$\displaystyle~{}\left(\partial^{\varphi}_{3}\bar{v}^{-}\cdot\overline{\nabla}\right)\partial_{t}^{7}v\cdot\mathbf{N}+\partial_{3}\left((\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)\right)\partial^{\varphi}_{3}\partial_{t}^{7}v\cdot\mathbf{N}.$
(3.124)

Thus, we find that, apart from the term
$(\partial_{3}\varphi)^{-1}\left[D_{t}^{\varphi,-}\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}\right]$,
all the other terms in $\mathfrak{C}_{i}^{*}(v_{i})$ include either a
tangential derivative falling on the leading order term or the term
$\partial^{\varphi}_{3}v\cdot\mathbf{N}$ (possibly with some derivatives) such
that $\mathcal{F}_{p}D_{t}^{\varphi}p$ and $\overline{\nabla}\cdot\bar{v}$ are
produced by invoking the continuity equation. Thus, when $\mathbf{Q}^{*}$ is
multiplied with these terms, its contribution in $Z^{*,\pm}$ can be directly
controlled without any loss of weights of Mach number.

It now remains to control $ZB^{*,\pm}_{0}+Z^{*,\pm}$ with
$Z_{0}^{*,\pm}:=\varepsilon^{16}\int_{\Omega^{\pm}}\mathbf{Q}^{*,\pm}(\partial_{3}\varphi)^{-1}\left[D_{t}^{\varphi,-}\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}\right]\,\mathrm{d}\mathcal{V}_{t}$.
Using $\,\mathrm{d}\mathcal{V}_{t}=\partial_{3}\varphi\,\mathrm{d}x$ and
Gauss-Green formula, we have

$\displaystyle
ZB^{*,\pm}_{0}+Z_{0}^{*,\pm}=\mp\varepsilon^{16}\int_{\Sigma}\mathbf{Q}^{*,\pm}\,\left[D_{t}^{\varphi,-}\partial_{t}^{7},N_{i},v_{i}^{\pm}\right]\,\mathrm{d}x^{\prime}+\varepsilon^{16}\int_{\Omega^{\pm}}\mathbf{Q}^{*,\pm}\left[D_{t}^{\varphi,-}\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}^{\pm}\right]\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{1}\sum_{k=1}^{6}\varepsilon^{16}\binom{7}{k}\left(\pm\int_{\Sigma}\mathbf{Q}^{*,\pm}(D_{t}^{\varphi,-})^{j}\partial_{t}^{k}v_{i}^{\pm}\,(D_{t}^{\varphi,-})^{1-j}\partial_{t}^{6-k}N_{i}\,\mathrm{d}x^{\prime}+\int_{\Omega^{\pm}}\mathbf{Q}^{*,\pm}(D_{t}^{\varphi,-})^{j}\partial_{t}^{k}\partial_{3}v_{i}^{\pm}\,(D_{t}^{\varphi,-})^{1-j}\partial_{t}^{6-k}\mathbf{N}_{i}\,\mathrm{d}x\right)$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\sum_{j=0}^{1}\sum_{k=1}^{6}\varepsilon^{16}\binom{7}{k}\int_{\Omega^{\pm}}\partial^{\varphi}_{3}\mathbf{Q}^{*,\pm}(D_{t}^{\varphi,-})^{j}\partial_{t}^{k}v_{i}^{\pm}\,(D_{t}^{\varphi,-})^{1-j}\partial_{t}^{6-k}\mathbf{N}_{i}\,\mathrm{d}\mathcal{V}_{t}.$
(3.125)

Recall that
$\mathbf{Q}^{*}=D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}-D_{t}^{\varphi,-}\partial_{t}^{7}\varphi\partial^{\varphi}_{3}q$,
we can integrate by parts this $D_{t}^{\varphi,-}$ under time integral and
invoke the momentum equation to reduce $\partial^{\varphi}_{3}\mathbf{Q}^{*}$
to $-\rho
D_{t}^{\varphi}\mathbf{V}^{*}+(b\cdot\nabla^{\varphi})\mathbf{B}^{*}+$lower
order terms. Note that
$[\partial^{\varphi}_{3},D_{t}^{\varphi,-}]\partial_{t}^{7}q=\partial^{\varphi}_{3}v_{j}^{-}\partial^{\varphi}_{j}q=(\partial^{\varphi}_{3}\bar{v}^{-}\cdot\overline{\nabla})\partial_{t}^{7}q+(\partial^{\varphi}_{3}v^{-}\cdot\mathbf{N})\partial_{t}^{7}\partial_{3}q$,
so one can still convert the normal derivative $\partial q$ to a tangential
derivative of $v,b$. Thus, we have

$\displaystyle\int_{0}^{t}ZB^{*,\pm}_{0}+Z_{0}^{*,\pm}\,\mathrm{d}\tau\stackrel{{\scriptstyle
L}}{{=}}$
$\displaystyle\sum_{j=0}^{1}\sum_{k=1}^{6}\varepsilon^{16}\binom{7}{k}\int_{0}^{t}\int_{\Omega^{\pm}}\partial^{\varphi}_{3}\mathbf{Q}^{\sharp,\pm}D_{t}^{\varphi,-}\left((D_{t}^{\varphi,-})^{j}\partial_{t}^{k}v_{i}^{\pm}\cdot,(D_{t}^{\varphi,-})^{1-j}\partial_{t}^{6-k}\mathbf{N}\right)\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle+\varepsilon^{16}\binom{7}{k}\int_{0}^{t}\int_{\Omega^{\pm}}\partial^{\varphi}_{3}\mathbf{Q}^{\sharp,\pm}\left((D_{t}^{\varphi,-})^{j}\partial_{t}^{k}v_{i}^{\pm}\cdot,(D_{t}^{\varphi,-})^{1-j}\partial_{t}^{6-k}\mathbf{N}\right)\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}$
$\displaystyle\lesssim$
$\displaystyle~{}\delta\|\partial^{\varphi}_{3}\mathbf{Q}^{\sharp,\pm}\|_{0}^{2}+P(E^{\kappa}(0))+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}~{}\forall\delta\in(0,1).$
(3.126)

Combining this with the control of remainder terms and commutators, we can
easily obtain that

$\displaystyle\int_{0}^{t}ZB^{*,\pm}+Z^{*,\pm}\,\mathrm{d}\tau\lesssim\delta
E_{8}^{\kappa}(0)+P(E^{\kappa}(0))+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}~{}\forall\delta\in(0,1).$
(3.127)

###  3.5 Tangential estimates: general cases and summary

Let
$\mathcal{T}^{\gamma}=(\omega(x_{3})\partial_{3})^{\gamma_{4}}\partial_{t}^{\gamma_{0}}\overline{\partial}{}_{1}^{\gamma_{1}}\overline{\partial}{}_{2}^{\gamma_{2}}$
be a tangential derivative with length of the multi-index
$\langle\gamma\rangle:=\gamma_{0}+\gamma_{1}+\gamma_{2}+2\times 0+\gamma_{4}$.
Section 3.3.1-Section 3.4 are devoted to the control of full spatial
derivatives ($\gamma_{1}+\gamma_{2}=\langle\gamma\rangle$) and full time
derivatives ($\gamma_{0}=\langle\gamma\rangle$). Now we analyze how to handle
the general case.

#### Space-time mixed derivatives: $\gamma_{0}>0$ and
$\gamma_{1}+\gamma_{2}>0$

Let us temporarily assume $\gamma_{4}=0$. In this case, the tangential
derivatives that we need to consider have the form
$\overline{\partial}{}^{4-l-k}\partial_{t}^{k}\mathcal{T}^{\alpha}$ with
$\langle\alpha\rangle=2l,~{}\alpha_{4}=0$ and weights of Mach number
$\varepsilon^{2l}$. That is, we need to consider
$\varepsilon^{2l}\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{4+l-k-\alpha_{0}}$-estimates.
Following the previous paper [55] by Luo and the author, the control of space-
time mixed tangential derivatives ($0<k+\alpha_{0}<4+l$) is the same as the
control of purely spatial tangential derivatives. In particular, compared with
the one-phase fluid problem [55], we only need one spatial derivative to do
integration by parts in order for the control of the extra problematic term

$\text{
VS}:=\varepsilon^{4l}\int_{\Sigma}\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{4+l-k-\alpha_{0}}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{4+l-k-\alpha_{0}}\psi\,\mathrm{d}x^{\prime}$

in which we need to integrate by parts $\overline{\partial}{}^{1/2}$ and seek
for the control of
$\varepsilon^{2l}\left|\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{4+l-k-\alpha_{0}}\right|_{1.5}$.
Mimicing the proof of Lemma 3.5, we can show that (replacing $k+\alpha_{0}$ by
$k$)

######  Lemma 3.7 (Elliptic estimate for the time derivatives of the free
interface).

Fix $l\in\\{0,1,2,3,4\\}$. For $0<k<4+l$, we have the following uniform-
in-$(\varepsilon,\kappa)$ inequality, in which the first term on the right
side disappears when $\kappa=0$.

$\displaystyle\left|\varepsilon^{2l}\partial_{t}^{k}\psi\right|_{5.5+l-k}\leq$
$\displaystyle~{}\left|\varepsilon^{2l}\partial_{t}^{k}\psi(0)\right|_{5.5+l-k}+\sigma^{-1}\left|\varepsilon^{2l}\partial_{t}^{k}\left\llbracket
q\right\rrbracket\right|_{3.5+l-k}$
$\displaystyle+P\left(\sigma^{-1},|\overline{\nabla}\psi|_{L^{\infty}},\sum_{j=0}^{l}E_{4+j}^{\kappa}(t)\right)\left(\left|\varepsilon^{2l}\partial_{t}^{k}\psi\right|_{4.5+l-k}+\left|\varepsilon^{2l}\partial_{t}^{k-1}\psi\right|_{5.5+l-k}\right).$

#### Weighted normal derivatives: $\gamma_{4}>0$

In the most general case, $\mathcal{T}^{\gamma}$ may contain weighted normal
derivative $\omega(x_{3})\partial_{3}$, so we have to analyze the commutator
involving $[\mathcal{T}^{\gamma},\partial_{3}]$ in $\mathfrak{C}(f)$ and
$\mathfrak{D}(f)$ defined in (3.2)-(3.2). The problematic thing is that
$\partial_{3}$ may fall on $\omega(x_{3})$ which converts a “tangential”
derivative $\omega(x_{3})\partial_{3}$ (a first-order derivative) to a normal
derivative $\partial_{3}$ (considered to be second-order under the setting of
anisotropic Sobolev spaces). Such terms in $\mathfrak{D}(f)$ are

$(\partial_{3}\varphi)^{-1}(v\cdot\mathbf{N}-\partial_{t}\varphi)[\mathcal{T}^{\gamma},\partial_{3}]f+(v\cdot\mathbf{N}-\partial_{t}\varphi)\frac{\partial_{3}f}{(\partial_{3}\varphi)^{2}}[\mathcal{T}^{\gamma},\partial_{3}]\varphi.$

They can be directly controlled because an extra weight
$(v\cdot\mathbf{N}-\partial_{t}\varphi)$, which vanishes on $\Sigma$, is
automatically generated to compensate the possible loss of weight function. As
for $\mathfrak{C}(f)$, we notice that the terms involving
$[\mathcal{T}^{\gamma},\partial_{3}]$ can be written to be

$\frac{\mathbf{N}_{i}}{\partial_{3}\varphi}[\mathcal{T}^{\gamma},\partial_{3}]f-\frac{\mathbf{N}_{i}}{\partial_{3}\varphi}\partial^{\varphi}_{3}f[\mathcal{T}^{\gamma},\partial_{3}]\varphi,~{}~{}f=q\text{
or }v_{i}.$

The second term above is easy to control because
$\varphi(t,x)=x_{3}+\chi(x_{3})\psi(t,x^{\prime})$ implies the
$C^{\infty}$-regularity of $\varphi$ in $x_{3}$ direction. For the first term,
it may generate a term $\mathcal{T}^{\beta}\partial_{3}f\mathbf{N}_{i}$ with
$\beta_{i}=\gamma_{i}(i=0,1,2),~{}\beta_{4}=\gamma_{4}-1$, whose
$L^{2}(\Omega)$ norm may be not directly bounded. Luckily, for $f=q$ or
$v_{i}$, we can again invoke the momentum equation or the continuity equation
to reduce $-\partial^{\varphi}_{3}q$ and
$\partial^{\varphi}_{3}v\cdot\mathbf{N}$ to tangential derivatives $\rho
D_{t}^{\varphi}v-(b\cdot\nabla^{\varphi})b$ and
$-\mathcal{F}_{p}D_{t}^{\varphi}p-\overline{\nabla}\cdot\bar{v}$ respectively.
Therefore, there is no extra loss of derivative in the commutators
$\mathfrak{C}(f)$ and $\mathfrak{D}(f)$ when $\gamma_{4}>0$.

#### Summary of tangential estimates

Finally, we need to recover the estimates of
$\mathcal{T}^{\gamma}(v,b,S,\sqrt{\mathcal{F}_{p}}p)$ from the
$L^{2}$-estimates of their Alinhac good unknowns. By definition, we have

$\displaystyle\left\|\mathcal{T}^{\gamma}f^{\pm}\right\|_{0,\pm}^{2}\leq\left\|\mathbf{F}^{\gamma,\pm}\right\|_{0,\pm}^{2}+\left|\mathcal{T}^{\gamma}\psi\right|_{0}^{2}\|\partial^{\varphi}_{3}f^{\pm}\|_{L^{\infty}(\Omega^{\pm})}^{2},$

in which $\left\|\mathbf{F}^{\gamma,\pm}\right\|_{0,\pm}$ and
$\left|\mathcal{T}^{\gamma}\psi\right|_{0}$ have been controlled by $\delta
E^{\kappa}(t)+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau$.
When $\mathcal{T}^{\gamma}$ contains at least one spatial derivative, we can
use $-\mathcal{T}q\sim D_{t}^{\varphi}v+(b\cdot\nabla^{\varphi})b$ to get the
control of $\mathcal{T}q$ instead of $\sqrt{\mathcal{F}_{p}}\mathcal{T}q$. For
the full time derivatives, we use
$D_{t}^{\varphi,-}=\partial_{t}+(\bar{v}^{-}\cdot\overline{\nabla})+(\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}$
to convert the $\varepsilon^{2l}\partial_{t}^{4+l}$-estimate to
$\varepsilon^{2l}D_{t}^{\varphi,-}\partial_{t}^{3+l}$-estimate,
$\varepsilon^{2l}\overline{\partial}{}\partial_{t}^{3+l}$-estimate and
$\varepsilon^{2l}(\omega\partial_{3})\partial_{t}^{3+l}$-estimate, in the
second part of which the norm
$|\varepsilon^{2l}\partial_{t}^{3+l}\psi(0)|_{2.5}$ is needed to control the
VS term. Also, since $\omega(x_{3})=0$ on the interface,
$\mathcal{T}^{\gamma}$ can be expressed as
$\overline{\partial}{}^{4+l-k}\partial_{t}^{k}$ for $0\leq k\leq 4+l,~{}0\leq
l\leq 4$. Hence, we conclude the tangential estimates by the following
inequalities

$\displaystyle\sum_{\pm}\sum_{\langle\alpha\rangle=2l}\sum_{\begin{subarray}{c}0\leq
k\leq 4-l\\\
k+\alpha_{0}<4+l\end{subarray}}\left\|\left(\varepsilon^{2l}\overline{\partial}{}^{4-k-l}\mathcal{T}^{\alpha}\partial_{t}^{k}(v^{\pm},b^{\pm},S^{\pm},p^{\pm})\right)\right\|^{2}_{0,\pm}$
$\displaystyle+\sum_{k=0}^{4+l}\left|\sqrt{\sigma}\varepsilon^{2l}\partial_{t}^{k}\psi\right|^{2}_{5+l-k}+\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\psi\right|_{6+l-k}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k+1}\psi(\tau)\right|_{5+l-k}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\sum_{k=0}^{3+l}\left|\varepsilon^{2l}\partial_{t}^{k}\psi(0)\right|_{5.5+l-k}^{2}+\sum_{j=0}^{l}P\left(\sigma^{-1},E_{4+j}^{\kappa}(0)\right)$
$\displaystyle+P\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(t)\right)\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau$
(3.128)

and

$\displaystyle\sum_{\pm}\sum_{k=0}^{4-l}\left\|\left(\varepsilon^{2l}\partial_{t}^{4+l}(v^{\pm},b^{\pm},S^{\pm},(\mathcal{F}_{p})^{\frac{1}{2}}p^{\pm})\right)\right\|^{2}_{4-k-l,\pm}$
$\displaystyle+\left|\sqrt{\sigma}\varepsilon^{2l}\partial_{t}^{4+l}\psi\right|^{2}_{1}+\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{4+l}\psi\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{5+l}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\left|\varepsilon^{2l}\partial_{t}^{3+l}\psi(0)\right|_{2.5}^{2}+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(t)\right)\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau.$
(3.129)

These are exactly the desired uniform-in-$(\kappa,\varepsilon)$ energy
estimates in Proposition 3.3.

###  3.6 Div-Curl analysis and reduction of pressure

The tangential derivatives of the variables $(v,b,p)$ are analyzed in Section
3.3-Section 3.5. Here we show the reduction of normal derivatives of pressure
and the analysis for the divergence and vorticity. We use the div-curl
decomposition (cf. Lemma B.1) such that the normal derivatives of $(v,b)$ are
controlled via their divergence and curl parts. For $0\leq l\leq 3,~{}0\leq
k\leq 3-l,~{}\langle\alpha\rangle=2l,~{}\alpha_{3}=0$, we have

$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{4-k-l,\pm}^{2}\leq
C\bigg{(}$
$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{3-k-l,\pm}^{2}$
$\displaystyle+\left\|\varepsilon^{2l}\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{3-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\overline{\partial}{}^{4-k-l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}\bigg{)}$
(3.130)

with

$C=C\left(\sum_{j=0}^{l}\sum_{k=0}^{3+j}|\varepsilon^{2j}\partial_{t}^{j}\psi|_{4+l-j}^{2},|\overline{\nabla}\psi|_{W^{1,\infty}}\right)>0$

a positive continuious function linear in
$|\varepsilon^{2j}\partial_{t}^{j}\psi|_{4+l-j}^{2}$. The conclusion for the
div-curl analysis is

######  Proposition 3.8.

Fix $l\in\\{0,1,2,3\\}$. For any $0\leq k\leq l-1$, any multi-index $\alpha$
satisfying $\langle\alpha\rangle=2l$ and any constant $\delta\in(0,1)$, we can
prove the following estimates for the curl part

$\displaystyle\left\|\varepsilon^{2l}\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{3-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm}\right\|_{3-k-l,\pm}^{2}$
(3.131) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)+E_{l+1}^{\kappa}(\tau)\,\mathrm{d}\tau,$

and for the divergence part

$\displaystyle\left\|\varepsilon^{2l}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{3-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{3-k-l,\pm}^{2}$
(3.132) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau.$

####  3.6.1 Reduction of pressure and divergence

Let us start with $l=0$. The spatial derivative of $q$ is controlled by
invoking the momentum equation:

$\displaystyle-\partial_{3}q=$
$\displaystyle~{}(\partial_{3}\varphi)\left(\rho
D_{t}^{\varphi}v_{3}-(b\cdot\nabla^{\varphi})b_{3}\right);$ (3.133)
$\displaystyle-\overline{\partial}{}_{i}q=$
$\displaystyle-(\partial_{3}\varphi)^{-1}\overline{\partial}{}_{i}\varphi\,\partial_{3}q+\rho
D_{t}^{\varphi}v_{i}-(b\cdot\nabla^{\varphi})b_{i},~{}~{}i=1,2.$ (3.134)

Let $\mathcal{T}$ be $\partial_{t}$ or $\overline{\partial}{}$ or
$\omega(x_{3})\partial_{3}$. Then we have

$\displaystyle\|\partial_{t}^{k}\partial_{3}q\|_{3-k}\lesssim$
$\displaystyle~{}\|\partial_{t}^{k}(\rho\mathcal{T}v_{3})\|_{3-k}+\|\partial_{t}^{k}(b\mathcal{T}b_{3})\|_{3-k}$
(3.135)
$\displaystyle\|\partial_{t}^{k}\overline{\partial}{}_{i}q\|_{3-k}\lesssim$
$\displaystyle~{}\|\partial_{t}^{k}(\overline{\partial}{}_{i}\varphi\partial_{3}q)\|_{3-k}+\|\partial_{t}^{k}(\rho\mathcal{T}v_{i})\|_{3-k}+\|\partial_{t}^{k}(b\mathcal{T}b_{i})\|_{3-k},$
(3.136)

in which the leading order terms are
$\|\partial_{t}^{k}\mathcal{T}(v,b)\|_{3-k}$ and
$|\partial_{t}^{k}\psi|_{4-k}$. This shows that we can convert the control of
spatial derivative of $q$ to tangential estimates of $v$ and $b$.

Next we turn to the div-curl analysis for $v,b$. Let us first analyze
$E_{4}(t)$. For $0\leq k\leq 3$, we have

$\displaystyle\|\partial_{t}^{k}v\|_{4-k}^{2}\leq
C(|\psi|_{4-k},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|\partial_{t}^{k}v\|_{0}^{2}+\|\nabla^{\varphi}\cdot\partial_{t}^{k}v\|_{3-k}^{2}+\|\nabla^{\varphi}\times\partial_{t}^{k}v\|_{3-k}^{2}+\|\overline{\partial}{}^{4-k}\partial_{t}^{k}v\|_{0}^{2}\right),$
(3.137) $\displaystyle\|\partial_{t}^{k}b\|_{4-k}^{2}\leq
C(|\psi|_{4-k},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|\partial_{t}^{k}b\|_{0}^{2}+\|\nabla^{\varphi}\cdot\partial_{t}^{k}b\|_{3-k}^{2}+\|\nabla^{\varphi}\times\partial_{t}^{k}b\|_{3-k}^{2}+\|\overline{\partial}{}^{4-k}\partial_{t}^{k}b\|_{0}^{2}\right).$
(3.138)

For the divergence, we can directly invoke the continuity equation to convert
$\nabla^{\varphi}\cdot v$ to time derivative of $p$ together with square
weights of Mach number. When $k=0$, we have

$\displaystyle\|\nabla^{\varphi}\cdot
v\|_{3}^{2}=\|\mathcal{F}_{p}D_{t}^{\varphi}p\|_{3}^{2},$ (3.139)

which is further reduced to the tangential derivatives of $v$ and $b$ by using
the above reduction of $q$. Note that the magnetic tension term
$\frac{1}{2}|b|^{2}$ in the total pressure $q$ does not involve extra normal
derivatives thanks to $\mathcal{T}(\frac{1}{2}|b|^{2})=b\cdot\mathcal{T}b$.
Taking $\partial_{t}$ in the continuity equation and omitting lower order
terms, we have

$\nabla^{\varphi}\cdot\partial_{t}^{k}v\stackrel{{\scriptstyle
L}}{{=}}-\mathcal{F}_{p}\partial_{t}^{k}D_{t}^{\varphi}p+(\partial_{3}\varphi)^{-1}\overline{\partial}{}\partial_{t}^{k}\varphi\cdot\partial_{3}v,$

which gives

$\displaystyle\|\nabla^{\varphi}\cdot\partial_{t}^{k}v^{\pm}\|_{3-k,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left(\left\|\mathcal{F}_{p}^{\pm}\partial_{t}^{k}\mathcal{T}p^{\pm}\right\|_{3-k,\pm}^{2}+\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}\right)+\text{
lower order terms}.$ (3.140)

Again, this can be reduced to tangential derivatives of $v,b$ until there is
no spatial derivative falling on $p$. As for the divergence of magnetic
fields, we can invoke the div-free constraint to convert it to lower order
terms. Namely, using $\nabla^{\varphi}\cdot b=0$, we have

$\nabla^{\varphi}\cdot\partial_{t}^{k}b\stackrel{{\scriptstyle
L}}{{=}}\underbrace{\partial_{t}^{k}(\nabla^{\varphi}\cdot
b)}_{=0}+(\partial_{3}\varphi)^{-1}\overline{\partial}{}\partial_{t}^{k}\varphi\cdot\partial_{3}b$

and thus

$\displaystyle\|\nabla^{\varphi}\cdot\partial_{t}^{k}b^{\pm}\|_{3-k,\pm}^{2}\lesssim
C(\|b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}+\text{
lower order terms}.$ (3.141)

The term $\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}$ has been controlled in
tangential estimates of $E_{4}^{\kappa}(t)$. Combining the result of
tangential estimates in Proposition 3.3, the control of divergence of time
derivatives is concluded by

$\displaystyle\left\|\nabla^{\varphi}\cdot\partial_{t}^{k}(v^{\pm},b^{\pm})\right\|_{3-k,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\mathcal{F}_{p}^{\pm}\partial_{t}^{k}\mathcal{T}p^{\pm}\right\|_{3-k,\pm}^{2}+C(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}$
(3.142) $\displaystyle\lesssim$
$\displaystyle~{}C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\mathcal{F}_{p}^{\pm}\partial_{t}^{k}\mathcal{T}p^{\pm}\right\|_{3-k,\pm}^{2}+\delta
E_{4}^{\kappa}(t)+P(E_{4}^{\kappa}(0))+P(E_{4}^{\kappa}(t))\int_{0}^{t}P(E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau,$

where the term involving $p^{\pm}$ can be further reduced to
$\mathcal{T}(v^{\pm},b^{\pm})$ when $3-k>0$ so that one can further apply the
div-curl analysis to it.

####  3.6.2 Vorticity analysis for $E_{4}$

Taking $\nabla^{\varphi}\times$ in the momentum equation of $v$ and the
evolution equation of $b$, we get the evolution equation for the vorticity
$\nabla^{\varphi}\times v$ and the current density $\nabla^{\varphi}\times b$

$\displaystyle\rho D_{t}^{\varphi}(\nabla^{\varphi}\times
v)-(b\cdot\nabla^{\varphi})(\nabla^{\varphi}\times b)=$
$\displaystyle~{}-(\nabla^{\varphi}\rho)\times(D_{t}^{\varphi}v)-\rho(\nabla^{\varphi}v_{j})\times(\partial^{\varphi}_{j}v)+(\nabla^{\varphi}b_{j})\times(\partial^{\varphi}_{j}b),$
(3.143) $\displaystyle D_{t}^{\varphi}(\nabla^{\varphi}\times
b)-(b\cdot\nabla^{\varphi})(\nabla^{\varphi}\times
v)-b\times\nabla^{\varphi}(\nabla^{\varphi}\cdot v)=$
$\displaystyle-(\nabla^{\varphi}\times b)(\nabla^{\varphi}\cdot
v)-(\nabla^{\varphi}v_{j})\times(\partial^{\varphi}_{j}b)+(\nabla^{\varphi}b_{j})\times(\partial^{\varphi}_{j}v),$
(3.144)

and taking $\partial^{3}$ gives

$\displaystyle\rho D_{t}^{\varphi}(\partial^{3}\nabla^{\varphi}\times
v)-(b\cdot\nabla^{\varphi})(\partial^{3}\nabla^{\varphi}\times b)=$
$\displaystyle~{}RK_{v},$ (3.145) $\displaystyle
D_{t}^{\varphi}(\partial^{3}\nabla^{\varphi}\times
b)-(b\cdot\nabla^{\varphi})(\partial^{3}\nabla^{\varphi}\times
v)-b\times\partial^{3}\nabla^{\varphi}(\nabla^{\varphi}\cdot v)=$
$\displaystyle~{}RK_{b},$ (3.146)

where

$\displaystyle RK_{v}:=$ $\displaystyle-[\partial^{3},\rho
D_{t}^{\varphi}](\nabla^{\varphi}\times
v)+[\partial^{3},(b\cdot\nabla^{\varphi})](\nabla^{\varphi}\times
b)+\partial^{3}(\text{right side of }\eqref{curlveq}),$ $\displaystyle
RK_{b}:=$ $\displaystyle-[\partial^{3},D_{t}^{\varphi}](\nabla^{\varphi}\times
b)+[\partial^{3},(b\cdot\nabla^{\varphi})](\nabla^{\varphi}\times
v)+\partial^{3}(\text{right side of }\eqref{curlbeq}).$

Direct computation shows that the highest-order terms in $RK_{v},~{}RK_{b}$
only have 4 spatial derivatives and do not contain time derivative of $q$.
Therefore, we can prove the $H^{3}$-control of the voriticity and current
density by standard energy estimates.

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}\left|\partial^{3}(\nabla^{\varphi}\times
v^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}=\int_{\Omega^{\pm}}\rho^{\pm}D_{t}^{\varphi\pm}(\partial^{3}\nabla^{\varphi}\times
v^{\pm})\cdot(\partial^{3}\nabla^{\varphi}\times
v^{\pm})\,\mathrm{d}\mathcal{V}_{t}$ (3.147) $\displaystyle=$
$\displaystyle\int_{\Omega^{\pm}}(b^{\pm}\cdot\nabla^{\varphi})(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot(\partial^{3}\nabla^{\varphi}\times
v^{\pm})\,\mathrm{d}\mathcal{V}_{t}+\underbrace{\int_{\Omega^{\pm}}RK_{v}^{\pm}\cdot(\partial^{3}\nabla^{\varphi}\times
v^{\pm})\,\mathrm{d}\mathcal{V}_{t}}_{=:L_{1}^{\pm}}$
$\displaystyle\overset{(b^{\pm}\cdot\nabla^{\varphi})}{===}$
$\displaystyle-\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot D_{t}^{\varphi\pm}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\,\mathrm{d}\mathcal{V}_{t}+\underbrace{\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}(\nabla^{\varphi}\cdot
v^{\pm}))\right)\,\mathrm{d}\mathcal{V}_{t}}_{=:K_{1}^{\pm}}$
$\displaystyle+\underbrace{\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot
RK_{b}^{\pm}\,\mathrm{d}\mathcal{V}_{t}}_{L_{2}^{\pm}}+L_{1}^{\pm},$

where $L_{1}^{\pm},~{}L_{2}^{\pm}$ are directly controlled

$\displaystyle L_{1}^{\pm}+L_{2}^{\pm}\leq
P(\|(v^{\pm},b^{\pm})\|_{4,\pm},|\psi|_{4}).$ (3.148)

It remains to analyze the term $K_{1}^{\pm}$ in which there is a key
observation for the energy structure of compressible MHD system. We invoke the
continuity equation $\nabla^{\varphi}\cdot
v^{\pm}=\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}p^{\pm}$ and commute
$D_{t}^{\varphi\pm}$ with $\nabla^{\varphi}$ to get

$\nabla^{\varphi}(\nabla^{\varphi}\cdot
v^{\pm})=-\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}\nabla^{\varphi}p^{\pm}+\mathcal{F}_{p}^{\pm}(\nabla^{\varphi}v_{j}^{\pm})(\partial^{\varphi}_{j}p^{\pm}).$

Next, we rewrite the momentum equation to be
$\rho^{\pm}D_{t}^{\varphi\pm}v^{\pm}+b^{\pm}\times(\nabla^{\varphi}\times
b^{\pm})=-\nabla^{\varphi}p^{\pm}$ and plug it into the highest-order term
$-\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}\nabla^{\varphi}p^{\pm}$ to get

$\displaystyle-\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}\nabla^{\varphi}p^{\pm}=$
$\displaystyle~{}\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}(\rho^{\pm}D_{t}^{\varphi\pm}v^{\pm})+\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}(b^{\pm}\times(\nabla^{\varphi}\times
b^{\pm}))$ $\displaystyle=$
$\displaystyle~{}\mathcal{F}_{p}^{\pm}\rho^{\pm}(D_{t}^{\varphi\pm})^{2}v^{\pm}+\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}(b^{\pm}\times(\nabla^{\varphi}\times
b^{\pm}))+\mathcal{F}_{p}^{\pm}(D_{t}^{\varphi\pm}\rho^{\pm})(D_{t}^{\varphi\pm}v^{\pm}).$

Thus, the term $K_{1}^{\pm}$ becomes

$\displaystyle K_{1}^{\pm}=$
$\displaystyle\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot\left(b^{\pm}\times(\mathcal{F}_{p}^{\pm}\rho^{\pm}\partial^{3}(D_{t}^{\varphi\pm})^{2}v^{\pm})\right)\,\mathrm{d}\mathcal{V}_{t}$
(3.149)
$\displaystyle-\int_{\Omega^{\pm}}\mathcal{F}_{p}^{\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\cdot
D_{t}^{\varphi\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot RK_{p}^{\pm}\,\mathrm{d}\mathcal{V}_{t},$

where

$\displaystyle RK_{p}^{\pm}:=$
$\displaystyle~{}\mathcal{F}_{p}^{\pm}\partial^{3}\left((\nabla^{\varphi}v_{j}^{\pm})(\partial^{\varphi}_{j}p^{\pm})+(D_{t}^{\varphi\pm}\rho^{\pm})(D_{t}^{\varphi\pm}v^{\pm})\right)+[\partial^{3},\mathcal{F}_{p}^{\pm}\rho^{\pm}](D_{t}^{\varphi\pm})^{2}v^{\pm}$
$\displaystyle+\mathcal{F}_{p}^{\pm}[\partial^{3},D_{t}^{\varphi\pm}](b^{\pm}\times(\nabla^{\varphi}\times
b^{\pm}))+\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}\left([\partial^{3},b^{\pm}\times](\nabla^{\varphi}\times
b^{\pm})\right)$

consists of $\leq 4$ derivatives in each term and its contribution can be
directly controlled

$\displaystyle
L_{3}^{\pm}:=\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot RK_{p}^{\pm}\,\mathrm{d}\mathcal{V}_{t}\leq
P(\|b^{\pm},v^{\pm},\mathcal{F}_{p}^{\pm}p^{\pm}\|_{4,\pm},\|\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}(v^{\pm},b^{\pm},p^{\pm})\|_{3,\pm})$
(3.150)

Note that the second term on the right side of $K_{1}^{\pm}$ is obtained by
using the vector identity
$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=-\mathbf{c}\cdot(\mathbf{b}\times\mathbf{a})$:

$(\partial^{3}\nabla^{\varphi}\times b^{\pm})\cdot\left(b^{\pm}\times
D_{t}^{\varphi\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\right)=-D_{t}^{\varphi\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\cdot\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right).$

Therefore, we have

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}\left|\partial^{3}(\nabla^{\varphi}\times
v^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}$ (3.151) $\displaystyle=$
$\displaystyle-\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot D_{t}^{\varphi\pm}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}\mathcal{F}_{p}^{\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\cdot
D_{t}^{\varphi\pm}\left(b^{\pm}\times(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\right)\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\underbrace{\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\varphi}\times
b^{\pm})\cdot\left(b^{\pm}\times(\mathcal{F}_{p}\rho^{\pm}\partial^{3}(D_{t}^{\varphi\pm})^{2}v^{\pm})\right)\,\mathrm{d}\mathcal{V}_{t}}_{K^{\pm}}+L_{1}^{\pm}+L_{2}^{\pm}+L_{3}^{\pm}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\left|\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}+\mathcal{F}_{p}^{\pm}\left|b^{\pm}\times\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\frac{1}{2}\int_{\Omega^{\pm}}(\nabla^{\varphi}\cdot
v^{\pm})\left(\left|\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}+\mathcal{F}_{p}^{\pm}\left|b^{\pm}\times\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}\right)\,\mathrm{d}\mathcal{V}_{t}+K^{\pm}+L_{1}^{\pm}+L_{2}^{\pm}+L_{3}^{\pm},$

which further gives the control of vorticity and current density
simultaneously

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}\left|\partial^{3}(\nabla^{\varphi}\times
v^{\pm})\right|^{2}+\left|\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}+\mathcal{F}_{p}^{\pm}\left|b^{\pm}\times\partial^{3}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}$ (3.152)
$\displaystyle\lesssim$
$\displaystyle~{}P(E_{4}^{\kappa}(t))+P(|\psi|_{4})\|b^{\pm}\|_{4,\pm}\|b^{\pm}\rho^{\pm}\|_{L^{\infty}(\Omega^{\pm})}\left\|\varepsilon^{2}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{3}\leq
P(E_{4}^{\kappa}(t))+E_{5}^{\kappa}(t).$

Hence, the vorticity analysis for compressible ideal MHD cannot be closed in
standard Sobolev space because of the term
$\varepsilon^{2}\partial^{3}(D_{t}^{\varphi\pm})^{2}v^{\pm}$ in $K^{\pm}$.
Instead, the appearance of this term indicates us to trade one normal
derivative (in the curl operator) for two tangential derivatives
$(D_{t}^{\varphi\pm})^{2}$ together with square weights of Mach number
$\varepsilon^{2}$. Besides, the normal derivative part involving
$\partial^{3}D_{t}^{\varphi\pm}(\nabla^{\varphi}\times b^{\pm})$ contributes
to the energy of current density thanks to the special structure of Lorentz
force $-b^{\pm}\times(\nabla^{\varphi}\times b^{\pm})$. This is exactly the
motivation for us to define the energy functional $E(t)$ under the setting the
anisotropic Sobolev spaces instead of standard Sobolev spaces.

Similarly, the curl estimates for the time derivatives (in $E_{4}(t)$) can be
proven in the same way by replacing $\partial^{3}$ with
$\partial^{3-k}\partial_{t}^{k}~{}(1\leq k\leq 3)$. We omit the details and
list the conclusion

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}\left|\partial^{3-k}\partial_{t}^{k}(\nabla^{\varphi}\times
v^{\pm})\right|^{2}+\left|\partial^{3-k}\partial_{t}^{k}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}+\mathcal{F}_{p}^{\pm}\left|b^{\pm}\times\partial^{3-k}\partial_{t}^{k}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}$ (3.153)
$\displaystyle\lesssim$
$\displaystyle~{}P(E_{4}^{\kappa}(t))+\left\|\varepsilon^{2}\partial_{t}^{k}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{3-k,\pm}^{2}\leq
P(E_{4}^{\kappa}(t))+E_{5}^{\kappa}(t).$

Finally, we need to commute $\partial_{t}^{k}$ with $\nabla^{\varphi}\times$
when $k\geq 1$. We have

$(\nabla^{\varphi}\times\partial_{t}^{k}v)_{i}\stackrel{{\scriptstyle
L}}{{=}}\partial_{t}^{k}(\nabla^{\varphi}\times
v)_{i}+\epsilon_{ijl}(\partial_{3}\varphi)^{-1}(\overline{\partial}{}_{j}\partial_{t}^{k}\varphi)(\partial_{3}v_{l}),$

where $\epsilon_{ijl}$ is the sign of permutation $(ijl)\in S_{3}$. This gives

$\displaystyle\|\nabla^{\varphi}\times\partial_{t}^{k}v^{\pm}\|_{3-k,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left(\left\|\partial_{t}^{k}(\nabla^{\varphi}\times
v^{\pm})\right\|_{3-k,\pm}^{2}+\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}\right)+\text{
lower order terms},$ (3.154)

where both leading order terms have been controlled in tangential estimates of
$E_{4}^{\kappa}(t)$. The same result holds for $b^{\pm}$. Using the result of
tangential estimates of $E_{4}^{\kappa}(t)$, we have: for any
$k\in\\{0,1,2,3\\}$ and any $\delta\in(0,1)$

$\displaystyle\left\|\nabla^{\varphi}\times\partial_{t}^{k}v^{\pm}\right\|_{3-k,\pm}^{2}+\left\|\nabla^{\varphi}\times\partial_{t}^{k}b^{\pm}\right\|_{3-k,\pm}^{2}$
(3.155) $\displaystyle\lesssim$
$\displaystyle~{}P(E_{4}^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))+E_{5}^{\kappa}(\tau)\,\mathrm{d}\tau+P(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\partial_{t}^{k}\psi\right|_{4-k}^{2}$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4}^{\kappa}(t)+P(\sigma^{-1},E_{4}^{\kappa}(0))+P(E_{4}^{\kappa}(t))\int_{0}^{t}P(\sigma^{-1},E_{4}^{\kappa}(\tau))+E_{5}^{\kappa}(\tau)\,\mathrm{d}\tau,$

####  3.6.3 Further div-curl analysis for $E_{5}\sim E_{7}$

The vorticity analysis for $E_{4}(t)$ requires the control of
$\left\|\varepsilon^{2}\partial_{t}^{k}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{3-k}^{2}$
for $0\leq k\leq 3$. When $0\leq k\leq 2$, there are still normal derivatives
in this term. Thus, we shall do further div-curl analysis on
$\left\|\varepsilon^{2}\partial_{t}^{k}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{3-k}^{2}$
for $0\leq k\leq 2$. Let
$\mathcal{T}^{\alpha}=\partial_{t}^{\alpha_{0}}\overline{\partial}{}_{1}^{\alpha_{1}}\overline{\partial}{}_{2}^{\alpha_{2}}(\omega(x_{3})\partial_{3})^{\alpha_{4}}$
with $\langle\alpha\rangle=2$. The divergence part can be reduced in the same
way as in Section 3.6.1. We take $\partial_{t}^{k}\mathcal{T}^{\alpha}$ in the
continuity equation and omit the lower order terms to get

$\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}v\stackrel{{\scriptstyle
L}}{{=}}-\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}D_{t}^{\varphi}p+(\partial_{3}\varphi)^{-1}\overline{\partial}{}\partial_{t}^{k}\mathcal{T}^{\alpha}\varphi\cdot\partial_{3}v,$

which gives

$\displaystyle\|\varepsilon^{2}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\|_{2-k}^{2}\lesssim
C(\|v\|_{W^{1,\infty}})\left(\left\|\varepsilon^{4}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{2-k,\pm}^{2}+\left|\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{2-k}^{2}\right)+\text{
lower order terms}.$ (3.156)

######  Remark 3.3.

The term generated when commuting $\mathcal{T}^{\alpha}$ with
$\nabla^{\varphi}$ is actually of lower order. One can check that (see also
[85, (3.24)-(3.25)])

$[(\omega\partial_{3})^{m},\partial_{3}]f=\underbrace{(\omega\partial_{3})^{m}\partial_{3}f-\partial_{3}((\omega\partial_{3})^{m}f)}_{\text{both
are (m+1)-th order terms}}=\sum_{k\leq
m-1}c_{m,k}(\omega\partial_{3})^{k}\partial_{3}f=\sum_{k\leq
m-1}d_{m,k}\partial_{3}(\omega\partial_{3})^{k}f$

for some smooth functions $c_{m,k},d_{m,k}$ depending on $m,k$ and the
derivatives (up to order $m$) of $\omega$, and the right side only contains
$\leq m$-th order terms.

Similarly, using $\nabla^{\varphi}\cdot b=0$, we have

$\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}b\stackrel{{\scriptstyle
L}}{{=}}\underbrace{\partial_{t}^{k}\mathcal{T}^{\alpha}(\nabla^{\varphi}\cdot
b)}_{=0}+(\partial_{3}\varphi)^{-1}\overline{\partial}{}\partial_{t}^{k}\mathcal{T}^{\alpha}\varphi\cdot\partial_{3}b$

and thus

$\displaystyle\|\varepsilon^{2}\nabla^{\varphi}\cdot\partial_{t}^{k}b^{\pm}\|_{2-k,\pm}^{2}\lesssim
C(\|b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{2-k}^{2}+\text{
lower order terms}.$ (3.157)

The control of divergence part in the analysis of $E_{5}^{\kappa}(t)$ is
concluded by the following energy inequality. For any $k\in\\{0,1,2\\}$, any
multi-index $\alpha$ with $\langle\alpha\rangle=2$ and any $\delta\in(0,1)$

$\displaystyle\left\|\varepsilon^{2}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{2-k,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\varepsilon^{4}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{2-k,\pm}^{2}+C(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{3-k}^{2}$
(3.158) $\displaystyle\lesssim$
$\displaystyle~{}C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\varepsilon^{4}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{2-k,\pm}^{2}+\delta
E_{5}^{\kappa}(t)+P(E_{4}^{\kappa}(0),E_{5}^{\kappa}(0))+P(E_{4}^{\kappa}(t))\int_{0}^{t}P(E_{4}^{\kappa}(\tau),E_{5}^{\kappa}(\tau))\,\mathrm{d}\tau,$

where the term involving $\mathcal{T}p^{\pm}$ can be further reduced to
$\mathcal{T}(v^{\pm},b^{\pm})$ when $2-k>0$ so that one can further apply the
div-curl analysis to it.

As for the curl part, we can still mimic the proof in Section 3.6.2 to get the
control of
$\left\|\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}(\nabla^{\varphi}\times(v,b))\right\|_{2-k}$
for $0\leq k\leq 2$ and $\langle\alpha\rangle=2$ with $\alpha_{3}=0$

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}\left|\varepsilon^{2}\partial^{2-k}\partial_{t}^{k}\mathcal{T}^{\alpha}(\nabla^{\varphi}\times
v^{\pm})\right|^{2}+\left|\varepsilon^{2}\partial^{2-k}\partial_{t}^{k}\mathcal{T}^{\alpha}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}+\mathcal{F}_{p}^{\pm}\left|\varepsilon^{2}b^{\pm}\times\partial^{2-k}\partial_{t}^{k}\mathcal{T}^{\alpha}(\nabla^{\varphi}\times
b^{\pm})\right|^{2}\,\mathrm{d}\mathcal{V}_{t}$ (3.159)
$\displaystyle\lesssim$
$\displaystyle~{}P(E_{4}^{\kappa}(t),E_{5}^{\kappa}(t))+\left\|\varepsilon^{4}\partial_{t}^{k}\mathcal{T}^{\alpha}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{2-k}^{2}\leq
P(E_{4}^{\kappa}(t),E_{5}^{\kappa}(t))+E_{6}^{\kappa}(t).$

Then we commute $\partial^{2-k}\partial_{t}^{k}\mathcal{T}^{\alpha}$ with
$\nabla^{\varphi}\times$ to get: for any $k\in\\{0,1,2\\}$, any multi-index
$\alpha$ with $\langle\alpha\rangle=2$ and $\alpha_{3}=0$, and any
$\delta\in(0,1)$

$\displaystyle\left\|\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{2-k,\pm}^{2}+\left\|\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm}\right\|_{2-k,\pm}^{2}$
(3.160) $\displaystyle\lesssim$
$\displaystyle~{}P(E_{4}^{\kappa}(0),E_{5}^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(\tau),E_{5}^{\kappa}(\tau))+E_{6}^{\kappa}(\tau)\,\mathrm{d}\tau+P(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{2-k}^{2}$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{5}^{\kappa}(t)+P(\sigma^{-1},E_{4}^{\kappa}(0),E_{5}^{\kappa}(0))+P(E_{4}^{\kappa}(t))\int_{0}^{t}P(\sigma^{-1},E_{4}^{\kappa}(\tau),E_{5}^{\kappa}(\tau))+E_{6}^{\kappa}(\tau)\,\mathrm{d}\tau,$

where we use the result of tangential estimates to control
$\left|\varepsilon^{2}\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{2-k}$.
When $k\leq 1$ in the above energy estimate, we shall continue to apply the
div-curl analysis to
$\left\|\varepsilon^{4}\partial_{t}^{k}\mathcal{T}^{\alpha}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{2-k}^{2}$.

For $E_{6}^{\kappa}$ and $E_{7}^{\kappa}$, we have analogous div-curl
inequalities. For $l=2,3$, we continue to analyze the divergence and the curl
according to (3.130). Similarly as above, we have the following estimates for
any $k\in\\{0,1\\}$, any multi-index $\alpha$ with
$\langle\alpha\rangle=4,~{}\alpha_{3}=0$ and any $\delta\in(0,1)$

$\displaystyle\left\|\varepsilon^{4}\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{1-k,\pm}^{2}+\left\|\varepsilon^{4}\nabla^{\varphi}\times\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm}\right\|_{1-k,\pm}^{2}$
(3.161) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{6}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{l=0}^{2}E_{4+l}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{l=0}^{2}E_{4+l}^{\kappa}(\tau)\right)+E_{7}^{\kappa}(\tau)\,\mathrm{d}\tau,$

where this $E_{7}^{\kappa}$ term is contributed by
$\left\|\varepsilon^{6}\partial_{t}^{k}\mathcal{T}^{\alpha}(D_{t}^{\varphi\pm})^{2}v^{\pm}\right\|_{1-k,\pm}$.
When $k=0$ in this term, we again apply the div-curl analysis to it in order
to eliminate all normal derivatives falling on $v,b$. For any multi-index
$\alpha$ with $\langle\alpha\rangle=6$ and $\alpha_{3}=0$ and any
$\delta\in(0,1)$, we have

$\displaystyle\left\|\varepsilon^{6}\nabla^{\varphi}\times\mathcal{T}^{\alpha}v^{\pm}\right\|_{0,\pm}^{2}+\left\|\varepsilon^{6}\nabla^{\varphi}\times\mathcal{T}^{\alpha}b^{\pm}\right\|_{0,\pm}^{2}$
(3.162) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{7}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{l=0}^{3}E_{4+l}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{l=0}^{3}E_{4+l}^{\kappa}(\tau)\right)+E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau.$

The control of divergence part for $E_{6}^{\kappa}(t),~{}E_{7}^{\kappa}(t)$
also follows the same way as $E_{4}^{\kappa}(t),E_{5}^{\kappa}(t)$. For any
$k\in\\{0,1\\}$, any multi-index $\alpha$ with
$\langle\alpha\rangle=4,~{}\alpha_{3}=0$, we have

$\displaystyle\left\|\varepsilon^{4}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{1-k,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\varepsilon^{6}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{1-k,\pm}^{2}+C(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\partial_{t}^{k}\mathcal{T}^{\alpha}\psi\right|_{2-k}^{2}$
(3.163) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{6}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{l=0}^{2}E_{4+l}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{l=0}^{2}E_{4+l}^{\kappa}(\tau)\right)\,\mathrm{d}\tau,$

where the term
$\left\|\varepsilon^{6}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{1-k,\pm}^{2}$
does not appear because it can be converted to full tangential derivatives
(part of $E_{6}^{\kappa}(t)$) which has been controlled in Section 3.3-Section
3.5. For any multi-index $\alpha$ with
$\langle\alpha\rangle=6,~{}\alpha_{3}=0$, we have

$\displaystyle\left\|\varepsilon^{6}\nabla^{\varphi}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}\lesssim
C(\|v^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left\|\varepsilon^{8}\partial_{t}^{k}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{0,\pm}^{2}+C(\|v^{\pm},b^{\pm}\|_{W^{1,\infty}(\Omega^{\pm})})\left|\varepsilon^{6}\mathcal{T}^{\alpha}\psi\right|_{1}^{2}$
(3.164) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{7}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{l=0}^{3}E_{4+l}^{\kappa}(0)\right)+P(E_{4}^{\kappa}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{l=0}^{3}E_{4+l}^{\kappa}(\tau)\right)\,\mathrm{d}\tau,$

where the term
$\left\|\varepsilon^{8}\mathcal{T}^{\alpha}\mathcal{T}p^{\pm}\right\|_{0,\pm}^{2}$
does not appear because it has been included in tangential estimates for
$E_{7}^{\kappa}(t)$.

###  3.7 Uniform estimates for the nonlinear approximate system

####  3.7.1 Control of the entropy

It remains to control the full (anisotropic) Sobolev norms of the entropy
functions $S^{\pm}$. This can be easily proven thanks to
$D_{t}^{\varphi\pm}S^{\pm}=0$. In the control of $E_{4+l}^{\kappa}(t)$ for
fixed $0\leq l\leq 4$, we need to take the derivative
$\partial_{*}^{\alpha}:=\partial^{4-l-k}\partial_{t}^{k}\mathcal{T}^{\gamma}=\partial_{3}^{\gamma_{3}^{\prime}}(\omega\partial_{3})^{\gamma_{4}}\partial_{t}^{k+\gamma_{0}}\overline{\partial}{}_{1}^{\gamma_{1}+\gamma_{1}^{\prime}}\overline{\partial}{}_{2}^{\gamma_{2}+\gamma_{2}^{\prime}}$
with $\gamma_{0}+\gamma_{1}+\gamma_{2}+\gamma_{4}=2l$,
$\gamma_{1}^{\prime}+\gamma_{2}^{\prime}+\gamma_{3}^{\prime}=4-k-l$ and $0\leq
k\leq 4-l$ and also multiply the weight $\varepsilon^{2l}$. Then we can
introduce the Alinhac good unknown $\mathbf{S}^{\partial}$ with respect to
this general derivative $\partial_{*}^{\alpha}$ by

$\mathbf{S}^{\partial,\pm}:=\partial_{*}^{\alpha}S^{\pm}-\partial_{*}^{\alpha}\varphi\partial^{\varphi}_{3}S^{\pm},$

which satisfies the evolution equation
$D_{t}^{\varphi\pm}\mathbf{S}^{\partial,\pm}=\mathfrak{D}^{\partial}(S^{\pm})$
in $\Omega^{\pm}$ where $\mathfrak{D}^{\partial}(S^{\pm})$ is defined by (3.2)
after replacing $\mathcal{T}^{\gamma}$ with $\partial_{*}^{\alpha}$. We will
get

$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\gamma}S^{\pm}\right\|_{4-k-l,\pm}^{2}\lesssim\left\|\varepsilon^{2l}\mathbf{S}^{\partial,\pm}\right\|_{0,\pm}^{2}+\left|\partial_{*}^{\alpha}\psi\right|_{0}^{2}\|\partial_{3}S^{\pm}\|_{L^{\infty}(\Omega^{\pm})}^{2}$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+E_{4}^{\kappa}(t)\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau,$
(3.165)

where we again invoke the estimate of
$\left|\partial_{*}^{\alpha}\psi\right|_{0}$ that has been proven in Section
3.3-Section 3.5.

####  3.7.2 Uniform-in-$\kappa$ estimates for the nonlinear approximate system

Summarizing Proposition 3.2 ($L^{2}$-energy conservation), Proposition 3.3
(tangential estimates), Proposition 3.8 (div-curl estimates) and (3.165)
(entropy estimates), we conclude the estimates of the energy functional
$E^{\kappa}(t)$ for the nonlinear approximate system (3.1) by

$E^{\kappa}(t)\lesssim\delta
E^{\kappa}(t)+P(E^{\kappa}(0))+P(E^{\kappa}(t))\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}\forall\delta\in(0,1)$
(3.166)

Thus, choosing $\delta$ suitably small such that $\delta E^{\kappa}(t)$ can be
absorbed by the left side and then using Gronwall-type argument, we find that
there exists a time $T_{\sigma}>0$ that depends on $\sigma$ and the initial
data and is independent of $\kappa$ and $\varepsilon$, such that

$\displaystyle\sup\limits_{0\leq t\leq T_{\sigma}}E^{\kappa}(t)\leq
C(\sigma^{-1})P(E^{\kappa}(0)),$ (3.167)

which is exactly the conclusion of Proposition 3.1.

##  4 Well-posedness of the nonlinear approximate system

We already prove the uniform-in-$\kappa$ estimates for the nonlinear
approximate problem (3.1). If we can prove the well-posedness of (3.1) for
each fixed $\kappa>0$, then the uniform estimates allow us to take the limit
$\kappa\to 0_{+}$ and prove the local existence of system (1.33) for the
compressible current-vortex sheets with surface tension. Since there is no
loss of regularity in the estimate of $E^{\kappa}(t)$, we would like to use
Picard iteration to construct the solution to (3.1) for each fixed $\kappa$.

###  4.1 Construction of the linearized problem

We start with $\psi^{[-1]}=\psi^{[0]}=0$ and
$(v^{[0],\pm},b^{[0],\pm},\rho^{[0],\pm},S^{[0],\pm})=(\vec{0},\vec{0},\underline{\rho}^{\pm},0)$
for some constants $\underline{\rho}^{\pm}\geq\bar{\rho_{0}}$. Then for any
$n\geq 0,n\in\mathbb{N}$, given
$\\{(v^{[k],\pm},b^{[k],\pm},\rho^{[k],\pm},S^{[k],\pm})\\}_{k\leq n}$, we
define
$(v^{[n+1],\pm},b^{[n+1],\pm},q^{[n+1],\pm},S^{[n+1],\pm},\psi^{[n+1]})$ by
the following linear system with variable coefficients only depending on
$(v^{[n],\pm},b^{[n],\pm},q^{[n],\pm},S^{[n],\pm},\psi^{[n]},\psi^{[n-1]})$

$\begin{cases}\rho^{{[n]},\pm}D_{t}^{\varphi^{[n]},\pm}v^{{[n+1]},\pm}-(\bm{b}^{{[n]},\pm}\cdot\nabla^{\varphi^{[n]},\pm})b^{{[n+1]},\pm}+\nabla^{\varphi^{[n]},\pm}q^{{[n+1]},\pm}=0&~{}~{}\text{
in }[0,T]\times\Omega^{\pm},\\\
(\mathcal{F}_{p}^{\pm})^{[n]}D_{t}^{\varphi^{[n]},\pm}q^{{[n+1]},\pm}-(\mathcal{F}_{p}^{\pm})^{[n]}D_{t}^{\varphi^{[n]},\pm}b^{{[n+1]},\pm}\cdot\bm{b}^{[n],\pm}+\nabla^{\varphi^{[n]},\pm}\cdot
v^{{[n+1]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{[n]},\pm}b^{{[n+1]},\pm}-(\bm{b}^{{[n]},\pm}\cdot\nabla^{\varphi^{[n]},\pm})v^{{[n+1]},\pm}+\bm{b}^{[n],\pm}\nabla^{\varphi^{[n]},\pm}\cdot
v^{{[n+1]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{[n]},\pm}S^{{[n+1]},\pm}=0&~{}~{}\text{ in
}[0,T]\times\Omega^{\pm},\\\ \left\llbracket
q^{{[n+1]}}\right\rrbracket=\sigma\mathcal{H}(\psi^{[n]})-\kappa(1-\overline{\Delta}{})^{2}\psi^{{[n+1]}}-\kappa(1-\overline{\Delta}{})\partial_{t}\psi^{{[n+1]}}&~{}~{}\text{
on }[0,T]\times\Sigma,\\\ \partial_{t}\psi^{[n+1]}=v^{[n+1],\pm}\cdot
N^{[n]}&~{}~{}\text{ on }[0,T]\times\Sigma,\\\
v_{3}^{[n+1],\pm}=0,~{}~{}&~{}~{}\text{ on }[0,T]\times\Sigma^{\pm},\\\
(v^{{[n+1]},\pm},b^{{[n+1]},\pm},q^{{[n+1]},\pm},S^{{[n+1]},\pm},\psi^{\pm})|_{t=0}=(v_{0}^{\kappa,\pm},b_{0}^{\kappa,\pm},q_{0}^{\kappa,\pm},S_{0}^{\kappa,\pm},\psi_{0}^{\kappa}),\end{cases}$
(4.1)

where $\bm{b}_{i}^{[n],\pm}:=b_{i}^{[n],\pm}$ for $i=1,2$ and
$\bm{b}_{3}^{[n],\pm}$ is defined by

$\displaystyle\bm{b}_{3}^{[n],\pm}:=b_{3}^{[n],\pm}+\mathfrak{R}_{T}^{\pm}\left(b_{1}^{[n],\pm}\overline{\partial}{}_{1}\psi^{[n]}+b_{2}^{[n],\pm}\overline{\partial}{}_{2}\psi^{[n]}-b_{3}^{[n],\pm}\right)\big{|}_{\Sigma}$
(4.2)

where $\mathfrak{R}_{T}^{\pm}$ is the lifting operator defined in Lemma B.3.
The initial data
$(v_{0}^{\kappa,\pm},b_{0}^{\kappa,\pm},\rho_{0}^{\kappa,\pm},S_{0}^{\kappa,\pm},\psi_{0}^{\kappa})$
is the same as (3.1). The basic state
$(v^{[n],\pm},\bm{b}^{[n],\pm},\rho^{[n],\pm},S^{[n],\pm},\psi^{[n]},\psi^{[n-1]})$
satisfies

  1. 1.

(The hyperbolicity assumption) $\rho^{{[n]},\pm}>0$ is determined by the
equation of state (1.18) where $p^{{[n]},\pm}$ is defined by
$p^{{[n]},\pm}:=q^{{[n]},\pm}-\frac{1}{2}|b^{{[n]},\pm}|^{2}$. Then define
$\mathcal{F}^{[n]}=\log\rho^{[n]},~{}~{}\mathcal{F}_{p}^{{[n]},\pm}:=\frac{\partial\mathcal{F}^{{[n]},\pm}}{\partial
p}(p^{{[n+1]},\pm},S^{{[n+1]},\pm})>0.$

  2. 2.

(Tangential magnetic fields) $\bm{b}^{[n],\pm}\cdot N^{{[n]}}=0\text{ on
}\Sigma,$ and $\bm{b}^{[n],\pm}_{3}=0$ on $\Sigma^{\pm}$.

  3. 3.

(Linearized material derivatives and covariant derivatives)

$\displaystyle D_{t}^{\varphi^{[n]},\pm}:=$
$\displaystyle~{}\partial_{t}+\bar{v}^{[n]}\cdot\overline{\nabla}+\frac{1}{\partial_{3}\varphi^{[n]}}(v^{[n]}\cdot\mathbf{N}^{[n-1]}-\partial_{t}\varphi^{[n]})\partial_{3},$
(4.3) $\displaystyle\partial_{t}^{\varphi^{[n]}}:=$
$\displaystyle~{}\partial_{t}-\frac{\partial_{t}\varphi^{[n]}}{\partial_{3}\varphi^{[n]}}\partial_{3},~{}\nabla_{a}^{\varphi^{[n]}}=\partial_{a}^{\varphi^{[n]}}:=\partial_{a}-\frac{\partial_{a}\varphi^{[n]}}{\partial_{3}\varphi^{[n]}}\partial_{3},~{}~{}a=1,2,\quad\nabla_{a}^{\varphi^{[n]}}=\partial_{3}^{\varphi^{[n]}}:=\frac{1}{\partial_{3}\varphi^{[n]}}\partial_{3}$
(4.4)

where $N^{[n]}:=(-\partial_{1}\psi^{[n]},-\partial_{2}\psi^{[n]},1)^{\top}$
and $\mathbf{N}^{[n]}$ is the extension of $N^{[n]}$ with
$\varphi^{{[n]}}=x_{3}+\chi(x_{3})\psi^{{[n]}}(t,x^{\prime})$.

After solving the linear problem (4.1), we define
$p^{{[n+1]},\pm}=q^{{[n+1]},\pm}-\frac{1}{2}|b^{{[n+1]},\pm}|^{2}$ and use the
equation of state $p^{{[n+1]}}=p^{{[n+1]}}(\rho^{[n+1]},S^{[n+1]})$ to
determine the density $\rho^{[n+1]}>0$. We shall also define the “modified
magnteic fields” $\bm{b}^{[n+1],\pm}$ as follows in order to guarantee
$\bm{b}^{[n+1],\pm}\cdot N^{{[n+1]}}=0$ on $\Sigma$ and $\Sigma^{\pm}$:

$\displaystyle\bm{b}_{1}^{[n+1],\pm}=b_{1}^{{[n+1]},\pm},\quad\bm{b}_{2}^{[n+1],\pm}=b_{2}^{{[n+1]},\pm},$
$\displaystyle\bm{b}_{3}^{[n+1],\pm}=b_{3}^{[n+1],\pm}+\mathfrak{R}_{T}^{\pm}\left(b_{1}^{[n+1],\pm}\overline{\partial}{}_{1}\psi^{[n+1]}+b_{2}^{[n+1],\pm}\overline{\partial}{}_{2}\psi^{[n+1]}-b_{3}^{[n+1],\pm}\right)\big{|}_{\Sigma}.$
(4.5)

######  Remark 4.1 (The boundary constraint of magnetic fields).

The modified basic state $\mathring{\bm{b}}$ is necessary here, because the
quantity $b^{[n+1]}$ solved from (4.1) may not be tangential on $\Sigma$ and
so integrating $(b\cdot\nabla^{\varphi})$ by parts produces uncontrollable
boundary terms. When taking the limit $n\to\infty$, we can show that the limit
function $b^{[\infty]}$ also satisfy the boundary constraint
$b^{[\infty]}\cdot N^{[\infty]}|_{\Sigma}=0$ which then indicates that
$\bm{b}_{3}^{[\infty]}=b_{3}^{[\infty]}$ in order to recover the nonlinear
approximate system (3.1). We refer to Section 4.4.1 for details.

######  Remark 4.2 (The divergence constraint of magnetic fields).

Notice that the divergence-free condition for $b^{\pm}$ no longer propagates
from the initial data for the linear problem, but we will show that the
contribution of the divergence of part of $b^{\pm}$ is still controllable and
does not introduce extra substantial difficulty. After solving the nonlinear
problem (3.1) for each fixed $\kappa>0$, $\nabla^{\varphi}\cdot b^{\pm}=0$ in
(3.1) is automatically recovered from the initial constraint
$\nabla^{\varphi}\cdot b_{0}^{\kappa,\pm}=0$.

For simplicity of notations, given any $n\in\mathbb{N}$, we denote
$(v^{[n+1],\pm},b^{[n+1],\pm},q^{[n+1],\pm},p^{[n+1],\pm},\rho^{[n+1],\pm},S^{[n+1],\pm},\psi^{[n+1]})$,
$(v^{[n],\pm},b^{[n],\pm},\bm{b}^{[n],\pm},q^{[n],\pm},\rho^{[n],\pm},p^{[n],\pm},S^{[n],\pm},\psi^{[n]})$,
$\psi^{[n-1]}$ respectively by
$(v^{\pm},b^{\pm},q^{\pm},p^{\pm},\rho^{\pm},S^{\pm})$,
$(\mathring{v}^{\pm},\mathring{b}^{\pm},\mathring{\bm{b}}^{\pm},\mathring{q}^{\pm},\mathring{\rho}^{\pm},\mathring{p}^{\pm},\mathring{S}^{\pm},\mathring{\psi})$
and $\dot{\psi}$. Also, we denote $D_{t}^{\varphi^{[n]},\pm}$ and
$\partial_{i}^{\varphi^{[n]}},~{}\nabla_{i}^{\varphi^{[n]}}$ by
$D_{t}^{\mathring{\varphi}\pm}$ and
$\partial^{\mathring{\varphi}}_{i},~{}\nabla^{\mathring{\varphi}}$. Thus, the
linear problem above becomes

$\begin{cases}\mathring{\rho}D_{t}^{\mathring{\varphi}\pm}v^{\pm}-(\mathring{\bm{b}}\cdot\nabla^{\mathring{\varphi}})b^{\pm}+\nabla^{\mathring{\varphi}}q^{\pm}=0&~{}~{}\text{
in }[0,T]\times\Omega^{\pm},\\\
\mathring{\mathcal{F}}_{p}^{\pm}D_{t}^{\mathring{\varphi}\pm}q^{\pm}-\mathring{\mathcal{F}}_{p}^{\pm}D_{t}^{\mathring{\varphi}\pm}b^{\pm}\cdot\mathring{\bm{b}}^{\pm}+\nabla^{\mathring{\varphi}}\cdot
v^{\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\mathring{\varphi}\pm}b^{\pm}-(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})v^{\pm}+\mathring{\bm{b}}^{\pm}\nabla^{\mathring{\varphi}}\cdot
v^{\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\mathring{\varphi}\pm}S^{\pm}=0&~{}~{}\text{ in
}[0,T]\times\Omega^{\pm},\\\ \left\llbracket
q\right\rrbracket=\sigma\mathcal{H}(\mathring{\psi})-\kappa(1-\overline{\Delta}{})^{2}\psi-\kappa(1-\overline{\Delta}{})\partial_{t}\psi&~{}~{}\text{
on }[0,T]\times\Sigma,\\\
\partial_{t}\psi=v^{\pm}\cdot\mathring{N}&~{}~{}\text{ on
}[0,T]\times\Sigma,\\\ v_{3}^{\pm}=0&~{}~{}\text{ on
}[0,T]\times\Sigma^{\pm},\\\
(v^{\pm},b^{\pm},q^{\pm},S^{\pm},\psi)|_{t=0}=(v_{0}^{\kappa,\pm},b_{0}^{\kappa,\pm},q_{0}^{\kappa,\pm},S_{0}^{\kappa,\pm},\psi_{0}^{\kappa}),\end{cases}$
(4.6)

where
$D_{t}^{\mathring{\varphi}\pm}=\partial_{t}+\mathring{\bar{v}}\cdot\overline{\nabla}+\frac{1}{\partial_{3}\mathring{\varphi}}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\partial_{3}$
and
$\mathcal{H}(\mathring{\psi})=\overline{\nabla}\cdot(\overline{\nabla}\mathring{\psi}/|\mathring{N}|).$

###  4.2 Well-posedness of the linearized approximate problem

Although (4.6) is a first-order linear symmetric hyperbolic system with
characteristic boundary conditions, it is still not easy to apply the duality
argument in Lax-Phillips [46] to prove the well-posedness of (4.6) in
$L^{2}([0,T]\times\Omega^{\pm})$ because the boundary condition for the dual
system of (4.6) may not be explicitly calculated due to the appearance of the
regularization term $\kappa(1-\overline{\Delta}{})^{2}\psi$. Instead, we apply
the classical Galerkin’s method to construct the weak solution in
$L^{2}([0,T]\times\Omega^{\pm})$ to the linearized problem (4.6). Once this is
done, the weak solution is actually a strong solution by the argument in [60,
Chapter 2.2.3].

We assume the basic state
$(\mathring{v},\mathring{b},\mathring{q},\mathring{\rho},\mathring{p},\mathring{S},\mathring{\psi})$
and $\dot{\psi}$ satisfy the following bounds: There exists some
$\mathring{K}_{0}>0$ and a time $T_{\kappa}>0$ (depending on $\kappa>0$) such
that

$\displaystyle\sup_{0\leq t\leq
T_{\kappa}}\sum_{l=0}^{4}\bigg{(}\sum_{\pm}\sum_{\langle\alpha\rangle=2l}\sum_{k=0}^{4-l}\left\|\left(\varepsilon^{2l}\mathcal{T}^{\alpha}\partial_{t}^{k}(\mathring{v}^{\pm},\mathring{b}^{\pm},\mathring{S}^{\pm},(\mathring{\mathcal{F}}_{p}^{\pm})^{\frac{(k+\alpha_{0}-l-3)_{+}}{2}}\mathring{q}^{\pm})\right)\right\|^{2}_{4-k-l,\pm}$
(4.7)
$\displaystyle+\sum_{k=0}^{4+l}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\mathring{\psi}\right|^{2}_{6+l-k}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{5+l}\mathring{\psi}\right|^{2}_{1}\,\mathrm{d}\tau\bigg{)}$
$\displaystyle\leq\mathring{K}_{0},$

where
$\mathcal{T}^{\alpha}:=(\omega(x_{3})\partial_{3})^{\alpha_{4}}\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
with the multi-index
$\alpha=(\alpha_{0},\alpha_{1},\alpha_{2},0,\alpha_{4}),~{}\langle\alpha\rangle=\alpha_{0}+\alpha_{1}+\alpha_{2}+2\times
0+\alpha_{4}$. Moreover, we have

$\forall 0\leq T\leq
T_{\kappa},\quad\int_{0}^{T}\left\|\varepsilon^{2l}\mathcal{T}^{\alpha}\partial_{t}^{k}\mathring{\bm{b}}^{\pm}\right\|_{4-k-l,\pm}^{2}\,\mathrm{d}t\leq
C(\mathring{K}_{0}).$

######  Remark 4.3.

The $L_{t}^{2}$-type bound of $\mathring{\bm{b}}$ is obtained by using the
second part of Lemma B.3 and the $\sqrt{\kappa}$-weighted enhanced regularity
for the free interface. Indeed, the modification term
$\mathfrak{R}_{T}^{\pm}\left(\mathring{b}_{1}^{\pm}\overline{\partial}{}_{1}\mathring{\psi}+\mathring{b}_{2}^{\pm}\overline{\partial}{}_{2}\mathring{\psi}-\mathring{b}_{3}^{\pm}\right)\big{|}_{\Sigma}$
has vanishing initial value thank to $b_{0}^{\kappa,\pm}\cdot
N_{0}^{\kappa}=0$ on $\Sigma$. Thus, one can extend this function to
$(-\infty,T]\times\Omega^{\pm}$ and then apply the trace lemma for anisotropic
Sobolev spaces (cf. Trakhinin-Wang [81, Lemma 3.4] or Lemma B.3 in this paper)
to show that

$\displaystyle\int_{0}^{T}\left\|\mathcal{T}^{\alpha}\partial_{t}^{k}(\mathring{b}^{\pm}-\mathring{\bm{b}}^{\pm})\right\|_{4-k-l,\pm}^{2}\,\mathrm{d}t\leq\left\|\mathring{b}^{\pm}-\mathring{\bm{b}}^{\pm}\right\|_{8,*,T,\pm}^{2}\lesssim\left|\mathring{b}_{1}^{\pm}\overline{\partial}{}_{1}\mathring{\psi}+\mathring{b}_{2}^{\pm}\overline{\partial}{}_{2}\mathring{\psi}-\mathring{b}_{3}^{\pm}\right|_{7,T}^{2}$
$\displaystyle\lesssim$
$\displaystyle~{}\left\|\mathring{b}_{1}^{\pm}\overline{\partial}{}_{1}\mathring{\varphi}+\mathring{b}_{2}^{\pm}\overline{\partial}{}_{2}\mathring{\varphi}-\mathring{b}_{3}^{\pm}\right\|_{8,*,T,\pm}^{2}=\int_{0}^{T}\left\|\mathring{b}_{1}^{\pm}\overline{\partial}{}_{1}\mathring{\varphi}+\mathring{b}_{2}^{\pm}\overline{\partial}{}_{2}\mathring{\varphi}-\mathring{b}_{3}^{\pm}\right\|_{8,*,\pm}^{2}\,\mathrm{d}t\lesssim
T\mathring{K}_{0},~{}~{}\forall T\in[0,T_{\kappa}],$

where $\|\cdot\|_{m,*,T,\pm},|\cdot|_{m,T}$ norms are defined in Appendix B.
Notice that this $\sqrt{\kappa}$-weighted enhanced regularity is necessary
here, otherwise we lose the control of $|\overline{\partial}{}\psi(t)|_{8}$
and a loss of tangential derivative occurs as in lots of previous works [12,
80, 81, 82] and references therein.

####  4.2.1 Construction of Galerkin sequence

Since our domain $\Omega:=\mathbb{T}^{2}\times(-H,H)$ is bounded, there exists
an orthonormal basis $\\{\bm{e}_{j}\\}_{j=1}^{\infty}\subset
C^{\infty}(\Omega)$ for $L^{2}(\Omega)$ which is also an orthogonal basis of
$H_{0}^{1}(\Omega)$. To construct the Galerkin sequence, we first write the
linearized system (4.6) into a symmetric hyperbolic system of
$U^{\pm}:=(q^{\pm},v^{\pm},b^{\pm},S^{\pm})^{\top}\in\mathbb{R}^{8}$:

$\displaystyle
A_{0}(\mathring{U}^{\pm})\partial_{t}U^{\pm}+A_{1}(\mathring{U}^{\pm})\partial_{1}U^{\pm}+A_{2}(\mathring{U}^{\pm})\partial_{2}U^{\pm}+A_{3}(\mathring{U}^{\pm})\partial_{3}U^{\pm}=0\quad\text{
in }\Omega^{\pm}$ (4.8)

where the coefficient matrices are

$\displaystyle
A_{0}(\mathring{U}):=\begin{bmatrix}\mathring{\mathcal{F}}_{p}&\vec{0}^{\top}&-\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}^{\top}&0\\\
\vec{0}&\mathring{\rho}\bm{I}_{3}&\bm{O}_{3}&\vec{0}\\\
-\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}&\bm{O}_{3}&\bm{I}_{3}+\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\otimes\mathring{\bm{b}}&\vec{0}\\\
0&\vec{0}^{\top}&\vec{0}^{\top}&1\end{bmatrix},~{}A_{i}(\mathring{U}):=\begin{bmatrix}\mathring{\mathcal{F}}_{p}\mathring{\bar{v}}_{i}&\vec{e_{i}}^{\top}&-\mathring{\mathcal{F}}_{p}\mathring{\bar{v}}_{i}\mathring{\bm{b}}^{\top}&0\\\
\vec{e_{i}}&\mathring{\rho}\mathring{\bar{v}}_{i}\bm{I}_{3}&-\mathring{\bar{\bm{b}}}_{i}\bm{I}_{3}&\vec{0}\\\
-\mathring{\mathcal{F}}_{p}\mathring{\bar{v}}_{i}\mathring{\bm{b}}&-\mathring{\bar{\bm{b}}}_{i}\bm{I}_{3}&\mathring{\bar{v}}_{i}\bm{I}_{3}+\mathring{\mathcal{F}}_{p}\mathring{\bar{v}}_{i}(\mathring{\bm{b}}\otimes\mathring{\bm{b}})&\vec{0}\\\
0&\vec{0}^{\top}&\vec{0}^{\top}&\mathring{\bar{v}}_{i}\end{bmatrix}~{}(i=1,2),$
$\displaystyle
A_{3}(\mathring{U}):=\frac{1}{\partial_{3}\mathring{\varphi}}\begin{bmatrix}\mathring{\mathcal{F}}_{p}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})&\mathring{\mathbf{N}}^{\top}&-\mathring{\mathcal{F}}_{p}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\mathring{\bm{b}}^{\top}&0\\\
\mathring{\mathbf{N}}&\mathring{\rho}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\bm{I}_{3}&-(\mathring{\bm{b}}\cdot\mathring{\mathbf{N}})\bm{I}_{3}&\vec{0}\\\
-\mathring{\mathcal{F}}_{p}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\mathring{\bm{b}}&-(\mathring{\bm{b}}\cdot\mathring{\mathbf{N}})\bm{I}_{3}&(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\bm{I}_{3}+\mathring{\mathcal{F}}_{p}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})(\mathring{\bm{b}}\otimes\mathring{\bm{b}})&\vec{0}\\\
0&\vec{0}^{\top}&\vec{0}^{\top}&\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi}\end{bmatrix}.$

Also notice that the matrix $A_{3}(\mathring{U})$ is equal to the following
matrix on the boundary

$A_{3}(\mathring{U})|_{\Sigma,\Sigma^{\pm}}:=\begin{bmatrix}0&\mathring{\mathbf{N}}^{\top}&\bm{0}_{4}^{\top}&\\\
\mathring{\mathbf{N}}&\bm{O}_{3}&\\\ \bm{0}_{4}&&\bm{O}_{4}\end{bmatrix},$

which has constant rank 2 and has one negative eigenvalue and one positive
eigenvalue, so the correct number of boundary conditions to solve $U^{\pm}$ in
(4.8) is $1+1=2$ (the jump conditions for $q$ and
$v\cdot\mathring{\mathbf{N}}$), and we need one more equation, namely
$\partial_{t}\psi=v^{\pm}\cdot\mathring{N}$, to determine the free interface.

Given $2\leq m\in\mathbb{N}^{*}$, we introduce the Galerkin sequence
$\\{U^{m,\pm}(t,x),\psi^{m}(t,x^{\prime})\\}$ by

$\displaystyle U_{j}^{m,\pm}(t,x):=$
$\displaystyle\sum_{l=1}^{m}U_{lj}^{m,\pm}(t)\bm{e}_{l}(x)\quad 1\leq j\leq
8,$ (4.9) $\displaystyle\psi^{m}(t,x^{\prime}):=$
$\displaystyle\sum_{l=1}^{m}\psi_{l}^{m}(t)\bm{e}_{l}(x^{\prime},0).$ (4.10)

The Galerkin sequence is assumed to satisfy the boundary conditions

$\displaystyle\partial_{t}\psi^{m}=$
$\displaystyle~{}U_{4}^{m,\pm}-U_{2}^{m,\pm}\overline{\partial}{}_{1}\mathring{\psi}-U_{3}^{m,\pm}\overline{\partial}{}_{2}\mathring{\psi}$
(4.11) $\displaystyle\left\llbracket U_{1}^{m}\right\rrbracket=$
$\displaystyle~{}\sigma\mathcal{H}(\mathring{\psi})-\kappa(1-\overline{\Delta}{})^{2}\psi^{m}-\kappa(1-\overline{\Delta}{})\partial_{t}\psi^{m}$
(4.12)

Now we introduce an ODE system as the “truncated version” of (4.8) in
$\text{Span}\\{\bm{e}_{1},\cdots,\bm{e}_{m}\\}$ by testing the Galerkin
sequence by a vector field $\bm{\phi}:=(\phi_{1},\cdots,\phi_{8})^{\top}$ with

$\phi_{i}:=\sum_{l=1}^{m}\phi_{il}(t)\bm{e}_{l}(x)\in\text{Span}\\{\bm{e}_{1},\cdots,\bm{e}_{m}\\}.$
$\displaystyle\int_{\Omega^{\pm}}A_{0}^{ij}(\mathring{U}^{\pm})(\partial_{t}U_{j}^{m,\pm})\phi_{i}\,\mathrm{d}\mathring{\mathcal{V}}_{t}+\sum_{k=1}^{2}A_{k}^{ij}(\mathring{U}^{\pm})(\overline{\partial}{}_{k}U_{j}^{m,\pm})\phi_{i}\,\mathrm{d}\mathring{\mathcal{V}}_{t}+\int_{\Omega^{\pm}}A_{3}^{ij}(\mathring{U}^{\pm})(\partial_{3}U_{j}^{m,\pm})\phi_{i}\,\mathrm{d}\mathring{\mathcal{V}}_{t}=0$
(4.13)

where
$\,\mathrm{d}\mathring{\mathcal{V}}_{t}:=\partial_{3}\mathring{\varphi}\,\mathrm{d}x$
Integrating by parts in $\overline{\partial}{}_{k}$ and $\partial_{3}$, we get

$\displaystyle\int_{\Omega^{\pm}}A_{0}^{ij}(\mathring{U}^{\pm})(\partial_{t}U_{j}^{m,\pm})\phi_{i}-\sum_{k=1}^{3}U_{j}^{m,\pm}\partial_{k}(A_{k}^{ij}(\mathring{U}^{\pm})\phi_{i})\,\mathrm{d}\mathring{\mathcal{V}}_{t}\mp\int_{\Sigma}A_{3}^{ij}(\mathring{U}^{\pm})U_{j}^{m,\pm}\phi_{i}\,\mathrm{d}x^{\prime}=0.$
(4.14)

Plugging the Galerkin sequence into the above identity, we get

$\displaystyle\int_{\Omega^{\pm}}A_{0}^{ij}(\mathring{U}^{\pm})\bm{e}_{l}(x)\phi_{i}(U_{lj}^{m,\pm})^{\prime}(t)-\sum_{k=1}^{3}\partial_{k}(A_{k}^{ij}(\mathring{U}^{\pm})\phi_{i})\bm{e}_{l}(x)U_{lj}^{m,\pm}(t)\,\mathrm{d}\mathcal{V}_{t}=\pm\int_{\Sigma}A_{3}^{ij}(\mathring{U}^{\pm})U_{j}^{m,\pm}\phi_{i}\,\mathrm{d}x^{\prime}.$
(4.15)

Taking sum for the two parts in $\Omega^{\pm}$, setting
$\phi_{i}(x)=\bm{e}_{i}(x)$ and using the jump condition for $\left\llbracket
q\right\rrbracket$ and $v^{\pm}\cdot\mathring{\mathbf{N}}$, we obtain a first-
order linear ODE system for $\\{U_{lj}^{\pm}(t)\\}$

$\displaystyle\sum_{\pm}\left(\int_{\Omega^{\pm}}A_{0}^{ij}(\mathring{U}^{\pm})\bm{e}_{l}(x)\bm{e}_{i}(x)\,\mathrm{d}\mathring{\mathcal{V}}_{t}\right)(U_{lj}^{m,\pm})^{\prime}(t)-\left(\int_{\Omega^{\pm}}\partial_{k}(A_{k}^{ij}(\mathring{U}^{\pm})\bm{e}_{i}(x))\bm{e}_{l}(x)\,\mathrm{d}\mathring{\mathcal{V}}_{t}\right)U_{lj}^{m,\pm}(t)$
$\displaystyle=$ $\displaystyle\int_{\Sigma}\left\llbracket
q\right\rrbracket\underbrace{\left(-\bm{e}_{2}(x^{\prime},0)\overline{\partial}{}_{1}\mathring{\psi}(t,x^{\prime})-\bm{e}_{3}(x^{\prime},0)\overline{\partial}{}_{2}\mathring{\psi}(t,x^{\prime})+\bm{e}_{4}(x^{\prime},0)\right)}_{=:\phi_{v}\cdot\mathring{N}}\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle-\sigma\int_{\Sigma}(\overline{\nabla}\mathring{\psi}/|\mathring{N}|)\cdot\overline{\nabla}\left(\phi_{v}\cdot\mathring{N}\right)\,\mathrm{d}x^{\prime}-\kappa\int_{\Sigma}(1-\overline{\Delta}{})\psi^{m}\,(1-\overline{\Delta}{})\left(\phi_{v}\cdot\mathring{N}\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\kappa\int_{\Sigma}\partial_{t}\psi^{m}\left(\phi_{v}\cdot\mathring{N}\right)\,\mathrm{d}x^{\prime}-\kappa\int_{\Sigma}\overline{\nabla}\partial_{t}\psi^{m}\cdot\overline{\nabla}\left(\phi_{v}\cdot\mathring{N}\right)\,\mathrm{d}x^{\prime}.$
(4.16)

Since the basis $\\{\bm{e}_{l}\\}$ are smooth and the coefficients
$(\mathring{U}^{m,\pm},\mathring{\psi})$ are sufficiently regular, standard
ODE theory guarantees the local existence and uniqueness of the above ODE
system (4.16) with initial data

$U_{lj}^{m,\pm}(0):=\int_{\Omega^{\pm}}U_{j}^{m,\pm}(0,x)\bm{e}_{k}(x)\partial_{3}\mathring{\varphi}_{0}\,\mathrm{d}x.$

####  4.2.2 Existence of solutions to the linearized problem

The existence of weak solution is guaranteed by uniform-in-$m$ estimates for
the Galerkin sequence $\\{U^{m,\pm}(t,x),\psi^{m}(t,x^{\prime})\\}$. Now we
let the test function $\bm{\phi}=U^{m,\pm}$ in $\Omega^{\pm}$ respectively to
obtain the standard $L^{2}$-type energy estimates thanks to the symmetric
property of the coefficient matrices and the concrete form of
$A_{3}(\mathring{U})$ on the boundary

$\displaystyle\sum_{\pm}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\int_{\Omega^{\pm}}(U^{m,\pm})^{\top}\cdot
A_{0}(\mathring{U}^{\pm})U^{m,\pm}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$
$\displaystyle\sum_{\pm}\frac{1}{2}\int_{\Omega^{\pm}}(U^{m,\pm})^{\top}\cdot\partial_{t}(A_{0}(\mathring{U}^{\pm}))U^{m,\pm}\,\mathrm{d}\mathcal{V}_{t}+\frac{1}{2}\int_{\Omega^{\pm}}(U^{m,\pm})^{\top}\cdot\partial_{k}(A_{k}(\mathring{U}^{\pm}))U^{m,\pm}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\int_{\Sigma}\left\llbracket
U_{1}^{m}\right\rrbracket(U_{4}^{m,\pm}-U_{2}^{m,\pm}\overline{\partial}{}_{1}\mathring{\psi}-U_{3}^{m,\pm}\overline{\partial}{}_{2}\mathring{\psi})\,\mathrm{d}x^{\prime}$
(4.17)

where the interior term can be controlled directly by
$C(\mathring{K}_{0})\|U^{m,\pm}\|_{0,\pm}^{2}$. For the boundary term, using
the boundary conditions (4.11)-(4.12), we get the energy bounds under time
integral

$\displaystyle\int_{0}^{t}\int_{\Sigma}\left\llbracket
U_{1}^{m}\right\rrbracket(U_{4}^{m,\pm}-U_{2}^{m,\pm}\overline{\partial}{}_{1}\mathring{\psi}-U_{3}^{m,\pm}\overline{\partial}{}_{2}\mathring{\psi})\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\Sigma}\left(\sigma\mathcal{H}(\mathring{\psi})-\kappa(1-\overline{\Delta}{})^{2}\psi^{m}-\kappa(1-\overline{\Delta}{})\partial_{t}\psi^{m}\right)\partial_{t}\psi^{m}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle-\sigma\int_{0}^{t}\int_{\Sigma}(\overline{\nabla}\mathring{\psi}/|\mathring{\mathbf{N}}|)\cdot\overline{\nabla}\partial_{t}\psi^{m}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau-\kappa\int_{0}^{t}\int_{\Sigma}(1-\overline{\Delta}{})\psi^{m}\,(1-\overline{\Delta}{})\partial_{t}\psi_{m}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\int_{0}^{t}\int_{\Sigma}\partial_{t}\psi^{m}\partial_{t}\psi_{m}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau-\kappa\int_{0}^{t}\int_{\Sigma}\overline{\nabla}\partial_{t}\psi^{m}\cdot\overline{\nabla}\partial_{t}\psi_{m}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle~{}\frac{\sigma}{\sqrt{\kappa}}\int_{0}^{t}P(|\overline{\nabla}\mathring{\psi}|_{L^{\infty}})|\overline{\nabla}\mathring{\psi}|_{0}^{2}\,\mathrm{d}\tau+\delta|\sqrt{\kappa}\psi^{m}|_{L_{t}^{2}H_{x^{\prime}}^{1}}^{2}-\frac{1}{2}\left|\sqrt{\kappa}\psi^{m}\right|_{2}^{2}\bigg{|}^{t}_{0}-\int_{0}^{t}\left|\sqrt{\kappa}\partial_{t}\psi^{m}\right|_{1}^{2}\,\mathrm{d}\tau.$
(4.18)

We define

$\displaystyle\bm{N}^{m}(t):=\sum_{\pm}\left\|\left(\sqrt{\mathring{\mathcal{F}}_{p}^{\pm}}U_{1}^{m,\pm},U_{2}^{m,\pm},\cdots,U_{8}^{m,\pm}\right)\right\|_{0,\pm}^{2}+\left|\sqrt{\kappa}\psi^{m}\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\partial_{t}\psi^{m}\right|_{1}^{2}\,\mathrm{d}\tau.$
(4.19)

Since $A_{0}(\mathring{U}^{\pm})>0$, we obtain the uniform-in-$m$ estimate for
the Galerkin sequence $\\{U^{m,\pm}(t,x),\psi^{m}(t,x^{\prime})\\}$.

$\displaystyle\bm{N}^{m}(t)\leq\bm{N}^{m}(0)+\int_{0}^{t}C(\mathring{K}_{0},\kappa^{-1})\bm{N}^{m}(\tau)\,\mathrm{d}\tau,$
(4.20)

and thus there exists a time $T_{\bm{N}}>0$ (depending on $\kappa$ and
$\bm{N}^{m}(0)$, independent of $m$) such that

$\sup\limits_{0\leq t\leq T_{\bm{N}}}\bm{N}^{m}(t)\leq
C^{\prime}(\mathring{K}_{0},\kappa^{-1})\bm{N}^{m}(0).$

Because $L^{\infty}([0,T_{\bm{N}}];L^{2}(\Omega^{\pm}))$ is not reflexive, we
alternatively consider the weak convergence in
$L^{2}([0,T_{\bm{N}}];L^{2}(\Omega^{\pm}))$. By Eberlein-Šmulian theorem and
the uniqueness of expansion in Galerkin basis
$\\{\bm{e}_{l}\\}_{l=1}^{\infty}$, there exists a subsequence
$\\{U^{m_{k},\pm}(t,x),\psi^{m_{k}}(t,x^{\prime})\\}_{k=1}^{\infty}$ such that

$\displaystyle\left(\sqrt{\mathring{\mathcal{F}}_{p}^{\pm}}U_{1}^{m_{k},\pm},U_{2}^{m_{k},\pm},\cdots,U_{8}^{m_{k},\pm}\right)\rightharpoonup\left(\sqrt{\mathring{\mathcal{F}}_{p}^{\pm}}q^{\pm},v^{\pm},\cdots,b^{\pm},S^{\pm}\right)\text{
in }L^{2}([0,T_{\bm{N}}];L^{2}(\Omega^{\pm}))$ (4.21)
$\displaystyle\psi^{m_{k}}\rightharpoonup\psi\text{ in
}L^{2}([0,T_{\bm{N}}];H^{2}(\Sigma)),\quad\partial_{t}\psi^{m_{k}}\rightharpoonup\partial_{t}\psi\text{
in }L^{2}([0,T_{\bm{N}}];H^{1}(\Sigma)).$ (4.22)

This proves the existence of weak solution to (4.8) (and equivalently (4.6)).
The uniqueness easily follows from the estimate of $\bm{N}(t)$ and the
linearity of (4.8). The weak solution is actually a strong solution according
to the argument in [60, Chapter 2.2.3].

###  4.3 Higher-order estimates of the linearized approximate problem

To proceed the Picard iteration, we shall prove that the bounds (4.7) for the
coefficients $(\mathring{U},\mathring{\psi},\dot{\psi})$ can be preserved by
the solution to (4.6). Fix $\kappa>0$, we define the energy functional for
(4.6) to be

$\displaystyle\mathring{E}^{\kappa}(t):=$
$\displaystyle~{}\mathring{E}^{\kappa}_{4}(t)+\cdots+\mathring{E}^{\kappa}_{8}(t)$
(4.23) $\displaystyle\mathring{E}^{\kappa}_{4+l}(t):=$
$\displaystyle\sum_{\pm}\sum_{\langle\alpha\rangle=2l}\sum_{k=0}^{4-l}\left\|\left(\varepsilon^{2l}\mathcal{T}^{\alpha}\partial_{t}^{k}(v^{\pm},b^{\pm},S^{\pm},(\mathring{\mathcal{F}}_{p}^{\pm})^{\frac{(k+\alpha_{0}-l-3)_{+}}{2}}q^{\pm})\right)\right\|^{2}_{4-k-l,\pm}$
$\displaystyle+\sum_{k=0}^{4+l}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\psi\right|^{2}_{6+l-k}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{5+l}\psi\right|^{2}_{1}\,\mathrm{d}\tau$

where
$\mathcal{T}^{\alpha}:=(\omega(x_{3})\partial_{3})^{\alpha_{4}}\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
with the multi-index
$\alpha=(\alpha_{0},\alpha_{1},\alpha_{2},0,\alpha_{4}),~{}\langle\alpha\rangle=\alpha_{0}+\alpha_{1}+\alpha_{2}+2\times
0+\alpha_{4}$. We aim to prove that

######  Proposition 4.1.

There exists some $T_{\kappa}>0$ depending on $\kappa$ and $\mathring{K}_{0}$,
such that

$\sup_{0\leq t\leq T_{\kappa}}\mathring{E}^{\kappa}(t)\leq
C(\kappa^{-1},\mathring{K}_{0})\mathring{E}^{\kappa}(0).$

A large part of the proof of proposition (4.1) is similar to the analysis in
Section 3. Moreover, since $\kappa>0$ is fixed, we obtain higher boundary
regularity for the free interface $\psi$, which allows us to avoid some
technical steps (such as the analysis in Section 3.4). We will skip some
details for the part substantially similar to Section 3 and emphasize the
different part. Now we start with div-curl analysis.

####  4.3.1 Div-Curl analysis

We start with $\mathring{E}_{4}(t)$. Using (B.1) and the boundary conditions
for $v,b$, we get

$\displaystyle\left\|v^{\pm},b^{\pm}\right\|_{4,\pm}^{2}\lesssim
C(|\mathring{\psi}|_{4},|\overline{\nabla}\mathring{\psi}|_{W^{1,\infty}})\left(\|(v^{\pm},b^{\pm})\|_{0,\pm}^{2}+\|\nabla^{\mathring{\varphi}}\cdot(v^{\pm},b^{\pm})\|_{3,\pm}^{2}+\|\nabla^{\mathring{\varphi}}\times(v^{\pm},b^{\pm})\|_{3,\pm}^{2}+\|\overline{\partial}{}^{4}(v^{\pm},b^{\pm})\|_{0}^{2}\right).$
(4.24)

######  Remark 4.4.

Here we cannot use the div-curl inequality (B.2) to estimate the normal traces
because the boundary constraint $b\cdot\mathring{N}=0$ no longer holds for the
linearized problem.

The $L^{2}$-estimates are already proven in the uniform estimates of Galerkin
sequence, so we no longer repeat it. The treatment of
$\nabla^{\mathring{\varphi}}\cdot v$ is also the same as in Section 3.3.1,
that is, invoking the continuity equation. For
$\nabla^{\mathring{\varphi}}\cdot b$, we no longer have the div-free
constraint. Instead, we can take $\nabla^{\mathring{\varphi}}\cdot$ in the
linearized evolution equation of $b$ to get

$\displaystyle D_{t}^{\mathring{\varphi}\pm}(\nabla^{\mathring{\varphi}}\cdot
b^{\pm})=$
$\displaystyle~{}\partial^{\mathring{\varphi}}_{i}(\mathring{\bm{b}}_{j}^{\pm}\partial^{\mathring{\varphi}}_{j}v_{i}^{\pm})-\partial^{\mathring{\varphi}}_{i}(\mathring{\bm{b}}_{i}^{\pm}\partial^{\mathring{\varphi}}_{j}v_{j}^{\pm})+[D_{t}^{\mathring{\varphi}\pm},\nabla^{\mathring{\varphi}}\cdot]b^{\pm}$
(4.25) $\displaystyle=$
$\displaystyle~{}(\partial^{\mathring{\varphi}}_{i}\mathring{\bm{b}}_{j}^{\pm})(\partial^{\mathring{\varphi}}_{j}v_{i}^{\pm})-(\nabla^{\mathring{\varphi}}\cdot\bm{b}^{\pm})(\nabla^{\mathring{\varphi}}\cdot
v^{\pm})+[D_{t}^{\mathring{\varphi}\pm},\nabla^{\mathring{\varphi}}\cdot]b^{\pm}.$
(4.26)

Direct calculation shows that
$[D_{t}^{\mathring{\varphi}\pm},\partial^{\mathring{\varphi}}_{i}](\cdot)=-(\partial^{\mathring{\varphi}}_{i}\mathring{v}_{j})\partial^{\mathring{\varphi}}_{j}(\cdot)-(\partial^{\mathring{\varphi}}_{i}\partial_{t}(\mathring{\varphi}-\dot{\varphi}))\partial^{\mathring{\varphi}}_{3}(\cdot)$.
On the other hand, the $\kappa$-regularization term provides extra regularity
for $\varphi_{t},\mathring{\varphi}_{t},\dot{\varphi}_{t}$. Thus, standard
$H^{3}$ estimates give the control of divergence

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left\|\nabla^{\mathring{\varphi}}\cdot
b^{\pm}\right\|_{3,\pm}^{2}\lesssim
C(\mathring{K}_{0},\kappa^{-1})\left(\|b^{\pm}\|_{4,\pm}\|v^{\pm}\|_{4,\pm}\right)\leq
C(\mathring{K}_{0},\kappa^{-1})\mathring{E}^{\kappa}_{4}(t).$ (4.27)

The vorticity part is analyzed in a similar way as in Section 3.6. The
evolution equations are

$\displaystyle\mathring{\rho}D_{t}^{\mathring{\varphi}}(\nabla^{\mathring{\varphi}}\times
v)-(\mathring{\bm{b}}\cdot\nabla^{\mathring{\varphi}})(\nabla^{\mathring{\varphi}}\times
b)=$
$\displaystyle~{}(\nabla^{\mathring{\varphi}}\mathring{\rho})\times(D_{t}^{\mathring{\varphi}}v)-(\nabla^{\mathring{\varphi}}\mathring{\bm{b}}_{j})\times(\partial^{\mathring{\varphi}}_{j}b)$
$\displaystyle-\mathring{\rho}\left((\nabla^{\mathring{\varphi}}\mathring{v}_{j})\times(\partial^{\mathring{\varphi}}_{j}v)+\nabla^{\varphi}(\partial_{t}\mathring{\varphi}-\partial_{t}\dot{\varphi})\times\partial^{\mathring{\varphi}}_{3}v\right),$
$\displaystyle D_{t}^{\mathring{\varphi}}(\nabla^{\mathring{\varphi}}\times
b)-(\mathring{\bm{b}}\cdot\nabla^{\mathring{\varphi}})(\nabla^{\mathring{\varphi}}\times
v)-\mathring{\bm{b}}\times\nabla^{\mathring{\varphi}}(\nabla^{\mathring{\varphi}}\cdot
v)=$
$\displaystyle-(\nabla^{\mathring{\varphi}}\times\mathring{\bm{b}})(\nabla^{\mathring{\varphi}}\cdot
v)-(\nabla^{\mathring{\varphi}}\mathring{\bm{b}}_{j})\times(\partial^{\mathring{\varphi}}_{j}v)$
$\displaystyle-(\nabla^{\mathring{\varphi}}\mathring{v}_{j})\times(\partial^{\mathring{\varphi}}_{j}b)-\nabla^{\varphi}(\partial_{t}\mathring{\varphi}-\partial_{t}\dot{\varphi})\times\partial^{\mathring{\varphi}}_{3}b,$

on the right side of which the highest-order derivative is 1 (except the
mismatch term). Thus, we can still follow the analysis in Section 3.6.2 to get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\int_{\Omega^{\pm}}\mathring{\rho}^{\pm}\left|\partial^{3}\nabla^{\mathring{\varphi}}\times
v^{\pm}\right|^{2}+\left|\partial^{3}\nabla^{\mathring{\varphi}}\times
b^{\pm}\right|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}\leq
P(\mathring{E}^{\kappa}_{4}(t),\mathring{K}_{0})+\mathring{K}_{1}^{\pm},$
(4.28)

where

$\displaystyle\mathring{K}_{1}^{\pm}:=\int_{\Omega^{\pm}}(\partial^{3}\nabla^{\mathring{\varphi}}\times
b^{\pm})\cdot\left(\mathring{\bm{b}}^{\pm}\times(\partial^{3}\nabla^{\mathring{\varphi}}(\nabla^{\mathring{\varphi}}\cdot
v^{\pm}))\right)\,\mathrm{d}\mathring{\mathcal{V}}_{t}.$ (4.29)

Again, we invoke the continuity equation and the momentum equation to get

$\displaystyle\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}(\nabla^{\mathring{\varphi}}\cdot
v))\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}D_{t}^{\mathring{\varphi}}q)+\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\times((\partial^{3}\nabla^{\mathring{\varphi}}D_{t}^{\mathring{\varphi}}b_{j})\mathring{\bm{b}}_{j})$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\mathring{\mathcal{F}}_{p}\mathring{\rho}\mathring{\bm{b}}\times(\partial^{3}(D_{t}^{\mathring{\varphi}})^{2}v)-\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\times
D_{t}^{\mathring{\varphi}}(\mathring{b}_{j}\partial^{3}\partial^{\mathring{\varphi}}_{j}b)+\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\times
D_{t}^{\mathring{\varphi}}((\partial^{3}\nabla^{\mathring{\varphi}}b_{j})\mathring{\bm{b}}_{j})$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\mathring{\mathcal{F}}_{p}\mathring{\rho}\mathring{\bm{b}}\times(\partial^{3}(D_{t}^{\mathring{\varphi}})^{2}v)+\mathring{\mathcal{F}}_{p}\mathring{\bm{b}}\times
D_{t}^{\mathring{\varphi}}(\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}\times
b))$

where we use the vector identity
$(\mathbf{a}\times(\nabla^{\mathring{\varphi}}\times\mathbf{b}))_{i}=(\partial^{\mathring{\varphi}}_{i}\mathbf{b}_{j})\mathbf{a}_{j}-\mathbf{a}_{j}\partial^{\mathring{\varphi}}_{j}\mathbf{b}_{i}$,
and the omitted terms are directly controlled by
$P(\mathring{E}^{\kappa}_{4}(t),\mathring{K}_{0})$. Thus, we have

$\displaystyle\mathring{K}_{1}^{\pm}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\rho}(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\cdot\left(\mathring{\bm{b}}\times(\partial^{3}(D_{t}^{\mathring{\varphi}})^{2}v)\right)\,\mathrm{d}\mathring{\mathcal{V}}_{t}+\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\cdot\left(\mathring{\bm{b}}\times
D_{t}^{\mathring{\varphi}}(\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}b))\right)\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$
$\displaystyle\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\rho}(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\cdot\left(\mathring{\bm{b}}\times(\partial^{3}(D_{t}^{\mathring{\varphi}})^{2}v)\right)\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}D_{t}^{\mathring{\varphi}}\left(\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\right)\cdot\left(\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\right)\,\mathrm{d}\mathcal{V}_{t}$ $\displaystyle\lesssim$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}\left|\mathring{\bm{b}}\times(\partial^{3}\nabla^{\mathring{\varphi}}\times
b)\right|_{0}^{2}+P(\mathring{K}_{0})\mathring{E}^{\kappa}_{4}(t)+\mathring{E}^{\kappa}_{5}(t).$

So, we have

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathring{\rho}^{\pm}\left|\partial^{3}\nabla^{\mathring{\varphi}}\times
v^{\pm}\right|^{2}+\left|\partial^{3}\nabla^{\mathring{\varphi}}\times
b^{\pm}\right|^{2}+\mathcal{F}_{p}^{\pm}\left|\mathring{\bm{b}}^{\pm}\times(\partial^{3}\nabla^{\mathring{\varphi}}\times
b^{\pm})\right|_{0}^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle\lesssim$
$\displaystyle~{}P(\mathring{K}_{0})\mathring{E}_{4}^{\kappa}(t)+\mathring{E}_{5}^{\kappa}(t).$
(4.30)

Similarly as in Section 3.6.2 and Section 3.6.3, we can prove the div-curl
estimates for time-differentiated system and
$\mathcal{T}^{\alpha}$-differentiated system. For $0\leq l\leq 3,0\leq k\leq
3-l,\langle\alpha\rangle=2l,~{}\alpha_{3}=0$, we have

$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{4-l-k,\pm}^{2}\leq
C(|\mathring{\psi}|_{3})\bigg{(}$
$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{3-k-l,\pm}^{2}$
$\displaystyle+\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{3-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\overline{\partial}{}^{4-k-l}\partial_{t}^{k}\mathcal{T}^{\alpha}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}\bigg{)}.$
(4.31)

Then the curl part has the following control

$\displaystyle\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{3-l-k,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm}\right\|_{3-l-k,\pm}^{2}+\left\|\varepsilon^{2l}\mathring{\mathcal{F}}_{p}^{\pm}\mathring{b}\times(\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm})\right\|_{3-l-k,\pm}^{2}$
(4.32) $\displaystyle\lesssim$
$\displaystyle~{}P(\mathring{K}_{0})\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P(\mathring{K}_{0})\int_{0}^{t}\sum_{j=0}^{l}\mathring{E}_{4+j}^{\kappa}(\tau)+\mathring{E}_{4+l+1}^{\kappa}(\tau)\,\mathrm{d}\tau.$

Similarly, the divergence part is controlled by

$\displaystyle\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}v^{\pm}\right\|_{3-l-k,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\mathring{\varphi}}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}b^{\pm}\right\|_{3-l-k,\pm}^{2}$
(4.33) $\displaystyle\lesssim$
$\displaystyle~{}\left\|\varepsilon^{2l}\mathring{\mathcal{F}}_{p}^{\pm}\partial_{t}^{k}\mathcal{T}^{\alpha}D_{t}^{\mathring{\varphi}\pm}(q^{\pm},b^{\pm})\right\|_{3-l-k,\pm}^{2}+C(\mathring{K}_{0})\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P(\mathring{K}_{0})\int_{0}^{t}\sum_{j=0}^{l}\mathring{E}_{4+j}^{\kappa}(\tau)\,\mathrm{d}\tau,$

in which the first term will be controlled via tangential estimates.

For the pressure $q$, we still use the linearized momentum equation to convert
it to tangential derivatives of $v$ and $b$. This step is exactly the same as
Section 3.6.1, so we do not repeat the details here.

####  4.3.2 Tangential estimates

The tangential estimates are also proved in the same way as Section
3.3-Section 3.5. What’s more, the rather technical part in the estimates of
full time derivatives can be simplified a lot thanks to the
$\sqrt{\kappa}$-weighted extra regularity of the free interface. For
$\mathcal{T}^{\gamma}$-differentiated linearize system (4.6), we introduce the
corresponding Alinhac good unknown
$\mathring{\mathbf{F}}^{\gamma}:=\mathcal{T}^{\gamma}f-\mathcal{T}^{\gamma}\mathring{\varphi}\partial^{\mathring{\varphi}}_{3}f$
which satisfies

$\mathcal{T}^{\gamma}(\partial^{\mathring{\varphi}}_{i}f)=\partial^{\mathring{\varphi}}_{i}\mathring{\mathbf{F}}^{\gamma}+\mathring{\mathfrak{C}}^{\gamma}_{i}(f),\quad\mathcal{T}^{\gamma}(D_{t}^{\mathring{\varphi}}f)=D_{t}^{\mathring{\varphi}}\mathring{\mathbf{F}}^{\gamma}+\mathring{\mathfrak{D}}^{\gamma}(f),$

where

$\displaystyle\mathring{\mathfrak{C}}_{i}^{\gamma}(f)=$
$\displaystyle~{}(\partial^{\mathring{\varphi}}_{3}\partial^{\mathring{\varphi}}_{i}f)\mathcal{T}^{\gamma}\mathring{\varphi}+\left[\mathcal{T}^{\gamma},\frac{\mathring{\mathbf{N}}_{i}}{\partial_{3}\mathring{\varphi}},\partial_{3}f\right]+\partial_{3}f\left[\mathcal{T}^{\gamma},\mathring{\mathbf{N}}_{i},\frac{1}{\partial_{3}\mathring{\varphi}}\right]+\mathring{\mathbf{N}}_{i}\partial_{3}f\left[\mathcal{T}^{\gamma-\gamma^{\prime}},\frac{1}{(\partial_{3}\mathring{\varphi})^{2}}\right]\mathcal{T}^{\gamma^{\prime}}\partial_{3}\mathring{\varphi}$
$\displaystyle+\frac{\mathring{\mathbf{N}}_{i}}{\partial_{3}\mathring{\varphi}}[\mathcal{T}^{\gamma},\partial_{3}]f-\frac{\mathring{\mathbf{N}}_{i}}{(\partial_{3}\mathring{\varphi})^{2}}\partial_{3}f[\mathcal{T}^{\gamma},\partial_{3}]\mathring{\varphi},$
(4.34)

and

$\displaystyle\mathring{\mathfrak{D}}^{\gamma}(f)=$
$\displaystyle~{}(D_{t}^{\mathring{\varphi}}\partial^{\mathring{\varphi}}_{3}f)\mathcal{T}^{\gamma}\varphi+[\mathcal{T}^{\gamma},\mathring{\bar{v}}]\cdot\overline{\partial}{}f+\left[\mathcal{T}^{\gamma},\frac{1}{\partial_{3}\mathring{\varphi}}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi}),\partial_{3}f\right]+\left[\mathcal{T}^{\gamma},(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi}),\frac{1}{\partial_{3}\mathring{\varphi}}\right]\partial_{3}f$
$\displaystyle+\frac{1}{\partial_{3}\varphi}[\mathcal{T}^{\gamma},\mathring{v}]\cdot\dot{\mathbf{N}}\partial_{3}f-(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\partial_{3}f\left[\mathcal{T}^{\gamma-\gamma^{\prime}},\frac{1}{(\partial_{3}\mathring{\varphi})^{2}}\right]\mathcal{T}^{\gamma^{\prime}}\partial_{3}\mathring{\varphi}$
$\displaystyle+\frac{1}{\partial_{3}\mathring{\varphi}}(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})[\mathcal{T}^{\gamma},\partial_{3}]f+(\mathring{v}\cdot\dot{\mathbf{N}}-\partial_{t}\mathring{\varphi})\frac{\partial_{3}f}{(\partial_{3}\mathring{\varphi})^{2}}[\mathcal{T}^{\gamma},\partial_{3}]\mathring{\varphi}+\mathcal{T}^{\gamma}\partial_{t}(\dot{\varphi}-\mathring{\varphi})\partial^{\mathring{\varphi}}_{3}f$
(4.35)

with $\langle\gamma^{\prime}\rangle=1$. Since $\mathring{\mathbf{N}}_{3}=1$,
the third term in $\mathring{\mathfrak{C}}_{i}^{\gamma}(f)$ does not appear
when $i=3$. Under this setting, the $\mathcal{T}^{\gamma}$-differentiated
linearized system is reformulated as follows

$\displaystyle\mathring{\rho}^{\pm}D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{V}}^{\gamma,\pm}-(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{B}}^{\gamma,\pm}+\nabla^{\mathring{\varphi}}\mathring{\mathbf{Q}}^{\gamma,\pm}=\mathring{\mathcal{R}}_{v}^{\gamma,\pm}-\mathring{\mathfrak{C}}^{\gamma}(q^{\pm})~{}~{}$
$\displaystyle\text{ in }[0,T]\times\Omega^{\pm},$ (4.36)
$\displaystyle\mathring{\mathcal{F}}_{p}^{\pm}D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{Q}}^{\gamma,\pm}-\mathring{\mathcal{F}}_{p}^{\pm}D_{t}^{\mathring{\varphi}\pm}\mathbf{B}^{\gamma,\pm}\cdot\mathring{\bm{b}}^{\pm}+\nabla^{\mathring{\varphi}}\cdot\mathring{\mathbf{V}}^{\gamma,\pm}=\mathring{\mathcal{R}}_{p}^{\gamma,\pm}-\mathring{\mathfrak{C}}_{i}^{\gamma}(v_{i}^{\pm})~{}~{}$
$\displaystyle\text{ in }[0,T]\times\Omega^{\pm},$ (4.37) $\displaystyle
D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{B}}^{\gamma,\pm}-(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{V}}^{\gamma,\pm}+\mathring{b}^{\pm}(\nabla^{\mathring{\varphi}}\cdot\mathring{\mathbf{V}}^{\gamma,\pm})=\mathring{\mathcal{R}}_{b}^{\gamma,\pm}-\mathring{b}^{\pm}\mathring{\mathfrak{C}}_{i}^{\gamma}(v_{i}^{\pm})~{}~{}$
$\displaystyle\text{ in }[0,T]\times\Omega^{\pm},$ (4.38) $\displaystyle
D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{S}}^{\pm,\alpha}=\mathring{\mathfrak{D}}^{\gamma}(S^{\pm})~{}~{}$
$\displaystyle\text{ in }[0,T]\times\Omega^{\pm},$ (4.39)

with boundary conditions

$\displaystyle\left\llbracket\mathring{\mathbf{Q}}^{\gamma}\right\rrbracket=\sigma\mathcal{T}^{\gamma}\mathcal{H}(\mathring{\psi})-\kappa\mathcal{T}^{\gamma}(1-\overline{\Delta}{})^{2}\psi-\kappa\mathcal{T}^{\gamma}(1-\overline{\Delta}{})\partial_{t}\psi-\left\llbracket\partial_{3}q\right\rrbracket\mathcal{T}^{\gamma}\mathring{\psi}~{}~{}$
$\displaystyle\text{ on }[0,T]\times\Sigma,$ (4.40)
$\displaystyle\mathring{\mathbf{V}}^{\gamma,\pm}\cdot\mathring{N}=\partial_{t}\mathcal{T}^{\gamma}\psi+\bar{v}^{\pm}\cdot\overline{\nabla}\mathcal{T}^{\gamma}\mathring{\psi}-\mathcal{W}^{\gamma,\pm}~{}$
$\displaystyle\text{ on }[0,T]\times\Sigma,$ (4.41)

where
$\mathring{\mathcal{R}}_{v},\mathring{\mathcal{R}}_{p},\mathring{\mathcal{R}}_{b}$
terms consist of the following commutators

$\displaystyle\mathring{\mathcal{R}}_{v}^{\gamma,\pm}:=$
$\displaystyle~{}[\mathcal{T}^{\gamma},\mathring{\bm{b}}^{\pm}]\cdot\nabla^{\mathring{\varphi}}b^{\pm}-[\mathcal{T}^{\gamma},\mathring{\rho}^{\pm}]D_{t}^{\mathring{\varphi}\pm}v^{\pm}-\mathring{\rho}^{\pm}\mathring{\mathfrak{D}}^{\gamma}(v^{\pm})$
(4.42) $\displaystyle\mathring{\mathcal{R}}_{p}^{\gamma,\pm}:=$
$\displaystyle-[\mathcal{T}^{\gamma},\mathring{\mathcal{F}}_{p}^{\pm}]D_{t}^{\mathring{\varphi}\pm}q^{\pm}-\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\mathfrak{D}}^{\gamma}(q^{\pm})$
$\displaystyle+[\mathcal{T}^{\gamma},\mathring{\mathcal{F}}_{p}^{\pm}]D_{t}^{\mathring{\varphi}\pm}b^{\pm}\cdot\mathring{\bm{b}}^{\pm}+\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\mathfrak{D}}^{\gamma}(b^{\pm})\cdot\mathring{\bm{b}}^{\pm}+[\mathcal{T}^{\gamma},\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\bm{b}}^{\pm}]\cdot
D_{t}^{\mathring{\varphi}\pm}b^{\pm}$ (4.43)
$\displaystyle\mathring{\mathcal{R}}_{b}^{\gamma,\pm}:=$
$\displaystyle~{}[\mathcal{T}^{\gamma},\mathring{\bm{b}}^{\pm}]\cdot\nabla^{\mathring{\varphi}}v^{\pm}-\mathring{\mathfrak{D}}^{\gamma}(b^{\pm}),$
(4.44)

and the boundary term $\mathcal{W}^{\gamma,\pm}$ is

$\displaystyle\mathring{\mathcal{W}}^{\gamma,\pm}:=(\partial_{3}v^{\pm}\cdot\mathring{N})\mathcal{T}^{\gamma}\mathring{\psi}+[\mathcal{T}^{\gamma},\mathring{N}_{i},v_{i}^{\pm}].$
(4.45)

Given $0\leq l\leq 4$, we shall consider the tangential estimates for
$\overline{\partial}{}^{4-k-l}\partial_{t}^{k}\mathcal{T}^{\alpha}$ for $0\leq
k\leq 4-l$ and $\langle\alpha\rangle=2l,~{}\alpha_{3}=0$. Following the
analysis in Section 3.3-Section 3.5, using the linearized Reynolds transport
theorem (Lemma A.7), dropping $\gamma$ for simplicity of notations, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\int_{\Omega^{\pm}}\mathring{\rho}|\varepsilon^{2l}\mathring{\mathbf{V}}^{\pm}|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}=$
$\displaystyle\int_{\Omega^{\pm}}\varepsilon^{4l}(\mathring{b}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{V}}^{\pm}\cdot\nabla^{\mathring{\varphi}}\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle-\int_{\Omega^{\pm}}\varepsilon^{4l}(\mathring{\mathcal{R}}_{v}^{\pm}-\mathring{\mathfrak{C}}(q^{\pm}))\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle+\frac{1}{2}\int_{\Omega^{\pm}}\varepsilon^{4l}\left(D_{t}^{\mathring{\varphi}\pm}\mathring{\rho}+\mathring{\rho}\nabla^{\mathring{\varphi}}\cdot\mathring{v}^{\pm}+\mathring{\rho}\partial^{\mathring{\varphi}}_{3}(\mathring{\bar{v}}\cdot\overline{\nabla})(\mathring{\varphi}-\dot{\varphi})\right)|\mathring{\mathbf{V}}^{\pm}|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t},$
(4.46)

where the last two terms can be directly controlled by
$C(\mathring{K}_{0})\mathring{E}^{\kappa}(t)$. In the rest of this section, we
will use the notation $\stackrel{{\scriptstyle L}}{{=}}$ to skip some of these
remainder terms. We then analyze the first line. Integrating
$(\mathring{b}^{\pm}\cdot\nabla^{\mathring{\varphi}})$ and
$\nabla^{\mathring{\varphi}}$ by parts and using
$\mathring{b}\cdot\mathring{N}|_{\Sigma}=0$, we get

$\displaystyle\int_{\Omega^{\pm}}\varepsilon^{4l}(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}=-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{B}}^{\pm}\cdot(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\varepsilon^{4l}(\nabla^{\mathring{\varphi}}\cdot\mathring{\bm{b}}^{\pm})\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle=$
$\displaystyle-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{B}}^{\pm}\cdot
D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{B}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\varepsilon^{4l}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})(\nabla^{\mathring{\varphi}}\cdot\mathring{\mathbf{V}}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\varepsilon^{4l}(\nabla^{\mathring{\varphi}}\cdot\mathring{\bm{b}}^{\pm})\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}|\varepsilon^{2l}\mathring{\mathbf{B}}^{\pm}|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}(\varepsilon^{2l}\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})^{2}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathcal{F}}_{p}^{\pm}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
(4.47)

and

$\displaystyle-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{V}}^{\pm}\cdot\nabla^{\mathring{\varphi}}\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}=\underbrace{\pm\int_{\Sigma}\varepsilon^{4l}(\mathring{\mathbf{V}}^{\pm}\cdot\mathring{N})\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}x^{\prime}}_{=:\mathring{I}^{\pm}}+\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}(\nabla^{\mathring{\varphi}}\cdot\mathring{\mathbf{V}}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\mathring{I}^{\pm}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}(\varepsilon^{2l}\mathring{\mathbf{Q}}^{\pm})^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}+\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}D_{t}^{\mathring{\varphi}\pm}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}\underbrace{-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\mathring{\mathfrak{C}}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}}_{=:\mathring{Z}^{\pm}}.$
(4.48)

Notice that

$\displaystyle\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathcal{F}}_{p}^{\pm}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})D_{t}^{\mathring{\varphi}\pm}\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}+\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\mathbf{Q}}^{\pm}D_{t}^{\mathring{\varphi}\pm}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}\stackrel{{\scriptstyle
L}}{{=}}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathcal{F}}_{p}^{\pm}\mathring{\mathbf{Q}}^{\pm}(\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t},$
(4.49)

we find that

$\displaystyle\int_{\Omega^{\pm}}\varepsilon^{4l}(\mathring{\bm{b}}^{\pm}\cdot\nabla^{\mathring{\varphi}})\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\mathbf{V}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{V}}^{\pm}\cdot\nabla^{\mathring{\varphi}}\mathring{\mathbf{Q}}^{\pm}\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\mathring{I}^{\pm}+\mathring{Z}^{\pm}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}|\varepsilon^{2l}\mathring{\mathbf{B}}^{\pm}|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathring{\mathcal{F}}_{p}^{\pm}\left|\varepsilon^{2l}\left(\mathring{\mathbf{Q}}^{\pm}-\mathring{\mathbf{B}}^{\pm}\cdot\mathring{\bm{b}}^{\pm}\right)\right|^{2}\,\mathrm{d}\mathring{\mathcal{V}}_{t}.$
(4.50)

Thus, we already get the energy terms for
$\mathring{\mathbf{V}},\mathring{\mathbf{B}}$ and $\mathring{\mathbf{Q}}$, and
it remains to analyze the boundary term $\mathring{I}^{\pm}$. Again, following
the analysis in Section 3.3-Section 3.5, we have

$\displaystyle\mathring{I}^{+}+\mathring{I}^{-}=\mathring{\text{
ST}}+\mathring{\text{ ST}}^{\prime}+\mathring{\text{ VS}}+\mathring{\text{
RT}}+\mathring{\text{ RT}}^{+}+\mathring{\text{
RT}}^{-}+\mathring{ZB}^{+}+\mathring{ZB}^{-}$ (4.51)

where

$\displaystyle\mathring{\text{ ST}}:=$
$\displaystyle~{}\varepsilon^{4l}\int_{\Sigma}\mathcal{T}^{\gamma}\left\llbracket
q\right\rrbracket\,\partial_{t}\mathcal{T}^{\gamma}\psi\,\mathrm{d}x^{\prime},$
(4.52) $\displaystyle{\mathring{\text{ ST}}}^{\prime}:=$
$\displaystyle~{}\varepsilon^{4l}\int_{\Sigma}\mathcal{T}^{\gamma}\left\llbracket
q\right\rrbracket\,(\bar{v}^{+}\cdot\overline{\nabla})\mathcal{T}^{\gamma}\mathring{\psi}\,\mathrm{d}x^{\prime},$
(4.53) $\displaystyle\mathring{\text{ VS}}:=$
$\displaystyle~{}\varepsilon^{4l}\int_{\Sigma}\mathcal{T}^{\gamma}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathcal{T}^{\gamma}\mathring{\psi}\,\mathrm{d}x^{\prime},$
(4.54) $\displaystyle\mathring{\text{ RT}}:=$
$\displaystyle-\varepsilon^{4l}\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\mathcal{T}^{\gamma}\mathring{\psi}\,\partial_{t}\mathcal{T}^{\gamma}\psi\,\mathrm{d}x^{\prime},$
(4.55) $\displaystyle\mathring{\text{ RT}}^{\pm}:=$
$\displaystyle\mp\varepsilon^{4l}\int_{\Sigma}\partial_{3}q^{\pm}\,\mathcal{T}^{\gamma}\mathring{\psi}\,(\bar{v}^{\pm}\cdot\overline{\nabla})\mathcal{T}^{\gamma}\mathring{\psi}\,\mathrm{d}x^{\prime},$
(4.56) $\displaystyle\mathring{ZB}^{\pm}:=$
$\displaystyle\mp\varepsilon^{4l}\int_{\Sigma}\mathring{\mathbf{Q}}^{\pm}\mathring{\mathcal{W}}^{\pm}\,\mathrm{d}x^{\prime},~{}~{}\mathring{Z}^{\pm}=-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\mathring{\mathfrak{C}}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}.$
(4.57)

####  4.3.3 Analysis of the boundary integrals

Since the weight function $\omega(x_{3})$ vanishes on $\Sigma$, we can
alternatively write
$\mathcal{T}^{\alpha}=\partial_{t}^{\alpha_{0}}\overline{\partial}{}^{2l-\alpha_{0}}$
and
$\mathcal{T}^{\gamma}=\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{4+l-(k+\alpha_{0})}$.
Replacing $k+\alpha_{0}$ by $k$, it suffices to analyze the case
$\mathcal{T}^{\gamma}=\partial_{t}^{k}\overline{\partial}{}^{4+l-k}$ for
$0\leq k\leq 4+l,~{}0\leq l\leq 4$. First, there is no need to analyze
$\mathring{\text{ RT}}$ and $\mathring{\text{ RT}}^{\pm}$ because they can be
directly controlled by using the energy bounds (4.7) for the basic state. For
the term $\mathring{\text{ ST}}$, the boundary regularity is given by the
$\kappa$-regularization terms instead of the surface tension because we do not
need a uniform-in-$\kappa$ estimate for the linearized problem. Using the jump
conditions for $\left\llbracket q\right\rrbracket$ and integrating by parts,
we have

$\displaystyle\int_{0}^{t}\mathring{\text{ ST}}\,\mathrm{d}\tau=$
$\displaystyle~{}\sigma\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\left(\frac{\overline{\nabla}\mathring{\psi}}{\sqrt{1+|\overline{\nabla}\mathring{\psi}|^{2}}}\right)\cdot\overline{\nabla}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}(1-\overline{\Delta}{})\psi\,\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}(1-\overline{\Delta}{})\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau-\kappa\int_{0}^{t}\int_{\Sigma}\left|\varepsilon^{2l}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\langle\overline{\partial}{}\rangle\psi\right|^{2}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle-\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\psi\right|_{2}^{2}\bigg{|}^{t}_{0}-\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\psi\right|_{L_{t}^{2}H_{x^{\prime}}^{1}}^{2}+\delta\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\psi\right|_{L_{t}^{2}H_{x^{\prime}}^{1}}^{2}+\frac{\sigma}{\kappa}\int_{0}^{t}C(\mathring{K}_{0})\,\mathrm{d}\tau.$
(4.58)

For the term $\mathring{\text{ ST}}^{\prime}$, we have

$\displaystyle\int_{0}^{t}{\mathring{\text{ ST}}}^{\prime}\,\mathrm{d}\tau=$
$\displaystyle~{}\sigma\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\left(\frac{\overline{\nabla}\mathring{\psi}}{\sqrt{1+|\overline{\nabla}\mathring{\psi}|^{2}}}\right)\cdot\overline{\nabla}\left((\bar{v}^{+}\cdot\overline{\nabla})\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\right)\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}(1-\overline{\Delta}{})\psi\,(1-\overline{\Delta}{})\left((\bar{v}^{+}\cdot\overline{\nabla})\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\right)\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\langle\overline{\partial}{}\rangle\psi\,\langle\overline{\partial}{}\rangle\left((\bar{v}^{+}\cdot\overline{\nabla})\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\right)\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\sigma
C(\mathring{K}_{0},\kappa^{-1})t+C(\mathring{K}_{0})\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\psi\right|_{2}\|v^{+}\|_{4,\pm}\,\mathrm{d}\tau$
$\displaystyle+\delta\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k+1}\overline{\partial}{}^{4+l-k}\psi\right|_{L_{t}^{2}H_{x^{\prime}}^{1}}^{2}+C(\mathring{K}_{0})\int_{0}^{t}\|v^{+}\|_{4,\pm}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\sigma
C(\mathring{K}_{0},\kappa^{-1})t+C(\mathring{K}_{0})\int_{0}^{t}\mathring{E}_{4+l}^{\kappa}(\tau)+\mathring{E}_{4}^{\kappa}(\tau)\,\mathrm{d}\tau.$
(4.59)

The term $\mathring{\text{ VS}}$ can also be directly controlled even if
$\mathcal{T}^{\gamma}$ only contains time derivatives. When $k<4+l$, we can
use the $\kappa$-weighted energy to control it after integrating
$\overline{\partial}{}^{\frac{1}{2}}$ by parts and using Lemma B.4

$\displaystyle\mathring{\text{ VS}}=$
$\displaystyle\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{k}\overline{\partial}{}^{3.5+l-k}q^{-}\,\overline{\partial}{}^{\frac{1}{2}}\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\lesssim$
$\displaystyle~{}\|\varepsilon^{2l}\partial_{t}^{k}\overline{\partial}{}^{4+l-k}q^{-}\|_{0,-}^{\frac{1}{2}}\|\varepsilon^{2l}\partial_{t}^{k}\overline{\partial}{}^{3+l-k}\partial_{3}q^{-}\|_{0,-}^{\frac{1}{2}}|\bar{v}^{\pm}|_{W^{\frac{1}{2},\infty}}\left|\varepsilon^{2l}\partial_{t}^{k}\mathring{\psi}\right|_{5.5+l-k}$
$\displaystyle\lesssim$
$\displaystyle~{}(\mathring{E}^{\kappa}_{4}(t)+\mathring{E}^{\kappa}_{4+l}(t))C(\mathring{K}_{0}).$
(4.60)

When $k=4+l$, we can first integrate $\partial_{t}$ by parts and then
integrate $\overline{\partial}{}^{\frac{1}{2}}$ by parts

$\displaystyle\int_{0}^{t}\mathring{\text{
VS}}\,\mathrm{d}\tau\stackrel{{\scriptstyle
L}}{{=}}\int_{0}^{t}\int_{\Sigma}\varepsilon^{4l}\overline{\partial}{}^{\frac{1}{2}}(\left\llbracket\bar{v}\right\rrbracket\partial_{t}^{3+l}q^{-})\,\overline{\partial}{}^{\frac{1}{2}}\partial_{t}^{5+l}\mathring{\psi}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau+\int_{\Sigma}\varepsilon^{4l}\partial_{t}^{3+l}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{4+l}\mathring{\psi}\,\mathrm{d}x^{\prime}\bigg{|}^{t}_{0}$
$\displaystyle\lesssim$
$\displaystyle~{}\int_{0}^{t}\|\varepsilon^{2l}\overline{\partial}{}\partial_{t}^{3+l}q^{-}\|_{0,-}^{\frac{1}{2}}\|\varepsilon^{2l}\partial_{t}^{3+l}\partial_{3}q^{-}\|_{0,-}^{\frac{1}{2}}|\bar{v}^{\pm}|_{W^{\frac{1}{2},\infty}}\left|\varepsilon^{2l}\partial_{t}^{5+l}\mathring{\psi}\right|_{0.5}\,\mathrm{d}\tau$
$\displaystyle+\delta\|\varepsilon^{2l}\partial_{t}^{3+l}q^{-}\|_{1,-}^{2}+|\bar{v}^{\pm}|_{L^{\infty}}^{2}\left|\varepsilon^{2l}\partial_{t}^{4}\mathring{\psi}\right|_{1}^{2}+C(\mathring{K}_{0})\mathring{E}^{\kappa}(0)$
$\displaystyle\lesssim$
$\displaystyle~{}\delta\mathring{E}_{4+l}^{\kappa}(t)+C(\mathring{K}_{0},\kappa^{-1})\left(\mathring{E}^{\kappa}(0)+\int_{0}^{t}\mathring{E}^{\kappa}(\tau)\,\mathrm{d}\tau\right).$
(4.61)

For $\mathring{ZB}+\mathring{Z}$, the cancellation structure obtained in
Section 3.3.1 and Section 3.4 still holds. Following step 4 in Section 3.4, we
have

$\displaystyle\mathring{ZB}^{\pm}+\mathring{Z}^{\pm}=$
$\displaystyle~{}\mp\int_{\Sigma}\varepsilon^{4l}(\partial_{t}^{k}\overline{\partial}{}^{4+l-k}q^{\pm}-\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\partial_{3}q^{\pm})(\partial_{3}v^{\pm}\cdot\mathring{N})\,\partial_{t}^{k}\overline{\partial}{}^{4+l-k}\mathring{\psi}\,\mathrm{d}x^{\prime}$
$\displaystyle\mp\int_{\Sigma}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\left[\partial_{t}^{k}\overline{\partial}{}^{4+l-k},\mathring{N}_{i},v_{i}^{\pm}\right]\,\mathrm{d}x^{\prime}-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\mathring{\mathfrak{C}}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t},$
(4.62)

where the first line is controlled in the same way as$\mathring{\text{ VS}}$.
Mimicing the proof in step 4 in Section 3.4, we have

$\displaystyle\mp\int_{\Sigma}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\left[\partial_{t}^{k}\overline{\partial}{}^{4+l-k},\mathring{N}_{i},v_{i}^{\pm}\right]\,\mathrm{d}x^{\prime}-\int_{\Omega^{\pm}}\varepsilon^{4l}\mathring{\mathbf{Q}}^{\pm}\mathring{\mathfrak{C}}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathring{\mathcal{V}}_{t}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\int_{\Omega^{\pm}}\varepsilon^{4l}\partial^{\mathring{\varphi}}_{3}\mathring{\mathbf{Q}}^{\pm}\left[\partial_{t}^{k}\overline{\partial}{}^{4+l-k},\mathring{\mathbf{N}}_{i},v_{i}^{\pm}\right]\,\mathrm{d}\mathring{\mathcal{V}}_{t},$
(4.63)

whose time integral can be directly controlled by

$\delta\mathring{E}_{4+l}^{\kappa}(t)+C(\mathring{K}_{0},\kappa^{-1})\left(\mathring{E}^{\kappa}(0)+\int_{0}^{t}\mathring{E}^{\kappa}(\tau)\,\mathrm{d}\tau\right)$

after integrating by parts one tangential derivative in
$\partial_{t}^{k}\overline{\partial}{}^{4+l-k}$.

####  4.3.4 Uniform-in-$n$ estimates for the linearized approximate system

Summarizing the estimates obtained in Section 4.3.1-Section 4.3.3, we prove
that for any $\delta\in(0,1)$,

$\mathring{E}^{\kappa}(t)\lesssim\delta\mathring{E}^{\kappa}(t)+C(\mathring{K}_{0},\kappa^{-1})\left(\mathring{E}^{\kappa}(0)+\int_{0}^{t}\mathring{E}^{\kappa}(\tau)\,\mathrm{d}\tau\right).$

Choosing $\delta>0$ suitably small such that the $\delta$-term can be absorbed
by the left side and using Grönwall’s inequality, we find that there exists a
time $T_{\kappa}>0$ (independent of $\varepsilon$ and $n$), such that

$\sup_{0\leq t\leq T_{\kappa}}\mathring{E}^{\kappa}(t)\leq
C^{\prime}(\mathring{K}_{0},\kappa^{-1})\mathring{E}^{\kappa}(0)$

for some positive function $C^{\prime}$ continuous in its arguments. Following
the argument in remark 4.3, it is straightforward to show that

$\sum_{\pm}\sum_{l=0}^{4}\sum_{\langle\alpha\rangle=2l}\sum_{k=0}^{4-l}\int_{0}^{t}\left\|\varepsilon^{2l}\mathcal{T}^{\alpha}\partial_{t}^{k}\bm{b}^{\pm}\right\|_{4-k-l,\pm}^{2}\,\mathrm{d}\tau<P(\mathring{E}^{\kappa}(t))\quad\forall
t\in[0,T_{\kappa}].$

###  4.4 Picard iteration: well-posedness of the nonlinear approximate problem

We already establish the local existence of the linear system (4.1) for each
$n$ and the uniform-in-$n$ estimates for the solution to (4.1). It suffices to
prove
$\\{(v^{{[n]},\pm},b^{{[n]},\pm},\bm{b}^{{[n]},\pm},q^{{[n]},\pm},\psi^{{[n]}})\\}$
has a strongly convergent subsequence (in certain anisotropic Sobolev norms).

####  4.4.1 The way to recover the nonlinear approximate system

Let us first see how to construct the solution to the nonlinear system (3.1)
for fixed $\kappa>0$ if the strong convergence result has been proven. We
assume that the expected limit is denoted by
$(v^{[\infty],\pm},b^{[\infty],\pm},\bm{b}^{[\infty],\pm},q^{[\infty],\pm},\psi^{[\infty]})$.
Then the limit functions satisfy the following system

$\begin{cases}\rho^{[\infty],\pm}D_{t}^{\varphi^{[\infty]},\pm}v^{{[\infty]},\pm}-(\bm{b}^{[\infty],\pm}\cdot\nabla^{\varphi^{[\infty]}})b^{{[\infty]},\pm}+\nabla^{\varphi^{[\infty]}}q^{{[\infty]},\pm}=0&~{}~{}\text{
in }[0,T]\times\Omega^{\pm},\\\
(\mathcal{F}_{p}^{\pm})^{[\infty]}D_{t}^{\varphi^{{[\infty]}},\pm}q^{{[\infty]},\pm}-(\mathcal{F}_{p}^{\pm})^{[\infty]}D_{t}^{\varphi^{{[\infty]}},\pm}b^{{[\infty]},\pm}\cdot\bm{b}^{{[\infty]},\pm}+\nabla^{\varphi^{[\infty]}}\cdot
v^{{[\infty]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{{[\infty]}},\pm}b^{{[\infty]},\pm}-(\bm{b}^{[\infty],\pm}\cdot\nabla^{\varphi^{[\infty]}})v^{{[\infty]},\pm}+\bm{b}^{{[\infty]},\pm}\nabla^{\varphi^{[\infty]}}\cdot
v^{{[\infty]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{{[\infty]}},\pm}S^{{[\infty]},\pm}=0&~{}~{}\text{ in
}[0,T]\times\Omega^{\pm},\\\ \left\llbracket
q^{[\infty]}\right\rrbracket=\sigma\mathcal{H}(\psi^{{[\infty]}})-\kappa(1-\overline{\Delta}{})^{2}\psi^{{[\infty]}}-\kappa(1-\overline{\Delta}{})\partial_{t}\psi^{{[\infty]}}&~{}~{}\text{
on }[0,T]\times\Sigma,\\\
\partial_{t}\psi^{{[\infty]}}=v^{{[\infty]},\pm}\cdot
N^{{[\infty]}}&~{}~{}\text{ on }[0,T]\times\Sigma,\\\
v_{3}^{{[\infty]},\pm}=0&~{}~{}\text{ on }[0,T]\times\Sigma^{\pm},\\\
(v^{{[\infty]},\pm},b^{{[\infty]},\pm},q^{{[\infty]},\pm},S^{{[\infty]},\pm},\psi^{{[\infty]}})|_{t=0}=(v_{0}^{\kappa,\pm},b_{0}^{\kappa,\pm},q_{0}^{\kappa,\pm},S_{0}^{\kappa,\pm},\psi_{0}^{\kappa}),\end{cases}$
(4.64)

where $\rho^{[\infty]}$ is defined via the equation of state $\rho=\rho(p,S)$
and $p^{[\infty]}:=q^{[\infty]}-\frac{1}{2}|b^{[\infty]}|^{2}$. Also we have

$\displaystyle D_{t}^{\varphi^{[\infty]},\pm}=$
$\displaystyle~{}\partial_{t}+\bar{v}^{{[\infty]},\pm}\cdot\overline{\nabla}+\frac{1}{\partial_{3}\varphi^{[\infty]}}(v^{{[\infty]},\pm}\cdot\mathbf{N}^{{[\infty]}}-\partial_{t}\varphi^{[\infty]})\partial_{3},$
$\displaystyle\bm{b}^{[\infty],\pm}\cdot\nabla^{\varphi^{[\infty]}}=$
$\displaystyle~{}\bar{\bm{b}}^{{[\infty]},\pm}\cdot\overline{\nabla}+\frac{1}{\partial_{3}\varphi^{[\infty]}}(\bm{b}^{{[\infty]},\pm}\cdot
N^{[\infty]})\partial_{3}.$

We must prove that $b_{3}^{[\infty],\pm}=\bm{b}_{3}^{[\infty],\pm}$ in
$\Omega^{\pm}$. According to the definition of $\bm{b}^{[n]}$, the limit
function satisfies

$\bm{b}_{3}^{[\infty],\pm}=b_{3}^{[\infty],\pm}+\mathfrak{R}_{T}^{\pm}\left(b_{1}^{[\infty],\pm}\overline{\partial}{}_{1}\psi^{[\infty]}+b_{2}^{[\infty],\pm}\overline{\partial}{}_{2}\psi^{[\infty]}-b_{3}^{[\infty],\pm}\right)\big{|}_{\Sigma}\Rightarrow\bm{b}_{3}^{[\infty],\pm}\cdot
N^{{[\infty]}}|_{\Sigma}=0.$

Since Lemma B.3 implies that $\mathfrak{R}_{T}^{\pm}(0)=0$, then the remaining
step is to show $b^{[\infty],\pm}\cdot N^{[\infty]}|_{\Sigma}=0$ holds with in
the lifespan of the solution to (4.64) provided $b^{[\infty],\pm}\cdot
N^{[\infty]}|_{t=0}=0$ on $\Sigma$. On $\Sigma$, we compute that

$\displaystyle D_{t}^{\varphi^{{[\infty]}},\pm}(b^{{[\infty]},\pm}\cdot
N^{[\infty]})=D_{t}^{\varphi^{{[\infty]}},\pm}b^{{[\infty]},\pm}\cdot
N^{[\infty]}+b^{{[\infty]},\pm}\cdot
D_{t}^{\varphi^{{[\infty]}},\pm}N^{[\infty]}$ $\displaystyle=$
$\displaystyle\underbrace{(\bar{\bm{b}}^{{[\infty]},\pm}\cdot\overline{\nabla})}_{=\bar{b}^{{[\infty]},\pm}\cdot\overline{\nabla}}v^{{[\infty]},\pm}\cdot
N^{[\infty]}+\underbrace{(\bm{b}^{{[\infty]},\pm}\cdot
N^{[\infty]})}_{=0\text{ on }\Sigma}\partial_{3}v^{{[\infty]},\pm}\cdot
N^{[\infty]}+\underbrace{(\bm{b}^{{[\infty]},\pm}\cdot
N^{[\infty]})}_{=0\text{ on }\Sigma}(\nabla^{\varphi^{[\infty]}}\cdot
v^{{[\infty]},\pm})$
$\displaystyle-\bar{b}_{i}^{{[\infty]},\pm}\overline{\partial}{}_{i}\partial_{t}\psi^{{[\infty]}}-\bar{b}_{i}^{{[\infty]},\pm}\bar{v}_{j}^{{[\infty]},\pm}\overline{\partial}{}_{j}\overline{\partial}{}_{i}\psi^{{[\infty]}}$
$\displaystyle=$
$\displaystyle~{}(\bar{b}^{{[\infty]},\pm}\cdot\overline{\nabla})\underbrace{\left(v_{3}^{{[\infty]},\pm}-\bar{v}_{j}^{{[\infty]},\pm}\overline{\partial}{}_{j}\psi^{{[\infty]}}\right)}_{=\partial_{t}\psi^{{[\infty]}}}+\bar{b}_{i}^{{[\infty]},\pm}\bar{v}_{j}^{{[\infty]},\pm}\overline{\partial}{}_{j}\overline{\partial}{}_{i}\psi^{{[\infty]}}-\bar{b}_{i}^{{[\infty]},\pm}\overline{\partial}{}_{i}\partial_{t}\psi^{{[\infty]}}-\bar{b}_{i}^{{[\infty]},\pm}\bar{v}_{j}^{{[\infty]},\pm}\overline{\partial}{}_{j}\overline{\partial}{}_{i}\psi^{{[\infty]}}=0.$

Thus standard $L^{2}$ energy estimate shows that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\left|b^{{[\infty]},\pm}\cdot
N^{[\infty]}\right|^{2}\,\mathrm{d}x^{\prime}=$
$\displaystyle\int_{\Sigma}(\overline{\nabla}\cdot\bar{v}^{{[\infty]},\pm})\left|b^{{[\infty]},\pm}\cdot
N^{[\infty]}\right|^{2}\,\mathrm{d}x^{\prime}\leq|\overline{\partial}{}v^{{[\infty]},\pm}|_{L^{\infty}}\left|b^{{[\infty]},\pm}\cdot
N^{[\infty]}\right|_{0}^{2}.$ (4.65)

Since $b^{[\infty],\pm}\cdot N^{[\infty]}|_{t=0}=0$ on $\Sigma$, we conclude
that $b^{[\infty],\pm}\cdot N^{[\infty]}=0$ always holds on $\Sigma$ by using
Grönwall’s inequality. Plugging it back to the expression of
$\bm{b}_{3}^{[\infty],\pm}$, we find
$\bm{b}_{3}^{{[\infty]},\pm}=b_{3}^{{[\infty]},\pm}$ in $\Omega^{\pm}$ as
desired. Then we can replace $\bm{b}$ by $b$ in the limit system (4.64)

$\begin{cases}\rho^{[\infty],\pm}D_{t}^{\varphi^{[\infty]},\pm}v^{{[\infty]},\pm}-(b^{[\infty],\pm}\cdot\nabla^{\varphi^{[\infty]}})b^{{[\infty]},\pm}+\nabla^{\varphi^{[\infty]}}q^{{[\infty]},\pm}=0&~{}~{}\text{
in }[0,T]\times\Omega^{\pm},\\\
(\mathcal{F}_{p}^{\pm})^{[\infty]}D_{t}^{\varphi^{{[\infty]}},\pm}q^{{[\infty]},\pm}-(\mathcal{F}_{p}^{\pm})^{[\infty]}D_{t}^{\varphi^{{[\infty]}},\pm}b^{{[\infty]},\pm}\cdot
b^{{[\infty]},\pm}+\nabla^{\varphi^{[\infty]}}\cdot
v^{{[\infty]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{{[\infty]}},\pm}b^{{[\infty]},\pm}-(b^{[\infty],\pm}\cdot\nabla^{\varphi^{[\infty]}})v^{{[\infty]},\pm}+b^{{[\infty]},\pm}\nabla^{\varphi^{[\infty]}}\cdot
v^{{[\infty]},\pm}=0&~{}~{}\text{ in }[0,T]\times\Omega^{\pm},\\\
D_{t}^{\varphi^{{[\infty]}},\pm}S^{{[\infty]},\pm}=0&~{}~{}\text{ in
}[0,T]\times\Omega^{\pm},\\\ \left\llbracket
q^{[\infty]}\right\rrbracket=\sigma\mathcal{H}(\psi^{{[\infty]}})-\kappa(1-\overline{\Delta}{})^{2}\psi^{{[\infty]}}-\kappa(1-\overline{\Delta}{})\partial_{t}\psi^{{[\infty]}}&~{}~{}\text{
on }[0,T]\times\Sigma,\\\
\partial_{t}\psi^{{[\infty]}}=v^{{[\infty]},\pm}\cdot N^{{[\infty]}},\quad
b^{{[\infty]},\pm}\cdot N^{{[\infty]}}=0&~{}~{}\text{ on
}[0,T]\times\Sigma,\\\
v_{3}^{{[\infty]},\pm}=b_{3}^{{[\infty]},\pm}=0&~{}~{}\text{ on
}[0,T]\times\Sigma^{\pm},\\\
(v^{{[\infty]},\pm},b^{{[\infty]},\pm},q^{{[\infty]},\pm},S^{{[\infty]},\pm},\psi^{{[\infty]}})|_{t=0}=(v_{0}^{\kappa,\pm},b_{0}^{\kappa,\pm},q_{0}^{\kappa,\pm},S_{0}^{\kappa,\pm},\psi_{0}^{\kappa}),\end{cases}$
(4.66)

in which
$(v^{[\infty],\pm},b^{[\infty],\pm},q^{[\infty],\pm},\psi^{[\infty]})$ exactly
gives the solution to the nonlinear approximate system (3.1). Finally, the
divergence constraint $\nabla^{\varphi^{[\infty]}}\cdot b^{[\infty],\pm}=0$ in
$\Omega^{\pm}$ automatically holds thanks to the second equation, the fourth
equation in (4.66) and $\nabla^{\varphi_{0}^{\kappa}}\cdot
b_{0}^{\kappa,\pm}=0$ in $\Omega^{\pm}$.

####  4.4.2 Proof of strong convergence

For a function sequence $\\{f^{{[n]},\pm}\\}$, we define
$[f]^{{[n]},\pm}:=f^{{[n+1]},\pm}-f^{{[n]},\pm}$. Then we can write the linear
system of
$\\{([v]^{{[n]},\pm},[b]^{{[n]},\pm},[q]^{{[n]},\pm},[\psi]^{{[n]}})\\}$ as
follows

$\begin{cases}\rho^{{[n]},\pm}D_{t}^{\varphi^{[n]}}[v]^{{[n]},\pm}-(\bm{b}^{{[n]},\pm}\cdot\nabla^{\varphi^{[n]}})[b]^{{[n]},\pm}+\nabla^{\varphi^{[n]}}[q]^{{[n]},\pm}+\nabla^{[\varphi]^{[n-1]}}q^{{[n]},\pm}=-\mathring{f}_{v}^{{[n]},\pm}&\text{
in }[0,T]\times\Omega^{\pm}\\\
\mathcal{F}_{p}^{{[n]},\pm}D_{t}^{\varphi^{[n]}}[q]^{{[n]},\pm}-\mathcal{F}_{p}^{{[n]},\pm}D_{t}^{\varphi^{[n]}}[b]^{{[n]},\pm}\cdot\bm{b}^{{[n]},\pm}+\nabla^{\varphi^{[n]}}\cdot[v]^{{[n]},\pm}+\nabla^{[\varphi]^{[n-1]}}\cdot
v^{{[n]},\pm}=-\mathring{f}_{p}^{{[n]},\pm}&\text{ in
}[0,T]\times\Omega^{\pm}\\\
D_{t}^{\varphi^{[n]}}[b]^{{[n]},\pm}-(\bm{b}^{{[n]},\pm}\cdot\nabla^{\varphi^{[n]}})[v]^{{[n]},\pm}+\bm{b}^{{[n]},\pm}(\nabla^{\varphi^{[n]}}\cdot[v]^{{[n]},\pm}+\nabla^{[\varphi]^{[n-1]}}\cdot
v^{{[n]},\pm})=-\mathring{f}_{b}^{{[n]},\pm}&\text{ in
}[0,T]\times\Omega^{\pm}\\\
D_{t}^{\varphi^{[n]}}[S]^{{[n]},\pm}=-\mathring{f}_{S}^{{[n]},\pm}&\text{ in
}[0,T]\times\Omega^{\pm}\\\
\left\llbracket[q]^{{[n]},\pm}\right\rrbracket=\sigma(\mathcal{H}(\psi^{[n]})-\mathcal{H}(\psi^{[n-1]}))-\kappa(1-\overline{\Delta}{})^{2}[\psi]^{[n]}-\kappa(1-\overline{\Delta}{})\partial_{t}[\psi]^{[n]}&\text{
on }[0,T]\times\Sigma\\\ \partial_{t}[\psi]^{[n]}=[v]^{{[n]},\pm}\cdot
N^{[n]}+v^{{[n]},\pm}\cdot[N]^{{[n-1]}}&\text{ on }[0,T]\times\Sigma\\\
v_{3}^{{[n]},\pm}=v_{3}^{{[n-1]},\pm}=\bm{b}_{3}^{{[n]},\pm}=\bm{b}_{3}^{{[n-1]},\pm}=0&\text{
on }[0,T]\times\Sigma^{\pm}\\\
([v]^{[n]},[b]^{[n]},[q]^{[n]},[\psi]^{[n]})|_{t=0}=(\vec{0},\vec{0},0,0),\end{cases}$
(4.67)

where the source terms are defined by

$\displaystyle\mathring{f}_{v}^{{[n]},\pm}:=$
$\displaystyle~{}[\rho]^{{[n-1]},\pm}\partial_{t}v^{{[n]},\pm}+[\rho\bar{v}]^{{[n-1]},\pm}\cdot\overline{\nabla}v^{{[n]},\pm}+[\rho
V_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}v^{{[n]},\pm}$
$\displaystyle~{}-[\bm{b}]^{{[n-1]},\pm}\cdot\overline{\nabla}b^{{[n]},\pm}-[\bm{B}_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}b^{{[n]},\pm},$
(4.68) $\displaystyle\mathring{f}_{q}^{{[n]},\pm}:=$
$\displaystyle~{}[\mathcal{F}_{p}]^{{[n-1]},\pm}\partial_{t}q^{{[n]},\pm}+[\mathcal{F}_{p}\bar{v}]^{{[n-1]},\pm}\cdot\overline{\nabla}q^{{[n]},\pm}+[\mathcal{F}_{p}V_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}q^{{[n]},\pm}$
$\displaystyle-([\mathcal{F}_{p}]^{{[n-1]},\pm}\partial_{t}b^{{[n]},\pm}+[\mathcal{F}_{p}\bar{v}]^{{[n-1]},\pm}\cdot\overline{\nabla}b^{{[n]},\pm}+[\mathcal{F}_{p}V_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}b^{{[n]},\pm})\cdot\bm{b}^{{[n]},\pm}$
$\displaystyle-(\mathcal{F}_{p}^{\pm})^{{[n-1]}}D_{t}^{\varphi^{[n-1]}}b^{{[n]},\pm}\cdot[\bm{b}]^{{[n-1]},\pm},$
(4.69) $\displaystyle\mathring{f}_{p}^{{[n]},\pm}:=$
$\displaystyle~{}[\bar{v}]^{{[n-1]},\pm}\cdot\overline{\nabla}b^{{[n]},\pm}+[V_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}b^{{[n]},\pm}-[\bm{b}]^{{[n-1]},\pm}\cdot\overline{\nabla}v^{{[n]},\pm}-[\bm{B}_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}v^{{[n]},\pm}$
$\displaystyle+[\bm{b}]^{{[n-1]},\pm}(\nabla^{\varphi^{[n-1]}}\cdot
v^{{[n]},\pm}),$ (4.70) $\displaystyle\mathring{f}_{S}^{{[n]},\pm}:=$
$\displaystyle~{}[\bar{v}]^{{[n-1]},\pm}\cdot\overline{\nabla}S^{{[n]},\pm}+[V_{\mathbf{N}}]^{{[n-1]},\pm}\partial_{3}S^{{[n]},\pm},$
(4.71)

with

$\displaystyle
V_{\mathbf{N}}^{[n]}:=\frac{1}{\partial_{3}\varphi^{[n]}}(v^{[n]}\cdot\mathbf{N}^{[n-1]}-\partial_{t}\varphi^{[n]}),\quad\bm{B}_{\mathbf{N}}^{[n]}:=\frac{1}{\partial_{3}\varphi^{[n]}}(\bm{b}^{[n]}\cdot\mathbf{N}^{[n]}),\quad\nabla^{[\varphi]^{[n-1]}}f^{[n]}:=-[\mathbf{N}/\partial_{3}\varphi]^{[n-1]}\partial_{3}f^{[n]}$

For $1\leq n\in\mathbb{N}^{*}$, we define the energy for the linear system
(4.67) as follows

$\displaystyle[\mathring{E}^{\kappa}]^{[n]}(t):=$
$\displaystyle~{}[\mathring{E}^{\kappa}]_{3}^{[n]}(t)+\cdots+[\mathring{E}^{\kappa}]_{6}^{[n]}(t),$
$\displaystyle[\mathring{E}^{\kappa}]_{3+l}^{[n]}(t):=$
$\displaystyle\sum_{\pm}\sum_{k=0}^{3}\sum_{\langle\alpha\rangle=2l}\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm},[q]^{{[n]},\pm},[S]^{{[n]},\pm})\right\|_{3-k-l}^{2}$
$\displaystyle+\sum_{k=0}^{3+l}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}[\psi]^{[n]}\right|_{5+l-k}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{4+l}[\psi]^{[n]}\right|_{1}^{2}\,\mathrm{d}\tau,~{}~{}0\leq
l\leq 3,$ (4.72)

where
$\mathcal{T}^{\alpha}:=(\omega(x_{3})\partial_{3})^{\alpha_{4}}\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
with the multi-index
$\alpha=(\alpha_{0},\alpha_{1},\alpha_{2},0,\alpha_{4}),~{}\langle\alpha\rangle=\alpha_{0}+\alpha_{1}+\alpha_{2}+2\times
0+\alpha_{4}$. It should be noted that the initial value of
$[\mathring{E}^{\kappa}]^{[n]}$. Thus, we shall prove the following
proposition in order for the strong convergence.

######  Proposition 4.2.

There exists a time $T_{\kappa}^{\prime}>0$ depending on $\kappa$ and
$\mathring{K}_{0}$, such that

$\displaystyle\forall 2\leq n\in\mathbb{N}^{*},\quad\sup_{0\leq t\leq
T_{\kappa}^{\prime}}[\mathring{E}^{\kappa}]^{{[n]}}(t)\leq\frac{1}{4}\left(\sup_{0\leq
t\leq T_{\kappa}^{\prime}}[\mathring{E}^{\kappa}]^{{[n-1]}}(t)+\sup_{0\leq
t\leq T_{\kappa}^{\prime}}[\mathring{E}^{\kappa}]^{{[n-2]}}(t)\right).$ (4.73)

The proof of proposition is substantially similar to the estimates of
$\mathring{E}(t)$ in Section 4.3, so we will not go into every detail but only
write the sketch of the proof.

#### Step 1: Div-Curl analysis and reduction of pressure

The reduction of pressure follows in the same way as in Section 3.6.1.
Invoking the momentum equation, we have

$-(\partial_{3}\varphi^{{[n]}})^{-1}\partial_{3}[q]^{{[n]},\pm}=\rho^{{[n]},\pm}D_{t}^{\varphi^{[n]}}[v]^{{[n]},\pm}-(\bm{b}^{{[n]},\pm}\cdot\nabla^{\varphi^{[n]}})[b]^{{[n]},\pm}+\mathring{f}_{v}^{{[n]},\pm}+(\partial_{3}\varphi^{{[n]}})^{-1}\partial_{3}q^{{[n]},\pm}.$

Then using
$\partial^{\mathring{\varphi}}_{i}=\overline{\partial}{}_{i}-\overline{\partial}{}_{i}\mathring{\varphi}\partial^{\mathring{\varphi}}_{3}$,
we can convert $\partial q$ to a spatial derivative of $v$ and $b$ plus the
given term $\partial_{3}q^{{[n]},\pm}$.

For the div-curl analysis, using (B.1), we have for $0\leq l\leq 2,~{}0\leq
k\leq 2-l$

$\displaystyle\left\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm})\right\|_{3-k-l,\pm}^{2}$
$\displaystyle\leq C(\mathring{K}_{0})$
$\displaystyle\bigg{(}\|\varepsilon^{2l}\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm})\|_{0,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\varphi^{[n]}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm})\right\|_{2-k-l,\pm}^{2}$
$\displaystyle+\left\|\varepsilon^{2l}\nabla^{\varphi^{[n]}}\cdot\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm})\right\|_{2-k-l,\pm}^{2}+\left\|\overline{\partial}{}^{3-k-l}\partial_{t}^{k}\mathcal{T}^{\alpha}([v]^{{[n]},\pm},[b]^{{[n]},\pm})\right\|_{0,\pm}^{2}\bigg{)}.$
(4.74)

The $L^{2}$ estimate is straightforward, so we skip the proof. For the curl
part, we again analyze the evolution equations of vorticity and current

$\displaystyle\rho^{[n]}D_{t}^{\varphi^{[n]}}(\nabla^{\varphi^{[n]}}\times[v]^{[n]})-(\bm{b}^{[n]}\cdot\nabla^{\varphi^{[n]}})(\nabla^{\varphi^{[n]}}\times[b]^{[n]})$
$\displaystyle=$
$\displaystyle-\nabla^{\varphi^{[n]}}\times\mathring{f}_{v}^{[n]}-\nabla^{\varphi^{[n]}}\rho^{[n]}\times
D_{t}^{\varphi^{[n]}}[v]^{[n]}+(\nabla^{\varphi^{[n]}}\bm{b}_{j}^{{[n]}})\times(\partial_{j}^{\varphi^{[n]}}[b]^{[n]})+\rho^{[n]}[D_{t}^{\varphi^{[n]}},\nabla^{\varphi^{[n]}}\times][v]^{[n]},$
(4.75) $\displaystyle
D_{t}^{\varphi^{[n]}}(\nabla^{\varphi^{[n]}}\times[b]^{[n]})-(\bm{b}^{[n]}\cdot\nabla^{\varphi^{[n]}})(\nabla^{\varphi^{[n]}}\times[v]^{[n]})-\bm{b}^{[n]}\times(\nabla^{\varphi^{[n]}}(\nabla^{\varphi^{[n]}}\cdot[v]^{[n]}))$
$\displaystyle=$
$\displaystyle-\nabla^{\varphi^{[n]}}\times\mathring{f}_{b}^{[n]}+[D_{t}^{\varphi^{[n]}},\nabla^{\varphi^{[n]}}\times][b]^{[n]}+(\nabla^{\varphi^{[n]}}\bm{b}_{j}^{{[n]}})\times(\partial_{j}^{\varphi^{[n]}}[v]^{[n]})$
$\displaystyle-\nabla^{\varphi^{[n]}}\times(\bm{b}^{[n]}\nabla^{[\varphi]^{[n-1]}}\cdot
v^{{[n]},\pm})-(\nabla^{\varphi^{[n]}}\times\bm{b}^{[n]})(\nabla^{\varphi^{[n]}}\cdot[v]^{[n]}).$
(4.76)

Mimicing the proof in Section 4.3.1 and using the vanishing initial value of
system (4.67), we can prove

$\displaystyle\left\|\varepsilon^{2l}\nabla^{\varphi^{[n]}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}[v]^{{[n]},\pm}\right\|_{2-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\nabla^{\varphi^{[n]}}\times\partial_{t}^{k}\mathcal{T}^{\alpha}[b]^{{[n]},\pm}\right\|_{2-k-l,\pm}^{2}+\left\|\varepsilon^{2l}\sqrt{(\mathcal{F}_{p}^{\pm})^{[n]}}\bm{b}^{{[n]},\pm}\times(\partial_{t}^{k}\mathcal{T}^{\alpha}[b]^{\pm})\right\|_{2-l-k,\pm}^{2}$
(4.77) $\displaystyle\lesssim$
$\displaystyle~{}C(\mathring{K}_{0})\int_{0}^{t}\sum_{j=0}^{l}[E]_{3+j}^{{[n]}}(\tau)+[E]_{3+l+1}^{{[n]}}(\tau)\,\mathrm{d}\tau.$

Similarly, the divergence of $[v]^{[n]}$ can be converted to tangential
derivatives of $[q]^{[n]}$ and $[b]^{[n]}$ by invoking the continuity
equation, and the evolution equation of