For $r_{1}$, we use Proposition (3.2) with estimate (3.4) and the defintion of
$\mathcal{J}_{\mu}$ with estimate (3.24) to deduce,

$\displaystyle r_{1}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|\mathcal{J}_{\mu}[{\varepsilon}\zeta]f^{\sharp}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)\big{(}|f^{\sharp}|_{\mathring{H}^{s+\frac{k}{2}}}+|(\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}^{+}f^{\sharp}|_{\mathring{H}^{s+\frac{1}{2}}}\big{)}$
$\displaystyle\leq{\varepsilon}\mu^{-\frac{1}{4}}M(t_{0}+3)|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}}$

For the estimate on $r_{2}$ we use (3.19) and (3.24)

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

For $r_{3}$ we simply use (B.30) and (3.24):

$\displaystyle
r_{3}\leq{\varepsilon}\mu^{-\frac{1}{4}}M|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}}.$

Gathering all these estimates implies

$\displaystyle|R_{2}|_{H^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{-1}M(t_{0}+3)|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{H^{s}},$

and adding the estimates for $R_{1}$ and $R_{2}$ completes the proof.

∎

##  4\. Quasilinearization of the internal water wave system

In this section, we will put the internal water waves system (1.13) in a
quasilinear form. This is done by applying time and space derivatives to the
system and taking care of the principal terms. In particular, on need shape
derivative formulas for the Dirichlet-Neumann operator (2.1), where we prove
in Lemma A.23 that

$\mathrm{d}_{\zeta}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi=-{\varepsilon}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\big{(}h(\underline{w}^{+}-\gamma\underline{w}^{-})\big{)}-{\varepsilon}\mu\mathcal{I}[\mathbf{U}]h,$

where $\underline{w}^{\pm}$ is defined below together with the operator
$\mathcal{I}[\mathbf{U}]h$. In fact, the operator $\mathcal{I}[\mathbf{U}]h$
is one of the main quantities that we need to understand, and is where we will
use the symbolic descriptions from the previous section. Before stating the
main result, we formally introduce the main operators involved, and that will
be studied in detail later.

######  Definition 4.1.

Let the functions $\psi^{\pm}$ serve as Dirichlet data for the elliptic
problems (1.6) and (1.4), then we define the horizontal components of the
velocities at the surface by

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

We define the vertical component of the velocity at the surface by

$\underline{V}^{\pm}=\partial_{x}\psi^{\pm}-{\varepsilon}(\underline{w}^{\pm}\psi)\partial_{x}\zeta,$

and

$[\\![\underline{V}^{\pm}]\\!]=\underline{V}^{+}-\underline{V}^{-}.$

We refer to Corollary A.18 for the precise definition. Moreover, we define the
quantity related to the Rayleigh-Taylor criterion by

$\mathfrak{a}=\Big{(}(1-\gamma)+{\varepsilon}\big{(}(\partial_{t}+{\varepsilon}\underline{V}^{+}\partial_{x})\underline{w}^{+}-\gamma(\partial_{t}+{\varepsilon}\underline{V}^{-}\partial_{x})\underline{w}^{-}\big{)}\Big{)},$
(4.1)

and a quantity related to the presence of surface tension:

$\mathcal{K}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]\bullet=(1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2})^{-\frac{3}{2}}\bullet.$

From these quantities, we let $\mathbf{U}=(\zeta,\psi)^{T}$ and define the
linear operator, $\mathcal{I}[\mathbf{U}]$, of order one by,

$\mathcal{I}[\mathbf{U}]\bullet=\partial_{x}(\bullet\underline{V}^{+})+\gamma\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}\Big{(}\bullet[\\![\underline{V}^{\pm}]\\!]]\Big{)},$
(4.2)

and its adjoint reads,

$\mathcal{I}[\mathbf{U}]^{\ast}\bullet=-\underline{V}^{+}\partial_{x}\bullet-\gamma[\\![\underline{V}^{\pm}]\\!]\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\bullet\big{)}.$
(4.3)

Moreover, we define the instability operator:

$\mathfrak{Ins}[\mathbf{U}]\bullet=\mathfrak{a}\bullet-(1-\gamma)\gamma{\varepsilon}^{2}\mu[\\![\underline{V}^{\pm}]\\!]\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{(}\bullet[\\![\underline{V}^{\pm}]\\!]\big{)}-\mathrm{bo}^{-1}\partial_{x}\mathcal{K}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]\partial_{x}\bullet,$
(4.4)

where

$\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\bullet=\partial_{x}\circ\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\partial_{x}\bullet.$
(4.5)

Then to put the internal water waves system in matrix form, it is convenient
to introduce the notation

$\mathcal{A}[U]=\begin{pmatrix}0&-\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\\\
\mathfrak{Ins}[\mathbf{U}]&0\end{pmatrix},\qquad\mathcal{B}[\mathbf{U}]=\begin{pmatrix}{\varepsilon}\mathcal{I}[\mathbf{U}]&0\\\
0&-{\varepsilon}\mathcal{I}[\mathbf{U}]^{\ast}\end{pmatrix},$

where $\mathcal{A}$ and $\mathcal{B}$ corresponds to the principal part of the
system. To account for surface tension, one also needs to track the dependence
in the sub-principal part, which is defined for
$\alpha=(\alpha_{1},\alpha_{2})\in{\mathbb{N}}^{2}$:

$\mathcal{C}_{\alpha}[U]=\begin{pmatrix}0&-\frac{1}{\mu}\mathcal{G}_{\mu,(\alpha)}[{\varepsilon}\zeta]\\\
\mathrm{bo}^{-1}\mathcal{K}_{(\alpha)}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]&0\end{pmatrix},$

where we let $(\partial_{1},\partial_{2})=(\partial_{x},\partial_{t})$,
$F=(f_{1},f_{2})$ to define

$\mathcal{G}_{\mu,(\alpha)}[{\varepsilon}\zeta]F=\sum\limits_{j=1}^{2}\
\alpha_{j}\mathrm{d}\mathcal{G}(\partial_{j}\zeta)f_{j},$

and we have:

$\mathcal{K}_{(\alpha)}[\partial_{x}\zeta]F=-\partial_{x}\Big{(}\sum\limits_{j=1}^{2}\mathrm{d}\mathcal{K}[\partial_{x}\partial_{j}\zeta]\partial_{x}f_{j}+\mathcal{K}[\partial_{x}f_{j}]\partial_{x}\partial_{j}\zeta\Big{)}.$

With these formulas, we can state the main result of the section.

######  Proposition 4.2.

Let $T>0$, $t_{0}\geq 1$ and $N\in{\mathbb{N}}$ be such that $N\geq 5$.
Furthermore, let
$\mathbf{U}=(\zeta,\psi)^{T}\in\mathscr{E}_{\mathrm{bo},T}^{N}$ be such that
(1.14) is satisfied on $[0,T]$. Also, for any
$\alpha=(\alpha^{1},\alpha^{2})\in{\mathbb{N}}^{2}$,
$\check{\alpha}^{j}=\alpha-\mathbf{e}_{j}$, with $1\leq|\alpha|\leq N$ we
define

$\partial_{x,t}^{\alpha}=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\partial_{x}^{\alpha_{1}}\partial_{t}^{\alpha_{2}},$

and let $\underline{w}^{\pm}$ be defined as in (A.41). Moreover, define
$\underline{w}$ by

$\displaystyle\underline{w}$
$\displaystyle=\underline{w}^{+}-\gamma\underline{w}^{-},$

and

$\zeta_{(\alpha)}=\partial_{x,t}^{\alpha}\zeta,\quad\psi_{(\alpha)}=\partial_{x,t}^{\alpha}\psi-{\varepsilon}\underline{w}\partial_{x,t}^{\alpha}\zeta,\quad\psi_{\langle\check{\alpha}\rangle}=(\psi_{(\check{\alpha}^{1})},\psi_{(\check{\alpha}^{2})})^{T}.$

Then for $\mathbf{U}_{(\alpha)}=(\zeta_{(\alpha)},\psi_{(\alpha)})^{T}$ and
$\mathbf{U}_{(\check{\alpha})}=(\zeta_{(\alpha)},\psi_{(\check{\alpha})})^{T}$,
there holds

$\displaystyle\text{if}\>\>1\leq|\alpha|<N:$ $\displaystyle\quad\hskip
110.96556pt\partial_{t}\mathbf{U}_{(\alpha)}+\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)}={\varepsilon}(R_{\alpha},S_{\alpha})^{T},$
(4.6) $\displaystyle\text{if}\qquad\>|\alpha|=N:$
$\displaystyle\quad\partial_{t}\mathbf{U}_{(\alpha)}+\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)}+\mathcal{B}[\mathbf{U}]\mathbf{U}_{(\alpha)}+\mathcal{C}[\mathbf{U}]\mathbf{U}_{(\check{\alpha})}={\varepsilon}(R_{\alpha},S_{\alpha})^{T}$
(4.7)

The rest functions satisfy the estimates

$|R_{\alpha}|_{H^{1}_{\gamma,\mathrm{bo}}}+|S_{\alpha}|_{\dot{H}^{\frac{1}{2}}_{\mu}}\leq
C\mathcal{E}^{N}(\mathbf{U})\big{(}1+{\varepsilon}\mu^{\frac{1}{4}}|[\\![\underline{V}^{\pm}]\\!]|_{L^{\infty}}|\zeta|_{<N+\frac{1}{2}>}\big{)},$
(4.8)

for some $C>0$ and where $|\zeta|_{<N+\frac{1}{2}>}$ is defined by

$|\zeta|_{<N+\frac{1}{2}>}=\sum\limits_{\alpha\in{\mathbb{N}}^{2},|\alpha|=N}|\partial_{x,t}^{\alpha}\zeta|_{\mathring{H}^{\frac{1}{2}}}.$

For the proof, we need to carefully track the dependencies in the small
parameters. However, this part of is very similar to the one in [51] and can
therefore be considered a technical point. We give the details in the Appendix
$\mathrm{A}$, Section A.5. Now, before we proceed with the energy estimates,
we need to give a rigorous meaning to the operators given in Definition 4.1.
This will be done in separate subsections.

####  4.0.1. Properties of $\mathcal{I}[\mathbf{U}]\bullet$

First, we study the operator $\mathcal{I}[\mathbf{U}]\bullet$ which appears
later in the shape derivative formulas for $\mathcal{G}_{\mu}$.

######  Proposition 4.3.

Let $t_{0}\geq 1$ and $\mathbf{U}=(\zeta,\psi)^{T}$, with $\zeta\in
H^{t_{0}+3}({\mathbb{R}})$ satisfying (1.14) and
$\psi\in\dot{H}^{t_{0}+3}({\mathbb{R}})$. Then we may define the operator

$\mathcal{I}[\mathbf{U}]\bullet=\partial_{x}(\bullet\underline{V}^{+})+\gamma\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}\Big{(}\bullet[\\![\underline{V}^{\pm}]\\!]\Big{)},$

and it satisfies the following properties:

  * 1.

For all $0\leq s\leq t_{0}$, there holds,

$|\mathcal{I}[\mathbf{U}]f|_{H^{s-\frac{1}{2}}}\leq
M|f|_{H^{s+\frac{1}{2}}}|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}},$ (4.9)

and

$|\mathcal{I}[\mathbf{U}]f|_{H^{s-\frac{1}{2}}}\leq
M|f|_{H^{t_{0}+\frac{1}{2}}}|\partial_{x}\psi|_{H^{s+\frac{1}{2}}}.$ (4.10)

  * 2.

Let $\mathfrak{a}=1+\mathfrak{b}$, with $\mathfrak{b}\in
H^{t_{0}+1}({\mathbb{R}})$, and for all $f\in L^{2}({\mathbb{R}})$ there
holds,

$\big{(}\mathfrak{a}\mathcal{I}[\mathbf{U}]f,f\big{)}_{L^{2}}\leq
M(t_{0}+3)(1+|\mathfrak{b}|_{H^{t_{0}+1}})|\partial_{x}\psi|_{H^{t_{0}+1}}|f|^{2}_{L^{2}}.$
(4.11)

  * 3.

Let $K\in H^{t_{0}+1}({\mathbb{R}})$, then for all $f\in H^{1}({\mathbb{R}})$
there holds,

$\big{(}\partial_{x}(K\partial_{x}\mathcal{I}[\mathbf{U}]f),f\big{)}_{L^{2}}\leq
M(t_{0}+3)|K|_{H^{t_{0}+1}}|\partial_{x}\psi|_{H^{t_{0}+1}}|f|^{2}_{H^{1}}.$

  * 4.

For all $f\in\dot{H}^{\frac{1}{2}}({\mathbb{R}})$, $g\in
H^{\frac{1}{2}}({\mathbb{R}})$, one has

$\big{(}\mathcal{I}[\mathbf{U}]^{\ast}f,g\big{)}_{L^{2}}\leq
M|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}}|f|_{\dot{H}^{\frac{1}{2}}_{\mu}}(|g|_{L^{2}}+\mu^{\frac{1}{4}}|g|_{H^{\frac{1}{2}}}).$

###### Proof.

We prove each point in separate steps.

Step 1. To prove the first point, we observe that the first part of
$\mathcal{I}[\mathbf{U}]f$ is estimated by the product estimate (B.6) and
(A.45):

$\displaystyle|\partial_{x}(f\underline{V}^{+})|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq|f|_{H^{s+\frac{1}{2}}}|\underline{V}^{+}|_{H^{\max\\{s+\frac{1}{2},t_{0}\\}}}$
$\displaystyle\leq
M|f|_{H^{s+\frac{1}{2}}}|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}}.$

To estimate the remaining part we first estimate $\mathcal{G}_{\mu}$ with
(2.2) to find that

$\displaystyle|\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!])\big{)}|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!])\big{)}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

Then use (A.34), the product estimate (B.6), and (A.45) to see that

$\displaystyle\mu^{\frac{3}{4}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!])\big{)}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{1}{2}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!])\big{)}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq|f[\\![\underline{V}^{\pm}]\\!]|_{H^{s+\frac{1}{2}}}$
$\displaystyle\leq
M|f|_{H^{s+\frac{1}{2}}}|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}}.$

For the proof of (4.10), it is proved similarly where we only need to modify
the part when we apply the product estimate.

Step 2. In this estimate, we note that the instability operator is a first-
order differential operator acting on $f\mapsto\mathcal{I}[\mathbf{U}]f$. But
the principal symbol is skew-adjoint. Indeed, for the first part, we use
integration by parts

$\displaystyle\big{(}\mathfrak{a}\partial_{x}(f\underline{V}^{+}),f\big{)}_{L^{2}}$
$\displaystyle=\big{(}\mathfrak{a}(\partial_{x}\underline{V}^{+})f,f\big{)}_{L^{2}}-\frac{1}{2}\big{(}\big{(}\partial_{x}(\mathfrak{a}\underline{V}^{+})\big{)}f,f\big{)}_{L^{2}}.$

Then Hölder’s inequality, the Sobolev embedding
$H^{t_{0}}({\mathbb{R}})\hookrightarrow L^{\infty}({\mathbb{R}})$, the product
estimate (B.6), and (A.44) gives

$\displaystyle|\big{(}\mathfrak{a}\partial_{x}(f\underline{V}^{+}),f\big{)}_{L^{2}}|$
$\displaystyle=(1+|b|_{H^{t_{0}+1}})|\underline{V}^{+}|_{H^{t_{0}+1}}|f|_{L^{2}}^{2}$
$\displaystyle\leq(1+|b|_{H^{t_{0}+1}})|\partial_{x}\psi|_{H^{t_{0}+1}}|f|_{L^{2}}^{2}.$

For the second part, we use integration by parts together with the fact that
$\mathcal{G}_{\mu}$ and $\mathcal{G}_{\mu}^{-}$ are symmetric to make the
following decomposition

$\displaystyle\big{(}\mathfrak{a}\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!])\big{)},f\big{)}_{L^{2}}$
$\displaystyle=-\big{(}f[\\![\underline{V}^{\pm}]\\!],\partial_{x}\Big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathfrak{a}f)\Big{)}\big{)}_{L^{2}}$
$\displaystyle=-\big{(}f[\\![\underline{V}^{\pm}]\\!],\partial_{x}\Big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\Big{)}(\mathfrak{a}f)\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!]),\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}(\mathfrak{a}f)\big{)}_{L^{2}}$
$\displaystyle=:R_{1}+R_{2}.$

For $R_{1}$ we use Cauchy-Schwarz and estimate (3.13) (with $s=\frac{1}{2}$)
to deduce that

$\displaystyle|R_{1}|$
$\displaystyle\leq|f[\\![\underline{V}^{\pm}]\\!]|_{L^{2}}|((\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)})(\mathfrak{a}f)|_{\mathring{H}^{1}}$
$\displaystyle\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)|f[\\![\underline{V}^{\pm}]\\!]|_{L^{2}}|\mathfrak{a}f|_{L^{2}}.$

Then use the product estimate (B.6) and (A.45) to get that

$|R_{1}|\leq{\varepsilon}\sqrt{\mu}M(t_{0}+3)(1+|b|_{H^{t_{0}}})|\partial_{x}\psi|_{H^{t_{0}}}|f|^{2}_{L^{2}}.$

For $R_{2}$, we will split it into several parts:

$\displaystyle R_{2}$
$\displaystyle=\big{(}\Big{(}\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}^{\ast}-\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\Big{)}\partial_{x}(f[\\![\underline{V}^{\pm}]\\!]),(\mathfrak{a}f)\big{)}_{L^{2}}+\big{(}\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}(f\partial_{x}[\\![\underline{V}^{\pm}]\\!]),(\mathfrak{a}f)\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}[\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)},[\\![\underline{V}^{\pm}]\\!]]\partial_{x}f,(\mathfrak{a}f)\big{)}_{L^{2}}+\big{(}[\\![\underline{V}^{\pm}]\\!]\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\partial_{x}f,(\mathfrak{a}f)\big{)}_{L^{2}}$
$\displaystyle=R_{2}^{1}+R_{2}^{2}+R_{2}^{3}+R_{2}^{4}.$

For $R_{2}^{1}$ we use estimate (B.29), for $R_{2}^{2}$ we use (B.27), for
$R_{2}^{3}$ we use (B.28), combined with (B.6) and (A.45) we get that

$|R_{2}^{1}|+|R_{2}^{2}|+|R_{2}^{3}|\leq\sqrt{\mu}M(1+|b|_{H^{t_{0}}})|\partial_{x}\psi|_{H^{t_{0}+1}}|f|^{2}_{L^{2}}.$

On the other hand, we can use integration by parts on $R_{2}$ to cancel
$R_{2}^{4}$. Indeed, we obtain that

$\displaystyle R_{2}$
$\displaystyle=-\big{(}(f[\\![\underline{V}^{\pm}]\\!]),\mathrm{Op}\Big{(}\partial_{x}\frac{S^{+}}{S^{-}S_{J}}\Big{)}(\mathfrak{a}f)\big{)}_{L^{2}}-\big{(}(f[\\![\underline{V}^{\pm}]\\!]),\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}(f\partial_{x}\mathfrak{b})\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}(f[\\![\underline{V}^{\pm}]\\!]),[\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)},\mathfrak{b}]\partial_{x}f\big{)}_{L^{2}}-\big{(}(f[\\![\underline{V}^{\pm}]\\!]),\mathfrak{a}\mathrm{Op}\Big{(}\frac{S^{+}}{S^{-}S_{J}}\Big{)}\partial_{x}f\big{)}_{L^{2}}$
$\displaystyle=R_{2}^{5}+R_{2}^{6}+R_{2}^{7}+R_{2}^{8}.$

Here we estimate $R_{2}^{5},R_{2}^{6},R_{2}^{7}$ as above where we also use
estimate (B.32) for $R_{2}^{5}$. Adding these two decompositions implies

$2|R_{2}|\leq\big{|}\sum\limits_{j=1}^{8}R_{2}^{j}\big{|}\leq\sqrt{\mu}M(1+|b|_{H^{t_{0}+1}})|\partial_{x}\psi|_{H^{t_{0}+1}}|f|^{2}_{L^{2}}.$

Step 3. Since $K$ is symmetric and $f\mapsto\mathcal{I}[\mathbf{U}]f$ is skew-
symmetric, we can absorb one derivative by integrating by parts as we did in
the previous step.

Step 4. From the definition (4.3) we have that

$\displaystyle\big{(}\mathcal{I}[\mathbf{U}]^{\ast}f,g\big{)}_{L^{2}}$
$\displaystyle=-\big{(}\underline{V}^{+}\partial_{x}f,g\big{)}_{L^{2}}-\gamma\big{(}[\\![\underline{V}^{\pm}]\\!]\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]f\big{)},g\big{)}_{L^{2}}$
$\displaystyle=A_{1}+\gamma A_{2}.$

For the first term, we introduce a commutator, then apply Hölder’s inequality,
Sobolev embedding, estimate (A.45), and (B.10) to find that

$\displaystyle|A_{1}|$
$\displaystyle\leq|[\underline{V}^{+},(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}]\frac{\partial_{x}}{(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}}f|_{L^{2}}|g|_{L^{2}}$
$\displaystyle\hskip
14.22636pt+|\underline{V}^{+}|_{L^{\infty}}|\frac{\partial_{x}}{(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}}f||(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}g|_{L^{2}}$
$\displaystyle\leq
M|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}}|f|_{\dot{H}^{\frac{1}{2}}_{\mu}}(|g|_{L^{2}}+\mu^{\frac{1}{4}}|g|_{H^{\frac{1}{2}}}).$

For the second term, we instead use commutator estimate (B.9) and (2.11) to
obtain

$\displaystyle|A_{2}|$
$\displaystyle\leq|\big{[}[\\![\underline{V}^{\pm}]\\!],|\mathrm{D}|^{\frac{1}{2}}\big{]}\mathcal{H}|\mathrm{D}|^{\frac{1}{2}}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]f|_{L^{2}}|g|_{L^{2}}$
$\displaystyle\hskip
14.22636pt+|[\\![\underline{V}^{\pm}]\\!]|_{L^{\infty}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}_{\mu}[{\varepsilon}\zeta]f|_{\mathring{H}^{\frac{1}{2}}}|g|_{\mathring{H}^{\frac{1}{2}}}$
$\displaystyle\leq
M\mu^{\frac{1}{4}}|\partial_{x}\psi|_{H^{t_{0}+\frac{1}{2}}}|f|_{\dot{H}^{\frac{1}{2}}_{\mu}}(|g|_{L^{2}}+|g|_{H^{\frac{1}{2}}}).$

∎

###  4.1. Properties of $\mathfrak{Ins}[\mathbf{U}]\bullet$.

The instability operator is defined in terms of $\mathfrak{E}_{\mu}$ which is
given by

$\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\bullet=\partial_{x}\circ\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\partial_{x}\bullet.$

Meaning we need to study its properties, which is the topic of the next
Proposition.

######  Proposition 4.4.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+2}({\mathbb{R}})$ be such that (1.14)
is satisfied. Then we have the following results:

  * 1.

There exist a constant $c\leq M$ such that for all $f\in
H^{\frac{1}{2}}({\mathbb{R}})$ there holds,

$0\leq\big{(}\mathfrak{E}_{\mu}[{\varepsilon}\zeta]f,f\big{)}_{L^{2}}\leq\frac{c}{\mu}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}^{2}.$
(4.12)

  * 2.

If suppose further that $\zeta$ is time dependent and satisfies (1.14)
uniformly in time, then for all $f\in H^{\frac{1}{2}}({\mathbb{R}})$ there
holds,

$|\big{(}\big{[}\partial_{t},\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{]}f,f\big{)}_{L^{2}}|\leq\frac{{\varepsilon}}{\mu}M|\partial_{t}\zeta|_{H^{t_{0}+1}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}^{2}.$
(4.13)

###### Proof.

We give the proof in two separate points.

Step 1. For the proof of the first point, we deduce the positivity by
integrating parts:

$\displaystyle\big{(}\mathfrak{E}_{\mu}[{\varepsilon}\zeta]f,f\big{)}_{L^{2}}$
$\displaystyle=-\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}f,\partial_{x}f\big{)}_{L^{2}}$
$\displaystyle=-\big{(}g,(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])\mathcal{J}_{\mu}[{\varepsilon}\zeta]g\big{)}_{L^{2}}$
$\displaystyle\geq 0,$

where we defined $g$ by

$g=\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}f,$

and used the fact that $\mathcal{G}^{-}_{\mu}$ is negative and
$\mathcal{J}_{\mu}$ is positive.

For the upper bound, we use Plancherel’s identity, and Cauchy-Schwarz
inequality with the estimates (2.23) and (A.34) to find that

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

Step 2. By direct computations, we need to control the following two terms:

$\displaystyle\big{[}\partial_{t},\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{]}f$
$\displaystyle=\partial_{x}\circ(\mathrm{d}_{\zeta}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\partial_{t}\zeta))(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\partial_{x}f$
$\displaystyle\hskip
14.22636pt+\partial_{x}\circ\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}\mathrm{d}_{\zeta}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}(\partial_{t}\zeta)\circ\partial_{x}f$
$\displaystyle=f_{1}+f_{2}.$

For the contribution of the first term, we use Planchrel’s identity, Cauchy-
Schwarz, and estimates (A.56) and (A.34):

$\displaystyle|\big{(}f_{1},f\big{)}_{L^{2}}|$
$\displaystyle\leq|\mathrm{d}_{\zeta}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\partial_{t}\zeta)(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\partial_{x}f|_{\dot{H}^{\frac{1}{2}}_{\mu}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}$
$\displaystyle\leq{\varepsilon}\mu^{-\frac{1}{4}}M|\partial_{t}\zeta|_{H^{t_{0}+1}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}f|_{\mathring{H}^{\frac{1}{2}}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}$
$\displaystyle\leq{\varepsilon}\mu^{-\frac{1}{4}}M|\partial_{t}\zeta|_{H^{t_{0}+1}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}^{2}.$

For the estimate on $f_{2}$ we use we use Plancherel’s identity, Cauchy-
Schwarz, (2.23), (A.37), and then (A.34) to obtain

$\displaystyle|\big{(}f_{2},f\big{)}_{L^{2}}|$
$\displaystyle\leq\mu^{-\frac{1}{4}}|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\mathrm{d}_{\zeta}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](\partial_{t}\zeta)\circ(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\partial_{x}f|_{\mathring{H}^{\frac{1}{2}}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}$
$\displaystyle\leq{\varepsilon}\mu^{-1}M|\partial_{t}\zeta|_{H^{t_{0}+1}}|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}f|_{L^{2}}^{2}.$

∎

######  Remark 4.5.

In the first point, we can define the smallest constant $c\leq M$ such that
(4.12) holds by

$\mathfrak{e}(\zeta)=\sup\limits_{f\in H^{\frac{1}{2}}({\mathbb{R}}),f\neq
0}\frac{\big{(}\mathcal{J}_{\mu}[{\varepsilon}\zeta]^{-1}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}f,\partial_{x}f\big{)}_{L^{2}}}{|1+\sqrt{\mu}|\mathrm{D}|^{\frac{1}{2}}f|_{L^{2}}^{2}}.$
(4.14)

For the next result, we will treat the properties of the instability operator.
In particular, we will show that under the stability criterion, we can have a
coercivity-type estimate. This is essential for the well-posedness theory to
work and relies on the surface tension parameter through the Bond number
$\mathrm{bo}^{-1}>0$. To be clear, we restate the stability criterion in the
introduction:

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

where

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

and $\mathfrak{a}$ is given by (4.16), $\mathfrak{e}$ is given in (4.14), and
we define

$\displaystyle\mathfrak{c}(\zeta)=\mathfrak{e}(\zeta)^{2}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}}^{2})^{\frac{3}{2}}.$

######  Proposition 4.6.

Let ${\varepsilon},\mu,\mathrm{bo}^{-1}\in(0,1)$, $T>0$, $t_{0}\geq 1$, and
$\mathbf{U}=(\zeta,\psi)\in\mathscr{E}^{N}_{\mathrm{bo},T}$ be such that the
non-cavitation condition (1.14) holds and satisfies the stability criterion
(1.18) on $[0,T]$. Then we have the following set of inequalities on the same
time interval:

  * 1.

For all $u\in H^{1}_{\gamma,\mathrm{bo}}({\mathbb{R}})$ and
$\mathfrak{Ins}(\mathbf{U})$ defined by (4.4), one has

$\big{(}u,\mathfrak{Ins}(\mathbf{U})u\big{)}_{L^{2}}\leq
C\mathcal{E}^{1}(\mathbf{U})|u|_{H^{1}_{\gamma,\mathrm{bo}}}^{2}.$ (4.15)

  * 2.

For $a(\mathbf{U})$ defined by

$a(\mathbf{U})=(1-\gamma)\gamma{\varepsilon}^{2}\mathfrak{e}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{L^{\infty}}^{2},$
(4.16)

and for there is some $C_{1}>0$ and $b(\mathbf{U})$ defined by

$b(\mathbf{U})=(1-\gamma)\gamma{\varepsilon}^{2}\sqrt{\mu}C_{1}\mathfrak{e}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{2},$
(4.17)

such that for $\mathfrak{d}$ defined by (1.18) there holds,

$\frac{1}{2}\mathfrak{d}(\mathbf{U})|u|_{H^{1}_{\gamma,\mathrm{bo}}}\leq\big{(}u,\mathfrak{Ins}(\mathbf{U})u\big{)}_{L^{2}}+a(\mathbf{U})|u|_{L^{2}}^{2}+b(\mathbf{U})|u|_{H^{-\frac{1}{2}}}.$
(4.18)

  * 3.

Lastly, there is a control on $u\in\mathring{H}^{\frac{1}{2}}({\mathbb{R}})$
through the inequality

${\varepsilon}\mu^{\frac{1}{4}}|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{2}|u|_{\mathring{H}^{\frac{1}{2}}}\leq{\varepsilon}C(\mathcal{E}^{1}(\mathbf{U}))|u|_{H^{1}_{\gamma,\mathrm{bo}}}.$
(4.19)

###### Proof.

The proof is similar to the one of Lemma $11$ in [51]. However, we give a
short proof for the convenience of the reader to track the constants that are
responsible for the definitions above. We divide the proof into three steps

Step 1. For the proof of (4.15), we have to deal with the terms:

$\displaystyle\big{(}u,\mathfrak{Ins}(\mathbf{U})u\big{)}_{L^{2}}$
$\displaystyle=\big{(}u,\mathfrak{a}u\big{)}_{L^{2}}-(1-\gamma)\gamma{\varepsilon}^{2}\mu\big{(}u,[\\![\underline{V}^{\pm}]\\!]\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{(}u[\\![\underline{V}^{\pm}]\\!]\big{)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\mathrm{bo}^{-1}\big{(}u,\partial_{x}\mathcal{K}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]\partial_{x}u\big{)}_{L^{2}}$
$\displaystyle=I_{1}+I_{2}+I_{3}.$

For $I_{1}$, we have that

$\big{(}\inf\limits_{x\in{\mathbb{R}}}\mathfrak{a}\big{)}|u|_{L^{2}}^{2}\leq
I_{1}\leq C\mathcal{E}^{1}(\mathbf{U})(1-\gamma)|u|_{L^{2}}^{2},$ (4.20)

where the upper bound is direct since the energy includes time derivatives.
For the estimate on $I_{2}$, we employ (4.12) to find that

$\displaystyle-(1-\gamma)\gamma{\varepsilon}^{2}\mathfrak{e}(\zeta)|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}(u[\\![\underline{V}^{\pm}]\\!])|_{L^{2}}^{2}\leq
I_{2}\leq 0.$

For the estimate on $I_{3}$, we have by integration by parts that

$\mathrm{bo}^{-1}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}})^{-\frac{3}{2}}|\partial_{x}u|_{L^{2}}^{2}\leq
I_{3}\leq\mathrm{bo}^{-1}|\partial_{x}u|_{L^{2}}^{2}.$ (4.21)

So the upper bound in (4.15) is proved.

Step 2. For the proof of (4.18), we need to work on the lower bound in
$I_{2}$. First, we can replace the lower bound using the commutator estimate
(B.8) and Hölder’s inequality to find that

$\displaystyle|\langle\sqrt{\mu}\mathrm{D}\rangle^{\frac{1}{2}}(u[\\![\underline{V}^{\pm}]\\!])|_{L^{2}}^{2}$
$\displaystyle\leq
C_{1}\sqrt{\mu}|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{2}|u|_{H^{-\frac{1}{2}}}^{2}+\sqrt{\mu}|[\\![\underline{V}^{\pm}]\\!]|_{L^{\infty}}^{2}||\mathrm{D}|^{\frac{1}{2}}u|_{L^{2}}^{2}$
$\displaystyle\hskip
14.22636pt+|[\\![\underline{V}^{\pm}]\\!]|^{2}_{L^{\infty}}|u|_{L^{2}}^{2},$
(4.22)

for some constant $C_{1}>0$. Then we may define $b(\mathbf{U})$ by (4.17) and
use the expression for $a(\mathbf{U})$ to find that

$\displaystyle
I_{3}\geq-b(\mathbf{U})|u|_{H^{-\frac{1}{2}}}-a(\mathbf{U})\big{(}|u|_{L^{2}}^{2}+\sqrt{\mu}|u|^{2}_{\mathring{H}^{\frac{1}{2}}}\big{)}.$

Then adding all these estimates, one finds that

$\displaystyle
a(\mathbf{U})|u|_{L^{2}}^{2}+b(\mathbf{U})|u|_{H^{-\frac{1}{2}}}$
$\displaystyle+\big{(}u,\mathfrak{Ins}(\mathbf{U})u\big{)}_{L^{2}}$
$\displaystyle=a(\mathbf{U})|u|_{L^{2}}^{2}+b(\mathbf{U})|u|_{H^{-\frac{1}{2}}}+I_{1}+I_{2}+I_{3}$
$\displaystyle\geq\big{(}\inf\limits_{x\in{\mathbb{R}}}\mathfrak{a}\big{)}|u|_{L^{2}}^{2}+\mathrm{bo}^{-1}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}})^{-\frac{3}{2}}|\partial_{x}u|_{L^{2}}^{2}-a(\mathbf{U})\sqrt{\mu}|u|^{2}_{\mathring{H}^{\frac{1}{2}}}.$

To conclude, we first use interpolation and Young’s inequality to find that

$\displaystyle\sqrt{\mu}a(\mathbf{U})|u|^{2}_{\mathring{H}^{\frac{1}{2}}}$
$\displaystyle\leq\sqrt{\mu}a(\mathbf{U})|u|_{L^{2}}|\partial_{x}u|_{L^{2}}$
$\displaystyle\leq\mathrm{bo}\frac{\mu}{2}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}})^{\frac{3}{2}}a(\mathbf{U})^{2}|u|_{L^{2}}^{2}+\frac{1}{2\mathrm{bo}}(1+{\varepsilon}^{2}\mu|\partial_{x}\zeta|_{L^{\infty}})^{-\frac{3}{2}}|\partial_{x}u|_{L^{2}}^{2}.$

Clearly, the proof is over if we have the following inequality

$\inf\limits_{x\in{\mathbb{R}}}\mathfrak{a}-\Upsilon\mathfrak{c}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{L^{\infty}}^{4}>0,$

which holds by (1.18) on $[0,T]$.

Step 3. To prove the third point, we again use interpolation and the full
stability criterion (1.18) to obtain

$\displaystyle{\varepsilon}^{2}\sqrt{\mu}|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{4}|u|^{2}_{\mathring{H}^{\frac{1}{2}}}$
$\displaystyle\leq\Upsilon\mathfrak{c}(\zeta)|[\\![\underline{V}^{\pm}]\\!]|_{H^{t_{0}+1}}^{4}|u|_{L^{2}}^{2}+\mathrm{bo}^{-1}(1+\mu|\partial_{x}\zeta|_{L^{\infty}})^{-\frac{3}{2}}|\partial_{x}u|_{L^{2}}^{2}$
$\displaystyle\leq{\varepsilon}^{2}M\big{(}(\inf\limits_{x\in{\mathbb{R}}}\mathfrak{a})|u|_{L^{2}}^{2}+\mathrm{bo}^{-1}|\partial_{x}u|_{L^{2}}^{2}\big{)}.$

∎

##  5\. A priori estimates

We are now in the position to derive energy estimates for the internal water
waves system (1.13). To define a natural energy to the system, we distinguish
between the cases $\alpha=0$ and $1\leq|\alpha|\leq N.$ For $\alpha=0$, we
have enough regularity on the data to control the solutions with the energy:

$E^{0}(\mathbf{U})=|\zeta|_{H^{1}_{\gamma,\mathrm{bo}}}^{2}+\frac{1}{\mu}\big{(}\psi,\mathcal{G}_{\mu}[0]\psi\big{)}_{L^{2}},$
(5.1)

where $\mathcal{G}[0]\psi$ is defined by formula (2.9):

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

For $1\leq|\alpha|\leq N$, we use Proposition 4.2 and exploit its quaslinear
structure to define a suitable symmetrizer. In particular, we will need to
cancel specific terms in the energy estimates, and it will be done by
introducing the symmetrizer:

$Q(\mathbf{U})=Q^{(1)}(\mathbf{U})+Q^{(2)}(\mathbf{U})=\begin{pmatrix}\mathfrak{Ins}[\mathbf{U}]&0\\\
0&\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\end{pmatrix}+\begin{pmatrix}a(\mathbf{U})+b(\mathbf{U})\Lambda^{-1}&0\\\
0&0\end{pmatrix}.$ (5.2)

where $a(U)$ and $b(U)$ are defined by (4.16) and (4.17). With this
symmetrizer at hand, we define energies by

$E_{\alpha}(\mathbb{U})=\big{(}\mathbb{U}_{(\alpha)},Q(\mathbb{U})\mathbb{U}_{(\alpha)}\big{)}_{L^{2}}.$
(5.3)

Finally, taking the sum over all $\alpha$ will be the main quantity used to
control the principal part of (1.13):

$E^{k}(\mathbb{U})=\sum\limits_{1\leq|\alpha|\leq k}E_{\alpha}(\mathbb{U}).$
(5.4)

Before we proceed, we make three comments, similar to the ones made in [51],
on the choice of the energy.

######  Remark 5.1.

ho

  * 1.

The term $Q_{1}[\mathbf{U}]$ in (5.2), is chosen specifically to cancel the
principal part:

$\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},Q_{1}(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}=0,$

which appears naturally in the energy estimates. See estimate of $B_{2}^{1}$
in the proof of Proposition 5.2 below.

  * 2.

The role of the term $Q_{2}[\mathbf{U}]$ is to make the energy equivalent to
the energy norm. In particular, for (5.4) to be coercive, we need
$Q_{2}[\mathbf{U}]$ and the stability criterion (1.18) to have a lower bound
on $\mathfrak{Ins}[\mathbf{U}]$ in the energy space.

  * 3.

To close the energy estimates, we actually need to modify (5.4). This is
because $\mathfrak{Ins}[\mathbf{U}]$ is a second-order operator and therefore
makes a contribution to the sub-principal part of the equation (when
$|\alpha|=N$). In particular, we correct the energy to cancel the terms

$\big{(}\mathcal{C}[\mathbf{U}]\mathbf{U}_{(\check{\alpha})},Q_{1}(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}},$

which will appear in the a priori estimates. To do so, we define the quantity

$\tilde{Q}(\mathbf{U})=\begin{pmatrix}\mathrm{bo}^{-1}\mathcal{K}_{(\alpha)}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]&0\\\
0&\frac{1}{\mu}\mathcal{G}_{\mu,(\alpha)}\end{pmatrix},$

where we define the modified energy for the internal water waves equation by

$E_{{\text{\tiny
IWW}}}^{N}(\mathbf{U})=E^{N}(\mathbf{U})+C_{2}CE^{N-1}(\mathbf{U})+\sum\limits_{|\alpha|=N}\big{(}\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}},$
(5.5)

for some constants $C,C_{2}>0$ to be fixed in the proof.

######  Proposition 5.2.

Let ${\varepsilon},\mu,\gamma\in(0,1)$,
$\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$, $t_{0}=1$, $N\geq 5$, and
$\mathbf{U}=(\zeta,\psi)^{T}\in\mathscr{E}_{\mathrm{bo},\gamma}^{N,t_{0}}$ be
a solution to (1.13) on a time interval $[0,T]$ for some $T>0$. Assume that
$\mathbf{U}$ satisfies the non-cavitation condition (1.14) and the stability
criterion (1.18) on $[0,T]$. Then, for the modified energy defined by (5.5)
there is a constant
$C=C(M(4),\frac{1}{h_{0}},\frac{1}{\mathfrak{d}(\mathbf{U})})>0$ such that,

$\frac{d}{dt}E_{{\text{\tiny
IWW}}}^{N}(\mathbf{U})\leq{\varepsilon}CE_{{\text{\tiny
IWW}}}^{N}(\mathbf{U}),$ (5.6)

for all $0<t<T$. Furthermore, for the energy defined by (5.4) we have for any
$0\leq k\leq N$ there is a universal constant $C>0$ such that

$\frac{1}{C}\mathcal{E}^{k}_{\mathrm{bo}}(\mathbf{U})\leq
E^{k}(\mathbf{U})\leq C\mathcal{E}^{k}_{\mathrm{bo}}(\mathbf{U}),$ (5.7)

and

$\frac{1}{C}\mathcal{E}^{N}_{\mathrm{bo}}(\mathbf{U})\leq E_{{\text{\tiny
IWW}}}^{N}(\mathbf{U})\leq C\mathcal{E}^{N}_{\mathrm{bo}}(\mathbf{U}),$ (5.8)

for all $0<t<T$.

###### Proof.

We first prove the equivalence between the different energy functionals before
turning to the proof of (5.6). For the proof of inequality (5.7), we divide it
into two cases.

Proof of (5.7) in the case $k=0$. Let $(\mathcal{G}_{\mu}[0])^{\frac{1}{2}}$
be the square root of the symbol associated to $\mathcal{G}_{\mu}[0]$ and then
use Plancherel’s identity to see that

$\frac{1}{\mu}\big{(}\psi,\mathcal{G}_{\mu}[0]\psi\big{)}_{L^{2}}=\frac{1}{\mu}|(\mathcal{G}_{\mu}[0])^{\frac{1}{2}}\psi|_{L^{2}}^{2}.$

Then use (B.2) to conclude that

$\frac{1}{C}\mathcal{E}^{0}_{\mathrm{bo}}(\mathbf{U})\leq
E^{0}(\mathbb{U})\leq C\mathcal{E}^{0}_{\mathrm{bo}}(\mathbf{U}).$

Proof of (5.7) in the case $1\leq k\leq N$. For the remaining cases, we use
the definition of the individual energies (5.3) to find that

$\displaystyle E_{\alpha}(\mathbf{U})$
$\displaystyle=\big{(}\zeta_{(\alpha)},\mathfrak{Ins}[U]\zeta_{(\alpha)}\big{)}_{L^{2}}+\big{(}\zeta_{(\alpha)},(a(\mathbf{U})+b(\mathbf{U})\Lambda^{-1})\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{\mu}\big{(}\psi,\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi\big{)}_{L^{2}}$

The first two terms are controlled by (4.18) and (4.15). While the last term
is controlled by (2.6) and (2.8), which implies the result

$\frac{1}{C}\mathcal{E}^{k}_{\mathrm{bo}}(\mathbf{U})\leq
E^{k}(\mathbb{U})\leq C\mathcal{E}^{k}_{\mathrm{bo}}(\mathbf{U}).$

Proof of (5.8). Next, we prove (5.8). To estimate the additional term, we
first observe that

$\displaystyle\frac{1}{\mu}|\big{(}\psi_{(\alpha)},\mathcal{G}_{{\mu},(\alpha)}\psi_{(\check{\alpha})}\big{)}_{L^{2}}|$
$\displaystyle\leq\frac{1}{\mu}\sum\limits_{j=1}^{2}\alpha_{j}\Big{(}|\big{(}\psi_{(\alpha)},\mathrm{d}\mathcal{G}_{\mu}^{+}(\partial_{j}\zeta)\mathcal{J}_{\mu}^{-1}\psi_{(\check{\alpha})}\big{)}_{L^{2}}|$
$\displaystyle\hskip
14.22636pt+|\big{(}\psi_{(\alpha)},\mathcal{G}_{\mu}^{+}\mathcal{J}_{\mu}^{-1}(\mathrm{d}\mathcal{J}_{\mu}(\partial_{j}\zeta))\mathcal{J}_{\mu}^{-1}\psi_{(\check{\alpha})}\big{)}_{L^{2}}|\Big{)}.$

For the first term we use (A.16) with (2.23), while the for the second term we
also use (A.6), (2.23), and then (A.55) to find that

$\frac{1}{\mu}|\big{(}\psi_{(\alpha)},\mathcal{G}_{{\mu},(\alpha)}\psi_{(\check{\alpha})}\big{)}_{L^{2}}|\leq{\varepsilon}M|\psi_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}|\psi_{(\check{\alpha})}|_{\dot{H}^{\frac{1}{2}}_{\mu}}$
(5.9)

Then since the surface tension term is the same as in [51, 52] and is
straightforward to estimate, we have by Young’s inequality that

$\displaystyle\sum\limits_{|\alpha|=N}|\big{(}\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}|$
$\displaystyle\leq\sum\limits_{|\alpha|=N}\mathrm{bo}^{-1}|\big{(}\zeta_{(\alpha)},\mathcal{K}_{(\alpha)}\zeta_{(\check{\alpha})}\big{)}_{L^{2}}|+\frac{1}{\mu}|\big{(}\psi_{(\alpha)},\mathcal{G}_{{\mu},(\alpha)}\psi_{(\check{\alpha})}\big{)}_{L^{2}}|$
$\displaystyle\leq\frac{1}{2C_{2}}\mathcal{E}^{N}(\mathbf{U})+\frac{C_{2}}{2}\mathcal{E}^{N-1}(\mathbf{U}),$

for any $C_{2}>0$. Then let $C_{2}\geq C$ and use (5.8) with the definition of
the modified energy (5.5) to see that

$\displaystyle\frac{1}{2}E^{N}(\mathbf{U})\leq E_{IWW}^{N}(\mathbf{U})\leq
C_{2}CE^{N}(\mathbf{U}).$

From these inequalities we can conclude by using (5.7) with $k=N$.

Lastly, we will prove energy estimate (5.6) by considering two cases.

An energy estimate in the case $k=0$. We first use (1.13) to make the
decomposition

$\displaystyle\partial_{t}\zeta$
$\displaystyle=\frac{1}{\mu}\mathcal{G}_{\mu}[0]\psi+{\varepsilon}\mathcal{N}_{1}(\mathbf{U})$
$\displaystyle\partial_{t}\psi$
$\displaystyle=-\big{(}(1-\gamma)-\mathrm{bo}^{-1}\partial_{x}^{2}\big{)}\zeta+{\varepsilon}\mathcal{N}_{2}(\mathbf{U}),$

where

$\mathcal{N}_{1}(\mathbf{U})=\frac{1}{\mu{\varepsilon}}(\mathcal{G}_{\mu}[{\varepsilon}\zeta]-\mathcal{G}_{\mu}[0])\psi,$

and

$\displaystyle\mathcal{N}_{2}(\mathbf{U})$
$\displaystyle=\frac{\sqrt{\mu}}{{\varepsilon}\mathrm{bo}}\partial_{x}\Big{(}\big{(}\frac{1}{\sqrt{1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2}}}-1\big{)}\partial_{x}\zeta\Big{)}-\frac{1}{2}\big{(}(\partial_{x}\psi^{+})^{2}+\gamma(\partial_{x}\psi^{-})^{2}\big{)}-{\varepsilon}\mathcal{N}(\mathbf{U}).$

The decomposition emphasizes the terms of order ${\varepsilon}$, while the
linear terms are canceled by our choice of the energy:

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}E^{0}(U)$
$\displaystyle=\big{(}\partial_{t}\zeta,\big{(}(1-\gamma)-\mathrm{bo}^{-1}\partial_{x}^{2}\big{)}\zeta\big{)}_{L^{2}}+\frac{1}{\mu}\big{(}\partial_{t}\psi,\mathcal{G}_{\mu}[0]\psi\big{)}_{L^{2}}$
$\displaystyle={\varepsilon}\big{(}\mathcal{N}_{1}(\mathbf{U}),\big{(}(1-\gamma)-\mathrm{bo}^{-1}\partial_{x}^{2}\big{)}\zeta\big{)}_{L^{2}}+\frac{{\varepsilon}}{\mu}\big{(}\mathcal{N}_{2}(\mathbf{U}),\mathcal{G}_{\mu}[0]\psi\big{)}_{L^{2}}.$
$\displaystyle=A_{1}+A_{2}.$

For the estimate on $A_{1}$, we use integration by parts, Cauchy-Schwarz
inequality, and (3.21) to find that

$\displaystyle|A_{1}|$
$\displaystyle\leq{\varepsilon}(1+\mathrm{bo}^{-\frac{1}{2}})M(t_{0}+2)|\partial_{x}\psi|_{H^{2}}|\zeta|_{H^{1}_{\gamma,\mathrm{bo}}}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the estimate on $A_{2}$, we integrate by parts, Hölder’s inequality, and
the Sobolev embedding to find that

$\displaystyle|A_{2}|\leq\frac{{\varepsilon}}{\mu}\mathrm{bo}^{-1}M|\partial_{x}\zeta|_{L^{2}}|\mathcal{G}_{\mu}[0]\psi|_{H^{1}}+\frac{{\varepsilon}}{\mu}(|\partial_{x}\psi^{+}|_{L^{2}}+|\partial_{x}\psi^{-}|_{L^{2}}+|\mathcal{N}(\mathbf{U})|_{L^{2}})|\mathcal{G}_{\mu}[0]\psi|_{L^{2}}$

where we recall the definition

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

To conclude, we simply use (B.1) to control
$\frac{1}{\mu}\mathcal{G}_{\mu}[0]\psi$, while for the remaining quantities we
use Corollary A.18 and estimates (A.40)

$|\partial_{x}\psi^{+}|_{L^{2}}+|\partial_{x}\psi^{-}|_{L^{2}}\leq
M|\partial_{x}\psi|_{L^{2}}.$

Moreover, to deal with the terms in $\mathcal{N}(\mathbf{U})$ we use the
product estimate (B.6) and estimate (A.42) to obtain

$\displaystyle|\mathcal{N}(\mathbf{U})|_{L^{2}}$ $\displaystyle\leq
M(\frac{\gamma}{\mu}|w^{-}|^{2}_{H^{1}}+\frac{1}{\mu}|w^{+}|_{H^{1}}^{2})$
$\displaystyle\leq M|\partial_{x}\psi|_{H^{2}}^{2}.$

Gathering all these estimates, we find that

$\frac{\mathrm{d}}{\mathrm{d}t}E^{0}(U)\leq{\varepsilon}M\mathcal{E}^{N}(\mathbf{U}).$
(5.10)

An energy estimate in the case $1\leq k\leq N$. We first derive an estimate on
each $E_{\alpha}(\mathbf{U})$ defined by (5.3) for $1\leq|\alpha|\leq N$. To
do so, we use the self-adjointness of $Q(\mathbf{U}_{(\alpha)})$ to find that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}E_{\alpha}(\mathbf{U})$
$\displaystyle=\frac{1}{2}\big{(}\mathbf{U}_{(\alpha)},\big{(}\partial_{t}Q(\mathbf{U})\big{)}\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}+\big{(}\partial_{t}\mathbf{U}_{(\alpha)},Q(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{1}+B_{2}.$

Control of $B_{1}$. By definition of $Q(\mathbf{U})$ given by (5.2), we find
that

$\displaystyle B_{1}$
$\displaystyle=\frac{1}{2}\big{(}[\partial_{t},\mathfrak{Ins}(\mathbf{U})]\zeta_{(\alpha)},\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{2\mu}\big{(}[\partial_{t},\mathcal{G}]\psi_{(\alpha)},\psi_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\frac{1}{2}\big{(}(\partial_{t}a)\zeta_{(\alpha)},\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{2}\big{(}(\partial_{t}b)\zeta_{(\alpha)},\Lambda^{-1}\zeta_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{1}^{1}+B_{1}^{2}+B_{1}^{3}+B_{1}^{4}.$

For the estimate on $B_{1}^{1}$, we have by definition and integration by
parts that

$\displaystyle B_{1}^{1}$
$\displaystyle=\frac{1}{2}\big{(}\zeta_{(\alpha)},\big{(}\partial_{t}\mathfrak{a}\big{)}\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{2\mathrm{bo}}\big{(}\partial_{x}\zeta_{(\alpha)},\big{[}\partial_{t},\mathcal{K}[{\varepsilon}\sqrt{\mu}\partial_{x}\zeta]\big{]}\partial_{x}\zeta_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-(1-\gamma)\gamma{\varepsilon}^{2}\mu\big{(}\big{(}\partial_{t}[\\![\underline{V}^{\pm}]\\!]\big{)}\zeta_{(\alpha)},\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{(}\zeta_{(\alpha)}[\\![\underline{V}^{\pm}]\\!]\big{)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\frac{1}{2}(1-\gamma)\gamma{\varepsilon}^{2}\mu\big{(}[\\![\underline{V}^{\pm}]\\!]\zeta_{(\alpha)},\big{[}\partial_{t},\mathfrak{E}_{\mu}[{\varepsilon}\zeta]\big{]}\big{(}\zeta_{(\alpha)}[\\![\underline{V}^{\pm}]\\!]\big{)}\big{)}_{L^{2}}$
$\displaystyle=B_{1}^{1,1}+B_{1}^{1,2}+B_{1}^{1,3}+B_{1}^{1,4}.$

For the estimate of $B_{1}^{1,1}$, we have by definition of $\mathfrak{a}$
given by (4.1) and estimates: Hölder’s inequality, Sobolev embedding, and
(B.6) to find that

$\displaystyle|B_{1}^{1,1}|$
$\displaystyle\leq\frac{1}{2}|\partial_{t}\mathfrak{a}|_{H^{t_{0}}}|\zeta_{(\alpha)}|_{L^{2}}^{2}$
$\displaystyle\leq{\varepsilon}(|\partial_{t}^{2}\underline{w}^{+}|_{H^{t_{0}}}+|\underline{V}^{+}|_{H^{t_{0}}}|\partial_{t}\partial_{x}\underline{w}^{+}|_{H^{t_{0}}})|\zeta_{(\alpha)}|_{L^{2}}^{2}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}(|\partial_{t}^{2}\underline{w}^{-}|_{H^{t_{0}}}+|\underline{V}^{-}|_{H^{t_{0}}}|\partial_{t}\partial_{x}\underline{w}^{-}|_{H^{t_{0}}})|\zeta_{(\alpha)}|_{L^{2}}^{2}.$

For $\partial_{t}^{2}\underline{w}^{+}$ we can easily prove that it satisfies
estimate (A.12) and use it with (A.42) and (A.40) to find that

$\displaystyle|\partial_{t}^{2}\underline{w}^{\pm}|_{H^{t_{0}}}$
$\displaystyle\leq\sum\limits_{j=1}^{2}\mu^{\frac{3}{4}}|\partial_{t}^{j}\zeta|_{H^{t_{0}+1}}|\partial_{t}^{2-j}\psi^{+}|_{\dot{H}_{\mu}^{t_{0}+1}}$
$\displaystyle\leq C\mathcal{E}^{4}(\mathbf{U}).$

The same can be done for $\underline{w}^{-}$, and combine it (A.45) we
conclude that

$\displaystyle|B_{1}^{1,1}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the estimate on $B_{1}^{1,2}$, it only depends on $\mathcal{K}_{(\alpha)}$
which is treated in [51, 52] and satisfies

$\displaystyle|B_{1}^{1,2}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the estimate on $B_{1}^{1,3}$, we use estimates (4.12), (4.1), and (4.19):

$\displaystyle B_{1}^{1,3}$
$\displaystyle\leq(1-\gamma)\gamma{\varepsilon}^{2}\mu|(1+\sqrt{\mu}|\mathrm{D}|)^{\frac{1}{2}}(\zeta_{(\alpha)}[\\![\underline{V}^{\pm}]\\!])|_{L^{2}}^{2}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Likewise, the estimate on $B_{1}^{1,4}$ we simply use (4.13) instead of (4.12)
to obtain the same bound and we deduce that

$\displaystyle|B_{1}^{1}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Clearly, the estimates on $B_{1}^{3}$ and $B_{1}^{1,4}$ is estimated similarly
to $B_{1}^{1,1}$, while we estimate $B_{1}^{2}$ wwe have by direct
computations that

$\displaystyle|B_{1}^{1,2}|\leq\frac{1}{2\mu}\big{(}[\partial_{t},\mathcal{G}_{\mu}^{+}]\mathcal{J}_{\mu}^{-1}\psi_{(\alpha)},\psi_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{2\mu}\big{(}\mathcal{G}_{\mu}^{+}\mathcal{J}_{\mu}^{-1}(\mathrm{d}\mathcal{J}_{\mu}(\partial_{t}\zeta))\mathcal{J}_{\mu}^{-1}\psi_{(\alpha)},\psi_{(\alpha)}\big{)}_{L^{2}}.$

For the first term we use Proposition $3.6$ of [8] with (2.23), while the for
the second term we also use (A.6), (2.23), and then (A.55) to find that

$\displaystyle|B_{1}^{1,2}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Collecting each estimate so far gives

$\displaystyle|B_{1}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Control of $B_{2}$. We use system (4.7) to find that

$\displaystyle B_{2}$
$\displaystyle=-\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},Q(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}-\big{(}\mathcal{B}[\mathbf{U}]\mathbf{U}_{(\alpha)},Q(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}\mathcal{C}[\mathbf{U}]\mathbf{U}_{(\check{\alpha})},Q(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}+{\varepsilon}\big{(}(R_{\alpha},S_{\alpha})^{T},Q(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{2}^{1}+B_{2}^{2}+B_{2}^{3}+B_{2}^{4}.$

Control of $B_{2}^{1}$. From Definition 4.1 and (5.2) we have that

$\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},Q_{1}(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}=0.$

We may therefore decompose $B_{2}^{1}$ into two pieces:

$\displaystyle B_{2}^{1}$
$\displaystyle=-\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},Q_{2}(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=a(\mathbf{U})\big{(}\frac{1}{\mu}\mathcal{G}_{\mu}\psi_{(\alpha)},\zeta_{(\alpha)}\big{)}_{L^{2}}+b(\mathbf{U})\big{(}\frac{1}{\mu}\mathcal{G}_{\mu}\psi_{(\alpha)},\Lambda^{-1}\zeta_{(\alpha)}\big{)}_{L^{2}}.$

Both terms are estimated directly by (2.8) and then use (4.19):

$\displaystyle|B_{2}^{1}|$ $\displaystyle\leq
M|\psi_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}\big{(}a(\mathbf{U})|\zeta_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}+b(\mathbf{U})|\Lambda^{-1}\zeta_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}\big{)}$
$\displaystyle\leq
M|\psi_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}\big{(}{\varepsilon}^{2}|\partial_{x}\zeta_{(\alpha)}|_{L^{2}}+b(\mathbf{U})|\zeta_{(\alpha)}|_{L^{2}}\big{)}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}),$

where we used that
${\varepsilon}^{2}\leq{\varepsilon}({\varepsilon}\sqrt{\mu})^{\frac{1}{2}}={\varepsilon}\mathrm{bo}^{-\frac{1}{2}}$.

Control of $B_{2}^{2}$. We first decompose $B_{2}^{2}$ into four parts

$\displaystyle B_{2}^{2}$
$\displaystyle={\varepsilon}\big{(}\mathcal{I}[\mathbf{U}]\zeta_{(\alpha)},\mathfrak{Ins}[\mathbf{U}]\zeta_{(\alpha)}\big{)}_{L^{2}}-\frac{{\varepsilon}}{\mu}\big{(}\mathcal{I}^{\ast}[\mathbf{U}]\psi_{(\alpha)},\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}a(\mathbf{U})\big{(}\mathcal{I}[\mathbf{U}]\zeta_{(\alpha)},\zeta_{(\alpha)}\big{)}_{L^{2}}+{\varepsilon}b(\mathbf{U})\big{(}\mathcal{I}[\mathbf{U}]\zeta_{(\alpha)},\Lambda^{-1}\zeta_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{2}^{2,1}+B_{2}^{2,2}+B_{2}^{2,3}+B_{2}^{2,4}.$

We first observe that the two last terms are easily treated with an estimate
of the type (4.11) to get that

$\displaystyle|B_{2}^{2,3}|+|B_{2}^{2,4}|$
$\displaystyle\leq{\varepsilon}M(4)|\partial_{x}\psi|_{H^{t_{0}+1}}|\zeta_{(\alpha)}|_{L^{2}}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the remaining two terms, we will need to work some more.

Control of $B_{2}^{2,1}$. Clearly, by (4.19) it is enough to prove that

$\displaystyle|B_{2}^{2,1}|\leq\mathcal{E}^{N}(\mathbf{U})\big{(}1+{\varepsilon}^{2}\sqrt{\mu}|[\\![\underline{V}^{\pm}]\\!]|_{W^{1,\infty}}^{2}|\zeta|^{2}_{<N+\frac{1}{2}>}\big{)}.$
(5.11)

Since the quantities involved in this depend on
$(\mathcal{G}^{-}_{\mu})^{-1}$, we need to adapt the proof of [51]. However,
the proof relies on the symbolic expression of
$\mathcal{J}_{\mu}^{-1}(\mathcal{G}^{-}_{\mu})^{-1}\partial_{x}$, and this
estimate is provided in Corollary 3.8 which has the same outcome as the one in
[51]. Using this expression, we can prove (5.11) following the proof provided
in [51]. In particular, we refer the reader to the proof of estimate $(5.15)$
in this paper.

Control of $B_{2}^{2,2}$. This estimate is similar to estimate $(5.16)$ in
[51], but we give the details here to account for the difference in
$\mathcal{G}^{-}_{\mu}$. In particular, we have by definition (4.3), that we
must provide an estimate on the terms

$\displaystyle B_{2}^{2,2}$
$\displaystyle=\frac{{\varepsilon}}{\mu}\big{(}\underline{V}^{+}\partial_{x}\psi_{(\alpha)},\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)}_{L^{2}}+\gamma\frac{{\varepsilon}}{\mu}\big{(}[\\![\underline{V}^{\pm}]\\!]\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu})^{-1}\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)},\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{2}^{2,2,1}+B_{2}^{2,2,2}.$

We let $g=\mathcal{J}_{\mu}^{-1}\psi_{(\alpha)}$ and use the definition
$\mathcal{G}_{\mu}=\mathcal{G}_{\mu}^{+}\mathcal{J}_{\mu}^{-1}$ to find that

$\displaystyle B_{2}^{2,2,1}$
$\displaystyle=\frac{{\varepsilon}}{\mu}\big{(}\underline{V}^{+}\partial_{x}\mathcal{J}_{\mu}g,\mathcal{G}^{+}_{\mu}g\big{)}_{L^{2}}$
$\displaystyle=\frac{{\varepsilon}}{\mu}\big{(}\underline{V}^{+}\partial_{x}g,\mathcal{G}^{+}_{\mu}g\big{)}_{L^{2}}-\gamma\frac{{\varepsilon}}{\mu}\big{(}\underline{V}^{+}\partial_{x}(\mathcal{G}^{-}_{\mu})^{-1}\mathcal{G}^{+}_{\mu}g,\mathcal{G}^{+}_{\mu}g\big{)}_{L^{2}}$
$\displaystyle=B_{2}^{2,2,1,1}+B_{2}^{2,2,1,2}.$

For the estimate on $B_{2}^{2,2,1,1}$, we use (A.10), combined with Sobolev
embedding, (2.23), and (A.45) to obtain,

$\displaystyle|B_{2}^{2,2,1,1}|$
$\displaystyle\leq{\varepsilon}M|\underline{V}^{+}|_{W^{1,\infty}}|g|_{\dot{H}^{\frac{1}{2}}_{\mu}}^{2}$
$\displaystyle\leq{\varepsilon}M|\underline{V}^{+}|_{H^{t_{0}+1}}|\psi_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}_{\mu}}^{2}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the estimate on $B_{2}^{2,2,1,2}$, we let
$\tilde{g}=(\mathcal{G}^{-}_{\mu})^{-1}\mathcal{G}^{+}_{\mu}\psi_{(\alpha)}$
and use (A.27). It is then straightforward to obtain the desired bound using
also (2.11):

$\displaystyle|B_{2}^{2,2,1,1}|$
$\displaystyle\leq\gamma\frac{{\varepsilon}}{\mu}|\big{(}\underline{V}^{+}\partial_{x}\tilde{g},\mathcal{G}^{-}_{\mu}\tilde{g}\big{)}_{L^{2}}|$
$\displaystyle\leq{\varepsilon}\mu^{-\frac{1}{2}}M|\underline{V}^{+}|_{W^{1,\infty}}|\tilde{g}|_{\mathring{H}^{\frac{1}{2}}}^{2}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Lastly, for the estimate on $B_{2}^{2,2,2}$ we let
$g^{\sharp}=(\mathcal{G}^{-}_{\mu})^{-1}\mathcal{G}_{\mu}\psi_{(\alpha)}$, and
again use (A.27) with (2.4) and (2.23):

$\displaystyle|B_{2}^{2,2,2}|$
$\displaystyle=\gamma\frac{{\varepsilon}}{\mu}|\big{(}[\\![\underline{V}^{\pm}]\\!]\partial_{x}g^{\sharp},\mathcal{G}^{-}_{\mu}g^{\sharp}\big{)}_{L^{2}}|$
$\displaystyle\leq{\varepsilon}\mu^{-\frac{1}{2}}M|(\mathcal{G}^{-}_{\mu})^{-1}\mathcal{G}^{+}_{\mu}\mathcal{J}_{\mu}^{-1}\psi_{(\alpha)}|_{\mathring{H}^{\frac{1}{2}}}^{2}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

From these estimates, we conclude the bound on $B_{2}^{2}$ where we find that

$\displaystyle|B_{2}^{2}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Control of $B_{2}^{3}$. To get an estimate of order ${\varepsilon}$ we need to
cancel the following term (when $k=N$):

$\displaystyle
B_{2}^{3,1}=\big{(}\mathcal{C}[\mathbf{U}]\mathbf{U}_{(\check{\alpha})},Q_{1}(\mathbf{U})\mathbf{U}_{(\alpha)}\big{)}_{L^{2}}=-\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}.$

We will cancel $B_{2}^{3,1}$ later by introducing $\tilde{Q}$ into the energy
(see Remark 5.1). Therefore, we may decompose $(B_{2}^{3}-B_{2}^{3,1})$ into
two parts:

$\displaystyle(B_{2}^{3}-B_{2}^{3,1})$
$\displaystyle=\frac{1}{\mu}a(\mathbf{U})\big{(}\mathcal{G}_{(\alpha)}\psi_{(\check{\alpha})},\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{1}{\mu}b(\mathbf{U})\big{(}\mathcal{G}_{(\alpha)}\psi_{(\check{\alpha})},\Lambda^{-1}\zeta_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle=B_{2}^{3,2}+B_{2}^{3,3}.$

However, these terms are easily dealt with using estimates on
$\mathcal{G}^{+}_{\mu}$ and $\mathcal{J}^{-1}_{\mu}$ (similar to (5.9)) which
yields the following estimate
$\frac{1}{\mu}|\mathcal{G}_{(\alpha)}\psi_{(\check{\alpha})}|_{L^{2}}\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U})$.
Thus, by Cauchy-Schwarz we obtain

$\displaystyle|B_{2}^{3,2}|+|B_{2}^{3,3}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

Control of $B_{2}^{4}$. For $B_{2}^{4}$ we have to estimate the following
terms,

$\displaystyle B_{2}^{4}$
$\displaystyle={\varepsilon}\big{(}R_{\alpha},\mathfrak{Ins}[\mathbf{U}]\zeta_{(\alpha)}\big{)}_{L^{2}}+\frac{{\varepsilon}}{\mu}\big{(}S_{\alpha},\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}a(\mathbf{U})\big{(}R_{\alpha},\zeta_{(\alpha)}\big{)}_{L^{2}}+{\varepsilon}b(\mathbf{U})\big{(}R_{\alpha},\Lambda^{-1}\zeta_{(\alpha)}\big{)}_{L^{2}}.$

For the first term, we use (4.15) and (4.8) to obtain,

$\displaystyle{\varepsilon}|\big{(}R_{\alpha},\mathfrak{Ins}[\mathbf{U}]\zeta_{(\alpha)}\big{)}_{L^{2}}|$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{1}(\mathbf{U})|R_{\alpha}|_{H^{1}_{\gamma,\mathrm{bo}}}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

For the second term is treated by (2.8), (4.8), and (4.19) to get

$\displaystyle\frac{{\varepsilon}}{\mu}|\big{(}S_{\alpha},\mathcal{G}_{\mu}\psi_{(\alpha)}\big{)}_{L^{2}}|$
$\displaystyle\leq{\varepsilon}M|S_{\alpha}|_{\dot{H}^{\frac{1}{2}}}|\psi_{(\alpha)}|_{\dot{H}^{\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

The remaining two estimates satisfy the same upper bound, and so gathering all
these estimates, we find that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}E_{\alpha}(\mathbf{U})-B_{1}^{3,1}\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$
(5.12)

As pointed out above, we need to cancel $B_{1}^{3,1}$ and is done by modifying
the energy by

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\big{(}\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}$
$\displaystyle=\frac{1}{2}\big{(}\mathbf{U}_{(\alpha)},\big{(}\partial_{t}\tilde{Q}(\mathbf{U})\big{)}\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}-\big{(}\mathcal{A}[\mathbf{U}]\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}\mathcal{B}[\mathbf{U}]\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}-\big{(}\mathcal{C}[\mathbf{U}]\mathbf{U}_{(\check{\alpha})},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+{\varepsilon}\big{(}(R_{\alpha},S_{\alpha})^{T},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}$
$\displaystyle=D_{1}+D_{2}^{1}+D_{2}^{2}+D_{2}^{3}+D_{2}^{4}.$

Where we observe that $D_{2}^{1}=-B_{2}^{3,1}$ and gives the desired
cancellation (when $k=N$), while the remaining terms are treated as above and
satisfy

$|D_{1}|+|D_{2}^{2}|+|D_{2}^{3}|+|D_{2}^{4}|\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}).$

We therefore find that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}E_{\alpha}(\mathbf{U})+\big{(}\mathbf{U}_{(\alpha)},\tilde{Q}(\mathbf{U})\mathbf{U}_{(\check{\alpha})}\big{)}_{L^{2}}\Big{)}\leq{\varepsilon}C\mathcal{E}^{N}(\mathbf{U}),$
(5.13)

where we recall that $\tilde{Q}=0$ when $|\alpha|=N$.

Proof of estimate (5.6). To conclude, we use (5.10) combined with (5.13) where
we sum over $0\leq|\alpha|\leq N$ and then apply estimate (5.8) to find that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}E_{{\text{\tiny
IWW}}}^{N}(\mathbf{U})\leq{\varepsilon}CE_{{\text{\tiny
IWW}}}^{N}(\mathbf{U}).$

∎

##  6\. Proof of Theorem 1.7

The strategy for the proof of Theorem 1.7 is classical, and we refer the
reader to [52] Chapter $4$ for a detailed proof of the water waves equations.
See also Chapter $9$ in the same book in the case of surface tension. We will
now give the main steps involved.

###### Proof.

The first step is to regularize the internal water waves equations (1.13), and
use the Fixed Point Theorem to deduce a solution. In fact, the solution is
smooth. However, the existence time depends on the regularization parameter
and shrinks to zero when taking the limit. To extend the existence time, one
applies Proposition 5.2 and then uses compactness to deduce a limit. For
uniqueness, one needs to have an estimate of the difference between two
solutions, which can be proved with estimates similar to the ones used for the
proof of Proposition 5.2. ∎

##  7\. Proof of Theorem 1.14

Since the long-time well-posedness of the unidirectional model (1.25) is
classical (see Remark 1.16) we only give the proof for (1.27). The strategy of
the proof is the same as for (1.13). It relies on the energy method, where we
need to find a suitable symmetrizer for the system. For simplicity, let
$\mathbf{U}=(\zeta,v)^{T}$, and we write (1.27) on the compact form:

$\partial_{t}\mathbb{U}+\mathcal{M}(\mathbb{U})\mathbb{U}=\mathbb{0},$ (7.1)

with

$\mathcal{M}(\mathbb{U})=\begin{pmatrix}{\varepsilon}v\partial_{x}&(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}\\\
(1-\gamma)\partial_{x}&{\varepsilon}v\partial_{x}\end{pmatrix}.$ (7.2)

where $h=1+{\varepsilon}\zeta$. To define an energy, we introduce the
symmetrizer

$\mathcal{Q}(\mathbb{U})=\begin{pmatrix}(1-\gamma)&0\\\
0&(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\end{pmatrix}.$ (7.3)

Then the energy associated with the weakly dispersive BO system (1.27) can be
rewritten as

$E_{\text{\tiny
BOs}}^{s}(\mathbb{U})=\big{(}\Lambda^{s}\mathbb{U},\mathcal{Q}(\mathbb{U})\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}.$

We will use this energy to deduce a bound on the solutions of (7.2). As noted
in the proof Theorem 1.7, this is not enough. One also needs a uniqueness type
estimate to establish the well-posedness with the energy method. This estimate
is derived similarly, but since we will need it for the proof of the full
justification, we also state it in the next result. For convenience, we let
two solutions of (7.2) be given by $\mathbf{U}_{1}=(\zeta_{1},v_{1})^{T}$ and
$\mathbf{U}_{2}=(\zeta_{2},v_{2})^{T}$ where we define
$(\eta,w)=(\zeta_{1}-\zeta_{2},u_{1}-u_{2})$. Then $\mathbf{W}=(\eta,w)^{T}$
solves

$\partial_{t}\mathbb{W}+\mathcal{M}(\mathbb{U}_{1},D)\mathbb{W}=\mathbb{F},$
(7.4)

with $\mathcal{M}$ defined as in (7.2) and the source term is given by

$\mathbb{F}=-\big{(}\mathcal{M}(\mathbb{U}_{1})-\mathcal{M}(\mathbb{U}_{2})\big{)}\mathbb{U}_{2}.$
(7.5)

The energy associated to (7.4) is given in terms of the symmetrizer
$\mathcal{Q}(\mathbb{U}_{1})$ defined in (7.3) and reads

$\tilde{E}^{s}_{\text{\tiny
BOs}}(\mathbb{W}):=\big{(}J^{s}\mathbb{W},\mathcal{Q}(\mathbf{U}_{1})J^{s}\mathbb{W}\big{)}_{L^{2}}.$
(7.6)

From these energies, we have the following a priori estimate and an estimate
of the difference between two solutions.

######  Proposition 7.1.

Let ${\varepsilon},\mu,\gamma\in(0,1)$, $s>\frac{3}{2}$, and $(\zeta,v)\in
C([0,T];H^{s}({\mathbb{R}})\times H^{s}(\mathbb{R}))$ be a solution to (7.1)
on a time interval $[0,T]$ for some $T>0$. Moreover, assume that (1.24) holds
uniformly in time and define
$K(s)=C_{1}(h_{\min,\gamma}^{-1},|(\zeta,v)|_{H^{s}\times H^{s}})>0$. Then,
for the energy given in Definition 5.4, there holds,

  * 1.

For all $0<t<T$, we have that

$\frac{\mathrm{d}}{\mathrm{d}t}E^{s}_{\text{\tiny
BOs}}(\mathbb{U})\leq{\varepsilon}K(s)E_{s}(\mathbb{U}),$ (7.7)

  * 2.

For all $0<t<T$, there exist $C_{2}>0$ such that

$C_{2}^{-1}|(\zeta,v)|^{2}_{H^{s}\times H^{s}}\leq E^{s}_{\text{\tiny
BOs}}(\mathbb{U})\leq C_{2}|(\zeta,v)|^{2}_{H^{s}\times H^{s}},$ (7.8)

Moreover, let $(\zeta_{1},v_{1}),(\zeta_{1},v_{1})\in
C([0,T];H^{s}({\mathbb{R}})\times H^{s}(\mathbb{R}))$ be solutions to (7.1) on
a time interval $[0,T]$ and satisfying the condition (1.24) uniformly in time.
Define the difference to be
$\mathbb{W}=(\psi,w)=(\zeta_{1}-\zeta_{2},v_{1}-v_{2})$ and let
$\tilde{K}(s)=C_{3}(h_{\min,\gamma}^{-1},|(\zeta_{i},v_{i})|_{H^{s}\times
H^{s}})>0$ for $i=1,2$. Then, for the energy defined by (7.6), there holds

  * 3.

For all $0<t<T$, we have that

$\frac{\mathrm{d}}{\mathrm{d}t}\tilde{E}^{0}_{\text{\tiny
BOs}}(\mathbb{W})\leq{\varepsilon}\tilde{K}(s)\tilde{E}^{0}_{\text{\tiny
BOs}}(\mathbb{W}).$ (7.9)

  * 4.

For all $0<t<T$, we have that

$\frac{\mathrm{d}}{\mathrm{d}t}\tilde{E}^{s}_{\text{\tiny
BOs}}(\mathbb{W})\leq{\varepsilon}\tilde{K}(s+1)\tilde{E}^{s}_{\text{\tiny
BOs}}(\mathbb{W}).$ (7.10)

  * 5.

For all $0<t<T$ and $r\geq 0$, there exist $C_{3}>0$ such that

$C_{3}^{-1}|(\eta,w)|^{2}_{H^{r}\times H^{r}}\leq\tilde{E}^{r}_{\text{\tiny
BOs}}(\mathbf{W})\leq C_{3}|(\eta,w)|^{2}_{H^{r}\times H^{r}}.$ (7.11)

######  Remark 7.2.

In estimate (7.10), we observe that there is a loss of derivative. This is not
sufficient for the proof of the continuous dependence with respect to initial
data. In fact, one would have to refine this estimate using a Bona-Smith
argument [16]. Since we are not concerned with the details of the well-
posedness of this system, we simply allow this rough estimate and use it to
deduce a the convergence estimate when comparing its solution with the ones of
the internal water waves system.

###### Proof of Proposition 7.1.

The proof of point $3-5$ is similar to the proof of the first two points. We
will therefore only prove (7.7) and (7.8).

We first prove estimate (7.8). By definition, we have that

$\displaystyle E^{s}_{\text{\tiny
BOs}}(\mathbb{U})=(1-\gamma)|\Lambda^{s}\zeta|_{L^{2}}^{2}+\big{(}\Lambda^{s}v,(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\Lambda^{s}v\big{)}_{L^{2}}.$

Then, as a result of the $\gamma$-dependent surface condition (1.24) and
Plancherel’s identity

$\displaystyle\big{(}\Lambda^{s}v,(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\Lambda^{s}v\big{)}_{L^{2}}$
$\displaystyle\geq h_{\min,\gamma}|v|_{H^{s}}^{2},$

since $0\leq\tanh(\sqrt{\mu}|\xi|)\leq 1$. The reverse inequality is a
consequence of Hölder’s inequality, the Sobolev embedding with
$s>\frac{3}{2}$:

$\displaystyle E^{s}_{\text{\tiny BOs}}(\mathbb{U})$
$\displaystyle\leq|\zeta|_{H^{s}}^{2}+|\sqrt{\tanh(\sqrt{\mu}|\mathrm{D}|)}v|_{H^{s}}^{2}+(1+{\varepsilon}|\zeta|_{L^{\infty}})|v|_{H^{s}}^{2}$
$\displaystyle\leq K(s)|(\zeta,v)|_{H^{s}\times H^{s}}^{2}.$

Next, we prove (7.7). By using (7.1) and the fact that
$\mathcal{Q}(\mathbb{U})$ is self-adjoint, we compute

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}E^{s}_{\text{\tiny
BOs}}(\mathbb{U})$
$\displaystyle=-\big{(}\Lambda^{s}\mathcal{M}(\mathbb{U})\mathbb{U},\mathcal{Q}(\mathbb{U})\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}+\frac{1}{2}\big{(}\Lambda^{s}\mathbb{U},(\partial_{t}\mathcal{Q}(\mathbb{U}))\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=-I+II.$

Control of $I$. We may write

$\displaystyle I$
$\displaystyle=\big{(}[\Lambda^{s},\mathcal{M}(\mathbb{U})]\mathbb{U},\mathcal{Q}(\mathbb{U})\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}+\big{(}\mathcal{Q}(\mathbb{U})\mathcal{M}(\mathbb{U})\Lambda^{s}\mathbb{U},\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=:I_{1}+I_{2}.$

Control of $I_{1}$. It follows from the Cauchy-Schwarz inequality that

$|I_{1}|\leq|[\Lambda^{s},\mathcal{M}(\mathbb{U})]\mathbb{U}|_{L^{2}}|\mathcal{Q}(\mathbb{U})\Lambda^{s}\mathbb{U}|_{L^{2}},$

where the first term is treated using the commutator estimate (B.8) yields

$\displaystyle|[\Lambda^{s},\mathcal{M}(\mathbb{U})]\mathbb{U}|_{L^{2}}$
$\displaystyle\leq{\varepsilon}|[\Lambda^{s},v]\partial_{x}\zeta|_{L^{2}}+{\varepsilon}|[\Lambda^{s},\zeta]\partial_{x}v|_{L^{2}}+{\varepsilon}|[\Lambda^{s},v]\partial_{x}v|_{L^{2}}$
$\displaystyle\leq{\varepsilon}K(s)|(\zeta,v)|_{H^{s}\times H^{s}}.$

The second term is easily treated by Hölder’s inequality, the Sobolev
embedding with $s>\frac{1}{2}$:

$\displaystyle|\mathcal{Q}(\mathbb{U})\Lambda^{s}\mathbb{U}|_{L^{2}}\lesssim|\Lambda^{s}\zeta|_{L^{2}}+{\varepsilon}|\zeta|_{L^{\infty}}|\Lambda^{s}v|_{L^{2}}\leq
K(s)|(\zeta,v)|_{H^{s}\times H^{s}}.$

The desired bound on $I_{1}$ then follows:

$|I_{1}|\leq{\varepsilon}K(s)|(\zeta,v)|_{H^{s}\times H^{s}}^{2}.$

Control of $I_{2}$. By definition, we must estimate the following terms,

$\displaystyle\big{(}\mathcal{Q}(\mathbb{U},D)\mathcal{M}(\mathbb{U})\Lambda^{s}\mathbb{U},\Lambda^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=(1-\gamma){\varepsilon}\big{(}v\partial_{x}\Lambda^{s}\zeta,\Lambda^{s}\zeta\big{)}_{L^{2}}+(1-\gamma)\big{(}(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}\Lambda^{s}v,\Lambda^{s}\zeta\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+(1-\gamma)\big{(}(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}\Lambda^{s}\zeta,\Lambda^{s}v\big{)}_{L^{2}}+\big{(}(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\Lambda^{s}v,\Lambda^{s}v\big{)}_{L^{2}}$
$\displaystyle=:I_{1}^{1}+I_{1}^{2}+I_{1}^{3}+I_{2}^{4}.$

Control of $I_{1}^{1}$. Using integration by part and the Sobolev embedding
yields

$\displaystyle|I_{1}^{1}|$
$\displaystyle\leq(1-\gamma)\frac{{\varepsilon}}{2}\big{|}\big{(}\partial_{x}v\Lambda^{s}\zeta,\Lambda^{s}\zeta\big{)}_{L^{2}}\big{|}\leq{\varepsilon}K(s)|(\zeta,v)|_{H^{s}\times
H^{s}}^{3}.$

Control of $I_{1}^{1}+I_{1}^{2}$. Observe, after integration by parts that

$\displaystyle I_{1}^{1}$
$\displaystyle=-I_{1}^{2}-{\varepsilon}\big{(}(\partial_{x}\zeta)\Lambda^{s}v,\Lambda^{s}v\big{)}_{L^{2}}.$

The first term cancels with $I_{1}^{2}$, while the Sobolev embedding easily
controls the remaining part,

${\varepsilon}|\big{(}(\partial_{x}\zeta)\Lambda^{s}v,\Lambda^{s}v\big{)}_{L^{2}}|\leq{\varepsilon}K(s)|(\zeta,v)|_{H^{s}\times
H^{s}}^{3}.$

Control of $II$. Using equation (7.1) gives us the following terms to
estimate,

$\displaystyle II$
$\displaystyle={\varepsilon}\big{(}\Lambda^{s}v,(\partial_{t}\zeta)\Lambda^{s}v\big{)}_{L^{2}}$
$\displaystyle=-{\varepsilon}\big{(}\Lambda^{s}v,\big{(}(h-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}v\big{)}\Lambda^{s}v\big{)}_{L^{2}}-{\varepsilon}^{2}\big{(}\Lambda^{s}v,(\partial_{x}(\zeta
v))\Lambda^{s}v\big{)}_{L^{2}}.$

Consequently, the desired estimate follows from Hölder’s inequality and the
Sobolev embedding

$\displaystyle\ |II|\leq{\varepsilon}K(s)|(\zeta,v)|_{H^{s}\times H^{s}}^{2}.$
(7.12)

Adding together all the estimates, combined with (7.8) yields,

$\frac{\mathrm{d}}{\mathrm{d}t}E^{s}_{\text{\tiny
BOs}}(\mathbb{U})\leq{\varepsilon}K(s)E^{s}_{\text{\tiny BOs}}(\mathbb{U}),$

and completes the proof of Proposition 7.1.

∎

###### Proof of Theorem 1.14.

The proof is an application of the energy method where we use the estimates in
Proposition 7.1 and Remark 7.2. We refer the reader to [65] for a similar
proof. ∎

##  8\. Proof of Theorem 1.17

For the derivation of (1.1), we follow the strategy given in [15], where we
formulated (1.13) in terms of the $(\zeta,v)$ solving system (1.22). The
convenience of this formulation is apparent since we can use the shallow water
expansion of $\mathcal{G}_{\mu}^{+}$ derived by Emerald in [24]444The
classical expansion of $\mathcal{G}_{\mu}^{+}$ is provided by Proposition
$3.8$ in [8]:
$|\mathcal{G}_{\mu}^{+}[{\varepsilon}\zeta]\psi^{+}-\partial_{x}\big{(}-\mu(1+{\varepsilon}\zeta)\partial_{x}\psi^{+}\big{)}|_{H^{s}}\leq\mu^{2}M(s+3)|\partial_{x}\psi|_{H^{s+3}}.$
(8.1) :

$|\mathcal{G}_{\mu}^{+}[{\varepsilon}\zeta]\psi^{+}-\partial_{x}\big{(}-\mu(1+{\varepsilon}\zeta)\mathrm{T}(\mathrm{D})\partial_{x}\psi^{+}\big{)}|_{H^{s}}\leq\mu^{2}{\varepsilon}M(s+3)|\partial_{x}\psi|_{H^{s+3}}.$
(8.2)

where
$\mathrm{T}(\mathrm{D})=\frac{\tanh(\sqrt{\mu}|\mathrm{D}|)}{\sqrt{\mu}|\mathrm{D}|}$
and for $s\geq 0$. Finally, the last ingredient before we turn to the proof is
an expansion of the interface operator.

######  Proposition 8.1.

Let $t_{0}\geq 1$, $s\geq 0$, and $\zeta\in H^{s+3}(\mathbb{R})$ such that it
satisfies (1.14). Then for all $\partial_{x}\psi^{+}\in H^{s+3}(\mathbb{R})$
there holds,

$\displaystyle\Big{|}\mathbb{H}[{\varepsilon}\zeta]\psi^{+}+\sqrt{\mu}|\mathrm{D}|\partial_{x}\psi^{+}\Big{|}_{H^{s-\frac{1}{2}}}\leq(\mu^{\frac{3}{2}}+\sqrt{\mu}{\varepsilon})M(s+3)|\partial_{x}\psi|_{H^{s+3}}.$
(8.3)

######  Remark 8.2.

The expansion of $\mathbb{H}[{\varepsilon}\zeta]$ is the same as the one
mentioned in [15] (see Remark $20$ of this paper). Even though the expansion
they gave was formal, it is straightforward to adapt their method in finite
depth to the current configuration.

###### Proof of Proposition 8.1.

The proof relies on making an approximate solution of
$\phi^{-}=\Phi^{-}\circ\Sigma^{-}$ solving (1.20). In particular, if we
simplify $\mathcal{G}^{+}_{\mu}$, then we want a solution to the problem

$\begin{cases}(\mu\partial_{x}^{2}+\partial_{z}^{2})\phi_{\mathrm{app}}^{-}=0,\\\
\partial_{z}\phi_{\mathrm{app}}^{-}\>{}_{|_{z=0}}=-\mu\partial_{x}^{2}\psi^{+},\end{cases}$
(8.4)

which is given by

$\displaystyle\phi_{\mathrm{app}}^{-}=-\sqrt{\mu}|\mathrm{D}|\mathrm{e}^{-z\sqrt{\mu}|\mathrm{D}|}\psi^{+}.$

Then defining the difference $u=\phi^{-}-\phi_{\mathrm{app}}^{-}$, we have
that it solves

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}u={\varepsilon}\mu
r_{1}\quad\text{in}\quad\mathcal{S}^{-}\\\
\partial_{n}^{P^{-}}u|_{z=0}=\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}+\mu\partial_{x}^{2}\psi^{+}+{\varepsilon}\mu
r_{2},\quad\lim\limits_{z\rightarrow\infty}|\nabla^{\mu}_{x,z}u|=0,\end{cases}$

where $r_{1}$ reads

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

and the second rest is given by

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

Then arguing as in Step 2. of the proof of Proposition 2.4, we find that

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

For the estimate on $r_{1}$ we use (3.11) to deal with the integrability on
$\mathcal{S}^{-}$, while the estimate for $r_{2}$ is straightforward since we
allow for loss of derivatives. Lastly, for we use (8.1) to approximate
$\mathcal{G}^{+}_{\mu}$. From these estimates, we get the following bound on
the gradient

$\displaystyle\|\nabla_{x,z}^{\mu}\Lambda^{s}u\|_{L^{2}(\mathcal{S}^{-})}\leq(\mu^{2}+\mu{\varepsilon})|\partial_{x}\psi^{+}|_{H^{s+3}}.$

To conclude, we use trace estimate (2.15) and the construction to find that

$\displaystyle|u|_{z=0}|_{\mathring{H}^{s+{\frac{1}{2}}}}$
$\displaystyle\leq\mu^{-\frac{1}{2}}\|\Lambda^{s}\nabla_{x,z}^{\mu}u\|_{L^{2}(\mathcal{S}^{-})}$
$\displaystyle\leq(\mu^{\frac{3}{2}}+\sqrt{\mu}{\varepsilon})|\partial_{x}\psi^{+}|_{H^{s+3}}.$

∎

Having these two expansions at hand, we can turn to the main proof of the
section.

###### Proof of Theorem 1.17.

We first observe that since
$|\partial_{x}\psi|_{H^{s}}\leq|\psi|_{\dot{H}^{s+1}_{\mu}}$, that Theorem 1.7
with estimate (1.19) implies

$\sup_{t\in[0,{\varepsilon}^{-1}T]}\big{(}|\zeta|_{H^{N-\frac{1}{2}}}^{2}+|v|_{H^{N-\frac{1}{2}}}^{2}\big{)}\leq
C(\mathcal{E}^{N}(\mathbf{U}^{0})).$ (8.5)

With this in mind, we give the proof in two steps where we first derive (1.)
in the sense of consistency.

Step 1. To derive (1.) we let $R$ be some generic function satisfying

$|R|_{H^{N-5}}\leq C(\mathcal{E}^{N}(\mathbf{U}_{0})).$ (8.6)

Then we can simplify (1.22). In particular, for the second equation in (1.13),
we first estimate the nonlinear terms. For
$\mathcal{N}[{\varepsilon}\zeta,\psi^{\pm}]$ we apply estimates (B.6), (A.25),
(A.7), and (1.19) to obtain,

$|\mathcal{N}[{\varepsilon}\zeta,\psi^{\pm}]|_{H^{N-5}}\leq\mu
C(\mathcal{E}^{N}(\mathbf{U}_{0})).$

For the first equation, we use (8.2), and we have that

$\begin{cases}\partial_{t}\zeta+\partial_{x}^{2}\mathrm{T}(\mathrm{D})\psi^{+}+{\varepsilon}\partial_{x}(\zeta\partial_{x}\mathrm{T}(\mathrm{D})\psi^{+})=\mu
R\\\
\partial_{t}v+\big{(}(1-\gamma)-\mathrm{bo}^{-1}\partial_{x}^{2}\big{)}\partial_{x}\zeta+\frac{{\varepsilon}}{2}\partial_{x}\big{(}(\partial_{x}\psi^{+})^{2}-\gamma(\mathbb{H}[{\varepsilon}\zeta]\psi^{+})^{2}\big{)}=(\mu+\mu{\varepsilon})R,\end{cases}$
(8.7)

To relate the equation to $v$, we observe by (8.3) that

$\displaystyle\partial_{x}\psi^{+}$
$\displaystyle=\gamma\mathbb{H}[{\varepsilon}\zeta]\psi^{+}+v$
$\displaystyle=-\gamma\sqrt{\mu}|\mathrm{D}|\partial_{x}\psi^{+}+v+(\mu^{\frac{3}{2}}+\sqrt{\mu}{\varepsilon})R$
$\displaystyle=(1-\gamma\sqrt{\mu}|\mathrm{D}|)v+(\mu^{\frac{3}{2}}+\sqrt{\mu}{\varepsilon})R.$

From this relation, we can express the first equation in terms of $\zeta$ and
$v$ also using the expansions of $\mathcal{G}^{+}_{\mu}$ given by (8.2) and
the expansion of $\mathrm{T}(\mathrm{D})$:

$|(\mathrm{T}(\mathrm{D})-1)f|_{H^{s}}\leq\mu C|\partial_{x}f|_{H^{s+1}},$

which is a direct consequence of Plancherel’s identity and a Taylor expansion.
Indeed, from these observations there holds,

$\displaystyle\partial_{x}^{2}\mathrm{T}(\mathrm{D})\psi^{+}+{\varepsilon}\partial_{x}(\zeta\partial_{x}\mathrm{T}(\mathrm{D})\psi^{+})$
$\displaystyle=(1-\gamma\sqrt{\mu}|\mathrm{D}|\mathrm{T}(\mathrm{D}))\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)$ $\displaystyle\hskip
14.22636pt+(\mu+{\varepsilon}\sqrt{\mu}+{\varepsilon}\mu)R.$

For the second equation, we have that

$(\partial_{x}\psi^{+})^{2}=v^{2}-2\gamma v\sqrt{\mu}|\mathrm{D}|v+\mu R,$

and

$\displaystyle(\mathbb{H}[{\varepsilon}\zeta]\psi^{+})^{2}$ $\displaystyle=\mu
R.$

Consequently, if we let $\mathrm{bo}^{-1}={\varepsilon}\sqrt{\mu}$ we find the
desired system

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

Step 2. To derive (1.25) we can want to make an approximation $v^{\text{\tiny
wBO}}$ in terms of $\zeta^{\text{\tiny wBO}}$ at order
$\mathcal{O}(\mu+{\varepsilon}^{2})$ solving (1.25). Then we will show that
this solution is consistent with the weakly dispersive BO system (1.27). The
proof is given in two steps, where we first make formal computations and then
prove the consistency once we have constructed $v^{\text{\tiny wBO}}$. To
simplify the presentation further, we let $c=(1-\gamma)^{\frac{1}{2}}$ and
introduce the change of variable $x=c\tilde{x}$ and $v=c\tilde{v}$ to obtain:

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

For simplicity, we also omit the tilde notation in the proof.

Step 2.1. Formal derivation: To construct the lowest order approximation of
$v^{\text{\tiny wBO}}$ in terms of the solution $\zeta^{\text{\tiny wBO}}$, we
let $\sqrt{\mu}=\mathcal{O}({\varepsilon})$ and first consider system (1.27)
at order ${\varepsilon}$:

$\begin{cases}\partial_{t}\zeta^{\text{\tiny wBO}}+\partial_{x}v^{\text{\tiny
wBO}}=0\\\ \partial_{t}v^{\text{\tiny wBO}}+\partial_{x}\zeta^{\text{\tiny
wBO}}=0.\end{cases}$

The system is reduced to a wave equation with speed one. We therefore choose
the right-moving solution where $v^{\text{\tiny wBO}}$ is equal to
$\zeta^{\text{\tiny wBO}}$ at first order. Then having an approximation at
first order we can now make a correction at higher order:

$v^{\text{\tiny wBO}}=\zeta^{\text{\tiny wBO}}+\sqrt{\mu}A+{\varepsilon}B,$

for some functions $A$ and $B$ depending on the solution $\zeta^{\text{\tiny
wBO}}$. In particular, we construct $A$ and $B$ by plugging the ansatz into
(1.27):

$\begin{cases}\partial_{t}\zeta^{\text{\tiny
wBO}}+\partial_{x}\zeta^{\text{\tiny
wBO}}+\big{(}\sqrt{\mu}\partial_{x}A-\gamma\tanh(\sqrt{\mu}|\mathrm{D}|)\partial_{x}\zeta^{\text{\tiny
wBO}}\big{)}+{\varepsilon}\big{(}\partial_{x}B+\partial_{x}((\zeta^{\text{\tiny
wBO}})^{2})\big{)}=0\\\ \partial_{t}\zeta^{\text{\tiny
wBO}}+\partial_{x}\zeta^{\text{\tiny
wBO}}+\sqrt{\mu}\partial_{t}A+{\varepsilon}\big{(}\partial_{t}B+\zeta^{\text{\tiny
wBO}}\partial_{x}\zeta^{\text{\tiny wBO}}\big{)}=0.\end{cases}$

Now using the transport equations to see that $\partial_{x}A=-\partial_{t}A$
and $\partial_{x}B=-\partial_{t}B$, we find the equations:

$\begin{cases}2\sqrt{\mu}\partial_{x}A=\gamma\tanh(\sqrt{\mu}|\mathrm{D}|))\partial_{x}\zeta^{\text{\tiny
wBO}}\\\ 2\partial_{x}B=-\zeta^{\text{\tiny
wBO}}\partial_{x}\zeta^{\text{\tiny wBO}}.\end{cases}$

We therefore let $v^{\text{\tiny wBO}}$ be given by

$v^{\text{\tiny
wBO}}=(1+\frac{\gamma}{2}\tanh(\sqrt{\mu}|\mathrm{D}|))\zeta^{\text{\tiny
wBO}}-\frac{{\varepsilon}}{4}(\zeta^{\text{\tiny wBO}})^{2}.$ (8.8)

Step 2.2. Rigorous derivation: For any $(\zeta_{0},v_{0})\in
H^{N-\frac{1}{2}}({\mathbb{R}})\times H^{N-\frac{1}{2}}({\mathbb{R}})$ where
$v_{0}$ satisfies

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

then there is a time $T_{1}>0$ and a unique solution $\zeta^{\text{\tiny
wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T_{1}]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$
to equation

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

Moreover, we can define $v^{\text{\tiny wBO}}\in
C\big{(}[0,{\varepsilon}^{-1}T_{1}]\>:\>H^{N-\frac{1}{2}}({\mathbb{R}})\big{)}$
by (8.8). Then by Plancherel’s identity and a Taylor expansion:

$|\tanh(\sqrt{\mu}|\xi|)f-\sqrt{\mu}|\xi|f|_{H^{s}}\leq\mu^{\frac{3}{2}}C|\partial_{x}^{3}f|_{H^{s}},$
(8.9)

we deduce from (1.26) that

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

for $R$ satisfying (8.6). Now, rescaling the equation back to its original
variables concludes the proof of this step.

Step 3. For the consistency result in the case of the BO equation we simply
use (8.9).

∎

##  9\. Proof of Theorem 1.18

###### Proof.

First, we let $N\geq 7$ and take initial data $(\zeta_{0},\psi_{0})$
satisfying the assumptions of Theorem 1.7. Then using (8.5), we can define the
solutions of the internal water waves equations (1.22) with variables

$(\zeta,v)\in
C([0,{\varepsilon}^{-1}T_{1}];H^{N-\frac{1}{2}}({\mathbb{R}})\times
H^{N-\frac{1}{2}}({\mathbb{R}})),$

from the data $(\zeta_{0},\partial_{x}\psi_{0})\in
H^{N-\frac{1}{2}}({\mathbb{R}})\times H^{N-\frac{1}{2}}({\mathbb{R}})$ for
some $T_{1}>0$. Next, we use Theorem 1.17 and the matrix notation (7.2) to say
that on the same time interval the functions $\mathbf{U}=(\zeta,v)^{T}$ solves

$\partial_{t}\mathbb{U}+\mathcal{M}(\mathbb{U})\mathbb{U}=(\mu+{\varepsilon}^{2})(R,R)^{T}$

for any $t\in[0,{\varepsilon}^{-1}T_{1}]$ and where the rest satisfies

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

We will now use this to prove estimates (1.30), (1.31), and (1.31) in three
separate steps.

Step 1. Proof of estimate (1.30). We take the same data $(\zeta_{0},v_{0})$
with $v_{0}=\partial_{x}\psi_{0}$, and use Theorem 1.27 to deduce the
existence of $T_{2}>0$ such that

$\mathbf{U}^{\text{\tiny BOs}}=(\zeta^{\text{\tiny BOs}},v^{\text{\tiny
BOs}})\in C([0,{\varepsilon}^{-1}T_{2}];H^{N-\frac{1}{2}}({\mathbb{R}})\times
H^{N-\frac{1}{2}}({\mathbb{R}})),$

solves system (7.1):

$\partial_{t}\mathbf{U}^{\text{\tiny BOs}}+\mathcal{M}(\mathbf{U}^{\text{\tiny
BOs}})\mathbf{U}^{\text{\tiny BOs}}=\mathbf{0},$

for any $t\in[0,{\varepsilon}^{-1}T_{2}]$. We may now take the difference
between the two solutions

$\mathbf{W}=(\eta,w)^{T}=\mathbf{U}-\mathbf{U}^{\text{\tiny BOs}},$

to find that

$\partial_{t}\mathbb{W}+\mathcal{M}(\mathbb{U})\mathbb{W}=\mathbb{F}+(\mu+{\varepsilon}^{2})(R,R)^{T},$

for any $t\in[0,{\varepsilon}^{-1}\min\\{T_{1},T_{2}\\}]$ where $\mathbb{F}$
is defined by (7.5). Then using the estimate (7.10) with $N-5>\frac{3}{2}$ and
adding the rest term we find that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tilde{E}_{N-5}(\mathbb{W})$
$\displaystyle\leq(\mu+{\varepsilon}^{2})|\big{(}\Lambda^{N-5}(R,R)^{T},\mathcal{Q}(\mathbf{U}_{1})\Lambda^{N-5}\mathbb{W}\big{)}_{L^{2}}|+\tilde{K}(N-4)\tilde{E}^{N-5}(\mathbf{W})$

However, by definition of $\mathcal{Q}(\mathbf{U}_{1})$, the Hölder
inequality, the Sobolev embedding, and (7.11) we obtain

$\frac{\mathrm{d}}{\mathrm{d}t}\tilde{E}_{N-5}(\mathbb{W})\leq(\mu+{\varepsilon}^{2})C(\mathcal{E}^{N}(\mathbf{U}_{0}))(\tilde{E}_{N-5}(\mathbf{W}))^{\frac{1}{2}}+{\varepsilon}\tilde{K}(N-4)\tilde{E}_{N-5}(\mathbf{W}).$

Then Grönwall’s inequality and (7.11) yields

$|(\eta,w)|_{H^{N-5}\times
H^{N-5}}\leq(\mu+{\varepsilon}^{2})C(\mathcal{E}^{N}(\mathbf{U}_{0}))e^{{\varepsilon}\tilde{K}(N-4)t}.$

Finally, we observe that

${\varepsilon}\tilde{K}(N-4)t\leq
C\mathcal{E}^{N}(\mathbf{U}_{0})\min\\{T_{1},T_{2}\\},$

and use it together with the embedding $H^{N-5}({\mathbb{R}})\hookrightarrow
L^{\infty}({\mathbb{R}})$ for $N\geq 7$ to find:

$\displaystyle|\mathbf{U}-\mathbf{U}^{\text{\tiny
BOs}}|_{L^{\infty}([0,t];{\mathbb{R}})}$
$\displaystyle\lesssim|(\eta,w)|_{L^{\infty}([0,t]:H^{N-5}\times H^{N-5})}$
$\displaystyle\leq(\mu+{\varepsilon}\sqrt{\mu})tC(\mathcal{E}^{N}(\mathbf{U}_{0})),$

for all $t\in[0,{\varepsilon}^{-1}\min\\{T_{1},T_{2}\\}]$. This completes the
proof of estimate (1.30).

Step 2. Proof of estimate (1.30). By Theorem 1.27 we deduce the existence of
$T_{3}>0$ such that

$\zeta^{\text{\tiny wBO}}\in
C([0,{\varepsilon}^{-1}T_{3}];H^{N-\frac{1}{2}}({\mathbb{R}})),$

solves (1.25) and from it we define

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

Moreover, by estimate (1.26) we have that

$\sup\limits_{t\in[0,{\varepsilon}^{-1}T_{3}]}|(\zeta^{\text{\tiny
wBO}},v^{\text{\tiny wBO}})|_{H^{N-\frac{1}{2}}\times H^{N-\frac{1}{2}}}\leq
C(|\zeta_{0}|_{H^{N-\frac{1}{2}}}).$

Then if we define $\mathbf{U}^{\text{\tiny wBO}}=(\zeta^{\text{\tiny
wBO}},v^{\text{\tiny wBO}})^{T}$ we can use Theorem 1.17 and argue as above to
find that

$\displaystyle|\mathbf{U}-\mathbf{U}^{\text{\tiny
wBO}}|_{L^{\infty}([0,t];{\mathbb{R}})}$
$\displaystyle\leq|\mathbf{U}-\mathbf{U}^{\text{\tiny
BOs}}|_{L^{\infty}([0,t];{\mathbb{R}})}+|\mathbf{U}^{\text{\tiny
BOs}}-\mathbf{U}^{\text{\tiny wBO}}|_{L^{\infty}([0,t];{\mathbb{R}})}$
$\displaystyle\leq\mu tC(\mathcal{E}^{N}(\mathbf{U}_{0})),$

for all $t\in[0,{\varepsilon}^{-1}\min\\{T_{1},T_{2},T_{3}\\}]$.

Step 3. Proof of estimate (1.30). Let $T=\min\\{T_{1},T_{2},T_{3}\\}$. Then
from the data $\zeta_{0}\in H^{N-\frac{1}{2}}({\mathbb{R}})$ there exist a
unique solution $\zeta^{\text{\tiny BO}}\in
C([0,{\varepsilon}^{-1}T];H^{N-\frac{1}{2}}({\mathbb{R}}))$ that is bounded by
its initial data. Moreover, by Theorem 1.17 the solution satisfies

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

Then if we define the difference $\tilde{\eta}=(\zeta^{\text{\tiny
wBO}}-\zeta^{\text{\tiny BO}})$ it is straight forward to deduce that

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|\tilde{\eta}|_{H^{N-5}}^{2}\leq{\varepsilon}C(|\zeta_{0}|_{H^{N-4}})|\tilde{\eta}|_{H^{N-5}}^{2}+\mu|R|_{H^{N-5}}|\tilde{\eta}|_{H^{N-5}}.$

The result is then a direct consequence of Grönwall’s inequality and the
previous steps. ∎

## Acknowledgements

This research was supported by a Trond Mohn Foundation grant. The material is
also based on discussions with David Lannes at Institut Mittag-Leffler in
Djursholm, Sweden during the program “Order and Randomness in Partial
Differential Equations” in Fall, 2023 that was supported by the Swedish
Research Council under grant no. 2016-06596.

I also would like to thank Jean-Claude Saut for suggesting the problem,
Vincent Duchêne for some important comments on the introduction, and my
advisor, Didier Pilod, for his comments on the introduction, support, and
friendship throughout the research period.

##  Appendix A Properties of the Dirichlet-Neumann operators

In this section we will give several results on
$\mathcal{G}^{\pm}_{\mu}[{\varepsilon}\zeta]$ and
$\mathcal{G}_{\mu}[{\varepsilon}\zeta]$. Then we will use these results to
share some details on the proof Proposition 4.2. But as we will shortly see,
the main quantities in (1.13) can be estimated in terms of the principal
unknown $(\zeta,\psi)$ where the estimates are very similar to the ones in
[51]. We will therefore give more details when there is a difference, and when
the estimates are the same, we will simply refer to the results in [51].

###  A.1. Properties of $\mathcal{G}_{\mu}^{+}$

We start this section by giving a precise definition of the positive
Dirichlet-Neumann operator $\mathcal{G}^{+}_{\mu}$.

######  Definition A.1.

Let $t_{0}>\frac{1}{2}$,
$\psi^{+}\in\dot{H}_{\mu}^{\frac{3}{2}}({\mathbb{R}})$, and $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ be such that (1.14) is satisfied. Let $\Phi^{+}$ be
the unique solution in $\dot{H}^{2}(\Omega_{t}^{+})$ of the boundary value
problem

$\begin{cases}\Delta^{\mu}_{x,z}\Phi^{+}=0\hskip
36.98866pt\qquad\text{in}\quad\Omega_{t}^{+}\\\ \Phi^{+}=\psi^{+}\hskip
72.55461pt\text{on}\quad z=\varepsilon\zeta\\\ \partial_{z}\Phi^{+}=0\hskip
71.41626pt\text{on}\quad z=-1,\end{cases}$

then $\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\in
H^{\frac{1}{2}}({\mathbb{R}})$ is defined by

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

######  Remark A.2.

Under the provisions of Definition A.1 and let $\phi^{+}=\Phi\circ\Sigma^{+}$
where $\Sigma^{+}$ is the diffeomorphisim from the fixed domain
$\mathcal{S}^{+}$ onto $\Omega^{+}_{t}$ given in Definition 1.20, then we have
that

$\begin{cases}\nabla^{\mu}_{x,z}\cdot
P(\Sigma^{+})\nabla_{x,z}^{\mu}\phi^{+}=0\hskip
19.91684pt\qquad\text{in}\quad\mathcal{S}^{+}\\\ \phi^{+}=\psi^{+}\hskip
115.23373pt\text{on}\quad z=0,\\\ \partial_{n}^{P^{+}}\phi^{+}=0\hskip
105.2751pt\text{on}\quad z=-1,\end{cases}$ (A.1)

where

$\displaystyle\partial_{n}^{P^{+}}=\mathbf{e}_{z}\cdot
P(\Sigma^{+})\nabla^{\mu}_{x,z}.$

Moreover, we have an equivalent expression of $\mathcal{G}^{+}[\zeta]\psi^{+}$
given by

$\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\psi^{+}=\partial_{n}^{P^{+}}\phi^{+}|_{z=0}.$
(A.2)

######  Remark A.3.

For ${\varepsilon}=0$ we have that $\mathcal{G}^{+}$ becomes

$\mathcal{G}^{-}_{\mu}[0]\psi^{+}=\sqrt{\mu}|\mathrm{D}|\tanh(\sqrt{\mu}|\mathrm{D}|)\psi^{+}.$
(A.3)

Next, we will state several results on the Dirichlet-Neumann operator that are
taken from [52, 51].

######  Proposition A.4.

Let $t_{0}>\frac{1}{2}$, $s\in[0,t_{0}+1]$ and $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ be such that (1.14) is satisfied. Then we have the
following properties:

  * 1.

For $\psi^{+}\in\dot{H}^{s+{\frac{1}{2}}}_{\mu}({\mathbb{R}})$ there is a
(variational) solution of (A.1) satisfying the estimates

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

and

$\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi^{+}\|_{L^{2}(\mathcal{S}^{+})}\leq\sqrt{\mu}M|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$
(A.5)

  * 2.

We may extend definition A.1 for

$\displaystyle\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]:\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}}),$

where for all
$\psi_{1},\psi_{2}\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$, there
holds,

$\int_{{\mathbb{R}}}\psi_{1}\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\psi_{2}\>\mathrm{d}x=\int_{\mathcal{S}^{+}}\nabla_{x,z}^{\mu}\phi_{1}^{+}\cdot
P^{+}(\Sigma^{+})\nabla_{x,z}^{\mu}\phi_{2}^{+}\>\mathrm{d}x\mathrm{d}z.$

  * 3.

For all $\psi_{1},\psi_{2}\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$,
there holds,

$\displaystyle\big{(}\Lambda^{s}\psi_{1},\Lambda^{s}\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\psi_{2}\big{)}_{L^{2}}\leq\mu
M|\psi_{1}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}|\psi_{2}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$
(A.6)

  * 4.

For $\psi^{+}\in H^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$ the following
estimates hold

$\displaystyle\forall
s\in[0,t_{0}+\frac{3}{2}],\quad|\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\psi^{+}|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
(A.7) $\displaystyle\forall
s\in[0,t_{0}+1],\>\quad|\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\psi^{+}|_{H^{s-\frac{1}{2}}}\leq\mu
M|\psi|_{\dot{H}_{\mu}^{s+1}}.$ (A.8)

  * 5.

For all $\psi_{1}\in\dot{H}_{\mu}^{s-\frac{1}{2}}({\mathbb{R}})$ and
$\psi_{2}\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$, there holds

$\big{(}\big{[}\Lambda^{s},\mathcal{G}^{+}_{\mu}[\varepsilon\zeta]\big{]}\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\leq\mu{\varepsilon}M|\psi_{1}|_{\dot{H}_{\mu}^{s-\frac{1}{2}}}|\psi_{2}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$
(A.9)

  * 6.

For all $V\in H^{t_{0}+1}({\mathbb{R}})$ and $\psi^{+}\in
H_{\mu}^{\frac{1}{2}}({\mathbb{R}})$ there holds,

$\big{(}(V\partial_{x}\psi^{+}),\frac{1}{\mu}\mathcal{G}_{\mu}^{+}[{\varepsilon}\zeta]\psi^{+}\big{)}_{L^{2}}\leq
M|V|_{W^{1,\infty}}|\psi^{+}|_{\dot{H}^{\frac{1}{2}}_{\mu}}$ (A.10)

######  Remark A.5.

The regularity on $\zeta$ here is not optimal. Specifically, the dependence on
$|\zeta|_{H^{t_{0}+2}}$ in the definition of $M$ can be lowered in the
estimates above. However, we do not give these estimates since we will in
other instances require more regularity on the free surface (as an example,
see estimate (3.3)).

######  Remark A.6.

The last estimate (A.9) is taken from [51] (see equation number $(2.24)$).
However, in $(2.24)$, there is an ${\varepsilon}$ missing due to a typo and
has been added here [53].

The next result concerns the shape derivative of $\mathcal{G}_{\mu}^{+}$. The
result is found in [52, 51].

######  Proposition A.7.

Let $t_{0}>\frac{1}{2}$, $s\in[0,t_{0}+1]$, and for any $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ satisfying (1.14) we have the following properties:

  * 1.

For $\psi^{+}\in\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$ the shape
derivative of $\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]$ at $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ in the direction of $h\in
H^{t_{0}+2}({\mathbb{R}})$ is given by the formula

$\mathrm{d}_{\zeta}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)\psi^{+}=-{\varepsilon}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h\underline{w}^{+})-{\varepsilon}\mu\partial_{x}(h\underline{V}^{+}).$
(A.11)

  * 2.

For all $0\leq s\leq t_{0}+1$, $j\geq 1$ there holds

$|\mathrm{d}^{j}_{\zeta}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)\psi^{+}|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
(A.12)

  * 3.

For all $0\leq s\leq t_{0}+\frac{1}{2}$, $j\geq 1$, and
$\psi^{+}\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$, there holds,

$|\mathrm{d}^{j}_{\zeta}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)\psi^{+}|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu
M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s+\frac{1}{2},t_{0}\\}+1}}|\psi^{+}|_{\dot{H}_{\mu}^{s+1}},$
(A.13)

  * 4.

For all $0\leq s\leq t_{0}+\frac{1}{2}$, $j\geq 1$,
$\psi^{+}\in\dot{H}^{\max\\{s,t_{0}\\}+1}_{\mu}({\mathbb{R}})$, there holds,

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)\psi^{+}|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu
M|h_{k}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi|_{\dot{H}_{\mu}^{\max\\{s,t_{0}\\}+1}}$
(A.14)

  * 5.

For all $0\leq s\leq t_{0}$, $j\geq 1$,
$\psi^{+}\in\dot{H}^{\max\\{s+\frac{1}{2},t_{0}\\}+1}_{\mu}({\mathbb{R}})$,
there holds,

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)\psi^{+}|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu
M|h_{k}|_{H^{s+1}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s+\frac{1}{2},t_{0}\\}+\frac{3}{2}}}|\psi^{+}|_{\dot{H}_{\mu}^{\max\\{s+\frac{1}{2},t_{0}\\}+1}}$
(A.15)

  * 6.

For all $0\leq s\leq t_{0}+1$, $j\geq 1$,
$\psi_{1},\psi_{2}\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$, there
holds,

$\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq{\varepsilon}^{j}\mu
M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi_{1}|_{\dot{H}^{s+\frac{1}{2}}}|\psi_{2}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
(A.16)

  * 7.

For for all $0\leq s\leq t_{0}+\frac{1}{2}$, $j\geq 1$,
$\psi_{1}\in\dot{H}^{\max\\{s,t_{0}\\}+1}_{\mu}({\mathbb{R}}),\psi_{2}\in\dot{H}^{s+\frac{1}{2}}_{\mu}({\mathbb{R}})$,
there holds,

$\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq{\varepsilon}^{j}\mu
M|h_{l}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
l}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi_{1}|_{\dot{H}^{\max\\{s,t_{0}\\}+1}}|\psi_{2}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
(A.17)

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

In this section will define and give the main properties of the negative
Dirichlet-Neumann operator.

######  Definition A.8.

Let $t_{0}\geq 1$, $\psi\in\mathring{H}^{\frac{3}{2}}({\mathbb{R}})$, and
$\zeta\in H^{t_{0}+2}({\mathbb{R}})$ be such that (1.14) is satisfied. Let
$\Phi^{-}$ be the unique solution in $\dot{H}^{2}(\Omega_{t}^{-})$ of the
boundary value problem

$\begin{cases}\Delta_{x,z}^{\mu}\Phi^{-}=0\hskip
36.98866pt\qquad\text{in}\quad\Omega_{t}^{-}\\\ \Phi^{-}=\psi\hskip
79.6678pt\text{on}\quad z={\varepsilon}\zeta,\end{cases}$

then $\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}\in
H^{\frac{1}{2}}({\mathbb{R}})$ is defined by

$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}=(\partial_{z}\Phi^{-}-{\varepsilon}\mu\partial_{x}\zeta\partial_{x}\Phi^{-}){|_{z={\varepsilon}\zeta}}.$
(A.18)

######  Remark A.9.

As noted in Remark $3.50$ $(2)$ in [52], we can define the negative Dirichlet-
Neumann operator by formula (A.18) (or formula (A.21) below) for
$\psi^{-}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ if $t_{0}>\frac{1}{2}$
and $s\geq\max\\{0,1-t_{0}\\}$, where the restriction is a consequence of
(B.6). We therefore put $t_{0}\geq 1$ for simplicity.

######  Remark A.10.

The scaling for $\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta]$ is different from
the one used in [52], where the author used the scaling natural for infinite
depth. In that case one would change $\mu=1$ and ${\varepsilon}$ by
$\epsilon=\frac{a}{\lambda}$. In our case, the internal water wave model
depends on both scales and is explained in detail in Subsection 1.2.1.

######  Remark A.11.

For ${\varepsilon}\zeta=0$ we have that $\mathcal{G}^{-}_{\mu}$ becomes

$\mathcal{G}_{\mu}^{-}[0]\psi^{-}=\sqrt{\mu}|\mathrm{D}|\psi^{-}.$ (A.19)

######  Remark A.12.

Under the provisions of Definition A.8 and let $\phi^{-}=\Phi\circ\Sigma^{-}$
where $\Sigma^{+}$ is the diffeomorphisim from the fixed domain
$\mathcal{S}^{-}$ onto $\Omega^{-}_{t}$ given in Definition 1.20, then we have
that

$\begin{cases}\nabla_{x,z}^{\mu}\cdot
P(\Sigma^{-})\nabla_{x,z}^{\mu}\phi^{-}=0\hskip
19.91684pt\qquad\text{in}\quad\mathcal{S}^{-}\\\ \phi^{-}=\psi\hskip
122.34692pt\text{on}\quad z=0,\end{cases}$ (A.20)

and we have an equivalent expression of
$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}$ given by

$\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}=\partial_{n}^{P^{-}}\phi^{-}|_{z=0}.$
(A.21)

Here we use the results in [52] that was given in the “infinite depth” case,
but is easily adapted to the current scaling.

######  Proposition A.13.

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. We have the following properties:

  * 1.

For $\psi\in\mathring{H}^{s+{\frac{1}{2}}}({\mathbb{R}})$ there is a
(variational) solution of (A.1) satisfying the estimate

$\sqrt{\mu}|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}\leq
M\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})},$ (A.22)

and

$\|\Lambda^{s}\nabla_{x,z}^{\mu}\phi^{-}\|_{L^{2}(\mathcal{S}^{-})}\leq\mu^{\frac{1}{4}}M|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.23)

  * 2.

By remark A.9 we may extend Definition A.8 for

$\displaystyle\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]:\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}}),$

where for all
$\psi_{1},\psi_{2}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$, there holds,

$\int_{{\mathbb{R}}}\psi_{1}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi_{2}\>\mathrm{d}x=-\int_{\mathcal{S}^{-}}\nabla_{x,z}^{\mu}\phi_{1}\cdot
P^{-}(\Sigma^{-})\nabla_{x,z}^{\mu}\phi_{2}\>\mathrm{d}x\mathrm{d}z.$

  * 3.

For all $\psi_{1},\psi_{2}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$,
there holds,

$\displaystyle\big{(}\Lambda^{s}\psi_{1},\Lambda^{s}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi_{2}\big{)}_{L^{2}}\leq\sqrt{\mu}M|\psi_{1}|_{\mathring{H}^{s+\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.24)

  * 4.

For $\psi^{-}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ the following
estimates hold

$|\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi^{-}|_{H^{s-\frac{1}{2}}}\leq\sqrt{\mu}M|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.25)

  * 5.

For all $\psi_{1}\in\mathring{H}^{s-\frac{1}{2}}({\mathbb{R}})$ and
$\psi_{2}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$, there holds,

$\big{(}\big{[}\Lambda^{s},\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\big{]}\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\leq{\varepsilon}\sqrt{\mu}M|\psi_{1}|_{\mathring{H}^{s-\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.26)

  * 6.

For all $V\in H^{t_{0}+1}({\mathbb{R}})$ and
$\psi^{-}\in\mathring{H}^{\frac{1}{2}}({\mathbb{R}})$ there holds,

$\big{(}(V\partial_{x}\psi^{-}),\frac{1}{\mu}\mathcal{G}_{\mu}^{-}[{\varepsilon}\zeta]\psi^{-}\big{)}_{L^{2}}\leq\mu^{-\frac{1}{2}}M|V|_{W^{1,\infty}}|\psi^{-}|^{2}_{\mathring{H}^{\frac{1}{2}}}.$
(A.27)

######  Remark A.14.

If we compare the estimates for $\mathcal{G}_{\mu}^{+}$ with the ones above,
we note that there is a $\sqrt{\mu}$ missing. However, this is a simple
consequence of the functional setting where there is an additional gain in
$\mu$ from the observation that

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

######  Remark A.15.

The inequality (A.26) is a direct extension of inequality $(2.24)$ in [51].
While the last inequality (A.27) is the extension of Proposition $3.30$ in
[52] to infinite depth. The extension is straightforward and is explained on
page $88$ of this book. Also, consult the previous remark for the scaling.

Lastly, we will need a shape derivative formula for $\mathcal{G}_{\mu}^{-}$
and estimates on higher order shape derivatives [52]:

######  Proposition A.16.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$, and take $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$. Then we have the following properties:

  * 1.

For $\psi^{-}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ the shape
derivative of $\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]$ at $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ in the direction of $h\in
H^{t_{0}+2}({\mathbb{R}})$ is given by the formula

$\mathrm{d}_{\zeta}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](h)\psi^{-}=-{\varepsilon}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](h\underline{w}^{-})-{\varepsilon}\mu\partial_{x}(h\underline{V}^{-}).$
(A.28)

  * 2.

For all $0\leq s\leq t_{0}+1$, $j\geq 1$ and
$\psi_{1},\psi_{2}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$, there holds,

$\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq{\varepsilon}^{j}\sqrt{\mu}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi_{1}|_{\mathring{H}^{s+\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}$
(A.29)

  * 3.

For all $0\leq s\leq t_{0}+\frac{1}{2}$, $j\geq 1$,
$\psi_{1}\in\mathring{H}^{\max\\{s,t_{0}\\}+\frac{3}{2}}({\mathbb{R}}),\psi_{2}\in\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$,
there holds,

$\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq{\varepsilon}^{j}\sqrt{\mu}M|h_{l}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
l}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi_{1}|_{\mathring{H}^{\max\\{s,t_{0}\\}+1}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.30)

###  A.3. Corollaries from the results in Section 2

We will give an important generalization of Proposition 2.4 where we prove
that we can trade $\mathcal{G}_{\mu}^{+}$ with any operator
$\mathrm{Op}(A):X\rightarrow H^{s-\frac{1}{2}}({\mathbb{R}})$ satisfying

$\big{|}\big{(}\Lambda^{s}\mathrm{Op}(A)\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq
MM_{A}(\psi_{1})\sqrt{\mu}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}},$ (A.31)

where $M_{A}(\psi)$ is some positive number defined by the norm on $X$. This
can be seen from Step $1.1$ in the proof, which is the only place where we use
that $(\mathcal{G}_{\mu}^{-})^{-1}$ is composed with $\mathcal{G}_{\mu}^{+}$.
We have the following result.

######  Corollary A.17.

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 if $\mathrm{Op}(A)$ satisfies the
condition (A.31), the mapping

$(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{Op}(A)\>:\>\begin{cases}X&\rightarrow\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})\\\
\psi&\mapsto(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{Op}(A)\psi\end{cases}$
(A.32)

is well-defined and satisfies

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{Op}(A)\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq
MM_{A}(\psi).$ (A.33)

Moreover, we have the particular cases of $\mathrm{Op}(A)$:

  * 1.

If $\mathrm{Op}(A)=\partial_{x}$, then
$X=\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ and there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq\mu^{-\frac{1}{2}}M|\psi|_{\mathring{H}^{s+\frac{1}{2}}}.$
(A.34)

  * 2.

If
$\mathrm{Op}(A)=\mathrm{d}_{\zeta}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]$
for $j\geq 1$, then $X=\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})$ and for
$\mathbf{h}=(h_{1},...,h_{j})\in H^{t_{0}+2}({\mathbb{R}})^{j}$ there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{d}_{\zeta}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}^{j}\mu^{\frac{1}{4}}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
(A.35)

and for $X=\dot{H}_{\mu}^{\max\\{s,t_{0}\\}+1}({\mathbb{R}})$ there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{d}_{\zeta}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}^{j}\mu^{\frac{1}{4}}M|h_{k}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi|_{\dot{H}_{\mu}^{\max\\{s,t_{0}\\}+1}}.$
(A.36)

  * 3.

If $\mathrm{Op}(A)=\mathrm{d}^{j}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]$
for $j\geq 1$, then $X=\mathring{H}^{s+\frac{1}{2}}({\mathbb{R}})$ and for
$\mathbf{h}=(h_{1},...,h_{j})\in H^{t_{0}+2}({\mathbb{R}})^{j}$ there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{d}^{j}_{\zeta}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta]\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}^{j}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\mathring{H}_{\mu}^{s+\frac{1}{2}}},$
(A.37)

and for $X=\mathring{H}^{\max\\{s,t_{0}\\}+1}({\mathbb{R}})$ there holds,

$|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathrm{d}^{j}_{\zeta}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi|_{\mathring{H}^{s+\frac{1}{2}}}\leq{\varepsilon}^{j}M|h_{k}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi|_{\mathring{H}^{\max\\{s,t_{0}\\}+1}}.$
(A.38)

###### Proof.

As explained above, we only need to consider the specific cases of
$\mathrm{Op}(A)$. Moreover, since we have (A.33) at hand, we simply verify
inequality (A.31) and identify the constant $M_{A}(\psi_{1})$.

Step 1. For the proof of (A.34) we use that

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

where $\mathcal{H}$ is the Hilbert transform and then use Plancherel’s
identity together with Cauchy-Schwarz inequality to deduce the bound

$\displaystyle\big{|}\big{(}\Lambda^{s}\partial_{x}\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}\leq
M|\psi_{1}|_{\mathring{H}^{s+\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}.$

Step 2. For the proof of (A.35), we use estimate (A.16) and(B.3) to get the
estimate

$\displaystyle\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}$
$\displaystyle\leq{\varepsilon}^{j}\mu
M|\psi_{1}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}|\psi_{2}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}$
$\displaystyle\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M|\psi_{1}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}.$

While the proof of (A.36) is obtained by (A.17).

Step 3. This step is a direct consequence of estimates (A.29) and (A.30),
where estimate (A.29) implies

$\displaystyle\big{|}\big{(}\Lambda^{s}\mathrm{d}^{j}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](\mathbf{h})\psi_{1},\Lambda^{s}\psi_{2}\big{)}_{L^{2}}\big{|}$
$\displaystyle\leq{\varepsilon}^{j}\sqrt{\mu}M|\psi_{1}|_{\mathring{H}^{s+\frac{1}{2}}}|\psi_{2}|_{\mathring{H}^{s+\frac{1}{2}}}\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}.$

∎

###  A.4. Basic definitions

We start this section by defining the main quantities involved in (1.13) and
relating them to the primary variables $(\zeta,\psi)$.

######  Corollary A.18.

Let $t_{0}\geq 1$, $s\in[0,t_{0}+1]$ , and $\zeta\in
H^{t_{0}+2}({\mathbb{R}})$ satisfying (1.14). Moreover define $\psi^{\pm}$ as
in Remark 2.11 and as the trace of $\phi^{\pm}$ satisfying (2.21). Then there
holds:

  * 1.

The tangential velocity is given by the mapping

$\mathcal{V}^{\pm}_{\parallel}\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}})\\\
\psi&\mapsto\partial_{x}\psi^{\pm}\end{cases}$ (A.39)

is well-defined and satisfies

$\forall s\in[0,t_{0}+1],\quad|\mathcal{V}^{\pm}_{\parallel}|_{H^{s}}\leq
M|\partial_{x}\psi|_{H^{s}}.$ (A.40)

  * 2.

The horizontal component of the velocity is given by the mappings

$\underline{w}^{\pm}\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}})\\\
\psi&\mapsto\frac{\mathcal{G}^{\pm}_{\mu}[{\varepsilon}\zeta]\psi^{+}+{\varepsilon}\mu\partial_{x}\zeta\partial_{x}\psi^{\pm}}{1+{\varepsilon}^{2}\mu(\partial_{x}\zeta)^{2}},\end{cases}$
(A.41)

is well-defined and satisfies

$\displaystyle
s\in[0,t_{0}+1],\quad|\underline{w}^{\pm}\psi|_{H^{s-\frac{1}{2}}}\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}},$
(A.42)

and

$\displaystyle
s\in[0,t_{0}+\frac{1}{2}],\quad|\underline{w}^{\pm}\psi|_{H^{s-\frac{1}{2}}}\leq\mu
M|\psi|_{\dot{H}^{s+1}_{\mu}}.$ (A.43)

  * 3.

The vertical component of the velocity is given by the mappings

${\underline{V}}^{\pm}\>:\>\begin{cases}\dot{H}_{\mu}^{s+\frac{1}{2}}({\mathbb{R}})&\rightarrow
H^{s-\frac{1}{2}}({\mathbb{R}})\\\
\psi&\mapsto\mathcal{V}^{\pm}_{\parallel}-{\varepsilon}(\underline{w}^{\pm}\psi)\partial_{x}\zeta\end{cases}$
(A.44)

is well-defined and satisfies

$s\in[0,t_{0}+\frac{1}{2}],\quad|\underline{V}^{\pm}|_{H^{s}}\leq
M|\partial_{x}\psi|_{H^{s}}.$ (A.45)

###### Proof.

We prove each point in separate steps.

Step 1. For $\mathcal{V}^{+}_{\parallel}$ we will use formula
$\psi^{+}=\mathcal{J}_{\mu}^{-1}\psi$ and tn Proposition 2.10 with estimate
(2.23) to get

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

For $\mathcal{V}^{-}_{\parallel}$ we use formula
$\psi^{-}=(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}\psi^{+}$, then
Proposition 2.4 with estimate (2.12), and the above estimates to deduce:

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

Step 2. We consider the first estimate of $\underline{w}^{+}$, where we use
estimate (A.7), the product estimates (B.6) (since $s-\frac{1}{2}\geq-t_{0}$),
and definition of $\psi^{+}$ with (2.23) to get:

$\displaystyle|\underline{w}^{+}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq|\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{H^{s-\frac{1}{2}}}+\mu{\varepsilon}|\partial_{x}\zeta|_{H^{\max\\{t_{0},s-\frac{1}{2}\\}}}|\partial_{x}\psi^{+}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{3}{4}}M|\psi^{+}|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}$
$\displaystyle\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}.$

For the second estimate on $w^{+}$, we argue similarly using (A.8) instead of
(A.7).

For $\underline{w}^{-}$ we first use the definition of
$\psi^{-}=(\mathcal{G}_{\mu}^{-})^{-1}\mathcal{G}_{\mu}^{+}\psi^{+}$ and apply
the same estimates, combined with (2.11):

$\displaystyle|\underline{w}^{-}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq|\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{H^{s-\frac{1}{2}}}+{\varepsilon}\mu|\partial_{x}\zeta|_{H^{\max\\{t_{0},s-\frac{1}{2}\\}}}|\partial_{x}\big{(}(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}\big{)}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}+{\varepsilon}\mu
M|(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta]\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{3}{4}}M|\psi|_{\dot{H}^{s+\frac{1}{2}}_{\mu}}.$

For the second estimate on $\underline{w}^{-}$, we argue as above, where we
use (A.8) instead of (A.7).

Step 3. We prove first (A.45). For the first part of $V^{\pm}$ we use
estimates (A.40), while the second part is estimated by the product estimate
(B.6) and then (A.42) to see that

$\displaystyle|V^{\pm}|_{H^{s}}$
$\displaystyle\leq|\mathcal{V}_{\parallel}^{\pm}|_{H^{s}}+{\varepsilon}|\underline{w}^{\pm}\partial_{x}\zeta|_{H^{s}}$
$\displaystyle\leq
M|\partial_{x}\psi|_{H^{s}}+{\varepsilon}|\partial_{x}\zeta|_{H^{\max\\{t_{0},s\\}}}|\underline{w}^{\pm}|_{H^{(s+\frac{1}{2})-\frac{1}{2}}}$
$\displaystyle\leq
M(|\partial_{x}\psi|_{H^{s}}+\mu^{\frac{3}{4}}|\psi|_{\dot{H}_{\mu}^{s+1}}),$
$\displaystyle\leq M(|\partial_{x}\psi|_{H^{s}}.$

∎

###  A.5. Proof of Proposition 4.2

We will in this section give the details on the proof of Proposition 4.2.
Since the strategy in the proof is exactly the same as in [51] and the main
quantities involved satisfy the same estimates (see Corollary A.18), we only
prove the steps that are unique to the current regime. The first step is to
derive linearization formulas. for the Dirichlet-Neumann operator
$\mathcal{G}_{\mu}$.

####  A.5.1. Linearization formulas

The main step in the quasilinearisation of the internal water waves system
(1.13) is to get linearization formulas for

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

Here it will be important to get an explicit formula for the shape derivative
of $\mathcal{G}_{\mu}$ with respect to $\zeta$ and track the dependencies in
the small parameters.

######  Proposition A.19.

Let $T>0$, $t_{0}\geq 1$ and $N\in{\mathbb{N}}$ be such that $N\geq 5$.
Furthermore, let $U=(\zeta,\psi)^{T}\in\mathscr{E}_{\mathrm{bo},T}^{N,t_{0}}$
be such that (1.14) is satisfied on $[0,T]$. For all
$\alpha=(\alpha^{1},\alpha^{2})\in{\mathbb{N}}^{2}$ with $1\leq|\alpha|\leq N$
and define

$\partial_{x,t}^{\alpha}=\sum_{\alpha_{1}+\alpha_{2}=|\alpha|}\partial_{x}^{\alpha_{1}}\partial_{t}^{\alpha_{2}},$

and let $\underline{w}^{\pm}$ be as defined in (A.41). Moreover, define
$\underline{w}$ by

$\displaystyle\underline{w}$
$\displaystyle=\underline{w}^{+}-\gamma\underline{w}^{-},$

and

$\zeta_{(\alpha)}=\partial_{x,t}^{\alpha}\zeta,\quad\psi_{(\alpha)}=\partial_{x,t}^{\alpha}\psi-{\varepsilon}\underline{w}\partial_{x,t}^{\alpha}\zeta,\quad\psi_{\langle{\alpha}\rangle}=(\psi_{(\alpha^{1})},\psi_{(\alpha^{2})}).$

Then for all $1\leq|\alpha|\leq N$, one has

$\displaystyle\text{if}\>\>\alpha<N:$
$\displaystyle\quad\frac{1}{\mu}\partial_{x,t}^{\alpha}(\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi)=\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi_{(\alpha)}+{\varepsilon}R_{(\alpha)},$
$\displaystyle\text{if}\>\>\alpha\leq N:$
$\displaystyle\quad\frac{1}{\mu}\partial_{x,t}^{\alpha}(\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi)=\frac{1}{\mu}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi_{(\alpha)}-{\varepsilon}\mathcal{I}[U]\zeta_{(\alpha)}+\frac{1}{\mu}\mathcal{G}_{\mu,(\alpha)}[{\varepsilon}\zeta]\psi_{(\check{\alpha})}+{\varepsilon}R_{\alpha},$

where the linear operators $\mathcal{I}[U]$ and $\mathcal{G}_{\mu,(\alpha)}$
are given in Definition 4.1, while $R_{\alpha}$ is a function that satisfies
the estimate

$|R_{\alpha}(t)|_{H^{1}_{\gamma,\mathrm{bo}}}\leq
C\mathcal{E}^{N}(\mathbf{U}(t)),$ (A.46)

for some $C>0$ and for all $t\in[0,T]$.

######  Remark A.20.

The principal part of the linearization formula for $\mathcal{G}_{\mu}$ is
given in terms of $\mathcal{G}_{\mu}\psi_{(\alpha)}$ and
$\mathcal{I}[U]\zeta_{\alpha}$. While the additional term
$\mathcal{G}_{\mu,(\alpha)}\psi_{(\check{\alpha})}$ is sub-principal that
offers no difficulty in the proof, but is needed to deal with surface tension
in the energy estimates (see the third point in Remark 5.1).

We will now give the main ingredients in proving the linearization formulas in
Lemma A.19. These formulas involve the precise formulation of the directional
derivative of $\mathcal{G}_{\mu}$, and we therefore give the definition.

######  Definition A.21 (Directional derivative).

Let $\zeta\in H^{t_{0}+2}({\mathbb{R}})$, then the directional derivative of
$\zeta\mapsto\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi$ in the direction of
$h\in H^{t_{0}+2}({\mathbb{R}})$ is given by

$\mathrm{d}_{\zeta}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi=\lim\limits_{\nu\rightarrow
0}\frac{\mathcal{G}_{\mu}[{\varepsilon}\zeta+\nu
h]\psi-\mathcal{G}_{\mu}[{\varepsilon}\zeta]\psi}{\nu}.$

######  Remark A.22.

Since $\psi\mapsto\mathcal{G}\psi$ is linear we have that
$\mathrm{d}_{\psi}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi=\mathcal{G}_{\mu}[{\varepsilon}\zeta]h$.

######  Lemma A.23.

Let $t_{0}\geq 1$ and $\zeta\in H^{t_{0}+2}({\mathbb{R}})$ be such that (1.14)
is satisfied.

  * 1.

For all $\psi\in\dot{H}_{\mu}^{\frac{1}{2}}({\mathbb{R}})$, $h\in
H^{t_{0}+2}({\mathbb{R}})$, and $U=(\zeta,\psi)^{T}$, one has

$\mathrm{d}_{\zeta}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi=-{\varepsilon}\mathcal{G}_{\mu}[{\varepsilon}\zeta]\big{(}h(\underline{w}^{+}-\gamma\underline{w}^{-})\big{)}-{\varepsilon}\mu\mathcal{I}[U]h.$
(A.47)

where $\mathcal{I}[U]h$ is defined by

$\mathcal{I}[U]h=\partial_{x}(h\underline{V}^{+})+\gamma\mathcal{G}_{\mu}[{\varepsilon}\zeta](\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\partial_{x}\Big{(}h(\underline{V}^{+}-\underline{V}^{-})\Big{)}.$

  * 2.

For all $0\leq s\leq t_{0}+1$, $j\geq 1$, there holds,

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
(A.48)

  * 3.

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

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu
M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s+\frac{1}{2},t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+1}}$
(A.49)

  * 4.

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

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M|h_{k}|_{H^{s+\frac{1}{2}}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+\frac{3}{2}}}|\psi|_{\dot{H}_{\mu}^{\max\\{s,t_{0}\\}+1}}$
(A.50)

  * 5.

For all $0\leq s\leq t_{0}$, $j\geq 1$, there holds,

$|\mathrm{d}_{\zeta}^{j}\mathcal{G}_{\mu}[{\varepsilon}\zeta](h)\psi|_{H^{s-\frac{1}{2}}}\leq{\varepsilon}^{j}\mu
M|h_{k}|_{H^{s+1}}\prod\limits_{m\neq
k}^{j}|h_{m}|_{H^{\max\\{s+\frac{1}{2},t_{0}\\}+\frac{3}{2}}}|\psi|_{\dot{H}_{\mu}^{\max\\{s+\frac{1}{2},t_{0}\\}+1}}$
(A.51)

###### Proof.

The proof is similar to the one in [51], but we have to track the dependence
in the small parameters for the current scaling and adapt them to a different
functional setting. We prove each point separately.

Step 1. We do the details here, where we note that it is important not to
compose the inverse of $\mathcal{G}_{\mu}^{+}$ with $\mathcal{G}_{\mu}^{-}$
(see Remark 2.5). With this in mind, we observe by direct computations
(suppressing the argument in $\zeta$) that

$\displaystyle\mathrm{d}(\mathcal{G}_{\mu}\circ\mathcal{J}_{\mu})(h)$
$\displaystyle=\mathrm{d}\mathcal{G}_{\mu}(h)\circ\mathcal{J}_{\mu}+\mathcal{G}_{\mu}\circ\mathrm{d}\mathcal{J}_{\mu}(h).$

Then use definition (2.1) and compose the above identity with
$\mathcal{J}_{\mu}^{-1}$ from the right to obtain that

$\mathrm{d}\mathcal{G}_{\mu}(h)=\mathrm{d}\mathcal{G}_{\mu}^{+}(h)\circ\mathcal{J}_{\mu}^{-1}-\mathcal{G}_{\mu}\circ\mathrm{d}\mathcal{J}_{\mu}(h)\circ\mathcal{J}_{\mu}^{-1}.$
(A.52)

Then we note by Proposition 2.11 that $\mathcal{J}_{\mu}^{-1}\psi=\psi^{+}$
allowing us to use formula (A.11) and express the first term by

$\displaystyle\mathrm{d}\mathcal{G}_{\mu}^{+}(h)\mathcal{J}_{\mu}^{-1}\psi$
$\displaystyle=\mathrm{d}\mathcal{G}_{\mu}^{+}(h)\psi^{+}$
$\displaystyle=-{\varepsilon}\mathcal{G}_{\mu}^{+}(h\underline{w}^{+})-{\varepsilon}\mu\partial_{x}(h\underline{V}^{+}).$

For the second term of (A.52), we first make the observation

$\displaystyle(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathrm{d}\mathcal{G}^{+}_{\mu}(h)$
$\displaystyle=(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathrm{d}(\mathcal{G}^{-}_{\mu}\circ\big{(}(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathcal{G^{+}_{\mu}}\big{)})(h)$
$\displaystyle=(\mathcal{G}^{-}_{\mu})^{-1}\circ\Big{(}\mathrm{d}\mathcal{G}^{-}_{\mu}(h)\circ\big{(}(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathcal{G}^{+}_{\mu}\big{)}-\gamma^{-1}\mathcal{G}^{-}_{\mu}\circ\mathrm{d}\mathcal{J}_{\mu}(h)\Big{)}$
$\displaystyle=(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathrm{d}\mathcal{G}^{-}_{\mu}(h)\circ(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathcal{G}^{+}_{\mu}+\gamma^{-1}\mathrm{d}\mathcal{J}_{\mu}(h),$

so that

$\mathrm{d}\mathcal{J}_{\mu}(h)=\gamma(\mathcal{G}^{-}_{\mu})^{-1}\circ\Big{(}\mathrm{d}\mathcal{G}^{+}_{\mu}(h)-\mathrm{d}\mathcal{G}^{-}_{\mu}(h)\circ(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathcal{G}^{+}_{\mu}\Big{)}.$

Then by Remark 2.11 we have that
$\psi^{-}=(\mathcal{G}_{\mu}^{-})^{-1}\circ\mathcal{G}_{\mu}^{+}\psi^{+}$,
together with the shape derivative formulas (A.11) and (A.28), we deduce that

$\displaystyle\mathrm{d}\mathcal{J}_{\mu}(h)\circ\mathcal{J}_{\mu}^{-1}\psi$
$\displaystyle=\mathrm{d}\mathcal{J}_{\mu}(h)\psi^{+}$
$\displaystyle=\gamma(\mathcal{G}^{-}_{\mu})^{-1}\Big{(}\mathrm{d}\mathcal{G}^{+}_{\mu}(h)\psi^{+}-\mathrm{d}\mathcal{G}^{-}_{\mu}(h)\psi^{-}\Big{)}$
$\displaystyle=-{\varepsilon}\gamma(\mathcal{G}^{-}_{\mu})^{-1}\Big{(}\mathcal{G}^{+}_{\mu}(hw^{+})-\mathcal{G}^{-}_{\mu}(hw^{-})\Big{)}-\gamma{\varepsilon}\mu(\mathcal{G}^{-}_{\mu})^{-1}\Big{(}\partial_{x}(hV^{+})-\partial_{x}(hV^{-})\Big{)}.$

From (A.52) and definition of $\mathcal{I}[U]h$ given by (4.2), we may collect
the above identities and factorize the leading terms together to find that

$\displaystyle\mathrm{d}\mathcal{G}_{\mu}(h)\psi$
$\displaystyle=-{\varepsilon}\mathcal{G}^{+}_{\mu}(hw^{+})-{\varepsilon}\gamma\mathcal{G}_{\mu}\circ(\mathcal{G}^{-}_{\mu})^{-1}\Big{(}\mathcal{G}^{+}_{\mu}(hw^{+})-\mathcal{G}^{-}_{\mu}(hw^{-})\Big{)}$
$\displaystyle\hskip
14.22636pt-{\varepsilon}\mu\Big{(}\partial_{x}(hV^{+})+\gamma\mathcal{G}_{\mu}\circ(\mathcal{G}^{-})^{-1}\Big{(}\partial_{x}(hV^{+})-\partial_{x}(hV^{-})\Big{)}\Big{)}$
$\displaystyle=-{\varepsilon}\Big{(}1+\gamma\mathcal{G}_{\mu}\circ(\mathcal{G}^{-})^{-1}\Big{)}\circ\mathcal{G}^{+}_{\mu}(hw^{+})+{\varepsilon}\gamma\mathcal{G}_{\mu}(hw^{-})-\mu{\varepsilon}\mathcal{I}(U)h.$

Clearly, the proof is a consequence of the following relation

$\displaystyle\Big{(}1+\gamma\mathcal{G}_{\mu}\circ(\mathcal{G}_{\mu}^{-})^{-1}\Big{)}\circ\mathcal{G}^{+}_{\mu}=\mathcal{G}_{\mu}.$
(A.53)

and can be seen in the identity

$\displaystyle\mathcal{G}^{+}_{\mu}$
$\displaystyle=\mathcal{G}_{\mu}\circ\mathcal{J}_{\mu}$
$\displaystyle=\mathcal{G}_{\mu}-\gamma\mathcal{G}_{\mu}\circ(\mathcal{G}^{-}_{\mu})^{-1}\circ\mathcal{G}^{+}_{\mu}.$

Step 2. We first use the previous step and (A.52) to write

$\displaystyle\mathrm{d}\mathcal{G}_{\mu}(h)\psi$
$\displaystyle=\mathrm{d}\mathcal{G}^{+}_{\mu}(h)\psi^{+}-\mathcal{G}\circ\mathrm{d}\mathcal{J}_{\mu}(h)\psi^{+}.$
(A.54)

Then for the first term, we use (A.12) and Proposition (2.10) with estimate
(2.23) to get that

$\displaystyle|\mathrm{d}^{j}\mathcal{G}^{+}_{\mu}(h)\psi^{+}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\mathcal{J}_{\mu}^{-1}\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}},$
$\displaystyle\leq{\varepsilon}^{j}\mu^{\frac{3}{4}}M\prod\limits_{m=1}^{j}|h_{m}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

For the second term, we first let $j=1$ and observe by direct calculations (or
see previous step) that

$\displaystyle|\mathrm{d}\mathcal{J}_{\mu}\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq|(\mathcal{G}_{\mu}^{-})^{-1}\mathrm{d}\mathcal{G}_{\mu}^{+}(h_{1})\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}+|(\mathcal{G}_{\mu}^{-})^{-1}\mathrm{d}\mathcal{G}_{\mu}^{-}(h_{1})\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle=:R_{1}+R_{2}.$

Here we treat $R_{1}$ by (A.35) and then (2.23) to get that

$R_{1}\leq{\varepsilon}\mu^{\frac{1}{4}}M|h_{1}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

For $R_{2}$ we use (A.37), then the relation
$\psi^{-}=(\mathcal{G}_{\mu}^{-1})\mathcal{G}_{\mu}^{+}\psi^{+}$ together with
estimates (2.11) and (2.23) to obtain,

$\displaystyle R_{2}$
$\displaystyle\leq{\varepsilon}M|h_{1}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi^{-}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{1}{4}}M|h_{1}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

As a result, for $j=1$, we have by (2.2) (for $0\leq s\leq t_{0}+1$) and the
above estimates that

$\displaystyle|\mathcal{G}\circ\mathrm{d}\mathcal{J}(h)\psi^{+}|_{H^{s-\frac{1}{2}}}$
$\displaystyle\leq\mu^{\frac{3}{2}}M|\mathrm{d}\mathcal{J}(h)\psi^{+}|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}$
$\displaystyle\leq\sqrt{\mu}M|\mathrm{d}\mathcal{J}(h)\psi^{+}|_{\mathring{H}^{s+\frac{1}{2}}}$
$\displaystyle\leq{\varepsilon}\mu^{\frac{3}{4}}M|h_{1}|_{H^{\max\\{s,t_{0}\\}+1}}|\psi|_{\dot{H}_{\mu}^{s+\frac{1}{2}}}.$

The remaining cases follow recursively and can be proved by induction.

Step 3$-$5. The proof follows similarly, where for the first term in (A.54) we
instead use (A.13) or (A.14), or (A.15). While for the second term, we can
gain precision in $\mu$ using (2.4).

∎

######  Remark A.24.

In Step $1.$ of the proof, we found a formula for the shape derivative of
$\mathcal{J}_{\mu}$:

$\mathrm{d}_{\zeta}\mathcal{J}_{\mu}[{\varepsilon}\zeta](h)=\gamma(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\Big{(}\mathrm{d}_{\zeta}\mathcal{G}^{+}_{\mu}[{\varepsilon}\zeta](h)-\mathrm{d}_{\zeta}\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta](h)\circ(\mathcal{G}^{-}_{\mu}[{\varepsilon}\zeta])^{-1}\circ\mathcal{G^{+}_{\mu}[{\varepsilon}\zeta]}\Big{)}.$