# Deep-water limit of the intermediate long wave equation in $L^{2}$

Andreia Chapouto, Guopeng Li, Tadahiro Oh, and Didier Pilod  Andreia Chapouto,
School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Guopeng Li, School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Tadahiro Oh, School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Didier Pilod
Department of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen,
Norway<EMAIL_ADDRESS>
###### Abstract.

We study the well-posedness issue of the intermediate long wave equation (ILW)
on both the real line and the circle. By applying the gauge transform for the
Benjamin-Ono equation (BO) and adapting the $L^{2}$ well-posedness argument
for BO by Molinet and the fourth author (2012), we prove global well-posedness
of ILW in $L^{2}$ on both the real line and the circle. In the periodic
setting, this provides the first low regularity well-posedness of ILW. We then
establish convergence of the ILW dynamics to the BO dynamics in the deep-water
limit at the $L^{2}$-level.

###### Key words and phrases:

intermediate long wave equation; Benjamin-Ono equation; deep-water limit

###### 2020 Mathematics Subject Classification:

35Q35, 76B55

###### Contents

  1. 1 Introduction
  2. 2 Notations, function spaces, and preliminary estimates
  3. 3 Gauge transform and nonlinear estimates
    1. 3.1 Gauge transform
    2. 3.2 Product estimates
    3. 3.3 Nonlinear estimates
  4. 4 Well-posedness of ILW in $L^{2}$
    1. 4.1 A priori estimates for smooth solutions
    2. 4.2 Difference estimates
    3. 4.3 Well-posedness
  5. 5 Deep-water limit of ILW in $L^{2}$

##  1\. Introduction

In this paper, we study the intermediate long wave equation (ILW):

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

where $\mathcal{M}=\mathbb{R}$ or $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$.
The equation (1.1), introduced in [26, 35], models the internal wave
propagation of the interface in a stratified fluid of finite depth $\delta>0$,
and the unknown $u:\mathbb{R}\times\mathcal{M}\to\mathbb{R}$ denotes the
amplitude of the internal wave at the interface. See also Remark 1.1 in [38].
The operator $\mathcal{G}_{\delta}$ in (1.1) is defined by

$\displaystyle\mathcal{G}_{\delta}=\mathcal{T}_{\delta}-\delta^{-1}\partial_{x}^{-1},$
(1.2)

where $\mathcal{T}_{\delta}$ is given by

$\displaystyle\widehat{\mathcal{T}_{\delta}f}(\xi)=-i\coth(\delta\xi)\widehat{f}(\xi),\quad\xi\in\widehat{\mathcal{M}}.$
(1.3)

Here, $\widehat{\mathcal{M}}$ denotes the Pontryagin dual of $\mathcal{M}$,
i.e. $\widehat{\mathcal{M}}=\mathbb{R}$ if $\mathcal{M}=\mathbb{R}$, and
$\widehat{\mathcal{M}}=\mathbb{Z}$ if $\mathcal{M}=\mathbb{T}$, and $\coth$
denotes the usual hyperbolic cotangent function:

$\displaystyle\coth(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\frac{e^{2x}+1}{e^{2x}-1},\quad
x\in\mathbb{R}\setminus\\{0\\}$ (1.4)

with the convention $\coth(x)-\frac{1}{x}=0$ for $x=0$. Note that
$\mathcal{T}_{\delta}$ is the inverse of the so-called Tilbert transform (with
an extra $-$ sign), appearing in the study of water waves of finite depth [6,
19, 2]. In particular, $\mathcal{T}_{\delta}$ is related to the Dirichlet-to-
Neumann map in a two-dimensional finite depth two-layer fluid, and in the
deep-water limit ($\delta\to\infty$), it converges to the Hilbert transform
$\mathcal{H}$ (with the Fourier multiplier $-i\textup{sgn}(\xi)$),
corresponding to the Dirichlet-to-Neumann map in infinite depth, which
indicates possible convergence of ILW (1.1) to the Benjamin-Ono equation (BO):

$\displaystyle\partial_{t}u-\mathcal{H}\partial_{x}^{2}u=\partial_{x}(u^{2}).$
(1.5)

Indeed, ILW (1.1) is an important physical model, providing a natural
connection between the deep-water regime (= the Benjamin-Ono regime) and the
shallow-water regime (= the KdV regime). As such, it has been studied
extensively from both the applied and theoretical points of view. See, for
example, a recent book [32, Chapter 3] by Klein and Saut for an overview of
the subject and the references therein. See also a survey [48]. While the ILW
equation has attracted increasingly more attention in recent years (see, for
example, [45, 30, 46, 36, 37, 38, 21, 12]), these two references indicate that
the rigorous mathematical study of ILW is still widely open, especially as
compared to BO and KdV. In particular, one of the fundamental, but challenging
questions is the convergence properties of ILW in the deep-water limit
($\delta\to\infty$) and in the shallow-water limit ($\delta\to 0$). See [36,
37, 38, 12] for recent progress on convergence problems from both the
deterministic and statistical points of view.

Our main goal in this paper is to present a unified argument for proving
$L^{2}$ global well-posedness of ILW (1.1) on both the real line and the
circle and then to establish convergence of the ILW dynamics to the BO
dynamics in the deep-water limit at the $L^{2}$-level. Before stating our main
result, let us recall a few conservation laws for ILW (1.1):

$\displaystyle\begin{split}\text{mean:
}&\int_{\mathcal{M}}udx,\qquad\text{mass: }M(u)=\int_{\mathcal{M}}u^{2}dx,\\\
\text{Hamiltonian:
}&E_{\delta}(u)=\frac{1}{2}\int_{\mathcal{M}}u\mathcal{G}_{\delta}\partial_{x}udx+\frac{1}{3}\int_{\mathcal{M}}u^{3}dx.\end{split}$
(1.6)

In particular, ILW (1.1) is a Hamiltonian PDE. Furthermore, it is known to be
completely integrable. We, however, do not make use of the completely
integrable structure in this paper.

We now state our first result.

######  Theorem 1.1.

Let $\mathcal{M}=\mathbb{R}$ or $\mathbb{T}$ and $0<\delta<\infty$. Then,
given any $s\geq 0$, ILW (1.1) is globally well-posed in $H^{s}(\mathcal{M})$.
Moreover, when $\mathcal{M}=\mathbb{T}$, by restricting to the subspace
$H^{s}_{0}(\mathbb{T})$ consisting of mean-zero functions, the solution map
for (1.1) is locally Lipschitz continuous from $H^{s}_{0}(\mathbb{T})$ to
$C(\mathbb{R};H^{s}_{0}(\mathbb{T}))$, where the latter is endowed with the
compact-open topology (in time).

In [44], it was shown that both BO and ILW are quasilinear in the sense that
the solution map is not smooth; see also [34]. This in particular implies that
we can not use a contraction argument for constructing solutions, making the
well-posedness question rather challenging. Nonetheless, the well-posedness
issue of the BO equation (1.5), corresponding to the infinite depth case
($\delta=\infty$), has been studied extensively on both the real line and the
circle; see [25, 33, 49, 23, 40, 10, 41, 42, 45, 22, 16, 31]. On the other
hand, there are only a handful of known well-posedness results for ILW (1.1);
see [1, 45, 43, 21]. In [45], Molinet and Vento proved local well-posedness of
(1.1) in $H^{\frac{1}{2}}(\mathcal{M})$, $\mathcal{M}=\mathbb{R}$ or
$\mathbb{T}$, via the improved energy method, which immediately extended to
global well-posedness thanks to the conservation laws in (1.6). In the real
line case, Molinet, Vento, and the fourth author [43] refined the improved
energy method in [45] (with the short-time Strichartz estimates as in [33, 27,
39] and a generalized Coifman-Meyer theorem) and proved local well-posedness
of (1.1) in $H^{s}(\mathbb{R})$ for $s>\frac{1}{4}$. In a recent work [21],
Ifrim and Saut studied (1.1) on the real line and proved its global well-
posedness in $L^{2}(\mathbb{R})$ (along with long-time behavior of small
solutions), based on an adaptation of the argument developed for the BO
equation (1.5) by Ifrim and Tataru [22] which combines the BO gauge transform
and the (quasilinear) norm form method. We point out that the argument in [21]
relies on dispersive estimates and thus is not applicable to the periodic
setting.

Our main strategy is to view (1.1) as a perturbation111after applying the
Galilean transform (3.1) to remove the term $\delta^{-1}\partial_{x}^{-1}$ in
(1.2). See (3.2). of (1.5) as in [21] and apply the BO gauge transform,
originally introduced by Tao [49]. We then adapt the argument in [42], where
Molinet and the fourth author introduced a simplified and unified approach to
the $L^{2}$ well-posedness of the BO equation (1.5) on both the circle and
real line, to the current ILW problem. See Sections 3 and 4 for details.

Next, we state our second result on convergence in the deep-water limit.

######  Theorem 1.2.

Let $\mathcal{M}=\mathbb{R}$ or $\mathbb{T}$ and $s\geq 0$. Given $u_{0}\in
H^{s}(\mathcal{M})$, let $u$ be the global solution to the BO equation (1.5)
with $u|_{t=0}=u_{0}$ constructed in [42]. Let
$\\{u_{0,\delta}\\}_{1\leq\delta<\infty}\subset H^{s}(\mathcal{M})$ such that
$u_{0,\delta}$ converges to $u_{0}$ in $H^{s}(\mathcal{M})$ as
$\delta\to\infty$, and let $u_{\delta}$ be the global solution to (1.1) with
$u_{\delta}|_{t=0}=u_{0,\delta}$ constructed in Theorem 1.1. Then, as
$\delta\to\infty$, $u_{\delta}$ converges to $u$ in
$C(\mathbb{R};H^{s}(\mathcal{M}))$, endowed with the compact-open topology (in
time).

Theorem 1.2 follows from a minor adaptation of the proof of Theorem 1.1. See
Section 5.

In [36], the second author proved the deep-water convergence on both the real
line and the circle for $s>\frac{1}{2}$. In view of the known results [45,
36], we restrict our attention to the range $0\leq s\leq\frac{1}{2}$ in the
remaining part of the paper. Furthermore, we present the proofs of Theorems
1.1 and 1.2 only in the periodic case, which is closely related to the study
of invariant measures; see Remark 1.5. The required modification for the real
line case is minor and straightforward in view of the presentation of the
current paper and [42], where the latter treated the real line case. In the
periodic setting, in order to avoid the problem at the zeroth frequency, it is
crucial to work with mean-zero functions. In view of the conservation of the
spatial mean, we perform a change of unknowns via the following Galilean
transform [14] (with $\mu(u_{0})=\frac{1}{2\pi}\int_{\mathbb{T}}u_{0}(x)dx$):

$\displaystyle\widetilde{u}(t,x)=\Gamma_{u_{0}}(u)(t,x)=u(t,x-2\mu(u_{0})t)-\mu(u_{0})$
(1.7)

to reduce the problem to the mean-zero case in proving Theorem 1.1. Lastly,
for simplicity of the presentation, we restrict our attention to positive
times in the remaining part of the paper.

######  Remark 1.3.

(i) The local Lipschitz continuity the solution map for (1.1) on $\mathbb{T}$
claimed in Theorem 1.1 indeed holds on any hyperplane of $H^{s}(\mathbb{T})$
of functions with a fixed mean. In [41, Theorem 1.2], Molinet proved that the
solution map to (1.5) on $H^{s}_{0}(\mathbb{T})$ is in fact real-analytic for
$s\geq 0$ (and fails to be smooth for $s<0$). While we expect the same results
to hold for ILW (1.1), we do not pursue this issue in this paper.

(ii) Proceeding as in [42], we can prove unconditional uniqueness of ILW (1.1)
in $H^{s}(\mathbb{R})$ for $\frac{1}{4}<s\leq\frac{1}{2}$ and
$H^{\frac{1}{2}}(\mathbb{T})$. (For $s>\frac{1}{2}$, it was shown in [45] on
both the real line and the circle.) Since the required modification is
straightforward, we omit details.

######  Remark 1.4.

Given $\delta>0$, consider the following scaling transformation [1]:

$\displaystyle v(t,x)=\tfrac{3}{\delta}u\big{(}\tfrac{3}{\delta}t,x\big{)},$
(1.8)

which transforms (1.1) to the following scaled ILW equation:

$\displaystyle\partial_{t}v-\frac{3}{\delta}\mathcal{G}_{\delta}\partial_{x}^{2}v=\partial_{x}(v^{2}).$
(1.9)

Then, under suitable assumptions, the scaled ILW (1.9) is known to converge to
the KdV equation in the shallow-water limit ($\delta\to 0$) [1, 36, 38]:

$\displaystyle\partial_{t}v+\partial_{x}^{3}v=\partial_{x}(v^{2}).$

It would be of interest to study the shallow-water convergence of ILW (1.1) at
the $L^{2}$-level. As mentioned above, our argument for proving Theorem 1.1
relies on the Galilean transform (3.1) to remove
$\delta^{-1}\partial_{x}^{-1}$ in (1.2) (which is crucial in the shallow-water
limit) which is not amenable to the shallow-water limit since the spatial
translation by $\delta^{-1}t$ diverges as $\delta\to 0$.

We point out that the scaling transformation (1.8) is consistent with the
physical setting. The ILW equation (1.1) describes the motion of the fluid
interface in a stratified fluid of depth $\delta>0$, where $u$ denotes the
amplitude of the internal wave at the interface. As $\delta\to 0$, the entire
fluid depth tends to $0$ and, in particular, the amplitude of the internal
wave at the interface is $O(\delta)$, which also tends to $0$. Hence, if we
want to observe any meaningful limiting behavior, we need to magnify the fluid
motion by a factor $\sim\frac{1}{\delta}$, which is exactly what the scaling
transformation (1.8) does.

######  Remark 1.5.

The ILW equation (1.1) and the BO equation (1.5) are known to be completely
integrable. In particular, they possess infinite sequences of conservation
laws $E_{\delta,\frac{k}{2}}(u)$ and $E_{\text{\rm BO},\frac{k}{2}}(u)$ at
regularities $\frac{k}{2}$, $k\in\mathbb{N}\cup\\{0\\}$. In [38, 12], the
first three authors (with Zheng) constructed the (formally invariant) measures
$\rho_{\delta,\frac{k}{2}}$ (with a density of the form
$\exp(-E_{\delta,\frac{k}{2}}(u))$ associated with these conservation laws for
the periodic ILW on $\mathbb{T}$, $k\in\mathbb{N}$, and proved their
convergence to the corresponding measures for the periodic BO, constructed by
Tzvetkov and Visciglia [51], in the deep-water limit.222In [38, 12],
convergence in the shallow-water limit was also studied, but we restrict our
discussion to the deep-water limit here. Moreover, when $k\geq 3$, they
established invariance of these measures $\rho_{\delta,\frac{k}{2}}$ under the
ILW dynamics (for each fixed $0<\delta<\infty$) and convergence of the
associated invariant ILW dynamics to the corresponding invariant BO dynamics
[52, 53] in the deep-water limit.

It is known that $\rho_{\delta,\frac{k}{2}}$ is supported on
$H^{\frac{k}{2}-\varepsilon}(\mathbb{T})\setminus H^{\frac{k}{2}}(\mathbb{T})$
for any $\varepsilon>0$. In particular, when $k=2$, the measure
$\rho_{\delta,1}$ is supported slightly outside $H^{\frac{1}{2}}(\mathbb{T})$,
where Theorem 1.1 now provides globally well-defined dynamics. Therefore, the
$k=2$ case is now within reach. In a forthcoming work, we will treat the $k=2$
case: invariance of the measure $\rho_{\delta,1}$ under the ILW dynamics (for
each fixed $0<\delta<\infty$) and deep-water convergence of the invariant
$\rho_{\delta,1}$-dynamics to the corresponding invariant BO dynamics due to
Deng, Tzvetkov, and Visciglia [15].

######  Remark 1.6.

In a recent preprint [11], with J. Forlano, we further exploited the idea of
viewing ILW (1.1) as a perturbation of BO (1.5) and established a global-in-
time a priori bound on the $H^{s}$-norm of a solution to (1.1) for
$s>-\frac{1}{2}$. More precisely, we showed that there exists an (explicit)
non-decreasing function
$C_{s}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that

$\|u(t)\|_{H^{s}}\leq C_{s}\big{(}T,\|u_{0}\|_{H^{s}}\big{)}$

for any finite $T>0$ and any smooth solution $u$ to (1.1) on
$\mathcal{M}=\mathbb{R}$ or $\mathbb{T}$, provided that $s>-\frac{1}{2}$. In
the same work, we also proved ill-posedness of (1.1) in $H^{s}(\mathcal{M})$
for $s<-\frac{1}{2}$, thus establishing sharpness (of the regularity range) of
the aforementioned a priori bound (modulo the endpoint $s=-\frac{1}{2}$,
corresponding to the “critical” regularity for ILW; see Remark 4.2 below for a
discussion on the scaling property of ILW).

##  2\. Notations, function spaces, and preliminary estimates

By $A\lesssim B$, we mean $A\leq CB$ for some constant $C>0$. We use $A\sim B$
to mean $A\lesssim B$ and $B\lesssim A$. We write $A\ll B$, if there is some
small $c>0$, such that $A\leq cB$.

Let $s\in\mathbb{R}$ and $1\leq p\leq\infty$. We define the $L^{2}$-based
Sobolev space $H^{s}(\mathbb{T})$ by the norm:

$\displaystyle\|f\|_{H^{s}}=\|\langle
n\rangle^{s}\widehat{f}(n)\|_{\ell^{2}_{n}},$

where $\langle x\rangle=(1+|x|^{2})^{\frac{1}{2}}$, and we set
$H^{s}_{0}(T)=\big{\\{}f\in H^{s}(\mathbb{T}):\widehat{f}(0)=0\big{\\}}$ as
the subspace consisting of mean-zero functions. We also define the
$L^{p}$-based Sobolev space $W^{s,p}(\mathbb{T})$ by the norm:

$\displaystyle\|f\|_{W^{s,p}}=\|\langle
D\rangle^{s}f\|_{L^{p}}=\big{\|}\mathcal{F}^{-1}(\langle
n\rangle^{s}\widehat{f}(n))\big{\|}_{L^{p}},$

where $D=-i\partial_{x}$ (with the Fourier multiplier $n$) and
$\mathcal{F}^{-1}$ denotes the inverse Fourier transform. When $p=2$, we have
$H^{s}(\mathbb{T})=W^{s,2}(\mathbb{T})$. In dealing with space-time function
spaces, we use short-hand notations such as $L^{p}_{T}H^{s}_{x}$ for
$L^{p}([0,T];H^{s}(\mathbb{T}))$.

Let $\eta:\mathbb{R}\to[0,1]$ be a smooth bump function supported on
$[-\frac{8}{5},\frac{8}{5}]$ and $\eta\equiv 1$ on
$\big{[}-\frac{5}{4},\frac{5}{4}\big{]}$. For $\xi\in\mathbb{R}$, we set
$\varphi_{0}(\xi)=\eta(|\xi|)$ and

$\displaystyle\varphi_{j}(\xi)=\eta\big{(}\tfrac{|\xi|}{2^{j}}\big{)}-\eta\big{(}\tfrac{|\xi|}{2^{j-1}}\big{)}$

for $j\in\mathbb{N}$. Then, for $j\in\mathbb{Z}_{\geq
0}:=\mathbb{N}\cup\\{0\\}$, we define the Littlewood-Paley projector
$\mathbf{Q}_{j}$ as the Fourier multiplier operator with a symbol
$\varphi_{j}$. Note that we have

$\displaystyle\sum_{j=0}^{\infty}\varphi_{j}(\xi)=1$

for each $\xi\in\mathbb{R}$. Thus, we have the following Littlewood-Paley
decomposition:

$f=\sum_{j=0}^{\infty}\mathbf{Q}_{j}f.$

Define the projectors $\mathbf{P}_{+}$ and $\mathbf{P}_{-}$ by setting

$\mathbf{P}_{+}f=\mathcal{F}^{-1}(\mathbf{1}_{\\{n\geq
1\\}}\widehat{f}(n))\qquad\text{and}\qquad\mathbf{P}_{-}f=\mathcal{F}^{-1}(\mathbf{1}_{\\{n\leq-1\\}}\widehat{f}(n)).$

We also define $\mathbf{P}_{\textup{hi}}$, $\mathbf{P}_{\textup{HI}}$,
$\mathbf{P}_{\textup{lo}}$, and $\mathbf{P}_{\textup{LO}}$ by setting

$\displaystyle\mathbf{P}_{\textup{hi}}=\sum_{j=1}^{\infty}\mathbf{Q}_{j},\quad\mathbf{P}_{\textup{HI}}=\sum_{j=3}^{\infty}\mathbf{Q}_{j},\quad\mathbf{P}_{\textup{lo}}=\textup{Id}-\mathbf{P}_{\textup{hi}},\quad\text{and}\quad\mathbf{P}_{\textup{LO}}=\textup{Id}-\mathbf{P}_{\textup{HI}},$
(2.1)

and set
$\mathbf{P}_{\pm,\textup{hi}}=\mathbf{P}_{\pm}\mathbf{P}_{\textup{hi}}$, etc.

Next, we recall the $X^{s,b}$-spaces introduced by Bourgain [7, 8] and their
variants defined by the following norms:

$\displaystyle\|u\|_{X^{s,b}}$ $\displaystyle=\|\langle
n\rangle^{s}\langle\tau-|n|n\rangle^{b}\widehat{u}(\tau,n)\|_{\ell^{2}_{n}L^{2}_{\tau}},$
$\displaystyle\|u\|_{Z^{s,b}}$ $\displaystyle=\|\langle
n\rangle^{s}\langle\tau-|n|n\rangle^{b}\widehat{u}(\tau,n)\|_{\ell^{2}_{n}L^{1}_{\tau}},$
$\displaystyle\|u\|_{\widetilde{Z}^{s,b}}$
$\displaystyle=\bigg{(}\sum_{j=0}^{\infty}\|\mathbf{Q}_{j}u\|_{Z^{s,b}}\bigg{)}^{\frac{1}{2}},$
$\displaystyle\|u\|_{Y^{s,b}}$
$\displaystyle=\|u\|_{X^{s,b}}+\|u\|_{\widetilde{Z}^{s,b-\frac{1}{2}}}.$

See [7, 8, 29, 13] for basic properties. Note that we have
$\widetilde{Z}^{s,b}\subset Z^{s,b}$. For $T>0$, we define the space
$X^{s,b}(T)$ to be the restriction of the $X^{s,b}$-space onto the time
interval $[0,T]$ via the norm:

$\|u\|_{X^{s,b}(T)}:=\inf\big{\\{}\|v\|_{X^{s,b}}:v|_{[0,T]}=u\big{\\}}.$
(2.2)

We also define the restriction spaces $\widetilde{Z}^{s,b}(T)$ and
$Y^{s,b}(T)$ in an analogous manner. Note that $Y^{s,b}(T)$ is complete.
Recall the following embedding:

$\displaystyle Y^{s,\frac{1}{2}}(T)\subset Z^{s,0}(T)\subset
C([0,T];H^{s}(\mathbb{T})).$ (2.3)

Let $S(t)$ denote the linear propagator for the BO equation (1.5) defined by

$\displaystyle\widehat{S(t)f}(n)=e^{it|n|n}\widehat{f}(n).$

Then, we have the following linear estimates; see [7, 29, 50, 42].

######  Lemma 2.1.

Let $s\in\mathbb{R}$. Then, we have

$\displaystyle\|\eta(t)S(t)f\|_{Y^{s,\frac{1}{2}}}$
$\displaystyle\lesssim\|f\|_{H^{s}},$
$\displaystyle\bigg{\|}\eta(t)\int_{0}^{t}S(t-t^{\prime})F(t^{\prime})dt^{\prime}\bigg{\|}_{Y^{s,\frac{1}{2}}}$
$\displaystyle\lesssim\|F\|_{Y^{s,-\frac{1}{2}}}.$

Next, we recall the periodic $L^{4}$-Strichartz estimate due to Bourgain [7];
see also [40, (6)].

######  Lemma 2.2.

The following estimate holds:

$\displaystyle\|u\|_{L^{4}_{t,x}}\lesssim\|u\|_{X^{0,\frac{3}{8}}}\lesssim\|u\|_{Y^{0,\frac{1}{2}}}.$

For $0<\delta\leq\infty$, define $\mathcal{Q}_{\delta}$ by

$\displaystyle\mathcal{Q}_{\delta}=(\mathcal{T}_{\delta}-\mathcal{H})\partial_{x}$
(2.4)

with the understanding that $\mathcal{Q}_{\infty}=0$ (corresponding to the BO
equation (1.5)), where $\mathcal{T}_{\delta}$ is as in (1.3). The operator
$\mathcal{Q}_{\delta}$ naturally appears in viewing ILW as a perturbation of
the BO equation; see (3.2) and (3.6) below. The next lemma establishes a
smoothing property of the operator $\mathcal{Q}_{\delta}$; see also Lemma 2.2
in [21].

######  Lemma 2.3.

Let $s\geq 0$. Then, we have

$\displaystyle\|\mathcal{Q}_{\delta}f\|_{H^{s}}\lesssim\delta^{-1}(1+\delta^{-s})\|f\|_{L^{2}}.$
(2.5)

In particular, we have

$\displaystyle\|\mathcal{Q}_{\delta}f\|_{H^{s}}\lesssim\|f\|_{L^{2}},$

uniformly in $\delta\geq 1$.

###### Proof.

With a slight abuse of notation, let $\widehat{\mathcal{Q}}_{\delta}(n)$
denotes the multiplier for the operator $\mathcal{Q}_{\delta}$:

$\displaystyle\widehat{\mathcal{Q}}_{\delta}(n)=n\big{(}\coth(\delta
n)-\textup{sgn}(n)\big{)}.$ (2.6)

From (1.4), we have

$\displaystyle\coth(\delta
n)-\textup{sgn}(n)=\textup{sgn}(n)\cdot\frac{2}{e^{2|\delta n|}-1}$ (2.7)

for $n\neq 0$. Then, by noting that $x^{s+1}\leq C_{s}(e^{2x}-1)$ for any
$x\geq 0$, it follows from (2.6) and (2.7) that

$\displaystyle\sup_{n\in\mathbb{Z}\setminus\\{0\\}}\langle
n\rangle^{s}|\widehat{\mathcal{Q}}_{\delta}(n)|\lesssim\delta^{-1}(1+\delta^{-s}),$

from which (2.5) follows. ∎

We now recall the following fractional Leibniz rule in the periodic setting;
see [5]. See also [24, Lemma 9.A.2] and [17, Lemma 3.4].

######  Lemma 2.4.

Let $s>0$ and $1<p_{j},q_{j},r\leq\infty$, $j=1,2$, such that
$\frac{1}{r}=\frac{1}{p_{j}}+\frac{1}{q_{j}}$. Then, we have

$\displaystyle\|J^{s}(fg)\|_{L^{r}(\mathbb{T})}\lesssim\|J^{s}f\|_{L^{p_{1}}(\mathbb{T})}\|g\|_{L^{q_{1}}(\mathbb{T})}+\|f\|_{L^{p_{2}}(\mathbb{T})}\|J^{s}g\|_{L^{q_{2}}(\mathbb{T})},$
(2.8)

where $J^{s}=\langle\partial_{x}\rangle^{s}$ denotes the Bessel potential of
order $-s$.

Note that the estimate (2.8) holds for $\frac{1}{2}\leq r\leq 1$ as well (with
additional assumptions such as $s>\frac{1}{r}-1$ or $s\in 2\mathbb{N}$); see
[5].

We conclude this section by recalling the basic property of the translation
operator $\tau$ defined by

$\displaystyle\tau_{h}(u)(t,x)=\tau(h,u)(t,x):=u(t,x+ht).$ (2.9)

######  Lemma 2.5.

Let $T>0$ and $s\geq 0$. Then, $\tau$ defined in (2.9) is continuous from
$\mathbb{R}\times C([0,T];H^{s}(\mathbb{T}))$ into
$C([0,T];H^{s}(\mathbb{T}))$, but is not uniformly continuous in every
neighborhood of the origin $(0,0)$. For fixed $h\in\mathbb{R}$, $\tau_{h}$ is
an isometry on $C([0,T];H^{s}(\mathbb{T}))$ (in particular, Lipschitz
continuous).

See [20, Propositions 3.2.1 and 3.2.2] for the first two claims. The second
claim is clear from
$\|\tau_{h}u(t)-\tau_{h}v(t)\|_{H^{s}_{x}}=\|u(t)-v(t)\|_{H^{s}_{x}}$ for any
$t\in\mathbb{R}$.

##  3\. Gauge transform and nonlinear estimates

In this section, by viewing ILW (1.1) as a perturbation of BO (1.5), we first
apply the gauge transform for BO to ILW. We then follow the argument in [42]
and establish a priori estimates.

###  3.1. Gauge transform

By applying the Galilean transform

$\displaystyle v(t,x)=\tau_{\delta^{-1}}(u)(t,x)=u(t,x+\delta^{-1}t),$ (3.1)

where $\tau_{\delta^{-1}}=\tau(\delta^{-1},\,\cdot\,)$ is as in (2.9), we can
rewrite (1.1) as the following renormalized ILW:

$\displaystyle\partial_{t}v-\mathcal{T}_{\delta}\partial_{x}^{2}v=\partial_{x}(v^{2})$
(3.2)

with $v|_{t=0}=u_{0}$, where $\mathcal{T}_{\delta}$ is as in (1.3).

Following the recent work [21], our main strategy is to view (3.2) as a
perturbation of the BO equation (1.5) as follows:

$\displaystyle\partial_{t}v-\mathcal{H}\partial_{x}^{2}v=\mathcal{Q}_{\delta}\partial_{x}v+\partial_{x}(v^{2}),$
(3.3)

where $\mathcal{Q}_{\delta}$ is as in (2.4), and apply the gauge transform for
the BO equation (1.5) (see (3.10) below) to (3.3). Lemma 2.3 shows that
$\mathcal{Q}_{\delta}$ has a smoothing property of any order, which allows us
to indeed view the first term on the right-hand side of (3.3) as a
perturbation in a suitable sense.

In a seminal work [49], Tao introduced a gauge transform, analogous to the
Hopf-Cole transform for the heat equation, in studying low regularity well-
posedness of BO (1.5). Let us now recall the BO gauge transform in the
periodic setting [41, 42] and apply it to (3.3). Let $F=F[v]$ be the mean-zero
primitive of $v$ defined by

$\displaystyle F=F[v]=\partial_{x}^{-1}\mathbf{P}_{\neq 0}v,$ (3.4)

where $\partial_{x}^{-1}$ denotes the Fourier multiplier operator with symbol
$\mathbf{1}_{\\{n\neq 0\\}}\cdot(in)^{-1}$ and $\mathbf{P}_{\neq 0}$ denotes
the projection onto the non-zero Fourier modes. In particular, for a mean-zero
function $v$, we have

$\displaystyle v=\mathbf{P}_{\neq 0}v=\partial_{x}F.$ (3.5)

By noting $\partial_{x}^{-1}\partial_{x}=\mathbf{P}_{\neq 0}$, it is easy to
see that, if $v$ is a solution to (3.3), then $F=F[v]$ satisfies the following
equation:

$\displaystyle\partial_{t}F-\mathcal{H}\partial_{x}^{2}F=\mathcal{Q}_{\delta}v+v^{2}-\mathbf{P}_{0}(v^{2}),$
(3.6)

where $\mathbf{P}_{0}=\textup{Id}-\mathbf{P}_{\neq 0}$ denotes the projection
onto the zeroth Fourier coefficient.

######  Remark 3.1.

For $j=1,2$, let $v_{j}$ be a function $C([0,T];L^{2}(\mathbb{T}))$ such that
$\widehat{v}_{j}(t,0)=0$ for any $t\in[0,T]$. Then, by letting $F_{j}$ be the
mean-zero primitive of $v_{j}$ defined by
$F_{j}=\partial_{x}^{-1}\mathbf{P}_{\neq 0}v_{j}$, it follows from Bernstein’s
inequality that

$\displaystyle\|F_{1}(0)-F_{2}(0)\|_{L^{\infty}}\lesssim\|v_{1}(0)-v_{2}(0)\|_{L^{2}}$
(3.7)

and

$\displaystyle\|F_{1}-F_{2}\|_{L^{\infty}_{T,x}}\lesssim\|v_{1}-v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}.$
(3.8)

These bounds allow us to prove local Lipschitz continuity of the solution map
to (3.2) in $H^{s}_{0}(\mathbb{T})$. We point out that the bounds (3.7) and
(3.8) do not hold on the real line due to the low-frequency issue. In order to
overcome this difficulty in the real line case, an extra assumption on the
low-frequency part of initial data was imposed in Section 4 of [42], namely,

$\displaystyle\mathbf{P}_{\textup{LO}}v_{1}(0)=\mathbf{P}_{\textup{LO}}v_{2}(0),$
(3.9)

See [42, Lemma 4.1], where the bounds (3.7) and (3.8) are established in the
real line case under the extra assumption (3.9).

We now introduce the gauge transform (for the periodic BO equation):333In the
current periodic setting, it is possible to define a gauge transform with
$\mathbf{P}_{+}$ instead of $\mathbf{P}_{+,\textup{hi}}$ (as in [40, 41]) and
carry out the analysis presented in this paper. We, however, keep the
definition of the gauge transform (3.10) with $\mathbf{P}_{+,\textup{hi}}$ so
that it is easier to compare the discussion in this paper with the
presentation in [42].

$\displaystyle W=\mathbf{P}_{+,\textup{hi}}(e^{iF}).$ (3.10)

By noting

$\displaystyle\textup{Id}=\mathbf{P}_{+,\textup{hi}}+\mathbf{P}_{\textup{lo}}+\mathbf{P}_{-,\textup{hi}}\qquad\text{and}\qquad\mathbf{P}_{+,\textup{hi}}(\mathbf{P}_{-,\textup{hi}}f\cdot\mathbf{P}_{-}g)=0,$

a direction computation with (3.10), (3.6), and (3.5) yields

$\displaystyle\begin{split}\partial_{t}W-\mathcal{H}\partial_{x}^{2}W&=\partial_{t}W+i\partial_{x}^{2}W\\\
&=i\mathbf{P}_{+,\textup{hi}}\big{(}e^{iF}(\partial_{t}F+i\partial_{x}^{2}F-(\partial_{x}F)^{2})\big{)}\\\
&=-2\mathbf{P}_{+,\textup{hi}}(W\mathbf{P}_{-}\partial_{x}v)-2\mathbf{P}_{+,\textup{hi}}(\mathbf{P}_{\textup{lo}}e^{iF}\mathbf{P}_{-}\partial_{x}v)\\\
&\quad+i\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}v)-i\mathbf{P}_{0}(v^{2})W.\end{split}$
(3.11)

Finally, we define $w=w(v)$ by setting

$\displaystyle
w=\partial_{x}W=\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF})=i\mathbf{P}_{+,\textup{hi}}(e^{iF}v).$
(3.12)

By differentiating (3.11), we then obtain

$\displaystyle\begin{split}\partial_{t}w-\mathcal{H}\partial_{x}^{2}w&=-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(W\mathbf{P}_{-}\partial_{x}v)-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(\mathbf{P}_{\textup{lo}}e^{iF}\mathbf{P}_{-}\partial_{x}v)\\\
&\quad+i\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}v)-i\mathbf{P}_{0}(v^{2})w\\\
&=:\mathcal{N}_{\delta}(w,v).\end{split}$ (3.13)

As compared to [42, (3-4)] (modulo constants), we have two extra terms on the
right-hand side of (3.13). The third term appears from the difference between
the ILW and BO equations, whereas the fourth term appears due to the current
periodic setting.

Let us conclude this discussion by expressing $v$ in terms of the gauged
function $w=w(v)$. From (3.5) and (3.12) with we can write $v$ as

$\displaystyle\begin{split}v&=\partial_{x}F=e^{-iF}\cdot
e^{iF}\partial_{x}F=-ie^{-iF}\cdot\partial_{x}e^{iF}\\\
&=-ie^{-iF}w+e^{-iF}\mathbf{P}_{\textup{lo}}(e^{iF}v)-ie^{-iF}\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF}.\end{split}$

By applying $\mathbf{P}_{+,\textup{HI}}$, we then obtain

$\displaystyle\begin{split}\mathbf{P}_{+,\textup{HI}}v&=-i\mathbf{P}_{+,\textup{HI}}(e^{-iF}w)+\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{hi}}e^{-iF}\cdot\mathbf{P}_{\textup{lo}}(e^{iF}v)\big{)}\\\
&\quad-i\mathbf{P}_{+,\textup{HI}}(\mathbf{P}_{+,\textup{HI}}e^{-iF}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF}),\end{split}$
(3.14)

where we have used
$\mathbf{P}_{+,\textup{HI}}\big{(}(\textup{Id}-\mathbf{P}_{+,\textup{hi}})f\cdot\mathbf{P}_{\textup{lo}}g\big{)}=\mathbf{P}_{+,\textup{HI}}\big{(}(\textup{Id}-\mathbf{P}_{+,\textup{HI}})f\cdot\mathbf{P}_{-,\textup{hi}}f\big{)}=0$.

###  3.2. Product estimates

We first recall the following product estimate; see [40, Lemma 3.2].

######  Lemma 3.2.

Let $s,s_{1},s_{2}\geq 0$ and $1<q,q_{1},q_{2}<\infty$ such that $s_{1}\geq
s$, $1+s=s_{1}+s_{2}$, and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.
Then, we have

$\displaystyle\|D^{s}\mathbf{P}_{+}(f\mathbf{P}_{-}\partial_{x}g)\|_{L^{q}}\lesssim\|D^{s_{1}}f\|_{L^{q_{1}}}\|D^{s_{2}}g\|_{L^{q_{2}}}.$

Next, we recall the following lemma on the multiplication with $e^{\pm iF}$ in
Sobolev spaces; see [42, Lemma 2.7]. See also [40, Lemma 3.1].

######  Lemma 3.3.

Let $2\leq q<\infty$ and $0\leq s\leq\frac{1}{2}$. For $j=1,2$, let $F_{j}$
denote the mean-zero primitive of a mean-zero real-valued function $f_{j}$ on
$\mathbb{T}$ such that $\partial_{x}F_{j}=f_{j}$. Then, we have

$\displaystyle\|J^{s}(e^{\pm
iF_{j}}g)\|_{L^{q}}\lesssim\big{(}1+\|f_{j}\|_{L^{2}}\big{)}\|J^{s}g\|_{L^{q}}$

and

$\displaystyle\big{\|}J^{s}\big{(}(e^{\pm iF_{1}}-e^{\pm
iF_{2}})g\big{)}\big{\|}_{L^{q}}$
$\displaystyle\quad\lesssim\Big{(}\|f_{1}-f_{2}\|_{L^{2}}+\|e^{\pm
iF_{1}}-e^{\pm
iF_{2}}\|_{L^{\infty}}\big{(}1+\|f_{1}\|_{L^{2}}\big{)}\Big{)}\|J^{s}g\|_{L^{q}}.$

In [42, Lemma 2.7], Lemma 3.3 was proven for $0\leq s\leq\frac{1}{q}$ but
their proof easily extends to the range $0\leq s\leq\frac{1}{2}$. In [42, the
last line on p. 372], the condition $s=\frac{1}{q}-\frac{1}{q_{1}}$ was used
to control $\|g\|_{L^{q_{1}}}$ by $\|J^{s}g\|_{L^{q}}$, which yielded the
restriction $s\leq\frac{1}{q}$. We can, however, use Sobolev’s inequality with
$s\geq\frac{1}{q}-\frac{1}{q_{1}}$; see [3]. In repeating the proof of [42,
Lemma 2.7], the restriction $s\leq\frac{1}{2}$ appears in the bound
$\|D^{s+\frac{1}{2}}\mathbf{P}_{\textup{hi}}e^{\pm
iF_{j}}\|_{L^{2}}\lesssim\|\partial_{x}e^{\pm iF_{j}}\|_{L^{2}}$ in [42, p.
373].

###  3.3. Nonlinear estimates

We now state nonlinear estimates which allow us to establish a priori bounds
on a solution $v$ to (3.2) and the associated gauged function $w$ in (3.12).
The following results follow from slight modifications of the corresponding
results for the $\delta=\infty$ case; see Propositions 3.2 and 3.4 in [42].
The first lemma controls a solution $v$ to (3.2) in terms of the gauged
function $w$ defined in (3.12).

######  Lemma 3.4.

Let $0\leq s\leq 1$ and $0\leq\kappa\leq 1$. Then, there exists $\theta>0$
such that

$\displaystyle\|v\|_{X^{s-\kappa,\kappa}(T)}\lesssim
T^{\theta}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}+\|v\|_{L^{4}_{T,x}}\|J^{s}v\|_{L^{4}_{T,x}}$
(3.15)

for any $0<\delta\leq\infty$, $0<T\leq 1$, and any smooth solution $v$ to
(3.2) on the time interval $[0,T]$. Moreover if $0\leq s\leq\frac{1}{2}$, we
have

$\displaystyle\begin{split}\|J^{s}v\|_{L^{p}_{T}L^{q}_{x}}&\lesssim\|v(0)\|_{L^{2}}+\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|w\|_{Y^{s,\frac{1}{2}}(T)}\\\
&\quad+\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|v\|_{L^{\infty}_{T}L^{2}_{x}}\|v\|_{L^{\infty}_{T}H^{s}_{x}}+T\delta^{-1}\|v\|_{L^{\infty}_{T}L^{2}_{x}}\end{split}$
(3.16)

for $(p,q)=(\infty,2)$ or $(4,4)$, where $w=w(v)$ is as in (3.12).

###### Proof.

Proceeding as in the proof of Proposition 3.2 in [42], we have

$\displaystyle\|\widetilde{v}\|_{X^{s-\kappa,\kappa}}\lesssim\|\partial_{t}v-\mathcal{H}\partial_{x}^{2}v\|_{L^{2}_{T}H^{s-1}_{x}}+T^{\theta}\|v\|_{L^{\infty}_{T}H^{s}_{x}}$
(3.17)

for any $0\leq\kappa\leq 1$, where $\widetilde{v}(t)=\eta(t)v(t)$ is an
extension of $v$ from $[0,T]$ to $\mathbb{R}$. By substituting (3.3) to the
first term on the right-hand side of (3.17) and applying Lemma 2.3 and the
fractional Leibniz rule (Lemma 2.4), we have

$\displaystyle\begin{split}\|\partial_{t}v-\mathcal{H}\partial_{x}^{2}v\|_{L^{2}_{T}H^{s-1}_{x}}&\lesssim\|\mathcal{Q}_{\delta}v\|_{L^{2}_{T}H^{s}_{x}}+\|v\|_{L^{4}_{T,x}}\|J^{s}v\|_{L^{4}_{T,x}}\\\
&\lesssim
T^{\theta}\delta^{-1}\|v\|_{L^{\infty}_{T}H^{s}_{x}}+\|v\|_{L^{4}_{T,x}}\|J^{s}v\|_{L^{4}_{T,x}}.\end{split}$
(3.18)

Hence, (3.15) follows from (3.17) and (3.18).

Note that the identity (3.14) does not involve the depth parameter $\delta$
(or an equation) and is the same as (3-6) in [42] (modulo constants). Thus,
provided that (i) $(p,q)=(\infty,2)$ and $0\leq s\leq\frac{1}{2}$ or (ii)
$(p,q)=(4,4)$ and $0\leq s\leq\frac{1}{4}$, we can proceed as in the proof of
Proposition 3.2 in [42]. Note, however, that in [42], the low-frequency
contribution was estimated by studying the Duhamel formulation of the BO
equation; see [42, (3-17) and (3-18)]. In our case, we instead need to study
the Duhamel formulation of (3.3), where the additional contribution as
compared to [42] is estimated via Lemma 2.3 as

$\displaystyle\bigg{\|}J^{s}\int_{0}^{t}S(t-t^{\prime})\mathbf{P}_{\textup{LO}}\mathcal{Q}_{\delta}\partial_{x}v(t^{\prime})dt^{\prime}\bigg{\|}_{L^{p}_{T}L^{q}_{x}}\lesssim
T\delta^{-1}\|v\|_{L^{\infty}_{T}L^{2}_{x}}.$ (3.19)

This yields the second estimate (3.16) except for the case $(p,q)=(4,4)$ and
$\frac{1}{4}<s\leq\frac{1}{2}$.

It remains to consider the case $(p,q)=(4,4)$ and
$\frac{1}{4}<s\leq\frac{1}{2}$. In view of (3.14), we write
$v=\mathbf{P}_{+}v+\overline{\mathbf{P}_{+}v}$ (recall that $v$ is real-
valued) and

$\displaystyle\mathbf{P}_{+}v$
$\displaystyle=\mathbf{P}_{+,\textup{LO}}v-i\mathbf{P}_{+,\textup{HI}}(e^{-iF}w)+\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{hi}}e^{-iF}\cdot\mathbf{P}_{\textup{lo}}(e^{iF}v)\big{)}$
$\displaystyle\quad-i\mathbf{P}_{+,\textup{HI}}(\mathbf{P}_{+,\textup{HI}}e^{-iF}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF})$
$\displaystyle=:A_{1}+A_{2}+A_{3}+A_{4}.$

Note that the estimates (3-18) (see also (3.19) above) and (3-14) in [42] on
$A_{1}$ and $A_{3}$, respectively, also hold for
$\frac{1}{4}<s\leq\frac{1}{2}$. As for $A_{2}$, by applying Lemmas 3.3 and
2.2, we have

$\displaystyle\begin{split}\|J^{s}A_{2}\|_{L^{4}_{T,x}}&=\|J^{s}\mathbf{P}_{+,\textup{HI}}(e^{-iF}w)\|_{L^{4}_{T,x}}\lesssim\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|J^{s}w\|_{L^{4}_{T,x}}\\\
&\lesssim\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|w\|_{Y^{s,\frac{1}{2}}(T)}\end{split}$
(3.20)

extending [42, (3-13)] to $\frac{1}{4}<s\leq\frac{1}{2}$.

Next, we treat $A_{4}$. In view of the frequency localizations, we have

$\displaystyle\begin{split}A_{4}=-i\sum_{j,j_{1}=3}^{\infty}\sum_{j_{2}=1}^{\infty}\mathbf{1}_{j_{1}\geq
j,j_{2}}\cdot\mathbf{Q}_{j}\mathbf{P}_{+}(\mathbf{Q}_{j_{1}}\mathbf{P}_{+}e^{-iF}\cdot\mathbf{Q}_{j_{2}}\partial_{x}\mathbf{P}_{-}e^{iF}).\end{split}$
(3.21)

In the following, we work for fixed time $t$. For simplicity of notations, we
suppress $t$-dependence. By the Littlewood-Paley theorem and Minkowski’s
integral inequality, we have

$\displaystyle\begin{split}\|J^{s}A_{4}\|_{L^{4}_{x}}&\sim\bigg{\|}\bigg{(}\sum_{j=3}^{\infty}|\mathbf{Q}_{j}J^{s}\mathbf{P}_{+}(\mathbf{P}_{+,\textup{HI}}e^{-iF}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF})|^{2}\bigg{)}^{\frac{1}{2}}\bigg{\|}_{L^{4}_{x}}\\\
&\lesssim\bigg{(}\sum_{j=3}^{\infty}2^{2(s-\frac{1}{4})j}\|\mathbf{Q}_{j}J^{\frac{1}{4}}\mathbf{P}_{+}(\mathbf{P}_{+,\textup{HI}}e^{-iF}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF})\|_{L^{4}_{x}}^{2}\bigg{)}^{\frac{1}{2}}.\end{split}$
(3.22)

We separately estimate the contributions to (3.22) by considering the
following two cases: $2^{j_{1}}\sim 2^{j}$ and $2^{j_{1}}\sim 2^{j_{2}}\gg
2^{j}$, where $j_{1}$ and $j_{2}$ are as in (3.21).

$\bullet$ Case 1: $2^{j_{1}}\sim 2^{j}$.
In this case, we have $|j_{1}-j|\leq 2$. Then, proceeding as in [42, (3-15)]
with Lemma 3.2, Sobolev’s inequality, and (3.5), we have

$\displaystyle\begin{split}\eqref{MP4b}&\lesssim\sum_{k=-2}^{2}\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}+k}D^{\frac{1}{4}}\mathbf{P}_{+}(\mathbf{Q}_{j_{1}}\mathbf{P}_{+}e^{-iF}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF})\|_{L^{4}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\\\
&\lesssim\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|D^{\frac{5}{8}}\mathbf{Q}_{j_{1}}\mathbf{P}_{+}e^{-iF}\|_{L^{8}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\|D^{\frac{5}{8}}\mathbf{P}_{-,\textup{hi}}e^{iF}\|_{L^{8}_{x}}\\\
&\lesssim\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}}\partial_{x}\mathbf{P}_{+}e^{-iF}\|_{L^{2}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\|\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF}\|_{L^{2}_{x}}\\\
&\lesssim\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}}\mathbf{P}_{+}(e^{-iF}v)\|_{L^{2}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\|v\|_{L^{2}_{x}}.\end{split}$
(3.23)

By Lemma 3.3 (recall that $\frac{1}{4}<s\leq\frac{1}{2}$), we have

$\displaystyle\begin{split}&\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}}\mathbf{P}_{+}(e^{-iF}v)\|_{L^{2}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\\\
&\hphantom{XXXXX}\lesssim\|e^{-iF}v\|_{H^{s-\frac{1}{4}}_{x}}\lesssim\big{(}1+\|v\|_{L^{2}_{x}}\big{)}\|v\|_{H^{s-\frac{1}{4}}_{x}}.\end{split}$
(3.24)

Hence, from (3.22), (3.23), and (3.24), we conclude that the contribution from
this case (after taking the $L^{4}_{T}$-norm) is bounded by the right-hand
side of (3.16).

$\bullet$ Case 2: $2^{j_{1}}\sim 2^{j_{2}}\gg 2^{j}$.
In this case, we have $|j_{1}-j_{2}|\leq 2$ and $j\leq j_{1}-3$. Proceeding as
in (3.23), we have

$\displaystyle\begin{split}\eqref{MP4b}&\lesssim\sum_{k=-2}^{2}\sum_{j_{1}=3}^{\infty}\bigg{(}\sum_{j=3}^{j_{1}-3}2^{2(s-\frac{1}{4})j}\\\
&\hphantom{XXXXX}\times\|D^{\frac{1}{4}}\mathbf{P}_{+,\textup{HI}}(\mathbf{Q}_{j_{1}}\mathbf{P}_{+}e^{-iF}\cdot\mathbf{Q}_{j_{1}+k}\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF})\|_{L^{4}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\\\
&\lesssim\sum_{k=-2}^{2}\sum_{j_{1}=3}^{\infty}2^{(s-\frac{1}{4})j_{1}}\|D^{\frac{5}{8}}\mathbf{Q}_{j_{1}}\mathbf{P}_{+}e^{-iF}\|_{L^{8}_{x}}\|\mathbf{Q}_{j_{1}+k}D^{\frac{5}{8}}\mathbf{P}_{-,\textup{hi}}e^{iF}\|_{L^{8}_{x}}\\\
&\lesssim\sum_{k=-2}^{2}\sum_{j_{1}=3}^{\infty}2^{(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}}\partial_{x}\mathbf{P}_{+}e^{-iF}\|_{L^{2}_{x}}\|\mathbf{Q}_{j_{1}+k}\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF}\|_{L^{2}_{x}}\\\
&\lesssim\bigg{(}\sum_{j_{1}=3}^{\infty}2^{2(s-\frac{1}{4})j_{1}}\|\mathbf{Q}_{j_{1}}\mathbf{P}_{+}(e^{-iF}v)\|_{L^{2}_{x}}^{2}\bigg{)}^{\frac{1}{2}}\|v\|_{L^{2}_{x}},\end{split}$
(3.25)

where we used Cauchy-Schwarz’s inequality (in $j_{1}$) in the last step.
Hence, from (3.22), (3.25), and (3.24), we conclude that the contribution from
this case (after taking the $L^{4}_{T}$-norm) is bounded by the right-hand
side of (3.16).

Putting everything together, we conclude (3.16) when $(p,q)=(4,4)$ and
$\frac{1}{4}<s\leq\frac{1}{2}$. ∎

The following proposition allows us to estimate the gauged function $w$ in
terms of $v$.

######  Proposition 3.5.

Let $0\leq s\leq\frac{1}{2}$. Then, there exists $\theta>0$ such that

$\displaystyle\begin{split}&\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}\\\
&\quad\lesssim\|v\|_{L^{4}_{T,x}}^{2}+\Big{(}T^{\theta}\|v\|_{L^{\infty}_{T}L^{2}_{x}}+\|v\|_{L^{4}_{T,x}}+\|v\|_{X^{-1,1}(T)}\Big{)}\|w\|_{X^{s,\frac{1}{2}}(T)}\\\
&\qquad+T^{\theta}\|v\|_{L^{\infty}_{T}L^{2}_{x}}^{2}\|w\|_{Y^{s,\frac{1}{2}}(T)}+T^{\theta}\delta^{-1}(1+\delta^{-1})\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}^{2}\|v\|_{L^{\infty}_{T}H^{s}_{x}}\end{split}$
(3.26)

for any $0<\delta\leq\infty$, $0<T\leq 1$, and any smooth solution $v$ to
(3.2) the time interval on $[0,T]$, where $w=w(v)$ is as in (3.12) and
$\mathcal{N}_{\delta}(w,v)$ is as in (3.13).

Note that the last term on the right-hand side of (3.26) contains a linear
term $\delta^{-1}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}$. In order to
implement a bootstrap argument, we need additional smallness. This is the
primary reason that we keep a small power of $T$ for this term. The factor
$T^{\theta}$ on the other terms are used in (4.4). See Section 4.

We now present the proof of Proposition 3.5.

###### Proof of Proposition 3.5.

By arguing as in [42, Proposition 3.5 and Lemma 3.7], we can bound the
contribution from the first two terms on the right-hand side of (3.13) by the
right-hand side of (3.26).

Let us first treat the fourth term on the right-hand side of (3.13). By (2.3),
we have

$\displaystyle\|\mathbf{1}_{[0,T]}(t)\cdot\mathbf{P}_{0}(v^{2})w\|_{L^{2}_{t}H^{s}_{x}}\lesssim
T^{\theta}\|v\|_{L^{\infty}_{T}L^{2}_{x}}^{2}\|w\|_{L^{\infty}_{T}H^{s}_{x}}\lesssim
T^{\theta}\|v\|_{L^{\infty}_{T}L^{2}_{x}}^{2}\|w\|_{Y^{s,\frac{1}{2}}(T)}.$
(3.27)

We now turn our attention to the third term in (3.13). Proceeding as in [42,
(3-47)] with $\partial_{x}F=v$ and Lemma 3.3, we have

$\displaystyle\begin{split}&\|\mathbf{1}_{[0,T]}(t)\cdot\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}v)\|_{Y^{s,-\frac{1}{2}}}\lesssim\|\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}v)\|_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad\leq\|\mathbf{P}_{+,\textup{hi}}(e^{iF}v\mathcal{Q}_{\delta}v)\|_{L^{1+\kappa}_{T}H^{s}_{x}}+\|\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}\partial_{x}v)\|_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad\lesssim
T^{\theta}\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{(}\|v\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T}H^{s}_{x}}+\|\mathcal{Q}_{\delta}\partial_{x}v\|_{L^{\infty}_{T}H^{s}_{x}}\Big{)}\end{split}$
(3.28)

for any $\kappa>0$. From the fractional Leibniz rule (Lemma 2.4), Sobolev’s
embedding, and Lemma 2.3, we have

$\displaystyle\begin{split}\|v\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T}H^{s}_{x}}&\lesssim\|v\|_{L^{\infty}_{T}H^{s}_{x}}\|\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T,x}}+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\|\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T}W^{s,\infty}_{x}}\\\
&\lesssim\|v\|_{L^{\infty}_{T}H^{s}_{x}}\|\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T}H^{1}_{x}}+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\|\mathcal{Q}_{\delta}v\|_{L^{\infty}_{T}H^{s+1}_{x}}\\\
&\lesssim\delta^{-1}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}\|v\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(3.29)

By Lemma 2.3, we also have

$\displaystyle\|\mathcal{Q}_{\delta}\partial_{x}v\|_{L^{\infty}_{T}H^{s}_{x}}\lesssim\delta^{-1}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}.$
(3.30)

Hence, from (3.28), (3.29), and (3.30), we obtain

$\displaystyle\begin{split}\|\mathbf{1}_{[0,T]}(t)\cdot\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF}\mathcal{Q}_{\delta}v)\|_{Y^{s,-\frac{1}{2}}}\lesssim
T^{\theta}\delta^{-1}(1+\delta^{-1})\big{(}1+\|v\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}^{2}\|v\|_{L^{\infty}_{T}H^{s}_{x}}.\end{split}$
(3.31)

Therefore, from (3.27) and (3.31), we conclude (3.26). ∎

######  Remark 3.6.

In the proof of Proposition 3.5, we handled the first two terms on the right-
hand side of (3.13) by simply invoking the bilinear estimates in [42,
Proposition 3.5 and Lemma 3.7]. We point out that these bilinear estimates
(which hold on both the real line and the circle) are the key ingredients for
the simplification and unification of the $L^{2}$ well-posedness argument for
the BO equation (1.5) presented in [42].

######  Remark 3.7.

We point out that the map
$T\mapsto\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}$
is continuous for smooth $w$ and $v$. Indeed, given $h>0$, it follows from the
triangle inequality that

$\displaystyle\Big{|}\|$
$\displaystyle\mathbf{1}_{[0,T+h]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}-\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}\Big{|}$
$\displaystyle\leq\|\mathbf{1}_{[T,T+h]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}$
$\displaystyle\lesssim\|\mathbf{1}_{[T,T+h]}\cdot\mathcal{N}_{\delta}(w,v)\|_{X^{s,0}}=\|\mathcal{N}_{\delta}(w,v)\|_{L^{2}([T,T+h];H^{s})}$
$\displaystyle\longrightarrow 0,$

as $h\to 0$. A similar computation holds for $h<0$, tending to $0$. See [4,
Lemma A.8], [9, Lemma 4.4], and [47, Remark 3.3] for the continuity property
(in $T$) of the $X^{s,b}$-norm and its variants under the multiplication by
$\mathbf{1}_{[0,T]}$. See also [28, Lemma 6.3] and [18, Lemma 8.1] for similar
statements in the context of the more complicated short-time Fourier
restriction norm method.

##  4\. Well-posedness of ILW in $L^{2}$

In this section, we present the proof of Theorem 1.1. With Lemma 3.4 and
Proposition 3.5 in hand, it essentially follows from the discussion in Section
4 of [42], and thus we keep our presentation brief. Our main goal is to prove
local well-posedness of the renormalized ILW (3.2) in $L^{2}(\mathbb{T})$.
Once we establish local well-posedness, global well-posedness follows from the
$L^{2}$-conservation.

Fix $0<\delta<\infty$ and $0\leq s\leq\frac{1}{2}$. We only discuss the case
where initial data has sufficiently small $L^{2}$-norm. More precisely, in the
remaining part of this section, we assume that all the initial data in the
following discussion have the $L^{2}$-norm less than some sufficiently small
$\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))>0$ (to be chosen later). See
Remark 4.2 for the general case.

###  4.1. A priori estimates for smooth solutions

Let us first derive a priori estimates for smooth solutions to (3.2). Let $v$
be a smooth global solution to (3.2) and $w=w(v)$ be the gauged function
defined in (3.12). Given $T>0$, set

$\displaystyle
N_{T}^{s}(v)=\max\Big{(}\|v\|_{L^{\infty}_{T}H^{s}_{x}},\|v\|_{L^{4}_{T}W^{s,4}_{x}},\|w(0)\|_{H^{s}},\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}\Big{)},$
(4.1)

where $\mathcal{N}_{\delta}(w,v)$ is as in (3.13). We first note that from the
Duhamel formulation of (3.13) with (2.2) (for $Y^{s,\frac{1}{2}}(T)$), Lemma
2.1, and (4.1), we have

$\displaystyle\|w\|_{Y^{s,\frac{1}{2}}(T)}\lesssim\|w(0)\|_{H^{s}}+\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}\lesssim
N^{s}_{T}(v).$ (4.2)

Thanks to the smoothness of $v$ (and $w$), we see that $N^{s}_{T}(v)$ is
continuous in $T$; see Remark 3.7. Moreover, from the definition,
$N^{s}_{T}(v)$ is non-decreasing in $T$. From (3.12) and Lemma 3.3, we have

$\displaystyle\|w(0)\|_{H^{s}}\lesssim\big{(}1+\|v(0)\|_{L^{2}}\big{)}\|v(0)\|_{H^{s}}.$
(4.3)

See also [42, (3-49)]. Hence, from (4.1) and (4.3), we have

$\displaystyle\lim_{T\to
0+}N_{T}^{s}(v)\lesssim\big{(}1+\|v(0)\|_{L^{2}}\big{)}\|v(0)\|_{H^{s}}.$
(4.4)

In (4.4), the control on
$\|\mathbf{1}_{[0,T]}\cdot\mathcal{N}_{\delta}(w,v)\|_{Y^{s,-\frac{1}{2}}}$
follows from (3.26) and (3.15). On the other hand, from Lemma 3.4 and
Proposition 3.5 with (4.2) and (4.3), we have444Note that the constants in
(4.5) depend on $s$. We, however, use (4.5) only at two regularities ($s$ and
$0$) and thus we can simply take the worse constants in (4.5).

$\displaystyle\begin{split}N^{s}_{T}(v)&\leq
C_{1}\big{(}1+\|v(0)\|_{L^{2}}\big{)}\|v(0)\|_{H^{s}}\\\
&\quad+C_{2}(1+\delta^{-1})^{2}N^{0}_{T}(v)\big{(}1+N^{0}_{T}(v)\big{)}N^{s}_{T}(v)\\\
&\quad+C_{3}T^{\theta}\delta^{-1}(1+\delta^{-1})N^{s}_{T}(v)\end{split}$ (4.5)

for any $0<T\leq 1$. We point out that there are linear terms on the right-
hand sides of (3.16) and (3.26): $\|w\|_{Y^{s,\frac{1}{2}}(T)}$,
$T\delta^{-1}\|v\|_{L^{\infty}_{T}L^{2}_{x}}$, and
$T^{\theta}\delta^{-1}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}$, where
the last two terms are bounded by the last term on the right-hand side of
(4.5). As for the linear term $\|w\|_{Y^{s,\frac{1}{2}}(T)}$ on the right-hand
side of (3.16), we can use (4.2) and (3.26) to replace it by
$T^{\theta}\delta^{-1}(1+\delta^{-1})\|v\|_{L^{\infty}_{T}H^{s}_{x}}$ (along
with the other terms on the right-hand side of (3.26)), which is in turn
bounded by the last term on the right-hand side of (4.5).

By choosing sufficiently small $T_{*}=T_{*}(\min(\delta,1))>0$ such that
$C_{3}T_{*}^{\theta}\delta^{-1}(1+\delta^{-1})\leq\frac{1}{2}$, we then have

$\displaystyle\begin{split}N^{s}_{T}(v)&\leq
2C_{1}\big{(}1+\|v(0)\|_{L^{2}}\big{)}\|v(0)\|_{H^{s}}\\\
&\quad+2C_{2}(1+\delta^{-1})^{2}N^{0}_{T}(v)\big{(}1+N^{0}_{T}(v)\big{)}N^{s}_{T}(v)\end{split}$
(4.6)

for any $0<T\leq T_{*}$. By a standard continuity argument, we have

$\displaystyle N^{0}_{T}(v)$ $\displaystyle\lesssim\varepsilon$ (4.7)

for any $0<T\leq T_{*}$, provided that
$\|v(0)\|_{L^{2}}=\varepsilon\leq\varepsilon_{0}$ for some sufficiently small
$\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))$. Then, by applying a
continuity argument with (4.6) and (4.7), we obtain

$\displaystyle N^{s}_{T}(v)$ $\displaystyle\lesssim\|v(0)\|_{H^{s}}$ (4.8)

for any $0<T\leq T_{*}$.

######  Remark 4.1.

From (4.5), we can choose $T_{*}=T_{*}(\min(\delta,1))\gtrsim 1$ and small
$\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))\geq\varepsilon_{1}>0$,
uniformly in $\delta\geq 1$. The same comment applies to the discussion below.

###  4.2. Difference estimates

Let $v_{j}$, $j=1,2$, be smooth global solutions to (3.2) with

$\displaystyle\|v_{j}(0)\|_{L^{2}}\leq\varepsilon\leq\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1)),$
(4.9)

and let $F_{j}$ be the mean-zero primitive of $v_{j}$ as defined in (3.4).
Then, as in (3.10) and (3.12), define the associated gauged functions $W_{j}$
and $w_{j}$, $j=1,2$, by setting

$\displaystyle
W_{j}=\mathbf{P}_{+,\textup{hi}}(e^{iF_{j}})\qquad\text{and}\qquad
w_{j}=\partial_{x}W_{j}.$

Then, from the discussion in the previous subsection (see (4.7) and (4.8) with
(4.2)), we have

$\displaystyle\begin{split}\max_{j=1,2}\|w\|_{Y^{0,\frac{1}{2}}(T)}&\lesssim
N^{0}_{T}(v_{j})\lesssim\varepsilon\leq\varepsilon_{0},\\\
\max_{j=1,2}\|w\|_{Y^{s,\frac{1}{2}}(T)}&\lesssim
N^{s}_{T}(v_{j})\lesssim\max_{j=1,2}\|v_{j}(0)\|_{H^{s}}=:M\end{split}$ (4.10)

for any $0<T\leq T_{*}=T_{*}(\min(\delta,1))$, $j=1,2$.

Let $\widetilde{v}=v_{1}-v_{2}$, $\widetilde{W}=W_{1}-W_{2}$, and
$\widetilde{w}=w_{1}-w_{2}$. By the mean value theorem with (3.7) and (3.8),
we have

$\displaystyle\begin{split}\|e^{\pm iF_{1}(0)}-e^{\pm
iF_{2}(0)}\|_{L^{\infty}}&\leq\|F_{1}(0)-F_{2}(0)\|_{L^{\infty}}\lesssim\|\widetilde{v}(0)\|_{L^{2}},\\\
\|e^{\pm iF_{1}}-e^{\pm
iF_{2}}\|_{L^{\infty}_{T,x}}&\leq\|F_{1}-F_{2}\|_{L^{\infty}_{T,x}}\lesssim\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.11)

Then, from (3.12), Lemma 3.3, and (4.11) with (4.9), we have

$\displaystyle\begin{split}\|\widetilde{w}(0)\|_{H^{s}}&\lesssim\big{(}1+\|v_{1}(0)\|_{L^{2}}\big{)}\|\widetilde{v}(0)\|_{H^{s}}\\\
&\quad+\Big{(}\|\widetilde{v}(0)\|_{L^{2}}+\|e^{iF_{1}(0)}-e^{iF_{2}(0)}\|_{L^{\infty}}\big{(}1+\|v_{1}(0)\|_{L^{2}}\big{)}\Big{)}\|v_{2}(0)\|_{H^{s}}\\\
&\lesssim\|\widetilde{v}(0)\|_{H^{s}}+\|v_{2}(0)\|_{H^{s}}\|\widetilde{v}(0)\|_{L^{2}}.\end{split}$
(4.12)

We now estimate $\widetilde{w}$. From (3.13), we have

$\displaystyle\begin{split}\partial_{t}&\widetilde{w}-\mathcal{H}\partial_{x}^{2}\widetilde{w}\\\
&=-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(W_{1}\mathbf{P}_{-}\partial_{x}\widetilde{v})-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(\widetilde{W}\mathbf{P}_{-}\partial_{x}v_{2})\\\
&\quad-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(\mathbf{P}_{\textup{lo}}e^{iF_{1}}\mathbf{P}_{-}\partial_{x}\widetilde{v})-2\partial_{x}\mathbf{P}_{+,\textup{hi}}\big{(}\mathbf{P}_{\textup{lo}}(e^{iF_{1}}-e^{iF_{2}})\mathbf{P}_{-}\partial_{x}v_{2}\big{)}\\\
&\quad+i\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF_{1}}\mathcal{Q}_{\delta}\widetilde{v})+i\partial_{x}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}v_{2}\big{)}\\\
&\quad-i\mathbf{P}_{0}(v_{1}^{2})\widetilde{w}-i\mathbf{P}_{0}(v_{1}^{2}-v_{2}^{2})w_{2}\\\
&=:B_{1}+\cdots+B_{8}.\end{split}$ (4.13)

Note that the first four terms on the right-hand side are already treated in
Section 4 of [42], which, in view of (4.10) and (4.11), are bounded by the
first two terms on the right-hand side of (4.20) below.555In fact, there is is
another term $\varepsilon\|\widetilde{w}\|_{X^{s,\frac{1}{2}}(T)}$,
controlling the second term on the right-hand side of (4.13); see [42,
Proposition 3.5]. In (4.20), this term is already hidden on the left-hand
side, which explains the reason for the factor 2 in front of the last term in
(4.20).  By a slight modification of (3.27) with (4.10), we have

$\displaystyle\begin{split}\|B_{7}+B_{8}\|_{Y^{s,-\frac{1}{2}}(T)}&\lesssim\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}^{2}\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}\\\
&\quad+\big{(}\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}+\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\|w_{2}\|_{Y^{s,\frac{1}{2}}(T)}\\\
&\lesssim\varepsilon^{2}\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}+\varepsilon\|w_{2}\|_{Y^{s,\frac{1}{2}}(T)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\end{split}$
(4.14)

for any $0<T\leq T_{*}$. Proceeding as in the proof of Proposition 3.5 with
(4.10), given an absolute small constant $c_{0}>0$ (to be chosen later), by
possibly making $T_{*}=T_{*}(\min(\delta,1))>0$ smaller, we can show that
there exists $\theta>0$ such that

$\displaystyle\begin{split}\|B_{5}\|_{Y^{s,-\frac{1}{2}}(T)}&\leq
CT^{\theta}\delta^{-1}(1+\delta^{-1})\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\\\
&\quad\times\Big{(}\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\\\
&\leq
c_{0}\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\end{split}$
(4.15)

for any $0<T\leq T_{*}$.

Let us now estimate $B_{6}$. By Cauchy-Schwarz’s inequality, Sobolev’s
inequality, and Minkowski’s integral inequality (see [42, (3-47)]), we have

$\displaystyle\begin{split}\|B_{6}\|_{Y^{s,-\frac{1}{2}}(T)}&\lesssim\big{\|}\partial_{x}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\leq\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}\partial_{x}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad+\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}\partial_{x}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\leq\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}e^{iF_{1}}(v_{1}-v_{2})\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad+\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})v_{2}\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad+\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}\partial_{x}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\end{split}$
(4.16)

for any $\kappa>0$. By applying Lemma 3.3 and proceeding as in (3.29) with
(4.10), we have

$\displaystyle\begin{split}&\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}e^{iF_{1}}(v_{1}-v_{2})\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad\lesssim\delta^{-1}(1+\delta^{-1})\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{(}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\\\
&\quad\lesssim\delta^{-1}(1+\delta^{-1})\Big{(}\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}.\end{split}$
(4.17)

By Lemma 3.3, (4.11), and (3.29) with (4.10), we have

$\displaystyle\begin{split}&\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})v_{2}\mathcal{Q}_{\delta}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad\lesssim\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{iF_{1}}-e^{iF_{2}}\|_{L^{\infty}_{T,x}}\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{)}\|v_{2}\mathcal{Q}_{\delta}v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\\\
&\quad\lesssim\delta^{-1}(1+\delta^{-1})\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\quad\lesssim\varepsilon\delta^{-1}(1+\delta^{-1})\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.18)

Similarly, by Lemma 3.3, (4.11), and Lemma 2.3 with (4.10), we have

$\displaystyle\begin{split}&\big{\|}\mathbf{P}_{+,\textup{hi}}\big{(}(e^{iF_{1}}-e^{iF_{2}})\mathcal{Q}_{\delta}\partial_{x}v_{2}\big{)}\big{\|}_{L^{1+\kappa}_{T}H^{s}_{x}}\\\
&\quad\lesssim\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{iF_{1}}-e^{iF_{2}}\|_{L^{\infty}_{T,x}}\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{)}\|\mathcal{Q}_{\delta}v_{2}\|_{L^{\infty}_{T}H^{s+1}_{x}}\\\
&\quad\lesssim\delta^{-1}(1+\delta^{-1})\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.19)

Therefore, from the Duhamel formulation of (4.13), Lemma 2.1, (4.14), (4.15),
(4.16), (4.17), (4.18), and (4.19) together with [42, the math display after
(4-4)], (4.11), and (4.10), we obtain

$\displaystyle\begin{split}\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}&\leq
C\|\widetilde{w}(0)\|_{H^{s}}\\\
&\quad+C\big{(}\varepsilon+\|w_{1}\|_{X^{s,\frac{1}{2}}(T)}\big{)}\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|\widetilde{v}\|_{L^{4}_{T,x}}+\|\widetilde{v}\|_{X^{-1,1}(T)}\Big{)}\\\
&\quad+C\varepsilon\|w_{2}\|_{Y^{s,\frac{1}{2}}(T)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\quad+C\delta^{-1}(1+\delta^{-1})\Big{(}\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\\\
&\quad+2c_{0}\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\end{split}$
(4.20)

for any $0<T\leq T_{*}$, where $c_{0}$ is as in (4.15).

From (3.3), we have

$\displaystyle\partial_{t}\widetilde{v}-\mathcal{H}\partial_{x}^{2}\widetilde{v}=\mathcal{Q}_{\delta}\partial_{x}\widetilde{v}+\partial_{x}\big{(}(v_{1}+v_{2})\widetilde{v}\big{)}.$
(4.21)

Then, by a slight modification of the proof of (3.15) in Lemma 3.4 with
(4.10), we have

$\displaystyle\|\widetilde{v}\|_{X^{-1,1}(T)}\lesssim
T^{\theta}(1+\delta^{-1})\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\varepsilon\|\widetilde{v}\|_{L^{4}_{T,x}}$
(4.22)

for any $0<T\leq T_{*}$.

In the following, we estimate the $L^{p}_{T}W^{s,q}_{x}$-norm of
$\widetilde{v}=\mathbf{P}_{+}\widetilde{v}+\overline{\mathbf{P}_{+}\widetilde{v}}$.
From (3.14) with $e^{iF}v=-i\partial_{x}e^{iF}$, we have

$\displaystyle\begin{split}\mathbf{P}_{+}\widetilde{v}&=\mathbf{P}_{+,\textup{LO}}\widetilde{v}+\mathbf{P}_{+,\textup{HI}}\widetilde{v}\\\
&=\mathbf{P}_{+,\textup{LO}}\widetilde{v}-i\mathbf{P}_{+,\textup{HI}}(e^{-iF_{1}}\widetilde{w})-i\mathbf{P}_{+,\textup{HI}}\big{(}(e^{-iF_{1}}-e^{-iF_{2}})w_{2}\big{)}\\\
&\quad-i\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{hi}}e^{-iF_{1}}\cdot\partial_{x}\mathbf{P}_{\textup{lo}}(e^{iF_{1}}-e^{iF_{2}})\big{)}\\\
&\quad-i\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{hi}}(e^{-iF_{1}}-e^{-iF_{2}})\cdot\partial_{x}\mathbf{P}_{\textup{lo}}e^{iF_{2}}\big{)}\\\
&\quad-i\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{HI}}e^{-iF_{1}}\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}(e^{iF_{1}}-e^{iF_{2}})\big{)}\\\
&\quad-i\mathbf{P}_{+,\textup{HI}}\big{(}\mathbf{P}_{+,\textup{HI}}(e^{-iF_{1}}-e^{-iF_{2}})\cdot\partial_{x}\mathbf{P}_{-,\textup{hi}}e^{iF_{2}}\big{)}\\\
&=:E_{1}+\cdots+E_{7}.\end{split}$ (4.23)

As pointed out in the proof of Lemma 3.4, (4.23) agrees with the corresponding
expression in [42, p.385] (modulo constants). In particular, when (i)
$(p,q)=(\infty,2)$ and $0\leq s\leq\frac{1}{2}$ or (ii) $(p,q)=(4,4)$ and
$0\leq s\leq\frac{1}{4}$, the $L^{p}_{T}W^{s,q}_{x}$-norm of
$\widetilde{v}=\mathbf{P}_{+}\widetilde{v}+\overline{\mathbf{P}_{+}\widetilde{v}}$
is already estimated in [42, (4-8)], except for the additional term in the
low-frequency contribution which is estimated as in (3.19):

$\displaystyle\bigg{\|}J^{s}\int_{0}^{t}S(t-t^{\prime})\mathbf{P}_{+,\textup{LO}}\mathcal{Q}_{\delta}\partial_{x}\widetilde{v}(t^{\prime})dt^{\prime}\bigg{\|}_{L^{p}_{T}L^{q}_{x}}\lesssim
T\delta^{-1}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.$ (4.24)

Hence, we only need to consider the case $(p,q)=(4,4)$ and
$\frac{1}{4}<s\leq\frac{1}{2}$.

In the following, we work with $T\in(0,T_{*}]$ such that (4.10) holds. By
applying $\mathbf{P}_{+,\textup{LO}}$ to the Duhamel formulation of (4.21) and
proceeding as in [42, (3-17) and (3-18)] with (4.10) and (4.24), given an
absolute small constant $c_{1}>0$ (to be chosen later), we have

$\displaystyle\begin{split}\|J^{s}E_{1}\|_{L^{4}_{T,x}}&\leq\|\widetilde{v}(0)\|_{L^{2}}+C\big{(}\varepsilon+T(1+\delta^{-1})\big{)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\leq\|\widetilde{v}(0)\|_{L^{2}}+(C\varepsilon+c_{1})\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}},\end{split}$
(4.25)

where the second step holds by possibly making $T_{*}=T_{*}(\min(\delta,1))>0$
smaller. Proceeding as in (3.20) with (4.10), we have

$\displaystyle\begin{split}\|J^{s}E_{2}\|_{L^{4}_{T,x}}&\lesssim\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}\lesssim\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}.\end{split}$
(4.26)

By Lemma 3.3, (4.11), Lemma 2.2 with (4.10), we have

$\displaystyle\begin{split}\|J^{s}E_{3}\|_{L^{4}_{T,x}}&\lesssim\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{-iF_{1}}-e^{-iF_{2}}\|_{L^{\infty}_{T,x}}\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{)}\|J^{s}w_{2}\|_{L^{4}_{T,x}}\\\
&\lesssim\|w_{2}\|_{Y^{s,\frac{1}{2}}(T)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.27)

Proceeding as in [42, (3-14)] with (4.11) and (4.10), we have

$\displaystyle\begin{split}\|J^{s}E_{4}\|_{L^{4}_{T,x}}&\lesssim\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\|e^{iF_{1}}v_{1}-e^{iF_{2}}v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\lesssim\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{iF_{1}}-e^{iF_{2}}\|_{L^{\infty}_{T,x}}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\\\
&\lesssim\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.28)

Similarly, proceeding as in [42, (3-14)] with (3.4), (4.11), and (4.10), we
have

$\displaystyle\begin{split}\|J^{s}E_{5}\|_{L^{4}_{T,x}}&\lesssim\|\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{-iF_{1}}-e^{-iF_{2}})\|_{L^{\infty}_{T}L^{2}_{x}}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\lesssim\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{-iF_{1}}-e^{-iF_{2}}\|_{L^{\infty}_{T,x}}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
\ &\lesssim\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.29)

Next, we treat $E_{6}$ and $E_{7}$. Proceeding as in (4.29), we have

$\displaystyle\begin{split}\|\partial_{x}\mathbf{P}_{-,\textup{hi}}(e^{iF_{1}}-e^{iF_{2}})\|_{L^{\infty}_{T}L^{2}_{x}}&\lesssim\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.30)

By Lemma 3.3 (recall that $\frac{1}{4}<s\leq\frac{1}{2}$) with (4.11) and
(4.10), we have

$\displaystyle\begin{split}\|&e^{-iF_{1}}v_{1}-e^{-iF_{2}}v_{2}\|_{L^{\infty}_{T}H^{s-\frac{1}{4}}_{x}}\leq\|e^{-iF_{1}}\widetilde{v}\|_{L^{\infty}_{T}H^{s-\frac{1}{4}}_{x}}+\|(e^{-iF_{1}}-e^{-iF_{2}})v_{2}\|_{L^{\infty}_{T}H^{s-\frac{1}{4}}_{x}}\\\
&\lesssim\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s-\frac{1}{4}}_{x}}\\\
&\quad+\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|e^{-iF_{1}}-e^{-iF_{2}}\|_{L^{\infty}_{T,x}}\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\Big{)}\|v_{2}\|_{L^{\infty}_{T}H^{s-\frac{1}{4}}_{x}}\\\
&\lesssim\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.31)

Thus, proceeding as in (3.22)-(3.25) with (4.30), (4.31), and (4.10), we
obtain

$\displaystyle\begin{split}&\|J^{s}E_{6}\|_{L^{4}_{T,x}}+\|J^{s}E_{7}\|_{L^{4}_{T,x}}\\\
&\quad\lesssim\big{(}1+\|v_{1}\|_{L^{\infty}_{T}L^{2}_{x}}\big{)}\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\quad\quad+\Big{(}\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\Big{)}\|v_{2}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\quad\lesssim\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\varepsilon\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}.\end{split}$
(4.32)

Therefore, putting (4.23), (4.25), (4.26), (4.27), (4.28), (4.29), and (4.32)
together with [42, (4-8)] and (4.11), we obtain

$\displaystyle\begin{split}\|J^{s}\widetilde{v}\|_{L^{p}_{T}L^{q}_{x}}&\lesssim\|\widetilde{v}(0)\|_{L^{2}}+\Big{(}\|w_{2}\|_{Y^{s,\frac{1}{2}}(T)}+\|v_{1}\|_{L^{\infty}_{T}H^{s}_{x}}+\varepsilon\|v_{2}\|_{L^{\infty}_{T}H^{s}_{x}}+c_{1}\Big{)}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}\\\
&\quad+\varepsilon\|\widetilde{v}\|_{L^{\infty}_{T}H^{s}_{x}}+\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}\end{split}$
(4.33)

for $0\leq s\leq\frac{1}{2}$, $(p,q)=(\infty,2)$ or $(4,4)$, and any $0<T\leq
T_{*}$, where $c_{1}$ is as in (4.25).

By collecting (4.12), (4.20), (4.22), and (4.33) for $s=0$ with (4.10), we
first obtain

$\displaystyle\begin{split}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|\widetilde{v}\|_{L^{4}_{T,x}}+\|\widetilde{w}\|_{Y^{0,\frac{1}{2}}(T)}+\|\widetilde{v}\|_{X^{-1,1}(T)}&\lesssim\|\widetilde{v}(0)\|_{L^{2}}\end{split}$
(4.34)

for any $0<T\leq T_{*}$, provided that we choose (i)
$\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))>0$ in (4.10) sufficiently
small such that

$\displaystyle\varepsilon_{0}\delta^{-1}(1+\delta^{-1})\ll 1$ (4.35)

and (ii) $c_{0}>0$ in (4.15) and (4.20) and $c_{1}>0$ in (4.25) sufficiently
small, which can be guaranteed by choosing $T_{*}=T_{*}(\min(\delta,1))>0$
sufficiently small. Finally, from (4.12), (4.20), (4.22), and (4.33) with
(4.34), we conclude that, with $M$ as in (4.10),

$\displaystyle\begin{split}\|J^{s}\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}+\|J^{s}\widetilde{v}\|_{L^{4}_{T,x}}+\|\widetilde{w}\|_{Y^{s,\frac{1}{2}}(T)}+\|\widetilde{v}\|_{X^{-1,1}(T)}&\leq
C(M)\|\widetilde{v}(0)\|_{H^{s}}\end{split}$ (4.36)

for any $0<T\leq T_{*}$, provided that $\varepsilon_{0},c_{0}>0$ are
sufficiently small as above.

###  4.3. Well-posedness

Once we have (4.36), local well-posedness of the renormalized ILW (3.2) (and
hence of the original ILW equation (1.1)) follows from a standard argument.
Given $0\leq s\leq\frac{1}{2}$, let $u_{0}\in H^{s}_{0}(\mathbb{T})$ with
$2\|u_{0}\|_{L^{2}}\leq\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))$ and
consider the sequence $\\{u_{0,k}\\}_{k\in\mathbb{N}}$ of smooth mean-zero
initial data $u_{0,k}=\Pi_{\leq k}u_{0}$, where $\Pi_{\leq k}$ is the
frequency projector onto the (spatial) frequencies $\\{n\in\mathbb{Z}:|n|\leq
k\\}$. Let $v_{k}$ be the smooth global solution to (3.2) with
$v_{k}|_{t=0}=u_{0,k}$. Then, it follows from (4.36) that $v_{k}$ converges to
a limit $v$ in $C([0,T_{*}];H^{s}(\mathbb{T}))\cap
L^{4}([0,T_{*}];W^{s,4}(\mathbb{T}))$ and that the associated gauged function
$w_{k}=w_{k}(v_{k})$ (defined as in (3.12)) converges to a limit $w$ in
$Y^{s,\frac{1}{2}}(T_{*})$, where $T_{*}=T_{*}(\min(\delta,1))>0$ is as in the
previous subsections. It is easy to verify that the limit $v$ satisfies the
equation (3.2) with $v|_{t=0}=u_{0}$ and that the limits $v$ and $w$ satisfy
(3.12). In the current periodic setting, uniqueness of solutions and local
Lipschitz continuity of the solution map for the renormalized ILW (3.2) follow
from (4.36). Furthermore, by noting that the local existence time
$T_{*}=T_{*}(\min(\delta,1))>0$ depends only on $\delta>0$ (as long as we
choose $\varepsilon_{0}$ (= the upper bound on the $L^{2}$-norm of initial
data) sufficiently small such that (4.35) is satisfied), we conclude global
well-posedness of (3.2) from the conservation of the $L^{2}$-norm. Finally, by
undoing the Galilean transform (3.1), we obtain global well-posedness of the
original ILW equation (1.1). This concludes the proof of Theorem 1.1.

######  Remark 4.2.

In this section, we proved local and global well-posedness of the renormalized
ILW (3.2) (and then of ILW (1.1)) with the small $L^{2}$-norm assumption. In
the general case, we apply a scaling argument and reduce the situation to the
small $L^{2}$-norm case as in [13, 40, 41, 42]. Given $\lambda\geq 1$, let
$\mathcal{S}_{\lambda}$ be the $\dot{H}^{-\frac{1}{2}}$-invariant scaling
operator defined by

$\displaystyle\mathcal{S}_{\lambda}(u)(t,x)=\lambda^{-1}u(\lambda^{-2}t,\lambda^{-1}x).$
(4.37)

Recall that the BO equation (1.5) on $\mathbb{R}$ in invariant under the
action of $\mathcal{S}_{\lambda}$. While the renormalized ILW (3.2) on
$\mathbb{R}$ does not enjoy any scaling symmetry, a direct computation shows
that if $v$ is a solution to (3.2) on $\mathbb{R}$ with the depth parameter
$\delta$, then $v_{\lambda}=\mathcal{S}_{\lambda}(v)$ is a solution to (3.2)
on $\mathbb{R}$ with the depth parameter $\lambda\delta$:

$\displaystyle\partial_{t}v_{\lambda}-\mathcal{T}_{\lambda\delta}\partial_{x}^{2}v_{\lambda}=\partial_{x}(v_{\lambda}^{2}).$
(4.38)

Namely, the family of the renormalized ILW equations with depth parameters
$0<\delta<\infty$ remains invariant under the action of
$\mathcal{S}_{\lambda}$. In the periodic setting, if $v$ satisfies (3.2) on
$\mathbb{T}$ then $v_{\lambda}=\mathcal{S}_{\lambda}(v)$ satisfies (4.38) on
the dilated torus $\mathbb{T}_{\lambda}=\mathbb{R}/(2\pi\lambda\mathbb{Z})$.
With $u_{0}=v(0)$, the scaled initial data for $v_{\lambda}$ satisfying (4.38)
is given by $u_{0,\lambda}(x)=\lambda^{-1}u_{0}(\lambda^{-1}x)$. Noting that

$\displaystyle\|u_{0,\lambda}\|_{L^{2}(\mathbb{T}_{\lambda})}=\lambda^{-\frac{1}{2}}\|u_{0}\|_{L^{2}(\mathbb{T})},$

we can choose $\lambda\gg 1$ such that
$\|u_{0,\lambda}\|_{L^{2}(\mathbb{T}_{\lambda})}\leq\varepsilon_{0}=\varepsilon_{0}(\min(\delta,1))$.
We point out that we can choose $\varepsilon_{0}(\min(\delta,1))$
independently of $\lambda\geq 1$ (since $\lambda\delta\geq\delta$). As pointed
out in [42, Section 7], all the estimates hold true on $\mathbb{T}_{\lambda}$
with constants independent of $\lambda\geq 1$. Here, the most important (and
well-known) point is that the periodic $L^{4}$-Strichartz estimate (Lemma 2.2)
holds true on $\mathbb{T}_{\lambda}$ with a constant independent of
$\lambda\geq 1$. With this observation, we can repeat the argument presented
in this section and prove global well-posedness of (4.38) in
$H^{s}(\mathbb{T}_{\lambda})$ with the small $L^{2}$-norm assumption. Then, by
undoing the scaling, we conclude global well-posedness of (3.2) (and hence of
(1.1)) in $H^{s}(\mathbb{T})$ without the small $L^{2}$-norm assumption.

######  Remark 4.3.

The bound (4.11), coming from (3.7) and (3.8), played a crucial role in the
argument on $\mathbb{T}$ presented above, yielding local Lipschitz continuity
of the solution map in the periodic case. As pointed out in Remark 3.1,
however, (3.7) and (3.8) (and hence (4.11)) do not hold in the real line case.
This is the primary reason for the failure of local uniform continuity of the
solution map for ILW on $\mathbb{R}$ constructed in Theorem 1.1. Nonetheless,
under the extra assumption (3.9), one can still prove (3.7) and (3.8) on
$\mathbb{R}$ (but the argument uses the equation; see [42, Lemma 4.1] in the
case of the BO equation), giving local Lipschitz continuity of the solution
map restricted to the class (3.9) (say, for fixed $v_{1}(0)$) consisting of
elements with the identical low-frequency parts. See Section 4 in [42] for a
further discussion.

##  5\. Deep-water limit of ILW in $L^{2}$

In this section, by slightly modifying the argument in Section 4, we present
the proof of Theorem 1.2. As in the previous sections, we only consider the
periodic case. Moreover, we only consider the case with sufficiently small
$L^{2}$-norms.

$\bullet$ Part 1: In this part, we consider the mean-zero case. Namely, given
$0\leq s\leq\frac{1}{2}$, fix a mean-zero function $u_{0}\in
H^{s}_{0}(\mathbb{T})$ and let $\\{u_{0,\delta}\\}_{1\leq\delta<\infty}$ be a
net in $H^{s}_{0}(\mathbb{T})$ (of mean-zero functions) such that
$u_{0,\delta}$ converges to $u_{0}$ in $H^{s}(\mathbb{T})$ as
$\delta\to\infty$.

We first establish convergence of solutions to the renormalized ILW (3.2) to
the associated solution to the BO equation (1.5). Let $u$ be the global
solution to the BO equation (1.5) with $u|_{t=0}=u_{0}$ constructed in [42],
and let $v_{\delta}$, $1\leq\delta<\infty$, be the global solution to (3.2)
with $v_{\delta}|_{t=0}=u_{0,\delta}$ constructed in Theorem 1.1. Recalling
that $\mathcal{Q}_{\infty}=0$, the renormalized ILW (3.2) formally reduces to
the BO equation (1.5) when $\delta=\infty$. In view of this, we set
$v_{\infty}=u$ and $u_{0,\infty}=u_{0}$. For each $1\leq\delta\leq\infty$, let
$F_{\delta}$ be the mean-zero primitive of $v_{\delta}$ as defined in (3.4),
and let $W_{\delta}$ and $w_{\delta}$ be the associated gauged functions as in
(3.10) and (3.12).

Let $N^{s}_{T}$ be as in (4.1). Then, the a priori bound (4.6) holds uniformly
in $1\leq\delta\leq\infty$ (including $\delta=\infty$):

$\displaystyle N^{s}_{T}(v_{\delta})$
$\displaystyle\lesssim\big{(}1+\|u_{0,\delta}\|_{L^{2}}\big{)}\|u_{0,\delta}\|_{H^{s}}+N^{0}_{T}(v_{\delta})\big{(}1+N^{0}_{T}(v_{\delta})\big{)}N^{s}_{T}(v_{\delta})$
(5.1)

for any $0<T\leq T_{*}$, where $T_{*}>0$ is now independent of
$1\leq\delta\leq\infty$. By assuming that
$\|u_{0,\infty}\|_{L^{2}}=\|u_{0}\|_{L^{2}}\leq\varepsilon$ for sufficiently
small $\varepsilon>0$, it follows from the convergence of $u_{0,\delta}$ to
$u_{0,\infty}$ in $H^{s}(\mathbb{T})$ that there exists
$\delta_{0}=\delta_{0}(\varepsilon)\geq 1$ such that
$\|u_{0,\delta}\|_{L^{2}}\lesssim\varepsilon$ for any $\delta\geq\delta_{0}$.
Then, from (5.1), we have

$\displaystyle
N^{0}_{T_{*}}(v_{\delta})\lesssim\varepsilon\qquad\text{and}\qquad
N^{s}_{T_{*}}(v_{\delta})$ $\displaystyle\lesssim\|u_{0,\delta}\|_{H^{s}}\leq
M$ (5.2)

for some $M>0$, uniformly in $\delta_{0}\leq\delta\leq\infty$.

Let $\widetilde{v}_{\delta}=v_{\delta}-v_{\infty}$,
$\widetilde{W}_{\delta}=W_{\delta}-W_{\infty}$, and
$\widetilde{w}_{\delta}=w_{\delta}-w_{\infty}$. The difference
$\widetilde{w}_{\delta}$ satisfies

$\displaystyle\begin{split}\partial_{t}&\widetilde{w}_{\delta}-\mathcal{H}\partial_{x}^{2}\widetilde{w}_{\delta}\\\
&=-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(W_{\delta}\mathbf{P}_{-}\partial_{x}\widetilde{v}_{\delta})-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(\widetilde{W}_{\delta}\mathbf{P}_{-}\partial_{x}v_{\infty})\\\
&\quad-2\partial_{x}\mathbf{P}_{+,\textup{hi}}(\mathbf{P}_{\textup{lo}}e^{iF_{\delta}}\mathbf{P}_{-}\partial_{x}\widetilde{v}_{\delta})-2\partial_{x}\mathbf{P}_{+,\textup{hi}}\big{(}\mathbf{P}_{\textup{lo}}(e^{iF_{\delta}}-e^{iF_{\infty}})\mathbf{P}_{-}\partial_{x}v_{\infty}\big{)}\\\
&\quad+i\partial_{x}\mathbf{P}_{+,\textup{hi}}(e^{iF_{\delta}}\mathcal{Q}_{\delta}v_{\delta})+0\\\
&\quad-i\mathbf{P}_{0}(v_{\delta}^{2})\widetilde{w}_{\delta}-i\mathbf{P}_{0}(v_{\delta}^{2}-v_{\infty}^{2})w_{\infty}.\end{split}$
(5.3)

Compare this with (4.13), where the difference appears only in the fifth term
and the sixth term (which is $0$ in (5.3)). In estimating the fifth term on
the right-hand side of (5.3), we instead use (3.31). Hence, from the
discussion in Subsection 4.2 leading to (4.20) with (5.2), we have

$\displaystyle\begin{split}\|\widetilde{w}_{\delta}\|_{Y^{s,\frac{1}{2}}(T_{*})}&\leq
C\|\widetilde{w}_{\delta}(0)\|_{H^{s}}\\\
&\quad+C\big{(}\varepsilon+\|w_{\delta}\|_{X^{s,\frac{1}{2}}(T_{*})}\big{)}\Big{(}\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}+\|\widetilde{v}_{\delta}\|_{L^{4}_{T_{*},x}}+\|\widetilde{v}_{\delta}\|_{X^{-1,1}(T_{*})}\Big{)}\\\
&\quad+C\varepsilon\|w_{\infty}\|_{Y^{s,\frac{1}{2}}(T_{*})}\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}+T_{*}^{\theta}M\delta^{-1},\end{split}$
(5.4)

uniformly in $\delta_{0}\leq\delta\leq\infty$. From (1.5) and (3.3), we have

$\displaystyle\partial_{t}\widetilde{v}_{\delta}-\mathcal{H}\partial_{x}^{2}\widetilde{v}_{\delta}=\mathcal{Q}_{\delta}\partial_{x}v_{\delta}+\partial_{x}\big{(}(v_{\delta}+v_{\infty})\widetilde{v}_{\delta}\big{)}.$
(5.5)

Then, by a slight modification of the proof of (3.15) in Lemma 3.4 with (5.2),
we have

$\displaystyle\begin{split}\|\widetilde{v}_{\delta}\|_{X^{-1,1}(T_{*})}&\lesssim\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}+\varepsilon\|\widetilde{v}_{\delta}\|_{L^{4}_{T_{*},x}}+\delta^{-1}\|v_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}\\\
&\lesssim\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}+\varepsilon\|\widetilde{v}_{\delta}\|_{L^{4}_{T_{*},x}}+\varepsilon\delta^{-1},\end{split}$
(5.6)

uniformly in $\delta_{0}\leq\delta\leq\infty$.

Lastly, note that (5.5) is different from (4.21) only on the first term on the
right-hand side of (5.5) (under the identification of $v_{1}$ and $v_{2}$ with
$v_{\delta}$ and $v_{\infty}$, respectively). Namely, the bound (4.33) holds
except for $c_{1}\|\widetilde{v}\|_{L^{\infty}_{T}L^{2}_{x}}$ in (4.33),
coming from (4.24) and (4.25). In this case, we instead apply (3.19) with
(5.2) to estimate the contribution from the first term on the right-hand side
of (5.5). This yields

$\displaystyle\begin{split}\|J^{s}\widetilde{v}_{\delta}\|_{L^{p}_{T_{*}}L^{q}_{x}}&\lesssim\|\widetilde{v}_{\delta}(0)\|_{L^{2}}+\Big{(}\|w_{\infty}\|_{Y^{s,\frac{1}{2}}(T_{*})}+\|v_{\delta}\|_{L^{\infty}_{T_{*}}H^{s}_{x}}+\varepsilon\|v_{\infty}\|_{L^{\infty}_{T_{*}}H^{s}_{x}}\Big{)}\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}\\\
&\quad+\varepsilon\|\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}H^{s}_{x}}+\|\widetilde{w}_{\delta}\|_{Y^{s,\frac{1}{2}}(T_{*})}+\varepsilon\delta^{-1}\end{split}$
(5.7)

for $0\leq s\leq\frac{1}{2}$ and $(p,q)=(\infty,2)$ or $(4,4)$, uniformly in
$\delta_{0}\leq\delta\leq\infty$, where the last term represents the
contribution from the first term on the right-hand side of (5.5).

Therefore, putting (5.4), (5.6), and (5.7) together with (4.12) (for
$\widetilde{w}_{\delta}(0)$ and $\widetilde{v}_{\delta}(0)$) and proceeding as
in Subsection 4.2 (first for $s=0$ as in (4.34) and then for $s$ as in
(4.36)), we obtain

$\displaystyle\begin{split}&\|J^{s}\widetilde{v}_{\delta}\|_{L^{\infty}_{T_{*}}L^{2}_{x}}+\|J^{s}\widetilde{v}\|_{L^{4}_{T_{*},x}}+\|\widetilde{w}_{\delta}\|_{Y^{s,\frac{1}{2}}(T_{*})}+\|\widetilde{v}_{\delta}\|_{X^{-1,1}(T_{*})}\\\
&\quad\leq
C(M)\|\widetilde{v}_{\delta}(0)\|_{H^{s}}+C(M)\delta^{-1},\end{split}$ (5.8)

uniformly in $\delta_{0}\leq\delta\leq\infty$, provided that $T_{*}>0$
sufficiently small. (Recall that $T_{*}>0$ is independent of
$1\leq\delta\leq\infty$.) Here, the last term on the right-hand side of (5.8)
comes from the last term in (5.6). Since the right-hand side of (5.8) tends to
0 as $\delta\to\infty$, we conclude that, as $\delta\to\infty$, the solution
$v_{\delta}$ to (3.2) converges to the solution $u=v_{\infty}$ to (1.5) in
$C([0,T_{*}];H^{s}(\mathbb{T}))\cap L^{4}([0,T_{*}];W^{s,4}(\mathbb{T}))$.
Given any $T_{0}\gg 1$, we can iterate this local-in-time convergence argument
and conclude convergence on the entire interval $[0,T_{0}]$.

Next, we transfer this convergence back to the original ILW equation (1.1).
Let $u_{\delta}$ be the global solution to (1.1) with
$u_{\delta}|_{t=0}=u_{0,\delta}$. Then, from (3.1), we have
$u_{\delta}=\tau_{-\delta^{-1}}v_{\delta}=\tau(-\delta^{-1},v_{\delta}).$
Letting $u$ be the global solution to the BO equation (1.5) with
$u|_{t=0}=u_{0}$ as above, it follows from the triangle inequality that

$\displaystyle\begin{split}\|u-u_{\delta}\|_{C_{T_{0}}H^{s}_{x}}&\leq\|u-\tau(-\delta^{-1},u)\|_{C_{T_{0}}H^{s}_{x}}+\|\tau(-\delta^{-1},u)-u_{\delta}\|_{C_{T_{0}}H^{s}_{x}}\\\
&\leq\|u-\tau(-\delta^{-1},u)\|_{C_{T_{0}}H^{s}_{x}}+\|u-v_{\delta}\|_{C_{T_{0}}H^{s}_{x}}.\end{split}$
(5.9)

By noting that $u=\tau(0,u)$, it follows from Lemma 2.5 that the first term on
the right-hand side of (5.9) tends to $0$ as $\delta\to\infty$, while the
second term tends to $0$ as $\delta\to\infty$ in view of the discussion above.
This proves convergence of $u_{\delta}$ to $u$ in
$C([0,T_{0}];H^{s}(\mathbb{T}))$ for any $T_{0}>0$.

$\bullet$ Part 2: In this step, we consider the general case without the mean-
zero assumption. Given $u_{0}\in H^{s}(\mathbb{T})$, let
$\\{u_{0,\delta}\\}_{1\leq\delta<\infty}$ be a net in $H^{s}(\mathbb{T})$ such
that $u_{0,\delta}$ converges to $u_{0}$ in $H^{s}(\mathbb{T})$ as
$\delta\to\infty$. Note that the spatial mean $\mu(u_{0,\delta})$ of
$u_{0,\delta}$ converges to the spatial mean $\mu(u_{0})$ of $u_{0}$ as
$\delta\to\infty$.

Let $u$ be the global solution to the BO equation (1.5) with $u|_{t=0}=u_{0}$
constructed in [42], and let $u_{\delta}$, $1\leq\delta<\infty$, be the global
solution to (1.1) with $u_{\delta}|_{t=0}=u_{0,\delta}$ constructed in Theorem
1.1. Then, by setting $\widetilde{u}=\Gamma_{u_{0}}(u)$ and
$\widetilde{u}_{\delta}=\Gamma_{u_{0,\delta}}(u_{\delta})$, where $\Gamma$ is
the Galilean transform defined in (1.7), we see that $\widetilde{u}$ (and
$\widetilde{u}_{\delta}$, respectively) is the global solution to (1.5) with
the mean-zero initial data
$\widetilde{u}|_{t=0}=\widetilde{u}_{0}=u_{0}-\mu(u_{0})$ (and to (1.1) with
the mean-zero initial data
$\widetilde{u}_{\delta}|_{t=0}=\widetilde{u}_{0,\delta}=u_{0,\delta}-\mu(u_{0,\delta})$,
respectively). Hence, from Part 1, we see that, given any $T_{0}>0$,
$\widetilde{u}_{\delta}$ converges to $\widetilde{u}$ in
$C([0,T_{0}];H^{s}(\mathbb{T}))$ as $\delta\to\infty$.

With $\tau_{h}$ as in (2.9), we have

$\displaystyle
u=\tau(2\mu(u_{0}),\widetilde{u})+\mu(u_{0})\qquad\text{and}\qquad
u_{\delta}=\tau(2\mu(u_{0,\delta}),\widetilde{u}_{\delta})+\mu(u_{0,\delta}).$

Therefore, in view of the convergence of $\mu(u_{0,\delta})$ and
$\widetilde{u}_{\delta}$ to $\mu(u_{0})$ and $\widetilde{u}$, respectively, we
conclude from the joint continuity of $\tau$ (Lemma 2.5) that $u_{\delta}$
converges to $u$ in $C([0,T_{0}];H^{s}(\mathbb{T}))$ as $\delta\to\infty$.
This concludes the proof of Theorem 1.2.

######  Acknowledgements.

This material is based upon work supported by the Swedish Research Council
under grant no. 2016-06596 while the first, third, and fourth authors were in
residence at Institut Mittag-Leffler in Djursholm, Sweden during the program
“Order and Randomness in Partial Differential Equations” in Fall, 2023. A.C.,
G.L., and T.O. were supported by the European Research Council (grant no.
864138 “SingStochDispDyn”). G.L. was also supported by the EPSRC New
Investigator Award (grant no. EP/S033157/1). D.P. was supported by a Trond
Mohn Foundation grant.

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