\tilde{\phi}^{app}_{0}}\|_{L^{2}_{w}},\end{split}$

which gives the claimed exponential growth of $\|{\tilde{\phi}(t)}\|_{L^{2}}$.
∎

######  Remark 2.3 (Illposedness in $H^{m}$ for $m>0$).

In fact, a small modification of the above proof shows that (2.2) is ill-posed
in $H^{m}$ for $m>0$. More precisely, the following growth occurs:

$\|{{\partial_{x}^{m}}\tilde{\phi}_{(\lambda)}}\|_{L^{2}}(t)\geq
c_{0}\exp({(1+2m)}|\lambda|t),\quad\mbox{for any }m\geq 0\mbox{ and }0<t<T,$

with $c_{0},T>0$ depending only on $g_{0}$ and $m$.

We now sketch the needed modification; see Section 2.5 for the complete proof.
We would like to modify the last part of the proof of Proposition 2.1 using
“differentiation by parts”: under the assumption that
$\partial_{x}^{-m}(|x|\tilde{\phi}^{app})\in L^{2}$,

$\begin{split}\|{\partial_{x}^{m}\tilde{\phi}}\|_{L^{2}}\|{\partial_{x}^{-m}(|x|\tilde{\phi}^{app})}\|_{L^{2}}\geq\operatorname{Re}\,\langle{\partial_{x}^{m}\tilde{\phi},\partial_{x}^{-m}(|x|\tilde{\phi}^{app})}\rangle(t)=\operatorname{Re}\,\langle{|x|^{\frac{1}{2}}\tilde{\phi},|x|^{\frac{1}{2}}\tilde{\phi}^{app}}\rangle(t).\end{split}$

Now the point is that
$\partial_{x}^{-1}=x\partial_{y}^{-1}=e^{y}\partial_{y}^{-1}$ and
$y\simeq-2|\lambda|t$ on the support of $\tilde{\phi}^{app}(t)$, which gives a
faster rate of degeneration
$\|{\partial_{x}^{-m}(|x|\tilde{\phi}^{app})}\|_{L^{2}}\lesssim
e^{-(2m+1)|\lambda|t}$. This gives the claimed lower bound for
$\|{\partial_{x}^{m}\tilde{\phi}}\|_{L^{2}}$. In general, there could be some
low frequency part of $|x|\tilde{\phi}^{app}$ which does not degenerate, and
for this reason we introduce a decomposition of $|x|\tilde{\phi}^{app}$ into
high and low frequency parts in the actual proof in Section 2.5.

######  Remark 2.4 (Additional ideas in the proof of Theorems 1.1 and 1.2).

For a general equation of the form (1.5), we do not have access to a
stationary solution with a linear degeneracy in general (furthermore, we shall
also require that $f_{0}$ be compactly supported, which rules out $xe^{2it}$,
too). Hence, we shall carry out the above analysis (degenerating wave packet
construction, modified energy estimate and duality) where the background
solution $f$ is merely a regular (most likely) _time-dependent_ solution to
(1.5), which has compactly supported initial data $f_{0}$ with a linear
degeneracy, in place of $xe^{2it}$.

Theorem 1.1 is proved by considering a perturbation $f+\widetilde{\phi}$ of
such an $f$, and arguing that if $f+\widetilde{\phi}$ exists as a regular
solution (i.e., if we are in the second case in Theorem 1.1), then the above
growth mechanism for $\widetilde{\phi}$ can be justified. To prove Theorem
1.2, we consider initial data $\widetilde{\phi}_{0}$ consisting of a
superposition of an infinitude of configurations as above (i.e.,
$\sum_{k}(f_{k,0}+\widetilde{\phi}_{k,0})$, where $f_{k,0}$ has a linear
degeneracy and $\widetilde{\phi}_{k,0}$ is a degenerating wave packet adapted
to $f_{k,0}$) with unbounded rates (i.e., the initial frequencies of the
degenerating wave packets are unbounded), disjoint supports (i.e.,
$\\{{\mathrm{supp}}\,f_{k,0}\cup{\mathrm{supp}}\,\phi_{k,0}\\}_{k}$ is
pairwise disjoint), yet with an $\epsilon$-small $C^{m_{0}}$-norm. Then we
perform a contradiction argument: if a regular solution $\phi$ to such initial
data exists, then we may justify the growth mechanism (as in Proposition 2.1),
which is absurd. For details, see Sections 2.5 and 2.6 below.

###  2.2 Properties of a regular linearly degenerate solution

We shall assume that there exists a smooth solution to (1.5) which is linearly
degenerate, and analyze its properties. To be precise, let
$f:[0,\delta]\times[-x_{1},x_{1}]\rightarrow\mathbb{C}$ be a
$L^{\infty}([0,\delta];C^{3,1}([{x_{0}}-x_{1},{x_{0}+}x_{1}]))$ solution to
(1.5) with some $x_{1},\delta>0$ satisfying

$\begin{split}f_{0}\in C^{3,1}([{x_{0}}-x_{1},{x_{0}}+x_{1}]),\quad
f_{0}(x_{0})=0,\quad|f^{\prime}_{0}(x_{0})|>0\end{split}$

at $t=0$ for some $x_{0}\in\mathbb{T}$.

Owing to the symmetries of (1.5) (translation and phase rotation), as well as
its behavior under the transformation $\phi\mapsto c\phi$, we may assume
without loss of generality that $x_{0}=0$ and $f^{\prime}_{0}(0)=1$. Then,
from the equation it is easy to see that, on the time interval $[0,\delta]$,

$\begin{split}f(t,0)=0\end{split}$

and

$\begin{split}i\frac{\mathrm{d}}{\mathrm{d}t}f^{\prime}(t,0)=-(\alpha_{1}+\beta_{1})|f^{\prime}(t,0)|^{2}f^{\prime}(t,0),\end{split}$

which implies in particular that

$|f^{\prime}(t,0)|=1,\quad\hbox{ and }|f(t,x)|=x+O(|x|^{2})\hbox{ uniformly in
$t$.}$

More generally, we have the following lemma.

######  Lemma 2.5.

Let $s\geq 2$ be an integer and let $f\in
C_{t}([0,\delta];C^{s-1,1}(\mathbb{T}))$ be a solution to (1.5). Then

  1. 1.

The zero set of $f(t,x)$ is preserved in time, i.e., $a\in\mathbb{T}$ is a
zero of $f(0,x)$ if and only if it is a zero for $f(t,x)$ for all
$t\in[0,\delta]$.

  2. 2.

Let $a\in\mathbb{T}$ be a zero of $f(0,x)$. Then
$\\{\partial_{x}^{k}f(t,a)\\}_{k=0}^{s-1}$ is determined by the initial data
at $x=a$, i.e., $\\{\partial_{x}^{k}f(0,a)\\}_{k=0}^{s-1}$.

Here, the important point is that, thanks to the regularity assumption,
$f(0,x)$ vanishes at least linearly at each zero $x=a$, which is _critical_
for (1.5) in the senses discussed in §1.3.

###### Proof.

By the regularity assumption (in particular, that $s\geq 2$), it follows from
(1.5) that $|{\partial_{t}f(t,x)}|\leq C|{f(t,x)}|$; hence the first statement
follows. To prove the second statement, consider a zero $a$ of $f(0,x)$.
Without any loss of generality, we may assume that $a=0$. By the assumption
and Taylor expansion, we have

$\displaystyle f(t,x)$
$\displaystyle=\sum_{k=1}^{s-1}\frac{1}{k!}\partial_{x}^{k}f(t,0)x^{k}+O(|{x}|^{s}),$
$\displaystyle f_{x}(t,x)$
$\displaystyle=\sum_{k=0}^{s-2}\frac{1}{k!}\partial_{x}^{k+1}f(t,0)x^{k}+O(|{x}|^{s-1}),$
$\displaystyle f_{xx}(t,x)$
$\displaystyle=\sum_{k=0}^{s-3}\frac{1}{k!}\partial_{x}^{k+2}f(t,0)x^{k}+O(|{x}|^{s-2}),$

where the implicit constants depend only on
$\|{f}\|_{L^{\infty}_{t}C^{s-1,1}_{x}}$ (and in particular are independent of
$(t,x)$). Plugging this into (1.5) and matching the coefficients of $x^{k}$
for $k=1,\ldots,s-1$, we formally obtain a determined system of first-order
ODE’s for $\\{\partial_{x}^{k}f(t,0)\\}_{k=1}^{s-1}$; here, the fact that
$f(t,x)$ vanishes at least linearly is crucially used to ensure that no
$\partial_{x}^{k}f(t,0)$ with $k>s-1$ arises. Indeed, these ODEs may be
justified in the sense of distributions by testing (1.5) against a test
function of the form $\eta(t)(-1)^{k}\partial_{x}^{k}\chi_{\epsilon}(x)$,
where $\eta\in C^{\infty}_{c}(0,\delta)$, $\chi\in
C^{\infty}_{c}(-\frac{1}{2},\frac{1}{2})$ with $\int\chi=1$ and
$\chi_{\epsilon}(x)=\epsilon^{-1}\chi(\epsilon^{-1}x)$. By the uniqueness of
this ODE system, the desired statement follows. ∎

From now on, given $f_{0}$ which are linearly degenerate at $x=0$ and
$f_{x}(0,0)$ positive real, we are going to take $0<x_{1}<1$ smaller if
necessary, so that

$\begin{split}\left(\sup_{x\in[-x_{1},x_{1}]}|f_{xx}(0,x)|\right)x_{1}<\frac{f_{x}(0,0)}{2}\end{split}$
(2.9)

holds. In particular, we have that

$\begin{split}\frac{1}{2}f_{x}(0,0)<|f_{x}(0,x)|<2f_{x}(0,0),\quad\mbox{for
all}\quad x\in[-x_{1},x_{1}].\end{split}$

######  Proposition 2.6.

Let $f\in L^{\infty}_{t}([0,\delta];C^{3,1}([-x_{1},x_{1}])$ be a solution to
(1.5), and set $M=\|{f}\|_{L^{\infty}([0,\delta];C^{3,1})}$. Then, we have the
pointwise bounds

$\begin{split}|f(t,x)|\leq|f_{0}(x)|\exp(CM^{2}t)\end{split}$ (2.10)

and

$\begin{split}\left|\partial_{t}(|f(t,x)|^{2})\right|\leq
CM\exp(CM^{2}\delta)\left(1+(f_{x}(0,0))^{-1}M\right)^{3}\left(|f_{0}(x)|^{3}+tM^{3}|f_{0}(x)|^{2}\right)\end{split}$
(2.11)

for all $t\in[0,\delta]$ and $|x|\leq x_{1}$.

###### Proof.

We first note directly from (1.5) that $\left|\partial_{t}|f(t,x)|\right|\leq
C\|{f}\|^{2}_{L^{\infty}_{t}C^{1,1}_{x}}|f(t,x)|$ holds, which gives (2.10).
Now note that $f\in L^{\infty}([0,\delta];C^{3,1}([-x_{1},x_{1}]))$ implies,
via (1.5), that

$\begin{split}|\partial_{t}f(t,x)|\leq
C|f(t,x)|\|{f}\|^{2}_{L^{\infty}_{t}C^{1,1}_{x}},\qquad|\partial_{tt}f(t,x)|&\leq
C|f(t,x)|^{2}\|{f}\|^{3}_{L^{\infty}_{t}C^{3,1}_{x}}.\end{split}$ (2.12)

Then, the Taylor expansion in time of $f(t,x)$ gives

$\begin{split}|f(t,x)|^{2}&=|f_{0}(x)|^{2}+2\operatorname{Re}\left(\overline{f_{0}(x)}\int_{0}^{t}(\partial_{t}f)(t^{\prime},x)\,\mathrm{d}t^{\prime}\right)+\left|\int_{0}^{t}(\partial_{t}f)(t^{\prime},x)\,\mathrm{d}t^{\prime}\right|^{2}.\end{split}$

Taking the time derivative,

$\begin{split}\partial_{t}(|f(t,x)|^{2})=2\operatorname{Re}\left(\overline{f_{0}(x)}(\partial_{t}f)(t,x)\right)+2\operatorname{Re}\left(\overline{(\partial_{t}f)(t,x)}\int_{0}^{t}(\partial_{t}f)(t^{\prime},x)\,\mathrm{d}t^{\prime}\right).\end{split}$
(2.13)

Using (2.10), the last term in (2.13) is bounded by

$\begin{split}C\|{f}\|^{4}_{L^{\infty}_{t}C^{3,1}_{x}}|f(t,x)|\int_{0}^{t}|f(t^{\prime},x)|\,\mathrm{d}t^{\prime}\leq
CM^{4}t\exp(CM^{2}\delta)|f_{0}(x)|^{2}.\end{split}$

For the other term in the right hand side of (2.13), we further rewrite as

$\begin{split}2\operatorname{Re}\left(\overline{f_{0}(x)}(\partial_{t}f)(t,x)\right)=2\operatorname{Re}\left(\overline{f_{0}(x)}(\partial_{t}f)(0,x)\right)+2\operatorname{Re}\left(\overline{f_{0}(x)}\int_{0}^{t}(\partial_{tt}f)(t^{\prime},x)\,\mathrm{d}t^{\prime}\right),\end{split}$

and note that the last term is bounded using (2.12) by
$C\exp(CM^{2}\delta)M^{3}t|f_{0}(x)|^{3}.$ On the other hand, the first term
on the right hand side equals

$\begin{split}&\operatorname{Im}\left(\overline{f_{0}(x)}\left(|{f_{0}(x)}|^{2}\partial_{xx}f_{0}(x)+\alpha_{1}f_{0}(x)|{\partial_{x}f_{0}(x)}|^{2}+\beta_{1}\overline{f_{0}(x)}(\partial_{x}f_{0}(x))^{2}+\mu_{1}|{f_{0}(x)}|^{2}f_{0}(x)\right)\right)\\\
&\qquad=\beta_{1}\operatorname{Im}\left(\overline{f_{0}(x)}^{2}(\partial_{x}f_{0}(x))^{2}\right)+O(\|{f_{0}}\|_{C^{1,1}})|f_{0}(x)|^{3},\end{split}$

and we see that the leading term in the Taylor expansion of
$\overline{f_{0}(x)}^{2}(\partial_{x}f_{0}(x))^{2}$ is purely real, with
remainder bounded by

$\begin{split}C(f_{x}(0,0)^{3}|x|^{3}\|{f_{0}}\|_{\dot{C}^{1,1}}+f_{x}(0,0)^{2}|x|^{4}\|{f_{0}}\|_{\dot{C}^{1,1}}^{2}+f_{x}(0,0)^{3}|x|^{5}\|{f_{0}}\|_{\dot{C}^{1,1}}^{2})\leq
C(1+f_{x}(0,0)^{-1}M)^{3}|f_{0}(x)|^{3},\end{split}$

where we have used $|x|<x_{1}$ and the smallness of $x_{1}$ from (2.9).
Collecting the bounds, we obtain the proposition. ∎

###  2.3 Degenerating wave packets for the linearized equation

In this subsection, our goal is to construct approximate solutions, called
degenerating wave packets, for the linearization of (1.5) around a (possibly
hypothetical) regular linearly degenerate solution, which possess the desired
degeneration property that is responsible for the illposedness of (1.5); see
Proposition 2.7 below.

####  2.3.1 Properties of degenerating wave packets

Given a smooth solution $f$ to (1.5), let us write $\phi=f+\tilde{\phi}$ where
$\phi$ is another smooth solution to (1.5). The equation for $\tilde{\phi}$ is
given by

$\begin{split}&i\partial_{t}\tilde{\phi}+|{f}|^{2}\partial_{xx}\tilde{\phi}+\alpha_{1}f(\overline{\partial_{x}f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})+2\beta_{1}\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}=Q_{f}[\tilde{\phi}],\end{split}$
(2.14)

with

$\begin{split}V_{f}&=\overline{f}\partial_{xx}f+\alpha_{1}|{\partial_{x}f}|^{2}+2\mu_{1}|{f}|^{2},\\\
W_{f}&=f\partial_{xx}f+\beta_{1}(\partial_{x}f)^{2}{+}\mu_{1}f^{2},\\\
Q_{f}[\tilde{\phi}]&=-(\overline{f}\tilde{\phi}+f\overline{\tilde{\phi}})\partial_{xx}\tilde{\phi}-\alpha_{1}\tilde{\phi}(\partial_{x}\overline{f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})-\alpha_{1}f|{\partial_{x}\tilde{\phi}}|^{2}-2\beta_{1}\overline{\tilde{\phi}}\partial_{x}f\partial_{x}\tilde{\phi}-\beta_{1}\overline{f}(\partial_{x}\tilde{\phi})^{2}\\\
&\mathrel{\phantom{=}}-|{\tilde{\phi}}|^{2}\partial_{xx}\tilde{\phi}-\alpha_{1}\tilde{\phi}|{\partial_{x}\tilde{\phi}}|^{2}-\beta_{1}\overline{\tilde{\phi}}(\partial_{x}\tilde{\phi})^{2}{-\mu_{0}|\tilde{\phi}|^{2}\tilde{\phi}-2\mu_{1}f|\tilde{\phi}|^{2}-\mu_{1}\overline{f}(\tilde{\phi})^{2}}.\end{split}$
(2.15)

Note that $Q_{f}[\tilde{\phi}]$ is at least quadratic in $\tilde{\phi}$ and
its derivatives. Dropping the right hand side, we obtain the linearized
equation around $f$:

$\begin{split}i\partial_{t}\tilde{\phi}+|{f}|^{2}\partial_{xx}\tilde{\phi}+\alpha_{1}f(\overline{\partial_{x}f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})+2\beta_{1}\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}=0.\end{split}$
(2.16)

We now state the key proposition of this section, which shows properties of
degenerating wave packets for (2.16). We introduce the notation

$\|{g}\|_{W^{s,p}_{(L)}}=\sum_{j=0}^{s}\|{(L\partial_{x})^{j}g}\|_{L^{p}_{x}}$

and write $H^{s}_{(L)}$ when $p=2$.

######  Proposition 2.7.

Let $f\in L^{\infty}([0,\delta];{C}^{s_{0}-1,1}([0,x_{1}]))$ be a solution to
(1.5) with $s_{0}\geq 4$ satisfying

$\begin{split}f(0,0)=0,\quad f^{\prime}(0,0)={A},\end{split}$ (2.17)

for some $A>0$. By taking $x_{1}<1$ small if necessary, assume that (2.9)
holds. Then, to any $\lambda\leq-1$ and a $C^{\infty}$-smooth complex-valued
profile $g_{0}$ supported in $(\frac{1}{2}x_{1},x_{1})$, we may associate a
function
$\tilde{\phi}^{app}_{(\lambda)}=\tilde{\phi}^{app}_{(\lambda)}[g_{0},f]$
defined in $[0,\delta]\times\mathbb{R}$ satisfying the following properties:

  * •

(linearity) the map $g_{0}\mapsto\tilde{\phi}^{app}_{(\lambda)}$ is (real)
linear;

  * •

(support property)
${\mathrm{supp}}\,(\tilde{\phi}^{app}_{(\lambda)}[g_{0}])\subset(0,e^{-|\lambda|A^{2}t}x_{1})$;

  * •

(initial data) for any $1\leq p\leq\infty$,

$\begin{split}\frac{1}{C}\|{g_{0}}\|_{L^{2}}\leq
A^{\sigma_{c}-1}\|{|f|^{-\sigma_{c}}\tilde{\phi}^{app}(0,x)}\|_{L^{2}}&\leq
C\|{g_{0}}\|_{L^{2}},\\\
A^{\sigma_{c}-1}\|{|f|^{\sigma_{c}}\tilde{\phi}^{app}(0,x)}\|_{L^{p}}&\leq
Cx_{1}^{\frac{1}{p}-\frac{1}{2}}\|{g_{0}}\|_{L^{p}};\end{split}$

  * •

(regularity) for $0\leq n\leq s_{0}-2$, we have

$\begin{split}\|{{|{f}|}^{-\sigma_{c}}({|{f}|}\partial_{x})^{n}\tilde{\phi}^{app}_{(\lambda)}(t,x)}\|_{L^{2}}\leq
C_{f,\delta}A^{-\sigma_{c}+n+1}|\lambda|^{n}\|{g_{0}}\|_{H^{n}_{(x_{1})}},\quad
t\leq\min\\{A^{-2}|\lambda|^{-\frac{1}{2}},\delta\\};\end{split}$ (2.18)

  * •

(degeneration) for any $1\leq p\leq 2$, ${0\leq s\leq s_{0}-2}$, and
$\gamma^{\prime}\geq-s-\frac{1}{p}+\frac{1}{2}$, we have

$|{f}|^{-\sigma_{c}+\gamma^{\prime}}\tilde{\phi}^{app}_{(\lambda)}=\partial_{x}^{s}\left(\frac{|{f}|^{\gamma^{\prime}+s-\frac{1}{2}}}{i^{s}\lambda^{s}(1+|{f}|\partial_{x}S)^{s}}\psi^{app}_{(\lambda)}\right)+|{f}|^{-\sigma_{c}+\gamma^{\prime}}\tilde{\phi}^{small}_{(\lambda)},$
(2.19)

for some $\psi^{app}_{(\lambda)}$, $\tilde{\phi}^{small}_{(\lambda)}$, and
$S$, where $\psi^{app}_{(\lambda)}$ is independent of $p$, $s$ and
$\gamma^{\prime}$, and

$\displaystyle\left\|\frac{|{f}|^{\gamma^{\prime}+s-\frac{1}{2}}}{\lambda^{s}(1+|{f}|\partial_{x}S)^{s}}\psi^{app}_{(\lambda)}(t,x)\right\|_{L^{p}}$
$\displaystyle\leq
C_{f,\delta}^{1+\gamma^{\prime}}A^{\gamma^{\prime}+s+\frac{1}{2}}{|\lambda|^{-s}}e^{-2|{\lambda}|(\gamma^{\prime}+s+\frac{1}{p}-\frac{1}{2})A^{2}t}\|{g_{0}}\|_{L^{2}},$
(2.20)
$\displaystyle\|{|{f}|^{-\sigma_{c}}\tilde{\phi}^{small}_{(\lambda)}(t,x)}\|_{L^{2}}$
$\displaystyle\leq
C_{f,\delta}A^{-\sigma_{c}+1}{|\lambda|^{-1}}\|{g_{0}}\|_{H^{s}_{(x_{1})}}$
(2.21)

for $t\leq\min\\{A^{-2}|{\lambda}|^{-\frac{1}{2}},\delta\\}$, after taking
$\delta>0$ smaller in a way that
$\delta\|{f}\|^{2}_{L^{\infty}([0,\delta];C^{1,1})}$ is small in terms of
$A^{-1}\|{f}\|_{L^{\infty}([0,\delta];C^{3,1})}$;

  * •

(error estimate) defining the error
$\boldsymbol{\epsilon}[\tilde{\phi}^{app}_{(\lambda)}]$ by the left hand side
of (2.16) with $\tilde{\phi}=\tilde{\phi}^{app}_{(\lambda)}$, we have the
estimate

$\begin{split}\|{|{f}|^{-\sigma_{c}}\boldsymbol{\epsilon}[\tilde{\phi}^{app}_{(\lambda)}](t)}\|_{L^{2}}\leq
C_{f,\delta}A^{-\sigma_{c}+3}\|{g_{0}}\|_{H^{2}_{(x_{1})}},\quad
t\leq\min\\{A^{-2}|\lambda|^{-\frac{1}{2}},\delta\\}.\end{split}$ (2.22)

In the above estimates, the constant $C_{f,\delta}$ satisfies

$\begin{split}C_{f,\delta}\leq
C_{0}\left(1+A^{-1}\|{f}\|_{L^{\infty}_{t}C^{s_{0}-1,1}}\right)^{N_{0}}\exp(C_{0}\|{f}\|_{L^{\infty}_{t}C^{s_{0}-1,1}}^{2}\delta)\end{split}$
(2.23)

for some $C_{0},N_{0}>0$ depending on $\alpha_{1},\beta_{1},\mu_{1}$ and
$s_{0}$ but not on $f$ and $x_{1}$.

We fix $A=1$ and prove Proposition 2.7 in the remainder of this subsection. In
the general case, given $f$ we can define $\tilde{f}(t,x):=A^{-1}f(A^{-2}t,x)$
which is another solution $L^{\infty}_{t}C^{s-1,1}$ solution to (1.5)
satisfying $\tilde{f}_{x}(0,0)=1$. Then, we simply define

$\begin{split}\tilde{\phi}^{app}_{(\lambda)}[g_{0},f](t,x):=\tilde{\phi}^{app}_{(\lambda)}[Ag_{0},\tilde{f}](A^{2}t,x),\end{split}$

and verify the claimed properties of $\tilde{\phi}^{app}_{(\lambda)}[g_{0},f]$
using those for $\tilde{\phi}^{app}_{(\lambda)}[Ag_{0},\tilde{f}]$. In the
proof, it will be seen that $|f|\partial_{x}S$ remains invariant under this
rescaling.

####  2.3.2 Renormalization and wave packet construction

With $x_{1}>0$ given in Proposition 2.7, we define the variable $y$ for
$t\in[0,\delta]$ and $x\in(0,x_{1}]$ by

$\begin{split}y(t,x)=-\int_{x}^{x_{1}}\frac{1}{|f(t,x^{\prime})|}\,\mathrm{d}x^{\prime}\leq
0.\end{split}$

For each $t\geq 0$, the inverse of $x\mapsto y(t,x)$ is denoted by $x=x(t,y)$.
From $|f(t,x)|=x+O(|x|^{2})$, we have that

$\begin{split}{y(t,x)}-{\ln\frac{x}{x_{1}}}=B(t,x),\qquad
x(t,y)=x_{1}e^{y-B},\qquad|B(t,x)|\leq
Cx_{1}\|{f}\|_{L^{\infty}_{t}C^{1,1}}.\end{split}$ (2.24)

Using $|f|\partial_{x}=\partial_{y}$, we rewrite (2.16) in
$(t,y)$-coordinates:

$i\partial_{t}\tilde{\phi}+ih\partial_{y}\tilde{\phi}+\partial_{y}^{2}\tilde{\phi}+\frac{\alpha_{1}f\overline{\partial_{y}f}+2\beta_{1}\overline{f}\partial_{y}f-|{f}|\partial_{y}|{f}|}{|{f}|^{2}}\partial_{y}\tilde{\phi}+\alpha_{1}\frac{f\partial_{y}f}{|{f}|^{2}}\partial_{y}\overline{\tilde{\phi}}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}=0.$
(2.25)

Here, we have introduced $h(t,y)=\partial_{t}y$ so that
$\partial_{t}\tilde{\phi}(t,x)=\partial_{t}\tilde{\phi}(t,y)+h(t,y)\partial_{y}\tilde{\phi}(t,y).$
Now define

$\begin{split}{G(t,y)=(-\sigma_{c}+\frac{1}{2})\ln|{f}|(t,y)},\quad(\partial_{y}G)(t,y)={(-\sigma_{c}+\tfrac{1}{2})}\frac{\operatorname{Re}(\overline{f}\partial_{y}f)}{|f|^{2}}(t,y)\end{split}$

and introducing the conjugation $\varphi=e^{G}\tilde{\phi},$ we obtain (recall
from (1.6) that $\sigma_{c}=-(\frac{\alpha_{1}}{2}+\beta_{1}-1)$)

$\begin{split}&i\partial_{t}\varphi+\partial_{yy}\varphi+{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\overline{\partial_{y}\varphi}+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\varphi=\mathcal{B}_{0}[\varphi]\end{split}$
(2.26)

with

$\begin{split}\mathcal{B}_{0}[\varphi]={i}(\partial_{t}G)\varphi+(\partial_{yy}G+(\partial_{y}G)^{2})\varphi{+\alpha_{1}\frac{f\partial_{y}f}{|{f}|^{2}}\partial_{y}G\overline{\varphi}+\left((-\alpha_{1}+2\beta_{1})\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i(\partial_{y}G)\varphi-
V_{f}\varphi-W_{f}\overline{\varphi}.}\end{split}$

Note that the terms in $\mathcal{B}_{0}$ do not contain derivatives of
$\varphi$. To handle the term containing $\overline{\partial_{y}\varphi}$ in
the left hand side of (2.26), we make yet another change of variables:
introducing formally

$\begin{split}\psi=\varphi+{\frac{\alpha_{1}}{2}}\frac{f\partial_{y}f}{|f|^{2}}\partial_{y}^{-1}\overline{\varphi},\end{split}$

we have that (2.26) turns into

$\begin{split}i\partial_{t}\psi+\partial_{yy}\psi+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+{h}\right)i\partial_{y}\psi=\cdots,\end{split}$
(2.27)

where the terms on the right hand side do not contain any derivatives of
$\psi$. Motivated by this computation, we construct a wave packet approximate
solution for (2.26) by starting with a wave packet for the preceding equation
for $\psi$, then coming back to $\varphi$. More precisely, given $g_{0}(x)$ as
in Proposition 2.7, we take

$\begin{split}a_{0}(y)=x_{1}^{\frac{1}{2}}g_{0}(x(0,y)),\end{split}$

which is supported in $y\in{(-\frac{1}{2}\ln 2,0)}$ by (2.24). For each
$\lambda<0$, we define

$\begin{split}\psi^{app}_{(\lambda)}(t,y):=e^{i\lambda(y-\lambda
t)}a_{(\lambda)}(t,y)\end{split}$ (2.28)

where $a_{(\lambda)}(t,y)$ is the unique solution to

$\begin{split}\partial_{t}a_{(\lambda)}+2\lambda\partial_{y}a_{(\lambda)}=\frac{\lambda}{i}\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)a_{(\lambda)}\end{split}$
(2.29)

with initial data $a_{(\lambda)}(0,y)=a_{0}(y)$. Given $\psi^{app}$, set

$\begin{split}\varphi^{app}_{(\lambda)}=\psi^{app}_{(\lambda)}+\frac{{\alpha_{1}}}{2i\lambda}\frac{f\partial_{y}f}{|f|^{2}}\overline{\psi^{app}_{(\lambda)}},\end{split}$
(2.30)

which will be shown to be an approximate solution to (2.26). Finally, the
degenerating wave packet is defined by

$\begin{split}\tilde{\phi}^{app}_{(\lambda)}[g_{0},f]=e^{-G}\varphi^{app}_{(\lambda)}=|f|^{\sigma_{c}-\frac{1}{2}}\varphi^{app}_{(\lambda)}.\end{split}$
(2.31)

####  2.3.3 Proof of Proposition 2.7

Now that we have defined the wave packet solution, let us proceed to confirm
the properties stated in Proposition 2.7.

Linearity and support property. From the definition, linearity is clear.
Furthermore, note from (2.31), (2.30), (2.28) that the support of
$\tilde{\phi}^{app}_{(\lambda)}(t,\cdot)$ coincides with that of
$a_{(\lambda)}(t,\cdot)$. (From now on, we shall suppress from writing out the
subscript $\lambda$.) On the other hand, note the following formulae for $a$:

$\displaystyle a(t,y)$ $\displaystyle=e^{i\lambda S(t,y)}a_{0}(y-2\lambda t),$
(2.32) $\displaystyle S(t,y)$
$\displaystyle=\int_{0}^{t}\left((-\alpha_{1}+2\beta_{1})\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|{f}|^{2}}+h\right)(t^{\prime},y-2\lambda(t-t^{\prime}))\,\mathrm{d}t^{\prime}.$
(2.33)

Since $\lambda<0$, the support of $a(t,\cdot)$ is contained in the interval
$(-\frac{1}{2}\ln 2+2\lambda t,2\lambda t)\subseteq(-\infty,0)$ for $t\geq 0$,
which verifies the support property of $\tilde{\phi}^{app}$ via (2.24).

Regularity estimates. To begin with, we obtain estimates on
$h:=\partial_{t}y$. Recalling (2.13), we have

$\begin{split}h&=-\int_{x}^{x_{1}}\partial_{t}\left(\frac{1}{|f(t,x^{\prime})|}\right)\,\mathrm{d}x^{\prime}=\int_{x}^{x_{1}}\frac{\partial_{t}(|f|^{2})}{|f|^{3}}\,\mathrm{d}x^{\prime},\qquad\partial_{y}h=|f|\partial_{x}h=-\frac{\partial_{t}(|f|^{2})}{|f|^{2}}.\end{split}$

Applying (2.10) and (2.11), we obtain the pointwise estimates

$\begin{split}|h|\leq
C_{f,\delta}(1+t\ln\frac{1}{x})x_{1},\qquad|\partial_{y}h|\leq
C_{f,\delta}(x+t).\end{split}$ (2.34)

We now estimate $a$. Observing that the right hand side in (2.29) is purely
imaginary,

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|a|^{2}(t,y)=-2\lambda\operatorname{Re}(\partial_{y}a\overline{a})(t,y),\quad\mbox{which
gives}\quad\frac{\mathrm{d}}{\mathrm{d}t}\|{a}\|_{L^{2}_{y}}^{{2}}=0.\end{split}$

Now, taking a $y$-derivative, we see that

$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\partial_{y}a_{(\lambda)}}\|_{L^{2}}^{2}\leq
C_{f,\delta}|\lambda|(e^{-|\lambda|t}+t)\|{a}\|_{L^{2}_{y}}\|{\partial_{y}a}\|_{L^{2}_{y}}\end{split}$

where we have used

$\begin{split}\left|\partial_{y}\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)\right|\leq
C_{f,\delta}(x(t,y)+t)\leq C_{f,\delta}(\exp(-|\lambda|t)+t)\end{split}$
(2.35)

on the support of $a(t,\cdot)$. This estimate follows from (2.34) and

$\begin{split}\left|\partial_{y}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}\right|\leq\left|\partial_{x}\partial_{y}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}\right|x.\end{split}$

Therefore, by integrating in time, we obtain

$\|{\partial_{y}a(t)}\|_{L^{2}_{y}}\leq C_{f,\delta}\|{a_{0}}\|_{H^{1}_{y}}$

uniformly in $\lambda$, for $(t,\lambda)$ satisfying
$t\leq{|\lambda|^{-\frac{1}{2}}}$. A similar argument applies to the estimate
of $\partial_{y}^{k}a$, as long as $k\leq s_{0}-2$; one can proceed by an
induction in $k$, using the bound

$\begin{split}\left|\partial_{y}^{k}\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)\right|\leq
C_{f,\delta}(x(t,y)+t)\leq C_{f,\delta}(\exp(-|\lambda|t)+t)\end{split}$

on the support of $a(t,\cdot)$. The estimate for $|\partial_{y}^{k}h|$ readily
follows from the explicit decomposition $\partial_{y}h=h_{1}+th_{2}$, where
$h_{1}$ and $h_{2}$ are $L^{\infty}_{t}C^{s_{0}-2,1}$-smooth functions defined
by

$\begin{split}h_{1}(t,x)=2|f|^{-2}\operatorname{Re}\left(\overline{f_{0}(x)}(\partial_{t}f)(t,x)\right),\quad
h_{2}(t,x)=2|f|^{-2}\operatorname{Re}\left(\overline{(\partial_{t}f)(t,x)}\,\frac{1}{t}\int_{0}^{t}(\partial_{t}f)(t^{\prime},x)\,\mathrm{d}t^{\prime}\right).\end{split}$

Hence we conclude

$\begin{split}\|{a(t)}\|_{H^{k}_{y}}\leq
C_{f,\delta}\|{a_{0}}\|_{H^{k}_{y}},\quad 0\leq
t\leq\min\\{{|\lambda|^{-\frac{1}{2}}},\delta\\}.\end{split}$ (2.36)

In what follows, we shall restrict the variable $t$ to
$[0,\min\\{|\lambda|^{-\frac{1}{2}},\delta\\}]$.

Initial data and regularity estimates. At the initial time, from
$a_{0}(y)=x_{1}^{\frac{1}{2}}g_{0}(x(0,y))$ we have that

$\begin{split}\int|a_{0}(y)|^{2}\,\mathrm{d}y=\int
x_{1}|f_{0}(x)|^{-1}|g_{0}(x)|^{2}\,\mathrm{d}x\end{split}$

and note that the right hand side is equivalent up to constants with
$\|{g_{0}}\|_{L^{2}_{x}}^{2}$. This gives the claimed initial data estimate in
the case $p=2$, and the case of general $p$ can be proved similarly. Next,
with $\partial_{y}=|f_{0}(x)|\partial_{x}$ at the initial time, we note the
bound

$\begin{split}|\partial_{y}^{k}a_{0}(x)|\leq
C_{k}x_{1}^{\frac{1}{2}}\left(\sum_{j=1}^{k}\|{f_{0}}\|_{C^{k-2,1}}^{k-j}|f_{0}(x)|^{j}|\partial_{x}^{j}g_{0}(x)|\right)\end{split}$

which gives

$\begin{split}\|{a_{0}}\|_{H^{k}_{y}}\leq
C_{k}(1+\|{f_{0}}\|_{C^{k-2,1}})^{k-1}\|{g_{0}}\|_{H^{k}_{(x_{1})}},\quad
k\leq s_{0}-1.\end{split}$ (2.37)

Let us now check the regularity estimate (2.18) in the case $n=0$: using
$|f|^{-\sigma_{c}+\frac{1}{2}}{=}e^{G}$,

$\begin{split}\|{|{f}|^{-\sigma_{c}}\tilde{\phi}^{app}_{(\lambda)}(t,x)}\|_{L^{2}}^{2}&=\int_{0}^{x_{1}}|\tilde{\phi}^{app}_{(\lambda)}(t,x)|^{2}|f(t,x)|^{-2\sigma_{c}}\,\mathrm{d}x=\int_{-\infty}^{0}|\tilde{\phi}^{app}_{(\lambda)}(t,y)|^{2}|f(t,y)|^{-2\sigma_{c}+1}\,\mathrm{d}y\\\
&=\int_{-\infty}^{0}|\varphi^{app}_{(\lambda)}(t,y)|^{2}\,\mathrm{d}y\leq
C(1+|\lambda|^{-1})\int_{-\infty}^{0}|\psi^{app}_{(\lambda)}(t,y)|^{2}\,\mathrm{d}y\\\
&\leq C\|{a(t)}\|_{L^{2}_{y}}^{2}\leq C\|{a_{0}}\|_{L^{2}_{y}}\leq
C\|{g_{0}}\|_{L^{2}}.\end{split}$

The cases $1\leq n\leq s_{0}-2$ can be handled similarly, using (2.36) and
(2.37).

Degeneration estimate. Next, we check the degeneration property (2.19). To
simplify the notation, we introduce the notation

$H=qO_{k}(a_{0})\quad\Leftrightarrow\quad\sup_{t\in[0,\min\\{\lambda^{-\frac{1}{2}},\delta\\}]}\left\|{|{f}|^{\frac{1}{2}}\frac{H}{q}}\right\|_{L^{2}_{y}}\leq
C_{f,\delta}\|{a_{0}}\|_{H^{k}_{y}}.$

Note that
$\|{|{f}|^{\frac{1}{2}}(\cdot)}\|_{L^{2}_{y}}=\|{\cdot}\|_{L^{2}_{x}}$ for
each $t$. The terms that are abbreviated as $\frac{1}{\lambda}O_{k}(a_{0})$
(for $k\leq s$) will constitute
$|{f}|^{-\sigma_{c}}\tilde{\phi}^{small}_{(\lambda)}$; the desired estimate
(2.21) would be an immediate consequence of the $L^{2}$ norm estimate embedded
in the $O_{k}(\cdot)$ notation. Recalling the definitions of
$\tilde{\phi}^{app}_{(\lambda)}$, $\varphi^{app}_{(\lambda)}$, and
$\psi^{app}_{(\lambda)}$, and arguing as in the proof of the regularity
estimate, we have

$\displaystyle|{f}|^{-\sigma_{c}+\gamma^{\prime}}\tilde{\phi}^{app}_{(\lambda)}$
$\displaystyle=|{f(t,y)}|^{\gamma^{\prime}-\frac{1}{2}}\psi^{app}_{(\lambda)}+\frac{|{f}|^{\gamma^{\prime}}}{\lambda}O_{0}(a_{0}).$

For the first term, we have

$\displaystyle\|{|{f(t,x)}|^{\gamma^{\prime}-\frac{1}{2}}\psi^{app}_{(\lambda)}(t,x)}\|_{L^{p}}^{p}$
$\displaystyle\leq
C\int_{-\infty}^{0}|{\psi^{app}_{(\lambda)}(t,y)}|^{p}|{f(t,y)}|^{p\gamma^{\prime}-\frac{p}{2}+1}\,\mathrm{d}y$
$\displaystyle\leq
C\left(\int_{-\infty}^{0}|{\psi^{app}_{(\lambda)}(t,y)}|^{2}\,\mathrm{d}y\right)^{\frac{p}{2}}\left(\int_{{\mathrm{supp}}\,\psi^{app}_{(\lambda)}(t,\cdot)}|{f(t,y)}|^{\frac{p}{1-\frac{p}{2}}\gamma^{\prime}+1}\,\mathrm{d}y\right)^{1-\frac{p}{2}}$
$\displaystyle\leq
C\|{a_{0}}\|_{L^{2}_{y}}^{p}\left(\int_{{\mathrm{supp}}\,\psi^{app}_{(\lambda)}(t,\cdot)}|{f(t,y)}|^{\frac{p}{1-\frac{p}{2}}\gamma^{\prime}+1}\,\mathrm{d}y\right)^{p(\frac{1}{p}-\frac{1}{2})},$

so it remains to estimate the last factor. Note that, since
$|{f}|^{-1}\partial_{y}|{f}|=\partial_{x}|{f}|=1+O(x)$ and $x\leq
x_{1}e^{\frac{y}{2}}$ for $y\in(-\infty,0)$, we have

$|{f(t,y)}|\leq Ce^{y}\quad\hbox{ for }y\in(-\infty,0).$

Using the support property
${\mathrm{supp}}\,\psi^{app}_{(\lambda)}(t,\cdot)\subseteq(-\infty,-2|{\lambda}|t)$,
we see that

$\displaystyle\left(\int_{{\mathrm{supp}}\,\psi^{app}_{(\lambda)}(t,\cdot)}|{f(t,y)}|^{\frac{p}{1-\frac{p}{2}}\gamma^{\prime}+1}\,\mathrm{d}y\right)^{\frac{1}{p}-\frac{1}{2}}\lesssim
e^{-2|{\lambda}|(\gamma^{\prime}+\frac{1}{p}-\frac{1}{2})t}.$

Hence, the desired estimate (2.19) in the case $s=0$ now follows.

To treat the cases $s>0$, we begin by recalling that
$\psi^{app}_{(\lambda)}=\psi^{app}=e^{i\lambda(y-\lambda
t+S(t,y))}a_{0}(y-2\lambda t)$. Note the identity

$\displaystyle e^{i\lambda(y-\lambda
t+S)}=\frac{|{f}|}{i\lambda(1+\partial_{y}S)}\left(\frac{1}{|{f}|}\partial_{y}\right)e^{i\lambda(y-\lambda
t+S)}.$

For the expression $\partial_{y}S$ in the denominator, recalling (2.33) and
(2.35), we have

$|{\partial_{y}S}|\leq C_{f,\delta}t{x},$

and in particular note that $1+\partial_{y}S\geq\frac{1}{2}$ when $t$ is
sufficiently small, which can be arranged by taking $\delta>0$ smaller.
Commuting $\frac{1}{|{f}|}\partial_{y}$ (which equals $\partial_{x}$ in the
$(t,x)$–coordinates) outside, we have

$\displaystyle|{f(t,y)}|^{\gamma^{\prime}-\frac{1}{2}}\psi^{app}_{(\lambda)}$
$\displaystyle=\frac{1}{|{f}|}\partial_{y}\left(\frac{|{f(t,y)}|^{\gamma^{\prime}+1-\frac{1}{2}}}{i\lambda(1+\partial_{y}S)}\psi^{app}_{(\lambda)}\right)+\frac{|{f}|^{\gamma^{\prime}}}{\lambda}O_{1}(a_{0}).$

By arguing as in the case of $s=0$, the expression inside the parentheses can
be shown to obey the degeneration bound (2.20). The cases $s>1$ are handled
similarly.

Error estimate. To begin with, at the level of $\psi^{app}_{(\lambda)}$, the
point of choosing $a(t,{y})$ as the solution of (2.29) is to have

$\begin{split}i\partial_{t}\psi^{app}_{(\lambda)}+\partial_{yy}\psi^{app}_{(\lambda)}+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\psi^{app}_{(\lambda)}=O_{2}(a_{0}),\end{split}$
(2.38)

which can be checked with a direct computation using (2.27). We now see that
$\varphi^{app}_{(\lambda)}$ is an approximate solution to (2.26), which is
motivated by the following heuristics:

$\begin{split}\varphi^{app}_{(\lambda)}\simeq\psi^{app}_{(\lambda)}-{\frac{\alpha_{1}}{2}}\frac{f\partial_{y}f}{|f|^{2}}\partial_{y}^{-1}\overline{\psi^{app}_{(\lambda)}}\simeq\psi^{app}_{(\lambda)}+\frac{{\alpha_{1}}}{2i\lambda}\frac{f\partial_{y}f}{|f|^{2}}\overline{\psi^{app}_{(\lambda)}}.\end{split}$

To this end, (2.38) gives

$\begin{split}-i\overline{\partial_{t}\psi^{app}_{(\lambda)}}+\overline{\partial_{yy}\psi^{app}_{(\lambda)}}-\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\overline{\partial_{y}\psi^{app}_{(\lambda)}}=O_{2}(a_{0})\end{split}$

and from this it is not difficult to see that

$\begin{split}\frac{1}{2i\lambda}[i\partial_{t}+\partial_{yy}]\overline{\psi^{app}_{(\lambda)}}=\frac{1}{i\lambda}\partial_{yy}\overline{\psi^{app}_{(\lambda)}}+O_{2}(a_{0})=-\partial_{y}\overline{\psi^{app}_{(\lambda)}}+O_{2}(a_{0}),\end{split}$
(2.39)

so that

$\begin{split}(i\partial_{t}+\partial_{yy})\left(\frac{\alpha_{1}}{2i\lambda}\frac{f\partial_{y}f}{|{f}|^{2}}\overline{\psi^{app}_{(\lambda)}}\right)=-\alpha_{1}\frac{f\partial_{y}f}{|{f}|^{2}}\partial_{y}\overline{\psi^{app}_{(\lambda)}}+O_{2}(a_{0}).\end{split}$

Using (2.38), (2.39), and

$\begin{split}\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\left(\frac{{\alpha_{1}}}{2i\lambda}\frac{f\partial_{y}f}{|f|^{2}}\overline{\psi^{app}_{(\lambda)}}\right)=O_{2}(a_{0}),\quad\frac{{\alpha_{1}^{2}}f\partial_{y}f}{2i\lambda|f|^{2}}\partial_{y}\left(\frac{\overline{f\partial_{y}f}}{|f|^{2}}\psi^{app}_{(\lambda)}\right)=O_{2}(a_{0}),\end{split}$

we simplify

$\begin{split}&\left[i\partial_{t}+\partial_{yy}+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\right]\varphi^{app}_{(\lambda)}+{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\overline{\partial_{y}\varphi^{app}_{(\lambda)}}\\\
&\quad=\left[i\partial_{t}+\partial_{yy}+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\right]\left(\psi^{app}_{(\lambda)}+\frac{{\alpha_{1}}}{2i\lambda}\frac{f\partial_{y}f}{|f|^{2}}\overline{\psi^{app}_{(\lambda)}}\right)\\\
&\quad\mathrel{\phantom{=}}+{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\overline{\partial_{y}\psi^{app}_{(\lambda)}}-\frac{{\alpha_{1}^{2}}f\partial_{y}f}{2i\lambda|f|^{2}}\partial_{y}\left(\frac{\overline{f\partial_{y}f}}{|f|^{2}}\psi^{app}_{(\lambda)}\right)\\\
&\quad=-{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\partial_{y}\overline{\psi^{app}_{(\lambda)}}+{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\partial_{y}\overline{\psi^{app}_{(\lambda)}}+O_{2}(a_{0})=O_{2}(a_{0}).\end{split}$

Moreover, it is easy to see that
$\mathcal{B}_{0}[\varphi^{app}_{(\lambda)}]=O_{2}(a_{0})$ and finally, the
error estimate (2.22) follows from

$\begin{split}\left[i\partial_{t}+\partial_{yy}+\left({(-\alpha_{1}+2\beta_{1})}\frac{\operatorname{Im}(\overline{f}\partial_{y}f)}{|f|^{2}}+h\right)i\partial_{y}\right]\varphi^{app}_{(\lambda)}+{\alpha_{1}}\frac{f\partial_{y}f}{|f|^{2}}\overline{\partial_{y}\varphi^{app}_{(\lambda)}}-\mathcal{B}_{0}[\varphi^{app}_{(\lambda)}]=O_{2}(a_{0})\end{split}$

and (2.37). This completes the proof of Proposition 2.7. ∎

###  2.4 Modified and generalized energy estimates

In §2.4.1 and §2.4.2, we establish the modified and generalized (or bilinear)
energy estimates, respectively, that we shall need the proofs of Theorems 1.1
and 1.2.

####  2.4.1 Modified energy estimate

Assume that $f$ and $\phi=f+\tilde{\phi}$ are solutions to (1.1) on some time
interval. Then, recall that $\tilde{\phi}$ solves

$\begin{split}{i\partial_{t}\tilde{\phi}+|{f}|^{2}\partial_{xx}\tilde{\phi}+\alpha_{1}f(\overline{\partial_{x}f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})+2\beta_{1}\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}=Q_{f}[\tilde{\phi}],}\end{split}$
(2.40)

where $V_{f}$, $W_{f}$ and $Q_{f}[\cdot]$ are defined in (2.15). For a
solution $\tilde{\phi}$ of (2.40), we have the following

######  Proposition 2.8.

Let $f\in L^{\infty}([0,\delta];C^{{s_{c}-1},1}))$ be a solution to (1.5) and
$\tilde{\phi}\in L^{\infty}([0,\delta^{\prime}];C^{{s_{c}-1},1}))$ to (2.40)
for some $0<\delta^{\prime}\leq\delta$. When $\sigma_{c}\geq\frac{1}{2}$,
assume furthermore that at every zero $a$ of $f_{0}$, we have
$\partial_{x}f_{0}(a)\neq 0$ and $\tilde{\phi}_{0}(x)$ vanishes up to order
$\lfloor\sigma_{c}-\frac{1}{2}\rfloor$ at $a$. Then, on
$t\in[0,\delta^{\prime}]$, we have

$\begin{split}\|{|f(t)|^{-\sigma_{c}}\tilde{\phi}(t)}\|_{L^{2}}\leq\|{|f_{0}|^{-\sigma_{c}}\tilde{\phi}_{0}}\|_{L^{2}}\exp\left(C(\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}+\|{\tilde{\phi}}\|_{L^{\infty}_{t}C^{1,1}}^{2})t\right)\end{split}$
(2.41)

where $\sigma_{c}$ is as in (1.6) and $C>0$ is an absolute constant.

###### Proof.

We first present a formal computation without worrying about the finiteness of
the modified energy and the validity of integration by parts, and discuss its
justification below. We compute that

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x=\int|\tilde{\phi}|^{2}\,\partial_{t}|f|^{-2\sigma_{c}}\,\mathrm{d}x+\int{\partial_{t}|\tilde{\phi}|^{2}}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x.\end{split}$

The first term can be bounded using the pointwise inequality

$\begin{split}\left|\frac{\partial_{t}|f|}{|f|}\right|\lesssim\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}.\end{split}$

To handle the second term, we write

$\begin{split}\partial_{t}|\tilde{\phi}|^{2}&=\operatorname{Re}\left(i|{f}|^{2}\partial_{xx}\tilde{\phi}\overline{\tilde{\phi}}\right)+\alpha_{1}\operatorname{Re}\left(if(\partial_{x}\overline{f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})\overline{\tilde{\phi}}\right)+2\beta_{1}\operatorname{Re}\left(i\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}\overline{\tilde{\phi}}\right)\\\
&\mathrel{\phantom{=}}+\operatorname{Re}\left(iV_{f}\tilde{\phi}\overline{\tilde{\phi}}\right)+\operatorname{Re}\left(iW_{f}(\overline{\tilde{\phi}})^{2}\right)-\operatorname{Re}\left(iQ_{f}[\tilde{\phi}]\overline{\tilde{\phi}}\right).\end{split}$
(2.42)

We multiply both sides by $|f|^{-2\sigma_{c}}$ and integrate in $x$. From the
first term on the right-hand side, we obtain, after an integration by parts,

$\displaystyle\int\operatorname{Re}\left(i|{f}|^{2}\partial_{xx}\tilde{\phi}\overline{\tilde{\phi}}\right)|{f}|^{-2\sigma_{c}}\,\mathrm{d}x$
$\displaystyle=-\int|{f}|^{2-2\sigma_{c}}\operatorname{Re}(i\partial_{x}\tilde{\phi}\partial_{x}\overline{\tilde{\phi}})\,\mathrm{d}x-(2-2\sigma_{c})\int|{f}|^{1-2\sigma_{c}}\partial_{x}|{f}|\operatorname{Re}(i\partial_{x}\tilde{\phi}\overline{\tilde{\phi}})$
(2.43)
$\displaystyle=-(2-2\sigma_{c})\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(\overline{f}\partial_{x}f)\operatorname{Re}(i\partial_{x}\tilde{\phi}\overline{\tilde{\phi}}).$

From the second and third terms on the right-hand side of (2.42), we have

$\displaystyle\int\left[\alpha_{1}\operatorname{Re}\left(if(\partial_{x}\overline{f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})\overline{\tilde{\phi}}\right)+2\beta_{1}\operatorname{Re}\left(i\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}\overline{\tilde{\phi}}\right)\right]|{f}|^{-2\sigma_{c}}\,\mathrm{d}x$
$\displaystyle=(\alpha_{1}+2\beta_{1})\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(\overline{f}\partial_{x}f)\operatorname{Re}(i\partial_{x}\tilde{\phi}\overline{\tilde{\phi}})\,\mathrm{d}x+\frac{\alpha_{1}-2\beta_{1}}{2}\int|{f}|^{-2\sigma_{c}}\operatorname{Im}(\overline{f}\partial_{x}f)\partial_{x}|{\tilde{\phi}}|^{2}\,\mathrm{d}x$
$\displaystyle\mathrel{\phantom{=}}+\frac{\alpha_{1}}{2}\int|{f}|^{-2\sigma_{c}}\operatorname{Re}\left(i{f\partial_{x}f}\partial_{x}(\overline{\tilde{\phi}})^{2}\right)\,\mathrm{d}x$
$\displaystyle=(\alpha_{1}+2\beta_{1})\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(\overline{f}\partial_{x}f)\operatorname{Re}(i\partial_{x}\tilde{\phi}\overline{\tilde{\phi}})\,\mathrm{d}x-\frac{\alpha_{1}-2\beta_{1}}{2}\int\left(\partial_{x}(|{f}|^{-2\sigma_{c}}\operatorname{Im}(\overline{f}\partial_{x}f))\right)|{\tilde{\phi}}|^{2}\,\mathrm{d}x$
$\displaystyle\mathrel{\phantom{=}}-\frac{\alpha_{1}}{4}\int\left(i\partial_{x}(|{f}|^{-2\sigma_{c}}{f\partial_{x}f})\right)(\overline{\tilde{\phi}})^{2}\,\mathrm{d}x+\frac{\alpha_{1}}{4}\int\left(i\partial_{x}(|{f}|^{-2\sigma_{c}}{\overline{f\partial_{x}f}})\right)(\tilde{\phi})^{2}\,\mathrm{d}x.$

By our choice of $\sigma_{c}$ in (1.6), the first term on the right-hand side
cancels exactly with (2.43). The remaining terms are estimated from the above
by
$C\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}\int|{\tilde{\phi}}|^{2}|{f}|^{-2\sigma_{c}}\,\mathrm{d}x$.
Next, it is easy to see that

$\displaystyle\left|\int\operatorname{Re}(iV_{f}\tilde{\phi}\overline{\tilde{\phi}})|{f}|^{-2\sigma_{c}}\,\mathrm{d}x\right|+\left|\int\operatorname{Re}(iW_{f}\overline{\tilde{\phi}}\overline{\tilde{\phi}})|{f}|^{-2\sigma_{c}}\,\mathrm{d}x\right|\lesssim\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x.$

It remains to estimate
$\int\operatorname{Re}(iQ_{f}[\tilde{\phi}]\overline{\tilde{\phi}})|{f}|^{-2\sigma_{c}}\,\mathrm{d}x$.
The contribution of any term with at least one factor of $\tilde{\phi}$
(without any derivatives) may be easily estimated by
$(\|{f}\|_{C^{1,1}}^{2}+\|{\tilde{\phi}}\|_{C^{1,1}}^{2})\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x$.
Recalling the expression for $Q_{f}$ from (2.15), we may estimate

$\displaystyle\left|\int\operatorname{Re}(iQ_{f}[\tilde{\phi}]\overline{\tilde{\phi}})|{f}|^{-2\sigma_{c}}\,\mathrm{d}x\right|$
$\displaystyle\lesssim(\|{f}\|_{C^{1,1}}\|{\tilde{\phi}}\|_{C^{1,1}}+\|{\tilde{\phi}}\|_{C^{1,1}}^{2})\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x$
$\displaystyle\mathrel{\phantom{\lesssim}}+\left(\int|{\partial_{x}\tilde{\phi}}|^{4}|{f}|^{2-2\sigma_{c}}\right)^{\frac{1}{2}}\left(\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x\right)^{\frac{1}{2}}.$

Integrating by parts and using Hölder’s inequality, we have

$\displaystyle\int(\partial_{x}\tilde{\phi})^{4}|{f}|^{2-2\sigma_{c}}$
$\displaystyle=\int\tilde{\phi}(\partial_{x}\tilde{\phi})^{2}(-3\partial_{xx}\tilde{\phi}|{f}|-(2-2\sigma_{c})\partial_{x}\tilde{\phi}\partial_{x}|{f}|)|{f}||{f}|^{-2\sigma_{c}}\,\mathrm{d}x$
$\displaystyle\leq
C\|{\tilde{\phi}}\|_{C^{1,1}}\|{f}\|_{C^{0,1}}\left(\int\tilde{\phi}^{2}|{f}|^{-2\sigma_{c}}\,\mathrm{d}x\right)^{\frac{1}{2}}\left(\int(\partial_{x}\tilde{\phi})^{4}|{f}|^{2-2\sigma_{c}}\,\mathrm{d}x\right)^{\frac{1}{2}}.$

Hence,

$\displaystyle\left|\int\operatorname{Re}(iQ_{f}[\tilde{\phi}]\overline{\tilde{\phi}})|{f}|^{-2\sigma_{c}}\,\mathrm{d}x\right|$
$\displaystyle\lesssim(\|{f}\|_{C^{1,1}}\|{\tilde{\phi}}\|_{C^{1,1}}+\|{\tilde{\phi}}\|_{C^{1,1}}^{2})\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x.$

Collecting all the terms, we conclude that

$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}t}\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x\right|\lesssim\left(\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}+\|{\tilde{\phi}}\|_{L^{\infty}_{t}C^{1,1}}^{2}\right)\int|\tilde{\phi}|^{2}\,|f|^{-2\sigma_{c}}\,\mathrm{d}x.\end{split}$

Integrating in time gives

$\begin{split}\|{|f(t)|^{-\sigma_{c}}\tilde{\phi}(t)}\|_{L^{2}}\leq\|{|f_{0}|^{-\sigma_{c}}\tilde{\phi}_{0}}\|_{L^{2}}\exp\left(C(\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}+\|{\tilde{\phi}}\|_{L^{\infty}_{t}C^{1,1}}^{2})t\right),\end{split}$
(2.44)

which is the desired conclusion.

We now sketch the observations needed to make the above computation rigorous.
Note that, in order for (2.41) to be nontrivial, the right-hand side must be
finite, i.e., $\|{|{f_{0}}|^{-\sigma_{c}}\tilde{\phi}_{0}}\|_{L^{2}}<+\infty$.
When $\sigma_{c}\geq\frac{1}{2}$, this implies the vanishing of
$\tilde{\phi}_{0}$ at each zero $a$ of $f$ (which is isolated by the
assumption in this case) up to order $\lfloor\sigma_{c}-\frac{1}{2}\rfloor$.
Applying Lemma 2.5 to the $L^{\infty}_{t}([0,\delta];C^{s_{c}-1,1}))$
solutions $f$ and $f+\tilde{\phi}$, it follows that the zero set of $f(t,x)$,
as well as the nonvanishing of $f^{\prime}(t,a)$ and the vanishing of
$\tilde{\phi}(t,x)$ up to order $\lfloor\sigma_{c}-\frac{1}{2}\rfloor$ at each
zero $a$ of $f$, is preserved in $t\in[0,\delta]$. As a consequence,
$\|{|{f}|^{-\sigma_{c}}\tilde{\phi}}\|_{L^{2}}<+\infty$ for every
$t\in[0,\delta]$ as well. Using the vanishing properties of $f$ and
$\tilde{\phi}$ (the latter is needed only when $\sigma_{c}\geq\frac{1}{2}$),
the above computation can then be justified. ∎

The above proposition motivates us to define the following hermitian product
and norm (which will be referred to as the modified energy): given some $f$,
we shall consider

$\begin{split}\langle{v,u}\rangle_{f}(t)=\int|f(t,\cdot)|^{-2\sigma_{c}}v(t,\cdot)\overline{u(t,\cdot)}\,\mathrm{d}x.,\qquad\|{v}\|_{f}^{2}(t)=\int|f(t,\cdot)|^{-2\sigma_{c}}|v(t,\cdot)|^{2}\,\mathrm{d}x.\end{split}$

####  2.4.2 Generalized (bilinear) energy estimate

We proceed to prove the generalized energy estimate.

######  Proposition 2.9.

Let $\tilde{\phi}$ to be a solution of

$\begin{split}[i\partial_{t}+\mathcal{L}_{f}]\tilde{\phi}=Q_{f}[\tilde{\phi}],\end{split}$

where $[i\partial_{t}+\mathcal{L}_{f}]\tilde{\phi}$ denotes the left-hand side
of (2.14), and let $\tilde{\phi}^{app}=\tilde{\phi}^{app}[g_{0},f]$ be the
degenerating wave packet constructed in Proposition 2.7. Then, we have the
following estimate on $t\in[0,\min\\{|\lambda|^{-\frac{1}{2}},\delta\\}]$:

$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Re}\langle{\tilde{\phi},\tilde{\phi}^{app}}\rangle_{f}\right|\leq\left(C(\|{f}\|_{L^{\infty}_{t}C^{1,1}_{x}}^{2}+\|{\tilde{\phi}}\|_{L^{\infty}_{t}C^{1,1}_{x}}^{2})\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}}+C_{f,\delta}A^{-\sigma_{c}+3}\|{g_{0}}\|_{H^{2}_{(x_{1})}}\right)\|{\tilde{\phi}}\|_{L^{2}_{f}}.\end{split}$
(2.45)

###### Proof.

In the proof, the time variable $t$ will be restricted to the interval
$[0,\min\\{|\lambda|^{-\frac{1}{2}},\delta\\}]$. Before we proceed, let us
recall that $\mathcal{L}_{f}$ is given by

$\begin{split}\mathcal{L}_{f}[\tilde{\phi}]&:=|{f}|^{2}\partial_{xx}\tilde{\phi}+\alpha_{1}f(\overline{\partial_{x}f}\partial_{x}\tilde{\phi}+\partial_{x}f\partial_{x}\overline{\tilde{\phi}})+2\beta_{1}\overline{f}\partial_{x}f\partial_{x}\tilde{\phi}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}\\\
&\mathrel{\phantom{:}}=|{f}|^{2}\partial_{xx}\tilde{\phi}+(\alpha_{1}+2\beta_{1})\operatorname{Re}(\overline{f}\partial_{x}f)\partial_{x}\tilde{\phi}+(-\alpha_{1}+2\beta_{1})i\operatorname{Im}(\overline{f}\partial_{x}f)\partial_{x}\tilde{\phi}+\alpha_{1}f\partial_{x}f\partial_{x}\overline{\tilde{\phi}}+V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}}\end{split}$

and that $\tilde{\phi}^{app}$ satisfies
$[i\partial_{t}+\mathcal{L}_{f}]\tilde{\phi}^{app}=\boldsymbol{\epsilon}_{\tilde{\phi}}.$
We compute666Here, since $\tilde{\phi}^{app}$ is smooth and compactly
supported away from the zeroes of $f$ at each $t$, there are no issues
whatsoever in justifying the computation below.

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Re}\langle{\tilde{\phi},\tilde{\phi}^{app}}\rangle_{f}&=\operatorname{Re}\left(\int{-2\sigma_{c}|{f}|^{-2\sigma_{c}-1}}\partial_{t}|{f}|\tilde{\phi}\overline{\tilde{\phi}^{app}}+\int
i|f|^{{-2\sigma_{c}}}(\mathcal{L}_{f}[\tilde{\phi}]{-}Q_{f}[\tilde{\phi}])\overline{\tilde{\phi}^{app}}-\int
i|f|^{{-2\sigma_{c}}}\tilde{\phi}(\overline{\mathcal{L}_{f}[\tilde{\phi}^{app}]{-}\boldsymbol{\epsilon}_{\tilde{\phi}}})\right).\end{split}$

Using the estimates for $|\partial_{t}|f||,Q_{f}[\tilde{\phi}]$, and
$\boldsymbol{\epsilon}_{\tilde{\phi}}$, we can bound

$\begin{split}\left|\operatorname{Re}\int{|{f}|^{-2\sigma_{c}-1}}\partial_{t}|{f}|\tilde{\phi}\overline{\tilde{\phi}^{app}}\right|\lesssim\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}\|{\tilde{\phi}}\|_{L^{2}_{f}}\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}},\end{split}$
$\begin{split}\left|\operatorname{Re}\int
i|f|^{{-2\sigma_{c}}}Q_{f}[\tilde{\phi}]\overline{\tilde{\phi}^{app}}\right|\lesssim\left(\|{f}\|_{L^{\infty}_{t}C^{1,1}}^{2}+\|{\tilde{\phi}}\|_{L^{\infty}_{t}C^{1,1}}^{2}\right)\|{\tilde{\phi}}\|_{L^{2}_{f}}\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}}\end{split}$

and

$\begin{split}\left|\operatorname{Re}\int
i|f|^{{-2\sigma_{c}}}\tilde{\phi}\,\overline{\boldsymbol{\epsilon}_{\tilde{\phi}}}\right|\leq
C_{f,\delta}\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}}A^{-\sigma_{c}+3}\|{g_{0}}\|_{H^{2}_{(x_{1})}}.\end{split}$

We now consider the remaining expression

$\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i\mathcal{L}_{f}[\tilde{\phi}]\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i\tilde{\phi}\overline{\mathcal{L}_{f}[\tilde{\phi}^{app}]})\,\mathrm{d}x.$

For the contribution of the principal term $|{f}|^{2}\partial_{xx}$, we obtain

$\displaystyle\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i|{f}|^{2}\partial_{xx}\tilde{\phi}\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i\tilde{\phi}\overline{|{f}|^{2}\partial_{xx}\tilde{\phi}^{app}})\,\mathrm{d}x$
$\displaystyle=-\int|{f}|^{2-2\sigma_{c}}\operatorname{Re}(i\partial_{x}\phi\overline{\partial_{x}\tilde{\phi}^{app}})\,\mathrm{d}x-\int(2-2\sigma_{c})|{f}|^{1-2\sigma_{c}}\partial_{x}|{f}|\operatorname{Re}(i\partial_{x}\phi\overline{\tilde{\phi}^{app}})\,\mathrm{d}x$
$\displaystyle\mathrel{\phantom{=}}+\int|{f}|^{2-2\sigma_{c}}\operatorname{Re}(i\partial_{x}\phi\overline{\partial_{x}\tilde{\phi}^{app}})\,\mathrm{d}x+\int(2-2\sigma_{c})|{f}|^{1-2\sigma_{c}}\partial_{x}|{f}|\operatorname{Re}(i\phi\overline{\partial_{x}\tilde{\phi}^{app}})\,\mathrm{d}x$
$\displaystyle=-\int(2-2\sigma_{c})|{f}|^{1-2\sigma_{c}}\partial_{x}|{f}|\left(\operatorname{Re}(i\partial_{x}\phi\overline{\tilde{\phi}^{app}})-\operatorname{Re}(i\phi\overline{\partial_{x}\tilde{\phi}^{app}})\right)\,\mathrm{d}x=:I.$

Since $|{f}|\partial_{x}|{f}|=\operatorname{Re}(\overline{f}\partial_{x}f)$,
this term cancels with some of the first-order terms, i.e.,

$\displaystyle\int(\alpha_{1}+2\beta_{1})|{f}|^{-2\sigma_{c}}\operatorname{Re}(\overline{f}\partial_{x}f)\operatorname{Re}(i\partial_{x}\tilde{\phi}\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int(\alpha_{1}+2\beta_{1})|{f}|^{-2\sigma_{c}}\operatorname{Re}(\overline{f}\partial_{x}f)\operatorname{Re}(i\tilde{\phi}\overline{\partial_{x}\tilde{\phi}^{app}})\,\mathrm{d}x=-I.$

For the remaining first-order terms, we have, after integrating by parts,

$\displaystyle-\int(-\alpha_{1}+2\beta_{1})|{f}|^{-2\sigma_{c}}\operatorname{Im}(\overline{f}\partial_{x}f)\operatorname{Re}(\partial_{x}\tilde{\phi}\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int(-\alpha_{1}+2\beta_{1})|{f}|^{-2\sigma_{c}}\operatorname{Im}(\overline{f}\partial_{x}f)\operatorname{Re}(\tilde{\phi}\overline{\partial_{x}\tilde{\phi}^{app}})\,\mathrm{d}x$
$\displaystyle=(-\alpha_{1}+2\beta_{1})\int\left(\partial_{x}(|{f}|^{-2\sigma_{c}}\operatorname{Im}(\overline{f}\partial_{x}f))\right)\operatorname{Re}(\tilde{\phi}\overline{\tilde{\phi}^{app}})\,\mathrm{d}x,$
$\displaystyle\int\alpha_{1}|{f}|^{-2\sigma_{c}}\operatorname{Re}(if\partial_{x}f\partial_{x}\overline{\tilde{\phi}}\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int\alpha_{1}|{f}|^{-2\sigma_{c}}\operatorname{Re}(i\phi\overline{f\partial_{x}f\partial_{x}\overline{\tilde{\phi}^{app}}})\,\mathrm{d}x$
$\displaystyle=\frac{\alpha_{1}}{2}\int|{f}|^{-2\sigma_{c}}\left(if\partial_{x}f\partial_{x}\overline{\tilde{\phi}}\overline{\tilde{\phi}^{app}}-i\overline{f\partial_{x}f}\partial_{x}\tilde{\phi}\tilde{\phi}^{app}\right)\,\mathrm{d}x-\frac{\alpha_{1}}{2}\int|{f}|^{-2\sigma_{c}}\left(i\overline{f\partial_{x}f}\phi\partial_{x}\tilde{\phi}^{app}-if\partial_{x}f\overline{\phi}\partial_{x}\overline{\tilde{\phi}^{app}}\right)\,\mathrm{d}x$
$\displaystyle=-\frac{\alpha_{1}}{2}\int\left(\partial_{x}(i|{f}|^{-2\sigma_{c}}f\partial_{x}f)\right)\overline{\tilde{\phi}}\overline{\tilde{\phi}^{app}}\,\mathrm{d}x+\frac{\alpha_{1}}{2}\int\left(\partial_{x}(i|{f}|^{-2\sigma_{c}}\overline{f\partial_{x}f})\right)\tilde{\phi}\tilde{\phi}^{app}\,\mathrm{d}x.$

Both expressions may be bounded from the above by
$C\|{f}\|_{C^{1,1}}^{2}\|{\tilde{\phi}}\|_{L^{2}_{f}}\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}}$.
Finally, for the zeroth order terms, we easily have

$\displaystyle\left|\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i(V_{f}\tilde{\phi}+W_{f}\overline{\tilde{\phi}})\overline{\tilde{\phi}^{app}})\,\mathrm{d}x-\int|{f}|^{-2\sigma_{c}}\operatorname{Re}(i\tilde{\phi}\overline{(V_{f}\tilde{\phi}^{app}+W_{f}\overline{\tilde{\phi}^{app}})}\,\mathrm{d}x\right|\lesssim\|{f}\|_{C^{1,1}}^{2}\|{\tilde{\phi}}\|_{L^{2}_{f}}\|{\tilde{\phi}^{app}}\|_{L^{2}_{f}}.$

This gives (2.45), which concludes the proposition. ∎

###  2.5 Proof of Theorem 1.1

We are now in a position to conclude the proof of Theorem 1.1 for the equation
(1.5). To begin with, let $f$ satisfy the assumptions of the theorem with
$f_{0}=f(t=0)$. We may assume that $f_{0}(0)=0$ and $f_{0}^{\prime}(0)=:A>0$
by translation and phase rotation if necessary. We also fix $x_{1}$ as in
Proposition 2.7.

Now let $\epsilon>0$, ${s_{0}}\geq s_{c}$ and $0<\delta^{\prime}\leq\delta$ be
given. We take some $\lambda\leq-1$ and $g_{0}$ satisfying the assumptions of
Proposition 2.7, and define $\tilde{\phi}_{0}$ by

$\begin{split}\tilde{\phi}_{0}=\epsilon\,c({s_{0}})|\lambda|^{-{s_{0}}}\tilde{\phi}^{app}_{(\lambda)}(t=0)[g_{0};f].\end{split}$

Here, $\tilde{\phi}^{app}_{(\lambda)}=\tilde{\phi}^{app}_{(\lambda)}[g_{0};f]$
is the degenerating wave packet constructed in Proposition 2.7 using $g_{0}$.
It is not difficult to check that $\tilde{\phi}^{app}_{(\lambda)}(t=0)\in
C^{\infty}_{c}$ and
$\|{\tilde{\phi}^{app}_{(\lambda)}(t=0)}\|_{C^{{s_{0}}}}\lesssim_{{s_{0}}}|\lambda|^{{s_{0}}}$.
Hence, by taking a sufficiently small $c({s_{0}})>0$, we can ensure that
$\|{\tilde{\phi}_{0}}\|_{C^{{s_{0}}}}\leq\epsilon$ uniformly for all
$\lambda\leq-1$, as required by the statement of the theorem. We observe that

$\begin{split}\operatorname{Re}\langle{\tilde{\phi}_{0},\tilde{\phi}^{app}_{(\lambda)}(t=0)}\rangle_{f}\geq
c_{0}\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}\|{\tilde{\phi}^{app}(t=0)}\|_{L^{2}_{f}}\end{split}$
(2.46)

for some $c_{0}>0$ independent of $\lambda$. To proceed, let us assume that
the first option in the theorem does not hold; namely, there exists a solution
$\phi$ to (1.5) satisfying
$\|{\phi-f}\|_{L^{\infty}([0,\delta^{\prime}];C^{s_{c}})}<+\infty$ and
$\phi(t=0)=f_{0}+\tilde{\phi}_{0}$. On $[0,\delta^{\prime}]$, we write
$\tilde{\phi}=\phi-f$ and set

$\begin{split}M_{{2}}=\sup_{t\in[0,\delta^{\prime}]}\left(\|{f(t)}\|_{C^{1,1}}+\|{\tilde{\phi}}\|_{C^{1,1}}\right).\end{split}$

We shall now establish the claimed norm inflation statement for
$\tilde{\phi}$, by taking $|\lambda|$ sufficiently large but in a way
depending only on $f$ and $\delta^{\prime}$.

On the time interval $[0,\delta^{\prime}]$, using Proposition 2.8 and (2.18)
we obtain that

$\begin{split}\|{\tilde{\phi}(t)}\|_{L^{2}_{f}}\leq\exp(CM_{{2}}^{2}t)\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}},\qquad\|{\tilde{\phi}^{app}(t)}\|_{L^{2}_{f}}\leq
C_{f,\delta}A^{-\sigma_{c}+1}\|{g_{0}}\|_{L^{2}}.\end{split}$

In particular, we note that $\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}<+\infty$ since
$\tilde{\phi}_{0}$ is supported away from the zeroes of $f$, and as discussed
in §2.4.1, $\tilde{\phi}(t)$ vanishes sufficiently fast at the zeroes of $f$
(ultimately due to Lemma 2.5) so that $\|{\tilde{\phi}(t)}\|_{L^{2}_{f}}$ is
well-defined and obeys the above bound. Applying (2.45), integrating in time
on the interval
$[0,\min\\{\delta^{\prime},cM_{2}^{-2},A^{-2}|\lambda|^{-\frac{1}{2}}\\}]$ for
a sufficiently small $c>0$ and using (2.46), we have

$\begin{split}\operatorname{Re}\langle{\tilde{\phi}(t),\tilde{\phi}^{app}(t)}\rangle_{f}\geq\frac{c_{0}}{2}\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}\|{g_{0}}\|_{L^{2}}\quad\hbox{
for
}|{t}|\leq\min\\{\delta^{\prime},cM_{2}^{-2},A^{-2}|\lambda|^{-\frac{1}{2}}\\}.\end{split}$
(2.47)

Next, applying (2.19)–(2.21) with $\gamma^{\prime}=-\sigma_{c}$ and $s=s_{c}$,
we have

$\displaystyle\operatorname{Re}\langle{\tilde{\phi}(t),\tilde{\phi}^{app}(t)}\rangle_{f}-C_{f,\delta}A^{-\sigma_{c}+1}|\lambda|^{-1}\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}\|{g_{0}}\|_{H^{|{s_{c}}|}_{(x_{1})}}$
$\displaystyle\leq\operatorname{Re}\left\langle{\tilde{\phi}(t),\partial_{x}^{s_{c}}\left(\frac{|{f}|^{-\sigma_{c}+s_{c}-\frac{1}{2}}}{i^{s_{c}}\lambda^{s_{c}}(1+|{f}|\partial_{x}S)^{s_{c}}}\psi^{app}(t)\right)}\right\rangle$
$\displaystyle\leq\|{\partial_{x}^{s_{c}}\tilde{\phi}(t)}\|_{L^{\infty}}|\lambda|^{-s_{c}}\|{(1+|{f}|\partial_{x}S)^{-s_{c}}|{f}|^{s_{c}-\sigma_{c}-\frac{1}{2}}\psi^{app}(t)}\|_{L^{1}}$
$\displaystyle\leq
C_{f,\delta}^{1-\sigma_{c}}A^{s_{c}-\sigma_{c}+\frac{1}{2}}|\lambda|^{-s_{c}}e^{-2|{\lambda}|(s_{c}-\sigma_{c}+\frac{1}{2})A^{2}t}\|{\partial_{x}^{s_{c}}\tilde{\phi}(t)}\|_{L^{\infty}}\|{g_{0}}\|_{L^{2}}.$

Taking $|{\lambda}|$ sufficiently large, we may ensure that

$\displaystyle
C_{f,\delta}A^{-\sigma_{c}+1}|\lambda|^{-1}\|{g_{0}}\|_{H_{(x_{1})}^{|{s_{c}}|}}\leq\frac{c_{0}}{4}\|{g_{0}}\|_{L^{2}}\qquad\mbox{and}\qquad
A^{-2}|\lambda|^{-\frac{1}{2}}<\delta^{\prime},$ (2.48)

which gives, after combining the previous two inequalities with (2.47),

$\displaystyle\frac{c_{0}}{4}C_{f,\delta}^{-\sigma_{c}+1}A^{-(s_{c}-\sigma_{c}+\frac{1}{2})}|\lambda|^{s_{c}}e^{2|{\lambda}|(s_{c}-\sigma_{c}+\frac{1}{2})A^{2}t}\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}\leq\|{\partial_{x}^{s_{c}}\tilde{\phi}(t)}\|_{L^{\infty}}\quad\hbox{
for }|{t}|\leq\min\\{cM_{2}^{-2},A^{-2}|\lambda|^{-\frac{1}{2}}\\}.$

For each $|\lambda|$ satisfying (2.48), there are two cases; either (i)
$cM_{2}^{-2}<A^{-2}|\lambda|^{-\frac{1}{2}}$ or (ii) $cM_{2}^{-2}\geq
A^{-2}|\lambda|^{-\frac{1}{2}}$. In the case (i), we obtain that
$M_{2}\gtrsim_{A}|\lambda|^{\frac{1}{4}}\gtrsim_{A}(\delta^{\prime})^{-\frac{1}{2}}$
using (2.48). Here, we could have assumed that $|\lambda|$ is sufficiently
large from the beginning so that
$\sup_{t\in[0,\delta^{\prime}]}\|{f(t)}\|_{C^{1,1}}\ll_{A}|\lambda|^{\frac{1}{4}}.$
Then,
$M_{2}\simeq\sup_{t\in[0,\delta^{\prime}]}\|{\tilde{\phi}(t)}\|_{C^{1,1}}$ and
the desired norm inflation follows simply from our assumption in (1.7) that
$s_{c}\geq 2$. In the case (ii), we simply take
$t=A^{-2}|{\lambda}|^{-\frac{1}{2}}$ in (2.47), which gives the claimed norm
inflation (actually, in this case we obtain a much stronger growth in terms of
$1/\delta^{\prime}$) using that $s_{c}>\sigma_{c}-\frac{1}{2}$. This finishes
the proof. ∎

######  Remark 2.10.

At the end of the above proof, observe that we could have followed the same
argument but have used (2.19)–(2.21) with $\gamma^{\prime}=-\sigma_{c}$,
$s=\sigma$ and $p=2$ to derive

$\frac{c_{0}}{4}C_{f,\delta}^{-\sigma_{c}+1}A^{-(\sigma-\sigma_{c})}|\lambda|^{s_{c}}e^{2|{\lambda}|(\sigma-\sigma_{c})A^{2}t}\|{\tilde{\phi}_{0}}\|_{L^{2}_{f}}\leq\|{\tilde{\phi}(t)}\|_{H^{\sigma}},$

for $t\leq\min\\{cM_{2}^{-2},A^{-2}|{\lambda}|^{-\frac{1}{2}}\\}$. This can be
used to prove the inflation of the $H^{\sigma}$-norm for any
$\sigma>\sigma_{c}$ in the second alternative of Theorem 1.1.

###  2.6 Proof of Theorem 1.2

Let us divide the proof of Theorem 1.2 into several steps.

Choice of background solution. Towards a contradiction, we shall assume that
there exist $\epsilon>0$ and $s_{0}\geq s_{c}+2$ such that for any
$\phi_{0}\in C^{\infty}(\mathbb{T})$ satisfying
$\|{\phi_{0}}\|_{C^{s_{0}}}<\epsilon$, there exist $\delta=\delta(\phi_{0})>0$
and a solution $\phi\in L^{\infty}([0,\delta];{C^{s_{c}+1,1}})$ to (1.5) with
initial data $\phi(t=0)=\phi_{0}$.

Under this assumption, let us fix a function $\mathring{f}_{0}\in
C^{\infty}(\mathbb{T})$ which is supported in $(-\frac{1}{2},\frac{1}{2})$ and
$\mathring{f}_{0}(x)=x$ in $[-\frac{1}{4},\frac{1}{4}]$. We then set

$\begin{split}f_{0}:=\sum_{k=k_{0}}^{\infty}f_{k,0}:=\sum_{k=k_{0}}^{\infty}A_{k}2^{-k}\mathring{f}_{0}(2^{k}(x-x_{k})),\quad
A_{k}=2^{-k^{2}},\quad x_{k}=2^{-\frac{k}{2}}\end{split}$

where $k_{0}=k_{0}(s_{0},\epsilon,\mathring{f}_{0})\geq 1$ is taken
sufficiently large to achieve $\|{f_{0}}\|_{C^{s_{0}}}<\frac{\epsilon}{2}$. It
is not difficult to see that $f_{0}\in C^{\infty}(\mathbb{T})$. Furthermore,
since the supports of $f_{k,0}$ are disjoint from each other, for each $k\geq
k_{0}$, we may choose $\chi_{k}\in C^{\infty}(\mathbb{T})$ to be a cutoff
function satisfying $\chi_{k}=1$ on ${\mathrm{supp}}\,(f_{k,0})$ and
$\chi_{k}=0$ on ${\mathrm{supp}}\,(f_{k^{\prime},0})$ for any $k^{\prime}\neq
k$. From the contradiction hypothesis, we have a solution $f(t)\in
L^{\infty}([0,\delta];{C^{{s_{c}+1},1}})$ to (1.5) with initial data
${f}_{0}$, for some $\delta>0$. The estimate $|\partial_{t}{f}|\lesssim|{f}|$
shows that ${\mathrm{supp}}\,({f}(t))={\mathrm{supp}}\,({f}_{0})$ on
$[0,\delta]$, and since $\chi_{k}$ equals either 0 or 1 on
${\mathrm{supp}}\,({f}(t))={\mathrm{supp}}\,({f}_{0})$, we have that

$\begin{split}\chi_{k}=\chi_{k}^{3},\qquad\partial_{x}\chi_{k}=0\end{split}$

on ${\mathrm{supp}}\,({f}(t))$ for any $k\geq k_{0}$. Using these
observations, it follows that for each $k\geq k_{0}$, we have that
$f_{k}:=\chi_{k}f$ is again a solution to (1.5) with initial data
$f_{k}(t=0)=f_{k,0}$. Furthermore, the
$L^{\infty}([0,\delta];{C^{{s_{c}+1},1}})$–norm of $f_{k}$ is bounded
uniformly in $k$.

In the following, we consider the weight $w(t,x)={|f(t,x)|^{-2\sigma_{c}}}$
and define

$\begin{split}\langle{a,b}\rangle_{w}(t)=\int_{\mathbb{T}}|w(t,x)|a(t,x)\overline{b(t,x)}\,\mathrm{d}x,\quad\|{a}\|_{L^{2}_{w}}^{2}=\langle{a,a}\rangle_{w}.\end{split}$

Choice of wave packet solutions. We now fix some nonzero function $g_{0}\in
C^{\infty}$ supported in $(\frac{1}{8},\frac{1}{4})$ and take
$g_{k}(x):={2^{\frac{k}{2}}}g_{0}(2^{k}(x-x_{k}))$. For some sequence
$\\{\lambda_{k}\\}_{k\geq k_{0}}$ to be determined (for now, we take
$-\lambda_{k}\geq A_{k}^{10}$), we consider the sequence of wave packet
solutions

$\begin{split}{\tilde{\phi}_{k}^{app}:=\tilde{\phi}^{app}_{(\lambda_{k})}[g_{k};f_{k}],}\end{split}$

where $\tilde{\phi}^{app}_{(\lambda_{k})}[g_{k};f_{k}]$ is the wave packet
solution from Proposition 2.7 with data $g_{k},\lambda_{k}$, adapted to the
linearly degenerate solution $f_{k}$, with $A=A_{k}$ and $x_{1}=2^{-k-2}$. We
define the corresponding error by

$\begin{split}[i\partial_{t}+\mathcal{L}_{f_{k}}]\tilde{\phi}^{app}_{k}={\boldsymbol{\epsilon}}_{k}\end{split}$
(2.49)

where the operator $[i\partial_{t}+\mathcal{L}_{f_{k}}]$ is obtained from
(2.14) by replacing $f$ with $f_{k}$. Applying Proposition 2.7, we obtain the
following bounds: with
$\delta_{k}:=\min\\{\delta,A_{k}^{-2}|\lambda_{k}|^{-\frac{1}{2}}\\}=A_{k}^{-2}|\lambda_{k}|^{-\frac{1}{2}}$
(by our choice of $-\lambda_{k}$ in the above),

  * •

$\|{\tilde{\phi}^{app}_{k}}\|_{L^{\infty}([0,\delta_{k}];L^{2}_{w})}\leq
C_{f_{k},\delta_{k}}\|{\tilde{\phi}^{app}_{k}}\|_{L^{2}_{w}}(t=0)\leq
C_{f_{k},\delta_{k}}A_{k}^{-\sigma_{c}+1}\|{g_{0}}\|_{L^{2}}$;

  * •

$\|{\boldsymbol{\epsilon}_{k}}\|_{L^{\infty}([0,\delta_{k}],L^{2}_{w})}\leq
C_{f_{k},\delta_{k}}A_{k}^{-\sigma_{c}+3}\|{g_{k}}\|_{H^{2}_{(2^{-k-2})}}\leq
C_{f_{k},\delta_{k}}A_{k}^{-\sigma_{c}+3}\|{g_{0}}\|_{H^{2}}$;

and

$\begin{split}|f|^{-2\sigma_{c}}\tilde{\phi}^{app}_{k}=\partial_{x}^{s_{c}}\left(\frac{|f|^{-\sigma_{c}+s_{c}-\frac{1}{2}}}{(i\lambda_{k})^{s_{c}}(1+|f|\partial_{x}S)^{s_{c}}}\psi^{app}_{k}\right)+|f|^{-2\sigma_{c}}\tilde{\phi}^{small}_{k}\end{split}$

with

$\begin{split}\left\|\frac{|f|^{-\sigma_{c}+s_{c}-\frac{1}{2}}}{(i\lambda_{k})^{s_{c}}(1+|f|\partial_{x}S)^{s_{c}}}\psi^{app}_{k}\right\|_{L^{1}}\leq
C_{f_{k},\delta_{k}}^{1-\sigma_{c}}A_{k}^{-\sigma_{c}+s_{c}+\frac{1}{2}}|\lambda_{k}|^{-s_{c}}\exp\left(-2|\lambda_{k}|(-\sigma_{c}+s_{c}+\frac{1}{2})A_{k}t\right)\|{g_{0}}\|_{L^{2}}\end{split}$

for $0\leq t\leq\delta_{k}$ and

$\begin{split}\|{\tilde{\phi}_{k}^{small}}\|_{L^{\infty}([0,\delta_{k}];L^{2}_{w})}\leq
C_{f_{k},\delta_{k}}A_{k}^{-\sigma_{c}+1}|\lambda_{k}|^{-1}\|{g_{k}}\|_{H^{s_{c}}_{(2^{-k-2})}}\leq
C_{f_{k},\delta_{k}}A_{k}^{-\sigma_{c}+1}|\lambda_{k}|^{-1}\|{g_{0}}\|_{H^{s_{c}}}.\end{split}$

From (2.23) we see that

$\begin{split}C_{f_{k},\delta_{k}}\lesssim(1+A_{k}^{-1}M)^{N_{0}}\exp(C_{0}M^{2}\delta_{k}),\qquad
M=\sup_{t\in[0,\delta]}\|{f(t,\cdot)}\|_{C^{s_{c}+1,1}}\end{split}$

where the implicit constant and $N_{0}$ depends on
$g_{0},\alpha_{1},\beta_{1},\mu_{1},s_{c}$, but not on $k$ and $\lambda_{k}$.
Then, simply using $\delta_{k}\leq\delta$ and recalling $A_{k}=2^{-k^{2}}$, we
see that $C_{f_{k},\delta_{k}}\lesssim 2^{N_{0}k^{2}}$ holds, where the
implicit constant depends further on $M,\delta$ but not on $k$ and
$\lambda_{k}$. In turn, this gives an upper bound on the constants in the
estimates above; for instance

$\begin{split}C_{f_{k},\delta_{k}}^{1-\sigma_{c}}A_{k}^{-\sigma_{c}+s_{c}+\frac{1}{2}}\lesssim
2^{N_{1}k^{2}}\end{split}$

with some $N_{1}>0$ depending additionally on $\sigma_{c},s_{c}$. In the
following, we shall write $\lesssim$ as long as the implicit constant does not
depend on $k$ and $\lambda_{k}$.

Choice of initial data. We now take

$\begin{split}\tilde{\phi}_{0}(x)=\sum_{k=k_{0}}^{\infty}\tilde{\phi}_{k,0}(x):=\sum_{k=k_{0}}^{\infty}\exp(-|\lambda_{k}|^{\frac{1}{4}})\tilde{\phi}_{k}^{app}(t=0,x),\end{split}$
(2.50)

which belongs to $C^{\infty}(\mathbb{T})$. By taking $k_{0}$ even larger if
necessary, we can guarantee that
$\|{\tilde{\phi}_{0}}\|_{C^{m}}<\frac{\epsilon}{2}$. Then we set
$\phi_{0}=f_{0}+\tilde{\phi}_{0},$ which verifies
$\|{\phi_{0}}\|_{C^{m}}<\epsilon$. Again from the contradiction hypothesis, we
have a $L^{\infty}_{t}C^{{s_{c}+1},1}$–solution $\phi(t)$ to (1.5) with
initial data $\phi_{0}$ on some time interval $[0,\delta^{\prime}]$. We may
assume that $0<\delta^{\prime}\leq\delta$, and define

$\begin{split}\tilde{\phi}(t):=\phi(t)-f(t),\quad\tilde{\phi}_{k}(t):=\chi_{k}\tilde{\phi}(t)\end{split}$

for all $k\geq k_{0}$. We have that
$\sum_{k=k_{0}}^{\infty}\tilde{\phi}_{k}=\tilde{\phi}$; this follows from
$\partial_{t}|f+\tilde{\phi}|\lesssim|f+\tilde{\phi}|$ and the uniform
pointwise estimate
$|f+\tilde{\phi}|(t,x)\lesssim|f_{0}+\tilde{\phi}_{0}|(x)\lesssim|f_{0}|(x).$
Then we see that $\tilde{\phi}_{k}$ solves

$\begin{split}[i\partial_{t}+\mathcal{L}_{f_{k}}]\tilde{\phi}_{k}=Q_{f_{k}}[\tilde{\phi}_{k}],\end{split}$

(which is (2.40) with $f$ and $\tilde{\phi}$ replaced with $f_{k}$ and
$\tilde{\phi}_{k}$, respectively). We note that the
$L^{\infty}([0,\delta^{\prime}];C^{s_{0}+1,1})$ norm is uniformly bounded for
$\\{f_{k}\\}_{k\geq k_{0}}$ and $\\{\tilde{\phi}_{k}\\}_{k\geq k_{0}}$.
Therefore, from Proposition 2.8, we obtain the estimate

$\begin{split}\|{\tilde{\phi}_{k}}\|_{L^{\infty}([0,\delta^{\prime}];L^{2}_{w})}\lesssim\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\end{split}$
(2.51)

uniformly in $k\geq k_{0}$. Now, combining this with the generalized energy
estimate (2.45) for $\tilde{\phi}_{k}$ and $\tilde{\phi}^{app}_{k}$, we obtain
that

$\begin{split}\left|\frac{\mathrm{d}}{\mathrm{d}t}\,\operatorname{Re}\langle{\tilde{\phi}_{k},\tilde{\phi}^{app}_{k}}\rangle_{w}\right|\lesssim
2^{N_{1}k^{2}}\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\|{g_{0}}\|_{H^{2}}\end{split}$
(2.52)

for $t\in[0,\delta_{k}]$. We shall now take $|\lambda_{k}|$ larger so that
$\delta_{k}=A_{k}^{-2}|\lambda_{k}|^{-\frac{1}{2}}$ satisfies
$2^{N_{1}k^{2}}\delta_{k}$ is very small with respect to the implicit
constants in (2.51) and (2.52). Then, since at $t=0$ we have

$\begin{split}\operatorname{Re}\langle{\tilde{\phi}_{k},\tilde{\phi}^{app}_{k}}\rangle_{w}(t=0)\geq\frac{1}{4}\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\|{\tilde{\phi}^{app}_{k,0}}\|_{L^{2}_{w_{0}}},\end{split}$

by integrating (2.52) in time from $t=0$ to $\delta_{k}$, we obtain

$\begin{split}\operatorname{Re}\langle{\tilde{\phi}_{k},\tilde{\phi}^{app}_{k}}\rangle_{w}(\delta_{k})\geq\frac{1}{8}\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\|{\tilde{\phi}^{app}_{k,0}}\|_{L^{2}_{w_{0}}}.\end{split}$
(2.53)

At $t=\delta_{k}$ we write

$\begin{split}\operatorname{Re}\langle{\tilde{\phi}_{k},\tilde{\phi}^{app}_{k}}\rangle_{w}(\delta_{k})=(-1)^{s_{c}}\operatorname{Re}\langle{\partial_{x}^{s_{c}}\tilde{\phi}_{k},\frac{|f|^{-\sigma_{c}+s_{c}-\frac{1}{2}}}{(i\lambda_{k})^{s_{c}}(1+|f|\partial_{x}S)^{s_{c}}}\psi^{app}_{k}}\rangle+\operatorname{Re}\langle{\tilde{\phi}_{k},\tilde{\phi}^{small}_{k}}\rangle_{w}\end{split}$

and then combining the estimates of the right hand side with (2.53), we get

$\begin{split}\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\|{\tilde{\phi}^{app}_{k,0}}\|_{L^{2}_{w_{0}}}\lesssim
2^{N_{1}k^{2}}\left(\|{\partial_{x}^{s_{c}}\tilde{\phi}_{k}}\|_{L^{\infty}}|\lambda_{k}|^{-s_{c}}\exp\left(-2|\lambda_{k}|^{\frac{1}{2}}(-\sigma_{c}+s_{c}+\frac{1}{2})\right)\|{g_{0}}\|_{L^{2}}+|\lambda_{k}|^{-1}\|{g_{0}}\|_{H^{s_{c}}}\right).\end{split}$

By taking $|\lambda_{k}|$ even larger if necessary, we can guarantee that
$|\lambda_{k}|^{-1}2^{N_{1}k^{2}}\|{g_{0}}\|_{H^{s_{c}}}\ll\|{\tilde{\phi}^{app}_{k,0}}\|_{L^{2}_{w_{0}}}$
holds ($\ll$ is defined in terms of the implicit constant in the previous
inequality), so that we deduce

$\begin{split}\|{\tilde{\phi}_{k,0}}\|_{L^{2}_{w_{0}}}\|{\tilde{\phi}^{app}_{k,0}}\|_{L^{2}_{w_{0}}}\lesssim
2^{N_{1}k^{2}}\|{\partial_{x}^{s_{c}}\tilde{\phi}_{k}}\|_{L^{\infty}}|\lambda_{k}|^{-s_{c}}\exp\left(-2|\lambda_{k}|^{\frac{1}{2}}(-\sigma_{c}+s_{c}+\frac{1}{2})\right)\|{g_{0}}\|_{L^{2}}\end{split}$

and then recalling the form of $\tilde{\phi}_{k,0}$ from (2.50),

$\begin{split}\|{\partial_{x}^{s_{c}}\tilde{\phi}_{k}}\|_{L^{\infty}}(t=\delta_{k})\gtrsim
2^{-N_{1}k^{2}}|\lambda_{k}|^{s_{c}}\exp\left(2|\lambda_{k}|^{\frac{1}{2}}(-\sigma_{c}+s_{c}+\frac{1}{2})-|\lambda_{k}|^{\frac{1}{4}}\right).\end{split}$

Note that the left hand side is bounded by
$\|{f}\|_{L^{\infty}([0,\delta^{\prime}];C^{s_{c}+1,1})}+\|{\phi}\|_{L^{\infty}([0,\delta^{\prime}];C^{s_{c}+1,1})}$
for all $k$ sufficiently large. This is a contradiction since the right hand
side diverges as $k\to\infty$. The proof is now complete. ∎

##  3 KdV-type equations

This section is organized as follows. After setting up some pieces of notation
in Section 3.1, we study the properties of regular cubically degenerate
solutions – typically denoted by $f$ – in Section 3.2. Then in Section 3.3, we
carry out the key construction of degenerating wave packets for the linearized
equation around $f$, and in Section 3.4, we establish a modified energy
estimate for the perturbation (solving the nonlinear difference equation)
around $f$. Finally, in Sections 3.5 and 3.6, we prove Theorems 1.4 and 1.5,
respectively.

###  3.1 Preliminaries

We introduce the following quantity defined for a $C^{2,1}$ function $f$ on an
interval $I$:

$\begin{split}{\|{f}\|_{Y(I)}=\|{f^{-\frac{2}{3}}\partial_{x}f}\|^{3}_{L^{\infty}(I)}+\|{f^{-\frac{1}{3}}\partial_{xx}f}\|_{L^{\infty}(I)}^{\frac{3}{2}}+\|{f}\|_{L^{\infty}(I)}+\|{\partial_{xxx}f}\|_{L^{\infty}(I)}.}\end{split}$

We shall write $f\in{Y(I)}$ if $\|{f}\|_{Y(I)}$ is finite. This quantity is
appropriate to handle solutions with degeneracies of order at least 3. For
convenience we set

$\begin{split}{\|{f}\|_{\widetilde{C}^{k,\alpha}(I)}=\|{f}\|_{C^{k,\alpha}(I)}+\|{f}\|_{Y(I)}.}\end{split}$

For $f$ depending on time, we say $f\in
L^{\infty}([0,\delta];\widetilde{C}^{k,\alpha}(I))$ (resp. $f\in
L^{\infty}([0,\delta];Y(I)$) if

$\begin{split}\|{f}\|_{L^{\infty}([0,\delta];\widetilde{C}^{k,\alpha}(I))}:=\sup_{t\in[0,\delta]}\|{f(t)}\|_{\widetilde{C}^{k,\alpha}(I)}<+\infty.\\\