# Justification of the Benjamin-Ono equation as an internal water waves model

Martin Oen Paulsen  Department of Mathematics
University of Bergen
Postbox 7800
5020 Bergen
Norway<EMAIL_ADDRESS>
###### Abstract.

In this paper, we give the first rigorous justification of the Benjamin-Ono
equation:

$\hskip
85.35826pt\partial_{t}\zeta+c(1-\frac{\gamma}{2}\sqrt{\mu}|\mathrm{D}|)\partial_{x}\zeta+c\frac{3{\varepsilon}}{2}\zeta\partial_{x}\zeta=0,\hskip
85.35826pt\text{(BO)}$

as an internal water wave model on the physical time scale. To be precise, we
first prove the existence of a solution to the internal water wave equations
for a two-layer fluid with surface tension, where one layer is of shallow
depth and the other is of infinite depth. The existence time is of order
$\mathcal{O}(\frac{1}{{\varepsilon}})$ for ${\varepsilon}\leq\sqrt{\mu}$ and
with a small amount of surface tension
$\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$. Here, ${\varepsilon}$ and $\mu$
denote the nonlinearity and shallowness parameters, and $\mathrm{bo}$ is the
Bond number. Then, we show that these solutions are close, on the same time
scale, to the solutions of the BO equation with a precision of order
$\mathcal{O}(\mu)$.

The long-time well-posedness of the two-layer fluid problem was first studied
by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where
both fluids have finite depth. Here, we adapt this work to the case where one
of the fluid domains is of finite depth, and the other one is of infinite
depth. The novelties of the proof are related to the geometry of the problem,
where the difference in domains alters the functional setting for the
Dirichlet-Neumann operators involved. In particular, we study the various
compositions of these operators that require a refined symbolic analysis of
the Dirichlet-Neumann operator on infinite depth and derive new pseudo-
differential estimates that might be of independent interest.

###### Key words and phrases:

Benjamin-Ono equation, rigorous justification; long time well-posedness

###### 2010 Mathematics Subject Classification:

Primary: 35Q35; Secondary: 76B55, 76B45

##  1\. Introduction

###  1.1. The Benjamin-Ono equation

The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the
unidirectional propagation of weakly nonlinear, long internal waves in a two-
layer fluid. The equation is given by

$\partial_{t}\zeta+c(1-\frac{\gamma}{2}\sqrt{\mu}|\mathrm{D}|)\partial_{x}\zeta+c\frac{3{\varepsilon}}{2}\zeta\partial_{x}\zeta=0,$
(1.1)

where $x\in{\mathbb{R}}$, $t>0$ and $\zeta=\zeta(x,t)$ denotes the free
surface which is a real valued function. Here, ${\varepsilon}$ is a small
parameter measuring the weak nonlinearity of the waves, $\mu$ is the
shallowness parameter, $c>0$ is the wave speed, and $\gamma\in(0,1)$ is the
ratio between the densities of the two fluids. The operator $|\mathrm{D}|$ is
a Fourier multiplier defined by

$|\mathrm{D}|f(x)=\mathcal{F}^{-1}\big{(}|\xi|\mathcal{F}(f)(\xi)\big{)}(x).$

The BO equation was introduced formally by Benjamin [12] in 1967 and at the
same time independently by Davis and Acrivos [22]. We also refer the reader to
the book by Klein and Saut [46], Chapter 3, for a detailed state of the art.
The studies in [12, 22] showed that the BO equation admits solitary waves with
mere algebraic decay, as opposed to the exponential decay exhibited for the
solitary waves of the KdV equation. Davis and Acrivos also gave experimental
results. The experiments were carried out in a wave tank with a stratified
solution of salt and water, where almost any disturbance to the surface would,
after a short time, produce a wave with a fixed shape that propagates stably.
It was later noted by Ono [64] that the ease with which they could generate
solitary waves indicates that they are solitons.

The paper by Ono sparked much interest in studying the dynamics of the BO
equation. It was proved that the solitary waves are unique (up to translation)
[11], and the stability of these objects is studied in [13, 73, 43] (see the
references for a precise definition). Moreover, the stability of these waves
is strong enough to preserve its own identity upon nonlinear interactions. The
strong interaction between several solitary waves is studied in [55, 62] and
relies on explicit formulas (see also [43] for the asymptotic stability of one
soliton and $N-$solitons).

The fact that explicit solutions like the soliton (or multi-solitons) exist is
a consequence of the complete integrability of the BO equation. Nakamura [61]
proved the existence of an infinite number of conserved quantities and
discovered a Lax pair structure (see also [14, 25, 29]). This insight is
proven to be crucial for the study of the dynamics of the BO equation and was
further developed by Gérard and Kapeller [29]. They constructed a nonlinear
Fourier transform for the BO equation on the torus, which has several
applications to low regularity well-posedness of the initial value problem,
the long-time behavior of solutions, and stability of traveling waves (see
[28] for a survey on this topic). More recently, Gérard [31] derived an
explicit formula for the BO equation based on the Lax pair structure with
remarkable consequences, for instance, the zero-dispersion limit problem [32]
(see also [27, 26]).

The Cauchy problem for BO has been extensively studied in the last 40 years.
It was first proved to be globally well-posed in $H^{s}({\mathbb{R}})$ for
$s>\frac{3}{2}$ using an energy method, see [1, 38]. We also refer to the
results [66, 47, 42] for an improvment by including the dispersive smoothing
effects into the energy estimates. One of the main difficulties in improving
the result further is that the flow map fails to be $C^{2}$ in any Sobolev
space $H^{s}({\mathbb{R}})$ [60] (see also [48]). A breakthrough was achieved
by Tao [71], where he introduced a clever change of variables (the gauge
transform) to improve the structure of the nonlinearity. As a consequence, he
obtained a global well-posedness result for data in $H^{1}({\mathbb{R}})$.
Several papers expanded on these ideas. We refer the interested reader to [17,
37, 59, 34] for results on the line and [57, 58, 59] in the periodic case
culminating in the global well-posedness in $L^{2}$. So far, the theory is
based on PDE methods. However, by actively using the integrable structure,
Gérard, Kappeler, and Topalov [30] proved the sharp global well-posedness
result in $H^{s}(\mathbb{T})$ for $s>-\frac{1}{2}$ on the torus. Also, still
relying on the integrability, Killip, Laurens, and Vişan [45] recently proved
the global well-posedness in $H^{s}(\mathbb{R})$ for $s>-\frac{1}{2}$ on the
line.

0.8

Figure 1. The blue line denotes the surface elevation $z={\varepsilon}\zeta$
and separates two fluids with density $0<\rho^{-}<\rho^{+}$.

####  1.1.1. Full justification

Despite the rich well-posedness theory for the BO equation, it is still an
open question whether its solutions are close to the ones of the original
physical system. In the rigorous derivation of any asymptotic model, it is
fundamental to know whether its solutions converge to the solutions of the
reference model from which it is derived. The BO equation’s reference model is
a coupled system of Euler equations for two fluids that are joined with an
interface, as in Figure 1. Under the irrotationality condition, we will call
the reference model the “internal water waves system”. To prove that BO is a
valid approximation, we shall compare their solutions to the physical
parameters:

${\varepsilon}=\frac{a}{\lambda}\quad,\quad\mu=\frac{H^{2}}{\lambda^{2}}\quad\text{and}\quad\mathrm{bo}=\frac{\rho^{+}g\lambda^{2}}{\sigma},$

where $a$ is the typical amplitude of the waves, $H$ is the still water depth
in the lower fluid, $\lambda$ is the typical wavelength, $\rho^{+}$ is the
density of the lower fluid, $g$ is the acceleration of gravity, $\sigma$ is
the surface tension parameter. Since surface tension is only relevant for
short waves, we will suppose that $\mathrm{bo}^{-1}$ is small. To be precise,
we will let ${\varepsilon}\lesssim\sqrt{\mu}$ and
$\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$, where we answer the following
three questions:

  1. 1.

The solutions of the internal water wave equations exist on the relevant time
scale $\mathcal{O}(\frac{1}{{\varepsilon}})$.

  2. 2.

The solutions of the BO equation exist (at least) on the time scale
$\mathcal{O}(\frac{1}{{\varepsilon}})$.

  3. 3.

Lastly, we must establish the consistency between the BO equation and the
internal water wave equations and then show that the error is of order
$\mathcal{O}(\mu t)$ when comparing the two solutions.

The first point is the most challenging step of this paper, where we need to
prove that the internal water waves equations are long-time well-posed for
regular initial data. Moreover, we are confined to the specific geometry where
one fluid layer is of shallow depth and the other of infinite depth. The main
obstacle to constructing such solutions is the tendency of internal waves to
break down due to Kelvin-Helmholtz instabilities. This issue was resolved in
the case of a single fluid (i.e., $\rho^{-}=0$), where stable solutions are
deduced by imposing the Rayleigh-Taylor criterion:

$-\partial_{z}P^{+}|_{z={\varepsilon}\zeta}>0,$ (1.2)

where $z$ is the vertical coordinate, and $P^{+}|_{z={\varepsilon}\zeta}$ is
the pressure at the surface. The physical relevance of this criterion can be
seen by considering the Euler equations for a trivial flow
$-\partial_{z}P^{+}|_{z={\varepsilon}\zeta}=\rho^{+}g$, where gravity $g>0$ is
the restoring force. In this sense, the criterion is a natural condition to
ensure that the pressure force is restoring and does not amplify the waves
[72]. From a mathematical perspective, the criterion ensures the hyperbolicity
of the water waves equation. Moreover, it is proved that under this condition,
the water waves system in finite depth is locally well-posed by Lannes [49]
and then long-time well-posed by Alvarez-Samaniego and Lannes [8]. We also
refer the reader to the pioneering work of Wu [74, 75] in the case of infinite
depth where one of the key observations was the use of the Rayleigh-Taylor
criterion (1.2) to remove a smallness condition on the data (see also the more
recent work on extended life span and improved regularity results [76, 77, 78,
7, 5, 6, 3, 2]).

In the case of two fluid systems, the internal water waves system produces
Kelvin-Helmholtz instabilities and becomes ill-posed unless there is surface
tension $\sigma>0$ [23, 36, 39]. There are several results on the well-
posedness of the internal water wave systems in different configurations of
the fluid domain where the time of existence depends on $\sigma$ [9, 10, 18,
69, 70]. On the other hand, the strength of surface tension is only relevant
for very small values in water waves theory. Therefore, we must add surface
tension to exploit its regularizing effect, but obtain a uniform local well-
posedness result with respect to $\sigma>0$, allowing it to be taken small.
Lannes solved this problem in [51] for two fluid layers of finite depth, where
he derived a new stability criterion depending on $\sigma$. A crucial point is
that surface tension could be taken small enough in the criterion such that it
does not affect the main dynamics of the equation. It is also noted in the
paper that the criterion depends strongly on the geometry of the problem. Many
of the technical difficulties in this paper are related to this observation.
In this work, we consider one of the layers to be of infinite depth. This is
the key point of the paper, which will require a symbolic analysis of the
Dirichlet-Neumann operator in a new functional setting. To achieve this goal,
we derive several pseudo-differential estimates for symbols with limited
smoothness.

The second point is already well-known since the BO equation is globally well-
posed for regular data (see the discussion above). However, we will consider
two intermediate models to derive the BO equation. We will first derive a
weakly dispersive BO-type system from the internal water waves system that is
consistent with a precision of order $\mathcal{O}(\mu)$. Then we will consider
unidirectional solutions of this system, with the same precision, to deduce a
weakly dispersive BO equation, which is also consistent with the BO equation.
To derive these models, we closely follow the work of Bona, Lannes, and Saut
in [15]. In this paper, they derive several shallow water models for internal
fluids and comment on the formal derivation of the BO equation.

Finally, we comment on several works that are closely related to the
derivation of the BO equation. In [19], Craig, Guyenne, and Kalisch used a
Hamiltonian perturbation approach to formally derive asymptotic models from
the two-layer system. Among the models is the BO equation. The benefit of this
approach is that the systems inherit the Hamiltonian structure, but as noted
in [46], the process could lead to ill-posed systems. In particular, the BO
system they derive, which links the BO equation, is linearly ill-posed. In
[33], Ifrim, Rowan, Tataru, and Wan show that the BO equation can also be
viewed as an asymptotic model from the water waves equations in infinite depth
in the case of constant vorticity. The approximation they obtain is rigorously
justified but, of course, not related to the asymptotic description of
internal waves. Lastly, in [63], Ohi and Iguchi proved the well-posedness of
the internal water waves for one fluid of infinite depth to derive the BO
equation. However, in their paper, the existence time is of order
$\mathcal{O}(\mathrm{bo}^{-\frac{1}{2}})$, which is too short to justify the
BO equation on the physical time scale. The technique is based on the one of
Wu [74], where the reference model is given in holomorphic coordinates. We
will instead use a version of the Zakharov-Craig-Sulem formulation and closely
follow the work of Lannes [51]. In particular, the goal of this paper is to
prove the long-time existence of the internal water waves equations with one
fluid of infinite depth and positive surface tension. Then we show that the
difference between two regular solutions of the internal water waves equations
and the BO equation, which evolves from the same initial datum, is bounded by
$\mathcal{O}(\mu)$ for all $0\leq t\lesssim{\varepsilon}^{-1}$ for any
${\varepsilon},\mu\in(0,1)$ such that ${\varepsilon}\lesssim\mu$ and
$\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$.

###  1.2. The governing equations

The basis of this study is the Euler equations for an irrotational two-layer
fluid written in the Zakharov-Craig-Sulem formulation [80, 20, 21]. For the
upper layer, we consider the following set of equations

$\begin{cases}\partial_{t}\zeta-\mathcal{G}^{-}[\zeta]\psi^{-}=0\\\
\rho^{-}\Big{(}\partial_{t}\psi^{-}+g\zeta+\frac{1}{2}(\partial_{x}\psi^{-})^{2}-\frac{1}{2}\frac{(\mathcal{G}^{-}[\zeta]\psi^{-}+\partial_{x}\zeta\partial_{x}\psi^{-})^{2}}{1+(\partial_{x}\zeta)^{2}}\Big{)}=-P^{-}|_{z=\zeta}.\end{cases}$
(1.3)

Here, the free surface elevation is the graph of $\zeta(t,x)\in{\mathbb{R}}$,
the function $P^{-}|_{z=\epsilon\zeta}$ is the pressure force at the free
surface. The function $\psi^{-}(t,x)\in{\mathbb{R}}$ is the trace at the
surface of the velocity potential solving the elliptic problem

$\begin{cases}(\partial_{x}^{2}+\partial_{z}^{2})\Phi^{-}=0\quad\text{for}\quad\Omega^{-}=\\{(x,z)\>:\>z>\zeta\\}\\\
\Phi^{-}|_{z=\zeta}=\psi^{-},\end{cases}$ (1.4)

and $\mathcal{G}^{-}$ is the negative Dirichlet-Neumann operator defined by

$\mathcal{G}^{-}[\zeta]\psi^{-}=(\partial_{z}\phi^{-}+\partial_{x}\zeta\partial_{x}\phi^{-})|_{z=\zeta}.$

For the fluid in the lower layer, the governing equations are given in terms
of $(\zeta,\psi^{+})$ and read

$\begin{cases}\partial_{t}\zeta-\mathcal{G}^{+}[\zeta]\psi^{+}=0\\\
\rho^{+}\Big{(}\partial_{t}\psi^{+}+g\zeta+\frac{1}{2}(\partial_{x}\psi^{+})^{2}-\frac{1}{2}\frac{(\mathcal{G}^{+}[\zeta]\psi^{+}+\partial_{x}\zeta\partial_{x}\psi^{+})^{2}}{1+(\partial_{x}\zeta)^{2}}\Big{)}=-P^{+}|_{z=\zeta},\end{cases}$
(1.5)

where the elliptic problem in the lower fluid is given by

$\begin{cases}(\partial_{x}^{2}+\partial_{z}^{2})\Phi^{+}=0\quad\text{for}\quad\Omega^{+}=\\{(x,z)\>:\>-H<z<\zeta\\}\\\
\Phi^{+}|_{z=\zeta}=\psi^{+}\quad\partial_{z}\Phi^{+}|_{z=-H}=0,\end{cases}$
(1.6)

and the positive Dirichlet-Neumann operator is defined by

$\mathcal{G}^{+}[\zeta]\psi^{+}=(\partial_{z}\phi^{+}+\partial_{x}\zeta\partial_{x}\phi^{+})|_{z=\zeta}.$

To ease the notation, we make the following simplifications

$\gamma=\frac{\rho^{-}}{\rho^{+}},\quad\rho^{-}<\rho^{+}=1,\quad g=1.$

Moreover, we recall that the difference in pressure at the interface is
proportional to the mean curvature of the interface:

$P^{+}|_{z=\zeta}-P^{-}|_{z=\zeta}=\sigma\kappa(\zeta),$

where $\sigma\in(0,1)$ is the surface tension parameter and $\kappa(\zeta)$ is
defined by

$\kappa(\zeta)=-\partial_{x}\Big{(}\frac{\partial_{x}\zeta}{\sqrt{1+(\partial_{x}\zeta)^{2}}}\Big{)}.$
(1.7)

We will now collect all these equations into one system, where we reduce the
number of unknowns by using the first equation in (1.5) and (1.3) to see that

$\mathcal{G}^{-}[\zeta]\psi^{-}=\mathcal{G}^{+}[\zeta]\psi^{+}.$ (1.8)

In particular, we will prove later that we can write $\psi^{-}$ as a function
of $\psi^{+}$ through the inverse relation

$\psi^{-}=(\mathcal{G}^{-}[\zeta])^{-1}\mathcal{G}^{+}[\zeta]\psi^{+}.$

Then, we can define the new variable $\psi$ by the formula

$\displaystyle\psi$ $\displaystyle=\psi^{+}-\gamma\psi^{-}$
$\displaystyle=\big{(}1-\gamma(\mathcal{G}^{-}[\zeta])^{-1}\mathcal{G}^{+}[\zeta]\big{)}\psi^{+}$
$\displaystyle=\mathcal{J}[\zeta]\psi^{+}.$

The unknowns $\zeta$ and $\psi$ are the primary variables. We follow the work
of Lannes [51] to show that we can use them to recover the velocity potentials
$\Phi^{\pm}$ through the transmission problem:

$\begin{cases}\Delta_{x,z}\Phi^{\pm}=0\hskip
14.22636pt\qquad\text{in}\quad\Omega^{\pm}\\\
\Phi^{+}|_{z=\zeta}-\gamma\Phi^{-}|_{z=\zeta}=\psi\\\
\partial_{n}\Phi^{-}|_{{z=\zeta}}=\partial_{n}\Phi^{+}|_{{z=\zeta}},\quad\partial_{z}\Phi^{+}|_{z=-H}=0,\end{cases}$
(1.9)

where $\psi^{\pm}=\Phi^{\pm}|_{z=\zeta}$ and the normal condition on $z=\zeta$
is the same as (1.8). From these relations, it will be possible to reduce the
two-fluid equations into a set of equations defined by $\zeta$ and $\psi$
where we formally define a new Dirichlet-Neumann operator that links the two
fluids through the relation,

$\mathcal{G}[\zeta]=\mathcal{G}^{+}[\zeta](\mathcal{J}[\zeta])^{-1}.$ (1.10)

From the above expressions, we have the main governing equations (in
dimensional form) that we will study throughout this paper:

$\begin{cases}\partial_{t}\zeta-\mathcal{G}[\zeta]\psi=0\\\
\partial_{t}\psi+(1-\gamma)\zeta+\frac{1}{2}\big{(}(\partial_{x}\psi^{+})^{2}-\gamma(\partial_{x}\psi^{-})^{2}\big{)}+\mathcal{N}[\zeta,\psi^{\pm}]=-\sigma\kappa(\zeta),\end{cases}$
(1.11)

where

$\mathcal{N}[\zeta,\psi^{\pm}]=\frac{\gamma(\mathcal{G}^{-}[\zeta]\psi^{-}+\partial_{x}\zeta\partial_{x}\psi^{-})^{2}-(\mathcal{G}^{+}[\zeta]\psi^{+}+\partial_{x}\zeta\partial_{x}\psi^{+})^{2}}{2(1+(\partial_{x}\zeta)^{2})}.$

####  1.2.1. Nondimensionalization of the equations

To derive an asymptotic model from (1.11), we will compare every variable and
function with physical characteristic parameters of the same dimension $H,a$,
or $\lambda$. Since the BO equation describes long waves, it is natural to
consider the scaling:

$x=\lambda x^{\prime},\quad\zeta=a\zeta^{\prime},$

where the prime notation denotes a nondimensional quantity. To identify the
remaining variables $\psi^{\prime}$ and
$t=\frac{\lambda}{c_{\mathrm{ref}}}t^{\prime}$ one needs information on the
reference velocity $c_{\mathrm{ref}}$. To do so, we look at the linearized
equations (with $\sigma=0$):

$\begin{cases}\partial_{t}\zeta-\mathcal{G}[0]\psi=0\\\
\partial_{t}\psi+(1-\gamma)\zeta=0,\end{cases}$ (1.12)

where $\mathcal{G}[0]$ is a Fourier multiplier given by111See Remark 2.3

$\mathcal{G}[0]\psi(x)=\mathcal{F}^{-1}\Big{(}|\xi|\frac{\mathrm{tanh}(H|\xi|)}{1+\gamma\mathrm{tanh}(H|\xi|)}\hat{\psi}(\xi)\Big{)}(x).$

For a wave with typical wavelength $\lambda$, the frequencies are concentrated
around $|\xi|=\frac{2\pi}{\lambda}$. Therefore, if we suppose that the depth
of lower fluid is small compared to the wavelength, then we have by a Taylor
expansion that

$\mathcal{G}[0]\psi=-H\partial_{x}^{2}\psi,$

up to higher order terms in $\mu$. From this simplification we can reduce
(1.12) to a wave equation where we make the identification
$c_{\mathrm{ref}}^{2}=H(1-\gamma)$, and from the second equation we find the
dimensions of $\psi$:

$\psi=\frac{a\lambda}{\sqrt{H}}\psi^{\prime}.$

Lastly, we also choose to scale the transverse variable with $H$ (i.e.
$z=Hz^{\prime}$) to have a reference domain in the lower fluid of unitary
depth. Then applying these changes of variables and dropping the prime
notation, we find that the nondimensional internal water waves system (1.11)
is given by:

$\begin{cases}\partial_{t}\zeta-\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi=0\\\
\partial_{t}\psi+(1-\gamma)\zeta+\frac{1}{2}\big{(}{\varepsilon}(\partial_{x}\psi^{+})^{2}-\gamma{\varepsilon}(\partial_{x}\psi^{-})^{2}\big{)}+{\varepsilon}\mathcal{N}[{\varepsilon}\zeta,\psi^{\pm}]=-\frac{1}{\mathrm{bo}}\frac{1}{{\varepsilon}\sqrt{\mu}}\kappa({\varepsilon}\sqrt{\mu}\zeta),\end{cases}$
(1.13)

where

$\mathcal{N}[{\varepsilon}\zeta,\psi^{\pm}]=\frac{1}{2\mu}\frac{\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}+{\varepsilon}\mu\partial_{x}\zeta\partial_{x}\psi^{-})^{2}-(\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}+{\varepsilon}\mu\partial_{x}\zeta\partial_{x}\psi^{+})^{2}}{(1+\varepsilon^{2}\mu(\partial_{x}\zeta)^{2})}.$

The operators $\mathcal{G}_{\mu}^{\pm}[{\varepsilon}\zeta]$ are defined by

$\mathcal{G}_{\mu}^{\pm}[{\varepsilon}\zeta]\psi^{\pm}=(\partial_{z}\Phi^{\pm}+{\varepsilon}\mu\partial_{x}\zeta\partial_{x}\Phi^{\pm})|_{z={\varepsilon}\zeta},$

through the solutions of the scaled Laplace equations:

$\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\Phi^{+}=0\quad\text{for}\quad\Omega^{+}=\\{(x,z)\>:\>-1<z<{\varepsilon}\zeta\\}\\\
\Phi^{+}|_{z={\varepsilon}\zeta}=\psi^{+}\quad\partial_{z}\Phi^{+}|_{z=-1}=0,\end{cases}$

and

$\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\Phi^{-}=0\quad\text{for}\quad\Omega^{-}=\\{(x,z)\>:\>z>{\varepsilon}\zeta\\}\\\
\Phi^{-}|_{z={\varepsilon}\zeta}=\psi^{-}.\end{cases}$

###  1.3. Main results

In this paper we will first prove the well-posedness of (1.13) on a time scale
$\mathcal{O}(\frac{1}{{\varepsilon}})$ for ${\varepsilon}\lesssim\sqrt{\mu}$
and $\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$. To state this result, there
are two fundamental assumptions that we need to make.

######  Definition 1.1 (Non-cavitation condition).

Let ${\varepsilon}\in(0,1)$, $s>\frac{1}{2}$ and take $\zeta_{0}\in
H^{s}({\mathbb{R}})$. We say $\zeta_{0}$ satisfies the “non-cavitation
condition” if there exist $h_{\min}\in(0,1)$ such that

$h=1+{\varepsilon}\zeta_{0}(x)\geq h_{\min},\quad\text{for
all}\>\>\>x\in\mathbb{R}.$ (1.14)

The second condition is to ensure the solutions do not break down due to
Kelvin-Helmholtz instabilities and is key to showing the long-time existence
for solutions of (1.13). The criterion is enforced for data in the energy
space, which we will now define.

######  Definition 1.2 (Energy space).

Let ${\varepsilon},\mu,\gamma,\mathrm{bo}^{-1}\in(0,1)$ and
$N\in{\mathbb{N}}$. Then we define the function space
$H^{N+1}_{\gamma,\mathrm{bo}}({\mathbb{R}})$ by

$H^{N+1}_{\gamma,\mathrm{bo}}({\mathbb{R}})=H^{N+1}({\mathbb{R}}),$

with norm

$|u|_{H^{N+1}_{\gamma,\mathrm{bo}}}^{2}=(1-\gamma)|u|_{H^{N}}^{2}+\mathrm{bo}^{-1}|\partial_{x}u|^{2}_{H^{N}}.$

We define $\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$ as a Beppo-Levi space

$\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})=\dot{H}^{s+\frac{1}{2}}({\mathbb{R}})=\\{u\in
L^{2}_{loc}({\mathbb{R}})\>:\>\partial_{x}u\in
H^{s-\frac{1}{2}}({\mathbb{R}})\\},$

endowed with

$|u|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}=\Big{|}\frac{|\mathrm{D}|}{(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}}u\Big{|}_{H^{s}}.$

Moreover, let $\alpha\in{\mathbb{N}}^{2}$ and define the “good unknowns” by

$\zeta_{(\alpha)}=\partial_{x,t}^{\alpha}\zeta,\quad\psi_{(\alpha)}=\partial_{x,t}^{\alpha}\psi-{\varepsilon}\underline{w}\partial_{x,t}^{\alpha}\zeta,\quad\psi_{(0)}=\psi,\quad\zeta_{(0)}=\zeta.$

Then the natural energy space $\mathscr{E}_{\mathrm{bo},T}^{N}$ associated to
(1.13) is defined for functions
$\mathbf{U}=(\zeta_{(\alpha)},\psi_{(\alpha)})$ in

$\mathscr{E}_{\mathrm{bo},T}^{N}=\\{\mathbf{U}\in
C([0,T];H^{N}({\mathbb{R}})\times\dot{H}^{t_{0}+3}({\mathbb{R}}),\quad\sup\limits_{t\in[0,T]}\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(U(t))<\infty\\},$
(1.15)

whose norm is the square root of

$\mathcal{E}^{0}_{\mathrm{bo},\mu}(\mathbf{U})=|\zeta|_{H^{1}_{\gamma,\mathrm{bo}}}^{2}+|\psi|^{2}_{\dot{H}^{\frac{1}{2}}_{\mu}},$
(1.16)

and for $N\geq 1$ we define it by

$\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U})=|\partial_{x}\psi|_{H^{t_{0}+2}}^{2}+\sum\limits_{\alpha\in{\mathbb{N}}^{2},|\alpha|\leq
N}|\zeta_{(\alpha)}|_{H^{1}_{\gamma,\mathrm{bo}}}^{2}+|\psi_{(\alpha)}|^{2}_{\dot{H}^{\frac{1}{2}}_{\mu}}.$
(1.17)

######  Remark 1.3.

Here, the energy depends on both time derivatives and spatial derivatives.
This is the method put forward by [67, 68, 56] to control the surface tension
term for the water wave equations (see Remark 5.1 for the specifics on this
point). This method was later used for the internal water waves equations with
surface tension in the case of two fluids of finite depth [51], which is one
of the primary references of this paper.

######  Definition 1.4 (Stability criterion).

Let $\mathbf{U}_{0}=(\zeta_{0},\psi_{0})\in
L^{2}({\mathbb{R}})\times\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$ and
$\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U}_{0})<\infty$. Then we
define the “stability criterion” by

$\qquad
0<\mathfrak{d}(\mathbf{U}):=\inf\limits_{{\mathbb{R}}}\mathfrak{a}-\Upsilon\mathfrak{c}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{4},\quad\text{at}\>\>\>t=0,$
(1.18)

where

$\Upsilon=\frac{\mathrm{bo}}{4}(1-\gamma)^{2}\gamma^{2}{\varepsilon}^{2}\mu,$

and

$\displaystyle\mathfrak{a}$
$\displaystyle=\Big{(}(1-\gamma)+{\varepsilon}\big{(}(\partial_{t}+{\varepsilon}\underline{V}^{+}\partial_{x})\underline{w}^{+}-\gamma(\partial_{t}+{\varepsilon}\underline{V}^{-}\partial_{x})\underline{w}^{-}\big{)}\Big{)}$
$\displaystyle\mathfrak{e}(\zeta)$ $\displaystyle=\sup\limits_{f\in
H^{\frac{1}{2}}({\mathbb{R}}),f\neq
0}\frac{\big{(}(\mathcal{J}_{\mu}[{\varepsilon}\zeta])^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}f,\partial_{x}f\big{)}_{L^{2}}}{|1+\sqrt{\mu}|\mathrm{D}|^{\frac{1}{2}}f|_{L^{2}}^{2}}$
$\displaystyle\mathfrak{c}(\zeta)$
$\displaystyle=\mathfrak{e}(\zeta)^{2}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}}^{2})^{\frac{3}{2}}.$

The quantities $\underline{V}^{\pm}$, $\underline{w}^{\pm}$ describe the
horizontal and vertical velocity field in the fluids and is given in
Definition 4.1. See also Corollary A.18 in the Appendix, where they are given
in terms of $\zeta$ and $\psi$.

######  Remark 1.5.

The stability criterion (1.18) can be seen as a two-layer generalization of
the Rayleigh-Taylor criterion where

$\mathfrak{a}=-\big{(}\partial_{z}P^{+}-\gamma\partial_{z}P^{-}\big{)}|_{z={\varepsilon}\zeta}>\Upsilon\mathfrak{c}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{4}.$

Moreover, we let $\mathrm{bo}^{-1}$ to be of order
$\mathcal{O}({\varepsilon}\sqrt{\mu})$. In this case, the size of the quantity
$\Upsilon$ is of order $\mathcal{O}({\varepsilon}\sqrt{\mu})$ and is neglected
in the BO regime.

######  Remark 1.6.

One key difference with the work of Lannes [51] for the internal water waves
on finite depth is in the symbolic analysis of
$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]$. The operator depend on the
solution of an elliptic problem on a domain with infinite depth. This alters
the functional setting, where we also need precise estimates depending on the
parameters ${\varepsilon},\mu\in(0,1)$.

######  Theorem 1.7.

Let $t_{0}=1$, $N\geq 5$, ${\varepsilon},\mu,\gamma\in(0,1)$ such that
${\varepsilon}\leq\sqrt{\mu}$ and $\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$.
Assume that $\mathbf{U}_{0}=(\zeta_{0},\psi_{0})^{T}\in
L^{2}({\mathbb{R}})\times\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$ such that
$\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U}_{0})<\infty$. Suppose
further that $\mathbf{U}_{0}$ satisfies the non-cavitation condition (1.14)
and the stability criterion (1.18). Then there exists a constant
$C=C(h_{0}^{-1},\gamma,\frac{1}{\mathfrak{d}(\mathbf{U}_{0})})>0$ and a time

$T=T(C\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U}_{0}))>0,$

which is a nonincreasing function of its argument, and a unique solution
$\mathbf{U}=(\zeta,\psi)^{T}\in\mathscr{E}_{\mathrm{bo},{\varepsilon}^{-1}T}^{N}$
of (1.13). Moreover, the solution satisfies

$\sup\limits_{t\in[0,{\varepsilon}^{-1}T]}\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U})\leq\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U}_{0}).$
(1.19)

######  Remark 1.8.

For notational convenience we shall write $\mathcal{E}^{N}(\mathbf{U})$
instead of $\mathcal{E}^{N,t_{0}}_{\mathrm{bo},\mu}(\mathbf{U})$. We also
consider the case of one horizontal dimension since our primary goal is to
justify the BO equation, which is a model that does not include transverse
effects. We will deal with the higher dimensional case in a forthcoming paper.

######  Remark 1.9.

The local well-posedness of (1.13) was first proved by Ohi and Iguchi [63].
However, in their paper, the existence time is of order
$\mathcal{O}(\mathrm{bo}^{-\frac{1}{2}})$, which is far too short to justify
the BO equation on the physical time scale. In fact, Theorem 1.7 is the first
proof of the long-time well-posedness of the internal water waves in the case
where one layer is of infinite depth.

######  Remark 1.10.

The surface tension term is regularizing and plays a fundamental role in the
well-posedness of (1.13). However, as noted in Remark 1.5, it does not affect
the dynamics of the BO equation.

Having defined a solution of the reference model (1.13) on a long time scale,
the next step is to derive the asymptotic models. Here we follow the road map
in [15], where they derived several internal water wave models in finite depth
and gave comments on the formal derivation of the BO equation. In particular,
it is convenient to write (1.13) in terms of
$\psi^{+}\in\dot{H}^{\frac{3}{2}}_{\mu}({\mathbb{R}})$ through the interface
operator:

$\mathbf{H}[{\varepsilon}\zeta]\psi^{+}=\partial_{x}\psi^{-}\in
H^{\frac{1}{2}}({\mathbb{R}}),$

where
$\psi^{-}=\Phi^{-}|_{z={\varepsilon}\zeta}\in\mathring{H}^{\frac{3}{2}}({\mathbb{R}})$
and $\Phi^{-}\in\dot{H}^{2}(\Omega^{-})$ is the unique solution222See
Proposition 2.4. (up to a constant) of

$\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\Phi^{-}=0\hskip
95.3169pt\qquad\text{in}\quad\Omega^{-}\\\
\partial_{n}\Phi^{-}=(1+{\varepsilon}^{2}(\partial_{x}\zeta)^{2})^{-\frac{1}{2}}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\hskip
35.56593pt\text{on}\quad z={\varepsilon}\zeta.\end{cases}$ (1.20)

Then we may define the the velocity variable

$\displaystyle v$ $\displaystyle=\partial_{x}\psi$ (1.21)
$\displaystyle=\partial_{x}\psi^{+}-\gamma\mathbb{H}[{\varepsilon}\zeta]\psi^{+},$

and apply a derivative to the second equation of (1.13) to find that

$\begin{cases}\partial_{t}\zeta-\frac{1}{\mu}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}=0\\\
\partial_{t}v+(1-\gamma)\partial_{x}\zeta+\frac{{\varepsilon}}{2}\partial_{x}\big{(}(\partial_{x}\psi^{+})^{2}-\gamma(\mathbb{H}[{\varepsilon}\zeta]\psi^{+})^{2}\big{)}+{\varepsilon}\partial_{x}\mathcal{N}[{\varepsilon}\zeta,\psi^{\pm}]=-\frac{1}{\mathrm{bo}}\frac{1}{{\varepsilon}\sqrt{\mu}}\partial_{x}\kappa({\varepsilon}\sqrt{\mu}\zeta).\end{cases}$
(1.22)

As noted in the introduction, we will first derive a system from (1.22), which
is consistent in the BO regime, and then use it to derive a weakly dispersive
BO equation from which the derivation of the BO equation is direct. In
particular, we will show that a solution of the internal water waves
equations, with regular data $(\zeta_{0},v_{0})$, solves a weakly dispersive
BO system:

$\begin{cases}\partial_{t}\zeta+(1-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=0\\\
\partial_{t}v+c^{2}\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=0,\end{cases}$

up to an error of order $\mathcal{O}(\mu+{\varepsilon}\sqrt{\mu})$. Then under
an additional assumption on the data (for right-moving waves), we will show
that we can approximate this system with the solutions of

$\partial_{t}\zeta+c\big{(}1-\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|)\big{)}\partial_{x}\zeta+c\frac{3{\varepsilon}}{2}\zeta\partial_{x}\zeta=0.$

######  Remark 1.11.

An alternative approach would be to rigorously derive the regularized BO
system given by:

$\begin{cases}\big{(}1+\alpha\sqrt{\mu}\gamma|\mathrm{D}|\big{)}\partial_{t}\zeta+\big{(}1+(\alpha-1)\gamma\sqrt{\mu}|\mathrm{D}|\big{)}\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=0\\\
\partial_{t}v+c^{2}\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=0,\end{cases}$
(1.23)

for $\alpha\geq 0$. This is the model that was formally derived in [15], and
moreover we can use it to derive the “regularized Benjamin-Ono equation” [21]:

$\big{(}1+\alpha\sqrt{\mu}\gamma|\mathrm{D}|\big{)}\partial_{t}\zeta+c\partial_{x}\zeta+(2\alpha-1)\frac{\gamma}{2}\sqrt{\mu}|\mathrm{D}|\partial_{x}\zeta+c\frac{3{\varepsilon}}{2}\zeta\partial_{x}\zeta=0.$

The rigorous derivation of these models is straightforward when having Theorem
1.7 at hand. However, the result would depend on $\alpha$ since the long-time
well-posedness of (1.23) requires $\alpha>1$ [79].

Before we proceed, we establish the long-time well-posedness of the weakly
dispersive models introduced above. To do so, we will have to sharpen the non-
cavitation condition (1.14) to define an energy associated with (1.).

######  Definition 1.12 ($\gamma-$dependent surface condition).

Let ${\varepsilon},\gamma\in(0,1)$ and $s>\frac{1}{2}$. We say the initial
surface elevation $\zeta_{0}\in H^{s}(\mathbb{R}^{d})$ satisfy the
“$\gamma-$dependent surface condition” if there exist
$h_{\min,\gamma}\in(0,1)$ such that

$1+{\varepsilon}\zeta_{0}(x)-\gamma\geq h_{\min,\gamma},\quad\text{for
all}\>\>\>x\in\mathbb{R}.$ (1.24)

######  Remark 1.13.

The main difference with (1.23) is that we can impose a physical constraint on
the data instead of a constraint on the parameter in the equation.

######  Theorem 1.14.

Let ${\varepsilon},\mu,\gamma,c\in(0,1)$ and $s>\frac{3}{2}$. Assume that
$(\zeta_{0},v_{0})\in H^{s}({\mathbb{R}})\times H^{s}({\mathbb{R}})$. Then
there exists a constant $C=C(h_{\min,\gamma}^{-1},\gamma)$ and a time

$T=T(C|(\zeta_{0},v_{0})|_{H^{s}\times H^{s}})>0,$

which is a nonincreasing function of its argument such that:

  * 1\.

There exists a unique solution $\zeta^{\text{\tiny wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{s}({\mathbb{R}})\big{)}$ to equation

$\partial_{t}\zeta^{\text{\tiny
wBO}}+c\big{(}1-\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|)\big{)}\partial_{x}\zeta^{\text{\tiny
wBO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
wBO}}\partial_{x}\zeta^{\text{\tiny wBO}}=0,$ (1.25)

that satisfies

$\sup\limits_{t\in[0,{\varepsilon}^{-1}T]}|\zeta^{\text{\tiny
wBO}}|_{H^{s}}\leq C|\zeta_{0}|_{H^{s}}.$ (1.26)

  * 2.

Suppose further that $\zeta_{0}$ satisfies the $\gamma-$dependent surface
condition (1.24). Then there exist a unique solution $(\zeta^{\text{\tiny
BOs}},v^{\text{\tiny BOs}})\in
C([0,{\varepsilon}^{-1}T]:H^{s}\big{(}\mathbb{R})\times
H^{s}({\mathbb{R}})\big{)}$ to system

$\begin{cases}\partial_{t}\zeta^{\text{\tiny
BOs}}+(1-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v^{\text{\tiny
BOs}}+{\varepsilon}\partial_{x}(\zeta^{\text{\tiny BOs}}v^{\text{\tiny
BOs}})=0\\\ \partial_{t}v^{\text{\tiny
BOs}}+c^{2}\partial_{x}\zeta^{\text{\tiny BOs}}+{\varepsilon}v^{\text{\tiny
BOs}}\partial_{x}v^{\text{\tiny BOs}}=0,\end{cases}$ (1.27)

that satisfies

$\sup\limits_{t\in[0,{\varepsilon}^{-1}T]}|(\zeta^{\text{\tiny
BOs}},v^{\text{\tiny BOs}})|_{H^{s}\times H^{s}}\leq
C|(\zeta_{0},v_{0})|_{H^{s}\times H^{s}}.$ (1.28)

######  Remark 1.15.

System (1.) and (1.25) are new and is chosen such that it is easy to deduce
the long-time existence. The choice was based on the observations made in
[65], where weakly dispersive shallow water models are considered and can, in
some cases, give rise to well-posed systems while their strongly dispersive
versions are not.

######  Remark 1.16.

For the data $\zeta_{0}\in H^{s}({\mathbb{R}})$ with $s>\frac{3}{2}$ the long-
time (global) well-posedness of BO is classical. The result can easily be
extended for (1.25) with ${\varepsilon}\in(0,1)$. See, for instance, [54] in
the case of the BO equation on a fixed time with ${\varepsilon}=1$.

With this result in hand, we can use it as a link to prove the consistency
between the BO equation and the internal water waves equations.

######  Theorem 1.17.

Let ${\varepsilon},\mu,\gamma\in(0,1)$, $c^{2}=(1-\gamma)$ such that
${\varepsilon}\leq\sqrt{\mu}$ and $\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$.
Assume that $\mathbf{U}_{0}=(\zeta_{0},\psi_{0})^{T}$ satisfies the
assumptions of Theorem 1.7 and define $v_{0}=\partial_{x}\psi_{0}$. Suppose
also that $\zeta_{0}$ satisfies the $\gamma-$dependent surface condition
(1.24). Then there exists a time $T>0$ such that for some generic function $R$
satisfying

$|R|_{H^{N-5}}\leq C(\mathcal{E}^{N}(\mathbf{U}_{0})),$

for all $t\in[0,\varepsilon^{-1}T]$ we have the following results:

  * 1.

There exists a unique solution $(\zeta,v)^{T}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\times
H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$ to (1.22), where $v=\partial_{x}\psi$.
Moreover, on the same time interval, the same solution also satisfies

$\begin{cases}\partial_{t}\zeta+(1-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=(\mu+{\varepsilon}\sqrt{\mu})R\\\
\partial_{t}v+c^{2}\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=(\mu+{\varepsilon}\sqrt{\mu})R.\end{cases}$

  * 2.

There exist a unique solution $\zeta^{\text{\tiny wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$
that solves

$\partial_{t}\zeta^{\text{\tiny
wBO}}+c\big{(}1-\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|)\big{)}\partial_{x}\zeta^{\text{\tiny
wBO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
wBO}}\partial_{x}\zeta^{\text{\tiny wBO}}=0.$

Suppose further that the data $v_{0}$ is given by

$v_{0}=(1+\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|))\zeta_{0}-\frac{{\varepsilon}}{4}\zeta_{0}^{2},$
(1.29)

and define $v^{\text{\tiny wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$ by

$v^{\text{\tiny
wBO}}=(1+\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|))\zeta^{\text{\tiny
wBO}}-\frac{{\varepsilon}}{4}(\zeta^{\text{\tiny wBO}})^{2}.$

Then the solution $(\zeta^{\text{\tiny wBO}},v^{\text{\tiny wBO}})$ also
satisfies

$\begin{cases}\partial_{t}\zeta^{\text{\tiny
wBO}}+(1-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v^{\text{\tiny
wBO}}+{\varepsilon}\partial_{x}(\zeta^{\text{\tiny wBO}}v^{\text{\tiny
wBO}})=\mu R\\\ \partial_{t}v^{\text{\tiny
wBO}}+c^{2}\partial_{x}\zeta^{\text{\tiny wBO}}+{\varepsilon}v^{\text{\tiny
wBO}}\partial_{x}v^{\text{\tiny wBO}}=\mu R.\end{cases}$

  * 3.

There exist a unique solution $\zeta^{\text{\tiny BO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$
that solves

$\partial_{t}\zeta^{\text{\tiny
BO}}+c\big{(}1-\frac{\gamma}{2}\sqrt{\mu}|\mathrm{D}|\big{)}\partial_{x}\zeta^{\text{\tiny
BO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
BO}}\partial_{x}\zeta^{\text{\tiny BO}}=0,$

and on the same time interval it satisfies

$\partial_{t}\zeta^{\text{\tiny
BO}}+c\big{(}1-\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|)\big{)}\partial_{x}\zeta^{\text{\tiny
BO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
BO}}\partial_{x}\zeta^{\text{\tiny BO}}=\mu R.$

A consequence of the above results is the full justification of the BO
equation.

######  Theorem 1.18.

Let ${\varepsilon},\mu,\gamma\in(0,1)$, $c^{2}=(1-\gamma)$ such that
${\varepsilon}\leq\sqrt{\mu}$ and $\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$.
Assume that $\mathbf{U}_{0}=(\zeta_{0},\psi_{0})^{T}$ satisfies the
assumptions of Theorem 1.7 with $N\geq 7$ and define
$v_{0}=\partial_{x}\psi_{0}$. Suppose further that $\zeta_{0}$ satisfies the
$\gamma-$dependent surface condition (1.24) and that $v_{0}$ satisfies (1.29).
Then there exists $T>0$ such that:

  * 1.

There exist a unique solution $(\zeta,v)\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\times
H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$ to (1.22) where $v=\partial_{x}\psi$.

  * 2.

From the same initial data:

    * 2.1.

There exists a unique solution $(\zeta^{\text{\tiny BOs}},v^{\text{\tiny
BOs}})\in C([0,{\varepsilon}^{-1}T]:H^{s}\big{(}\mathbb{R})\times
H^{s}({\mathbb{R}})\big{)}$ to the weakly dispersive BO system

$\begin{cases}\partial_{t}\zeta^{\text{\tiny
BOs}}+(1-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v^{\text{\tiny
BOs}}+{\varepsilon}\partial_{x}(\zeta^{\text{\tiny BOs}}v^{\text{\tiny
BOs}})=0\\\ \partial_{t}v^{\text{\tiny
BOs}}+c^{2}\partial_{x}\zeta^{\text{\tiny BOs}}+{\varepsilon}v^{\text{\tiny
BOs}}\partial_{x}v^{\text{\tiny BOs}}=0,\end{cases}$

and for any $t\in[0,{\varepsilon}^{-1}T]$ there holds,

$|(\zeta-\zeta^{\text{\tiny BOs}},v-v^{\text{\tiny
BOs}})|_{L^{\infty}([0,t]\times{\mathbb{R}})}\leq(\mu+{\varepsilon}\sqrt{\mu})tC(\mathcal{E}^{N}(\mathbf{U}_{0})).$
(1.30)

    * 2.2.

There exists a unique solution $\zeta^{\text{\tiny wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$ to
the weakly dispersive BO equation

$\partial_{t}\zeta^{\text{\tiny
wBO}}+c(1-\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}\zeta^{\text{\tiny
wBO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
wBO}}\partial_{x}\zeta^{\text{\tiny wBO}}=0,$

and for any $t\in[0,{\varepsilon}^{-1}T]$ there holds,

$|\zeta-\zeta^{\text{\tiny wBO}}|_{L^{\infty}([0,t]\times{\mathbb{R}})}\leq\mu
tC(\mathcal{E}^{N}(\mathbf{U}_{0})).$ (1.31)

    * 2.3.

There exists a unique solution $\zeta^{\text{\tiny BO}}\in
C\big{(}[0,{\varepsilon}^{-1}T]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$ to
the BO equation

$\partial_{t}\zeta^{\text{\tiny
BO}}+c(1-\frac{\gamma}{2}\sqrt{\mu}|\mathrm{D}|)\partial_{x}\zeta^{\text{\tiny
BO}}+c\frac{3{\varepsilon}}{2}\zeta^{\text{\tiny
BO}}\partial_{x}\zeta^{\text{\tiny BO}}=0,$

and for any $t\in[0,{\varepsilon}^{-1}T]$ there holds,

$|\zeta-\zeta^{\text{\tiny BO}}|_{L^{\infty}([0,t]\times{\mathbb{R}})}\leq\mu
tC(\mathcal{E}^{N}(\mathbf{U}_{0})).$ (1.32)

######  Remark 1.19.

Estimates (1.30) and (1.31) together with the well-posedness theory imply the
full justification of their respective systems as internal water waves
equations. These are new results, but their primary purpose is to serve as an
intermediate step for the derivation of the BO equation and are added here for
the sake of completeness.

####  1.3.1. Strategy and outline of the proofs

The main body of the paper is devoted to the proof of Theorem 1, which relies
on energy estimates similar to the ones provided by Lannes [51]. To do so, we
first need to prove that the operators involved in the main system (1.13) are
well-defined and can be formulated solely in terms of $\zeta$ and $\psi$. We
start by studying the operator $\mathcal{G}_{\mu}$ in Section 2, which will be
given by the expression:

$\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{(}1-\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\color[rgb]{0,0,0}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{)}^{-1}\psi.$
(1.33)

The main difference with the work of Lannes is that we have the composition of
two operators $\mathcal{G}^{+}_{\mu}$ and $\mathcal{G}^{-}_{\mu}$ that act on
different space. This is a consequence of having one fluid of finite depth and
the other of infinite depth. In particular, to define the composition
$(\mathcal{G}_{\mu}^{-})^{-1}$ with $\mathcal{G}_{\mu}^{+}$ we need to work on
a homogeneous333The function space with norm
$|f|_{\mathring{H}^{s+\frac{1}{2}}}=|\mathrm{D}^{\frac{1}{2}}f|_{H^{s}}$.
target space.

The next step is to give a symbolic description of the operators involved in
the expression of $\mathcal{G}_{\mu}$. This involves some of the key estimates
that will be used to close the energy estimates. To do so, we need a symbolic
description of each of the operators involved in (1.33). The key estimate,
which is proved in Section 3 reads,

$|\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}-\big{(}-\sqrt{\mu}|\mathrm{D}|\psi^{-}\big{)}|_{H^{s+\frac{1}{2}}}\leq{\varepsilon}\mu
C\big{(}|\zeta|_{H^{t_{0}+3}}\big{)}|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}.$

The symbolic approximation of $\mathcal{G}^{-}_{\mu}$ is well-known; see the
collection of work by Lannes, Alazard, Metivier, Burq, and Zuily [49, 7, 5, 6]
for similar estimates. Their results are sharper in the sense that they
require less regularity on $\zeta$. However, the symbolic description they
provide is without the parameters ${\varepsilon},\mu$, and more importantly,
is given on the domain $\psi^{-}\in H^{s+\frac{1}{2}}({\mathbb{R}})$. More
precisely, the main difference is that we need to account for the small
parameters and arrive on the homogeneous space
$\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ to close the estimates later.
This result seems new and could be of independent interest. We refer the
reader to Remark 3.3 for the novelties of the proof and possible extensions.

The last step before providing the a priori estimates is the
quasilinearization of the internal water wave system. This is done in Section
4, where we only give detailed proofs of the steps unique to the current
setting. In fact, we can reduce some of the proofs to the estimates performed
in [51]. We also provide details on the derivation of the stability criterion
in Proposition 4.6, where we obtain coercivity type estimates. From these
results, we establish a priori bounds on the solution in Section 5. Then we
use them to prove Theorem 1.7 in Section 6.

For the remainder of the paper, we give the details on the derivation of the
BO equation. We also provide a complete justification of the models that link
the BO equation with the internal water waves equations. In Section 8, we
follow the work of Bona, Lannes, and Saut [15] to derive the intermediate
systems provided in Theorem 1.17 and the BO equation. Then, in Section 7, we
provide a short proof of Theorem 1.14, which gives the long-time well-
posedness of a weakly-dispersive BO system. Lastly, in Section 9, we conclude
the paper by proving Theorem 1.18, i.e., the full justification of each of the
systems derived in Theorem 1.17.

###  1.4. Definition and notations

  * •

We define the gradient by

$\nabla^{\mu}_{x,z}=(\sqrt{\mu}\partial_{x},\partial_{z})^{T}$

and we introduce the scaled Laplace operator

$\Delta^{\mu}_{x,z}=\nabla^{\mu}_{x,z}\cdot\nabla^{\mu}_{x,z}=\mu\partial_{x}^{2}+\partial_{z}^{2}.$

  * •

We let $c$ denote a positive constant independent of $\mu,{\varepsilon}$, and
$\mathrm{bo}$ that may change from line to line. Also, as a shorthand, we use
the notation $a\lesssim b$ to mean $a\leq c\>b$.

  * •

Let $L^{2}({\mathbb{R}})$ be the usual space of square integrable functions
with norm $|f|_{L^{2}}=\sqrt{\int_{{\mathbb{R}}}|f(x)|^{2}\>dx}$. Also, for
any $f,g\in L^{2}({\mathbb{R}})$ we denote the scalar product by
$\big{(}f,g\big{)}_{L^{2}}=\int_{{\mathbb{R}}}f(x)\overline{g(x)}\>dx$.

  * •

Let $f:\mathbb{R}\to\mathbb{R}$ be a tempered distribution, let $\hat{f}$ or
$\mathcal{F}f$ be its Fourier transform. Let $m:\mathbb{R}\to\mathbb{R}$ be a
smooth function. Then the Fourier multiplier associated with $m(\xi)$ is
denoted $\mathrm{m}$ and defined by the formula:

$\displaystyle\mathcal{F}\big{(}\mathrm{F}(\mathrm{D})f(x)\big{)}(\xi)=F(\xi)\hat{f}(\xi).$

  * •

For any $s\in\mathbb{R}$ we call the multiplier
$\widehat{|\mathrm{D}|^{s}f}(\xi)=|\xi|^{s}\hat{f}(\xi)$ the Riesz potential
of order $-s$.

  * •

For any $s\in\mathbb{R}$ we call the multiplier
$\Lambda^{s}=(1+\mathrm{D}^{2})^{\frac{s}{2}}=\langle\mathrm{D}\rangle^{s}$
the Bessel potential of order $-s$.

  * •

The Sobolev space $H^{s}(\mathbb{R})$ is equivalent to the weighted
$L^{2}-$space with $|f|_{H^{s}}=|\Lambda^{s}f|_{L^{2}}$.

  * •

For any $s\geq 0$ we will denote $\mathring{H}^{s+\frac{1}{2}}(\mathbb{R})$
the homogeneous Sobolev space with
$|f|_{\mathring{H}^{s+\frac{1}{2}}}=|\mathrm{D}^{\frac{1}{2}}f|_{H^{s}}$. One
should note that $|\mathrm{D}|=\mathcal{H}\partial_{x}$, where
$\widehat{\mathcal{H}f}(\xi)=-i\>\text{sgn}(\xi)\hat{f}(\xi)$ is the Hilbert
transform.

  * •

For any $s\geq 0$ we will denote $\dot{H}^{s+1}(\mathbb{R})$ the Beppo-Levi
space with $|f|_{\dot{H}^{s+1}}=|\Lambda^{s}\partial_{x}f|_{L^{2}}$.

  * •

For any $s\geq 0$ we will denote
$\dot{H}_{\mu}^{s+\frac{1}{2}}(\mathbb{R})=\dot{H}^{s+\frac{1}{2}}(\mathbb{R})$
with $|f|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}=|\mathfrak{B}f|_{L^{2}}$ and where
$\mathfrak{B}$ is a Fourier multiplier defined in frequency by:

$\mathcal{F}(\mathfrak{B}f)(\xi)=\frac{|\xi|}{(1+\sqrt{\mu}|\xi|)^{\frac{1}{2}}}\hat{f}(\xi).$

  * •

We say $f$ is a Schwartz function $\mathscr{S}(\mathbb{R})$, if $f\in
C^{\infty}(\mathbb{R})$ and satisfies for all $j,k\in\mathbb{N}$,

$\sup\limits_{x}|x^{j}\partial_{x}^{k}f|<\infty.$

  * •

Let $a<b$ be real numbers and consider the domain
$\mathcal{S}=(a,b)\times{\mathbb{R}}$. Then the space
$\dot{H}^{s}(\mathcal{S})$ is endowed with the seminorm

$\|f\|^{2}_{\dot{H}^{s}(\mathcal{S})}=\int_{a}^{b}|\nabla_{x,z}f(\cdot,z)|_{H^{s-1}}^{2}\>\mathrm{d}z.$

  * •

If $A$ and $B$ are two operators, then we denote the commutator between them
to be $[A,B]=AB-BA$.

  * •

Let $t_{0}>\frac{1}{2}$, $s\geq 0$, and $h_{\min}\in(0,1)$. Then for
$C(\cdot)$ being a positive non-decreasing function of its argument, we define
the constants

$M=C(\frac{1}{h_{\min}},|\zeta|_{H^{t_{0}+2}}),$

and

$M(s)=C(M,|\zeta|_{H^{s}}).$

####  1.4.1. Diffeomorphisms

In many instances it is convenient to “straighten” the fluid domain. In
particular, instead of working on the fluid domain $\Omega^{\pm}_{t}$ we
introduce the two strips:

$\mathcal{S}^{+}=\\{(x,z)\in{\mathbb{R}}^{2}\>:\>-1<z<0\\}\quad\text{and}\quad\mathcal{S}^{-}=\\{(x,z)\in{\mathbb{R}}^{2}\>:\>0<z\\}.$

The mapping between $\mathcal{S}^{\pm}$ and $\Omega_{t}^{\pm}$ will be given
by the trivial diffeomorphisms defined below.

######  Definition 1.20.

Let $t_{0}>\frac{1}{2}$ and $\zeta\in H^{t_{0}+2}(\mathbb{R})$ such that the
non-cavitation assumptions (1.14) is satisfied. For the lower domain, we have
that:

  * 1.

We define the time-dependent diffeomorphism mapping the lower domain
$\mathcal{S}^{+}$ onto the water domain $\Omega_{t}^{+}$ through

$\Sigma^{+}:\>\begin{cases}\mathcal{S}^{+}\hskip
14.22636pt\longrightarrow&\Omega_{t}^{+}\\\
(x,z)\>\>\mapsto&(x,z(1+{\varepsilon}\zeta)+{\varepsilon}\zeta).\end{cases}$

  * 2.

The Jacobi matrix $J_{\Sigma^{+}}$ is given by

$J_{\Sigma^{+}}=\begin{pmatrix}1&0\\\
\varepsilon(1+z)\partial_{x}\zeta&(1+\varepsilon\zeta)\end{pmatrix},$

and is bounded on $\mathcal{S}^{+}$. Moreover the determinant is given by
$1+{\varepsilon}\zeta$ and is bounded below due to the non-cavitation
condition (1.14).

  * 3.

The matrix associated with the change of variable for the Laplace problem is
given by

$\displaystyle
P(\Sigma^{+})=\mathrm{det}(J_{\Sigma^{+}})(J^{-1}_{\Sigma^{+}})(J_{\Sigma^{+}}^{-1})^{T}=\begin{pmatrix}(1+{\varepsilon}\zeta)&-{\varepsilon}\sqrt{\mu}(z+1)\partial_{x}\zeta\\\
-{\varepsilon}\sqrt{\mu}(z+1)\partial_{x}\zeta&\frac{1+{\varepsilon}^{2}\mu(z+1)^{2}|\partial_{x}\zeta|^{2}}{1+{\varepsilon}\zeta}\end{pmatrix},$

and is uniformly coercive. In fact, one can verify that it satisfies for all
$\theta\in{\mathbb{R}}^{d+1}$ and any $(X,z)\in\mathcal{S}^{+}$ that

$\displaystyle
P(\Sigma^{+})\theta\cdot\theta\geq\frac{1}{1+M}|\theta|^{2}\quad\text{and}\quad\|P^{+}(\Sigma^{+})\|_{L^{\infty}}\leq
M.$ (1.34)

Similarly, for the upper domain, we have that:

  * 1.

We define the time-dependent diffeomorphism mapping the upper-half plane
$\mathcal{S}^{-}$ onto the water domain $\Omega_{t}^{-}$ through the
transformation

$\Sigma^{-}:\>\begin{cases}\mathcal{S}^{-}\hskip
14.22636pt\longrightarrow&\Omega_{t}^{+}\\\
(x,z)\>\>\mapsto&(x,z+{\varepsilon}\zeta).\end{cases}$

  * 2.

The Jacobi matrix $J_{\Sigma^{-}}$ is given by

$J_{\Sigma^{-}}=\begin{pmatrix}1&0\\\
{\varepsilon}\partial_{x}\zeta&1\end{pmatrix},$

and is bounded on $\mathcal{S}^{-}$. Moreover, the determinant is given by
$1$.

  * 3.

The matrix associated with the change of variable for the Laplace problem is
given by

$\displaystyle
P(\Sigma^{-})=\mathrm{det}(J_{\Sigma^{-}})(J^{-1}_{\Sigma^{-}})(J_{\Sigma^{-}}^{-1})^{T}=\begin{pmatrix}1&-{\varepsilon}\sqrt{\mu}\partial_{x}\zeta\\\
-{\varepsilon}\sqrt{\mu}\partial_{x}\zeta&1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2}\end{pmatrix},$

and is uniformly coercive. In fact, one can verify that it satisfies for all
$\theta\in{\mathbb{R}}^{2}$ and any $(x,z)\in\mathcal{S}^{-}$ that

$\displaystyle
P(\Sigma^{-})\theta\cdot\theta\geq\frac{1}{1+M}|\theta|^{2}\quad\text{and}\quad\|P^{-}(\Sigma^{-})\|_{L^{\infty}}\leq
M.$ (1.35)

##  2\. Properties of $\mathcal{G}_{\mu}$

In this section, we aim to give a rigorous meaning to the operator
$\mathcal{G}_{\mu}$ introduced formally in equation (1.10) and study its
properties. The main results in this section will now be stated.

######  Proposition 2.1.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$, and $\zeta\in H^{t_{0}+2}({\mathbb{R}})$
be such that (1.14) is satisfied. Then the mapping

$\mathcal{G}_{\mu}[{\varepsilon}\zeta]\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}})\\\
\psi&\mapsto\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{(}1-\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\color[rgb]{0,0,0}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{)}^{-1}\psi,\end{cases}$
(2.1)

is well-defined and satisfies the following properties:

  * 1.

For any $0\leq s\leq t_{0}+1$ ,there holds,

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
(2.2)

and

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{1}{2}}M|\partial_{x}\psi|_{H^{s-\frac{1}{2}}}.$
(2.3)

  * 2.

For any $0\leq s\leq t_{0}+\frac{1}{2}$ there holds,

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi|_{H^{s-\frac{1}{2}}}\leq\mu
M|\psi|_{\dot{H}_{\mu}^{s+1}},$ (2.4)

and

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}M|\partial_{x}\psi|_{H^{s+\frac{1}{2}}}.$
(2.5)

  * 3.

The operator is uniformly coercive on on
$\psi\in\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$,

$|\psi|_{\dot{H}_{\mu}^{\frac{1}{2}}}^{2}\leq
M\big{(}\psi,\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi\big{)}_{L^{2}}.$
(2.6)

  * 4.

The bilinear form is symmetric on
$\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})\times\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$,

$\big{(}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi,\psi\big{)}_{L^{2}}=\big{(}\psi,\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi\big{)}_{L^{2}}.$
(2.7)

  * 5.

For all $s\in[0,t_{0}+1]$ and
$f,g\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$ there holds,

$|\big{(}\Lambda^{s}\mathcal{G}_{\mu}[{\varepsilon}\zeta]f,\Lambda^{s}g\big{)}_{L^{2}}|\leq\mu
M|f|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}|g|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$
(2.8)

######  Remark 2.2.

Here we take $t_{0}\geq 1$ since we will apply the result at high regularity.
If one would take $t_{0}>\frac{1}{2}$ then we would need
$s\in[\max\\{0,1-t_{0}\\},t_{0}+\frac{1}{2}]$ to enforce Remark A.9, which
specifies the needed regularity to define $\mathcal{G}^{-}_{\mu}$.

######  Remark 2.3.

For the undisturbed case, we can use formulas (A.3) and (A.19) to find that

$\mathcal{G}_{\mu}[0]=\sqrt{\mu}|\mathrm{D}|\frac{\mathrm{tanh}(\sqrt{\mu}|\mathrm{D}|)}{1+\gamma\mathrm{tanh}(\sqrt{\mu}|\mathrm{D}|)}.$
(2.9)

We will follow the same strategy as in [51], where we study the operators
involved in the expression for $\mathcal{G}_{\mu}$ and prove that we can
define them by $\psi$ through the transition problem (1.9). We study each part
individually in separate subsections. The main difference from the previous
work is that we have to carefully track the dependence on the parameters for
the current regime. Moreover, the functional setting of the upper fluid is
fundamentally different from the one in the lower fluid domain.

###  2.1. Properties of $(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}$.

For the description of the operator
$(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}$ we will follow the proof
of Proposition $1$ in [51]. The main difference is that we have an interaction
of two operators that act on different scales, and this is seen in the
estimates given below:

######  Proposition 2.4.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$ and $\zeta\in H^{t_{0}+2}({\mathbb{R}})$
be such that (1.14) is satisfied. Then the mapping

$(\mathcal{G}_{\mu}[{\varepsilon}\zeta]^{-})^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})\\\
\psi^{+}&(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\end{cases}$
(2.10)

is well-defined and satisfies

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}\leq\mu^{\frac{1}{4}}M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
(2.11)

and

$|\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\big{)}|_{H^{s}}\leq
M|\partial_{x}\psi^{+}|_{H^{s}}.$ (2.12)

######  Remark 2.5.

If we change the role of $\pm$ the result is not true in general. Indeed,
consider the case ${\varepsilon}\zeta=0$, then we have by direct computations
that

$\displaystyle(\mathcal{G}_{\mu}^{+}[0])^{-1}\mathcal{G}_{\mu}^{-}[0]=(\tanh(\sqrt{\mu}|\mathrm{D}|)^{-1},$

and we obtain the estimate

$|(\mathcal{G}_{\mu}^{+}[0])^{-1}\mathcal{G}_{\mu}^{-}[0]\psi^{-}|_{\dot{H}_{\mu}^{\frac{1}{2}}}\lesssim\frac{1}{\sqrt{\mu}}|\psi^{-}|_{H^{\frac{1}{2}}},$

which is incompatible with having
$\psi^{-}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$.

Before we turn to the proof, we need a Lemma that will be used to justify some
of the computations that will be made.

######  Lemma 2.6.

Suppose the provisions of Proposition 2.4, and further that
$\phi^{-}\in\dot{H}^{s+1}(\mathcal{S}^{-})$ is a solution to the system

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}\phi^{-}=0\hskip
14.22636pt\qquad\text{in}\quad\mathcal{S}^{-}\\\
\partial_{n}^{P^{-}}\phi^{-}=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\hskip
59.75095pt\text{on}\quad z=0.\end{cases}$ (2.13)

Then there is a number $R>0$ such that for
$\mathcal{S}^{-}_{R}=\\{(x,z)\in\mathcal{S}^{-}\>:\>z>R\\}$ and any
$\alpha+\beta$ integer $>0$ there holds,

$\partial_{z}^{\alpha}\partial_{x}^{\beta}\phi^{-}\in
L^{2}(\mathcal{S}^{-}_{R})\quad\text{and}\quad\lim\limits_{z\rightarrow\infty}\sup\limits_{x\in{\mathbb{R}}}\big{|}\partial_{z}^{\alpha}\partial_{x}^{\beta}\phi^{-}(x,z)\big{|}=0.$
(2.14)

Moreover, we have the following trace inequalities:

$|\phi^{-}|_{z=0}|_{\mathring{H}^{s+\frac{1}{2}}}\leq\|\Lambda^{s}\nabla_{x,z}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})},$
(2.15)

and

$|\phi^{-}|_{z=0}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}\leq\mu^{-\frac{1}{4}}\|\Lambda^{s}\nabla_{x,z}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}.$
(2.16)

######  Remark 2.7.

For the proof of (2.15), we need to work on the upper-half plane to prove the
estimate, and is the technical reason why we do not study the reverse
composition: $(\mathcal{G}_{\mu}^{+})^{-1}\mathcal{G}_{\mu}^{-}$.

######  Remark 2.8.

The proof of estimate (2.14) is given in [4] for $\phi|_{z=0}\in
H^{s+\frac{1}{2}}({\mathbb{R}})$ and also holds for
$\phi|_{z=0}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$.

Assuming for a moment that Lemma 2.6 holds true, we can give the proof of
Proposition 2.4.

###### Proof of Proposition 2.4.

We divide the proof into four main steps.

Step 1. $(\mathcal{G}_{\mu}^{-})^{-1}\circ\mathcal{G}_{\mu}^{+}$ is well-
defined on $\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$. It is sufficient to
prove that there exists a unique variational solution
$\phi^{-}\in\dot{H}^{s+1}(\mathcal{S}^{-})$ to the system (2.13) for any
$\psi^{+}\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$. Indeed, assuming
there is such a solution, then by (2.15), we can define

$\psi^{-}=\phi^{-}|_{z=0}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}}),$

so that for any $\psi^{+}\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$ there
is a $\psi^{-}$ where Proposition A.13 implies

$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}.$

To prove the claim we first let $s=0$, and use (2.14) to define the
variational problem associated to (2.13) by

$\displaystyle a(\phi^{-},\varphi)$
$\displaystyle=\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}\phi^{-}\cdot\nabla_{x,z}^{\mu}\varphi\>\mathrm{d}x\mathrm{d}z$
(2.17)
$\displaystyle=-\int_{\\{z=0\\}}(\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+})\varphi\>\mathrm{d}x$
$\displaystyle=L(\varphi),$

for any $\varphi\in
C^{\infty}(\overline{\mathcal{S}^{-}})\cap\dot{H}^{1}(\mathcal{S}^{-})$. We
will now verify the assumptions of Lax-Milgram’s Theorem to deduce a
variational solution in $\dot{H}^{1}(\mathcal{S}^{-})$ in two steps (extending
the result to $\dot{H}^{s+1}(\mathcal{S}^{-})$ is classical).

The first step is to show that the application $\varphi\mapsto L(\varphi)$ is
continuous on $\dot{H}^{1}(\mathcal{S}^{+})$. To do so, we note by estimate
(A.6) that

$\displaystyle|L(\varphi)|$
$\displaystyle\leq|\big{(}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+},\varphi|_{z=0}\big{)}_{L^{2}({\mathbb{R}})}|$
$\displaystyle\leq\mu|\psi^{+}|_{\dot{H}^{\frac{1}{2}}_{\mu}}|\varphi|_{z=0}|_{\dot{H}^{\frac{1}{2}}_{\mu}}.$

Then use (2.16) to obtain the needed bound

$\mu^{\frac{3}{4}}|\varphi|_{z=0}|_{\dot{H}^{\frac{1}{2}}_{\mu}}\leq\|\nabla_{x,z}^{\mu}\varphi\|_{L^{2}(\mathcal{S}^{-})}.$
(2.18)

Lastly, The bilinear form $a(\cdot,\cdot)$ is continuous and coercive on
$\dot{H}^{1}(\mathcal{S}^{+})$ by using estimate (1.35). We may therefore
conclude that there is a unique solution
$\phi^{-}\in\dot{H}^{1}(\mathcal{S}^{-})$.

Step 2. Estimate (2.11) holds true. Let
$\psi^{-}=\phi^{-}|_{z=0}\in\mathring{H}^{s+\frac{1}{2}}$ defined by the
solution of (2.13). Then we can use estimate (A.22) to get that

$\displaystyle|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}$ $\displaystyle\leq
M\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{+})}.$

Thus, we need to verify

$\|\Lambda^{s}\nabla^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}\leq\mu^{\frac{1}{4}}|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

To this end, we apply $\Lambda^{s}$ to (2.13) and study:

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}\Lambda^{s}\phi^{-}=-\nabla_{x,z}[P(\Sigma^{-}),\Lambda^{s}]\nabla_{x,z}^{\mu}\phi^{-}\hskip
28.45274pt\text{in}\quad\mathcal{S}^{-}\\\
\partial_{n}^{P^{-}}\Lambda^{s}\phi^{-}=-\mathbf{e}_{z}\cdot[P(\Sigma^{-}),\Lambda^{s}]\nabla_{x,z}^{\mu}\phi^{-}+\Lambda^{s}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\hskip
21.33955pt\text{on}\quad z=0.\end{cases}$

We may therefore multiply the equation with $\Lambda^{s}\phi^{-}$ and
integrate by parts on a finite domain
$\mathcal{S}^{-}_{R}=\\{(x,z)\in\mathcal{S}^{-}\>:\>0<z<R\\}$ and use the
Dominated Convergence Theorem with (2.14) to obtain

$\displaystyle\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}\Lambda^{s}\phi^{-}\cdot\nabla_{x,z}^{\mu}\Lambda^{s}\phi^{-}\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=-\int_{\\{z=0\\}}\Lambda^{s}(\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+})\Lambda^{s}\phi^{-}\>\mathrm{d}x$
$\displaystyle\hskip
14.22636pt+\int_{\mathcal{S}^{-}}[P(\Sigma^{-}),\Lambda^{s}]\nabla_{x,z}^{\mu}\phi^{-}\cdot\nabla_{x,z}^{\mu}\Lambda^{s}\phi^{-}\>\mathrm{d}x\mathrm{d}z.$

Now, by the coercivity estimate (1.35), Cauchy-Schwarz, (A.6), (2.16), and the
commutator estimate (B.8) we get that

$\displaystyle\|\nabla_{x,z}^{\mu}\Lambda^{s}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}\leq
M\big{(}\mu^{\frac{1}{4}}|\psi^{+}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}+\|\Lambda^{s-1}\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}\big{)}.$
(2.19)

Also, using the estimates in Step $1$, we obtain the base case

$\displaystyle\|\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}\leq\mu^{\frac{1}{4}}M|\psi^{+}|_{\dot{H}^{\frac{1}{2}}_{\mu}}.$
(2.20)

We may now conclude the proof of this step by continuous induction.

Step 3. To prove estimate (2.12) we first observe that

$\langle\xi\rangle^{s}|\xi|\lesssim\langle\xi\rangle^{s+\frac{1}{2}}|\xi|^{\frac{1}{2}}\quad\text{and}\quad\langle\xi\rangle^{s+\frac{1}{2}}\frac{|\xi|}{(1+\sqrt{\mu}|\xi|)^{\frac{1}{2}}}\lesssim\mu^{-\frac{1}{4}}\langle\xi\rangle^{s}|\xi|.$

Then use Plancherel’s identity and (2.11) (with $s+\frac{1}{2}\leq t_{0}+1$)
together with the estimates

$\displaystyle|\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\big{)}|_{H^{s}}$
$\displaystyle\leq|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{(s+\frac{1}{2})+\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{1}{4}}M|\psi^{+}|_{\dot{H}_{\mu}^{(s+\frac{1}{2})+\frac{1}{2}}}$
$\displaystyle\leq M|\partial_{x}\psi^{+}|_{H^{s}}.$

∎

To close this subsection, we give the proof of Lemma 2.6.

###### Proof of Lemma 2.6.

For the proof of (2.15), we define a multiplier being a smooth cut-off
function in frequency $\chi:[0,\infty)\rightarrow[0,1]$ such that $\chi(0)=1$,
$\chi(\xi)=0$ for $\xi>1$, and $\chi,\chi^{\prime}\in
L^{\infty}({\mathbb{R}})$. Then by the Fundamental Theorem of Calculus and
Young’s inequality, we obtain that

$\displaystyle||\mathrm{D}|^{\frac{1}{2}}\phi|_{z=0}^{-}|_{L^{2}}^{2}$
$\displaystyle=\int_{{\mathbb{R}}}\int_{0}^{\frac{2}{|\xi|}}\partial_{z}\big{(}\chi(z|\xi|)\big{|}|\xi|^{\frac{1}{2}}\hat{\phi}^{-}\big{|}^{2}\big{)}\>\mathrm{d}z\mathrm{d}\xi$
$\displaystyle\lesssim|\chi^{\prime}|_{L^{\infty}}\int_{{\mathbb{R}}}\int_{0}^{\infty}\big{|}|\xi|\hat{\phi}^{-}\big{|}^{2}\>\mathrm{d}z\mathrm{d}\xi+|\chi|_{L^{\infty}}\int_{{\mathbb{R}}}\int_{0}^{\infty}\big{|}(|\xi|+\partial_{z})\hat{\phi}^{-}\big{|}^{2}\>\mathrm{d}z\mathrm{d}\xi.$

Then to conclude, we use Plancherel’s identity to obtain (2.15).

For the proof of (2.16) we let $\eta\in C_{b}^{1}([0,\infty))$ such that
$\eta(0)=1$ and $\eta(z)=0$ for $z>1$. Then use the Fundamental Theorem of
Calculus and Young’s inequality to get that

$\displaystyle\int_{{\mathbb{R}}}\frac{|\xi|^{2}}{1+\sqrt{\mu}|\xi|}|\hat{\phi}|_{z=0}|^{2}\>\mathrm{d}\xi\lesssim|\eta^{\prime}|_{L^{\infty}}\int_{{\mathbb{R}}}\int_{0}^{\infty}|\xi|^{2}|\hat{\phi}|^{2}\>\mathrm{d}z\mathrm{d}\xi+|\eta|_{L^{\infty}}\frac{1}{\sqrt{\mu}}\int_{{\mathbb{R}}}\int_{0}^{\infty}|(|\xi|+\partial_{z})\hat{\phi}|^{2}\>\mathrm{d}z\mathrm{d}\xi.$

The result follows from the use of Plancherel’s identity.

∎

We should note that there are several results that follow from Proposition
2.4, and that will be used throughout the paper. However, to ease the
presentation, we postponed these results for the Appendix in Section A.3 since
the proofs are technical and not needed in this section.

###  2.2. Properties of
$(1-\gamma(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+})^{-1}$.

Following the road map provided in [51], we can recover the velocity
potentials $\phi^{\pm}$ from the knowledge of $\zeta$ and a trace $\psi$
defined though the transmission problem:

$\begin{cases}\nabla_{x,z}^{\mu^{\pm}}\cdot
P(\Sigma^{\pm})\nabla_{x,z}^{\mu^{\pm}}\phi^{\pm}=0\hskip
14.22636pt\qquad\text{in}\quad\mathcal{S}^{\pm}\\\
\phi^{+}|_{z=0}-\gamma\phi^{-}|_{z=0}=\psi\\\
\partial_{n}^{P^{-}}\phi^{-}|_{z=0}=\partial_{n}^{P^{+}}\phi^{+}|_{z=0},\quad\partial_{n}^{P^{+}}\phi^{+}|_{z=-1}=0,\end{cases}$
(2.21)

where the solvability of this problem is ensured in the next result:

######  Proposition 2.9.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$ and $\zeta\in H^{t_{0}+2}({\mathbb{R}})$
be such that (1.14) is satisfied. Then for all
$\psi\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$, there exist a unique
solution $\phi^{\pm}\in\dot{H}^{s+1}(\mathcal{S}^{\pm})$ to (2.21) and that
satisfies

$\|\Lambda^{s}\nabla^{\mu}_{x,z}\phi^{\pm}\|_{L^{2}(\mathcal{S}^{+})}\leq\sqrt{\mu}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}.$

The proof of this result is a consequence of the following Lemma:

######  Lemma 2.10.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$. Then the mapping

$\mathcal{J}_{\mu}[{\varepsilon}\zeta]\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})\\\
\psi^{+}&\mapsto(1-\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta])\psi^{+}\end{cases}$
(2.22)

is one-to-one and onto. Moreover, it satisfies the estimate

$|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}\leq
M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$ (2.23)

######  Remark 2.11.

For any $\psi\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$ we can define
$\psi^{+}$ by

$\psi^{+}=\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}}),$

from which we define $\psi^{-}$ by

$\displaystyle\psi^{-}=(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}}).$

Also, note that from these identities and (2.11) that there holds,

$\displaystyle|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle=|\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
(2.24)
$\displaystyle\leq|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}+\gamma|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}+\gamma\mu^{-\frac{1}{4}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

###### Proof of Lemma 2.10.

To prove estimate (2.23), we first consider the case $s=0$. Then we use the
definition of $\mathcal{J}_{\mu}[{\varepsilon}\zeta]$, the construction
$\psi^{-}=\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}$,
and
$\mathcal{G}_{\mu}[{\varepsilon}\zeta]^{+}\psi^{+}=\mathcal{G}_{\mu}[{\varepsilon}\zeta]^{-}\psi^{-}$
to get that

$\displaystyle\int_{{\mathbb{R}}}\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathrm{d}x$
$\displaystyle=\int_{{\mathbb{R}}}\psi^{+}\>\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathrm{d}x-\gamma\int_{{\mathbb{R}}}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathcal{G}^{+}\psi^{+}\>\mathrm{d}x$
$\displaystyle=\int_{{\mathbb{R}}}\psi^{+}\>\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathrm{d}x-\gamma\int_{{\mathbb{R}}}\psi^{-}\>\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}\>\mathrm{d}x.$

Then apply Proposition A.4 and A.13 combined with the coercivity of
$P(\Sigma^{\pm})$ to obtain the estimate

$\displaystyle\int_{{\mathbb{R}}}\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+}\>\mathcal{G}^{+}_{\mu}\psi^{+}\>\mathrm{d}x$
$\displaystyle=\int_{\mathcal{S}^{+}}P(\Sigma^{+})\nabla^{\mu}_{x,z}\phi^{+}\cdot\nabla^{\mu}_{x,z}\phi^{+}\>\mathrm{d}x\mathrm{d}z$
$\displaystyle\hskip
14.22636pt+\gamma\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}\phi^{-}\cdot\nabla_{x,z}^{\mu}\phi^{-}\>\mathrm{d}x\mathrm{d}z$
$\displaystyle\geq\frac{1}{1+M}\|\nabla_{x,z}^{\mu}\phi^{+}\|_{L^{2}(\mathcal{S}^{+})}^{2}.$

Moreover, since
$\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+}\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})\subset\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$
we have (A.6) at hand. In particular, we obtain from the above estimates and
(A.4) that

$\displaystyle|\psi^{+}|_{\dot{H}_{\mu}^{\frac{1}{2}}}\leq
M|\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\dot{H}_{\mu}^{\frac{1}{2}}}.$
(2.25)

Equivalently, estimate (2.23) holds in the case $s=0$. We may also use this
estimate to prove the invertibility as in [51]. In particular, we have from
the lower bound and Proposition 2.4 with estimate (2.11) that
$\mathcal{J}_{\mu}$ is an injective and closed operator since,

$\gamma|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}\leq\mu^{-\frac{1}{4}}\gamma|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}\leq\gamma
M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

Moreover, from the lower bound, (2.25), on $\mathcal{J}_{\mu}$ we know that it
is also semi-Fredholm. Then since for small enough values of $\gamma\in(0,1)$
it is invertible by a Neumann series expansion, we have from the homotopic
invariance of the index that the operator is in fact Fredholm of index zero
[40]. Consequently, the operator is also surjective and therefore invertible.

For the general case of $s\in[0,t_{0}+1]$, the proof is the same as Lemma $2$
in [51]. ∎

We may now use Lemma 2.10 to prove Proposition 2.9.

###### Proof of Proposition 2.9.

We first consider the existence of a unique solution $\phi^{+}$ in the lower
domain. Since $\psi\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$ we can use
Proposition 2.10 to make the definition:

$\psi^{+}=\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})\quad\text{and}\quad\psi^{-}=(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})\subset\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}}),$

where we let $\phi^{\pm}|_{z=0}=\psi^{\pm}$. Then we can use the first point
of Proposition A.4 to deduce a unique solution $\phi^{+}$ in the lower domain,
where the estimate follows from (A.5) and (2.23):

$\displaystyle\|\Lambda^{s}\nabla^{\mu}\phi^{+}\|_{L^{2}(\mathcal{S}^{\pm})}$
$\displaystyle\leq\sqrt{\mu}M|\psi^{+}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
$\displaystyle=\sqrt{\mu}M|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
$\displaystyle\leq\sqrt{\mu}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}.$

For the upper half plane, we use Proposition 2.4 we use Remark A.9 together
with the first point of Proposition A.13 to deduce a unique solution
$\phi^{-}$. The estimate is a consequence of estimates (A.23), (2.11), and
then (2.23):

$\displaystyle\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq\mu^{\frac{1}{4}}M|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=\mu^{\frac{1}{4}}\gamma
M|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq\sqrt{\mu}M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq\sqrt{\mu}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}.$

∎

We may now give the proof of the main result of the section.

###### Proof of Proposition 2.1.

We prove each point individually in four separate steps.

Step 1. The proof of estimate (2.2) and (2.3) follows by (A.7), (2.23), and
Plancherel’s identity:

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi|_{H^{s-\frac{1}{2}}}=|\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}\leq\mu^{\frac{3}{4}}|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}\leq\sqrt{\mu}|\partial_{x}\psi|_{H^{s}}.$

Step 2. The proof of (2.4) and (2.5), is proved the same way as in Step $1$,
but we instead use (A.8).

Step 3. The coercivity estimate (2.6) follows by construction where use the
identities in Remark 2.11 to get that

$\displaystyle\big{(}\psi,\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi\big{)}_{L^{2}}=\big{(}\psi,\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi\big{)}_{L^{2}}=\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi^{+},\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\big{)}_{L^{2}}.$

Now, argue as in the proof Proposition 2.10 where estimates (A.4) and (2.24)
implies

$\displaystyle\big{(}\psi,\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi\big{)}_{L^{2}}\geq\frac{\mu}{1+M}|\psi^{+}|^{2}_{\dot{H}^{\frac{1}{2}}_{\mu}}\geq\frac{\mu}{1+M}|\psi|^{2}_{\dot{H}^{\frac{1}{2}}_{\mu}}.$

Step 4. The symmetry follows by the second point in Propostions A.4 and A.13.

Step 5. Finally, for the proof estimate (2.8) we use (A.6) and Lemma 2.10.
∎

##  3\. Symbolic analysis of the Dirichlet-Neumann operator

In this section, we will give a symbolic description of the operator
$\mathcal{G}_{\mu}$ defined given by (2.1). The estimates need to be precise
in terms of the parameters $\mu,{\varepsilon}$, and where we carefully track
the Sobolev regularity with respect to $\psi$ and $\zeta$. One reason for
these expressions is that we need to have an estimate of the type:

$|\big{(}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\circ(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(fg)\big{)},f\big{)}_{L^{2}({\mathbb{R}})}|\leq
M(t_{0}+3)|g|_{H^{t_{0}+1}}|f|^{2}_{L^{2}},$ (3.1)

which appears naturally in the energy estimates and the quasilinearisation of
the main equations. As we can see from (3.1), one needs to absorb a derivative
and be uniform with respect to the small parameters. Also, recall that

$\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{(}1-\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\color[rgb]{0,0,0}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\Big{)}^{-1}\psi,$

which means we need a good symbolic description of each of the operators
involved in the expression. This is the strategy that was implemented in [49,
51], where we first will consider the symbolic description of
$\mathcal{G}^{\pm}_{\mu}$, then we treat the inverse operators that are
involved.

###  3.1. Symbolic analysis of $\mathcal{G}_{\mu}^{\pm}$

For the symbolic description of $\mathcal{G}^{+}_{\mu}$, we know that the
operator coincides with the ones studied in [51]. In particular, we can use
Theorem $4$, in dimension one, which is one of the key estimates of the paper:

######  Proposition 3.1 (Theorem 4 in [51]).

Let $t_{0}>\frac{1}{2}$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$ be such that
(1.14) is satisfied. Then for all $0\leq s\leq t_{0}$ and
$\psi\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$, one can approximate the
positive Dirichelt-Neumann operator by

$\mathrm{Op}(S^{+})\psi(x)=\sqrt{\mu}\mathcal{F}^{-1}\Big{(}|\xi|\tanh(\sqrt{\mu}t(x,\xi)\hat{\psi}(\xi)\Big{)}(x),$

where we define the “tail” by the symbol

$t(X,\xi)=(1+{\varepsilon}\zeta)\frac{\arctan({\varepsilon}\sqrt{\mu}\partial_{x}\zeta)}{{\varepsilon}\sqrt{\mu}\partial_{x}\zeta}|\xi|.$
(3.2)

Moreover, for $k=0,1$, the opertor satisfies the estimates

$|\mathcal{G}^{+}\psi-\mathrm{Op}(S^{+})\psi|_{H^{s+\frac{k}{2}}}\leq{\varepsilon}\mu^{1-\frac{k}{4}}M(t_{0}+3)|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$
(3.3)

In the case of infinite depth, it is pointed out in Remark $17$ in [51] that
the tail effects vanish (formally) since the hyperbolic tangent is a bottom
effect. In particular, we have the following result that proves this fact.

######  Proposition 3.2.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$. Then for all
$0\leq s\leq t_{0}$ and $\psi\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$,
one can approximate the negative Dirichelt-Neumann operator by

$S^{-}(\mathrm{D})=-\sqrt{\mu}|\mathrm{D}|,$

where we have the following estimate

$|\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi-\mathrm{Op}(S^{-})\psi|_{H^{s+\frac{1}{2}}}\leq{\varepsilon}\mu
M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}.$ (3.4)

######  Remark 3.3.

The proof can be seen as a modified version of the proof presented in [51],
but where we need to make two approximations of an elliptic problem depending
on the frequencies and weighted estimates to deal with some integrability
issues in $\mathcal{S}^{-}={\mathbb{R}}\times[0,\infty)$. The weights and the
cut-off functions are adapted to the approximation and deal with separate
issues. We also note that the proof in [51] also extends to two dimensions. In
fact, Proposition 3.2 is mainly the result that restricts Theorem 1.7 to one
horizontal dimension in this paper. However, we expect the result to be true
for $X\in{\mathbb{R}}^{2}$ by modifying the ansatz in the proof and letting
$\mathrm{Op}(S^{-})$ be given by

$\mathrm{Op}(S^{-})\psi(X)=-\sqrt{\mu}\mathcal{F}^{-1}\Big{(}\sqrt{|\xi|^{2}+{\varepsilon}^{2}\mu\big{(}(|\nabla_{X}\zeta|^{2}|\xi|^{2})-(\nabla_{X}\zeta\cdot\xi)^{2}\big{)}}\hat{\psi}(\xi)\Big{)}(X),$

for $X,\xi\in{\mathbb{R}}^{2}$.

###### Proof.

The proof will be given in several steps, where we first decompose the
estimate into two main parts depending on the frequencies. In particular, let
$\chi_{1}$ be a Fourier multiplier with a smooth symbol and equal to one
around zero. Also, let $\chi_{2}=1-\chi_{1}$ be the high-frequency part. Then
we have by duality

$\displaystyle|\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi+\sqrt{\mu}|\mathrm{D}|\psi|_{H^{s+\frac{1}{2}}}$
$\displaystyle=\sup\limits_{|\varphi|_{L^{2}}=1}\Big{|}\int_{{\mathbb{R}}}\Lambda^{s+\frac{1}{2}}\big{(}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi+\sqrt{\mu}|\mathrm{D}|\psi\big{)}\chi_{1}(\mathrm{D})\varphi\>\mathrm{d}x$
$\displaystyle\hskip
56.9055pt+\int_{{\mathbb{R}}}\Lambda^{s+\frac{1}{2}}\big{(}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi+\sqrt{\mu}|\mathrm{D}|\psi\big{)}\chi_{2}(\mathrm{D})\varphi\>\mathrm{d}x\Big{|}$
$\displaystyle\leq\sup\limits_{|\varphi|_{L^{2}}=1}\Big{(}|I_{1}(\varphi)|+|I_{2}(\varphi)|\Big{)}.$
(3.5)

We will now make two approximations of the elliptic problem (A.20), where the
estimate on $I_{1}$ can be made using the Poisson kernel. On the other hand,
the estimate on $I_{2}$ requires an approximate solution of (A.20) that
accounts for the principal part of the elliptic operator and is “well-behaved”
in high frequency on weighted Sobolev spaces.

Step 1. Approximate solutions for $I_{1}$. We know that
$\phi_{\mathrm{app}}^{1}=e^{-z\sqrt{\mu}|\mathrm{D}|}\psi$ solves

$\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\phi_{\text{app}}^{1}=0\quad\text{in}\quad\mathcal{S}^{-}\\\
\phi_{\text{app}}^{1}|_{z=0}=\psi,\end{cases}$ (3.6)

and

$\displaystyle\partial_{z}\phi_{\mathrm{app}}^{1}|_{z=0}=-\sqrt{\mu}|\mathrm{D}|\psi.$

Now, let $\phi^{-}$ be the solution of (A.20) and define
$u_{1}=\phi^{-}-\phi_{\mathrm{app}}^{1}$. Then $u_{1}$ solves

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{1}={\varepsilon}\mu
r_{1}\quad\text{in}\quad\mathcal{S}^{-}\\\
u_{1}|_{z=0}=0,\quad\lim\limits_{z\rightarrow\infty}|\nabla^{\mu}_{x,z}u_{1}|=0,\end{cases}$
(3.7)

where $r_{1}$ reads

$\displaystyle r_{1}$
$\displaystyle=\partial_{x}\big{(}(\partial_{x}\zeta)\partial_{z}\phi_{\mathrm{app}}^{1}\big{)}+(\partial_{x}\zeta)\partial_{x}\partial_{z}\phi_{\mathrm{app}}^{1}-{\varepsilon}\mu(\partial_{x}\zeta)^{2}\partial_{z}^{2}\phi_{\mathrm{app}}^{1}.$

Moreover, we have by construction that

$\displaystyle\partial_{n}^{P^{-}}u_{1}|_{z=0}=\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi+\sqrt{\mu}|\mathrm{D}|\psi+{\varepsilon}\mu
r_{2},$

where the second rest is given by

$r_{2}=(\partial_{x}\zeta)\partial_{x}\psi+{\varepsilon}\mu(\partial_{x}\zeta)^{2}|\mathrm{D}|\psi.$

Step 1.1. Estimate on $I_{1}$. From the previous step, we deduce that

$\displaystyle|I_{1}(\varphi)|$
$\displaystyle\leq\Big{|}\int_{{\mathbb{R}}}(\Lambda^{s+\frac{1}{2}}\partial_{n}^{P^{-}}u_{1}|_{z=0})\chi_{1}(\mathrm{D})\varphi\>\mathrm{d}x\Big{|}+{\varepsilon}\mu\Big{|}\int_{{\mathbb{R}}}(\Lambda^{s+\frac{1}{2}}r_{2})\chi_{1}(\mathrm{D})\varphi\>\mathrm{d}x\Big{|}$
$\displaystyle=|I_{1}^{1}(\varphi)|+|I_{1}^{2}(\varphi)|.$

We will now estimate each piece separately. For the estimate on
$|I_{1}^{1}(\varphi)|$, we use the Divergence Theorem to deduce that

$\displaystyle I_{1}^{1}(\varphi)$
$\displaystyle=\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{1}\cdot(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{1}}})\>\mathrm{d}x\mathrm{d}z+{\varepsilon}\mu\int_{\mathcal{S}^{-}}r_{1}(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi^{\mathrm{ext_{1}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=I_{1}^{1,1}(\varphi)+I_{1}^{1,2}(\varphi).$

Here we let $\varphi^{\mathrm{ext_{1}}}$ be the extension of $\varphi$ onto
the upper half-plane defined by

$\varphi^{\mathrm{ext_{1}}}(x,z)=\eta(\sqrt{\mu}z)\varphi(x),$

where $\eta(s)$ is a positive smooth function supported on $[0,1]$. Then we
estimate each term separately. For the estimate on $I_{1}^{1,1}$, we first use
Hölder inequality and Sobolev embedding to deduce that

$\displaystyle|I_{1}^{1,1}(\varphi)|$
$\displaystyle\leq\sqrt{\mu}(|\eta|_{L^{\infty}}+|\eta^{\prime}|_{L^{\infty}})M(t_{0}+1)\|\nabla_{x,z}^{\mu}u_{1}\|_{L^{2}(\mathcal{S}^{-})}|\varphi|_{L^{2}}.$
(3.8)

To conclude this estimate, we use (1.35) and the Divergence Theorem to make
the following observation:

$\displaystyle\|\nabla_{x,z}^{\mu}u_{1}\|_{L^{2}(\mathcal{S}^{-})}^{2}$
$\displaystyle\leq
M\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{1}\cdot\nabla_{x,z}^{\mu}u_{1}\>\mathrm{d}x\mathrm{d}z$
$\displaystyle\leq{\varepsilon}\mu
M\Big{|}\int_{\mathcal{S}^{-}}r_{1}u_{1}\>\mathrm{d}x\mathrm{d}z\Big{|}.$

Then use the definition of $r_{1}$ and integration by parts to distribute the
derivatives in a convenient way:

$\displaystyle\|\nabla_{x,z}^{\mu}u_{1}\|_{L^{2}(\mathcal{S}^{-})}^{2}$
$\displaystyle\leq{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\partial_{x}\zeta)(\partial_{z}\phi_{\mathrm{app}}^{1})\partial_{x}u_{1}\>\mathrm{d}x\mathrm{d}z\Big{|}+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\partial_{x}\zeta)(\partial_{x}\phi_{\mathrm{app}}^{1})\partial_{z}u_{1}\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}^{2}\mu\Big{|}\int_{\mathcal{S}^{-}}(\partial_{x}\zeta)^{2}\partial_{z}\phi_{\mathrm{app}}^{1}\partial_{z}u_{1}\>\mathrm{d}x\mathrm{d}z\Big{|}.$

In particular, if we apply Hölder’s inequality, Sobolev embedding, and
integrate the definition of $\phi_{\mathrm{app}}^{1}$ in $z$, we find that

$\displaystyle\|\nabla_{x,z}^{\mu}u_{1}\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq{\varepsilon}\mu
M\|e^{-z\sqrt{\mu}|\mathrm{D}|}|\mathrm{D}|\psi\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M|\psi|_{\mathring{H}^{\frac{1}{2}}},$

and we use it to can contol (3.8) by

$\displaystyle|I_{1}^{1,1}(\varphi)|$ $\displaystyle\leq{\varepsilon}\mu
M|\psi|_{\mathring{H}^{\frac{1}{2}}}|\varphi|_{L^{2}}.$

Moreover, the estimate on $I_{1}^{1,2}(\varphi)$ contains $r_{1}$, and can be
estimated similarly by integrating only in $x$ and using the cut-off
$\chi_{1}$ to obtain that

$\displaystyle|I_{1}^{1,2}(\varphi)|$
$\displaystyle\leq{\varepsilon}\mu\Big{(}\Big{|}\int_{\mathcal{S}^{-}}(\partial_{x}\zeta)(\partial_{z}\phi_{\mathrm{app}}^{1})\partial_{x}(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi^{\mathrm{ext_{1}}})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+\Big{|}\int_{\mathcal{S}^{-}}(\partial_{z}\phi_{\mathrm{app}}^{1})\partial_{x}\big{(}(\partial_{x}\zeta)\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi^{\mathrm{ext_{1}}}\big{)}\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\partial_{x}e^{-z\sqrt{\mu}|\mathrm{D}|}\psi)\partial_{x}\big{(}(\partial_{x}\zeta)^{2}\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi^{\mathrm{ext_{1}}}\big{)}\>\mathrm{d}x\mathrm{d}z\Big{|}\Big{)}$
$\displaystyle\leq{\varepsilon}\mu
M|\psi|_{\mathring{H}^{\frac{1}{2}}}|\varphi|_{L^{2}}.$

Finally, we arrive at the estimate of $I_{1}^{2}(\varphi)$. Here we use the
definition of $r_{1}$ to find that

$\displaystyle|I_{1}^{2}(\varphi)|$
$\displaystyle\leq{\varepsilon}\mu\Big{|}\int_{{\mathbb{R}}}(\partial_{x}\zeta)(\partial_{x}\psi)(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi)\>\mathrm{d}x\Big{|}+{\varepsilon}^{2}\mu^{\frac{3}{2}}\Big{|}\int_{{\mathbb{R}}}(\partial_{x}\zeta)^{2}(|\mathrm{D}|\psi)(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi)\>\mathrm{d}x\Big{|}$
$\displaystyle $
$\displaystyle=|I_{1}^{2,1}(\varphi)|+|I_{1}^{2,2}(\varphi)|.$

Each term is estimated similarly, and for the estimate of the first term, we
recall

$\partial_{x}=-\mathcal{H}|\mathrm{D}|,$

where $\mathcal{H}$ is the Hilbert transform. Consequently, we have from
Hölder’s inequality, Sobolev embedding, and commutator estimate (B.9) that

$\displaystyle|I_{1}^{2,1}(\varphi)|$
$\displaystyle\leq{\varepsilon}\mu\Big{|}\int_{{\mathbb{R}}}[\partial_{x}\zeta,|\mathrm{D}|^{\frac{1}{2}}]\mathcal{H}|\mathrm{D}|^{\frac{1}{2}}\psi)(\Lambda^{s+\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi)\>\mathrm{d}x\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{{\mathbb{R}}}(\partial_{x}\zeta)(\mathcal{H}|\mathrm{D}|^{\frac{1}{2}}\psi)(\Lambda^{s+\frac{1}{2}}|\mathrm{D}|^{\frac{1}{2}}\chi_{1}(\mathrm{D})\varphi)\>\mathrm{d}x\Big{|}$
$\displaystyle\leq{\varepsilon}\mu
M|\psi|_{\mathring{H}^{\frac{1}{2}}}|\varphi|_{L^{2}}.$

We may now collect all these estimates to conclude this step:

$\displaystyle|I_{1}(\varphi)|\leq{\varepsilon}\mu
M|\psi|_{\mathring{H}^{\frac{1}{2}}}|\varphi|_{L^{2}}.$

Step 2. Approximate solutions for $I_{2}$. By using the approach in [49, 51]
we can show that the function:

$\displaystyle\phi_{\text{app}}^{2}(x,z)$
$\displaystyle=(\text{Op}(L)\psi)(x,z)$ (3.9)
$\displaystyle=\mathcal{F}^{-1}\Big{(}L(x,\xi,z)\hat{\psi}(\xi)\Big{)}(x),$

where $L$ is a symbol given by

$L(x,\xi,z)=e^{-z\big{(}\frac{\sqrt{\mu}|\xi|}{1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2}}-i\frac{{\varepsilon}\mu\partial_{x}\zeta\xi}{1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2}}\big{)}},$

provides a good approximation of the elliptic problem (A.20). Moreover, we
have by direct computations that

$(\partial_{n}^{P^{-}}\phi^{2}_{\text{app}})|_{z=0}=-\sqrt{\mu}|\mathrm{D}|\psi.$

In fact, the approximation $\phi_{\text{app}}^{2}$ is constructed from the
solution of the following ODE:

$(1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2})\partial_{z}^{2}L-2i{\varepsilon}\mu(\partial_{x}\zeta)\xi\partial_{z}L-\mu\xi^{2}L=0,$

which corresponds to the principal part of (A.20) with “frozen coefficients”.
As a result, we have that $u_{2}=\phi^{-}-\phi_{\mathrm{app}}^{2}$ solves

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{2}={\varepsilon}\mu
r_{3}\quad\text{in}\quad\mathcal{S}^{-}\\\
u_{2}|_{z=0}=0,\quad\lim\limits_{z\rightarrow\infty}\omega(z)|\nabla^{\mu}_{x,z}u_{2}|=0,\end{cases}$
(3.10)

where we simply take $\omega(z)=e^{-\frac{z}{2}}$ and the rest term can be
computed explicitly:

$\displaystyle
r_{3}=2\text{Op}((\partial_{x}\zeta)\partial_{x}\partial_{z}L)\psi-2i\frac{1}{{\varepsilon}}\text{Op}(\partial_{x}L\xi)\psi-\frac{1}{{\varepsilon}}\text{Op}(\partial_{x}^{2}L)\psi+\mathrm{Op}\big{(}(\partial_{x}^{2}\zeta)\partial_{z}L\big{)}\psi.$

One should note that derivatives in $x$ on $L$ give rise to ${\varepsilon}\mu$
and is polynomial in $z$. The polynomial dependence in $z$ will require
weighted estimates in high frequency to get an estimate for
$\psi\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$.

Step 2.2. Estimate on $I_{2}$. By construction and the Divergence Theorem, we
have that

$\displaystyle I_{2}(\varphi)$
$\displaystyle=\int_{{\mathbb{R}}}(\Lambda^{s+\frac{1}{2}}\partial_{n}^{P^{-}}u_{2}|_{z=0})\chi_{2}(\mathrm{D})\varphi\>\mathrm{d}x$
$\displaystyle=\int_{\mathcal{S}^{-}}\Lambda^{s+1}P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{2}\cdot(\Lambda^{-\frac{1}{2}}\chi_{2}(\mathrm{D})\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{2}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\int_{\mathcal{S}^{-}}(\Lambda^{s}r_{3})(\Lambda^{\frac{1}{2}}\chi_{2}(\mathrm{D})\varphi^{\mathrm{ext_{2}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=I_{2}^{1}(\varphi)+I_{2}^{2}(\varphi).$

We now need to incorporate a weight in the estimates. To do so, we let
$\varphi^{\mathrm{ext_{2}}}$ be the extension of $\varphi$ onto the upper
half-plane defined by

$\varphi^{\mathrm{ext_{2}}}(x,z)=(\omega_{\mu}(z))^{2}e^{-z\sqrt{\mu}|\mathrm{D}|}\varphi(x).$

Here we let $\omega_{\mu}(z)=e^{-\frac{\sqrt{\mu}z}{2M}}$. For the estimate on
$I_{2}^{1}(\varphi)$ we first make the decomposition

$\displaystyle I_{2}^{1}(\varphi)$
$\displaystyle=\int_{\mathcal{S}^{-}}\omega_{\mu}\chi_{2}(\mathrm{D})\Lambda^{s+1}P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{2}\cdot(\Lambda^{-\frac{1}{2}}(\omega_{\mu})^{-1}\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{2}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=\int_{\mathcal{S}^{-}}[\chi_{2}(\mathrm{D})\Lambda^{s+1},P(\Sigma^{-})]\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\cdot(\Lambda^{-\frac{1}{2}}(\omega_{\mu})^{-1}\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{2}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle\hskip
14.22636pt+\int_{\mathcal{S}^{-}}P(\Sigma^{-})\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\cdot(\Lambda^{-\frac{1}{2}}(\omega_{\mu})^{-1}\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{2}}})\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=I_{2}^{1,1}(\varphi)+I_{2}^{1,2}(\varphi).$

For the control of $I_{2}^{1,1}(\varphi)$, we use Cauchy-Schwarz, commutator
estimate (B.8), and the half a derivative smoothing of the Poisson kernel:

$\||\mathrm{D}|^{\frac{1}{2}}e^{-z\sqrt{\mu}|\mathrm{D}|}\varphi\|_{L^{2}(\mathcal{S}^{-})}\leq\mu^{-\frac{1}{4}}|\varphi|_{L^{2}},$
(3.11)

and the weight:

$\|\omega_{\mu}(z)\varphi\|_{L^{2}(\mathcal{S}^{-})}\leq\mu^{-\frac{1}{4}}M|\varphi|_{L^{2}},$

to get that:

$\displaystyle|I_{2}^{1,1}(\varphi)|$ $\displaystyle\leq{\varepsilon}\mu
M\|\omega_{\mu}\Lambda^{s}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}\|\Lambda^{-\frac{1}{2}}(\omega_{\mu})^{-1}\nabla_{x,z}^{\mu}\varphi^{\mathrm{ext_{2}}})\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq{\varepsilon}\mu
M\|\omega_{\mu}\Lambda^{s}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}\mu^{\frac{1}{4}}|\varphi|_{L^{2}}.$

We also use the definition $u_{2}=\phi^{-}-\phi_{\mathrm{app}}^{2}$, the decay
of the weight, estimates (A.23) and the ones provided in Lemma B.10 to obtain

$\displaystyle\|\omega_{\mu}\Lambda^{s}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi\|_{L^{2}(\mathcal{S}^{-})}+\|\frac{1}{z}\Lambda^{s}\mathrm{Op}(\partial_{x}L)\psi\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\hskip
14.22636pt+\sqrt{\mu}\|\Lambda^{s}\mathrm{Op}(L)\partial_{x}\psi\|_{L^{2}(\mathcal{S}^{-})}+\|\Lambda^{s}\mathrm{Op}(\partial_{z}L)\psi\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq\mu^{\frac{1}{4}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}.$

We therefore have

$\displaystyle|I_{2}^{1,1}(\varphi)|\leq{\varepsilon}\mu
M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}|\varphi|_{L^{2}}.$

For the estimate on $I_{2}^{1,2}(\varphi)$, we use Hölder’s inequality and
Sobolev embedding to find that

$\displaystyle|I_{2}^{1,2}(\varphi)|$
$\displaystyle\leq\mu^{\frac{1}{4}}M\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(S^{-})}|\varphi|_{L^{2}}.$

So we have again reduced the problem to an elliptic estimate. Using the
coercivity of $P(\Sigma^{-})$ and integration by parts we find that

$\displaystyle A$
$\displaystyle=\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(S^{-})}^{2}$
(3.12) $\displaystyle\leq
M\Big{|}\int_{\mathcal{S}^{-}}P(\Sigma^{-})\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\cdot\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\leq M\Big{|}\int_{\mathcal{S}^{-}}(\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+M\Big{|}\int_{\mathcal{S}^{-}}(\mathbf{e}_{z}\cdot
P(\Sigma^{-})\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega^{\prime}_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle=M\cdot(A_{1}+A_{2}).$

Before estimating $A_{1}$, we will decompose it into three pieces:

$\displaystyle A_{1}$
$\displaystyle=\Big{|}\int_{\mathcal{S}^{-}}(\nabla_{x,z}^{\mu}\cdot[P(\Sigma^{-}),\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\nabla_{x,z}^{\mu}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}r_{3})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s+1}\chi_{2}(\mathrm{D})(\omega_{\mu})^{\prime}\mathbf{e}_{z}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle=A_{1}^{1}+A_{1}^{2}+A_{1}^{3}.$

For the estimate on $A_{1}^{1}$, we write out each term explicitly to clearly
see the terms we need to treat. In particular, we have

$\displaystyle\nabla_{x,z}^{\mu}\cdot[P(\Sigma^{-}),\Lambda^{s+1}\chi_{2}$
$\displaystyle(\mathrm{D})]\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}$
$\displaystyle=-{\varepsilon}\mu\partial_{x}[\partial_{x}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2}-{\varepsilon}\mu\partial_{z}[\partial_{x}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{x}u_{2}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}^{2}\mu\partial_{z}[(\partial_{x}\zeta)^{2},\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2}.$

Then using this decomposition together with Plancherel’s identity, integration
by parts, commutator estimate (B.8), and the support of $\chi_{2}$, we find
that

$\displaystyle A_{1}^{1}$
$\displaystyle\leq{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{-1}\partial_{x}[\partial_{x}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2})(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}([\partial_{x}^{2}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}([\partial_{x}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\partial_{x}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}^{2}\mu\Big{|}\int_{\mathcal{S}^{-}}([\partial_{x}\zeta,\Lambda^{s+1}\chi_{2}(\mathrm{D})]\omega_{\mu}\partial_{z}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\partial_{z}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\leq{\varepsilon}\mu
M\|\Lambda^{s}\partial_{z}u_{2}\|_{L^{2}(\mathcal{S}^{-})}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}u_{2}\|_{L^{2}(S^{-})}.$

Then to conclude this part, we simply use that
$u_{2}=\phi^{-}-\phi_{\mathrm{app}}^{2}$, together with the estimates (A.23)
and Lemma B.10:

$\displaystyle A_{1}^{1}$
$\displaystyle\leq{\varepsilon}\mu^{1+\frac{1}{4}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}u_{2}\|_{L^{2}(S^{-})}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(S^{-})}.$

Next, we make an estimate on $A_{1}^{2}$ using the definition of $r_{3}$ in
the previous step:

$\displaystyle A_{1}^{2}$ $\displaystyle\leq
2{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}\text{Op}((\partial_{x}\zeta)\partial_{x}\partial_{z}L)\psi)(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+2\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}\text{Op}(\partial_{x}L)\partial_{x}\psi)(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}\mathrm{Op}\big{(}(\partial_{x}^{2}\zeta)\partial_{z}L\big{)}\psi)(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+\mu\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}\text{Op}(\partial_{x}^{2}L)\psi)(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle=A_{1}^{2,1}+A_{1}^{2,2}+A_{1}^{2,3}+A_{1}^{2,4}.$

The first three terms are easily estimated by Lemma B.10, with Remark B.11,
and the support of $\chi_{2}$ where we find that

$\displaystyle A_{1}^{2,1}+A_{1}^{2,2}+A_{1}^{2,3}$
$\displaystyle\leq{\varepsilon}\mu^{1+\frac{1}{4}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}u_{2}\|_{L^{2}(S^{-})}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(S^{-})}.$

For the last term, we use the weight to gain decay in $z$. Then apply the same
estimates as above to find that

$\displaystyle A_{1}^{2,4}$
$\displaystyle\leq\sqrt{\mu}\|\Lambda^{s}\chi_{2}(\mathrm{D})\text{Op}(\frac{1}{z}\partial_{x}^{2}L)\psi\|_{L^{2}(\mathcal{S}^{-})}\|\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}.$

As a result, we have the same estimate on $A_{1}^{2}$, and so we proceed with
the estimate on $A_{1}^{3}$. We observe by definition of $P(\Sigma^{-})$ that
we have one term without ${\varepsilon}$ that need special attention:

$\displaystyle A_{1}^{3}$
$\displaystyle\leq\frac{{\varepsilon}\mu^{\frac{3}{2}}}{2M}\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}(\partial_{x}\zeta)\partial_{x}u_{2})(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+\frac{{\varepsilon}^{2}\mu^{\frac{3}{2}}}{2M}\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s}\chi_{2}(\mathrm{D})\omega_{\mu}(\partial_{x}\zeta)^{2}\partial_{z}u_{2})(\Lambda^{s+2}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\hskip
14.22636pt+\frac{\sqrt{\mu}}{2M}\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\partial_{z}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle=A_{1}^{3,1}+A_{1}^{3,2}+A_{1}^{3,3}.$

Here we see that the first two terms are easily estimated by using the weight
to gain decay and then combining it with the estimates in Lemma B.10,

$\displaystyle
A_{1}^{3,1}+A_{1}^{3,2}\leq{\varepsilon}\mu^{\frac{3}{4}}M(t_{0}+1)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(\mathcal{S}^{-})}.$

Lastly, for $A_{1}^{3,3}$ we use integration by parts and the support of
$\chi_{2}$ to find that

$\displaystyle A_{1}^{3,3}$
$\displaystyle\leq\frac{\mu}{4M}\Big{|}\int_{\mathcal{S}^{-}}(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})(\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}u_{2})\>\mathrm{d}x\mathrm{d}z\Big{|}$
$\displaystyle\leq\frac{\mu}{4M}\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}u_{2}\|_{L^{2}(\mathcal{S}^{-})}^{2},$

and therefore can be reabsorbed into $A$. The only remaining estimate is now
on $A_{2}$. However, we note that it is similar to $A_{1}^{3}$, and the same
estimates apply. In conclusion, we may gather all these estimates and use
(3.12) to find that

$\displaystyle(1-\frac{1}{4})\|\Lambda^{s+1}\chi_{2}(\mathrm{D})\omega_{\mu}\nabla_{x,z}^{\mu}u_{2}\|_{L^{2}(S^{-})}\leq{\varepsilon}\mu^{\frac{3}{4}}M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}.$

Returning to $I_{2}^{1}$, we obtain the bound

$\displaystyle|I_{2}^{1}(\varphi)|\leq{\varepsilon}\mu
M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}|\varphi|_{L^{2}}.$

To conclude this step, it only remains to estimate $I_{2}^{2}(\varphi)$.
However, this is just a simple version of $A_{1}^{2}$ and the following
estimate is easy to deduce

$\displaystyle|I_{2}(\varphi)|$
$\displaystyle\leq|I_{2}^{1}(\varphi)|+|I_{2}^{2}(\varphi)|$
$\displaystyle\leq{\varepsilon}\mu
M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}|\varphi|_{L^{2}}.$

Step 3. Conclusion of proof. We have from Step $1$. and Step $2$. that the
result holds:

$\displaystyle|\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi+\sqrt{\mu}|\mathrm{D}|\psi|_{H^{s+\frac{1}{2}}}$
$\displaystyle\leq\sup\limits_{|\varphi|_{L^{2}}=1}\Big{(}|I_{1}(\varphi)|+|I_{2}(\varphi)|\Big{)}$
$\displaystyle\leq{\varepsilon}\mu
M(t_{0}+3)|\psi|_{\mathring{H}^{s+\frac{1}{2}}}.$

∎

###  3.2. Symbolic analysis of
$(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}$

The next step in studying the symbolic behaviour of $\mathcal{G}_{\mu}$ is to
understand the composition of $(\mathcal{G}_{\mu}^{-})^{-1}$ with
$\mathcal{G}_{\mu}^{+}$.

######  Corollary 3.4.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$ be such that (1.14)
is satisfied. Then for all $0\leq s\leq t_{0}$ and
$\psi\in\mathscr{S}({\mathbb{R}})$ one can approximate the opertor
$(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}$ by

$\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}}\Big{)}\psi(x)=-\mathcal{F}^{-1}\big{(}\tanh(\sqrt{\mu}t(x,\xi))\hat{\psi}(\xi)\big{)}(x),$

where $t$ is defined by (3.2). Morerover, there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$
(3.13)

###### Proof.

We define $\tilde{\psi}^{-}=\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}}\Big{)}\psi$
and it can be seen as an approximation of $\psi^{-}$. Moreover, if we let
$\tilde{\phi}^{-}$ be a solution of

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}\tilde{\phi}^{-}=0\hskip
19.91684pt\qquad\text{in}\quad\mathcal{S}^{-}\\\
\tilde{\phi}^{-}=\tilde{\psi}^{-}\hskip 115.23373pt\text{on}\quad
z=0,\end{cases}$

then we can make the definition

$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\tilde{\psi}^{-}=\partial_{n}\tilde{\phi}^{-}|_{z=0}.$

Now, let $\phi^{-}$ be defined by the solution of

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}\phi^{-}=0\hskip
14.22636pt\qquad\text{in}\quad\mathcal{S}^{-}\\\
\partial_{n}^{P^{-}}\phi^{-}=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi\hskip
65.44142pt\text{on}\quad z=0.\end{cases}$ (3.14)

Then the difference $u=\phi^{-}-\tilde{\phi}^{-}$ satisfies

$\begin{cases}\nabla_{x,z}^{\mu}\cdot P(\Sigma^{-})\nabla_{x,z}^{\mu}u=0\hskip
48.36958pt\qquad\text{in}\quad\mathcal{S}^{-}\\\
\partial_{n}^{P^{-}}u=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\tilde{\psi}^{-}\hskip
39.83368pt\text{on}\quad z=0.\end{cases}$ (3.15)

It is straightforward to prove that these problems are well-defined where both
$\phi^{-}$ and $\tilde{\phi}^{-}$ satisfy (2.14). Consequently, we can
consider the variational equation:

$\displaystyle\int_{\mathcal{S}^{-}}P(\Sigma^{-})\nabla_{x,z}^{\mu}\Lambda^{s}u\cdot\nabla_{x,z}^{\mu}\Lambda^{s}u\>\mathrm{d}x\mathrm{d}z$
$\displaystyle=-\int_{\\{z=0\\}}\Lambda^{s}(\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\tilde{\psi}^{-})\Lambda^{s}u\>\mathrm{d}x$
$\displaystyle\hskip
28.45274pt+\int_{\mathcal{S}^{-}}[\Lambda^{s},P(\Sigma^{-})]\nabla_{x,z}^{\mu}u\cdot\nabla_{x,z}^{\mu}\Lambda^{s}u\>\mathrm{d}x\mathrm{d}z,$

where
$u_{0}:=u|_{z=0}=\psi-\tilde{\psi}^{-}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$
using the trace inequality (2.15):

$\sqrt{\mu}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}\leq\|\Lambda^{s}\nabla_{x,z}^{\mu}u\|_{L^{2}(\mathcal{S}^{-})}.$

To conclude the proof, we need the following inequality

$\displaystyle\Big{|}\big{(}\Lambda^{s}(\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\tilde{\psi}^{-}),\Lambda^{s}u_{0}\big{)}_{L^{2}}\Big{|}\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(3.16)

Assuming the inequality holds, we can argue as in the proof of Proposition
2.4, Step $2$. to obtain the result.

To prove (3.16), we will decompose the left-hand side in several pieces. In
particular, we first observe that

$\displaystyle\big{(}\Lambda^{s}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi,\Lambda^{s}u_{0}\big{)}_{L^{2}}$
$\displaystyle=\big{(}\mathrm{Op}\Big{(}\frac{S^{+}}{|\mathrm{D}|}\Big{)}^{\ast}\Lambda^{s}\psi,\Lambda^{s}|\mathrm{D}|u_{0}\big{)}_{L^{2}}+R_{1}+R_{2}$

where the rest is given by

$\displaystyle R_{1}$
$\displaystyle=\big{(}\big{[}\Lambda^{s},\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\big{]}\psi,\Lambda^{s}u_{0}\big{)}_{L^{2}},$
$\displaystyle R_{2}$
$\displaystyle=\big{(}\Lambda^{s-\frac{1}{2}}\psi,\Lambda^{\frac{1}{2}}\big{(}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}(S^{+})\big{)}\Lambda^{s}u_{0}\big{)}_{L^{2}}.$

Here we estimate $R_{1}$ by Cauchy-Schwarz, (A.9), and (B.3) to get that

$\displaystyle|R_{1}|$ $\displaystyle\leq{\varepsilon}\mu
M|\psi|_{\dot{H}^{s-\frac{1}{2}}_{\mu}}|u_{0}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

While for $R_{2}$ we use Cauchy-Schwarz, Proposition 3.1 (in the case $s=0$,
$k=1$), and (B.3) to get that

$\displaystyle|R_{2}|$
$\displaystyle\leq|\psi|_{H^{s-\frac{1}{2}}}|\big{(}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}(S^{+})\big{)}\Lambda^{s}u_{0}|_{H^{\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

Next, we decompose the remaining part where we get that

$\displaystyle\big{(}\Lambda^{s}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\tilde{\psi}^{-},\Lambda^{s}u_{0}\big{)}_{L^{2}}$
$\displaystyle=\big{(}\mathrm{Op}\Big{(}\frac{S^{+}}{|\mathrm{D}|}\Big{)}^{\ast}\Lambda^{s}\psi,\Lambda^{s}|\mathrm{D}|u_{0}\big{)}_{L^{2}}+R_{3}+R_{4}+R_{5},$

where the rest terms are defined by

$\displaystyle R_{3}$
$\displaystyle=\big{(}\big{[}\Lambda^{s},\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\big{]}\tilde{\psi}^{-},\Lambda^{s}u_{0}\big{)}_{L^{2}},$
$\displaystyle R_{4}$
$\displaystyle=\big{(}\Lambda^{s-\frac{1}{2}}\tilde{\psi}^{-},\Lambda^{\frac{1}{2}}\big{(}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}(S^{-})\big{)}\Lambda^{s}u_{0}\big{)}_{L^{2}},$
$\displaystyle R_{5}$
$\displaystyle=\big{(}\Lambda^{\frac{1}{2}}[\Lambda^{s},\mathrm{Op}\big{(}\frac{S^{+}}{S^{-}}\big{)}]\psi,\Lambda^{-\frac{1}{2}}\mathrm{Op}(S^{-})\Lambda^{s}u_{0}\big{)}_{L^{2}}.$

For $R_{3}$, the commutator estimate (A.26), and (B.22) yields,

$\displaystyle|R_{3}|$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M|\tilde{\psi}^{-}|_{\mathring{H}^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

For $R_{4}$, we argue similarly but use Proposition 3.2 to get that

$\displaystyle|R_{4}|\leq{\varepsilon}\mu
M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

Finally, for $R_{5}$, we apply Cauchy-Schwarz, the commutator estimate (B.23)
to get that

$\displaystyle|R_{5}|$
$\displaystyle\leq|[\Lambda^{s},\mathrm{Op}\big{(}\frac{S^{+}}{S^{-}}\big{)}]\psi|_{H^{\frac{1}{2}}}|\Lambda^{s-\frac{1}{2}}|\mathrm{D}|u_{0}|_{L^{2}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

Gathering all these observations, we can estimate the left-hand side of (3.16)
by

$\mathrm{LHS}_{\eqref{claim ineq G+ -
G-}}\leq\Big{|}\big{(}\Lambda^{\frac{1}{2}}(\mathrm{Op}\big{(}\frac{S^{+}}{|\mathrm{D}|}\big{)}^{\ast}-\mathrm{Op}\big{(}\frac{S^{+}}{|\mathrm{D}|}\big{)})\Lambda^{s}\psi,\Lambda^{s-\frac{1}{2}}|\mathrm{D}|u_{0}\big{)}_{L^{2}}\Big{|}+\sum\limits_{i=1}^{5}|R_{i}|.$

Then to conclude the proof, we use Cauchy-Schwarz, the adjoint estimate (B.24)
to get that

$\mathrm{LHS}_{\eqref{claim ineq G+ -
G-}}\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}|u_{0}|_{\mathring{H}^{s+\frac{1}{2}}}.$

∎

###  3.3. Symbolic analysis of $(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}$

Next we will study the symbolic behaviour of
$(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}$. To do so, we recall that we
may write $\mathcal{G}_{\mu}=\mathcal{G}^{+}_{\mu}\mathcal{J}_{\mu}^{-1}$
where the symbolic behaviour of $\mathcal{J}_{\mu}$ is captured by the symbol

$S_{J}=1-\gamma\frac{S^{+}}{S^{-}}.$ (3.17)

######  Corollary 3.5.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$ be such that (1.14)
is satisfied. Then for all $0\leq s\leq t_{0}$ and
$\psi\in\mathscr{S}({\mathbb{R}})$ one can approximate the operator
$(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]$
by

$\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\psi(x)=-\mathcal{F}^{-1}\Big{(}\frac{\tanh(\sqrt{\mu}t(x,\xi))}{1+\gamma\tanh(\sqrt{\mu}t(x,\xi))}\hat{\psi}(\xi)\Big{)}(x),$

where there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$
(3.18)

######  Remark 3.6.

The estimate (3.18) is not given in a homogeneous space, but this is not
needed since we ultimately want to prove (3.1) where we apply the estimate to
a derivative of a function.

###### Proof.

We will give the proof in three steps. First, we will prove that $S_{J}$
provides a good symbolic description of $\mathcal{J}_{\mu}$. Then we will show
that $1/S_{J}$ describes the inverse, and lastly, we combine each step with
the previous results of this section to prove (3.18).

Step 1. We can approximate $\mathcal{J}_{\mu}$ with $\mathrm{Op}(S_{J})$.
Indeed, by definition and estimate (3.13) we get that

$\displaystyle|\mathcal{J}_{\mu}[{\varepsilon}\zeta]\psi-\mathrm{Op}(S_{J})\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=\gamma|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi-\mathrm{Op}\big{(}\frac{S^{+}}{S^{-}}\big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
(3.19)
$\displaystyle\leq\gamma{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$

Step 2. We can approximate $\mathcal{J}_{\mu}^{-1}$ with
$\mathrm{Op}(1/S_{J})$. To prove this fact, we simply employ Proposition 2.10
with inequality (2.23) to deduce that

$\displaystyle|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi-\mathrm{Op}(1/S_{J})\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle=|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\big{(}1-\mathcal{J}_{\mu}[{\varepsilon}\zeta]\mathrm{Op}(1/S_{J})\big{)}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq
M|\big{(}1-\mathcal{J}_{\mu}[{\varepsilon}\zeta]\mathrm{Op}(1/S_{J})\big{)}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq
M\big{(}|\big{(}1-\mathrm{Op}(S_{J})\mathrm{Op}(1/S_{J})\big{)}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\hskip
14.22636pt+|\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}(S_{J})\big{)}\mathrm{Op}(1/S_{J})\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle=:R_{1}+R_{2}.$

Here $R_{1}$ is estimated by (B.26):

$\displaystyle R_{1}$
$\displaystyle\leq\mu^{-\frac{1}{4}}|\big{(}1-\mathrm{Op}(S_{J})\mathrm{Op}(1/S_{J})\big{)}\psi|_{H^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{1}{4}}|\psi|_{H^{s-\frac{1}{2}}}.$

While $R_{2}$ we use the estimate in Step 1. and (B.25) to obtain

$\displaystyle R_{2}$
$\displaystyle\leq\mu^{-\frac{1}{4}}|\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}(S_{J})\big{)}\mathrm{Op}(1/S_{J})\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{1}{4}}M(t_{0}+3)||\mathrm{Op}(1/S_{J})\psi|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{1}{4}}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$

As a result, we have the following estimate

$\displaystyle|\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi-\mathrm{Op}(1/S_{J})\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{1}{4}}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$
(3.20)

Step 3. Estimate (3.18) holds true. To prove this, we first use the definition
$\mathcal{G}_{\mu}=\mathcal{G}_{\mu}^{+}\mathcal{J}_{\mu}^{-1}$ to make the
decomposition:

$\displaystyle\mathrm{LHS}_{\eqref{Est G-invG}}$
$\displaystyle:=|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}-\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}\big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\hskip
14.22636pt+|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}\psi-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=:R_{3}+R_{4}.$

For $R_{3}$ we use Proposition 2.4 with inequality (2.11) and then Step 2. to
get that

$\displaystyle R_{3}$
$\displaystyle\leq\mu^{\frac{1}{4}}M|\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}-\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}\big{)}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$

For $R_{4}$, we decompose it further:

$\displaystyle R_{4}$
$\displaystyle=|\Big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}}\Big{)}\Big{)}\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\hskip
14.22636pt+|\Big{(}\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}}\Big{)}\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\Big{)}\psi|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=:R_{5}+R_{6}.$

For $R_{5}$ we use (3.13) and (B.25) to get that

$\displaystyle R_{5}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\mathrm{Op}\Big{(}\frac{1}{S_{J}}\Big{)}\psi|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\psi|_{H^{s-\frac{1}{2}}}.$

While for $R_{6}$ we simply employ estimate (B.31) which yields

$\displaystyle R_{6}$
$\displaystyle={\varepsilon}\sqrt{\mu}M|\psi|_{H^{s-\frac{1}{2}}}.$

Gathering all these estimates concludes the proof.

∎

###  3.4. Symbolic description of $\mathcal{G}_{\mu}$

For the next result, we will give a simple approximation of
$\mathcal{G}_{\mu}$. We will allow for a loss of derivatives in the estimate.
However, the loss in regularity is translated into precision with respect to
the small parameters.

######  Corollary 3.7.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$ be such that (1.14)
is satisfied. Then for all $0\leq s\leq t_{0}$ and
$\psi\in\mathscr{S}({\mathbb{R}})$ one can approximate the operator
$\mathcal{G}_{\mu}[{\varepsilon}\zeta]$ by the Fourier multiplier
$\mathcal{G}_{\mu}[0]$ defined by

$\mathcal{G}_{\mu}[0]=\sqrt{\mu}|\mathrm{D}|\frac{\tanh(\sqrt{\mu}|\mathrm{D}|)}{1+\gamma\tanh(\sqrt{\mu}|\mathrm{D}|)},$

and it satisfies the estimate

$|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi-\mathcal{G}_{\mu}[0]\psi|_{H^{s}}\leq\mu
M|\partial_{x}\psi|_{H^{s+1}}.$ (3.21)

###### Proof.

For the proof of inequality (3.21), we let
$\mathcal{G}_{\mu}[0]=\mathcal{G}^{+}_{\mu}[0]\mathcal{J}_{\mu}[0]^{-1}$ and
make the following decomposition

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi-\mathcal{G}_{\mu}^{+}[0]\mathcal{J}_{\mu}[0]^{-1}\psi|_{H^{s}}$
$\displaystyle\leq|\big{(}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]-\mathcal{G}_{\mu}^{+}[0]\big{)}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\psi|_{H^{s}}$
$\displaystyle\hskip
14.22636pt+|\mathcal{G}_{\mu}^{+}[0]\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}-\mathcal{J}_{\mu}[0]^{-1}\big{)}\psi|_{H^{s}}$
$\displaystyle=B_{1}+B_{1}.$

For the estimate on $B_{1}$, we use the classical small amplitude expansion of
$\mathcal{G}^{+}_{\mu}$ provided by Proposition $3.44$ (with $n=1$, $k=1$,
$s^{\prime}=s+\frac{1}{2}$) in [52], and then (2.23):

$\displaystyle B_{1}$ $\displaystyle\leq\mu
M|\psi|_{\dot{H}_{\mu}^{s+\frac{3}{2}}}$ $\displaystyle\leq\mu
M|\partial_{x}\psi|_{H^{s+1}}.$

For an estimate on $B_{2}$, we first observe that

$\sqrt{\mu}|\xi|\tanh(\sqrt{\mu}|\xi|)\leq\mu\frac{|\xi|^{2}}{(1+\sqrt{\mu}|\xi|)^{\frac{1}{2}}}.$

We may therefore use the product rule, combined with (2.23) and the shape
derivative formulas (A.55) and (A.56) to find that

$\displaystyle B_{2}$
$\displaystyle\leq\mu|\partial_{x}\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(1-\mathcal{J}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[0]^{-1})\psi\big{)}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq\mu
M\Big{(}|\mathrm{d}_{\zeta}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\partial_{x}\zeta)(1-\mathcal{J}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[0]^{-1})\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\hskip
14.22636pt+|\mathrm{d}_{\zeta}\mathcal{J}_{\mu}[{\varepsilon}\zeta](\partial_{x}\zeta)\mathcal{J}_{\mu}[0]^{-1}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}+|\partial_{x}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}\Big{)}$
$\displaystyle\leq{\varepsilon}\mu M|\partial_{x}\psi|_{H^{s+1}}.$

∎

###  3.5. Symbolic analysis of
$\mathcal{J}_{\mu}^{-1}(\mathcal{G}_{\mu}^{-})^{-1}\partial_{x}$

We end this section with the study of an operator that appears in the
quasilinearisation of the equations. In particular, we will need the symbolic
description for the energy estimates later (see inequality (5.11)). The
operator is well-defined by the first point in Corollary A.17, and the
symbolic description is given in the next result.

######  Corollary 3.8.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+3}({\mathbb{R}})$ be such that (1.14)
is satisfied. Then for all $0\leq s\leq t_{0}$ and $f\in
H^{s+\frac{1}{2}}({\mathbb{R}})$, one can approximate the operator
$\mathfrak{B}^{2}\mathcal{J}_{\mu}^{-1}(\mathcal{G}_{\mu}^{-})^{-1}\partial_{x}$
by

$\mathrm{Op}\Big{(}\frac{\mathfrak{B}^{2}}{S_{J}S^{-}}\Big{)}\partial_{x}f(x)=-\frac{1}{\sqrt{\mu}}\mathcal{F}^{-1}\Big{(}\frac{|\xi|(1+\sqrt{\mu}|\xi|)^{-1}}{1+\gamma\tanh(\sqrt{\mu}t(x,\xi))}i\xi\hat{f}(\xi)\Big{)}(x),$

and for $k=0,1$, there holds,

$|\mathfrak{B}^{2}(\mathcal{J}_{\mu}[{\varepsilon}\zeta])^{-1}(\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta])^{-1}\partial_{x}f-\mathrm{Op}(\frac{\mathfrak{B}^{2}}{S_{J}S^{-}})\partial_{x}f|_{H^{s+\frac{k}{2}}}\leq{\varepsilon}\mu^{-1-\frac{k}{2}}M(t_{0}+3)|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}}.$
(3.22)

###### Proof.

We follow the proof of Corollary $5$ in [51], where we first decompose the
main operator into three pieces:

$\displaystyle\mathfrak{B}^{2}(\mathcal{J}_{\mu}[{\varepsilon}\zeta])^{-1}(\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta])^{-1}\partial_{x}f$
$\displaystyle=\mathrm{Op}(\frac{\mathfrak{B}^{2}}{S_{J}S^{-}})\partial_{x}f+R_{1}+R_{2},$
(3.23)

where we define
$f^{\sharp}=\mathcal{J}_{\mu}^{-1}(\mathcal{G}_{\mu}^{-})^{-1}\partial_{x}f$

$R_{1}=\Big{(}1-\mathrm{Op}(\frac{\mathfrak{B}^{2}}{S_{J}S^{-}})\mathrm{Op}(\frac{S_{J}S^{-}}{\mathfrak{B}^{2}})\Big{)}\mathfrak{B}^{2}f^{\sharp},$

and

$R_{2}=\mathrm{Op}(\frac{\mathfrak{B}^{2}}{S_{J}S^{-}})\big{(}\mathrm{Op}(S^{-}S_{J})-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[{\varepsilon}\zeta]\big{)}f^{\sharp}.$

For the estimate on $R_{1}$ we use (B.33) to get,

$\displaystyle|R_{1}|_{H^{s+\frac{k}{2}}}\leq{\varepsilon}\mu^{-\frac{k}{4}}M|\mathfrak{B}f^{\sharp}|_{H^{s}}.$

Then apply (2.23) and(A.34) to find that

$\displaystyle|\mathfrak{B}f^{\sharp}|_{H^{s}}$
$\displaystyle=|(\mathcal{J}_{\mu}[{\varepsilon}\zeta])^{-1}(\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta])^{-1}\partial_{x}f|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
(3.24)
$\displaystyle\leq\mu^{-\frac{1}{4}}M|(\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta])^{-1}\partial_{x}f|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq\mu^{-\frac{3}{4}}M|f|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq\mu^{-1}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}},$

which yields the following estimate on $R_{1}$:

$\displaystyle|R_{1}|_{H^{s+\frac{k}{2}}}\leq{\varepsilon}\mu^{-1-\frac{k}{2}}M|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}}.$

For the estimate on $R_{2}$ we simply employ (B.25) and (B.5) to obtain,

$\displaystyle|R_{2}|_{H^{s+\frac{k}{2}}}$
$\displaystyle\leq\mu^{-\frac{3}{4}}M|\big{(}\mathrm{Op}(S^{-}S_{J})-\mathcal{G}^{-}_{\mu}\mathcal{J}_{\mu}\big{)}f^{\sharp}|_{\dot{H}_{\mu}^{s+\frac{k}{2}}}$
$\displaystyle\leq\mu^{-\frac{3}{4}}\mu^{-\frac{k}{4}}M|\big{(}\mathrm{Op}(S^{-}S_{J})-\mathcal{G}^{-}_{\mu}\mathcal{J}_{\mu}\big{)}f^{\sharp}|_{H^{s+\frac{1}{2}}},$

where we make the decomposition

$\displaystyle|\big{(}\mathrm{Op}(S^{-}S_{J}$
$\displaystyle)-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\mathcal{J}_{\mu}[{\varepsilon}\zeta]\big{)}f^{\sharp}|_{H^{s+\frac{1}{2}}}$
$\displaystyle\leq|\Big{(}\big{(}\mathrm{Op}(S^{-})-\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\big{)}\mathcal{J}_{\mu}[{\varepsilon}\zeta]\Big{)}f^{\sharp}|_{H^{s+\frac{1}{2}}}+|\Big{(}\mathrm{Op}(S^{-})\big{(}\mathrm{Op}(S_{J})-\mathcal{J}_{\mu}[{\varepsilon}\zeta]\big{)}\Big{)}f^{\sharp}|_{H^{s+\frac{1}{2}}}$
$\displaystyle\hskip
14.22636pt+|\big{(}\mathrm{Op}(S^{-}S_{J})-\mathrm{Op}(S^{-})\mathrm{Op}(S_{J})\big{)}f^{\sharp}|_{H^{s+\frac{1}{2}}}$
$\displaystyle=r_{1}+r_{2}+r_{3}.$