The approximate solution $({\mathbf{U}}^{a},\varphi^{a})$ enables us to
reformulate the original problem (2.1) as a nonlinear problem with zero
initial data. Set

$\mathcal{F}^{a}:=\begin{cases}-\mathbb{L}({\mathbf{U}}^{a},\Psi^{a}),\;&t>0,\\\
~{}~{}~{}~{}0,\;&t<0.\\\ \end{cases}$ (8.10)

From $\tilde{{\mathbf{U}}}^{a}\in H^{\mu+2}(\Omega_{T})$ and $\varphi^{a}\in
H^{\mu+2}(\Gamma_{T}),$ we have $\mathcal{F}^{a}\in H^{\mu+1}(\Omega_{T})$.
Given the approximate solution $(\tilde{{\mathbf{U}}}^{a},\varphi^{a})$ of
Lemma 8.2 and $\mathcal{F}^{a}$ defined in (8.10), we see that
$({\mathbf{U}},\varphi)=({\mathbf{U}}^{a},\varphi^{a})+({\mathbf{V}},\psi)$ is
a solution of the original problem (2.1) if
${\mathbf{V}}=({\mathbf{V}}^{+},{\mathbf{V}}^{-})^{T},\Psi=(\Psi^{+},\Psi^{-})^{T},$
$\Psi|_{x_{1}=0}:=\psi$ satisfy the following problem:

$\begin{cases}\mathcal{L}({\mathbf{V}},\Psi):=\mathbb{L}({\mathbf{U}}^{a}+{\mathbf{V}},\Psi^{a}+\Psi)-\mathbb{L}({\mathbf{U}}^{a},\Psi^{a})=\mathcal{F}^{a},&\text{in
}\Omega_{T},\\\
\mathcal{B}({\mathbf{V}}|_{x_{1}=0},\psi):=\mathbb{B}({\mathbf{U}}^{a}|_{x_{1}=0}+{\mathbf{V}}|_{x_{1}=0},\varphi^{a}+\psi)=0,&\text{on
}\Gamma_{T},\\\ ({\mathbf{V}},\psi)=0,&\text{for }t<0.\end{cases}$ (8.11)

The original nonlinear problem on $[0,T]\times\mathbb{R}^{2}_{+}$ is thus
reformulated as a problem on $\Omega_{T}$ whose solutions vanish in the past.

##  9\. Nash-Moser Iteration

In this section, we recall the Nash-Moser iteration for reader’s convenience.
First, we introduce the smoothing operators $S_{\theta}$ and describe the
iterative scheme for problem (8.11). For more details refer to [5, 14, 39].

######  Lemma 9.1.

Let $\mu\in\mathbb{N},$ with $\mu\geq 4$.
$\mathcal{F}^{s}_{\ast}(\Omega_{T}):=\\{u\in
H^{s}_{\ast}(\Omega_{T}):u=0\text{ for }t<0\\}.$ Define a family of smoothing
operators $\\{S_{\theta}\\}_{{\theta\geq 1}}$ on the anisotropic Sobolev space
from $\mathcal{F}^{3}_{\ast}(\Omega_{T})$ to $\bigcap_{s\geq
3}\mathcal{F}^{s}_{\ast}(\Omega_{T})$, such that

$||S_{\theta}u||_{k,\ast,T}\leq C\theta^{(k-j)_{+}}||u||_{j,\ast,T},\text{ for
all }k,j\in\\{1,\cdots,\mu\\},$ (9.1) $||S_{\theta}u-u||_{k,\ast,T}\leq
C\theta^{k-j}||u||_{j,\ast,T},\text{ for all }1\leq k\leq j\leq\mu,$ (9.2)
$||\frac{d}{d\theta}S_{\theta}u||_{k,\ast,T}\leq
C\theta^{k-j-1}||u||_{j,\ast,T},\text{ for all }k,j\in\\{1,\cdots,\mu\\},$
(9.3)

where $C$ is positive constant and
$k,j\in\mathbb{N},(k-j)_{+}:=\max\\{0,k-j\\}.$ In particular, if $u=v$ on
$\Gamma_{T},$ then $S_{\theta}u=S_{\theta}v$ on $\Gamma_{T}$. The definition
of $\mathcal{F}^{s}(\Gamma_{T})$ is entirely similar.

Now, we begin to formulate the Nash-Moser iteration scheme.

The iteration scheme starts from
$({\mathbf{V}}_{0},\Psi_{0},\psi_{0})=(0,0,0),$ and
$({\mathbf{V}}_{i},\Psi_{i},\psi_{i})$ is given such that

$({\mathbf{V}}_{i},\Psi_{i},\psi_{i})|_{t<0}=0,\quad\Psi^{+}_{i}|_{x_{1}=0}=\Psi^{-}_{i}|_{x_{1}=0}=\psi_{i}.$
(9.4)

Let us consider

${\mathbf{V}}_{i+1}={\mathbf{V}}_{i}+\delta{\mathbf{V}}_{i},\;\Psi_{i+1}=\Psi_{i}+\delta\Psi_{i},\;\psi_{i+1}=\psi_{i}+\delta\psi_{i},$
(9.5)

where the differences $(\delta{{\mathbf{V}}}_{i},\delta\psi_{i})$ will be
determined below. First, we can obtain
$(\delta\dot{{\mathbf{V}}}_{i},\delta\psi_{i})$ by solving the effective
linear problem:

$\begin{cases}\displaystyle\mathbb{L}_{e}^{\prime}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})\delta\dot{{\mathbf{V}}}_{i}=f_{i}&\text{
in }\Omega_{T},\\\
\displaystyle\mathbb{B}_{e}^{\prime}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})(\delta\dot{{\mathbf{V}}}_{i},\delta\psi_{i})=g_{i}&\text{
on }\Gamma_{T},\\\
\displaystyle(\delta\dot{{\mathbf{V}}}_{i},\delta\psi_{i})=0&\text{ for
}t<0,\end{cases}$ (9.6)

where operators $\mathbb{L}_{e}^{\prime},\mathbb{B}_{e}^{\prime}$ are defined
in (4.15) and (4.16),

$\delta\dot{{\mathbf{V}}}_{i}:=\delta{\mathbf{V}}_{i}-\frac{\partial_{1}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}})}{\partial_{1}(\Phi^{a}+\Psi_{i+\frac{1}{2}})}\delta\Psi_{i}$
(9.7)

is the Alinhac “good unknown” and
$({\mathbf{V}}_{i+\frac{1}{2}},\Psi_{i+\frac{1}{2}})$ is a smooth modified
state such that
$({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})$
satisfies (4.2)–(4.8) and (4.10). The source terms $(f_{i},g_{i})$ will be
defined through the accumulated errors at step $i$. $S_{\theta_{i}}$ is the
smoothing operator with ${\theta_{i}}$ defined by

$\theta_{0}\geq 1,\;\theta_{i}=\sqrt{\theta^{2}_{0}+i}\,.$ (9.8)

The errors at step $i$ can be defined from the following decompositions:

$\begin{split}&\mathcal{L}({\mathbf{V}}_{i+1},\Psi_{i+1})-\mathcal{L}({\mathbf{V}}_{i},\Psi_{i})\\\
&=\mathbb{L}^{\prime}({\mathbf{U}}^{a}+{\mathbf{V}}_{i},\Psi^{a}+\Psi_{i})(\delta{\mathbf{V}}_{i},\delta\Psi_{i})+e^{\prime}_{i}\\\
&=\mathbb{L}^{\prime}({\mathbf{U}}^{a}+S_{\theta_{i}}{\mathbf{V}}_{i},\Psi^{a}+S_{\theta_{i}}\Psi_{i})(\delta{\mathbf{V}}_{i},\delta\Psi_{i})+e^{\prime}_{i}+e^{\prime\prime}_{i}\\\
&=\mathbb{L}^{\prime}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})(\delta{\mathbf{V}}_{i},\delta\Psi_{i})+e^{\prime}_{i}+e^{\prime\prime}_{i}+e^{\prime\prime\prime}_{i}\\\
&=\mathbb{L}^{\prime}_{e}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})\delta\dot{{\mathbf{V}}}_{i}+e^{\prime}_{i}+e^{\prime\prime}_{i}+e^{\prime\prime\prime}_{i}+D_{i+\frac{1}{2}}\delta\Psi_{i}\\\
\end{split}$ (9.9)

and

$\begin{split}&\mathcal{B}({\mathbf{V}}_{i+1}|_{x_{1}=0},\psi_{i+1})-\mathcal{B}({\mathbf{V}}_{i}|_{x_{1}=0},\psi_{i})\\\
&=\mathbb{B}^{\prime}(({\mathbf{U}}^{a}+{\mathbf{V}}_{i})|_{x_{1}=0},\varphi^{a}+\psi_{i})(\delta{\mathbf{V}}_{i}|_{x_{1}=0},\delta\psi_{i})+\tilde{e}^{\prime}_{i}\\\
&=\mathbb{B}^{\prime}(({\mathbf{U}}^{a}+S_{\theta_{i}}{\mathbf{V}}_{i})|_{x_{1}=0},\varphi^{a}+S_{\theta_{i}}\Psi_{i}|_{x_{1}=0})(\delta{\mathbf{V}}_{i}|_{x_{1}=0},\delta\psi_{i})+\tilde{e}^{\prime}_{i}+\tilde{e}^{\prime\prime}_{i}\\\
&=\mathbb{B}^{\prime}_{e}(({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}})|_{x_{1}=0},\varphi^{a}+\psi_{i+\frac{1}{2}})(\delta\dot{{\mathbf{V}}}_{i}|_{x_{1}=0},\delta\psi_{i})+\tilde{e}^{\prime}_{i}+\tilde{e}^{\prime\prime}_{i}+\tilde{e}^{\prime\prime\prime}_{i},\\\
\end{split}$ (9.10)

where we write

$D_{i+\frac{1}{2}}:=\frac{1}{\partial_{1}(\Phi^{a}+\Psi_{i+\frac{1}{2}})}\partial_{1}\mathbb{L}({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}}),$
(9.11)

and have used (4.14) to get the last identity in (9.9). Denote

$e_{i}:=e^{\prime}_{i}+e^{\prime\prime}_{i}+e^{\prime\prime\prime}_{i}+D_{i+\frac{1}{2}}\delta\Psi_{i},\quad\tilde{e}_{i}:=\tilde{e}^{\prime}_{i}+\tilde{e}^{\prime\prime}_{i}+\tilde{e}^{\prime\prime\prime}_{i}.$
(9.12)

We assume
$f_{0}:=S_{\theta_{0}}\mathcal{F}^{a},(E_{0},\tilde{E}_{0},g_{0}):=(0,0,0)$
and $(f_{k},g_{k},e_{k},\tilde{e}_{k})$ are already given and vanish in the
past for $k\in\\{0,\cdots,i-1\\}.$ We can calculate the accumulated errors at
step $i,i\geq 1$, by

$E_{i}:=\sum^{i-1}_{k=0}e_{k},\quad\tilde{E}_{i}:=\sum^{i-1}_{k=0}\tilde{e}_{k}.$
(9.13)

Then, we obtain $f_{i}$ and $g_{i}$ for $i\geq 1$ from the equations:

$\sum^{i}_{k=0}f_{k}+S_{\theta_{i}}E_{i}=S_{\theta_{i}}\mathcal{F}^{a},\quad\sum^{i}_{k=0}g_{k}+S_{\theta_{i}}\tilde{E}_{i}=0.$
(9.14)

Then, given suitable $({\mathbf{V}}_{i+\frac{1}{2}},\Psi_{i+\frac{1}{2}})$, we
can obtain $(\delta\dot{{\mathbf{V}}}_{i},\delta\psi_{i})$ as the solutions of
the linear problem (9.6), $\delta{\mathbf{V}}_{i}$ from (9.7),
$({\mathbf{V}}_{i+1},\Psi_{i+1},\psi_{i+1})$ from (9.5). Since
$S_{\theta_{i}}\rightarrow I$ as $i\rightarrow\infty,$ we can formally obtain
the solution to problem (8.11) from
$\mathcal{L}({\mathbf{V}}_{i},\Psi_{i})\rightarrow\mathcal{F}^{a},\mathcal{B}({\mathbf{V}}_{i}|_{x_{1}=0},\psi_{i})\rightarrow
0,$ as error terms $(e_{i},\tilde{e}_{i})\rightarrow 0.$

##  10\. Proof of the Main Result

Now, we prove the local existence of solutions to (8.11) by a modified
iteration scheme of Nash-Moser type. From the sequence $\\{\theta_{i}\\}$
defined in (9.8), we set $\Delta_{i}:=\theta_{i+1}-\theta_{i}.$ Then, the
sequence $\\{\Delta_{i}\\}$ is decreasing and tends to $0$ as $i$ goes to
infinity. Moreover, we have

$\frac{1}{3\theta_{i}}\leq\Delta_{i}=\sqrt{\theta^{2}_{i}+1}-\theta_{i}\leq\frac{1}{2\theta_{i}},\;\forall\,i\in\mathbb{N}.$

###  10.1. Inductive analysis

Given a small fixed $\delta>0$, and an integer $\tilde{\alpha}$ that will be
chosen later on, we assume that the following estimate holds:

$||\tilde{{\mathbf{U}}}^{a}||_{\tilde{\alpha}+6,\ast,T}+||\varphi^{a}||_{H^{\tilde{\alpha}+6}(\Gamma_{T})}+||\mathcal{F}^{a}||_{\tilde{\alpha}+4,\ast,T}\leq\delta.$
(10.1)

We may assume that (10.1) holds, by taking $T>0$ sufficiently small.

Given the integer $\alpha$, our inductive assumptions read

$\displaystyle(H_{i-1})\left\\{\begin{split}&\displaystyle(a)\quad||(\delta{\mathbf{V}}_{k},\delta\Psi_{k})||_{s,\ast,T}+||\delta\psi_{k}||_{H^{s}(\Gamma_{T})}\leq\delta\theta^{s-\alpha-1}_{k}\Delta_{k},\\\
&\qquad\forall k\in\\{0,\cdots,i-1\\},\quad\forall
s\in\\{6,\cdots,\tilde{\alpha}\\}.\\\
&\displaystyle(b)\quad||\mathcal{L}({\mathbf{V}}_{k},\Psi_{k})-\mathcal{F}^{a}||_{s,\ast,T}\leq
2\delta\theta^{s-\alpha-1}_{k},\\\ &\qquad\forall
k\in\\{0,\cdots,i-1\\},\quad\forall s\in\\{6,\cdots,\tilde{\alpha}-2\\}.\\\
&\displaystyle(c)\quad||\mathcal{B}({\mathbf{V}}_{k}|_{x_{1}=0},\psi_{k})||_{H^{s}(\Gamma_{T})}\leq\delta\theta^{s-\alpha-1}_{k},\\\
&\qquad\forall k\in\\{0,\cdots,i-1\\},\quad\forall
s\in\\{6,\cdots,\tilde{\alpha}-2\\}.\\\ \end{split}\right.$ (10.2)

Our goal is to show that $(H_{0})$ holds and $(H_{i-1})$ implies $(H_{i})$,
for a suitable choice of the parameters $\alpha$, $\tilde{\alpha}$, for
$\delta>0$ and $T>0$ sufficiently small, for $\theta_{0}\geq 1$ sufficiently
large. Then, we conclude that $(H_{i})$ holds for all $i\in\mathbb{N}.$

######  Lemma 10.1.

If $T>0$ is sufficiently small, then $(H_{0})$ holds.

###### Proof.

The proof follows as in [39, Lemma 17]. ∎

Now we prove that $(H_{i-1})$ implies $(H_{i})$. The hypothesis $(H_{i-1})$
yields the following lemma.

######  Lemma 10.2.

[39, Lemma 7], [14, Lemma 7] If $\theta_{0}$ is large enough, then, for each
$k\in\\{0,\cdots,i\\}$, and each integer $s\in\\{6,\cdots,\tilde{\alpha}\\},$

$||({\mathbf{V}}_{k},\Psi_{k})||_{s,\ast,T}+||\psi_{k}||_{H^{s}(\Gamma_{T})}\lesssim\begin{cases}\delta\theta^{(s-\alpha)_{+}}_{k},&\text{if
}s\neq\alpha,\\\ \delta\log\theta_{k},&\text{if }s=\alpha,\\\ \end{cases}$
(10.3)
$||(I-S_{\theta_{k}})({\mathbf{V}}_{k},\Psi_{k})||_{s,\ast,T}+||(I-S_{\theta_{k}})\psi_{k}||_{H^{s}(\Gamma_{T})}\lesssim\delta\theta^{s-\alpha}_{k}.$
(10.4)

Furthermore, for each $k\in\\{0,\cdots,i\\}$, and each integer
$s\in\\{6,\cdots,\tilde{\alpha}+8\\},$

$||(S_{\theta_{k}}{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k})||_{s,\ast,T}+||S_{\theta_{k}}\psi_{k}||_{H^{s}(\Gamma_{T})}\lesssim\begin{cases}\delta\theta^{(s-\alpha)_{+}}_{k},&\text{if
}s\neq\alpha,\\\ \delta\log\theta_{k},&\text{if }s=\alpha.\\\ \end{cases}$
(10.5)

###  10.2. Estimate of the error terms

To derive $(H_{i})$ from $(H_{i-1})$, we need to estimate the quadratic error
terms $e^{\prime}_{k}$ and $\tilde{e}^{\prime}_{k},$ the first substitution
error terms $e^{\prime\prime}_{k}$ and $\tilde{e}^{\prime\prime}_{k},$ the
second substitution error terms $e^{\prime\prime\prime}_{k}$ and
$\tilde{e}^{\prime\prime\prime}_{k}$ and the last error term
$D_{k+\frac{1}{2}}\delta\Psi_{k}$ (cfr. (9.9)-(9.11)).

First, we denote the quadratic error terms by

$e^{\prime}_{k}:=\mathcal{L}({\mathbf{V}}_{k+1},\Psi_{k+1})-\mathcal{L}({\mathbf{V}}_{k},\Psi_{k})-\mathcal{L}^{\prime}({\mathbf{V}}_{k},\Psi_{k})(\delta{\mathbf{V}}_{k},\delta\Psi_{k}),$
(10.6)
$\tilde{e}^{\prime}_{k}:=\mathcal{B}({\mathbf{V}}_{k+1}|_{x_{1}=0},\psi_{k+1})-\mathcal{B}({\mathbf{V}}_{k}|_{x_{1}=0},\psi_{k})-\mathcal{B}^{\prime}({\mathbf{V}}_{k}|_{x_{1}=0},\psi_{k})(\delta{\mathbf{V}}_{k}|_{x_{1}=0},\delta\psi_{k}).$
(10.7)

Then, we get

$e^{\prime}_{k}=\int^{1}_{0}\mathbb{L}^{\prime\prime}({\mathbf{U}}^{a}+{\mathbf{V}}_{k}+\tau\delta{\mathbf{V}}_{k},\Psi^{a}+\Psi_{k}+\tau\delta\Psi_{k})((\delta{\mathbf{V}}_{k},\delta\Psi_{k}),(\delta{\mathbf{V}}_{k},\delta\Psi_{k}))(1-\tau)d\tau,$
$\tilde{e}^{\prime}_{k}=\frac{1}{2}\mathbb{B}^{\prime\prime}((\delta{\mathbf{V}}_{k},\delta\psi_{k}),(\delta{\mathbf{V}}_{k},\delta\psi_{k})),$

where $\mathbb{L}^{\prime\prime},\mathbb{B}^{\prime\prime}$ denote the second
order derivatives of the operators $\mathbb{L}$ and $\mathbb{B}.$ To be more
precise, we define

$\mathbb{L}^{\prime\prime}(\hat{\mathbf{U}},\hat{\Psi})(({\mathbf{V}},\Psi),(\tilde{{\mathbf{V}}},\tilde{\Psi})):=\frac{d}{d\varepsilon}\mathbb{L}^{\prime}(\hat{\mathbf{U}}+\varepsilon\tilde{{\mathbf{V}}},\hat{\Psi}+\varepsilon\tilde{\Psi})({\mathbf{V}},\Psi)\Big{|}_{\varepsilon=0},$
$\mathbb{B}^{\prime\prime}(({\mathbf{V}},\psi),(\tilde{{\mathbf{V}}},\tilde{\psi})):=\frac{d}{d\varepsilon}\mathbb{B}^{\prime}(\hat{\mathbf{U}}+\varepsilon\tilde{{\mathbf{V}}},\hat{\varphi}+\varepsilon\tilde{\psi})({\mathbf{V}},\psi)\Big{|}_{\varepsilon=0},$

where $\mathbb{L}^{\prime}$ and $\mathbb{B}^{\prime}$ are defined in (4.11)
and (4.12). Simple calculations yield that

$\mathbb{B}^{\prime\prime}(({\mathbf{V}},\psi),(\tilde{{\mathbf{V}}},\tilde{\psi})):=\left[\begin{array}[]{c}\tilde{u}^{+}_{2}\partial_{2}\psi+\partial_{2}\tilde{\psi}u^{+}_{2}\\\
\tilde{u}^{-}_{2}\partial_{2}\psi+\partial_{2}\tilde{\psi}u^{-}_{2}\\\
\mathbf{H}^{+}\cdot\tilde{\mathbf{H}}^{+}-\mathbf{H}^{-}\cdot\tilde{\mathbf{H}}^{-}\end{array}\right]\,.$
(10.8)

To estimate the error terms, we need to estimate the operators
$\mathbb{L}^{\prime\prime}$ and $\mathbb{B}^{\prime\prime}.$ Applying the
Moser-type calculus inequalities in Lemma 3.1 and Lemma 3.2 and the explicit
forms of $\mathbb{L}^{\prime\prime}$ and $\mathbb{B}^{\prime\prime}$ , we can
obtain the necessary estimates. Omitting the detailed calculation, we have the
following Lemma 10.3:

######  Lemma 10.3.

Let $T>0,$ and $s\in\mathbb{N}$ with $s\geq 6.$ Assume that
$(\breve{\mathbf{U}},\hat{\Psi})\in H^{s+2}_{\ast}(\Omega_{T})$ satisfies

$||(\hat{{\mathbf{U}}},\hat{\Psi})||_{W^{2,\infty}_{\ast}(\Omega_{T})}\leq\tilde{K},$

(recall that $\hat{{\mathbf{U}}}=\bar{{\mathbf{U}}}+\breve{{\mathbf{U}}}$) for
some constant $\tilde{K}>0.$ Then, there exists a positive $C$ depending on
$\tilde{K},$ but not on $T,$ such that if $({\mathbf{V}}_{i},\Psi_{i})\in
H^{s+2}_{\ast}(\Omega_{T})$ and $(W_{i},\psi_{i})\in H^{s}(\Gamma_{T})\times
H^{s+1}(\Gamma_{T}),$ for $i=1,2,$ then

$\begin{split}&||\mathbb{L}^{\prime\prime}(\hat{\mathbf{U}},\hat{\Psi})(({\mathbf{V}}_{1},\Psi_{1}),({\mathbf{V}}_{2},\Psi_{2}))||_{s,\ast,T}\\\
&\quad\leq
C||(\breve{{\mathbf{U}}},\hat{\Psi})||_{s+2,\ast,T}||({\mathbf{V}}_{1},\Psi_{1})||_{W^{2,\infty}_{\ast}(\Omega_{T})}||({\mathbf{V}}_{2},\Psi_{2})||_{W^{2,\infty}_{\ast}(\Omega_{T})}\\\
&\quad\quad+C\sum_{i\neq
j}||({\mathbf{V}}_{i},\Psi_{i})||_{s+2,\ast,T}||({\mathbf{V}}_{j},\Psi_{j})||_{W^{2,\infty}_{\ast}(\Omega_{T})},\end{split}$

and

$\begin{split}&||\mathbb{B}^{\prime\prime}((W_{1},\psi_{1}),(W_{2},\psi_{2}))||_{H^{s}(\Gamma_{T})}\\\
&\quad\leq C\sum_{i\neq
j}\Big{(}||W_{i}||_{H^{s}(\Gamma_{T})}||\psi_{j}||_{W^{1,\infty}(\Gamma_{T})}+||W_{i}||_{L^{\infty}(\Gamma_{T})}||\psi_{j}||_{H^{s+1}(\Gamma_{T})}\\\
&\quad\quad+||W_{i}||_{H^{s}(\Gamma_{T})}||W_{j}||_{L^{\infty}(\Gamma_{T})}\Big{)}.\end{split}$

####  10.2.1. Estimate of the quadratic errors

We now apply Lemma 10.3 to prove the following estimate for the quadratic
error terms.

######  Lemma 10.4.

Let $\alpha\geq 7.$ There exist $\delta>0$ sufficiently small and
$\theta_{0}\geq 1$ sufficiently large such that, for all
$k\in\\{0,\cdots,i-1\\},$ and all integers
$s\in\\{6,\cdots,\tilde{\alpha}-2\\},$ we have

$||e^{\prime}_{k}||_{s,\ast,T}\lesssim\delta^{2}\theta^{L_{1}(s)-1}_{k}\Delta_{k},$
(10.9)
$||\tilde{e}^{\prime}_{k}||_{H^{s}(\Gamma_{T})}\lesssim\delta^{2}\theta^{L_{1}(s)-1}_{k}\Delta_{k},$
(10.10)

where $L_{1}(s):=\max\\{(s+2-\alpha)_{+}+10-2\alpha;s+6-2\alpha\\}.$

###### Proof.

Using (10.1), the hypothesis $(H_{i-1})$ and the estimate (10.3), we use the
Sobolev inequalities (3.10) to get

$||(\mathbf{U}^{a},{\mathbf{V}}_{k},\delta{\mathbf{V}}_{k},\Psi^{a},\Psi_{k},\delta\Psi_{k})||_{W^{2,\infty}_{\ast}(\Omega_{T})}\lesssim
1.$

Then, we apply Lemma 10.3 and use Sobolev inequalities (3.10), the assumption
(10.1) and the hypothesis $(H_{i-1})$ to give

$\begin{split}||e^{\prime}_{k}||_{s,\ast,T}&\lesssim\delta^{2}\theta^{10-2\alpha}_{k}\Delta^{2}_{k}(1+||({\mathbf{V}}_{k},\Psi_{k})||_{s+2,\ast,T}+\delta\theta^{s+1-\alpha}_{k}\Delta_{k})\\\
&\quad\quad+||(\delta{\mathbf{V}}_{k},\delta\Psi_{k})||_{s+2,\ast,T}||(\delta{\mathbf{V}}_{k},\delta\Psi_{k})||_{6,\ast,T}\\\
&\lesssim\delta^{2}\theta^{10-2\alpha}_{k}\Delta^{2}_{k}(1+||({\mathbf{V}}_{k},\Psi_{k})||_{s+2,\ast,T})+\delta^{2}\theta^{s+6-2\alpha}_{k}\Delta^{2}_{k},\end{split}$

for all $s\in\\{6,\cdots,\tilde{\alpha}-2\\}.$ If $s+2\neq\alpha,$ then it
follows from (10.3) and $2\theta_{k}\Delta_{k}\leq 1,$ that

$\begin{split}||e^{\prime}_{k}||_{s,\ast,T}&\lesssim\delta^{2}\Delta^{2}_{k}(\theta^{(s+2-\alpha)_{+}+10-2\alpha}_{k}+\theta^{s+6-2\alpha}_{k})\\\
&\lesssim\delta^{2}\theta^{L_{1}(s)-1}_{k}\Delta_{k}.\end{split}$

If $s+2=\alpha,$ then it follows from (10.3) and $\alpha\geq 7,$ that

$\begin{split}||e^{\prime}_{k}||_{s,\ast,T}&\lesssim\delta^{2}\Delta^{2}_{k}(\theta^{11-2\alpha}_{k}+\theta^{4-\alpha}_{k})\\\
&\lesssim\delta^{2}\theta^{L_{1}(\alpha-2)-1}_{k}\Delta_{k}.\end{split}$

Therefore, we obtain (10.9). Now, we prove (10.10). Using Lemma 10.3 and trace
Theorem A.1, we obtain

$\begin{split}||\tilde{e}^{\prime}_{k}||_{H^{s}(\Gamma_{T})}&\lesssim||\delta{\mathbf{V}}_{k}||_{H^{s}(\Gamma_{T})}||\delta\psi_{k}||_{W^{1,\infty}(\Gamma_{T})}+||\delta{\mathbf{V}}_{k}||_{L^{\infty}(\Gamma_{T})}||\delta\psi_{k}||_{H^{s+1}(\Gamma_{T})}\\\
&\quad+||\delta{\mathbf{V}}_{k}||_{H^{s}(\Gamma_{T})}||\delta{\mathbf{V}}_{k}||_{L^{\infty}(\Gamma_{T})}\\\
&\lesssim\delta^{2}\theta^{L_{1}(s)-1}_{k}\Delta_{k}.\end{split}$

This completes the proof of Lemma 10.4. ∎

####  10.2.2. Estimate of the first substitution errors

We can estimate the first substitution errors
$e^{\prime\prime}_{k},\tilde{e}^{\prime\prime}_{k}$ of the iteration scheme,
defined in (9.9) and (9.10). We rewrite

$e^{\prime\prime}_{k}:=\mathcal{L}^{\prime}({\mathbf{V}}_{k},\Psi_{k})(\delta{\mathbf{V}}_{k},\delta\Psi_{k})-\mathcal{L}^{\prime}(S_{\theta_{k}}{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k})(\delta{\mathbf{V}}_{k},\delta\Psi_{k}),$
(10.11)
$\tilde{e}^{\prime\prime}_{k}:=\mathcal{B}^{\prime}({\mathbf{V}}_{k}|_{x_{1}=0},\psi_{k})(\delta{\mathbf{V}}_{k}|_{x_{1}=0},\delta\psi_{k})-\mathcal{B}^{\prime}(S_{\theta_{k}}{\mathbf{V}}_{k}|_{x_{1}=0},S_{\theta_{k}}\psi_{k})(\delta{\mathbf{V}}_{k}|_{x_{1}=0},\delta\psi_{k}).$
(10.12)

######  Lemma 10.5.

Let $\alpha\geq 7.$ There exist $\delta>0$ sufficiently small and
$\theta_{0}\geq 1$ sufficiently large, such that for all
$k\in\\{0,\cdots,i-1\\}$ and for all integer
$s\in\\{6,\cdots,\tilde{\alpha}-2\\},$ we have

$||e^{\prime\prime}_{k}||_{s,\ast,T}\lesssim\delta^{2}\theta^{L_{2}(s)-1}_{k}\Delta_{k},$
(10.13)
$||\tilde{e}^{\prime\prime}_{k}||_{H^{s}(\Gamma_{T})}\lesssim\delta^{2}\theta^{L_{2}(s)-1}_{k}\Delta_{k},$
(10.14)

where $L_{2}(s):=\max\\{(s+2-\alpha)_{+}+12-2\alpha;s+8-2\alpha\\}.$

###### Proof.

In view of (10.11) and (10.12) we have

$\begin{split}e^{\prime\prime}_{k}&=\int^{1}_{0}\mathbb{L}^{\prime\prime}(\mathbf{U}^{a}+S_{\theta_{k}}{\mathbf{V}}_{k}+\tau(I-S_{\theta_{k}}){\mathbf{V}}_{k},\Psi^{a}+S_{\theta_{k}}\Psi_{k}+\tau(I-S_{\theta_{k}})\Psi_{k})\\\
&\quad\quad((\delta{\mathbf{V}}_{k},\delta\Psi_{k}),((I-S_{\theta_{k}}){\mathbf{V}}_{k},(I-S_{\theta_{k}})\Psi_{k}))d\tau,\\\
\tilde{e}^{\prime\prime}_{k}&=\mathbb{B}^{\prime\prime}((\delta{\mathbf{V}}_{k}|_{x_{1}=0},\delta\psi_{k}),((I-S_{\theta_{k}}){\mathbf{V}}_{k}|_{x_{1}=0},(I-S_{\theta_{k}})\psi_{k})).\end{split}$

Using (10.4) and (10.5), we have

$||(S_{\theta_{k}}{\mathbf{V}}_{k},{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k},\Psi_{k})||_{W^{2,\infty}_{\ast}(\Omega_{T})}\lesssim||(S_{\theta_{k}}{\mathbf{V}}_{k},{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k},\Psi_{k})||_{6,\ast,T}\lesssim
1.$

Next, we apply Lemma 10.3, use Sobolev inequalities (3.10), (10.1), the
Hypothesis $(H_{i-1})$ and (10.4) to get that

$||e^{\prime\prime}_{k}||_{s,\ast,T}\lesssim\delta^{2}\theta^{11-2\alpha}_{k}\Delta_{k}(1+||(S_{\theta_{k}}{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k})||_{s+2,\ast,T})+\delta^{2}\theta^{s+7-2\alpha}_{k}\Delta_{k},$

for all $s\in\\{6,\cdots,\tilde{\alpha}-2\\}.$ Similar to the proof of Lemma
10.4, we can discuss $s+2\neq\alpha$ and $s+2=\alpha$ separately. Hence, using
(10.5), we can obtain (10.13) and (10.14). The proof of Lemma 10.5 is
completed. ∎

####  10.2.3. Estimate of the modified state

We need to construct a smooth modified state
$({\mathbf{V}}_{i+\frac{1}{2}},\psi_{i+\frac{1}{2}})$ such that
$({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\varphi^{a}+\psi_{i+\frac{1}{2}})$
satisfies the nonlinear constraints (4.2)–(4.8), (4.10) and (6.1). In this
regard, we remark it is crucial that the sum of the approximate solution
$({\mathbf{U}}^{a},\varphi^{a})$ and the modified state
$({\mathbf{V}}_{i+\frac{1}{2}},\psi_{i+\frac{1}{2}})$, instead of the latter
two separately, satisfies the aforementioned nonlinear constraints; indeed it
is just this sum which plays the role of basic state around which we need to
linearize problem (2.1) in the iteration scheme leading to its solution, see
problem (9.6). In the construction of some components of the modified state we
follow the similar approach of [1, 14, 34], while for the magnetic field we
are inspired by [34, 39].

######  Lemma 10.6.

Let $\alpha\geq 10$. There exist some functions
${\mathbf{V}}_{i+\frac{1}{2}},\Psi_{i+\frac{1}{2}},\psi_{i+\frac{1}{2}}$
vanishing in the past, such that
$({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}},\varphi^{a}+\psi_{i+\frac{1}{2}})$
satisfy the constraints (4.2)– (4.8), (4.10) and (6.1); moreover,

$\Psi_{i+\frac{1}{2}}^{\pm}=S_{\theta_{i}}\Psi^{\pm}_{i}\,,\quad\psi_{i+\frac{1}{2}}=(S_{\theta_{i}}\Psi^{\pm}_{i})|_{x_{1}=0,},$
(10.15) $p^{\pm}_{i+\frac{1}{2}}=S_{\theta_{i}}p^{\pm}_{i}\,,\quad
u^{\pm}_{2,i+\frac{1}{2}}=S_{\theta_{i}}u^{\pm}_{2,i}\,,\quad
S^{\pm}_{i+\frac{1}{2}}=S_{\theta_{i}}S^{\pm}_{i},$ (10.16)
$||{\mathbf{V}}_{i+\frac{1}{2}}-S_{\theta_{i}}{\mathbf{V}}_{i}||_{s,\ast,T}\lesssim\delta\theta^{s+4-\alpha}_{i},\text{
for }s\in\\{6,\cdots,\tilde{\alpha}+4\\},$ (10.17)

for sufficiently small $\delta>0$ and $T>0,$ and sufficiently large
$\theta_{0}\geq 1.$

###### Proof.

To shortcut notation, in the proof the $\pm$ indices are omitted. Let us
define $\Psi_{i+1/2}$, $\psi_{i+1/2}$, $p_{i+1/2}$, $S_{i+1/2}$ and the
tangential component $u_{2,i+1/2}$ of the velocity as in (10.15), (10.16);
this is similar as in [1], [14, Proposition 7], [34, Proposition 28]. It is
easily checked that all these functions vanish in the past.

Construction of the modified normal velocity.

In order to construct the normal component $u_{1,i+1/2}$ of the velocity, we
follow the idea of [1, 14]. We first introduce the following function
$\mathcal{G}$:

$\begin{split}\mathcal{G}:=&\partial_{t}(\varphi^{a}+\psi_{i+1/2})-(u^{a}_{1}+S_{\theta_{i}}u_{1,i})|_{x_{1}=0}+u^{a}_{2}|_{x_{1}=0}\partial_{2}\varphi^{a}\\\
&+(u^{a}_{2}+u_{2,i+1/2})|_{x_{1}=0}\partial_{2}\psi_{i+1/2}+u_{2,i+1/2}|_{x_{1}=0}\partial_{2}\varphi^{a}\,.\end{split}$

The normal component of the velocity $u_{1,i+1/2}$ is defined by

$u_{1,i+1/2}:=S_{\theta_{i}}u_{1,i}+\mathcal{R}_{T}\mathcal{G}\,,$ (10.18)

where $\mathcal{R}_{T}$ is the lifting operator
$H^{s-1}(\Gamma_{T})\rightarrow H^{s}_{\ast}(\Omega_{T})$, $s>1$, see [27]. It
is easily checked that $u_{1,i+1/2}$ vanishes in the past.
Let us note that, by construction, $u_{1,i+1/2}$ satisfies the following
equation on the boundary

$\partial_{t}(\varphi^{a}+\psi_{i+1/2})-(u_{1}^{a}+u_{1,i+1/2})+(u_{2}^{a}+u_{2,i+1/2})\partial_{2}(\varphi^{a}+\psi_{i+1/2})=0\,,\quad\mbox{on}\,\,\,\\{x_{1}=0\\}\,.$
(10.19)

We prove the estimate (10.17) for the part regarding ${\mathbf{u}}_{i+1/2}$.
We have

$||\mathbf{u}_{i+1/2}-S_{\theta_{i}}{\mathbf{u}}_{i}||_{s,\ast,T}=||\mathcal{R}_{T}\mathcal{G}||_{s,\ast,T}\lesssim||\mathcal{G}||_{H^{s-1}(\Gamma_{T})}\,.$

Now we rewrite $\mathcal{G}$ in a more convenient form, by using the error
$\varepsilon^{i}$ defined by

$\begin{split}\varepsilon^{i}:=&\partial_{t}\psi_{i}-u_{1,i}|_{x_{1}=0}+(\partial_{t}\varphi^{a}-u_{1}^{a}|_{x_{1}=0}+u_{2}^{a}|_{x_{1}=0}\partial_{2}\varphi^{a})+(u^{a}_{2}+u_{2,i})|_{x_{1}=0}\partial_{2}\psi_{i}\\\
&+u_{2,i}|_{x_{1}=0}\partial_{2}\varphi^{a}=\mathcal{B}({\mathbf{V}}_{i}|_{x_{1}=0},\psi_{i})_{1}\,.\end{split}$
(10.20)

In view of (10.16), by means of $\varepsilon^{i}$ we may rewrite $\mathcal{G}$
as:

$\begin{split}\mathcal{G}&=S_{\theta_{i}}\varepsilon^{i}-[S_{\theta_{i}},\partial_{t}]\psi_{i}+(I-S_{\theta_{i}})(\partial_{t}\varphi^{a}-u^{a}_{1}+u^{a}_{2}\partial_{2}\varphi^{a})-S_{\theta_{i}}((u^{a}_{2}+u_{2,i})\partial_{2}\psi_{i})\\\
&\quad-
S_{\theta_{i}}(u_{2,i}\partial_{2}\varphi^{a})+(u^{a}_{2}+S_{\theta_{i}}u_{2,i})\partial_{2}\psi_{i+1/2}+S_{\theta_{i}}u_{2,i}\partial_{2}\varphi^{a}\\\
&=S_{\theta_{i}}\varepsilon^{i}-[S_{\theta_{i}},\partial_{t}]\psi_{i}+(I-S_{\theta_{i}})(\partial_{t}\varphi^{a}-u^{a}_{1}+u^{a}_{2}\partial_{2}\varphi^{a}))\\\
&\quad-\left(S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})-u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}\right)-\left(S_{\theta_{i}}u_{2,i}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u_{2,i}\partial_{2}\psi_{i})\right)\\\
&\quad-\left(S_{\theta_{i}}u_{2,i}\partial_{2}\varphi^{a}-S_{\theta_{i}}(u_{2,i}\partial_{2}\varphi^{a})\right).\end{split}$
(10.21)

To estimate the first term $S_{\theta_{i}}\varepsilon^{i}$ on the right-hand
side, we use the decomposition:

$\begin{split}\varepsilon^{i}&=\left(\mathcal{B}({\mathbf{V}}_{i},\psi_{i})_{1}-\mathcal{B}({\mathbf{V}}_{i-1},\psi_{i-1})_{1}\right)+\mathcal{B}({\mathbf{V}}_{i-1},\psi_{i-1})_{1}\\\
&=\partial_{t}\delta\psi_{i-1}-\delta
u_{1,i-1}+(u_{2}^{a}+u_{2,i-1})\partial_{2}\delta\psi_{i-1}+\delta
u_{2,i-1}\partial_{2}(\varphi^{a}+\delta\psi_{i-1}+\psi_{i-1})\\\
&\quad+\mathcal{B}({\mathbf{V}}_{i-1},\psi_{i-1})_{1}\,.\end{split}$

Then we exploit point (c) of ($H_{i-1}$) and the properties of smoothing
operators, to get

$||S_{\theta_{i}}\varepsilon^{i}||_{H^{s-1}(\Gamma_{T})}\lesssim||\varepsilon^{i}||_{H^{s-1}(\Gamma_{T})}\lesssim\delta\theta_{i}^{s-\alpha-1}\,.$
(10.22)

In order to make an estimate of the commutator term
$[S_{\theta_{i}},\partial_{t}]\psi_{i}$, we use different arguments for large
and small orders $s$.
For all $s\in\\{6,\dots,\alpha\\}$ we write the commutator as

$[\partial_{t},S_{\theta_{i}}]\psi_{i}=\partial_{t}(S_{\theta_{i}}-I)\psi_{i}+(I-S_{\theta_{i}})\partial_{t}\psi_{i}\,,$

then we use estimates (9.2), (10.3), (10.4) to get

$||[\partial_{t},S_{\theta_{i}}]\psi_{i}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad s\in\\{6,\dots,\alpha\\}\,.$ (10.23)

In order to get the similar estimate as (10.23) for
$s\in\\{\alpha+1,\dots,\tilde{\alpha}+6\\}$, we directly estimate the two
terms of the commutator
$[\partial_{t},S_{\theta_{i}}]\psi_{i}=\partial_{t}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}\partial_{t}\psi_{i}$,
using (9.1) and (10.5).
To estimate the third term in the right-hand side of (10.21), we proceed as
above by applying different arguments to small and large orders $s$.
For integers $s\in\\{6,\dots,\alpha\\}$, we apply (9.2) to get

$\begin{split}||(I-S_{\theta_{i}})&(\partial_{t}\varphi^{a}-u^{a}_{1}+u_{2}^{a}\partial_{2}\varphi^{a})||_{H^{s-1}(\Gamma_{T})}\leq
C\theta_{i}^{s-\alpha}||\partial_{t}\varphi^{a}-u^{a}_{1}+u_{2}^{a}\partial_{2}\varphi^{a}||_{H^{\alpha-1}(\Gamma_{T})}\\\
&\leq
C\theta_{i}^{s-\alpha}(||\varphi^{a}||_{H^{\alpha-1}(\Gamma_{T})}+||u^{a}||_{\alpha,\ast,T})\leq
C\delta\theta_{i}^{s-\alpha}\,,\end{split}$

in view of (10.1).
For integers $s\in\\{\alpha+1,\dots.\tilde{\alpha}+6\\}$, we use (9.1) and
(10.1) to estimate directly

$\begin{split}||(I-S_{\theta_{i}})&(\partial_{t}\varphi^{a}-u^{a}_{1}+u_{2}^{a}\partial_{2}\varphi^{a})||_{H^{s-1}(\Gamma_{T})}\leq||E^{a}||_{H^{s-1}(\Gamma_{T})}+||S_{\theta_{i}}E^{a}||_{H^{s-1}(\Gamma_{T})}\\\
&\leq||E^{a}||_{H^{s-1}(\Gamma_{T})}+C\theta_{i}^{(s-1-(\alpha-1))_{+}}||E^{a}||_{H^{\alpha-1}(\Gamma_{T})}\leq
C\theta_{i}^{s-\alpha}||E^{a}||_{H^{s-1}(\Gamma_{T})}\\\ &\leq
C\theta_{i}^{s-\alpha}(||\varphi^{a}||_{H^{s}(\Gamma_{T})}+||u^{a}||_{s,\ast,T})\leq
C\delta\theta_{i}^{s-\alpha}\,,\end{split}$

where
$E^{a}:=\partial_{t}\varphi^{a}-u^{a}_{1}+u_{2}^{a}\partial_{2}\varphi^{a}$
has been set.
Let us now estimate the fourth term
$-\left(S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})-u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}\right)$;
once again we need to argue separately on different values of $s$.
For small integers $s\in\\{6,\dots,\tilde{\alpha}\\}$ we rewrite the above in
the form

$u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})=u_{2}^{a}\partial_{2}(S_{\theta_{i}}-I)\psi_{i}+(I-S_{\theta_{i}})(u_{2}^{a}\partial_{2}\psi_{i})\,,$
(10.24)

which takes advantage of the appearing of the difference $I-S_{\theta_{i}}$;
thus estimate (10.4) and Moser-type calculus inequalities of Lemma 3.1 yield

$\begin{split}||u_{2}^{a}\partial_{2}(S_{\theta_{i}}-I)&\psi_{i}||_{H^{s-1}(\Gamma_{T})}\lesssim||u^{a}_{2}||_{L^{\infty}(\Gamma_{T})}||(I-S_{\theta_{i}})\psi_{i}||_{H^{s}(\Gamma_{T})}\\\
&+||u^{a}||_{s,\ast,T}||(I-S_{\theta_{i}})\psi_{i}||_{H^{2}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}\\}\,.\end{split}$
(10.25)

As for the second term in the right-hand side of (10.24), a further splitting
of the range of $s$ covered by estimate (10.25) is required. For integers $s$
such that $6\leq s\leq\alpha+1$, we apply (9.2), Moser-type calculus
inequalities of Lemma 3.1 and (10.3) to obtain

$\begin{split}||(I-S_{\theta_{i}})&(u_{2}^{a}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C\theta_{i}^{s-1-(\alpha+1)}||u_{2}^{a}\partial_{2}\psi_{i}||_{H^{\alpha+1}(\Gamma_{T})}\\\
&\leq
C\theta_{i}^{s-\alpha-2}\left\\{||u^{a}||_{L^{\infty}(\Gamma_{T})}||\psi_{i}||_{H^{\alpha+1}(\Gamma_{T})}+||u^{a}||_{\alpha+2,\ast,T}||\psi_{i}||_{H^{3}(\Gamma_{T})}\right\\}\\\
&\leq C\theta_{i}^{s-\alpha-2}(\delta\theta_{i}+\delta)\leq
C\delta\theta_{i}^{s-\alpha-1}\,,\quad\mbox{for}\,\,\,6\leq
s\leq\alpha+1\,.\end{split}$ (10.26)

On the other hand for integers $s$ such that $\alpha+1<s\leq\tilde{\alpha}$,
we use (9.1), Moser-type calculus inequalities and (10.3) to get

$\begin{split}||(I-S_{\theta_{i}})&(u_{2}^{a}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C||u_{2}^{a}\partial_{2}\psi_{i}||_{H^{s-1}(\Gamma_{T})}\\\ &\leq
C\left\\{||u^{a}||_{L^{\infty}(\Gamma_{T})}||\psi_{i}||_{H^{s}(\Gamma_{T})}+||u^{a}||_{s,\ast,T}||\psi_{i}||_{H^{3}(\Gamma_{T})}\right\\}\\\
&\leq C\theta_{i}^{(s-\alpha)_{+}}+C\delta\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,\alpha+1<s\leq\tilde{\alpha}\,.\end{split}$
(10.27)

Gathering (10.25)–(10.27) we end up with

$||u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}\\}\,.$
(10.28)

For higher integers $s\in\\{\tilde{\alpha}+1,\dots,\tilde{\alpha}+6\\}$, we
estimate separately the two terms of the difference
$u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})$;
using again estimates (9.1), (10.3), (10.5) and Moser-type calculus
inequalities, we obtain

$\begin{split}||u^{a}_{2}\partial_{2}S_{\theta_{i}}\psi_{i}||_{H^{s-1}(\Gamma_{T})}\lesssim||u^{a}_{2}||_{L^{\infty}(\Gamma_{T})}||S_{\theta_{i}}\psi_{i}||_{H^{s}(\Gamma_{T})}+||u^{a}_{2}||_{s,\ast,T}||S_{\theta_{i}}\psi_{i}||_{H^{3}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha}\,;\end{split}$
$\begin{split}||S_{\theta_{i}}&(u^{a}_{2}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C\theta_{i}^{s-1-\alpha}||u^{a}_{2}\partial_{2}\psi_{i}||_{H^{\alpha}(\Gamma_{T})}\\\
&\leq
C\theta_{i}^{s-1-\alpha}\left\\{||u^{a}_{2}||_{L^{\infty}(\Gamma_{T})}||\psi_{i}||_{H^{\alpha+1}(\Gamma_{T})}+||u^{a}_{2}||_{\alpha+1,\ast,T}||\psi_{i}||_{H^{3}(\Gamma_{T})}\right\\}\\\
&\leq C\theta_{i}^{s-1-\alpha}(\delta\theta_{i}+C\delta)\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,\tilde{\alpha}+1\leq
s\leq\tilde{\alpha}+6\,.\end{split}$

Adding the last two inequalities we end up with

$||u_{2}^{a}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u^{a}_{2}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{\tilde{\alpha}+1,\dots,\tilde{\alpha}+6\\}\,.$
(10.29)

Gathering estimates (10.28) and (10.29) provide the estimate for all integers
$s\in\\{6,\dots,\tilde{\alpha}+6\\}$.
For the last two terms in the right-hand side of (10.21) we use the same
arguments as above, where we still separate small and large orders $s$; in the
case of small $s$ we manage to rewrite the expression in order to make
advantage from the boundedness properties (9.2), (10.4) of $I-S_{\theta_{i}}$;
for large $s$ we estimate separately each term of the difference using Moser-
type calculus inequalities and (9.1), (10.3), (10.5). Doing so, we derive:

$||S_{\theta_{i}}u_{2,i}\partial_{2}S_{\theta_{i}}\psi_{i}-S_{\theta_{i}}(u_{2,i}\partial_{2}\psi_{i})||_{H^{s-1}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha+1}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+6\\}\,;$
$||S_{\theta_{i}}u_{2,i}\partial_{2}\varphi^{a}-S_{\theta_{i}}(u_{2,i}\partial_{2}\varphi^{a})||_{H^{s-1}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+6\\}\,.$

Gathering all the previously found estimates we end up with

$||\mathbf{u}_{i+1/2}-S_{\theta_{i}}{\mathbf{u}}_{i}||_{s,\ast,T}\lesssim||\mathcal{G}||_{H^{s-1}(\Gamma_{T})}\leq
C\delta\theta_{i}^{s-\alpha+1}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+6\\}\,.$
(10.30)

In the above estimate it is fundamental that $s\leq\tilde{\alpha}+6$ in order
to prove the following (10.40). Let us recall that the above estimate holds
under the smallness assumption (10.1).

Construction of the modified magnetic field.

Let us see now how to define the modified magnetic field $\mathbf{H}_{i+1/2}$,
following somehow [34, Proposition 28], [39, Proposition 12].
Let us denote the nonlinear equation satisfied by the magnetic field in (2.1)
by

$\mathbb{L}_{H}({\mathbf{u}},\mathbf{H},\Psi)=0\,,\quad\mbox{in}\,\,\,\Omega_{T}\,.$

The field $\mathbf{H}_{i+1/2}$ should be such that
$\mathbf{H}^{a}+\mathbf{H}_{i+1/2}$ satisfies (4.8), that is

$\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},\mathbf{H}^{a}+\mathbf{H}_{i+1/2},\Psi^{a}+\Psi_{i+1/2})=0\,,\quad\mbox{in}\,\,\,\Omega_{T}\,.$
(10.31)

We note that equation (10.31) is linear in $\mathbf{H}^{a}+\mathbf{H}_{i+1/2}$
and does not need to be supplemented with any boundary condition; in fact, the
coefficient of $\partial_{1}(\mathbf{H}^{a}+\mathbf{H}_{i+1/2})$ is zero along
the boundary because of (10.19) (the left-hand side of (10.19) is nothing but
$w_{1}|_{x_{1}=0}$, computed for ${\mathbf{u}}^{a}+{\mathbf{u}}_{i+1/2}$ and
$\Psi^{a}+\Psi_{i+1/2}$ instead of $\mathbf{u}$ and $\Psi$ respectively).
Therefore, for given ${\mathbf{u}}_{i+1/2}$, $\Psi_{i+1/2}$,
${\mathbf{U}}^{a}$ and $\Psi^{a}$, (10.31) has a unique solution
$\mathbf{H}^{\prime}$, from which we derive the existence of a unique
$\mathbf{H}_{i+1/2}=\mathbf{H}^{\prime}-\mathbf{H}^{a}$, vanishing in the
past.
In order to estimate ${\mathbf{H}}_{i+1/2}-S_{\theta_{i}}{\mathbf{H}}_{i}$, we
first observe that (10.31) yields

$\begin{split}\mathbb{L}_{H}&({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}_{i+1/2}-S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i+1/2})\\\
&=\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}^{a}+{\mathbf{H}}_{i+1/2}-S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i+1/2})\\\
&\quad\quad-\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}^{a},\Psi^{a}+\Psi_{i+1/2})\\\
&=-\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}^{a}+S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+S_{\theta_{i}}\Psi_{i})\,.\end{split}$

Then $\mathbf{H}_{i+1/2}-S_{\theta_{i}}\mathbf{H}_{i}$ solves the equation

$\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}_{i+1/2}-S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i+1/2})=F^{i+1/2}_{H}\,,$
(10.32)

where

$F_{H}^{i+1/2}:=\Delta_{1}+\Delta_{2}-S_{\theta_{i}}\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i},{\mathbf{H}}^{a}+{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i})\,;$
(10.33)
$\Delta_{1}:=S_{\theta_{i}}\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i},{\mathbf{H}}^{a}+{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i})-\mathbb{L}_{H}({\mathbf{u}}^{a}+S_{\theta_{i}}\mathbf{u}_{i},{\mathbf{H}}^{a}+S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+S_{\theta_{i}}\Psi_{i})\,;$
$\begin{split}\Delta_{2}&:=\mathbb{L}_{H}({\mathbf{u}}^{a}+S_{\theta_{i}}\mathbf{u}_{i},{\mathbf{H}}^{a}+S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+S_{\theta_{i}}\Psi_{i})\\\
&\qquad\qquad\qquad-\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},{\mathbf{H}}^{a}+S_{\theta_{i}}{\mathbf{H}}_{i},\Psi^{a}+S_{\theta_{i}}\Psi_{i})\,.\end{split}$

Let us first write the explicit form of $\Delta_{1}$. Using the definition of
the nonlinear operator $\mathbb{L}_{H}({\bf u},{\bf H},\Psi)$ we have

$\begin{split}\Delta_{1}=&S_{\theta_{i}}\Big{\\{}\partial_{t}({\bf H}^{a}+{\bf
H}_{i})+\frac{1}{\partial_{1}(\Phi^{a}+\Psi_{i})}\big{(}({\bf w}[{\bf
u}^{a}+{\bf u}_{i}]\cdot\nabla)({\bf H}^{a}+{\bf H}_{i})\\\ &-({\bf h}[{\bf
H}^{a}+{\bf H}_{i}]\cdot\nabla)({\bf u}^{a}+{\bf u}_{i})+({\bf H}^{a}+{\bf
H}_{i}){\rm div}({\bf v}[{\bf u}^{a}+{\bf u}_{i}]))\big{)}\Big{\\}}\\\
&-\Big{\\{}\partial_{t}({\bf H}^{a}+S_{\theta_{i}}{\bf
H}_{i})+\frac{1}{\partial_{1}(\Phi^{a}+S_{\theta_{i}}\Psi_{i})}\left({\bf
w}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]\cdot\nabla\right)({\bf
H}^{a}+S_{\theta_{i}}{\bf H}_{i})\\\ &-({\bf h}[{\bf H}^{a}+S_{\theta_{i}}{\bf
H}_{i}]\cdot\nabla)({\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i})+({\bf
H}^{a}+S_{\theta_{i}}{\bf H}_{i}){\rm div}({\bf v}[{\bf
u}^{a}+S_{\theta_{i}}{\bf u}_{i}]))\Big{\\}}\,,\end{split}$

where

$\begin{split}&{\bf v}[{\bf u}^{a}+{\bf u}_{i}]:=\Bigg{(}({\bf u}^{a}+{\bf
u_{i}})\cdot\Big{(}1,-\partial_{2}(\Psi^{a}+\Psi_{i})\Big{)},({\bf u}^{a}+{\bf
u}_{i})_{2}\partial_{1}(\Phi^{a}+\Psi_{i})\Bigg{)}\,,\\\ &{\bf w}[{\bf
u}^{a}+{\bf u}_{i}]:={\bf v}[{\bf u}^{a}+{\bf
u}_{i}]-\Big{(}\partial_{t}(\Psi^{a}+\Psi_{i}),0\Big{)}\,,\\\ &{\bf h}[{\bf
H}^{a}+{\bf H}_{i}]:=\Bigg{(}({\bf H}^{a}+{\bf
H_{i}})\cdot\Big{(}1,-\partial_{2}(\Psi^{a}+\Psi_{i})\Big{)},({\bf H}^{a}+{\bf
H}_{i})_{2}\partial_{1}(\Phi^{a}+\Psi_{i})\Bigg{)}\,.\\\ \end{split}$

The vectors ${\bf v}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]$, ${\bf w}[{\bf
u}^{a}+S_{\theta_{i}}{\bf u}_{i}]$, ${\bf h}[{\bf H}^{a}+S_{\theta_{i}}{\bf
H}_{i}]$ are defined by completely similar expressions with
$S_{\theta_{i}}{\bf u}_{i}$, $S_{\theta_{i}}{\bf H}_{i}$,
$S_{\theta_{i}}\Psi_{i}$ instead of ${\bf u}_{i}$, ${\bf H}_{i}$, $\Psi_{i}$.

We split the range of $s$ into small and large values (in order to take
advantage of the continuity estimates of $I-S_{\theta_{i}}$, see (9.2) and
(10.4)). For small values of $s$, by using (10.3), (10.4), the smallness
assumption (10.1) and Moser-type calculus inequalities we get

$||{\bf v}[{\bf u}^{a}+{\bf u}_{i}]||_{s,\ast,T}\leq
C\delta^{2}\begin{cases}\theta_{i}^{(s+2-\alpha)_{+}}\quad&s+2\not=\alpha,\\\
\theta_{i}\quad&s+2=\alpha,\end{cases}\qquad
s\in\\{6,\dots,\tilde{\alpha}-2\\},$ (10.34) $||{\bf v}[{\bf
u}^{a}+S_{\theta_{i}}{\bf u}_{i}]-{\bf v}[{\bf u}^{a}+{\bf
u}_{i}]||_{s,\ast,T}\leq C\delta\theta_{i}^{s+2-\alpha},\qquad
s\in\\{6,\dots,\tilde{\alpha}-2\\},\,\,\alpha\geq 9.$ (10.35)

Adding (10.34), (10.35) gives an estimate of ${\bf v}[{\bf
u}^{a}+S_{\theta_{i}}{\bf u}_{i}]$. We obtain similar estimates for ${\bf
w}[{\bf u}^{a}+{\bf u}_{i}],{\bf h}[{\bf H}^{a}+{\bf H}_{i}],{\bf w}[{\bf
u}^{a}+S_{\theta_{i}}{\bf u}_{i}]$ and ${\bf h}[{\bf H}^{a}+S_{\theta_{i}}{\bf
H}_{i}]$.

We decompose $\Delta_{1}$ as sum of terms with differences $I-S_{\theta_{i}}$
put in evidence. Making repeated use of estimates (9.1), (9.2), (10.3)–(10.5),
the smallness assumption (10.1), Moser-type calculus inequalities of Lemmata
3.1 , 3.2, we get for $\alpha\geq 9$

$||\Delta_{1}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+4-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\alpha-3\\}\,.$
(10.36)

For large values $s\geq{\alpha}-3$ we use (9.1), (10.5) to obtain

$||{\bf v}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]||_{s,\ast,T}\leq
C\delta^{2}\begin{cases}\theta_{i}^{(s+2-\alpha)_{+}}\quad&s+2\not=\alpha,\\\
\theta_{i}\quad&s+2=\alpha\end{cases}\qquad
s\in\\{6,\dots,\tilde{\alpha}+6\\},$ (10.37)

with similar estimates for ${\bf w}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]$
and ${\bf h}[{\bf H}^{a}+S_{\theta_{i}}{\bf H}_{i}]$. With a direct estimate
of the terms in $\Delta_{1}$ we extend (10.36) to the cases $s\geq{\alpha}-2$
and finally obtain

$||\Delta_{1}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+4-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+4\\}\,.$
(10.38)

In the above estimate (10.37) it is fundamental that $s\leq\tilde{\alpha}+6$.
This is used for the estimate (10.38) of $\Delta_{1}$, where the estimate of
the normal derivative of $\mathbf{v}[\cdot]$ in the anisotropic space
$H^{s}_{\ast}$, for $s\leq\tilde{\alpha}+4$, requires an estimate of
$\mathbf{v}[\cdot]$ in $H^{s+2}_{\ast}$.
As for $\Delta_{2}$, we have the explicit expression

$\begin{split}\Delta_{2}=&\frac{1}{\partial_{1}(\Phi^{a}+\Psi_{i+1/2})}\left\\{({\bf
w}^{i+1/2}\cdot\nabla)({\bf H}^{a}+S_{\theta_{i}}{\bf H}_{i})-({\bf h}[{\bf
H}^{a}+S_{\theta_{i}}{\bf H}_{i}]\cdot\nabla)(S_{\theta_{i}}{\bf u}_{i}-{\bf
u}_{i+1/2})\right.\\\ &\left.\qquad+({\bf H}^{a}+S_{\theta_{i}}{\bf
H}_{i}){\rm div}({\bf v}[S_{\theta_{i}}{\bf u}_{i}-{\bf
u}_{i+1/2}])\right\\}\,,\end{split}$

where

$\begin{split}&{\bf v}[S_{\theta_{i}}{\bf u}_{i}-{\bf u}_{i+1/2}]:={\bf
v}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]-{\bf v}[{\bf u}^{a}+{\bf
u}_{i+1/2}]=\\\ &=\Bigg{(}(S_{\theta_{i}}{\bf u_{i}}-{\bf
u}_{i+1/2})\cdot\Big{(}1,-\partial_{2}(\Psi^{a}+\Psi_{i+1/2})\Big{)},(S_{\theta_{i}}{\bf
u}_{i}-{\bf u}_{i+1/2})_{2}\partial_{1}(\Phi^{a}+\Psi_{i+1/2})\Bigg{)}\,,\\\
&{\bf w}^{i+1/2}:={\bf w}[{\bf u}^{a}+S_{\theta_{i}}{\bf u}_{i}]-{\bf w}[{\bf
u}^{a}+{\bf u}_{i+1/2}]={\bf v}[S_{\theta_{i}}{\bf u}_{i}-{\bf
u}_{i+1/2}]\,.\end{split}$ (10.39)

From the definition (10.39), using Moser-type calculus inequalities, (10.1),
(10.5), (10.15), and the estimate (10.30), we obtain

$||{\bf v}[S_{\theta_{i}}{\bf u}_{i}-{\bf u}_{i+1/2}]||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+1-\alpha},\qquad
s\in\\{6,\dots,\tilde{\alpha}+6\\},\,\,\alpha\geq 9.$ (10.40)

Again in (10.40) we need $s\leq\tilde{\alpha}+6$ in order to get the following
estimate (10.41) of $\Delta_{2}$, where the estimate of the normal derivative
of $\mathbf{v}[\cdot]$ in the anisotropic space $H^{s}_{\ast}$, for
$s\leq\tilde{\alpha}+4$, requires an estimate of $\mathbf{v}[\cdot]$ in
$H^{s+2}_{\ast}$.

Making repeated use of Moser-type calculus inequalities, (10.1), (10.5),
(10.15), the estimates (10.30) and (10.40), we get for $\alpha\geq 9$

$||\Delta_{2}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+3-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+4\\}\,.$
(10.41)

To estimate the last term of $F_{H}^{i+1/2}$ we write

$\begin{split}&S_{\theta_{i}}\mathbb{L}_{H}({\mathbf{u}}^{a}+\mathbf{u}_{i},{\mathbf{H}}^{a}+{\mathbf{H}}_{i},\Psi^{a}+\Psi_{i})=S_{\theta_{i}}\big{(}\mathcal{L}_{H}(\mathbf{u}_{i},{\mathbf{H}}_{i},\Psi_{i})+\mathbb{L}_{H}({\mathbf{u}}^{a},{\mathbf{H}}^{a},\Psi^{a})\big{)}\\\
&=S_{\theta_{i}}\big{(}\mathcal{L}_{H}(\mathbf{u}_{i},{\mathbf{H}}_{i},\Psi_{i})-\mathcal{F}^{a}_{H}\big{)}\\\
&=S_{\theta_{i}}\big{(}\mathcal{L}_{H}(\mathbf{u}_{i-1}+\delta\mathbf{u}_{i-1},{\mathbf{H}}_{i-1}+\delta{\mathbf{H}}_{i-1},\Psi_{i-1}+\delta\Psi_{i-1})-\mathcal{L}_{H}(\mathbf{u}_{i-1},{\mathbf{H}}_{i-1},\Psi_{i-1})\big{)}\\\
&\quad+S_{\theta_{i}}\big{(}\mathcal{L}_{H}(\mathbf{u}_{i-1},{\mathbf{H}}_{i-1},\Psi_{i-1})-\mathcal{F}^{a}_{H}\big{)},\end{split}$
(10.42)

where $\mathcal{L}_{H},\mathcal{F}^{a}_{H}$ are the $\mathbf{H}$-component in
(8.11), (8.10), respectively. From (9.1), $(H_{i-1})(b)$ we readily obtain

$||S_{\theta_{i}}\big{(}\mathcal{L}_{H}(\mathbf{u}_{i-1},{\mathbf{H}}_{i-1},\Psi_{i-1})-\mathcal{F}^{a}_{H}\big{)}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s-\alpha-1}\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}-2\\}\,.$
(10.43)

For the first difference in (10.42) let us denote

$\begin{split}\Delta_{3}:=&\mathcal{L}_{H}(\mathbf{u}_{i-1}+\delta\mathbf{u}_{i-1},{\mathbf{H}}_{i-1}+\delta{\mathbf{H}}_{i-1},\Psi_{i-1}+\delta\Psi_{i-1})\\\
&-\mathcal{L}_{H}(\mathbf{u}_{i-1},{\mathbf{H}}_{i-1},\Psi_{i-1}).\end{split}$

Hence we have

$\begin{split}||S_{\theta_{i}}\Delta_{3}||_{s,\ast,T}\leq
C\theta_{i}^{s-6}||\Delta_{3}||_{6,\ast,T}\leq
C\theta_{i}^{s-6}\cdot\delta\theta_{i}^{8-\alpha}=C\delta\theta_{i}^{s+2-\alpha},\quad\mbox{for}\,\,\,s\geq
6,\end{split}$ (10.44)

where in particular for the estimate of $||\Delta_{3}||_{6,\ast,T}$ we have
used $(H_{i-1})(a)$, (10.3), (10.34) and

$||{\bf v}[{\bf u}^{a}+\mathbf{u}_{i-1}+\delta\mathbf{u}_{i-1}]-{\bf v}[{\bf
u}^{a}+{\bf u}_{i-1}]||_{s,\ast,T}\leq
C\delta\theta_{i}^{s-\alpha},\qquad\,\,\,s\in\\{6,\dots,\tilde{\alpha}-2\\}\,,$

used in the cases $s=6$, $s=8$ and a similar estimate for ${\bf w}[{\bf
u}^{a}+\mathbf{u}_{i-1}+\delta\mathbf{u}_{i-1}]-{\bf w}[{\bf u}^{a}+{\bf
u}_{i-1}]$ in case $s=6$. Collecting (10.38), (10.41), (10.43), (10.44) we get

$||F_{H}^{i+1/2}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+4-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+4\\}\,.$
(10.45)

Equation (10.32) solved by $\mathbf{H}_{i+1/2}-S_{\theta_{i}}\mathbf{H}_{i}$
has the form

$\partial_{t}Y+\sum_{j=1}^{2}\mathcal{D}_{j}(b)\partial_{j}Y+\mathcal{Q}(b)Y=F_{H}^{i+1/2},$
(10.46)

for $Y=\mathbf{H}_{i+1/2}-S_{\theta_{i}}\mathbf{H}_{i}$,
$b=({\mathbf{u}}^{a}+\mathbf{u}_{i+1/2},\Psi^{a}+\Psi_{i+1/2})$, and where
$\mathcal{D}_{j}$ and $\mathcal{Q}$ are some matrices. The matrices
$\mathcal{D}_{j}$ are diagonal and, more important, $\mathcal{D}_{1}$ vanishes
at the boundary. This yelds that system (10.46) does not need any boundary
condition. A standard energy argument applied to (10.46) gives in view of
(10.45)

$||\mathbf{H}_{i+1/2}-S_{\theta_{i}}\mathbf{H}_{i}||_{s,\ast,T}\leq
C||F_{H}^{i+1/2}||_{s,\ast,T}\leq
C\delta\theta_{i}^{s+4-\alpha}\,,\quad\mbox{for}\,\,\,s\in\\{6,\dots,\tilde{\alpha}+4\\}\,.$
(10.47)

Collecting (10.30), (10.47) gives (10.17).

Using (8.9), (10.5), (10.17) for $s=6$, and taking $\delta>0$ sufficiently
small we have that $\varphi^{a}+\psi_{i+\frac{1}{2}}$ and
$\tilde{{\mathbf{U}}}^{a}+\mathbf{V}_{i+\frac{1}{2}}$ are sufficiently small.
Then, $\varphi^{a}+\psi_{i+\frac{1}{2}}$ satisfies (4.7) and, recalling Remark
3.1, ${{\mathbf{U}}}^{a}+\mathbf{V}_{i+\frac{1}{2}}$ satisfies (4.2), (4.4),
(4.6), (6.1). ${{\mathbf{H}}}^{a}+\mathbf{H}_{i+\frac{1}{2}}$ satifies (4.8)
by construction, see (10.31); the initial value at $t=0$ of
${{\mathbf{H}}}^{a}+\mathbf{H}_{i+\frac{1}{2}}$ satisfies (4.10) since
${{\mathbf{H}}}^{a}_{|t=0}={\mathbf{H}}_{0}$ satisfies (4.10) by assumption,
and ${{\mathbf{H}}}_{i+\frac{1}{2}}=0$ for $t\leq 0$ by continuity. In
conclusion, ${{\mathbf{U}}}^{a}+\mathbf{V}_{i+\frac{1}{2}}$ satisfies all the
constraints (4.2)–(4.8), (4.10) and (6.1) for the background state. ∎

####  10.2.4. Estimate of the second substitution errors

In the following Lemma, we can estimate the second substitution errors
$e^{\prime\prime\prime}_{k},\tilde{e}^{\prime\prime\prime}_{k}$ of the
iterative scheme. We define

$e^{\prime\prime\prime}_{k}:=\mathcal{L}^{\prime}(S_{\theta_{k}}{\mathbf{V}}_{k},S_{\theta_{k}}\Psi_{k})(\delta{\mathbf{V}}_{k},\delta\Psi_{k})-\mathcal{L}^{\prime}({\mathbf{V}}_{k+\frac{1}{2}},\Psi_{k+\frac{1}{2}})(\delta{\mathbf{V}}_{k},\delta\Psi_{k}),$
(10.48)
$\begin{split}\tilde{e}^{\prime\prime\prime}_{k}:=&\mathcal{B}^{\prime}(S_{\theta_{k}}{\mathbf{V}}_{k}|_{x_{1}=0},S_{\theta_{k}}\psi_{k})((\delta{\mathbf{V}}_{k})|_{x_{1}=0},\delta\psi_{k})\\\
&-\mathcal{B}^{\prime}({\mathbf{V}}_{k+\frac{1}{2}}|_{x_{1}=0},\psi_{k+\frac{1}{2}})((\delta{\mathbf{V}}_{k})|_{x_{1}=0},\delta\psi_{k}).\\\
\end{split}$ (10.49)

We can write (10.48) and (10.49) as follows:

$\begin{split}e^{\prime\prime\prime}_{k}&=\int^{1}_{0}\mathbb{L}^{\prime\prime}(\mathbf{U}^{a}+{\mathbf{V}}_{k+\frac{1}{2}}+\tau(S_{\theta_{k}}{\mathbf{V}}_{k}-{\mathbf{V}}_{k+\frac{1}{2}}),\Psi^{a}+S_{\theta_{k}}\Psi_{k})\\\
&\quad((\delta{\mathbf{V}}_{k},\delta\Psi_{k}),(S_{\theta_{k}}{\mathbf{V}}_{k}-{\mathbf{V}}_{k+\frac{1}{2}},0))d\tau,\\\
&\tilde{e}^{\prime\prime\prime}_{k}=\mathbb{B}^{\prime\prime}((\delta{\mathbf{V}}_{k}|_{x_{1}=0},\delta\psi_{k}),((S_{\theta_{k}}{\mathbf{V}}_{k}-{\mathbf{V}}_{k+\frac{1}{2}})|_{x_{1}=0},0)).\end{split}$

######  Lemma 10.7.

Let $\alpha\geq 10$. There exist $\delta>0$ sufficiently small and
$\theta_{0}\geq 1$ sufficiently large such that, for all
$k\in\\{0,\cdots,i-1\\}$ and for all integers
$s\in\\{6,\cdots,\tilde{\alpha}-2\\},$ we have

$||e^{\prime\prime\prime}_{k}||_{s,\ast,T}\lesssim\delta^{2}\theta^{L_{3}(s)-1}_{k}\Delta_{k},$
(10.50)
$||\tilde{e}^{\prime\prime\prime}_{k}||_{H^{s}(\Gamma_{T})}\lesssim\delta^{2}\theta^{L_{3}(s)-1}_{k}\Delta_{k},$
(10.51)

where $L_{3}(s):=\max\\{(s+2-\alpha)_{+}+16-2\alpha;s+12-2\alpha\\}.$

###### Proof.

Using (10.17) and Lemma 10.3, similar to the proof of Lemmata 10.4 and 10.5,
we obtain (10.50) and (10.51). Here, we can calculate the explicit form of
$\tilde{e}^{\prime\prime\prime}_{k}$ as

$\tilde{e}^{\prime\prime\prime}_{k}:=\left[\begin{array}[]{c}0\\\ 0\\\ \delta
H^{+}_{k}\cdot(S_{\theta_{k}}H^{+}_{k}-H^{+}_{k+\frac{1}{2}})-\delta
H^{-}_{k}\cdot(S_{\theta_{k}}H^{-}_{k}-H^{-}_{k+\frac{1}{2}})\end{array}\right]\,.$
(10.52)

∎

####  10.2.5. Estimate of the last error term

We now estimate the last error term (9.11):

$D_{k+\frac{1}{2}}\delta\Psi_{k}=\frac{\delta\Psi_{k}}{\partial_{1}(\Phi^{a}+\Psi_{k+\frac{1}{2}})}R_{k},$
(10.53)

where
$R_{k}:=\partial_{1}[\mathbb{L}({\mathbf{U}}^{a}+{\mathbf{V}}_{k+\frac{1}{2}},\Psi^{a}+\Psi_{k+\frac{1}{2}})].$
It is noted that

$|\partial_{1}(\Phi^{a}+\Psi_{k+\frac{1}{2}})|=|\pm
1+\partial_{1}(\Psi^{a}+\Psi_{k+\frac{1}{2}})|\geq\frac{1}{2},$

for $\delta>0$ sufficiently small.
The following Lemma 10.8 can be proved by direct calculations.

######  Lemma 10.8.

Let $\alpha\geq 10,\tilde{\alpha}\geq\alpha-4.$ There exist $\delta>0$
sufficiently small and $\theta_{0}\geq 1$ sufficiently large, such that, for
all $k\in\\{0,\cdots,i-1\\}$ and for all integers
$s\in\\{6,\cdots,\tilde{\alpha}-2\\}$, we have

$||D_{k+\frac{1}{2}}\delta\Psi_{k}||_{s,\ast,T}\lesssim\delta^{2}\theta^{L_{4}(s)-1}_{k}\Delta_{k},$
(10.54)

where $L_{4}(s):=s+14-2\alpha$.

###### Proof.

Using (3.7) and (3.9), we obtain that

$\begin{split}||D_{k+\frac{1}{2}}\delta\Psi_{k}||_{s,\ast,T}&\lesssim||\delta\Psi_{k}||_{4,\ast,T}||R_{k}||_{s,\ast,T}+||\delta\Psi_{k}||_{s,\ast,T}||R_{k}||_{4,\ast,T}\\\
&\quad+||\delta\Psi_{k}||_{4,\ast,T}||R_{k}||_{4,\ast,T}||\Psi^{a}+\Psi_{k+\frac{1}{2}}||_{s+2,\ast,T}.\end{split}$
(10.55)

Using (8.10) and (8.11), we can write

$R_{k}=\partial_{1}(\mathcal{L}({\mathbf{V}}_{k},\Psi_{k})-\mathcal{F}^{a}+\mathcal{I}),\text{
for }t>0,$ (10.56)

where

$\mathcal{I}:=\int^{1}_{0}\mathbb{L}^{\prime}(\mathbf{U}^{a}+{\mathbf{V}}_{k}+\tau({\mathbf{V}}_{k+\frac{1}{2}}-{\mathbf{V}}_{k}),\Psi^{a}+\Psi_{k}+\tau(\Psi_{k+\frac{1}{2}}-\Psi_{k}))({\mathbf{V}}_{k+\frac{1}{2}}-{\mathbf{V}}_{k},\Psi_{k+\frac{1}{2}}-\Psi_{k})d\tau.$

If $s\in\\{4,\cdots,\tilde{\alpha}-4\\},$ then using Hypothesis $(H_{i-1}),$
we obtain that

$||\mathcal{L}({\mathbf{V}}_{k},\Psi_{k})-\mathcal{F}^{a}||_{s+2,\ast,T}\leq
2\delta\theta^{s+1-\alpha}_{k}.$ (10.57)

Using (10.3),(10.4),(10.15)-(10.17), (10.56),(10.57), Sobolev inequalities
(3.10) and Moser-type calculus inequalities in Lemma 3.2, we have

$||R_{k}||_{s,\ast,T}\lesssim\delta(\theta^{s+8-\alpha}_{k}+\theta^{(s+4-\alpha)_{+}+10-\alpha}_{k}),\text{
for }s\in\\{4,\cdots,\tilde{\alpha}-4\\}.$ (10.58)

If $s\in\\{\tilde{\alpha}-3,\tilde{\alpha}-2\\}$, then, for
$\tilde{\alpha}\geq\alpha-4$, using (10.3), (10.5) and (10.15)-(10.17), we can
deduce directly from (10.56) that

$||R_{k}||_{s,\ast,T}\lesssim\delta\theta^{s+8-\alpha}_{k}.$

Therefore, (10.58) holds for all $s\in\\{4,\cdots,\tilde{\alpha}-2\\}.$ Using
Hypothesis $(H_{i-1}),$ (10.58), (10.5), (10.3) and (10.15)-(10.17) into
(10.55), we can obtain (10.54). ∎

Using Lemmata 10.4 \- 10.8, we can conclude the following estimates of the
error terms $e_{k},\tilde{e}_{k},$ defined by (9.12).

######  Lemma 10.9.

Let $\alpha\geq 10,\tilde{\alpha}\geq\alpha-4.$ There exist $\delta>0$
sufficiently small and $\theta_{0}\geq 1$ sufficiently large, such that for
all $k\in\\{0,\cdots,i-1\\}$ and for all integers
$s\in\\{6,\cdots,\tilde{\alpha}-2\\}$, we have

$||e_{k}||_{s,\ast,T}+||\tilde{e}_{k}||_{H^{s}(\Gamma_{T})}\lesssim\delta^{2}\theta^{L_{4}(s)-1}_{k}\Delta_{k},$
(10.59)

where $L_{4}(s)$ is defined in Lemma 10.8.

From Lemma 10.9, we obtain the estimate of the accumulated errors
$E_{i},\tilde{E}_{i},\hat{E}_{i},$ which are defined in (9.13).

######  Lemma 10.10.

Let $\alpha\geq 16,\tilde{\alpha}=\alpha+5.$ There exist $\delta>0$
sufficiently small and $\theta_{0}\geq 1$ sufficiently large, such that

$||E_{i}||_{\alpha+3,\ast,T}+||\tilde{E}_{i}||_{H^{\alpha+3}(\Gamma_{T})}\lesssim\delta^{2}\theta_{i}.$
(10.60)

###### Proof.

Using $L_{4}(\alpha+3)\leq 1$ if $\alpha\geq 16$, it follows from (10.59) that

$||E_{i}||_{\alpha+3,\ast,T}\lesssim\sum^{i-1}_{k=0}||e_{k}||_{\alpha+3,\ast,T}\lesssim\sum^{i-1}_{k=0}\delta^{2}\Delta_{k}\lesssim\delta^{2}\theta_{i}\,,$

if $\alpha+3\leq\tilde{\alpha}-2$. Similar arguments also hold for
$||\tilde{E}_{i}||_{H^{\alpha+3}(\Gamma_{T})}$. The minimal possible
$\tilde{\alpha}$ is $\alpha+5$. ∎

###  10.3. Convergence of the iteration scheme

We still need to estimate the source terms $f_{i},g_{i}.$

######  Lemma 10.11.

Let $\alpha\geq 16$ and $\tilde{\alpha}=\alpha+5$. There exist $\delta>0$
sufficiently small and $\theta_{0}\geq 1$ sufficiently large, such that for
all integers $s\in\\{6,\cdots,\tilde{\alpha}+2\\},$

$||f_{i}||_{s,\ast,T}\lesssim\Delta_{i}\Big{(}\theta^{s-\alpha-3}_{i}(||\mathcal{F}^{a}||_{\alpha+2,\ast,T}+\delta^{2})+\delta^{2}\theta^{L_{4}(s)-1}_{i}\Big{)},$
(10.61)
$||g_{i}||_{H^{s}(\Gamma_{T})}\lesssim\delta^{2}\Delta_{i}\Big{(}\theta^{s-\alpha-3}_{i}+\theta^{L_{4}(s)-1}_{i}\Big{)}\,.$
(10.62)

In the above inequalities we need the exponent $s-\alpha-3$ of $\theta_{i}$ to
compensate the loss of 2 derivatives for the data in (10.64), in order to
recover the exponent $s-\alpha-1$ in the corresponding terms of (10.68).

###### Proof.

As in [14, 34, 39], using (9.1)-(9.3), (9.14), (10.59), (10.60), we obtain
that

$\begin{split}||f_{i}||_{s,\ast,T}&\leq||(S_{\theta_{i}}-S_{\theta_{i-1}})\mathcal{F}^{a}-(S_{\theta_{i}}-S_{\theta_{i-1}})E_{i-1}-S_{\theta_{i}}e_{i-1}||_{s,\ast,T}\\\
&\lesssim\Delta_{i}\theta^{s-\alpha-3}_{i}(||\mathcal{F}^{a}||_{\alpha+2,\ast,T}+\theta^{-1}_{i}||E_{i-1}||_{\alpha+3,\ast,T})+||S_{\theta_{i}}e_{i-1}||_{s,\ast,T}\\\
&\lesssim\Delta_{i}\\{\theta^{s-\alpha-3}_{i}(||\mathcal{F}^{a}||_{\alpha+2,\ast,T}+\delta^{2})+\delta^{2}\theta^{L_{4}(s)-1}_{i}\\}.\end{split}$

Using (10.59) and (10.60) we can obtain (10.62). ∎

Similar to the proof of [14, 34, 39], we can obtain the estimate of
$(\delta{\mathbf{V}}_{i},\delta\Psi_{i})$ by (10.17) and the tame estimate
(7.1) applied to problem (9.6).

######  Lemma 10.12.

Let $\alpha\geq 16$ and $\tilde{\alpha}=\alpha+5$. If $\delta>0$ and
$||\mathcal{F}^{a}||_{\alpha+2,\ast,T}/\delta$ are sufficiently small and
$\theta_{0}\geq 1$ is sufficiently large, then for all integers
$s\in\\{6,\cdots,\tilde{\alpha}\\}$,

$||(\delta{\mathbf{V}}_{i},\delta\Psi_{i})||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}\leq\delta\theta^{s-\alpha-1}_{i}\Delta_{i}.$
(10.63)

###### Proof.

Let us consider problem (9.6) that will be solved by applying Theorem 7.1. We
first notice that
$({\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}},\varphi^{a}+\psi_{i+\frac{1}{2}})$
satisfy the constraints (4.2)–(4.8), (4.10), (6.1). Thus we may apply our tame
estimate (7.1) and obtain

$\begin{split}&||\delta\dot{{\mathbf{V}}_{i}}||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}\\\
&\leq C\Big{(}||{f}_{i}||_{s+2,\ast,T}+||{g}_{i}||_{H^{s+2}(\Gamma_{T})}\\\
&\quad+(||{f}_{i}||_{8,\ast,T}+||{g}_{i}||_{H^{8}(\Gamma_{T})})||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\nabla(\Psi^{a}+\Psi_{i+\frac{1}{2}}))||_{s+4,\ast,T}\Big{)}.\end{split}$
(10.64)

On the other hand, from (9.7) it follows

$\begin{split}&||\delta{{\mathbf{V}}_{i}}||_{s,\ast,T}\leq||\delta\dot{{\mathbf{V}}_{i}}||_{s,\ast,T}+C||\delta{{\mathbf{\Psi}}_{i}}||_{s,\ast,T}||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})||_{6,\ast,T}\\\
&\quad\qquad+C||\delta{{\mathbf{\Psi}}_{i}}||_{4,\ast,T}||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})||_{s+2,\ast,T}.\end{split}$
(10.65)

From

$||\delta{{\mathbf{\Psi}}_{i}}||_{s,\ast,T}\leq||\delta\psi_{i}||_{H^{s}(\Gamma_{T})},$
(10.66)

(10.64) for $s=4$ and (6.1) we have

$\begin{split}&||\delta{{\mathbf{\Psi}}_{i}}||_{4,\ast,T}\leq||\delta\psi_{i}||_{H^{4}(\Gamma_{T})}\leq
C\Big{(}||{f}_{i}||_{6,\ast,T}+||{g}_{i}||_{H^{6}(\Gamma_{T})}\\\
&\quad+(||{f}_{i}||_{8,\ast,T}+||{g}_{i}||_{H^{8}(\Gamma_{T})})||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\nabla(\Psi^{a}+\Psi_{i+\frac{1}{2}}))||_{8,\ast,T}\Big{)}\\\
&\leq C(||{f}_{i}||_{8,\ast,T}+||{g}_{i}||_{H^{8}(\Gamma_{T})}).\end{split}$

Then, from (10.64) and (10.65) we obtain

$\begin{split}&||\delta{{\mathbf{V}}_{i}}||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}\\\
&\leq C\Big{(}||{f}_{i}||_{s+2,\ast,T}+||{g}_{i}||_{H^{s+2}(\Gamma_{T})}\\\
&\quad+(||{f}_{i}||_{8,\ast,T}+||{g}_{i}||_{H^{8}(\Gamma_{T})})||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\nabla(\Psi^{a}+\Psi_{i+\frac{1}{2}}))||_{s+4,\ast,T}\Big{)}\\\
&\quad+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\Psi^{a}+\Psi_{i+\frac{1}{2}})||_{6,\ast,T}.\end{split}$

Using (8.9), (10.5) and (10.17) for $s=6$, taking $\delta>0$ sufficiently
small, we can absorb the last term in the right-hand side above into the left-
hand side to get

$\begin{split}&||\delta{{\mathbf{V}}_{i}}||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}\\\
&\leq C\Big{(}||{f}_{i}||_{s+2,\ast,T}+||{g}_{i}||_{H^{s+2}(\Gamma_{T})}\\\
&\quad+(||{f}_{i}||_{8,\ast,T}+||{g}_{i}||_{H^{8}(\Gamma_{T})})||(\tilde{\mathbf{U}}^{a}+{\mathbf{V}}_{i+\frac{1}{2}},\nabla(\Psi^{a}+\Psi_{i+\frac{1}{2}}))||_{s+4,\ast,T}\Big{)}.\end{split}$
(10.67)

The remaining part of the work is to estimate the right-hand side of (10.67).
Using Lemma 10.11, (10.5), Lemma 10.6 and (10.66), (10.67) becomes

$\begin{split}&||(\delta{{\mathbf{V}}_{i}},\delta{{\mathbf{\Psi}}_{i}})||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})}\\\
&\leq
C\Delta_{i}\Big{(}\theta_{i}^{s-\alpha-1}(||\mathcal{F}^{a}||_{\alpha+2,\ast,T}+\delta^{2})+\delta^{2}\theta_{i}^{L_{4}(s+2)-1}\Big{)}\\\
&\quad+C\Delta_{i}\Big{(}\theta_{i}^{5-\alpha}(||\mathcal{F}^{a}||_{\alpha+2,\ast,T}+\delta^{2})+\delta^{2}\theta_{i}^{21-2\alpha}\Big{)}\Big{(}\delta\theta_{i}^{s+8-\alpha}+\delta\theta_{i}^{(s+6-\alpha)_{+}}\Big{)}.\end{split}$
(10.68)

One checks that, for $\alpha\geq 16$ and $6\leq s\leq\tilde{\alpha}$, the
following inequalities hold true:

$\begin{split}&L_{4}(s+2)\leq s-\alpha,\qquad 5-\alpha+(s+6-\alpha)_{+}\leq
s-\alpha-1,\\\ &s+13-2\alpha\leq s-\alpha-1,\qquad 21-2\alpha+s+8-\alpha\leq
s-\alpha-1,\\\ &21-2\alpha+(s+6-\alpha)_{+}\leq s-\alpha-1.\end{split}$

From (10.68) we thus obtain (10.63), provided $\delta>0$ and
$||\mathcal{F}^{a}||_{\alpha+2,\ast,T}/\delta$ are sufficiently small and
$\theta_{0}\geq 1$ is sufficiently large. ∎

Finally, similar to the proof of [14, 34, 39], we can obtain the remaining
inequalities in $(H_{i}).$

######  Lemma 10.13.

Let $\alpha\geq 16$. If $\delta>0$ and
$||\mathcal{F}^{a}||_{\alpha+2,\ast,T}/\delta$ are sufficiently small and if
$\theta_{0}\geq 1$ is sufficiently large, then for all integers
$s\in\\{6,\cdots,\tilde{\alpha}-2\\}$

$||\mathcal{L}({\mathbf{V}}_{i},\Psi_{i})-\mathcal{F}^{a}||_{s,\ast,T}\leq
2\delta\theta^{s-\alpha-1}_{i}.$ (10.69)

Moreover, for all integers $s\in\\{6,\cdots,\tilde{\alpha}-2\\}$

$||\mathcal{B}({\mathbf{V}}_{i}|_{x_{1}=0},\Psi_{i})||_{H^{s}(\Gamma_{T})}\leq\delta\theta^{s-\alpha-1}_{i}.$
(10.70)

###### Proof.

Recall that, by summing the relations (9.9), we have

$\mathcal{L}({\mathbf{V}}_{i},\Psi_{i})-\mathcal{F}^{a}=(S_{\theta_{i-1}}-I)\mathcal{F}^{a}+(I-S_{\theta_{i-1}})E_{i-1}+e_{i-1}.$
(10.71)

The proof of (10.69) then follows by applying (9.2), (10.59), (10.60),
provided that $\delta>0$ and $||\mathcal{F}^{a}||_{\alpha+2,\ast,T}/\delta$
are sufficiently small and $\theta_{0}\geq 1$ is sufficiently large. The proof
of (10.70) is similar. ∎

We are now in the position to prove the main theorem for the existence of the
solution to the nonlinear problem (2.1).

Proof of Theorem 3.1: Let the initial data
$({\mathbf{U}}^{\pm}_{0},\varphi_{0})$ satisfy all the assumptions of Theorem
3.1. Let $\alpha=m+1\geq 16$, $\tilde{\alpha}=\alpha+5$, $\mu=m+10$. Then the
initial data ${\mathbf{U}}^{\pm}_{0}\in H^{\mu+1.5}(\mathbb{R}^{2}_{+})$ and
$\varphi_{0}\in H^{\mu+1.5}(\mathbb{R})$ are compatible up to order $\mu$ and
there exists an approximate solution $(\tilde{\mathbf{U}}^{a},\varphi^{a})\in
H^{\mu+2}(\Omega_{T})\times H^{\mu+2}(\Gamma_{T})$ to problem (2.1). Observe
that $\mu+2=\tilde{\alpha}+6$ as required in (10.1). We choose $\delta>0$,
$T>0$ sufficiently small, $\theta_{0}\geq 1$ sufficiently large as in the
previous lemmata. We also assume $T>0$ small enough so that
$||\mathcal{F}^{a}||_{\alpha+2,\ast,T}/\delta$ is sufficiently small. Then in
view of Lemmata 10.1, 10.12, 10.13, property ($H_{i}$) holds for all integers
$i$. In particular, we have

$\sum^{\infty}_{i=0}(||(\delta{\mathbf{V}}_{i},\delta\Psi_{i})||_{s,\ast,T}+||\delta\psi_{i}||_{H^{s}(\Gamma_{T})})\lesssim\sum^{\infty}_{i=0}\theta^{s-\alpha-2}_{i}<\infty,\quad\text{for}\,\,\,6\leq
s\leq\alpha-1.$

Thus, the sequence $({\mathbf{V}}_{i},\Psi_{i})$ converges to some limit
$({\mathbf{V}},\Psi)$ in $H^{\alpha-1}_{\ast}(\Omega_{T}),$ and sequence
$\psi_{i}$ converges to some limit $\psi$ in $H^{\alpha-1}(\Gamma_{T}).$
Passing to the limit in (10.69) and (10.70) for $s=m=\alpha-1$, we obtain
(8.11). Therefore,
$({\mathbf{U}},\Phi)=({\mathbf{U}}^{a}+{\mathbf{V}},\Phi^{a}+\Psi)$ is a
solution on $\Omega_{T}$ of nonlinear system (2.1).

##  Appendix A Trace theorem in anisotropic space

In this Appendix let us recall the following trace theorem in the anisotropic
space by Ohno, Shizuta, Yanagisawa [27].

######  Theorem A.1 ([27]).

Let $s\geq 2$ be an integer. Then the mapping

$C_{0}^{\infty}(\overline{\mathbb{R}^{2}_{+}})\ni
u\mapsto\\{\partial^{j}_{1}u|_{x_{1}=0}\,,\,\,j=0,\dots[s/2]-1\\}\in\underbrace{C^{\infty}_{0}(\mathbb{R})\times\dots
C^{\infty}_{0}(\mathbb{R})}_{[s/2]\,\,\,\mbox{times}}$

extends by continuity to a continuous linear mapping of

$H^{s}_{\ast}(\mathbb{R}^{2}_{+})\rightarrow\prod\limits_{j=0}^{[s/2]-1}H^{s-2j-1}(\mathbb{R})\,.$

This mapping is surjective and there exists a continuous linear right-inverse

$(v_{0},\dots,v_{[s/2]-1})\mapsto\mathcal{R}(v_{0},\dots,v_{[s/2]-1})$

of

$\prod\limits_{j=0}^{[s/2]-1}H^{s-2j-1}(\mathbb{R})\rightarrow
H^{s}_{\ast}(\mathbb{R}^{2}_{+})\,$

such that

$\partial^{j}_{1}(\mathcal{R}(v_{0},\dots,v_{[s/2]-1}))|_{x_{1}=0}=v_{j}\,,\quad
j=0,\dots,[s/2]-1\,.$

##  Appendix B Proof of Theorem 5.1

Here we focus only on the proof of the energy estimate (5.33); the existence
and uniqueness of the solution can be shown by standard methods.

Let ${\mathbf{V}}={\mathbf{V}}(t,{\bf
x})=({\mathbf{V}}^{+},{\mathbf{V}}^{-})$, $\varphi=\varphi(t,x_{2})$, where
${\mathbf{V}}^{\pm}:=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{u}_{2}^{\pm},\dot{H}^{\pm}_{n},\dot{H}^{\pm}_{2},\dot{S}^{\pm})$,
be sufficiently smooth vector fields respectively on $\Omega_{T}$ and
$\Gamma_{T}$, satisfying the linear system (5.3), or its equivalent form
(5.4), together with the boundary and the “initial” conditions from (4.31),
that is

$\partial_{t}\varphi+\hat{u}^{\pm}_{2}\partial_{2}\varphi-\dot{u}^{\pm}_{N}\mp\varphi\partial_{1}\hat{u}^{\pm}_{N}=0\,,\quad\dot{q}^{+}-\dot{q}^{-}+\varphi[\partial_{1}\hat{q}]=0\quad\mbox{on}\,\,\,\Gamma_{T}\,,$
(B.1) $({\mathbf{V}},\varphi)=(0,0)\quad\mbox{for}\,\,\,t<0\,.$ (B.2)

Recall from Section 4 that $\hat{\Psi}^{\pm}=\hat{\Psi}^{\pm}(t,{\bf x})$ are
defined through the basic front function
$\hat{\varphi}=\hat{\varphi}(t,x_{2})$ by

$\hat{\Psi}^{\pm}(t,{\bf x}):=\chi(\pm
x_{1})\hat{\varphi}(t,x_{2})\,,\quad\forall\,{\bf
x}\in\mathbb{R}^{2}_{+}\,\,\,t\in(-\infty,T]\,,$

where $\chi\in C^{\infty}_{0}({\mathbb{R}})$ satisfies $\chi\equiv 1$ on
$[-1,1]$, so that $\partial_{1}\hat{\Psi}^{\pm}|_{x_{1}=0}=0$. We set

$\dot{u}^{\pm}_{n}:=\dot{u}_{1}^{\pm}-\partial_{2}\hat{\Psi}^{\pm}\dot{u}^{\pm}_{2}\,,\quad\dot{H}^{\pm}_{n}:=\dot{H}_{1}^{\pm}-\partial_{2}\hat{\Psi}^{\pm}\dot{H}^{\pm}_{2}\,;$

notice in particular

$\dot{u}^{\pm}_{n}\equiv\dot{u}^{\pm}_{N}:=\dot{u}_{1}^{\pm}-\partial_{2}\hat{\varphi}\dot{u}^{\pm}_{2}\,,\quad\dot{H}^{\pm}_{n}\equiv\dot{H}^{\pm}_{N}:=\dot{H}_{1}^{\pm}-\partial_{2}\hat{\varphi}\dot{H}^{\pm}_{2}\,,\quad\mbox{on}\,\,\,\Gamma_{T}\,.$

Recall that $({\mathbf{V}},\varphi)$ as above must also satisfy the interior
and boundary constraints

$\mathrm{div}\dot{\mathbf{h}}^{\pm}=0\text{ in
}\Omega_{T}\quad\mbox{and}\quad\hat{H}^{\pm}_{2}\partial_{2}\varphi-\dot{H}^{\pm}_{N}\mp\varphi\partial_{1}\hat{H}^{\pm}_{N}=0\text{
on }\Gamma_{T}\,,$ (B.3)

where

$\dot{\mathbf{h}}^{\pm}:=(\dot{H}^{\pm}_{n},\partial_{1}\hat{\Phi}^{\pm}\dot{H}^{\pm}_{2})\quad\mbox{and}\quad\hat{\Phi}^{\pm}(t,{\bf
x}):=\pm x_{1}+\hat{\Psi}^{\pm}(t,{\bf x})\,.$

In the forthcoming calculations we will make use of the following shortcut
notation

$\displaystyle
I(t):=\|{\mathbf{V}}^{+}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\|{\mathbf{V}}^{-}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}\,,$
(B.4) $\displaystyle
I_{0}(t):=\|\partial_{t}{\mathbf{V}}^{+}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\|\partial_{t}{\mathbf{V}}^{-}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}\,,$
(B.5)
$\displaystyle\begin{split}I_{1,n}(t):=\|\partial_{1}{\mathbf{V}}^{+}_{n}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\|\partial_{1}{\mathbf{V}}^{-}_{n}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2},\end{split}$
(B.6) $\displaystyle
I_{\sigma}(t):=\|\sigma\partial_{1}{\mathbf{V}}^{+}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\|\sigma\partial_{1}{\mathbf{V}}^{-}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}\,,$
(B.7) $\displaystyle
I_{2}(t):=\|\partial_{2}{\mathbf{V}}^{+}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\|\partial_{2}{\mathbf{V}}^{-}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}\,,$
(B.8)

and we set

$I_{1,\ast}(t):=I(t)+I_{0}(t)+I_{\sigma}(t)+I_{2}(t)\,.$ (B.9)

In (B.6), ${\mathbf{V}}^{\pm}_{n}$ denote the noncharacteristic part of the
solution ${\mathbf{V}}^{\pm}$, that is

${\mathbf{V}}^{\pm}_{n}=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{H}^{\pm}_{n})\,.$

For all integers $m\geq 0$, we will also write $C_{m}$ to denote a generic
positive constant depending (nonlinearly) on
$\|\hat{\mathbf{U}}^{\pm}\|_{W^{m,\infty}(\Omega_{T})}$ and
$\|\nabla_{t,x_{2}}\hat{\varphi}\|_{W^{m,\infty}(\Gamma_{T})}$ and the
positive number $k$ from (4.2) and (4.4), that is

$C_{m}=C_{m}(\|\hat{\mathbf{U}}^{\pm}\|_{W^{m,\infty}(\Omega_{T})},\|\nabla_{t,x_{2}}\hat{\varphi}\|_{W^{m,\infty}(\Gamma_{T})},k)\,,$

that might possibly be different from an occurrence to another even within the
same sequence of inequalities.

From the notation above and in view of (B.2), it straightforwardly turns out
that

$\begin{split}&\frac{1}{C_{1}}\|\dot{\mathbf{U}}(t)\|_{H^{1}_{\ast}(\mathbb{R}^{2}_{+})}^{2}\leq
I(t)+I_{\sigma}(t)+I_{2}(t)\leq
C_{1}\|\dot{\mathbf{U}}(t)\|_{H^{1}_{\ast}(\mathbb{R}^{2}_{+})}^{2}\,,\\\
&\frac{1}{C_{1}}|||\dot{\mathbf{U}}(t)|||_{1,\ast}^{2}\leq I_{1,\ast}(t)\leq
C_{1}|||\dot{\mathbf{U}}(t)|||_{1,\ast}^{2}\,,\\\
&\frac{1}{C_{1}}\|\dot{\mathbf{U}}\|_{1,\ast,t}^{2}\leq\int_{0}^{t}I_{1,\ast}(s)ds\leq
C_{1}\|\dot{\mathbf{U}}\|_{1,\ast,t}^{2}\,,\end{split}$ (B.10)

for all $t\in(0,T]$, see (3.3), (3.4) and (5.1).

The estimate of Theorem 5.1 will be obtained by applying classical arguments
from the energy method in order to get a control of the $L^{2}$-norm of the
solution ${\mathbf{V}}^{\pm}$ and the front $\varphi$, as well as its
tangential space time derivatives, corresponding to the different expressions
listed in (B.4)–(B.8).

###  B.1. Estimate of $I(t)$

We scalarly multiply both sides of system (5.4) and integrate by parts in
$\Omega_{t}$ to get the energy identity

$\begin{split}\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}\cdot{\mathbf{V}})(t)d{\bf
x}&-\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}\,dx_{2}ds=2\int_{\Omega_{t}}\tilde{\mathcal{F}}\cdot{\mathbf{V}}d{\bf
x}ds\\\
&+\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}){\mathbf{V}}\cdot{\mathbf{V}}d{\bf
x}ds\,,\end{split}$ (B.11)

where, to shorcut notation, we write hereafter $\mathcal{B}_{k}$ instead of
$\mathcal{B}_{k}(\hat{{\mathbf{U}}},\hat{\Psi})$ for $k=0,1,2,3$.
From Cauchy-Schwarz and Young inequality, the right-hand side above is
estimated by

$\begin{split}&2\int_{\Omega_{t}}\tilde{\mathcal{F}}\cdot{\mathbf{V}}d{\bf
x}ds+\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}){\mathbf{V}}\cdot{\mathbf{V}}d{\bf
x}ds\\\ &\leq
2\|\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}\|_{L^{\infty}(\Omega_{t})}\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}^{2}\\\
&\leq\|\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}^{2}+\|\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}\|_{L^{\infty}(\Omega_{t})}\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}^{2}\\\
&\leq\|\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}^{2}\leq
C_{0}\|\tilde{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\int_{0}^{t}I(s)ds\\\
&\leq
C_{0}\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\int_{0}^{t}I(s)ds\,,\end{split}$
(B.12)

where

$\tilde{\mathcal{F}}=J^{T}\tilde{\mathbf{F}}\quad\mbox{and}\quad\tilde{\mathbf{F}}=\mathcal{S}(\hat{\mathbf{U}}){\mathbf{F}}$
(B.13)

are used, together with (B.4), to make the last inequality above.
Now we need to get an estimate of the quadratic form
$(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})$ under the second boundary
integral in the left-hand side of (B.11).

From (5.7), a direct calculation gives (5.8), (5.9), where we have made use of
the boundary conditions in (B.1) and (B.3). In (5.8) the initials “l.o.t” are
used to mean “lower order terms” with respect to the leading part
$[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}]\dot{q}^{+}\partial_{2}\varphi$ of the
boundary quadratic form (5.8). All terms appearing in (5.9) are products of
the form $\varphi\,\partial_{i}\varphi$, with $i=0,2$, or
$v_{j}^{\pm}|_{x_{1}=0}\,\varphi$, with $j=1,\dots,6$, up to some
coefficients. To make uniform notation, here we have set
$\partial_{0}\equiv\partial_{t}$ and
${\mathbf{V}}^{\pm}:=(v_{1}^{\pm},v^{\pm}_{2},v^{\pm}_{3},v^{\pm}_{4},v^{\pm}_{5},v^{\pm}_{6})$.
As already shown in Sect. 5, the major advantage to settle the functions
$\hat{\lambda}^{\pm}=\lambda(\hat{\mathbf{U}}^{\pm})$ as prescribed in Lemma
5.1 is making the leading term in the boundary quadratic form
$(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}$ to be identically
zero. Under this choice, the latter reduces indeed to

$\begin{split}\frac{1}{2}(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}&=\mbox{l.o.t.}=-[\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{1}\hat{H}_{N}]\dot{q}^{+}\varphi-[\partial_{1}\hat{q}]\varphi\partial_{t}\varphi\\\
&\quad-[\partial_{1}\hat{q}](\hat{u}^{-}_{2}-\hat{\lambda}\hat{H}^{-}_{2})\varphi\partial_{2}\varphi-[\partial_{1}\hat{q}](\partial_{1}\hat{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{1}\hat{H}^{-}_{N})\varphi^{2}\,.\end{split}$
(B.14)

We now focus on the estimate of the boundary integral of the different
quadratic terms in (B.14). Because the explicit form of the coefficients
involved in the different quadratic terms appearing in (B.14) is useless,
hereafter we adopt the custom to denote as $\hat{c}=\hat{c}(t,x_{2})$ a
generic function on $\Gamma_{T}$ represented by some nonlinear smooth function
of $\hat{\mathbf{U}}^{\pm}|_{x_{1}=0}$ and $\nabla_{t,x_{2}}\hat{\varphi}$
only, that may be possibly different from a line to another within the same
formula. From (B.14)

$\begin{split}\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}\,dx_{2}ds&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\dot{q}^{+}\varphi\,dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi\partial_{t}\varphi\,dx_{2}ds\\\
&\quad+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi\partial_{2}\varphi\,dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi^{2}\,dx_{2}ds\end{split}$

follows at once.
Hence Cauchy–Schwarz and Young’s inequalities together with Leibniz’s rule and
integration by parts yields

$\begin{split}&\left|\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\dot{q}^{+}\varphi\,dx_{2}ds\right|\leq\|\hat{c}\|_{L^{\infty}(\Gamma_{t})}\left\\{\|\dot{q}^{+}|_{x_{1}=0}\|_{L^{2}(\Gamma_{t})}^{2}+\|\varphi\|_{L^{2}(\Gamma_{t})}^{2}\right\\}\,;\\\
\\\
&\left|\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi\partial_{t}\varphi\,dx_{2}ds\right|=\left|\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\partial_{t}(\varphi^{2})\,dx_{2}ds\right|=\left|\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{t}(\hat{c}\varphi^{2})\,dx_{2}ds-\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{t}\hat{c}\varphi^{2}\,dx_{2}ds\right|\\\
&\qquad=\left|\frac{1}{2}\int_{\mathbb{R}}\\!\\!\hat{c}(t)\varphi^{2}(t)\,dx_{2}-\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{t}\hat{c}\varphi^{2}\,dx_{2}ds\right|\\\
&\qquad\leq\frac{1}{2}\|\hat{c}(t)\|_{L^{\infty}(\mathbb{R})}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}+\frac{1}{2}\|\partial_{t}\hat{c}\|_{L^{\infty}(\Gamma_{t})}\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\,;\\\
\\\
&\left|\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi\partial_{2}\varphi\,dx_{2}ds\right|=\left|\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\partial_{2}(\varphi^{2})\,dx_{2}ds\right|=\left|-\frac{1}{2}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{2}\hat{c}\varphi^{2}\,dx_{2}ds\right|\\\
&\qquad\quad\leq\frac{1}{2}\|\partial_{2}\hat{c}\|_{L^{\infty}(\Gamma_{t})}\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\,;\\\
\\\
&\left|\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\varphi^{2}\,dx_{2}ds\right|\leq\|\hat{c}\|_{L^{\infty}(\Gamma_{t})}\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\,.\end{split}$

Then we sum up the preceding estimates to get

$\begin{split}\left|\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}\,dx_{2}ds\right|\leq
C_{1}\left\\{\|\dot{q}^{+}|_{x_{1}=0}\|_{L^{2}(\Gamma_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}\right\\}+C_{2}\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\,.\end{split}$
(B.15)

Because functions $\hat{\lambda}^{\pm}$ chosen in Lemma 5.1 make
$\mathcal{B}_{0}$ to be positive definite, we also have

$\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}\cdot{\mathbf{V}})(t)d{\bf
x}\geq c_{0}\|{\mathbf{V}}(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}=c_{0}I(t)\,,$
(B.16)

with some constant $c_{0}>0$ (depending on the number $k$ in (4.2) and (4.4)).
Combining the latter with (B.11), (B.12) and (B.15) then leads to

$\begin{split}I(t)\leq\frac{1}{c_{0}}\left|\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})|_{x_{1}=0}\,dx_{2}ds\right|+C_{0}\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\int_{0}^{t}I(s)ds\\\
\leq
C_{0}\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\left\\{\|\dot{q}^{+}|_{x_{1}=0}\|_{L^{2}(\Gamma_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}+\int_{0}^{t}I(s)ds\right\\}+C_{2}\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\,.\end{split}$
(B.17)

Notice that in the right-hand side of (B.17) the $L^{2}$-norm of the trace of
$\dot{q}^{+}$ on the boundary $\Gamma_{t}$ needs for a control of the normal
derivative of $\dot{q}^{+}$ in the interior of the domain $\Omega_{t}$:
indeed, we apply to $\dot{q}^{+}$ the following trace type inequality

$\|f|_{x_{1}=0}\|^{2}_{L^{2}(\Gamma_{t})}\leq\|f\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{1}f\|^{2}_{L^{2}(\Omega_{t})}$
(B.18)

which holds true for an arbitrary sufficiently smooth scalar/vector-valued
function $f=f(t,{\bf x})$ over $\Omega_{t}$. The above estimate (B.18) follows
by passing from a boundary integral on $\mathbb{R}$ to a spatial volume one on
$\mathbb{R}^{2}_{+}$, at fixed $t$, as follows:

$\begin{split}&\int_{\mathbb{R}}|f(t)|_{x_{1}=0}|^{2}dx_{2}=-\int_{\mathbb{R}}\int_{0}^{+\infty}\partial_{1}\left(|f(t)|_{x_{1}=0}|^{2}\right)dx_{1}dx_{2}\\\
&=-\int_{\mathbb{R}^{2}_{+}}2f\partial_{1}f(t)\,dx\leq\|f(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\|\partial_{1}f(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\,,\end{split}$
(B.19)

from which (B.18) follows at once from integrating over $(-\infty,t)$.

This clearly shows that inequality (B.17) cannot provide a closed
$L^{2}$-estimate of the vector unknown ${\mathbf{V}}$, since a control of the
normal derivative of $\dot{q}^{+}$ on $\Omega_{t}$ is needed in the right-hand
side. Fortunately, the component $\dot{q}^{+}$ of ${\mathbf{V}}$ belongs to
the noncharacteristic components ${\bf
V}^{\pm}_{n}=(\dot{q}^{\pm},\dot{u}_{n}^{\pm},\dot{H}^{\pm}_{n})$, whose
normal derivative can be expressed directly from linear system (5.3) as a
function of tangential derivatives of ${\bf V}$ only and the source term
$\mathcal{F}$. Thus estimating the $L^{2}-$norm of
$\partial_{1}{\mathbf{V}}^{\pm}_{n}$ reduces to estimate the $L^{2}-$norm of
tangential derivatives of ${\bf V}$ and the $L^{2}-$norm of $\mathcal{F}$, as
it will be detailed in the next section.
Applying (B.18) to $f=\dot{q}^{+}$, from (B.17)

$\begin{split}I(t)\leq
C_{0}\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\left\\{\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}+\int_{0}^{t}(I+I_{1,n})(s)ds\right\\}+C_{2}\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\end{split}$
(B.20)

follows at once, see (B.6).

###  B.2. Estimate of $I_{1,n}(t)$

In this section we manage to find an “explicit” expression of the normal
derivative of the noncharacteristic component ${\mathbf{V}}^{\pm}_{n}$ of the
unknowns directly from linear system (5.3). The key step is taking advantage
of the form of the normal derivative coefficient $\mathcal{A}_{1}$ in (5.5)
and noticing that from $\mathcal{A}_{(0)}|_{x_{1}=0}=0$ the identity

$\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}}=\mathcal{H}_{(0)}\sigma\partial_{1}{\mathbf{V}}$
(B.21)

easily follows, where the matrix coefficient
$\mathcal{H}_{(0)}=\mathcal{H}_{(0)}(t,{\bf
x}):=\displaystyle\frac{\mathcal{A}_{(0)}(t,{\bf x})}{\sigma(x_{1})}\in
L^{\infty}(\Omega_{T})$ provided that
$\mathcal{A}_{(0)},\partial_{1}\mathcal{A}_{(0)}\in L^{\infty}(\Omega_{T})$
and obeys the estimate

$\|\mathcal{H}_{(0)}\|_{L^{\infty}(\Omega_{t})}\leq\|\partial_{1}\mathcal{A}_{(0)}\|_{L^{\infty}(\Omega_{t})}\,,$

see [24, Lemma B.9] for details.
After (B.21), from (5.3) we derive

$\mathcal{A}\partial_{1}{\mathbf{V}}=\mathcal{F}-\mathcal{A}_{0}\partial_{t}{\mathbf{V}}-\mathcal{H}_{(0)}\sigma\partial_{1}{\mathbf{V}}-\mathcal{A}_{2}\partial_{2}{\mathbf{V}}-\mathcal{A}_{3}{\mathbf{V}}\quad\mbox{in}\,\,\,\Omega_{T}\,,$
(B.22)

where the matrix $\mathcal{A}$ involved in the left-hand side above only
applies to components $\dot{q}^{\pm}$, $\dot{u}^{\pm}_{n}$ of the
noncharacteristic part ${\mathbf{V}}_{n}$ of ${\mathbf{V}}$, in view of (5.5);
indeed an explicit calculation gives

$\begin{split}&\mathcal{A}\partial_{1}{\mathbf{V}}=\left(\frac{1}{\partial_{1}\hat{\Phi}^{+}}\mathcal{E}_{12}\partial_{1}{\mathbf{V}}^{+},\frac{1}{\partial_{1}\hat{\Phi}^{-}}\mathcal{E}_{12}\partial_{1}{\mathbf{V}}^{-}\right)\\\
&\frac{1}{\partial_{1}\hat{\Phi}^{\pm}}\mathcal{E}_{12}\partial_{1}{\mathbf{V}}^{\pm}=\left(\frac{1}{\partial_{1}\hat{\Phi}^{\pm}}\partial_{1}\dot{u}^{\pm}_{n},\frac{1}{\partial_{1}\hat{\Phi}^{\pm}}\partial_{1}\dot{q}^{\pm},0,0,0,0\right)\,.\end{split}$
(B.23)

By gathering (B.22), (B.23) we derive the announced explicit form of normal
derivatives of the noncharacteristic components
$(\dot{q}^{\pm},\dot{u}_{n}^{\pm})$ of ${\mathbf{V}}^{\pm}$ as a function of
space time tangential derivatives of ${\mathbf{V}}^{\pm}$ and
$\mathcal{F}^{\pm}$ alone, namely

$\begin{split}&\partial_{1}\dot{q}^{\pm}=\partial_{1}\hat{\Phi}^{\pm}\left(\mathcal{F}^{\pm}-\mathcal{A}^{\pm}_{0}\partial_{t}{\mathbf{V}}^{\pm}-\mathcal{H}_{(0)}^{\pm}\sigma\partial_{1}{\mathbf{V}}^{\pm}-\mathcal{A}_{2}^{\pm}\partial_{2}{\mathbf{V}}^{\pm}-\mathcal{A}_{3}^{\pm}{\mathbf{V}}^{\pm}\right)_{1}\\\
&\partial_{1}\dot{u}^{\pm}_{n}=\partial_{1}\hat{\Phi}^{\pm}\left(\mathcal{F}^{\pm}-\mathcal{A}^{\pm}_{0}\partial_{t}{\mathbf{V}}^{\pm}-\mathcal{H}_{(0)}^{\pm}\sigma\partial_{1}{\mathbf{V}}^{\pm}-\mathcal{A}_{2}^{\pm}\partial_{2}{\mathbf{V}}^{\pm}-\mathcal{A}_{3}^{\pm}{\mathbf{V}}^{\pm}\right)_{2}\quad\mbox{in}\,\,\,\Omega_{T}\,,\end{split}$
(B.24)

where the subindices $1,2$ appearing above are referred to the first and the
second components of the vectors.
As regards to the normal derivative of the noncharacteristic components
$\dot{H}^{\pm}_{n}$, they can be still derived from the first condition in
(B.3) as a known function of tangential derivatives of ${\mathbf{V}}^{\pm}$

$\partial_{1}\dot{H}^{\pm}_{n}=-\partial_{2}(\dot{H}^{\pm}_{2}\partial_{1}\hat{\Phi}^{\pm})=-\partial_{2}\dot{H}^{\pm}_{2}\partial_{1}\hat{\Phi}^{\pm}-\dot{H}^{\pm}_{2}\partial_{2}\partial_{1}\hat{\Phi}^{\pm}\quad\mbox{in}\,\,\,\Omega_{T}\,.$
(B.25)

The estimate of $I_{1,n}(t)$ then follows at once from (B.24) and (B.25); we
get

$\begin{split}I_{1,n}(t)&\leq
C_{0}\|{\mathbf{F}}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+C_{1}\left\\{I(t)+I_{0}(t)+I_{\sigma}(t)+I_{2}(t)\right\\}\\\
&=C_{0}\|{\mathbf{F}}(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+C_{1}I_{1,\ast}(t)\,.\end{split}$
(B.26)

After estimate (B.26), it appears that the $L^{2}-$estimate of the normal
derivative of the noncharacteristic part ${\mathbf{V}}_{n}^{\pm}$ of the
solution is reducted to control the $L^{2}-$norm of the tangential space time
derivatives of ${\mathbf{V}}$, that is $I_{\sigma}(t)$, $I_{2}(t)$ and
$I_{0}(t)$, which naturally leads to establish an $H^{1}_{\ast}-$estimate for
${\mathbf{V}}$. The subsequent sections will be devoted to obtain the
$H^{1}_{\ast}-$estimate.

###  B.3. Estimate of $I_{\sigma}(t)$

We set for shortness ${\mathbf{V}}_{\sigma}:=\sigma\partial_{1}{\mathbf{V}}$.
Applying the conormal derivative $\sigma\partial_{1}$ to both sides of system
(5.4) we get a similar linear system satisfied by ${\mathbf{V}}_{\sigma}$. We
compute

$\begin{split}\mathcal{B}_{0}\partial_{t}{\mathbf{V}}_{\sigma}+\mathcal{B}_{1}\sigma\partial_{1}\partial_{1}{\mathbf{V}}+\sigma\partial_{1}\mathcal{B}_{1}\partial_{1}{\mathbf{V}}+\mathcal{B}_{2}\partial_{2}{\mathbf{V}}_{\sigma}+\mathcal{B}_{3}{\mathbf{V}}_{\sigma}\\\
=\sigma\partial_{1}\tilde{\mathcal{F}}-\sigma\partial_{1}\mathcal{B}_{0}\partial_{t}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{2}\partial_{2}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{3}{\mathbf{V}}.\end{split}$
(B.27)

From Leibniz’s rule

$\partial_{1}{\mathbf{V}}_{\sigma}=\partial_{1}(\sigma\partial_{1}{\mathbf{V}})=\sigma\partial_{1}\partial_{1}{\mathbf{V}}+\sigma^{\prime}\partial_{1}{\mathbf{V}}\,;$

plugging the latter into (B.27) and rewriting
$\sigma\partial_{1}\mathcal{B}_{1}\partial_{1}{\mathbf{V}}=\partial_{1}\mathcal{B}_{1}{\mathbf{V}}_{\sigma}$
gives

$\mathcal{B}_{0}\partial_{t}{\mathbf{V}}_{\sigma}+\mathcal{B}_{1}\partial_{1}{\mathbf{V}}_{\sigma}+\mathcal{B}_{2}\partial_{2}{\mathbf{V}}_{\sigma}+(\mathcal{B}_{3}+\partial_{1}\mathcal{B}_{1}){\mathbf{V}}_{\sigma}=\tilde{\mathcal{F}}_{\sigma}\,,$
(B.28)

where

$\tilde{\mathcal{F}}_{\sigma}:=\sigma\partial_{1}\tilde{\mathcal{F}}-\sigma\partial_{1}\mathcal{B}_{0}\partial_{t}{\mathbf{V}}+\sigma^{\prime}\mathcal{B}_{1}\partial_{1}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{2}\partial_{2}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{3}{\mathbf{V}}\,.$
(B.29)

Performing on system (B.28) the same standard energy arguments as done for
system (5.4) in Sect. B.1, leads to the following energy identity for
${\mathbf{V}}_{\sigma}$

$\begin{split}\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}_{\sigma}\cdot{\mathbf{V}}_{\sigma})(t)d{\bf
x}&-\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}_{\sigma}\cdot{\mathbf{V}}_{\sigma})|_{x_{1}=0}\,dx_{2}ds=2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{\sigma}\cdot{\mathbf{V}}_{\sigma}d{\bf
x}ds\\\
&+\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{2}\mathcal{B}_{2}-\partial_{1}\mathcal{B}_{1}-\mathcal{B}_{3}){\mathbf{V}_{\sigma}}\cdot{\mathbf{V}_{\sigma}}d{\bf
x}ds\,;\end{split}$

however because $\sigma|_{x_{1}=0}=0$, the quadratic form
$(\mathcal{B}_{1}{\mathbf{V}}_{\sigma}\cdot{\mathbf{V}}_{\sigma})$ under the
boundary integral in the left-hand side above vanishes and the energy identity
reduces to

$\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}_{\sigma}\cdot{\mathbf{V}}_{\sigma})(t)d{\bf
x}=2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{\sigma}\cdot{\mathbf{V}}_{\sigma}d{\bf
x}ds+\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{2}\mathcal{B}_{2}-\partial_{1}\mathcal{B}_{1}-\mathcal{B}_{3}){\mathbf{V}_{\sigma}}\cdot{\mathbf{V}_{\sigma}}d{\bf
x}ds\,.$ (B.30)

The second integral in the right-hand side of the identity above is trivially
estimated as in (B.12)

$\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{2}\mathcal{B}_{2}-\partial_{1}\mathcal{B}_{1}-\mathcal{B}_{3}){\mathbf{V}_{\sigma}}\cdot{\mathbf{V}_{\sigma}}d{\bf
x}ds\leq C_{1}\|{\mathbf{V}_{\sigma}}\|_{L^{2}(\Omega_{t})}^{2}\,.$

Now we focus on the estimate of the first integral

$2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{\sigma}\cdot{\mathbf{V}}_{\sigma}d{\bf
x}ds$ (B.31)

in the right-hand side of (B.30). Substituting the explicit form (B.29) of
$\tilde{F}_{\sigma}$, using that, similarly to $A_{1}$ (see (5.5))

$\mathcal{B}_{1}=\mathcal{B}^{(0)}_{1}+\mathcal{B}\,,\quad\mbox{where}\,\,\,\mathcal{B}^{(0)}_{1}|_{x_{1}=0}=0\,,$
(B.32)

while $\mathcal{B}$ only “applies” to the noncharacteristic components ${\bf
V}^{\pm}_{n}$ of ${\mathbf{V}}^{\pm}$, and denoting
$\mathcal{K}^{(0)}_{1}=\frac{\mathcal{B}^{(0)}_{1}}{\sigma},$ we have from
Cauchy–Schwarz and Young’s inequalities

$\begin{split}&\left|2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{\sigma}\cdot{\mathbf{V}}_{\sigma}d{\bf
x}ds\right|\\\
&=\left|2\int_{\Omega_{t}}\left(\sigma\partial_{1}\tilde{\mathcal{F}}-\sigma\partial_{1}\mathcal{B}_{0}\partial_{t}{\mathbf{V}}+\sigma^{\prime}\mathcal{K}^{(0)}_{1}\sigma\partial_{1}{\mathbf{V}}+\sigma^{\prime}\mathcal{B}\partial_{1}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{2}\partial_{2}{\mathbf{V}}-\sigma\partial_{1}\mathcal{B}_{3}{\mathbf{V}}\right)\cdot{\mathbf{V}}_{\sigma}d{\bf
x}ds\right|\\\ &\leq
2\left\\{\|\sigma\partial_{1}\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}+C_{1}\left(\|\partial_{t}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|{\mathbf{V}}_{\sigma}\|_{L^{2}(\Omega_{t})}+\|\partial_{1}{\mathbf{V}}_{n}\|_{L^{2}(\Omega_{t})}\right.\right.\\\
&\left.\left.\qquad\qquad+\|\partial_{2}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}\right)\right\\}\|{\mathbf{V}}_{\sigma}\|_{L^{2}(\Omega_{t})}\\\
&\leq\|\sigma\partial_{1}\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}^{2}+C_{1}\int_{0}^{t}(I+I_{0}+I_{\sigma}+I_{1,n}+I_{2})(s)ds\\\
&=C_{1}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\sigma\partial_{1}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.\end{split}$

At last, estimating from below the space integral in the left-hand side of
(B.30) as in (B.16) (with ${\mathbf{V}}_{\sigma}$ instead of ${\mathbf{V}}$)
we end up with

$I_{\sigma}(t)\leq
C_{1}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\sigma\partial_{1}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.$
(B.33)

###  B.4. Estimate of $I_{2}(t)$

We set for shortness ${\mathbf{V}}_{x_{2}}:=\partial_{2}{\mathbf{V}}$.
Applying $\partial_{2}$ to both sides of system (5.4) we get a similar linear
system satisfied by ${\mathbf{V}}_{x_{2}}$, that is

$\mathcal{B}_{0}\partial_{t}{\mathbf{V}}_{x_{2}}+\mathcal{B}_{1}\partial_{1}{\mathbf{V}}_{x_{2}}+\mathcal{B}_{2}\partial_{2}{\mathbf{V}}_{x_{2}}+(\partial_{2}\mathcal{B}_{2}+\mathcal{B}_{3}){\mathbf{V}}_{x_{2}}=\tilde{\mathcal{F}}_{2}\,,$
(B.34)

where

$\tilde{\mathcal{F}}_{2}:=\partial_{2}\tilde{\mathcal{F}}-\partial_{2}\mathcal{B}_{0}\partial_{t}{\mathbf{V}}-\partial_{2}\mathcal{B}_{1}\partial_{1}{\mathbf{V}}-\partial_{2}\mathcal{B}_{3}{\mathbf{V}}\,.$

As usual, from the above linear system we derive, by scalar multiplication by
${\mathbf{V}}_{x_{2}}$ and integration by parts in $\Omega_{t}$, the energy
identity

$\begin{split}\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}})(t)d{\bf
x}&-\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}})|_{x_{1}=0}\,dx_{2}ds=2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{2}\cdot{\mathbf{V}}_{x_{2}}d{\bf
x}ds\\\
&+\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}-2\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}){\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}}d{\bf
x}ds\,.\end{split}$ (B.35)

The second integral in the right-hand side of the above identity is estimated
as usual as

$\int_{\Omega_{t}}(\partial_{t}\mathcal{B}_{0}+\partial_{1}\mathcal{B}_{1}-2\partial_{2}\mathcal{B}_{2}-\mathcal{B}_{3}){\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}}d{\bf
x}ds\leq
C_{1}\|{\mathbf{V}}_{x_{2}}\|_{L^{2}(\Omega_{t})}^{2}=C_{1}\int_{0}^{t}I_{2}(s)ds\,,$
(B.36)

whereas to estimate the first integral in the right-hand side we still use the
decomposition of $\mathcal{B}_{1}$ in (B.32) and repeat the same arguments
used in the estimate of (B.31) and use (B.13), to get

$\begin{split}&2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{2}\cdot{\mathbf{V}}_{x_{2}}d{\bf
x}ds\\\ &\leq
2\left\\{\|\partial_{2}\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}+C_{1}\left(\|\partial_{t}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|\sigma\partial_{1}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|\partial_{1}{\mathbf{V}}_{n}\|_{L^{2}(\Omega_{t})}\right.\right.\\\
&\left.\left.\qquad\qquad+\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}\right)\right\\}\|{\mathbf{V}}_{x_{2}}\|_{L^{2}(\Omega_{t})}\\\
&\leq
C_{1}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{2}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.\end{split}$
(B.37)

Now we need to get an estimate of the quadratic form
$(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}})$ under the
second boundary integral in the left-hand side of (B.35). The explicit
expression of this quadratic form is in principle the same as the one for
${\mathbf{V}}$ in (5.7), that is

$(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}},{\mathbf{V}}_{x_{2}})|_{x_{1}=0}=2[\partial_{2}\dot{q}(\partial_{2}\dot{u}_{N}-\hat{\lambda}\partial_{2}\dot{H}_{N})],$
(B.38)

with $\hat{\lambda}:=\lambda(\hat{\mathbf{U}})$.
As done to treat the quadratic form (5.7) for ${\mathbf{V}}$, now we make use
of the boundary conditions in (B.1) and the boundary constraint in (B.3),
differentiated with respect to $x_{2}$, to rewrite (B.38) as the sum of the
same leading part as in (5.8), vanishing as a consequence of the choice of
$\hat{\lambda}^{\pm}$, and lower order terms. We compute

$\begin{split}(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}},{\mathbf{V}}_{x_{2}})|_{x_{1}=0}=2[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}]\partial_{2}\dot{q}^{+}\partial_{2}\partial_{2}\varphi+\mbox{l.o.t}\,,\end{split}$
(B.39)

where

$\begin{split}\mbox{l.o.t}:=&-2[\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{1}\hat{H}_{N}-\partial_{2}\hat{u}_{2}+\hat{\lambda}\partial_{2}\hat{H}_{2}]\partial_{2}\dot{q}^{+}\partial_{2}\varphi-2[\partial_{2}\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{2}\partial_{1}\hat{H}_{N}]\partial_{2}\dot{q}^{+}\varphi\\\
&-2[\partial_{1}\hat{q}]\partial_{2}\varphi(\partial_{2}\dot{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{2}\dot{H}^{-}_{N})-2\partial_{2}([\partial_{1}\hat{q}])\varphi(\partial_{2}\dot{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{2}\dot{H}^{-}_{N})\,.\end{split}$
(B.40)

As already announced the leading quadratic term
$2[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}]\dot{q}^{+}_{x_{2}}\partial_{2}\varphi$
in (B.39) vanishes because of the chosen $\hat{\lambda}^{\pm}$. We now focus
on the estimate of the boundary integral of the different lower order
quadratic terms in (B.40). Here below we denote again by
$\hat{c}=\hat{c}(t,x_{2})$ the different coefficients of those boundary lower
order terms, which are all smooth functions of
$\hat{\mathbf{U}}^{\pm}|_{x_{1}=0}$ and $\nabla_{t,x_{2}}\hat{\varphi}$ and
their derivatives and whose explicit form is useless.
From (B.40), we get

$\begin{split}\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}})|_{x_{1}=0}\,dx_{2}ds&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\partial_{2}\varphi
dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\varphi\,dx_{2}ds\\\
&\quad+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{u}^{-}_{N}\partial_{2}\varphi\,dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{H}^{-}_{N}\partial_{2}\varphi\,dx_{2}ds\\\
&\quad+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{u}^{-}_{N}\varphi\,dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{H}^{-}_{N}\varphi\,dx_{2}ds\,.\end{split}$
(B.41)

To estimate the boundary integrals above, we follow similar arguments to those
of [38].
The first step is to write $\partial_{2}\varphi$ as a linear combination of
$\dot{H}^{\pm}_{N}|_{x_{1}=0}$ and $\varphi$; this can be done by making use
of boundary constraints (B.3) and exploiting that $\hat{H}_{2}^{\pm}$ are
never simultaneously zero on the boundary as a consequence of the stability
condition (4.4), see Remark 4.3. From the boundary conditions (B.3) we have

$\hat{H}^{+}_{2}\partial_{2}\varphi=\dot{H}^{+}_{N}+\varphi\partial_{1}\hat{H}^{+}_{N}\quad\mbox{and}\quad\hat{H}^{-}_{2}\partial_{2}\varphi=\dot{H}^{-}_{N}-\varphi\partial_{1}\hat{H}^{-}_{N}\qquad\mbox{on}\,\,\,\Gamma_{T}\,.$

Then multiplying the first one by $\hat{H}^{+}_{2}|_{x_{1}=0}$ and the second
one by $\hat{H}^{-}_{2}|_{x_{1}=0}$, then adding the results we get

$\begin{split}\partial_{2}\varphi&=\frac{\hat{H}^{+}_{2}\dot{H}^{+}_{N}+\hat{H}^{-}_{2}\dot{H}^{-}_{N}+(\hat{H}^{+}_{2}\partial_{1}\hat{H}^{+}_{N}-\hat{H}^{-}_{2}\partial_{1}\hat{H}^{-}_{N})\varphi}{(\hat{H}^{+}_{2})^{2}+(\hat{H}^{-}_{2})^{2}}\\\
&=\hat{d}_{1}\dot{H}^{+}_{N}+\hat{d}_{2}\dot{H}^{-}_{N}+\hat{d}_{3}\varphi\,,\quad\mbox{on}\,\,\,\Gamma_{T}\,,\end{split}$
(B.42)

where $\hat{d}_{i}=\hat{d}_{i}(t,x_{2})$ are suitable functions depending only
on the boundary values of $\hat{H}^{\pm}$, $\partial_{1}\hat{H}^{\pm}$ and
second order derivatives of $\hat{\varphi}$, whose esplicit form could be
easily deduced from above.

Let us start to estimate the first term in the right-hand side of (B.41).
Inserting the expression of $\partial_{2}\varphi$ provided by (B.42), we find

$\begin{split}\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\partial_{2}\varphi
dx_{2}ds&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\dot{H}^{+}_{N}dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\dot{H}^{-}_{N}dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\varphi\,dx_{2}ds\\\
&=\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}\,.\end{split}$

Let us estimate $\mathcal{I}_{1}$. Here the trick of passing from a boundary
integral over $\mathbb{R}$ to a volume integral over $\mathbb{R}^{2}_{+}$ is
used as already done to get the trace inequality (B.18). In the following we
will adopt the notation $\hat{\mathbf{c}}=\hat{\mathbf{c}}(t,{\bf x})$ to mean
a suitable lifting from $\Gamma_{t}$ to $\Omega_{t}$ of a boundary coefficient
$\hat{c}=\hat{c}(t,x_{2})$. Then we have

$\begin{split}\mathcal{I}_{1}&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\dot{H}^{+}_{N}dx_{2}ds=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\partial_{1}(\hat{\mathbf{c}}\,\partial_{2}\dot{q}^{+}\dot{H}^{+}_{n})d{\bf
x}ds\\\
&=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\partial_{1}\hat{\mathbf{c}}\,\partial_{2}\dot{q}^{+}\dot{H}^{+}_{n}d{\bf
x}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\hat{\mathbf{c}}\,\partial_{1}\partial_{2}\dot{q}^{+}\dot{H}^{+}_{n}d{\bf
x}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\hat{\mathbf{c}}\,\partial_{2}\dot{q}^{+}\partial_{1}\dot{H}^{+}_{n}d{\bf
x}ds\\\ &=\mathcal{I}_{1,1}+\mathcal{I}_{1,2}+\mathcal{I}_{1,3}\,.\end{split}$

$\mathcal{I}_{1,1}$ and $\mathcal{I}_{1,3}$ can be easily estimated by Cauchy-
Schwarz and Young’s inequalities by

$\begin{split}|\mathcal{I}_{1,1}+\mathcal{I}_{1,3}|&\leq
C_{2}\|\partial_{2}\dot{q}^{+}\|_{L^{2}(\Omega_{t})}\|\dot{H}^{+}_{n}\|_{L^{2}(\Omega_{t})}+C_{1}\|\partial_{2}\dot{q}^{+}\|_{L^{2}(\Omega_{t})}\|\partial_{1}\dot{H}^{+}_{n}\|_{L^{2}(\Omega_{t})}\\\
&\leq C_{2}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\end{split}$
(B.43)

recall that
${\mathbf{V}}_{n}^{\pm}=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{H}^{\pm}_{n})$
are the noncharacteristic components of the solution ${\mathbf{V}}^{\pm}$, see
also (B.6).
As regards to $\mathcal{I}_{1,2}$, differently from above, we cannot
immediately end up by Cauchy-Schwarz and Young’s inequality, because this
should require a control of the $L^{2}-$norm of the second order derivative of
$\dot{q}^{+}$, preventing to close the $H^{1}_{\ast}-$estimate. Instead, here
integration by parts with respect to the tangential space variable $x_{2}$ and
Leibiniz’s rule are used to further rewrite $\mathcal{I}_{1,2}$ as

$\begin{split}\mathcal{I}_{1,2}&=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\hat{\mathbf{c}}\,\partial_{1}\partial_{2}\dot{q}^{+}\dot{H}^{+}_{n}d{\bf
x}ds=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\partial_{2}(\hat{\mathbf{c}}\dot{H}^{+}_{n})\partial_{1}\dot{q}^{+}d{\bf
x}ds\\\
&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\partial_{2}\hat{\mathbf{c}}\dot{H}^{+}_{n}\partial_{1}\dot{q}^{+}d{\bf
x}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\\!\\!\hat{\mathbf{c}}\partial_{2}\dot{H}^{+}_{n}\partial_{1}\dot{q}^{+}d{\bf
x}ds\,;\end{split}$ (B.44)

then we observe that the last two integral above are similar to
$\mathcal{I}_{1,1}$ and $\mathcal{I}_{1,3}$ and therefore can be estimated in
the same way by

$\begin{split}|\mathcal{I}_{1,2}|&\leq
C_{2}\|\partial_{1}\dot{q}^{+}\|_{L^{2}(\Omega_{t})}\left(\|\dot{H}^{+}_{n}\|_{L^{2}(\Omega_{t})}+\|\partial_{2}\dot{H}^{+}_{n}\|_{L^{2}(\Omega_{t})}\right)\\\
&\leq
C_{2}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.\end{split}$

Adding (B.43) and (B.44) gives the estimate of $\mathcal{I}_{1}$

$|\mathcal{I}_{1}|\leq
C_{2}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.$

It is clear that
$\mathcal{I}_{2}:=\displaystyle\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\dot{H}^{-}_{N}dx_{2}ds$
can estimated by repeating exactly the same arguments applied to
$\mathcal{I}_{1}$.
Concerning
$\mathcal{I}_{3}=\displaystyle\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\varphi\,dx_{2}ds$,
we are reduced to apply the same arguments as above by first integrating by
parts and then using Leibniz’s rule and replacing once again
$\partial_{2}\varphi$ by the expression in the right-hand side of (B.42):

$\begin{split}\mathcal{I}_{3}&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\dot{q}^{+}\varphi\,dx_{2}ds=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{2}(\hat{c}\varphi)\dot{q}^{+}dx_{2}ds\\\
&=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\partial_{2}\hat{c}\,\varphi\dot{q}^{+}dx_{2}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\partial_{2}\varphi\dot{q}^{+}dx_{2}ds\\\
&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\varphi\dot{q}^{+}dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\dot{H}^{+}_{N}\dot{q}^{+}dx_{2}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\\!\\!\hat{c}\,\dot{H}^{-}_{N}\dot{q}^{+}dx_{2}ds\,.\end{split}$

Since all the boundary traces under the integral are of noncharacteristic
components of ${\mathbf{V}}$, we end up by Cauchy-Schwarz, Young’s
inequalities and trace type inequality (B.18) to get

$|\mathcal{I}_{3}|\leq
C_{2}\left\\{\int_{0}^{t}(I+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|_{L^{2}(\mathbb{R})}^{2}ds\right\\}\,.$
(B.45)

To complete the estimate of the remaining boundary integrals involved in the
right-hand side of (B.41), it is then sufficient to notice that the second
boundary integral is exactly the same as $\mathcal{I}_{3}$, while the other
boundary integrals are the same as the first and second ones, where
$\partial_{2}\dot{q}^{+}|_{x_{1}=0}$ is replaced by
$\partial_{2}\dot{u}^{-}_{N}|_{x_{1}=0}$ or by
$\partial_{2}\dot{H}^{-}_{N}|_{x_{1}=0}$ (but $\dot{u}^{-}_{n}$ and
$\dot{H}^{-}_{n}$ are still noncharacteristic components of the vector
solution ${\mathbf{V}}$, so that they are treated along the same arguments as
to $\dot{q}^{+}$). Therefore we end up with

$\left|\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}_{x_{2}}\cdot{\mathbf{V}}_{x_{2}})|_{x_{1}=0}\,dx_{2}ds\right|\leq
C_{2}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|_{L^{2}(\mathbb{R})}^{2}ds\right\\}\,.$
(B.46)

Using (B.36), (B.37) and (B.46) together with the counterpart of (B.16) with
${\mathbf{V}}_{x_{2}}$ instead of ${\mathbf{V}}$, from (B.35) we obtain

$I_{2}(t)\leq
C_{1}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{2}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}\right\\}+C_{2}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|_{L^{2}(\mathbb{R})}^{2}ds\right\\}\,.$
(B.47)

###  B.5. Estimate of $I_{0}(t)$

We set for shortness ${\mathbf{V}}_{t}:=\partial_{t}{\mathbf{V}}$. Applying
$\partial_{t}$ to both sides of system (5.4) we get a similar linear system
satisfied by ${\mathbf{V}}_{t}$, that is

$\mathcal{B}_{0}\partial_{t}{\mathbf{V}}_{t}+\mathcal{B}_{1}\partial_{1}{\mathbf{V}}_{t}+\mathcal{B}_{2}\partial_{2}{\mathbf{V}}_{t}+(\partial_{t}\mathcal{B}_{0}+\mathcal{B}_{3}){\mathbf{V}}_{t}=\tilde{\mathcal{F}}_{t}\,,$
(B.48)

where

$\tilde{\mathcal{F}}_{t}:=\partial_{t}\tilde{\mathcal{F}}-\partial_{t}\mathcal{B}_{1}\partial_{1}{\mathbf{V}}-\partial_{t}\mathcal{B}_{2}\partial_{2}{\mathbf{V}}-\partial_{t}\mathcal{B}_{3}{\mathbf{V}}\,.$

As usual, from the above linear system we derive, by scalar multiplication by
${\mathbf{V}}_{t}$ and integration by parts in $\Omega_{t}$, the energy
identity

$\begin{split}\int_{\mathbb{R}^{2}_{+}}(\mathcal{B}_{0}{\mathbf{V}}_{t}\cdot{\mathbf{V}}_{t})(t)d{\bf
x}&-\int_{\Gamma_{t}}(\mathcal{B}_{1}{\mathbf{V}}_{s}\cdot{\mathbf{V}}_{s})|_{x_{1}=0}\,dx_{2}ds=2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{s}\cdot{\mathbf{V}}_{s}d{\bf
x}ds\\\
&+\int_{\Omega_{t}}(\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\partial_{t}\mathcal{B}_{0}-\mathcal{B}_{3}){\mathbf{V}}_{s}\cdot{\mathbf{V}}_{s}d{\bf
x}ds\,.\end{split}$ (B.49)

The second integral in the right-hand side of the above identity is estimated
as usual as

$\int_{\Omega_{t}}(\partial_{1}\mathcal{B}_{1}+\partial_{2}\mathcal{B}_{2}-\partial_{t}\mathcal{B}_{0}-\mathcal{B}_{3}){\mathbf{V}}_{s}\cdot{\mathbf{V}}_{s}d{\bf
x}ds\leq
C_{1}\|{\mathbf{V}}_{s}\|_{L^{2}(\Omega_{t})}^{2}=C_{1}\int_{0}^{t}I_{0}(s)ds\,,$
(B.50)

whereas to estimate the first integral in the right-hand side we still use the
decomposition of $\mathcal{B}_{1}$ in (B.32) and repeat the same arguments
used in the estimates (B.31), (B.37) and use (B.13), to get

$\begin{split}&2\int_{\Omega_{t}}\tilde{\mathcal{F}}_{s}\cdot{\mathbf{V}}_{s}d{\bf
x}ds\\\ &\leq
2\left\\{\|\partial_{s}\tilde{\mathcal{F}}\|_{L^{2}(\Omega_{t})}+C_{1}\left(\|\sigma\partial_{1}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|\partial_{1}{\mathbf{V}}_{n}\|_{L^{2}(\Omega_{t})}\right.\right.\\\
&\left.\left.\qquad\qquad+\|\partial_{2}{\mathbf{V}}\|_{L^{2}(\Omega_{t})}+\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}\right)\right\\}\|{\mathbf{V}}_{t}\|_{L^{2}(\Omega_{t})}\\\
&\leq
C_{1}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{s}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\right\\}\,.\end{split}$
(B.51)

Now we need to get an estimate of the quadratic form
$(\mathcal{B}_{1}{\mathbf{V}}_{t}\cdot{\mathbf{V}}_{t})$ under the second
boundary integral in the left-hand side of (B.49), whose explicit expression
is

$(\mathcal{B}_{1}{\mathbf{V}}_{t},{\mathbf{V}}_{t})|_{x_{1}=0}=2[\partial_{t}\dot{q}(\partial_{t}\dot{u}_{N}-\hat{\lambda}\partial_{t}\dot{H}_{N})],$
(B.52)

with $\hat{\lambda}:=\lambda(\hat{\mathbf{U}})$.
As in the case of the quadratic form (B.38), we make use of the boundary
conditions in (B.1), differentiated with respect to $t$, to rewrite (B.52) as
the sum of the same leading part as in (B.39), vanishing as a consequence of
the choice of $\hat{\lambda}^{\pm}$, and lower order terms. We compute

$\begin{split}(\mathcal{B}_{1}{\mathbf{V}}_{t},{\mathbf{V}}_{t})|_{x_{1}=0}=2[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}]\partial_{t}\dot{q}^{+}\partial_{t}\partial_{2}\varphi+\mbox{l.o.t}\,,\end{split}$
(B.53)

where

$\begin{split}\mbox{l.o.t}:=&-2[\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{1}\hat{H}_{N}]\partial_{t}\dot{q}^{+}\partial_{t}\varphi+2[\partial_{t}\hat{u}_{2}-\hat{\lambda}\partial_{t}\hat{H}_{2}]\partial_{2}\varphi\partial_{t}\dot{q}^{+}\\\
&-2[\partial_{t}\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{t}\partial_{1}\hat{H}_{N}]\partial_{t}\dot{q}^{+}\varphi-2[\partial_{1}\hat{q}]\partial_{t}\varphi(\partial_{t}\dot{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{t}\dot{H}^{-}_{N})\\\
&-2\partial_{t}([\partial_{1}\hat{q}])\varphi(\partial_{t}\dot{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{t}\dot{H}^{-}_{N})\\\
&=\hat{c}\,\partial_{t}\dot{q}^{+}\partial_{t}\varphi+\hat{c}\,\partial_{2}\varphi\partial_{t}\dot{q}^{+}+\hat{c}\,\partial_{t}\dot{q}^{+}\varphi+\hat{c}\,\partial_{t}\varphi\partial_{t}\dot{u}^{-}_{N}+\hat{c}\,\partial_{t}\varphi\partial_{t}\dot{H}^{-}_{N}\\\
&+\hat{c}\,\varphi\partial_{t}\dot{u}^{-}_{N}+\hat{c}\,\varphi\partial_{t}\dot{H}^{-}_{N}\,.\end{split}$
(B.54)

We solve the first boundary condition in (B.1) (we choose the $+$ side) with
respect to $\partial_{t}\varphi$ and replace $\partial_{2}\varphi$ by (B.42)
to get

$\partial_{t}\varphi=\dot{u}^{+}_{N}+\hat{D}_{1}\dot{H}^{+}_{N}+\hat{D}_{2}\dot{H}^{-}_{N}+\hat{D}_{3}\varphi\,,$
(B.55)

with suitable coefficients $\hat{D}_{i}=\hat{D}_{i}(t,x_{2})$, $i=1,2,3$,
smoothly depending on the boundary traces of $\hat{u}^{+}$, $\hat{H}^{\pm}$,
$\partial_{1}\hat{H}^{\pm}$, $\partial_{1}\hat{u}^{+}$ and second order
derivatives of $\hat{\varphi}$, whose explicit form is useless for the
subsequent calculations.
Now we insert (B.55) and (B.42) in the expression (B.54) to rewrite the latter
as

$\begin{split}\mbox{l.o.t}:=&\hat{c}\,\partial_{t}\dot{q}^{+}(\dot{u}^{+}_{N}+\hat{D}_{1}\dot{H}^{+}_{N}+\hat{D}_{2}\dot{H}^{-}_{N}+\hat{D}_{3}\varphi)+\hat{c}\,(\hat{d}_{1}\dot{H}^{+}_{N}+\hat{d}_{2}\dot{H}^{-}_{N}+\hat{d}_{3}\varphi)\partial_{t}\dot{q}^{+}\\\
&+\hat{c}\,\partial_{t}\dot{q}^{+}\varphi+\hat{c}\,(\dot{u}^{+}_{N}+\hat{D}_{1}\dot{H}^{+}_{N}+\hat{D}_{2}\dot{H}^{-}_{N}+\hat{D}_{3}\varphi)\partial_{t}\dot{u}^{-}_{N}\\\
&+\hat{c}\,(\dot{u}^{+}_{N}+\hat{D}_{1}\dot{H}^{+}_{N}+\hat{D}_{2}\dot{H}^{-}_{N}+\hat{D}_{3}\varphi)\partial_{t}\dot{H}^{-}_{N}+\hat{c}\,\varphi\partial_{t}\dot{u}^{-}_{N}+\hat{c}\,\varphi\partial_{t}\dot{H}^{-}_{N}\,.\end{split}$
(B.56)

From the resulting expression above, it appears that the lower order terms
above are reduced to a sum of two types of quadratic terms, namely

$\mbox{l.o.t}\cong\hat{c}\,\partial_{t}\alpha\,\beta+\hat{c}\,\partial_{t}\alpha\,\varphi\,,\quad\mbox{on}\,\,\,\Gamma_{t}\,,$

where $\alpha=\alpha(t,{\bf x})$ and $\beta=\beta(t,{\bf x})$ are used to mean
any component of the noncharacteristic part
${\mathbf{V}}^{\pm}_{n}=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{H}^{\pm}_{n})$
of the solution, while $\hat{c}=\hat{c}(t,x_{2})$ denotes, as usual, suitable
functions on $\Gamma_{t}$, smoothly depending on space-time derivatives of
$\hat{\mathbf{U}}^{\pm}$ and $\nabla_{t,x_{2}}\hat{\varphi}$ up to second
order.

In view of the preceding manipulations, to get an estimate of the boundary
integral of the quadratic form (B.52) we only need to estimate the following
types of boundary integrals:

$\mathcal{J}_{1}:=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\partial_{s}\alpha\,\beta\,dx_{2}ds\quad\mbox{and}\quad\mathcal{J}_{2}:=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\partial_{s}\alpha\,\varphi\,dx_{2}ds\,.$

As it was done in Sect. B.4, in $\mathcal{J}_{1}$ we pass to a volume integral
over $\mathbb{R}^{2}_{+}$ and use Leibniz’s rule to get

$\begin{split}\mathcal{J}_{1}=&-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\partial_{1}(\hat{\mathbf{c}}\,\partial_{s}\alpha\,\beta)d{\bf
x}ds=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\partial_{1}\hat{\mathbf{c}}\,\partial_{s}\alpha\,\beta\,d{\bf
x}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}\,\partial_{1}\partial_{s}\alpha\,\beta\,d{\bf
x}ds\\\
&-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}\,\partial_{s}\alpha\,\partial_{1}\beta\,d{\bf
x}ds=\mathcal{J}_{1,1}+\mathcal{J}_{1,2}+\mathcal{J}_{1,3}\,.\end{split}$

By Cauchy–Schwarz and Young’s inequalities, $\mathcal{J}_{1,1}$ and
$\mathcal{J}_{1,3}$ are estimated by

$\begin{split}|\mathcal{J}_{1,1}+\mathcal{J}_{1,3}|&\leq
C_{3}\left\\{\|{\mathbf{V}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{s}{\mathbf{V}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{1}{\mathbf{V}}_{n}\|^{2}_{L^{2}(\Omega_{t})}\right\\}\\\
&\leq C_{3}\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\,.\end{split}$ (B.57)

The middle integral $\mathcal{J}_{1,2}$ is the most involved one, because the
second order derivative $\partial_{1}\partial_{t}\alpha$ prevents from
directly estimating $\mathcal{J}_{1,2}$ similarly to (B.57).
From Leibniz’s rule with respect to time (notice that $\alpha|_{t=0}\equiv 0$
and $\beta|_{t=0}\equiv 0$), we rewrite $\mathcal{J}_{1,2}$ as

$\begin{split}\mathcal{J}_{1,2}&=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\partial_{s}(\hat{\mathbf{c}}\,\partial_{1}\alpha\,\beta)\,d{\bf
x}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\partial_{s}\hat{\mathbf{c}}\,\partial_{1}\alpha\,\beta\,d{\bf
x}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}\,\partial_{1}\alpha\,\partial_{s}\beta\,d{\bf
x}ds\\\
&=-\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}(t)\,\partial_{1}\alpha(t)\,\beta(t)\,d{\bf
x}+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\partial_{s}\hat{\mathbf{c}}\,\partial_{1}\alpha\,\beta\,d{\bf
x}ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}\,\partial_{1}\alpha\,\partial_{s}\beta\,d{\bf
x}ds\,.\end{split}$ (B.58)

The last two integrals above are estimated exactly by the same right-hand side
of (B.57).
Concerning, instead, the first spatial integral

$-\displaystyle\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}(t)\,\partial_{1}\alpha(t)\,\beta(t)\,d{\bf
x}\,,$

in view of estimates (B.20), (B.26) (recall that $\alpha$ is a
noncharacteristic component of ${\mathbf{V}}$), the use of Cauchy–Schwarz and
weighted Young’s inequalities gives

$\begin{split}&\left|-\int_{\mathbb{R}^{2}_{+}}\hat{\mathbf{c}}(t)\,\partial_{1}\alpha(t)\,\beta(t)\,d{\bf
x}\right|\leq
C_{2}\|\partial_{1}\alpha(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}\|\beta(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}\\\
&\leq\varepsilon\|\partial_{1}\alpha(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}+\frac{C_{2}}{\varepsilon}\|\beta(t)\|_{L^{2}(\mathbb{R}^{2}_{+})}^{2}\leq\varepsilon
I_{1,n}(t)+\frac{C_{2}}{\varepsilon}\left\\{\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}\right.\\\
&\quad\left.+\int_{0}^{t}(I+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\,,\end{split}$
(B.59)

where $\varepsilon>0$ will be chosen to be small enough.
Adding (B.59) and the analogous of (B.57) for the second and third integral in
the right-hand side of (B.58) finally yields

$\begin{split}|\mathcal{J}_{1,2}|\leq&\varepsilon
I_{1,n}(t)+\frac{C_{2}}{\varepsilon}\left\\{\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\\\
&+\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\end{split}$
(B.60)

and adding the latter and (B.57) we get

$\begin{split}|\mathcal{J}_{1}|\leq&\varepsilon
I_{1,n}(t)+\frac{C_{2}}{\varepsilon}\left\\{\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\\\
&+\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds\,.\end{split}$
(B.61)

The boundary integral $\mathcal{J}_{2}$ is treated along the same lines as
$\mathcal{J}_{1,2}$. After Leibniz’s rule with respect to time, we first
rewrite

$\begin{split}\mathcal{J}_{2}&=\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\partial_{s}(\hat{c}\,\alpha\,\varphi\,)dx_{2}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\partial_{s}\hat{c}\,\alpha\,\varphi\,dx_{2}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\alpha\,\partial_{s}\varphi\,dx_{2}ds\\\
&=\int_{\mathbb{R}}\hat{c}(t)\,\alpha(t)\,\varphi(t)dx_{2}-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\partial_{s}\hat{c}\,\alpha\,\varphi\,dx_{2}ds-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\alpha\,\partial_{s}\varphi\,dx_{2}ds\\\
&=\mathcal{J}_{2,1}+\mathcal{J}_{2,2}+\mathcal{J}_{2,3}\,.\end{split}$

The middle integral $\mathcal{J}_{2,2}$ is estimated at once by
Cauchy–Schwarz, Young’s inqualities and the trace type inequality (B.18)

$\begin{split}|\mathcal{J}_{2,2}|=\left|-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\partial_{s}\hat{c}\,\alpha\,\varphi\,dx_{2}ds\right|&\leq
C_{3}\left\\{\|{\mathbf{V}}\|_{L^{2}(\Omega_{t})}^{2}+\|\partial_{1}{\mathbf{V}}_{n}\|_{L^{2}(\Omega_{t})}^{2}+\|\varphi\|^{2}_{L^{2}(\Gamma_{t})}\right\\}\\\
&=C_{3}\left\\{\int_{0}^{t}(I(s)+I_{1,n}(s))ds+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\,.\end{split}$
(B.62)

Concerning $\mathcal{J}_{2,3}$, we substitute $\partial_{t}\varphi$ with the
right-hand side of (B.55) to rewrite it as

$\mathcal{J}_{2,3}=-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\alpha\,(\dot{u}^{+}_{N}+\hat{D}_{1}\dot{H}^{+}_{N}+\hat{D}_{2}\dot{H}^{-}_{N}+\hat{D}_{3}\varphi)\,dx_{2}ds\cong\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\alpha\,\beta\,dx_{2}\,ds+\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\hat{c}\,\alpha\,\varphi\,dx_{2}\,ds$

where the second type of boundary integral above is exactly the same as
$\mathcal{J}_{2,2}$ and the first one is trivially estimated by using once
again the trace type inequality, hence we get the same estimate (B.62)

$\begin{split}|\mathcal{J}_{2,3}|\leq
C_{3}\left\\{\int_{0}^{t}(I(s)+I_{1,n}(s))ds+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\,.\end{split}$
(B.63)

Concerning, instead, the first spatial integral $\mathcal{J}_{2,1}$, the use
of Cauchy–Schwarz and weighted Young’s inequalities and the trace type
inequality (B.19) gives

$\begin{split}|\mathcal{J}_{2,1}|&\leq\varepsilon\|\alpha(t)|_{x_{1}=0}\|_{L^{2}(\mathbb{R})}^{2}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\\
&\leq\varepsilon\left\\{\|\alpha(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\|\partial_{1}\alpha(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\\
&\leq\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\,.\end{split}$
(B.64)

Gathering (B.64), (B.62) and (B.63) we get

$\begin{split}|\mathcal{J}_{2}|\leq&\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\\
&+C_{3}\left\\{\int_{0}^{t}(I(s)+I_{1,n}(s))ds+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\,.\end{split}$
(B.65)

Summing the estimate of $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ above, we end
up with the following estimate of the boundary integral of the quadratic form
(B.52)

$\begin{split}&\Big{|}\int_{0}^{t}\int_{\mathbb{R}}(\mathcal{B}_{1}{\mathbf{V}}_{s},{\mathbf{V}}_{s})|_{x_{1}=0}dx_{2}\,ds\Big{|}\cong|\mathcal{J}_{1}+\mathcal{J}_{2}|\\\
&\quad\leq\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\left\\{\|{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}\right\\}\\\
&\qquad+\frac{C_{3}}{\varepsilon}\left\\{\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}ds\right\\}\,.\end{split}$
(B.66)

Using (B.50), (B.51) and (B.66) together with the counterpart of (B.16) with
${\mathbf{V}}_{t}$ instead of ${\mathbf{V}}$, from (B.49) we obtain

$\begin{split}I_{0}(t)\leq&\frac{C_{3}}{\varepsilon}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{s}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\int_{0}^{t}(I_{1,\ast}+I_{1,n})(s)ds+\int_{0}^{t}\|\varphi(s)\|_{L^{2}(\mathbb{R})}^{2}ds\right\\}\\\
&+\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}\,.\end{split}$
(B.67)

###  B.6. Estimate of the front $\varphi$

To estimate the $L^{2}-$norm of the front we multiply by $\varphi$ the first
boundary condition in (B.1) (we choose the $+$ side) and integrate by parts
over $\Gamma_{t}$ to get

$\int_{\mathbb{R}}|\varphi(t)|^{2}dx_{2}-\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}(\partial_{2}\hat{u}^{+}_{2})\varphi^{2}\,dx_{2}\,ds-2\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}\dot{u}^{+}_{N}\varphi\,dx_{2}\,ds-2\int_{0}^{t}\\!\\!\\!\int_{\mathbb{R}}(\partial_{1}\hat{u}^{+}_{N})\varphi^{2}\,dx_{2}\,ds=0\,,$
(B.68)

from which, by Cauchy–Schwarz and Young’s inequalities, and the trace type
inequality (B.18)

$\begin{split}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}&\leq
C_{1}\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}\,ds+\|\dot{u}^{+}_{N}|_{x_{1}=0}\|^{2}_{L^{2}(\Gamma_{t})}\\\
&\leq
C_{1}\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}\,ds+\|\partial_{1}{\mathbf{V}}_{n}^{+}\|^{2}_{L^{2}(\Omega_{t})}+\|{\mathbf{V}}^{+}\|^{2}_{L^{2}(\Omega_{t})}\\\
&\leq
C_{1}\int_{0}^{t}\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})}\,ds+\int_{0}^{t}(I_{1,n}+I)(s)ds\,.\end{split}$
(B.69)

Using the expressions of $\partial_{2}\varphi$ and $\partial_{t}\varphi$ in
(B.42) and (B.55) we can recover an estimate of the $L^{2}-$norms of those
derivatives from the estimate (B.69) and (B.18). Precisely, integrating over
$\mathbb{R}$ (B.42) and (B.55) and using (B.18) we get

$\begin{split}\|\partial_{2}\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}&\leq
C_{2}\\{\|{\mathbf{V}}_{n}(t)|_{x_{1}=0}\|^{2}_{L^{2}(\mathbb{R})}+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\}\\\
&\leq
C_{2}\\{I(t)+I_{1,n}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\}\,,\end{split}$
(B.70)

see (B.4), (B.6), and analogously

$\begin{split}\|\partial_{t}\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq
C_{2}\\{I(t)+I_{1,n}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\}\,.\end{split}$
(B.71)

###  B.7. $H^{1}_{\ast}-$estimate

We add estimates (B.20), (B.33), (B.47), (B.67) and (B.69) to get

$\begin{split}I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}&=I(t)+I_{0}(t)+I_{\sigma}(t)+I_{2}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\\\
&\leq\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}(s)+I_{1,n}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds\\\
&\quad+\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}\\\
&\quad+\frac{C_{3}}{\varepsilon}\left\\{\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\sigma\partial_{1}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\partial_{2}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}+\|\partial_{s}{\mathbf{F}}\|_{L^{2}(\Omega_{t})}^{2}\right\\}\\\
&\leq\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}(s)+I_{1,n}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds\\\
&\quad+\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{2}}{\varepsilon}\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}+\frac{C_{3}}{\varepsilon}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}\,.\end{split}$

Then using (B.69) to estimate $\|\varphi(t)\|_{L^{2}(\mathbb{R})}^{2}$ in the
right-hand side above, the previous inequality reduces to

$\begin{split}I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq&\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}(s)+I_{1,n}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds\\\
&+\varepsilon\left\\{I(t)+I_{1,n}(t)\right\\}+\frac{C_{3}}{\varepsilon}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}\,.\end{split}$

Then we use (B.26) to get a control of the normal derivatives of
${\mathbf{V}}^{\pm}_{n}$ involved in the term $I_{1,n}$ in the right-hand side
above to get

$\begin{split}I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq&\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds\\\
&+C_{1}\varepsilon\left\\{I_{1,\ast}(t)+\|{\mathbf{F}}(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\right\\}+\frac{C_{3}}{\varepsilon}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}\,.\end{split}$

In order to estimate the spatial $L^{2}-$norm of the source ${\mathbf{F}}(t)$
in the right-hand side above we use the following argument

$\begin{split}\|{\mathbf{F}}(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}&=\int_{\mathbb{R}^{2}_{+}}|{\mathbf{F}}(t)|^{2}d{\bf
x}=\int_{\mathbb{R}^{2}_{+}}\int_{0}^{t}\partial_{s}(|{\mathbf{F}}(s)|^{2})ds\,d{\bf
x}=\int_{\mathbb{R}^{2}_{+}}\int_{0}^{t}2{\mathbf{F}}(s)\partial_{s}{\mathbf{F}}(s)\,ds\,d{\bf
x}\\\
&\leq\int_{\mathbb{R}^{2}_{+}}\int_{0}^{t}|{\mathbf{F}}(s)|^{2}\,ds\,d{\bf
x}+\int_{\mathbb{R}^{2}_{+}}\int_{0}^{t}|\partial_{s}{\mathbf{F}}(s)|^{2}\,ds\,d{\bf
x}=\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}+\|\partial_{s}{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{t})}\,,\end{split}$

so that

$\begin{split}I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq&\frac{C_{3}}{\varepsilon}\int_{0}^{t}(I_{1,\ast}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds+C_{1}\varepsilon
I_{1,\ast}(t)+\frac{C_{3}}{\varepsilon}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}\,,\end{split}$

hence taking $\varepsilon>0$ small enough in order to absorb $C_{1}\varepsilon
I_{1,\ast}(t)$ in the left-hand side above, we end up with

$\begin{split}I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq&C_{3}\int_{0}^{t}(I_{1,\ast}(s)+\|\varphi(s)\|^{2}_{L^{2}(\mathbb{R})})ds+C_{3}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}\,.\end{split}$

Applying Grönwall’s lemma, we obtain

$I_{1,\ast}(t)+\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}\leq
C_{3}\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{t})}e^{C_{3}t}$

and integrating on $t$ over $(0,T)$

$\|{\mathbf{V}}\|^{2}_{1,\ast,T}+\|\varphi\|^{2}_{L^{2}(\Gamma_{T})}\leq
C\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{T})}\,,$ (B.72)

where $C$ is a positive constant depending only on $T$ and $K$ from (4.6).
From the use of estimates (B.70), (B.71) we can manage to include the
$L^{2}(\Gamma_{T})-$norms of $\nabla_{t,x_{2}}\varphi$ in the left-hand side
of (B.72), so as to get a control of the $H^{1}(\Gamma_{T})-$norm of the front
$\varphi$.
From integration of (B.70), (B.71) with respect to $t$ over $(0,T)$ and the
use of (B.26) we get

$\begin{split}\|\nabla_{t,x_{2}}\varphi\|^{2}_{L^{2}(\Gamma_{T})}&\leq
C_{2}\left\\{\int_{0}^{T}(I(t)+I_{1,n}(t))dt+\int_{0}^{T}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}dt\right\\}\\\
&\leq
C_{2}\left\\{\int_{0}^{T}I_{1,\ast}(t)\,dt+\int_{0}^{T}\|{\mathbf{F}}(t)\|^{2}_{L^{2}(\mathbb{R}^{2}_{+})}dt+\int_{0}^{T}\|\varphi(t)\|^{2}_{L^{2}(\mathbb{R})}dt\right\\}\\\
&\leq
C_{2}\left\\{\|{\mathbf{V}}\|^{2}_{1,\ast,T}+\|{\mathbf{F}}\|^{2}_{L^{2}(\Omega_{T})}+\|\varphi\|^{2}_{L^{2}(\Gamma_{T})}\right\\}\,,\end{split}$

where, in the last inequality, it is used that

$\int_{0}^{T}I_{1,\ast}(t)\,dt\cong\|{\mathbf{V}}\|^{2}_{1,\ast,T}\,.$

Using (B.72) to estimate the right-hand side above, we end up with

$\begin{split}\|\nabla_{t,x_{2}}\varphi\|^{2}_{L^{2}(\Gamma_{T})}\leq
C\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{T})}\,.\end{split}$ (B.73)

Then adding the latter to (B.72) we get

$\|{\mathbf{V}}\|^{2}_{1,\ast,T}+\|\varphi\|^{2}_{H^{1}(\Gamma_{T})}\leq
C\|{\mathbf{F}}\|^{2}_{H^{1}_{\ast}(\Omega_{T})}\,,$ (B.74)

which is just the estimate (5.33), in view of (5.1) (recalling that to
shortcut notation we have set $\dot{\mathbf{U}}=\dot{\mathbf{U}}^{\natural}$
and $\dot{\mathbf{U}}=J{\mathbf{V}}$, see (5.1)).

## Acknowledgments

The authors are grateful to the anonymous referee for her/his comments and
suggestions, that have contributed to improve this paper.

The research of A. Morando, P. Secchi, P. Trebeschi was supported in part by
the Italian MUR Project PRIN prot. 20204NT8W4. D. Yuan was supported by NSFC
Grant No.12001045 and China Postdoctoral Science Foundation No.2020M680428,
No.2021T140063. D. Yuan thanks the Department of Mathematics of the University
of Brescia for its kind hospitality.

## Statements and Declarations

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