$\displaystyle\nabla^{\varphi}\cdot b^{\pm}=0~{}~{}$ $\displaystyle\text{ in
}[0,T]\times\Omega^{\pm},$ (3.14) $\displaystyle
D_{t}^{\varphi\pm}\mathbf{S}^{\pm,\alpha}=\mathfrak{D}^{\gamma}(S^{\pm})~{}~{}$
$\displaystyle\text{ in }[0,T]\times\Omega^{\pm},$ (3.15)

with boundary conditions

$\displaystyle\left\llbracket\mathbf{Q}^{\gamma}\right\rrbracket=\sigma\mathcal{T}^{\gamma}\mathcal{H}-\kappa\mathcal{T}^{\gamma}(1-\overline{\Delta}{})^{2}\psi-\kappa\mathcal{T}^{\gamma}(1-\overline{\Delta}{})\partial_{t}\psi-\left\llbracket\partial_{3}q\right\rrbracket\mathcal{T}^{\gamma}\psi~{}~{}$
$\displaystyle\text{ on }[0,T]\times\Sigma,$ (3.16)
$\displaystyle\mathbf{V}^{\gamma,\pm}\cdot
N=\partial_{t}\mathcal{T}^{\gamma}\psi+\bar{v}^{\pm}\cdot\overline{\nabla}\mathcal{T}^{\gamma}\psi-\mathcal{W}^{\gamma,\pm}~{}$
$\displaystyle\text{ on }[0,T]\times\Sigma,$ (3.17) $\displaystyle
b^{\pm}\cdot N=0~{}$ $\displaystyle\text{ on }[0,T]\times\Sigma,$ (3.18)
$\displaystyle
b_{3}^{\pm}=v_{3}^{\pm}=\mathbf{B}_{3}^{\pm}=\mathbf{V}_{3}^{\pm}=0~{}$
$\displaystyle\text{ on }[0,T]\times\Sigma^{\pm},$ (3.19)

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

$\displaystyle\mathcal{R}_{v}^{\gamma,\pm}:=$
$\displaystyle~{}[\mathcal{T}^{\gamma},b^{\pm}]\cdot\nabla^{\varphi}b^{\pm}-[\mathcal{T}^{\gamma},\rho^{\pm}]D_{t}^{\varphi\pm}v^{\pm}-\rho^{\pm}\mathfrak{D}^{\gamma}(v^{\pm})$
(3.20) $\displaystyle\mathcal{R}_{p}^{\gamma,\pm}:=$
$\displaystyle-[\mathcal{T}^{\gamma},\mathcal{F}_{p}^{\pm}]D_{t}^{\varphi\pm}p^{\pm}-\mathcal{F}_{p}^{\pm}\mathfrak{D}^{\gamma}(p^{\pm})$
(3.21) $\displaystyle\mathcal{R}_{b}^{\gamma,\pm}:=$
$\displaystyle~{}[\mathcal{T}^{\gamma},b^{\pm}]\cdot\nabla^{\varphi}v^{\pm}-\mathfrak{D}^{\gamma}(b^{\pm}),$
(3.22)

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

$\displaystyle\mathcal{W}^{\gamma,\pm}:=(\partial_{3}v^{\pm}\cdot
N)\mathcal{T}^{\gamma}\psi+[\mathcal{T}^{\gamma},N_{i},v_{i}^{\pm}],$ (3.23)

Note that $\omega(x_{3})=0$ on $\Sigma\cup\Sigma^{\pm}$, so all boundary
conditions are vanishing when $\gamma_{4}>0$. Thus, $\mathcal{T}^{\gamma}$ can
be written as
$\partial_{t}^{k+\alpha_{0}}\overline{\partial}{}^{(4+l)-(k+\alpha_{0})}$ on
$\Sigma$. We can replace $k+\alpha_{0}$ by $k$ ($0\leq k\leq 4+l$) in the
boundary energy terms.

In the rest of Section 3, we aim to prove the following tangential estimates

######  Proposition 3.3 (Tangential estimates for the approximate system).

For fixed $l\in\\{0,1,2,3,4\\}$ and any $\delta\in(0,1)$, the following
uniform-in-$(\kappa,\varepsilon)$ energy inequalities hold:

$\displaystyle\sum_{\pm}\sum_{\langle\alpha\rangle=2l}\sum_{\begin{subarray}{c}0\leq
k\leq 4-l\\\
k+\alpha_{0}<4+l\end{subarray}}\left\|\left(\varepsilon^{2l}\overline{\partial}{}^{4-k-l}\mathcal{T}^{\alpha}\partial_{t}^{k}(v^{\pm},b^{\pm},S^{\pm},p^{\pm})\right)\right\|^{2}_{0,\pm}$
$\displaystyle+\sum_{k=0}^{3+l}\left|\sqrt{\sigma}\varepsilon^{2l}\partial_{t}^{k}\psi\right|^{2}_{5-k-l}+\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k}\psi\right|_{6+k-l}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{k+1}\psi(\tau)\right|_{5+k-l}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\sum_{k=0}^{3+l}\left|\varepsilon^{2l}\partial_{t}^{k}\psi(0)\right|_{5.5+l-k}^{2}+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(t)\right)\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau$
(3.24)

and

$\displaystyle\sum_{\pm}\sum_{k=0}^{4-l}\left\|\left(\varepsilon^{2l}\partial_{t}^{4+l}(v^{\pm},b^{\pm},S^{\pm},(\mathcal{F}_{p})^{\frac{1}{2}}p^{\pm})\right)\right\|^{2}_{4-k-l,\pm}+\left|\sqrt{\sigma}\varepsilon^{2l}\partial_{t}^{4+l}\psi\right|^{2}_{1}+\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{4+l}\psi\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\partial_{t}^{5+l}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\left|\varepsilon^{2l}\partial_{t}^{3+l}\psi(0)\right|_{2.5}^{2}+P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(0)\right)+P\left(\sum_{j=0}^{l}E_{4+j}^{\kappa}(t)\right)\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}E_{4+j}^{\kappa}(\tau)\right)\,\mathrm{d}\tau.$
(3.25)

Here the first inequality represents the case when there are at least one
spatial tangential derivatives and the second inequality represents the case
of full time derivatives. Moreover, the term
$\left|\varepsilon^{2l}\partial_{t}^{k}\psi(0)\right|_{5.5+l-k}^{2}$ on the
right side does not appear when $\kappa=0$.

###  3.3 Tangential estimates: full spatial derivatives

We first study the case when all tangential derivatives are spatial
derivatives $\overline{\partial}{}_{1}$ and $\overline{\partial}{}_{2}$,
namely $\gamma_{0}=\gamma_{4}=0$ in
$\mathcal{T}^{\gamma}:=(\omega(x_{3})\partial_{3})^{\gamma_{4}}\partial_{t}^{\gamma_{0}}\partial_{1}^{\gamma_{1}}\partial_{2}^{\gamma_{2}}$.
In view of the definition of $E(t)$ and the div-curl decomposition, we need to
prove the $L^{2}$ estimates for the
$\varepsilon^{2l}\overline{\partial}{}^{4+l}$-differentiated system $(0\leq
l\leq 4)$. We now consider the case $l=0$, that is, the
$\overline{\partial}{}^{4}$-estimate for the approximate system (3.1) and aim
to prove the following estimate

######  Proposition 3.4.

Fix $l\in\\{0,1,2,3,4\\}$. For the tangential derivative
$\mathcal{T}^{\gamma}=\overline{\partial}{}^{4+l},~{}(\gamma_{0}+\gamma_{4}=0,~{}\gamma_{1}+\gamma_{2}=4+l)$,
the $\varepsilon^{2l}\overline{\partial}{}^{4+l}$-differentiated approximate
system admits the following uniform-in-$(\kappa,\varepsilon)$ estimate: For
any $0<\delta<1$

$\displaystyle\left\|\varepsilon^{2l}\left(\mathbf{V}^{\gamma,\pm},\mathbf{B}^{\gamma,\pm},\mathbf{S}^{\gamma,\pm},\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\gamma,\pm}\right)(t)\right\|_{0}^{2}+\left|\sqrt{\sigma}\varepsilon^{2l}\overline{\partial}{}^{4+l}\psi(t)\right|_{1}^{2}+\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\psi(t)\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\partial_{t}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
(3.26) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\left|\varepsilon^{2l}\psi_{0}\right|_{5.5+l}^{2}+\sum_{j=0}^{l}\int_{0}^{t}P(\sigma^{-1},E_{4+j}^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}~{}0\leq
l\leq 4.$

####  3.3.1 The case $l=0$: $\overline{\partial}{}^{4}$-estimates

As stated in Section 2.2, we introduce the Alinhac good unknowns for
$\mathcal{T}^{\gamma}=\overline{\partial}{}^{4}$ and drop the script $\gamma$
for simplicity of notations

$\mathbf{V}^{\pm}:=\overline{\partial}{}^{4}v^{\pm}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}v^{\pm},~{}\mathbf{B}^{\pm}:=\overline{\partial}{}^{4}b^{\pm}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}b^{\pm},~{}\mathbf{P}^{\pm}:=\overline{\partial}{}^{4}p^{\pm}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}p^{\pm},~{}\mathbf{Q}^{\pm}:=\overline{\partial}{}^{4}q^{\pm}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}q^{\pm}.$

Note that we have

$\mathbf{Q}^{\pm}=\mathbf{P}^{\pm}+b\cdot\mathbf{B}^{\pm}+\underbrace{\sum_{k=1}^{3}c_{k}\overline{\partial}{}^{k}b^{\pm}\cdot\overline{\partial}{}^{4-k}b^{\pm}}_{=:\mathcal{R}_{q}^{\gamma,\pm}}$

for some constants $c_{k}\in\mathbb{N}^{*}$.

#### Step 1: Interior energy structure.

We test the equation (3.11) by $\mathbf{V}^{\pm}$ in $\Omega^{\pm}$ and
integrate by parts to get one boundary term and several interior terms

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\rho^{\pm}|\mathbf{V}^{\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}=\int_{\Omega^{\pm}}\rho^{\pm}D_{t}^{\varphi\pm}\mathbf{V}^{\pm}\cdot\mathbf{V}^{\pm}\,\mathrm{d}\mathcal{V}_{t}$
(3.27) $\displaystyle=$
$\displaystyle\int_{\Omega^{\pm}}(b^{\pm}\cdot\nabla^{\varphi})\mathbf{B}^{\pm}\cdot\mathbf{V}^{\pm}\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}\mathbf{V}^{\pm}\cdot\nabla^{\varphi}\mathbf{Q}^{\pm}\,\mathrm{d}\mathcal{V}_{t}+\underbrace{\int_{\Omega^{\pm}}\mathbf{V}^{\pm}\cdot(\mathcal{R}_{v}^{\pm}-\mathfrak{C}(q^{\pm}))\,\mathrm{d}\mathcal{V}_{t}}_{=:R_{1}^{\pm}}$
$\displaystyle=$
$\displaystyle-\int_{\Omega^{\pm}}\mathbf{B}^{\pm}\cdot(b^{\pm}\cdot\nabla^{\varphi})\mathbf{V}^{\pm}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}b^{\pm}\cdot\mathbf{B}^{\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{P}^{\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle\pm\int_{\Sigma}\mathbf{Q}^{\pm}(\mathbf{V}^{\pm}\cdot
N)\,\mathrm{d}x^{\prime}+R_{1}^{\pm}+\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}.$

Invoking the equation (3.13) for the evolution of $\mathbf{B}$ in the first
integral above, the energy of $\mathbf{B}^{\pm}$ is produced.

$\displaystyle-\int_{\Omega^{\pm}}\mathbf{B}^{\pm}\cdot(b^{\pm}\cdot\nabla^{\varphi})\mathbf{V}^{\pm}\,\mathrm{d}\mathcal{V}_{t}$
(3.28) $\displaystyle=$
$\displaystyle-\int_{\Omega^{\pm}}\mathbf{B}^{\pm}\cdot
D_{t}^{\varphi\pm}\mathbf{B}^{\pm}\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}(\mathbf{B}^{\pm}\cdot
b^{\pm})(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{B}^{\pm}\cdot\mathcal{R}_{b}^{\pm}\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}(\mathbf{B}^{\pm}\cdot
b^{\pm})\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}|\mathbf{B}^{\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}\underbrace{-\frac{1}{2}\int_{\Omega^{\pm}}(\nabla^{\varphi}\cdot
v^{\pm})|\mathbf{B}^{\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{B}^{\pm}\cdot\mathcal{R}_{b}^{\pm}\,\mathrm{d}\mathcal{V}_{t}}_{R_{2}^{\pm}}$
$\displaystyle-\int_{\Omega^{\pm}}(\mathbf{B}^{\pm}\cdot
b^{\pm})(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}(\mathbf{B}^{\pm}\cdot
b^{\pm})\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t},$

where the first term in the last line is cancelled with the second integral in
(3.27), and the analysis of the second term in the last line will be
postponed.

The third term in (3.27) produces the energy of
$(\mathcal{F}_{p}^{\pm})^{\frac{1}{2}}\mathbf{P}^{\pm}$ with the help of
equation (3.12).

$\displaystyle\int_{\Omega^{\pm}}\mathbf{P}^{\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}$
(3.29) $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\mathcal{F}_{p}^{\pm}(\mathbf{P}^{\pm})^{2}\,\mathrm{d}\mathcal{V}_{t}\underbrace{-\frac{1}{2}\int_{\Omega^{\pm}}(D_{t}^{\varphi\pm}\mathcal{F}_{p}^{\pm}+\mathcal{F}_{p}^{\pm}\nabla^{\varphi}\cdot
v^{\pm})|\mathbf{P}^{\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{P}^{\pm}\mathcal{R}_{p}^{\pm}\,\mathrm{d}\mathcal{V}_{t}}_{R_{3}^{\pm}}-\int_{\Omega^{\pm}}\mathbf{P}^{\pm}\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t}.$

The last term in (3.27) can be controlled by inserting again the continuity
equation and integrating $D_{t}^{\varphi\pm}$ by parts. We have

$\displaystyle\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{\pm})\,\mathrm{d}\mathcal{V}_{t}=-\int_{\Omega^{\pm}}\mathcal{F}_{p}^{\pm}\mathcal{R}_{q}^{\pm}D_{t}^{\varphi\pm}\mathbf{P}^{\pm}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}\mathcal{R}_{p}^{\pm}\,\mathrm{d}\mathcal{V}_{t}-\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t}$
(3.30) $\displaystyle=$
$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega^{\pm}}\left(\sqrt{\mathcal{F}_{p}^{\pm}}\mathcal{R}_{q}^{\pm}\right)\left(\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\pm}\right)\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\left(\sqrt{\mathcal{F}_{p}^{\pm}}D_{t}^{\varphi\pm}\mathcal{R}_{q}^{\pm}\right)\left(\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\pm}\right)\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}\mathcal{R}_{p}^{\pm}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle-\int_{\Omega^{\pm}}\mathcal{R}_{q}^{\pm}\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t},$

where the first term on the right side is controlled under time integral by

$\delta\left\|\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\pm}(t)\right\|_{0}^{2}+P(E^{\kappa}_{4}(0))+\int_{0}^{t}P(E^{\kappa}_{4}(\tau))\,\mathrm{d}\tau,~{}~{}\forall
0<\delta\ll 1$

and the second term, the third term on the right side can be both controlled
by $P(E_{4}(t))$ via direct computation because $\mathcal{R}_{q}$ only
contains 3-rd order tangential derivative of $b$.

The entropy is directly bounded by testing the transport equation of
$\mathbf{S}^{\pm}$ with $\mathbf{S}^{\pm}$ itself

$\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\int_{\Omega^{\pm}}\rho^{\pm}(\mathbf{S}^{\pm})^{2}\,\mathrm{d}\mathcal{V}_{t}=\int_{\Omega^{\pm}}\rho^{\pm}\mathfrak{D}(S^{\pm})\,\mathbf{S}^{\pm}\,\mathrm{d}\mathcal{V}_{t}\leq\|\mathbf{S}^{\pm}\|_{0}\|\rho^{\pm}\|_{L^{\infty}}\,\sqrt{E_{4}^{\kappa}(t)}.$
(3.31)

The remainder terms are controlled by direct computation. For the commutator
$\mathfrak{C},\mathfrak{D}$, we have
$\|\mathfrak{C}(f^{\pm})\|_{0,\pm}\lesssim C(|\psi|_{4})\|f^{\pm}\|_{0,\pm}$
and $\|\mathfrak{D}(f^{\pm})\|_{0,\pm}\lesssim
C(|\psi|_{4},|\partial_{t}\psi|_{L^{\infty}}|)\|f^{\pm}\|_{0,\pm}$ when
$\mathcal{T}^{\gamma}=\overline{\partial}{}^{4}$ by straightforward
computation. Note that the initial data is well-prepared in the sense that
$\partial_{t}v|_{t=0}=O(1)$ with respect to Mach number, so there is no loss
of $\varepsilon$-weight in $\mathcal{R}_{v}$ term. We have

$\displaystyle R_{1}^{\pm}+R_{2}^{\pm}+R_{3}^{\pm}\leq P(E_{4}^{\kappa}(t)).$
(3.32)

#### Step 2: The boundary regularity contributed by surface tension.

We denote
$Z^{\pm}:=-\int_{\Omega^{\pm}}(\mathbf{P}^{\pm}+b^{\pm}\cdot\mathbf{B}^{\pm}+\mathcal{R}_{q}^{\pm})\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t}=-\int_{\Omega^{\pm}}\mathbf{Q}^{\pm}\mathfrak{C}_{i}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t}$
to be the remaining interior terms presented above which should be controlled
together with some boundary terms involving $\mathcal{W}^{\pm}$. Now we
analyze the boundary integral in (3.27). The sum of two boundary integrals can
be written as

$\displaystyle\int_{\Sigma}\mathbf{Q}^{+}(\mathbf{V}^{+}\cdot
N)\,\mathrm{d}x^{\prime}-\int_{\Sigma}\mathbf{Q}^{-}(\mathbf{V}^{-}\cdot
N)\,\mathrm{d}x^{\prime}$ (3.33) $\displaystyle=$
$\displaystyle\int_{\Sigma}(\overline{\partial}{}^{4}q^{+}-\overline{\partial}{}^{4}\psi\partial_{3}q^{+})(\partial_{t}\overline{\partial}{}^{4}\psi+(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi-\mathcal{W}^{+})\,\mathrm{d}x^{\prime}$
$\displaystyle-\int_{\Sigma}(\overline{\partial}{}^{4}q^{-}-\overline{\partial}{}^{4}\psi\partial_{3}q^{-})(\partial_{t}\overline{\partial}{}^{4}\psi+(\bar{v}^{-}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi-\mathcal{W}^{-})\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle\int_{\Sigma}\overline{\partial}{}^{4}\left\llbracket
q\right\rrbracket\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}+\int_{\Sigma}\overline{\partial}{}^{4}\left\llbracket
q\right\rrbracket(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}+\int_{\Sigma}\overline{\partial}{}^{4}q^{-}(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle-\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\overline{\partial}{}^{4}\psi\partial_{t}\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\int_{\Sigma}\partial_{3}q^{+}\overline{\partial}{}^{4}\psi(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}+\int_{\Sigma}\partial_{3}q^{-}\overline{\partial}{}^{4}\psi(\bar{v}^{-}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle-\int_{\Sigma}\mathbf{Q}^{+}\mathcal{W}^{+}\,\mathrm{d}x^{\prime}+\int_{\Sigma}\mathbf{Q}^{-}\mathcal{W}^{-}\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle\text{ ST}+\text{ ST}^{\prime}+\text{
VS}+\text{ RT}+\text{ RT}^{+}+\text{ RT}^{-}+ZB^{+}+ZB^{-}.$

We will see that the term ST gives the $\sqrt{\sigma}$-weighted boundary
regularity (contributed by surface tension) and the $\sqrt{\kappa}$-weighted
boundary regularity (contributed by the two regularization terms) which help
us control the terms$\text{ ST}^{\prime}$, VS, RT,$\text{ RT}^{\pm}$. The
terms $ZB^{\pm}$ will be controlled together with $Z^{\pm}$ by using Gauss-
Green formula. Do note that the slip conditions imply
$\mathbf{V}_{3}^{\pm}=\mathbf{B}_{3}^{\pm}=\overline{\partial}{}\psi=0$ on
$\Sigma^{\pm}$, which eliminates all integrals on $\Sigma^{\pm}$.

Inserting the jump condition $\left\llbracket
q\right\rrbracket=\sigma\mathcal{H}-\kappa(1-\overline{\Delta}{})^{2}\psi-\kappa(1-\overline{\Delta}{})\psi_{t}$
into the term ST, we get

$\displaystyle\text{ ST}=$
$\displaystyle~{}\sigma\int_{\Sigma}\overline{\partial}{}^{4}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}-\int_{\Sigma}\kappa(1-\overline{\Delta}{})^{2}\overline{\partial}{}^{4}\psi\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}-\int_{\Sigma}\kappa(1-\overline{\Delta}{})\overline{\partial}{}^{4}\partial_{t}\psi\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}$
(3.34) $\displaystyle=$
$\displaystyle~{}\sigma\int_{\Sigma}\overline{\partial}{}^{4}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\left|\sqrt{\kappa}\langle\overline{\partial}{}\rangle^{2}\overline{\partial}{}^{4}\psi\right|^{2}\,\mathrm{d}x^{\prime}-\int_{\Sigma}\left|\sqrt{\kappa}\overline{\partial}{}^{4}\langle\overline{\partial}{}\rangle\psi_{t}\right|^{2}\,\mathrm{d}x^{\prime}.$

Integrating by parts in the mean curvature term and using

$\overline{\partial}{}\left(\frac{1}{|N|}\right)=\frac{\overline{\nabla}\psi\cdot\overline{\nabla}\overline{\partial}{}\psi}{|N|^{3}},~{}~{}~{}|N|=\sqrt{1+|\overline{\nabla}\psi|^{2}},$

we get

$\displaystyle~{}\sigma\int_{\Sigma}\overline{\partial}{}^{4}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\overline{\partial}{}^{4}\partial_{t}\psi\,\mathrm{d}x^{\prime}$
(3.35) $\displaystyle=$
$\displaystyle-\sigma\int_{\Sigma}\frac{\overline{\partial}{}^{4}\overline{\nabla}\psi}{|N|}\cdot\partial_{t}\overline{\nabla}\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}+\sigma\int_{\Sigma}\frac{\overline{\partial}{}\psi\cdot\overline{\nabla}\overline{\partial}{}^{4}\psi}{|N|^{3}}\overline{\nabla}\psi\cdot\partial_{t}\overline{\nabla}\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle\underbrace{-\sigma\int_{\Sigma}\left(\left[\overline{\partial}{}^{3},\frac{1}{|N|}\right]\overline{\partial}{}\overline{\nabla}_{i}\psi+\left[\overline{\partial}{}^{3},\frac{1}{|N|^{3}}\right](\overline{\nabla}_{k}\psi\cdot\overline{\partial}{}\overline{\nabla}_{k}\psi\overline{\nabla}_{i}\psi)-\frac{1}{|N|^{3}}[\overline{\partial}{}^{3},\overline{\nabla}_{i}\psi\overline{\nabla}_{k}\psi]\overline{\partial}{}\overline{\nabla}_{k}\psi\right)\cdot\partial_{t}\overline{\nabla}_{i}\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}}_{=:\text{
ST}_{1}^{R}}$

which is further equal to

$\displaystyle-\frac{\sigma}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\frac{|\overline{\partial}{}^{4}\overline{\nabla}\psi|^{2}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}-\frac{|\overline{\nabla}\psi\cdot\overline{\partial}{}^{4}\overline{\nabla}\psi|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\,\mathrm{d}x^{\prime}$
(3.36)
$\displaystyle+\underbrace{\frac{\sigma}{2}\int_{\Sigma}\partial_{t}\left(\frac{1}{|N|}\right)\left|\overline{\partial}{}^{4}\overline{\nabla}\psi\right|^{2}-\partial_{t}\left(\frac{1}{|N|^{3}}\right)\left|\overline{\nabla}\psi\cdot\overline{\nabla}\overline{\partial}{}^{4}\psi\right|^{2}\,\mathrm{d}x^{\prime}}_{=:\text{
ST}_{2}^{R}}+\text{ ST}_{1}^{R}$

The control of$\text{ ST}_{1}^{R},\text{ ST}_{2}^{R}$ is straightforward which
has been analyzed in the author’s previous paper [55, (4.77)-(4.78)], so we
only record the result here

$\text{ ST}_{1}^{R}+\text{ ST}_{2}^{R}\lesssim
P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}\left|\sqrt{\sigma}\overline{\partial}{}^{4}\overline{\nabla}\psi\right|_{0}\left|\sqrt{\sigma}\partial_{t}\overline{\partial}{}^{4}\psi\right|_{0}\leq
P(E_{4}^{\kappa}(t)).$

Using Cauchy’s inequality

$\forall\mathbf{a}\in\mathbb{R}^{2},~{}~{}\frac{|\mathbf{a}|^{2}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}-\frac{|\overline{\nabla}\psi\cdot\mathbf{a}|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\geq\frac{|\mathbf{a}|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}},$
(3.37)

we obtain the $\sqrt{\sigma}$-weighted boundary regularity

$\displaystyle\int_{0}^{t}\text{
ST}\,\mathrm{d}\tau+\frac{\sigma}{2}\int_{\Sigma}\frac{|\overline{\nabla}\overline{\partial}{}^{4}\psi|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\,\mathrm{d}x^{\prime}+\int_{\Sigma}\left|\sqrt{\kappa}\langle\overline{\partial}{}\rangle^{2}\overline{\partial}{}^{4}\psi\right|^{2}\,\mathrm{d}x^{\prime}+\int_{0}^{t}\int_{\Sigma}\left|\sqrt{\kappa}\overline{\partial}{}^{4}\langle\overline{\partial}{}\rangle\psi_{t}\right|^{2}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
(3.38) $\displaystyle\leq$ $\displaystyle\int_{0}^{t}\text{ ST}_{1}^{R}+\text{
ST}_{2}^{R}\,\mathrm{d}x^{\prime}\leq\int_{0}^{t}P(E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau.$

So far, we already obtain the boundary regularity $\sqrt{\sigma}\psi\in
H^{5}(\Sigma),~{}\sqrt{\kappa}\psi\in H^{6}(\Sigma)$ and
$\sqrt{\kappa}\psi_{t}\in L_{t}^{2}H_{x^{\prime}}^{5}([0,T]\times\Sigma)$.
Using this, we can easily control$\text{ ST}^{\prime}$ term in (3.33).
Invoking again the boundary condition for $\left\llbracket q\right\rrbracket$,
we get

$\text{
ST}^{\prime}=\int_{\Sigma}\sigma\mathcal{H}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\kappa\int_{\Sigma}(1-\overline{\Delta}{})^{2}\overline{\partial}{}^{4}\psi~{}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\kappa\int_{\Sigma}(1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi_{t}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}.$
(3.39)

Integrating by parts $1-\overline{\Delta}{}$ in the second term and
$\langle\overline{\partial}{}\rangle=\sqrt{1-\overline{\Delta}{}}$ in the
third term above, we can easily use the $\sqrt{\kappa}$-weighted energy to
control the last two terms.

$\displaystyle-\kappa\int_{\Sigma}(1-\overline{\Delta}{})^{2}\overline{\partial}{}^{4}\psi~{}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$
(3.40) $\displaystyle=$
$\displaystyle-\kappa\int_{\Sigma}\left((1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi\right)(\bar{v}^{+}\cdot\overline{\nabla})(1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\kappa\int_{\Sigma}\left((1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi\right)~{}[1-\overline{\Delta}{},\bar{v}^{+}\cdot\overline{\nabla}]\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime},$

where the first term is controlled by
$|\bar{v}^{+}|_{W^{1,\infty}}|\sqrt{\kappa}(1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi|_{0}^{2}$
after integrating $\bar{v}^{+}\cdot\overline{\nabla}$ by parts and using the
symmetry, and the second term is directly controlled by
$|\bar{v}^{+}|_{W^{2,\infty}}|\sqrt{\kappa}(1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi|_{0}|\sqrt{\kappa}\overline{\partial}{}^{4}\psi|_{2}$.
Similarly, we have for any $\delta\in(0,1)$

$\displaystyle-\kappa\int_{0}^{t}\int_{\Sigma}(1-\overline{\Delta}{})\overline{\partial}{}^{4}\psi_{t}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau=-\kappa\int_{0}^{t}\int_{\Sigma}\langle\overline{\partial}{}\rangle\overline{\partial}{}^{4}\psi_{t}~{}\langle\overline{\partial}{}\rangle((\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi)\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
(3.41) $\displaystyle\leq$
$\displaystyle~{}\delta\left|\sqrt{\kappa}\langle\overline{\partial}{}\rangle\overline{\partial}{}^{4}\psi_{t}\right|_{L_{t}^{2}L_{x^{\prime}}^{2}}^{2}+\frac{1}{4\delta}\int_{0}^{t}\int_{\Sigma}|\bar{v}^{+}|_{W^{1,\infty}}^{2}\left|\sqrt{\kappa}\overline{\partial}{}^{4}\psi\right|_{2}^{2}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau\leq\delta
E_{4}^{\kappa}(t)+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau.$

Picking $\delta>0$ to be sufficiently small, the $\delta$-term can be absorbed
by $E_{4}^{\kappa}(t)$. The first term in${\text{ ST}}^{\prime}$ is controlled
in the same way if we integrating $\overline{\nabla}\cdot$ by parts. Here we
only list the result and refer the details to [55, (4.87)-(4.89)]

$\int_{\Sigma}\sigma\mathcal{H}(\bar{v}^{+}\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}\leq
P(|\overline{\nabla}\psi|_{W^{1,\infty}})|\bar{v}^{+}|_{W^{1,\infty}}\left|\sqrt{\sigma}\overline{\nabla}\overline{\partial}{}^{4}\psi\right|_{0}^{2}\leq
P(E_{4}^{\kappa}(t)).$ (3.42)

Next we control the terms RT and$\text{ RT}^{\pm}$ in (3.33). Note that we do
not have the Rayleight-Taylor sign condition
$\left\llbracket\partial_{3}q\right\rrbracket|_{\Sigma}\geq c_{0}>0$, so we
have to use the $\sqrt{\sigma}$-weighted energy to control these terms, we
have

$\text{
RT}\leq|\partial_{3}q|_{L^{\infty}}|\psi|_{4}|\psi_{t}|_{4}\leq\sigma^{-1}P(E_{4}^{\kappa}(t)).$
(3.43)

Similarly, integrating $\bar{v}^{\pm}\cdot\nabla$ by parts in$\text{
RT}^{\pm}$ and using symmetry, the terms$\text{ RT}^{\pm}$ can be directly
controlled

$\text{
RT}^{\pm}\leq|\bar{v}^{\pm}\partial_{3}q|_{W^{1,\infty}}|\psi|_{4}^{2}\leq\sigma^{-1}P(E_{4}^{\kappa}(t)).$
(3.44)

#### Step 3: The crucial term for vortex sheets problem.

Now we study the term VS in (3.33) which appears to be the most problematic
term for the vortex sheets problem. Note that we do not have any boundary
condition for $q^{\pm}$ individually. Thus, we may alternatively integrate
$\overline{\partial}{}^{1/2}$ by parts and use (B.5) to control VS.

$\displaystyle\text{
VS}=\int_{\Sigma}\overline{\partial}{}^{4}q^{-}(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}\lesssim\|\overline{\partial}{}^{4}q^{-}\|_{0,-}^{\frac{1}{2}}\|\partial_{3}\overline{\partial}{}^{3}q^{-}\|_{0,-}^{\frac{1}{2}}\|\bar{v}^{\pm}\|_{2,\pm}|\overline{\nabla}\overline{\partial}{}^{4}\psi|_{1/2}\leq
P(E_{4}^{\kappa}(t))|\psi|_{5.5},$ (3.45)

where we have used the Kato-Ponce inequality (cf. Lemma B.6) for
$s=1/2,~{}p_{1}=2,~{}p_{2}=\infty,~{}q_{1}=q_{2}=4$ and Sobolev embedding
$H^{1/2}(\mathbb{T}^{2})\hookrightarrow L^{4}(\mathbb{T}^{2})$. Now we need to
control $|\psi|_{5.5}$ via the jump condition of $\left\llbracket
q\right\rrbracket$. Without the $\kappa$-regularization terms, we may use the
ellipticity of the mean curvature operator to control $|\psi|_{5.5}$ by
$\sigma^{-1}|\left\llbracket q\right\rrbracket|_{3.5}$. Now, we can still
prove analogous result for the $\kappa$-regularized jump condition.

######  Lemma 3.5 (Elliptic estimate for the free interface).

For any $s\geq 0.5$ and $\kappa>0$, we have the uniform-in-$\kappa$ estimate

$|\psi|_{s+1.5}\leq|\psi_{0}|_{s+1.5}+\sigma^{-1}\left(P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\overline{\partial}{}\psi|_{s-0.5}+|\left\llbracket
q\right\rrbracket|_{s-0.5}\right).$

Moreover, when $\kappa=0$, $|\psi_{0}|_{s+1.5}$ is not needed

$\displaystyle|\sigma\psi|_{s+1.5}\leq~{}P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\sigma\overline{\partial}{}\psi|_{s-0.5}+|\left\llbracket
q\right\rrbracket|_{s-0.5}.$ (3.46)

###### Proof.

We take $\langle\overline{\partial}{}\rangle^{s+0.5}$ in the jump condition to
get

$-\langle\overline{\partial}{}\rangle^{s+0.5}\left\llbracket
q\right\rrbracket=-\sigma\langle\overline{\partial}{}\rangle^{s+0.5}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)+\kappa(1-\overline{\Delta}{})^{2}\langle\overline{\partial}{}\rangle^{s+0.5}\psi+\kappa(1-\overline{\Delta}{})\langle\overline{\partial}{}\rangle^{s+0.5}\psi_{t}.$

Testing this equation with $\langle\overline{\partial}{}\rangle^{s+0.5}\psi$
in $L^{2}(\Sigma)$, we get

$-\int_{\Sigma}\langle\overline{\partial}{}\rangle^{s+0.5}\left\llbracket
q\right\rrbracket~{}\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}\leq|\langle\overline{\partial}{}\rangle^{s-0.5}\left\llbracket
q\right\rrbracket|_{0}|\langle\overline{\partial}{}\rangle^{s+1.5}\psi|_{0}.$

For the right side, we can mimic the treatment of ST term to obtain the
boundary regularity. The two regularization terms can be directly controlled

$\int_{\Sigma}\kappa(1-\overline{\Delta}{})^{2}\langle\overline{\partial}{}\rangle^{s+0.5}\psi~{}\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}=\int_{\Sigma}\kappa(1-\overline{\Delta}{})\langle\overline{\partial}{}\rangle^{s+0.5}\psi~{}(1-\overline{\Delta}{})\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}=\left|\sqrt{\kappa}\psi\right|_{s+2.5}^{2},$
$\int_{\Sigma}\kappa(1-\overline{\Delta}{})\langle\overline{\partial}{}\rangle^{s+0.5}\psi_{t}~{}\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}\overset{\langle\overline{\partial}{}\rangle}{=}\frac{\mathrm{d}}{\mathrm{d}t}\left|\sqrt{\kappa}\psi\right|_{s+1.5}^{2}.$

The term involving surface tension is controlled as follows

$\displaystyle-\sigma\int_{\Sigma}\langle\overline{\partial}{}\rangle^{s+0.5}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}=\sigma\int_{\Sigma}\langle\overline{\partial}{}\rangle^{s+0.5}\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\cdot\langle\overline{\partial}{}\rangle^{s+0.5}\overline{\nabla}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle~{}\sigma\int_{\Sigma}\frac{|\langle\overline{\partial}{}\rangle^{s+0.5}\overline{\nabla}\psi|^{2}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}-\frac{|\overline{\nabla}\psi\cdot\langle\overline{\partial}{}\rangle^{s+0.5}\overline{\nabla}\psi|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\,\mathrm{d}x^{\prime}$
$\displaystyle+\sigma\int_{\Sigma}\left(\left[\langle\overline{\partial}{}\rangle^{s-0.5},\frac{1}{|N|}\right]\langle\overline{\partial}{}\rangle\overline{\nabla}_{i}\psi+\left[\langle\overline{\partial}{}\rangle^{s-0.5},\frac{1}{|N|^{3}}\right](\overline{\nabla}_{k}\psi\cdot\langle\overline{\partial}{}\rangle\overline{\nabla}_{k}\psi\overline{\nabla}_{i}\psi)-\frac{1}{|N|^{3}}[\langle\overline{\partial}{}\rangle^{s-0.5},\overline{\nabla}\psi]\langle\overline{\partial}{}\rangle\overline{\nabla}_{i}\psi\right)\cdot\overline{\nabla}_{i}\langle\overline{\partial}{}\rangle^{s+0.5}\psi\,\mathrm{d}x^{\prime}$

Using Kato-Ponce commutator estimate (cf. (B.8) in Lemma B.6), the commutators
in the last line of the above identity are controlled by
$P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\overline{\partial}{}\psi|_{s-0.5}.$
Using again Cauchy’s inequality (3.37), we conclude the elliptic estimate by

$\displaystyle\sigma|\psi|_{s+1.5}^{2}+\kappa\left|\psi\right|_{s+2.5}^{2}+\kappa\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi\right|_{s+1.5}^{2}\leq~{}\left(P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\sigma\overline{\partial}{}\psi|_{s-0.5}+|\left\llbracket
q\right\rrbracket|_{s-0.5}\right)|\psi|_{s+1.5}.$

In particular, Lemma B.7 suggests that we have

$|\psi|_{s+1.5}\leq|\psi_{0}|_{s+1.5}+\sigma^{-1}\left(P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\sigma\overline{\partial}{}\psi|_{s-0.5}+|\left\llbracket
q\right\rrbracket|_{s-0.5}\right).$

Moreover, when $\kappa=0$, $|\psi_{0}|_{s+1.5}$ no longer appears as we do not
need Lemma B.7

$\displaystyle|\sigma\psi|_{s+1.5}\leq~{}P(|\overline{\nabla}\psi|_{L^{\infty}})|\overline{\nabla}\psi|_{W^{1,\infty}}|\sigma\overline{\partial}{}\psi|_{s-0.5}+|\left\llbracket
q\right\rrbracket|_{s-0.5}.$ (3.47)

∎

Now we can easily obtain the control for the problematic term VS by setting
$s=4$ in Lemma 3.5

$\text{ VS}\lesssim|\psi_{0}|_{5.5}+\sigma^{-1}P(E_{4}^{\kappa}(t)).$ (3.48)

#### Step 4: A cancellation structure for the incompressible limit.

It remains to control the term $Z^{\pm}$ and $ZB^{\pm}$. In
$\overline{\partial}{}^{4}$-estimates, each of these terms can be directly
controlled. However, in the control of $E_{8}^{\kappa}(t)$ and the control of
full time derivatives, there will be extra technical difficulties due to the
loss of Mach number or the anisotropy of the function spaces. Thus, we would
like to present a robust approach to control these terms. We take
$Z^{-}+ZB^{-}$ as an example and the “+” case is controlled in the same way by
reversing the sign when integrating by parts. Recall that
$\mathbf{Q}^{-}=\overline{\partial}{}^{4}q^{-}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}q^{-}$,
so we have

$\displaystyle ZB^{-}=$
$\displaystyle\int_{\Sigma}\overline{\partial}{}^{4}q^{-}(\partial_{3}v^{-}\cdot
N)\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\int_{\Sigma}\overline{\partial}{}^{4}\psi\partial_{3}q^{-}(\partial_{3}v^{-}\cdot
N)\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$ (3.49)
$\displaystyle+\sum_{k=1}^{3}\int_{\Sigma}\binom{4}{k}\mathbf{Q}^{-}\overline{\partial}{}^{4-k}v^{-}\cdot\overline{\partial}{}^{k}N\,\mathrm{d}x^{\prime}.$

The first two terms in $ZB^{-}$ can be directly controlled

$\displaystyle\int_{\Sigma}\overline{\partial}{}^{4}q^{-}(\partial_{3}v^{-}\cdot
N)\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}-\int_{\Sigma}\overline{\partial}{}^{4}\psi\partial_{3}q^{-}(\partial_{3}v^{-}\cdot
N)\overline{\partial}{}^{4}\psi\,\mathrm{d}x^{\prime}$ $\displaystyle\leq$
$\displaystyle~{}\left(|\overline{\partial}{}^{7/2}q^{-}|_{0}|\psi|_{4.5}+|\psi|_{4}^{2}\right)|\partial_{3}v\cdot
N|_{1.5}\leq\left(\|q^{-}\|_{4,-}|\psi|_{4.5}+|\psi|_{4}^{2}\right)\|\partial
v^{-}\|_{2,-}|\psi|_{2.5}\leq P(\sigma^{-1},E_{4}^{\kappa}(t)).$

The last term in $ZB^{-}$ is controlled together with
$Z^{-}:=-\int_{\Omega^{-}}\mathbf{Q}^{-}\mathfrak{C}_{i}(v_{i}^{-})\,\mathrm{d}\mathcal{V}_{t}$.
Recall that

$\displaystyle\mathfrak{C}_{i}(v_{i}^{-})=$
$\displaystyle~{}(\partial^{\varphi}_{3}\partial^{\varphi}_{i}v_{i}^{-})\overline{\partial}{}^{4}\varphi-\left[\overline{\partial}{}^{4},\frac{\partial_{i}\varphi}{\partial_{3}\varphi},\partial_{3}v_{i}^{-}\right]-\partial_{3}v_{i}^{-}\left[\overline{\partial}{}^{4},\partial_{i}\varphi,\frac{1}{\partial_{3}\varphi}\right]+\partial_{i}\varphi\partial_{3}v_{i}^{-}\left[\overline{\partial}{}^{3},\frac{1}{(\partial_{3}\varphi)^{2}}\right]\overline{\partial}{}\partial_{3}\varphi,~{}i=1,2$

and

$\displaystyle\mathfrak{C}_{3}(f)=$
$\displaystyle~{}(\partial^{\varphi}_{3})^{2}v_{3}^{-}\overline{\partial}{}^{4}\varphi+\left[\overline{\partial}{}^{4},\frac{1}{\partial_{3}\varphi},\partial_{3}v_{3}^{-}\right]-\partial_{3}v_{3}^{-}\left[\overline{\partial}{}^{3},\frac{1}{(\partial_{3}\varphi)^{2}}\right]\overline{\partial}{}\partial_{3}\varphi.$

Note that $\mathbf{N}_{i}=-\partial_{i}\varphi$ for $i=1,2$, so we have

$\displaystyle-\left[\overline{\partial}{}^{4},\frac{\partial_{i}\varphi}{\partial_{3}\varphi},\partial_{3}v_{i}^{-}\right]=\left[\overline{\partial}{}^{4},\frac{\mathbf{N}_{i}}{\partial_{3}\varphi},\partial_{3}v_{i}^{-}\right]=\sum_{k=1}^{3}\binom{4}{k}\overline{\partial}{}^{k}\left(\frac{\mathbf{N}_{i}}{\partial_{3}\varphi}\right)\overline{\partial}{}^{4-k}\partial_{3}v_{i}^{-}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\binom{4}{k}\overline{\partial}{}^{k}\mathbf{N}_{i}\partial^{\varphi}_{3}\overline{\partial}{}^{4-k}v_{i}^{-}-\binom{4}{k}\left[\overline{\partial}{}^{k},\frac{1}{\partial_{3}\varphi}\right]\partial_{i}\varphi\partial_{3}\overline{\partial}{}^{4-k}v_{i}^{-},$

where the contribution of the first term above gives us (using Gauss-Green
formula)

$\displaystyle
ZB^{-}-\sum_{k=1}^{3}\int_{\Omega^{-}}\binom{4}{k}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\partial^{\varphi}_{3}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}\mathcal{V}_{t}$
(3.50) $\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\binom{4}{k}\left(\int_{\Sigma}\mathbf{Q}^{-}\overline{\partial}{}^{4-k}v^{-}\cdot\overline{\partial}{}^{k}N\,\mathrm{d}x^{\prime}-\int_{\Omega^{-}}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\partial_{3}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x\right)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\binom{4}{k}\left(\int_{\Omega^{-}}\partial_{3}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x+\int_{\Omega^{-}}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\partial_{3}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x-\int_{\Omega^{-}}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\partial_{3}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x\right)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\binom{4}{k}\int_{\Omega^{-}}\partial_{3}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x.$

Now invoking
$\mathbf{Q}^{-}=\overline{\partial}{}^{4}q^{-}-\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}q^{-}$
and integrating one $\overline{\partial}{}$ by parts, we find that

$\sum_{k=1}^{3}\binom{4}{k}\int_{\Omega^{-}}\partial_{3}\mathbf{Q}^{-}\overline{\partial}{}^{k}\mathbf{N}_{i}\overline{\partial}{}^{4-k}v_{i}^{-}\,\mathrm{d}x\lesssim(\|\overline{\partial}{}^{3}\partial_{3}q^{-}\|_{0,-}+|\psi|_{4}\|\partial_{3}q^{-}\|_{L^{\infty}(\Omega^{-})})|\psi|_{4}\|v_{i}^{-}\|_{4,-}.$
(3.51)

Among other terms in $\mathfrak{C}_{i}(v_{i}^{-})$, we shall focus on the case
when there are 4 derivatives falling on $v_{i}^{-}$ and $\varphi$, and the
control of these terms (lised below) appears to be easier.

$\displaystyle-\int_{\Omega^{-}}\mathbf{Q}^{-}\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}\quad\text{from the first term in
}\mathfrak{C}_{i}(v_{i}^{-})$ (3.52) $\displaystyle
4\sum_{i=1}^{3}\int_{\Omega^{-}}\mathbf{Q}^{-}\partial_{3}\overline{\partial}{}\varphi~{}\partial^{\varphi}_{3}\overline{\partial}{}^{3}v^{-}\cdot\mathbf{N}\,\mathrm{d}x\quad\text{from
the second term in }\mathfrak{C}_{i}(v_{i}^{-})\text{ when
}\overline{\partial}{}^{3}\text{ falls on }\partial_{3}v_{i}^{-}.$

Note that $\partial^{\varphi}_{3}v^{-}\cdot N=\nabla^{\varphi}\cdot
v^{-}-\overline{\nabla}\cdot\bar{v}^{-}$, we have

$-\int_{\Omega^{-}}\mathbf{Q}^{-}\overline{\partial}{}^{4}\varphi\partial^{\varphi}_{3}(\nabla^{\varphi}\cdot
v)\,\mathrm{d}\mathcal{V}_{t}\lesssim\left\|\sqrt{\mathcal{F}_{p}^{-}}\mathbf{Q}^{-}\right\|_{0,-}\left\|\sqrt{\mathcal{F}_{p}^{-}}\partial_{3}D_{t}^{\varphi,-}p^{-}\right\|_{0,-}\left|\psi\right|_{4},$
(3.53)

and

$\displaystyle
4\int_{\Omega^{-}}\mathbf{Q}^{-}\partial_{3}\overline{\partial}{}\varphi~{}\partial^{\varphi}_{3}\overline{\partial}{}^{3}v^{-}\cdot\mathbf{N}\,\mathrm{d}x\overset{L}{=}$
$\displaystyle~{}4\int_{\Omega^{-}}\mathbf{Q}^{-}\partial_{3}\overline{\partial}{}\varphi~{}\overline{\partial}{}^{3}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}x-4\int_{\Omega^{-}}\mathbf{Q}^{-}\partial_{3}\overline{\partial}{}\varphi~{}\overline{\partial}{}^{3}(\overline{\nabla}\cdot\bar{v}^{-})\,\mathrm{d}x$
(3.54) $\displaystyle\lesssim$
$\displaystyle~{}|\overline{\partial}{}\psi|_{L^{\infty}}\left(\left\|\sqrt{\mathcal{F}_{p}^{-}}\mathbf{Q}^{-}\right\|_{0,-}\left\|\sqrt{\mathcal{F}_{p}^{-}}\overline{\partial}{}^{3}D_{t}^{\varphi,-}p^{-}\right\|_{0,-}+\|\overline{\nabla}\overline{\partial}{}^{3}\mathbf{Q}^{-}\|_{0,-}\|\overline{\partial}{}^{4}v^{-}\|_{0,-}\right).$

Thus, combining the estimates in the above four steps, we conclude the
$\overline{\partial}{}^{4}$-estimate by: For the tangential derivative
$\mathcal{T}^{\gamma}=\overline{\partial}{}^{4}~{}(\gamma_{0}=\gamma_{4}=0,~{}\gamma_{1}+\gamma_{2}=4)$
and for any $0<\delta<1$, we have

$\displaystyle\left\|\left(\mathbf{V}^{\gamma,\pm},\mathbf{B}^{\gamma,\pm},\mathbf{S}^{\gamma,\pm},\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\gamma,\pm}\right)(t)\right\|_{0}^{2}+\left|\sqrt{\sigma}\varepsilon^{2l}\overline{\partial}{}^{4+l}\overline{\nabla}\psi(t)\right|_{0}^{2}+\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\psi(t)\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\partial_{t}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
(3.55) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\left|\varepsilon^{2l}\psi_{0}\right|_{5.5+l}^{2}+\sum_{j=0}^{l}\int_{0}^{t}P(\sigma^{-1},E_{4+j}^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}~{}0\leq
l\leq 4.$

######  Remark 3.2.

It should be noted that we only have the $L^{2}$ control of
$\mathbf{V},\mathbf{B},\mathbf{S}$ and
$(\mathcal{F}_{p})^{\frac{1}{2}}\mathbf{P}$ in the tangential estimates, but
the term $\mathbf{Q}$ without $\mathcal{F}_{p}$-weight does appear in
tangential estimates. When $\mathcal{T}^{\gamma}$ contains at least one
spatial derivative, that is, $\gamma_{0}<\langle\gamma\rangle$, one can invoke
the momentum equation to replace $\mathcal{T}q$ by $D_{t}^{\varphi}v$ and
$(b\cdot\nabla^{\varphi})b$ to avoid the loss of Mach number. This also
suggests that we can actually control $\|\mathbf{P}\|_{0}$ instead of only
$\|\mathcal{F}_{p}^{1/2}\mathbf{P}\|_{0}$ when there is at least one spatial
derivatives. However, when $\mathcal{T}^{\gamma}$ only consists of time
derivatives, we cannot do such substitution any longer. Thus, we have to use
the above cancellation structure between $ZB$ and $Z$ to control these two
terms together.

####  3.3.2 The case $l>0$: No loss of regularity or weights of Mach number

Next we consider the tangential estimates for $\varepsilon$-weighted spatial
derivatives, namely $\varepsilon^{2l}\overline{\partial}{}^{4+l}$ for $1\leq
l\leq 4$. The proof is parallel to the case
$\mathcal{T}^{\gamma}=\overline{\partial}{}^{4}$, but we have to check the
following aspects

  * a.

We have to guarantee that there is no loss of $\mathcal{F}_{p}$-weight in
various commutators, especially those involving $q$.

  * b.

When $l=4$, we only have tangential regularity for 8 derivatives. Due to the
anisotropy of the function space $H_{*}^{8}$, we have to put extra efforts to
reduce the terms involving the derivative
$\overline{\partial}{}^{7}\partial_{3}$.

We only show the detailed modifications for the case $l=4$, that is, the
$\varepsilon^{8}\overline{\partial}{}^{8}$-estimate. When $1\leq l\leq 3$,
similar modifications can be made in the same way.

#### Commutators of type
$\varepsilon^{8}[\overline{\partial}{}^{8},f]\mathcal{T}g$ for
$\mathcal{T}=\overline{\partial}{}$ or $D_{t}^{\varphi}$

This type of commutator includes the following terms

$\displaystyle-[\mathcal{T}^{\gamma},\rho]D_{t}^{\varphi}v\text{ in
}\mathcal{R}_{v},~{}~{}-[\mathcal{T}^{\gamma},\mathcal{F}_{p}]D_{t}^{\varphi}p\text{
in }\mathcal{R}_{p},$
$\displaystyle[\mathcal{T}^{\gamma},\bar{v}]\cdot\overline{\partial}{}f\text{
and }\partial^{\varphi}_{3}f[\mathcal{T}^{\gamma},v]\cdot\mathbf{N}\text{ in
}\mathfrak{D}^{\gamma}(f)$

It is controlled directly by expanding the commutator. We have

$\displaystyle\varepsilon^{8}[\overline{\partial}{}^{8},f]\mathcal{T}g=$
$\displaystyle~{}(\varepsilon^{8}\overline{\partial}{}^{8}f)\mathcal{T}g+8(\varepsilon^{6}\overline{\partial}{}^{7}f)(\varepsilon^{2}\overline{\partial}{}\mathcal{T}g)+28(\varepsilon^{6}\overline{\partial}{}^{6}f)(\varepsilon^{2}\overline{\partial}{}^{2}\mathcal{T}g)+56(\varepsilon^{6}\overline{\partial}{}^{5}f)(\varepsilon^{2}\overline{\partial}{}^{3}\mathcal{T}g)$
$\displaystyle+70(\varepsilon^{2}\overline{\partial}{}^{4}f)(\varepsilon^{6}\overline{\partial}{}^{4}\mathcal{T}g)+56(\varepsilon^{2}\overline{\partial}{}^{3}f)(\varepsilon^{6}\overline{\partial}{}^{5}\mathcal{T}g)+28(\varepsilon^{2}\overline{\partial}{}^{2}f)(\varepsilon^{6}\overline{\partial}{}^{6}\mathcal{T}g)+8(\overline{\partial}{}f)(\varepsilon^{8}\overline{\partial}{}^{7}\mathcal{T}g),$

whose $L^{2}(\Omega)$ norm is controlled by

$\displaystyle\|\varepsilon^{8}\overline{\partial}{}^{8}f\|_{0}\|\mathcal{T}g\|_{L^{\infty}}+8\varepsilon^{2}\|\varepsilon^{6}\overline{\partial}{}^{7}f\|_{0}\|\overline{\partial}{}\mathcal{T}g\|_{L^{\infty}}+28\|\varepsilon^{6}\overline{\partial}{}^{6}f\|_{L^{6}}\|\varepsilon^{2}\overline{\partial}{}^{2}\mathcal{T}g\|_{L^{3}}+56\|\varepsilon^{6}\overline{\partial}{}^{5}f\|_{L^{6}}\|\varepsilon^{2}\overline{\partial}{}^{3}\mathcal{T}g\|_{L^{3}}$
$\displaystyle+$
$\displaystyle~{}70\|\varepsilon^{2}\overline{\partial}{}^{4}f\|_{L^{3}}\|\varepsilon^{6}\overline{\partial}{}^{4}\mathcal{T}g\|_{L^{6}}+56\|\varepsilon^{6}\overline{\partial}{}^{3}f\|_{L^{3}}\|\varepsilon^{2}\overline{\partial}{}^{5}\mathcal{T}g\|_{L^{6}}+28\|\varepsilon^{6}\overline{\partial}{}^{6}g\|_{L^{6}}\|\varepsilon^{2}\overline{\partial}{}^{2}\mathcal{T}f\|_{L^{3}}+8\varepsilon^{2}\|\varepsilon^{6}\overline{\partial}{}^{7}g\|_{0}\|\overline{\partial}{}\mathcal{T}f\|_{L^{\infty}}$
$\displaystyle\lesssim$
$\displaystyle~{}(1+\varepsilon^{2})\left(\sqrt{E_{8}^{\kappa}(t)E_{4}^{\kappa}(t})+\sqrt{E_{7}^{\kappa}(t)E_{4}^{\kappa}(t)}+\sqrt{E_{7}^{\kappa}(t)E_{5}^{\kappa}(t)}\right),$

where we use the Sobolev embedding $H^{1}\hookrightarrow L^{6}$ and
$H^{1}\hookrightarrow H^{1/2}\hookrightarrow L^{3}$ in 3D. In 2D case, we can
replace $(L^{6},L^{3})$ by $(L^{4},L^{4})$ and use Ladyženskaya’s inequality
$\|f\|_{L^{4}}^{2}\leq\|f\|_{L^{2}}\|\partial f\|_{L^{2}}\leq\|f\|_{1}^{2}$ to
obtain the same bound.

#### Commutator
$\varepsilon^{8}[\overline{\partial}{}^{8},b]\cdot\nabla^{\varphi}f$ for
$f=b,v$

This term appears when we commute $\mathcal{T}^{\gamma}$ with
$(b\cdot\nabla^{\varphi})$. Note that we can rewrite the directional
derivative to be
$(b\cdot\nabla^{\varphi})=\bar{b}\cdot\overline{\nabla}+(\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N})\partial_{3}$.
When commuting $\overline{\partial}{}^{8}$ with
$\bar{b}\cdot\overline{\nabla}$, the estimate is exactly the same as
$\varepsilon^{8}[\overline{\partial}{}^{8},f]\mathcal{T}g$. For the commutator
$[\overline{\partial}{}^{8},(\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N})]\partial_{3}f$,
we just need to put extra effort on the term
$8\overline{\partial}{}((\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N}))\overline{\partial}{}^{7}\partial_{3}f$
because the length of the multi-index exceeds 8 when $|x_{3}|\lesssim 1$.
(Recall that the weight function $\omega(x_{3})$ is comparable to $|x_{3}|$
when $x_{3}\lesssim 1$ and is comparable to 1 when $|x_{3}|\gg 1$.) In this
case, we notice that $b\cdot\mathbf{N}|_{\Sigma}=0$, and thus its interior
value can be expressed via the fundamental theorem of calculus

$(\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N})(x^{\prime},x_{3})=0+\int_{0}^{x_{3}}\partial_{3}\left((\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N})(x^{\prime},\xi_{3})\right)\mathrm{d}\xi_{3},$

whose $L^{\infty}(\Omega)$ norm is controlled by
$C\omega(x_{3})\|\partial_{3}(b\cdot\mathbf{N})\|_{L^{\infty}(\Omega)}$.

#### Commutator $\mathfrak{D}(f)$ for $f=v,~{}p,~{}b,~{}S$

Among all terms in (3.2), we need to further analyze the third term, that is,
the commutator
$\varepsilon^{8}\left[\overline{\partial}{}^{8},\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi),\partial_{3}f\right]$
for $f=v,b,p$. The problem is the same as above, that is,
$\overline{\partial}{}^{7}$ may fall on $\partial_{3}f$ which is not directly
controllable. Again, we notice that there is only one $\overline{\partial}{}$
falling on
$\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi)$ and
$(v\cdot\mathbf{N}-\partial_{t}\varphi)|_{\Sigma}=0$, so we can use the same
method (as in the control of
$\varepsilon^{8}[\overline{\partial}{}^{8},b]\cdot\nabla^{\varphi}f$) to
control this commutator.

#### Commutator $\mathfrak{C}(q)$

The problematic term is
$-8(\partial_{3}\varphi)^{-1}(\overline{\partial}{}\mathbf{N}_{i})(\overline{\partial}{}^{7}\partial_{3}q)$
arising from
$[\mathcal{T}^{\gamma},\mathbf{N}_{i}/\partial_{3}\varphi,\partial_{3}q]$. To
control this term, we can invoke the third component of the momentum equation
to convert $\partial_{3}q$ to tangential derivatives of other quantities

$-\partial_{3}q=(\partial_{3}\varphi)\left(\rho
D_{t}^{\varphi}v_{3}-(b\cdot\nabla^{\varphi})b_{3}\right),$

where
$D_{t}^{\varphi}=\partial_{t}+\bar{v}\cdot\overline{\nabla}+(\partial_{3}\varphi)^{-1}(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}$
and
$(b\cdot\nabla^{\varphi})=\bar{b}\cdot\overline{\nabla}+(\partial_{3}\varphi)^{-1}(b\cdot\mathbf{N})\partial_{3}$
are both tangential derivatives. Also, there is no loss of weight of Mach
number in this term because one can always replace $\overline{\partial}{}q$ by
$D_{t}^{\varphi}v$ and $(b\cdot\nabla^{\varphi})b$.

#### Commutator $\mathfrak{C}_{i}(v_{i})$

The problematic term is
$-8(\partial_{3}\varphi)^{-1}(\overline{\partial}{}\mathbf{N}_{i})(\overline{\partial}{}^{7}\partial_{3}v_{i})$
arising from
$[\mathcal{T}^{\gamma},\mathbf{N}_{i}/\partial_{3}\varphi,\partial_{3}v_{i}]$.
In fact, this term may not be controlled independently, but its contribution
only appears in
$-\int_{\Omega}\mathbf{Q}\mathfrak{C}_{i}(v_{i})\,\mathrm{d}\mathcal{V}_{t}$
which has been analyzed in step 4 of Section 3.3.1. Specifically, its
contribution in the term $Z$, after combining it with $ZB$ term, is

$8\varepsilon^{16}\int_{\Omega}\partial_{3}(\overline{\partial}{}^{8}q-\overline{\partial}{}^{8}\varphi\partial^{\varphi}_{3}q)~{}\overline{\partial}{}\mathbf{N}_{i}~{}\overline{\partial}{}^{7}v_{i}\,\mathrm{d}x,$

which is controlled by
$(\|\varepsilon^{8}\overline{\partial}{}^{7}\partial_{3}q\|_{0}+|\varepsilon^{8}\overline{\partial}{}^{8}\psi|_{0}\|\partial
q\|_{L^{\infty}})|\overline{\partial}{}\psi|_{W^{1,\infty}}\|\varepsilon^{8}\overline{\partial}{}^{8}v\|_{0}$
after integrating one $\overline{\partial}{}$ by parts. Then we convert
$\partial_{3}q$ to tangential derivatives of other quantities via the momentum
equation, which has been presented in the control of $\mathfrak{C}(q)$.

Based on the above analysis, we can follow the same method as in
$\overline{\partial}{}^{4}$-estimate to prove the following inequality for
$\varepsilon^{2l}\overline{\partial}{}^{4+l}$-estimates $(1\leq l\leq 4)$ for
the nonlinear $\kappa$-approximate problem (3.1): For any $0<\delta<1$ and
fixed $l\in\\{1,2,3,4\\}$.

$\displaystyle\left\|\varepsilon^{2l}\left(\mathbf{V}^{\gamma,\pm},\mathbf{B}^{\gamma,\pm},\mathbf{S}^{\gamma,\pm},\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\gamma,\pm}\right)(t)\right\|_{0,\pm}^{2}+\left|\sqrt{\sigma}\varepsilon^{2l}\overline{\partial}{}^{4+l}\overline{\nabla}\psi(t)\right|_{0}^{2}+\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\psi(t)\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}\overline{\partial}{}^{4+l}\partial_{t}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
(3.56) $\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\left|\varepsilon^{2l}\psi_{0}\right|_{5.5+l}^{2}+\sum_{j=0}^{l}\int_{0}^{t}P(\sigma^{-1},E_{4+j}^{\kappa}(\tau))\,\mathrm{d}\tau,$

where
$\left(\mathbf{V}^{\gamma,\pm},\mathbf{B}^{\gamma,\pm},\mathbf{S}^{\gamma,\pm},\mathbf{P}^{\gamma,\pm}\right)$
represent that Alinhac good unknowns of $(v^{\pm},b^{\pm},S^{\pm},p^{\pm})$
with respect to $\overline{\partial}{}^{4+l}$.

###  3.4 Tangential estimates: full time derivatives

Now we control the full time derivatives, that is, the
$\varepsilon^{2l}\partial_{t}^{4+l}$ estimates for $0\leq l\leq 4$. We will
take the most difficult case $l=4$ for an example, that is, the
$\varepsilon^{8}\partial_{t}^{8}$-estimate. The other cases $(0\leq l\leq 3)$
can be treated in the same way.

####  3.4.1 Replacing one time derivative by a material derivative

Following the analysis in Section 3.3.1 and Section 3.3.2, we expect to
control the following norms

$\left\|\varepsilon^{8}\left(\mathbf{V}^{\pm},\mathbf{B}^{\pm},\sqrt{\mathcal{F}_{p}^{\pm}}\mathbf{P}^{\pm},\mathbf{S}^{\pm}\right)\right\|_{0,\pm}^{2}+\left|\varepsilon^{8}\sqrt{\sigma}\partial_{t}^{8}\psi\right|_{1}^{2}+\left|\varepsilon^{8}\sqrt{\kappa}\partial_{t}^{8}\psi\right|_{2}^{2}+\left|\varepsilon^{8}\sqrt{\kappa}\partial_{t}^{9}\psi\right|_{L_{t}^{2}H_{x^{\prime}}^{1}}^{2},$

which further gives the control of
$\left\|\varepsilon^{8}\partial_{t}^{8}(v^{\pm},b^{\pm},\sqrt{\mathcal{F}_{p}^{\pm}}p^{\pm},S^{\pm})\right\|_{0}^{2}$.
However, there are several extra difficulties that may make our previous
method invalid.

  * a.

We cannot substitute $\partial q$ by $\mathcal{T}(v,b)$ because there is no
spatial derivative.

  * b.

$\partial_{t}^{4+l}p$ has weight
$\sqrt{\mathcal{F}_{p}^{\pm}}\varepsilon^{2l}=O(\varepsilon^{1+2l})$ instead
of $\varepsilon^{2l}$. There might be a loss of $\varepsilon$-weight.

  * c.

$\sqrt{\mathcal{F}_{p}^{\pm}}\varepsilon^{2l}\partial_{t}^{4+l}q$ only has
$L^{2}(\Omega)$ regularity, so the trace lemma is no longer valid.

  * d.

We cannot integrate by parts for “half-order time derivative”
$\partial_{t}^{1/2}$. Thus, the control of VS term will be rather different.

To overcome the abovementioned difficulties, especially (c) and (d) in the
control of the crucial boundary term VS, we would like to replace the full-
time derivative $\partial_{t}^{4+l}$ by $D_{t}^{\varphi,-}\partial_{t}^{3+l}$
where
$D_{t}^{\varphi,-}=\partial_{t}+\bar{v}^{-}\cdot\overline{\nabla}+(\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}$
and $v^{-}|_{\Omega^{+}}$ is defined to be the Sobolev extension of $v^{-}$ in
$\Omega^{+}$. We aim to prove the following estimates.

######  Proposition 3.6.

Fix $l\in\\{0,1,2,3,4\\}$. For
$\varepsilon^{2l}D_{t}^{\varphi,-}\partial_{t}^{3+l}$-differentiated
approximate system $(0\leq l\leq 4)$, we have the following uniform-
in-$(\kappa,\varepsilon)$ estimate for any $0<\delta<1$

$\displaystyle\left\|\varepsilon^{2l}\left(\mathbf{V}^{*,\gamma,\pm},\mathbf{B}^{*,\gamma,\pm},\mathbf{S}^{*,\gamma,\pm},(\mathcal{F}_{p}^{\pm})^{1/2}\mathbf{P}^{*,\gamma,\pm}\right)(t)\right\|_{0}^{2}+\left|\sqrt{\sigma}\varepsilon^{2l}D_{t}^{\varphi,-}\partial_{t}^{3+l}\overline{\nabla}\psi(t)\right|_{0}^{2}$
(3.57)
$\displaystyle+\left|\sqrt{\kappa}\varepsilon^{2l}D_{t}^{\varphi,-}\partial_{t}^{3+l}\psi(t)\right|_{2}^{2}+\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{2l}D_{t}^{\varphi,-}\partial_{t}^{3+l}\partial_{t}\psi(\tau)\right|_{1}^{2}\,\mathrm{d}\tau$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{4+l}^{\kappa}(t)+\sum_{j=0}^{l}P(E_{4+j}^{\kappa}(0))+\int_{0}^{t}P(\sigma^{-1},E_{4+j}^{\kappa}(\tau))\,\mathrm{d}\tau,~{}~{}~{}0\leq
l\leq 4,$

where
$\left(\mathbf{V}^{*,\gamma,\pm},\mathbf{B}^{*,\gamma,\pm},\mathbf{S}^{*,\gamma,\pm},(\mathcal{F}_{p}^{\pm})^{1/2}\mathbf{P}^{*,\gamma,\pm}\right)$
represent the Alinhac good unknowns of $(v^{\pm},b^{\pm},S^{\pm},p^{\pm})$
with respect to $D_{t}^{\varphi,-}\partial_{t}^{3+l}$, that is,
$\mathbf{F}^{*,\gamma,\pm}=D_{t}^{\varphi,-}\partial_{t}^{3+l}f^{\pm}-(D_{t}^{\varphi,-}\partial_{t}^{3+l}\varphi)\partial^{\varphi}_{3}f^{\pm}$.

For the case $l=4$, we introduce the Alinhac good unknowns with respect to
$D_{t}^{\varphi,-}\partial_{t}^{7}$

$(\mathbf{V}^{*,\pm},\mathbf{B}^{*,\pm},\mathbf{P}^{*,\pm},\mathbf{Q}^{*,\pm},\mathbf{S}^{*,\pm}):=D_{t}^{\varphi,-}\partial_{t}^{7}(v^{\pm},b^{\pm},p^{\pm},q^{\pm},S^{\pm})-(D_{t}^{\varphi,-}\partial_{t}^{7}\varphi)\partial^{\varphi}_{3}(v^{\pm},b^{\pm},p^{\pm},q^{\pm},S^{\pm}).$

They satisfy

$D_{t}^{\varphi,-}\partial_{t}^{7}\partial^{\varphi}_{i}f^{\pm}=\partial^{\varphi}_{i}\mathbf{F}^{*,\pm}+\mathfrak{C}_{i}^{*}(f^{\pm}),~{}~{}D_{t}^{\varphi,-}\partial_{t}^{7}D_{t}^{\varphi,-}f^{\pm}=D_{t}^{\varphi,-}\mathbf{F}^{*,\pm}+\mathfrak{D}_{i}^{*}(f^{\pm})f,$

where $\mathfrak{C}^{*}(f),\mathfrak{D}^{*}(f)$ are defined in the same way as
(3.2)-(3.2) by replacing $\mathcal{T}^{\gamma}$ with
$D_{t}^{\varphi,-}\partial_{t}^{7}$. The boundary conditions of these good
unknowns are

$\displaystyle\left\llbracket\mathbf{Q}^{*}\right\rrbracket=\sigma\overline{D_{t}^{-}}\partial_{t}^{7}\mathcal{H}-\kappa\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})^{2}\psi-\kappa\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\partial_{t}\psi-\left\llbracket\partial_{3}q\right\rrbracket\overline{D_{t}^{-}}\partial_{t}^{7}\psi$
(3.58) $\displaystyle\mathbf{V}^{*,\pm}\cdot
N=\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi+(\bar{v}^{\pm}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi-\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\psi-\mathcal{W}^{*,\pm},$
(3.59)

with

$\displaystyle\mathcal{W}^{*\pm}=(\partial_{3}v^{\pm}\cdot
N)\overline{D_{t}^{-}}\partial_{t}^{7}\psi+[D_{t}^{\varphi,-}\partial_{t}^{7},N_{i},v_{i}^{\pm}],$
(3.60)

where we use the fact that
$D_{t}^{\varphi\pm}|_{\Sigma}=\overline{D_{t}^{\pm}}=\partial_{t}+\bar{v}^{\pm}\cdot\overline{\nabla}$.
Note that $\overline{D_{t}^{-}}$ does not directly commute with $\partial_{t}$
or $\partial_{i}$, so there is an extra term
$-\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\psi$
in the expression of $\mathbf{V}^{\pm,*}\cdot N$.

####  3.4.2 Analysis of the interior commutators

Since we replaced $\partial_{t}^{8}$ with $D_{t}^{\varphi,-}\partial_{t}^{7}$
and $D_{t}^{\varphi,-}$ does not directly commute with $\partial_{3}$, we need
to further analyze the commutators $\mathfrak{C}_{i}(f)$ for $f=q$ and $v_{i}$
and $\mathfrak{D}(f)$ for $f=v,b,p,S$. The problematic thing is that
$\partial_{3}$ may fall on
$(\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)$ (in
$D_{t}^{\varphi,-}$) and produce a normal derivative without a weight function
that vanishes on $\Sigma$, which may introduce a second-order derivative in
the setting of anisotropic Sobolev space. This problem does not appear in
$\mathfrak{D}(f)$, as we find that such commutator has the form
$(\partial_{3}\varphi)^{-1}(v\cdot\mathbf{N}-\partial_{t}\varphi)[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]f$
which already includes a weight $(v\cdot\mathbf{N}-\partial_{t}\varphi)$ that
vanishes on $\Sigma$. In $\mathfrak{C}_{i}(f)$, according (3.2), we need to
further analyze the term
$\mathbf{N}_{i}(\partial_{3}\varphi)^{-1}[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]f$
for $f=q,v_{i}$. Using
$D_{t}^{\varphi,-}=\partial_{t}+\bar{v}^{-}\cdot\overline{\nabla}+(\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}$,
we have

$\displaystyle\mathbf{N}_{i}(\partial_{3}\varphi)^{-1}[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]f=\mathbf{N}_{i}(\partial_{3}\varphi)^{-1}[D_{t}^{\varphi,-},\partial_{3}]\partial_{t}^{7}f$
$\displaystyle=$
$\displaystyle-\mathbf{N}_{i}(\partial_{3}\varphi)^{-1}\partial_{3}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}f+\mathbf{N}_{i}\partial_{3}\left((\partial_{3}\varphi)^{-1}(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)\right)\partial^{\varphi}_{3}\partial_{t}^{7}f.$

The first term above can be directly controlled in $L^{2}$ because only
tangential derivative falls on $\partial_{t}^{7}f$. For the second term, we
can invoke the momentum equation and the continuity equation to convert this
normal derivative to a tangential derivative.

  * •

When $f=q$, we use $-\partial^{\varphi}_{3}q=\rho
D_{t}^{\varphi}v_{3}-(b\cdot\nabla^{\varphi})b_{3}$.

  * •

When $f=v_{i}$, using $\nabla^{\varphi}\cdot
v=\overline{\nabla}\cdot\bar{v}+\partial^{\varphi}_{3}v\cdot\mathbf{N}$, the
continuity equation becomes
$\partial^{\varphi}_{3}v\cdot\mathbf{N}=-\varepsilon^{2}D_{t}^{\varphi}p-\overline{\nabla}\cdot\bar{v}$.
Thus we have
$\partial_{t}^{7}\partial^{\varphi}_{3}v\cdot\mathbf{N}=-\partial_{t}^{7}(\varepsilon^{2}D_{t}^{\varphi}p+\overline{\nabla}\cdot\bar{v})+[\partial_{t}^{7},\mathbf{N}]\cdot\partial_{3}v$
in which both terms can be directly controlled in $\|\cdot\|_{8,*}$ norm.

Also note that there is no extra loss of Mach number even if
$\partial_{t}^{8}p$ requires one more $\varepsilon$-weight. In fact, the only
term in the commutators $\mathfrak{C},\mathfrak{D}$ that contains
$\partial_{t}^{8}p$ is $\mathcal{R}_{p}$, but there is an extra weight
$\mathcal{F}_{p}=O(\varepsilon^{2})$ multiplying on it. Therefore, we can
follow the same strategy presented in Section 3.3.1 and Section 3.3.2 to
analyze the interior part. We can prove the following energy identity

$\displaystyle\sum_{\pm}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\varepsilon^{16}}{2}\int_{\Omega^{\pm}}\rho^{\pm}|\mathbf{V}^{*,\pm}|^{2}+|\mathbf{B}^{*,\pm}|^{2}+\mathcal{F}_{p}^{\pm}(\mathbf{P}^{*,\pm})^{2}+\rho^{\pm}(\mathbf{S}^{*,\pm})^{2}\,\mathrm{d}\mathcal{V}_{t}$
(3.61) $\displaystyle=$ $\displaystyle\text{ ST}^{*}+{\text{
ST}^{*}}^{\prime}+\text{ VS}^{*}+\text{ RT}^{*}+\sum_{\pm}\text{
RT}^{*,\pm}+ZB^{*,\pm}+Z^{*,\pm}+R_{\Sigma}^{*,\pm}+R_{\Omega}^{*,\pm},$

where

$\displaystyle\text{ ST}^{*}:=$
$\displaystyle~{}\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}\left\llbracket
q\right\rrbracket\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.62) $\displaystyle{\text{ ST}^{*}}^{\prime}:=$
$\displaystyle~{}\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}\left\llbracket
q\right\rrbracket\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.63) $\displaystyle\text{ VS}^{*}:=$
$\displaystyle~{}\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.64) $\displaystyle\text{ RT}^{*}:=$
$\displaystyle-\varepsilon^{16}\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.65) $\displaystyle\text{ RT}^{*,\pm}:=$
$\displaystyle\mp\varepsilon^{16}\int_{\Sigma}\partial_{3}q^{\pm}\,\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,(\bar{v}^{\pm}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.66) $\displaystyle R_{\Sigma}^{*,\pm}:=$
$\displaystyle\pm\varepsilon^{16}\int_{\Sigma}\mathbf{Q}^{*,\pm}\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime},$
(3.67) $\displaystyle ZB^{*,\pm}:=$
$\displaystyle\mp\varepsilon^{16}\int_{\Sigma}\mathbf{Q}^{*,\pm}\mathcal{W}^{*,\pm}\,\mathrm{d}x^{\prime},~{}~{}Z^{*,\pm}=-\int_{\Omega^{\pm}}\varepsilon^{16}\mathbf{Q}^{\pm,*}\mathfrak{C}_{i}^{*}(v_{i}^{\pm})\,\mathrm{d}\mathcal{V}_{t},$
(3.68)

and $R_{\Omega}^{*,\pm}$ represents the controllable terms in the interior
containing the analogues of $R_{1}^{\pm},R_{2}^{\pm},R_{3}^{\pm}$.
Specifically, we have

$\displaystyle\varepsilon^{-16}R_{\Omega}^{*,\pm}=$
$\displaystyle\int_{\Omega^{\pm}}\mathbf{V}^{*,\pm}\cdot(\mathcal{R}_{v}^{*,\pm}-\mathfrak{C}^{*}(q^{\pm}))\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathcal{R}_{q}^{*,\pm}(\nabla^{\varphi}\cdot\mathbf{V}^{*,\pm})\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{B}^{*,\pm}\cdot\mathcal{R}_{b}^{*,\pm}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\mathbf{P}^{*,\pm}\mathcal{R}_{p}^{*,\pm}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle-\frac{1}{2}\int_{\Omega^{\pm}}(\nabla^{\varphi}\cdot
v^{\pm})|\mathbf{B}^{*,\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}-\frac{1}{2}\int_{\Omega^{\pm}}(D_{t}^{\varphi\pm}\mathcal{F}_{p}^{\pm}+\mathcal{F}_{p}^{\pm}\nabla^{\varphi}\cdot
v^{\pm})|\mathbf{P}^{*,\pm}|^{2}\,\mathrm{d}\mathcal{V}_{t}+\int_{\Omega^{\pm}}\rho^{\pm}\mathfrak{D}^{*}(S^{\pm})\,\mathbf{S}^{*,\pm}\,\mathrm{d}\mathcal{V}_{t}$
(3.69)

where

$\displaystyle\mathcal{R}_{v}^{*,\pm}:=$
$\displaystyle~{}[D_{t}^{\varphi,-}\partial_{t}^{7},b^{\pm}]\cdot\nabla^{\varphi}b^{\pm}-[D_{t}^{\varphi,-}\partial_{t}^{7},\rho^{\pm}]D_{t}^{\varphi\pm}v^{\pm}-\rho^{\pm}\mathfrak{D}^{*}(v^{\pm}),$
$\displaystyle\mathcal{R}_{p}^{*,\pm}:=$
$\displaystyle-[D_{t}^{\varphi,-}\partial_{t}^{7},\mathcal{F}_{p}^{\pm}]D_{t}^{\varphi\pm}p^{\pm}-\mathcal{F}_{p}^{\pm}\mathfrak{D}^{*}(p^{\pm}),$
$\displaystyle\mathcal{R}_{b}^{*,\pm}:=$
$\displaystyle~{}[D_{t}^{\varphi,-}\partial_{t}^{7},b^{\pm}]\cdot\nabla^{\varphi}v^{\pm}-\mathfrak{D}^{*}(b^{\pm}),\quad\mathcal{R}_{q}^{*,\pm}:=~{}\mathbf{Q}^{*,\pm}-\mathbf{P}^{*,\pm}-b^{\pm}\cdot\mathbf{B}^{*,\pm}.$

These terms can be directly controlled in the same way as presented in Section
3.3.1, so we omit the details

$\displaystyle\int_{0}^{t}R_{\Omega}^{*,\pm}\,\mathrm{d}\tau\lesssim
P(E^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(t))E_{8}^{\kappa}(t)\,\mathrm{d}\tau.$
(3.70)

####  3.4.3 Analysis of the boundary integrals

Similarly as in Section 3.3.1, we can decompose the control of these terms in
the following steps.

#### Step 1: Boundary regularity of full time derivatives given by surface
tension.

Invoking the boundary condition (3.58) for
$\left\llbracket\mathbf{Q}^{*}\right\rrbracket$, the term ST becomes

$\displaystyle\text{ ST}^{*}=$
$\displaystyle~{}\sigma\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\cdot\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
(3.71)
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})^{2}\psi\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\partial_{t}\psi\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle\text{ ST}_{0}^{*}+\text{
ST}_{1,\kappa}^{*}+\text{ ST}_{2,\kappa}^{*}.$

Commuting $\overline{\nabla}\cdot$ with $\overline{D_{t}^{-}}$, we have

$\text{
ST}_{0}^{*}=\sigma\varepsilon^{16}\int_{\Sigma}\overline{\nabla}\cdot\overline{D_{t}^{-}}\partial_{t}^{7}(\overline{\nabla}\psi/|N|)\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}+\underbrace{\sigma\varepsilon^{16}\int_{\Sigma}\overline{\partial}{}_{i}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\partial_{t}^{7}(\overline{\partial}{}_{i}\psi/|N|)\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}}_{\text{
ST}_{0}^{*,R}}$ (3.72)

Integrating $\overline{\nabla}\cdot$ by parts in the mean curvature term, we
get an analogous energy term contributed by surface tension as in Section
3.3.1

$\displaystyle~{}\sigma\varepsilon^{16}\int_{\Sigma}\overline{\nabla}\cdot\overline{D_{t}^{-}}\partial_{t}^{7}\left(\frac{\overline{\nabla}\psi}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\right)\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
(3.73) $\displaystyle=$
$\displaystyle-\sigma\varepsilon^{16}\int_{\Sigma}\frac{\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi}{|N|}\cdot\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi\,\mathrm{d}x^{\prime}+\sigma\varepsilon^{16}\int_{\Sigma}\frac{\overline{\nabla}\psi\cdot\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi}{|N|^{3}}\overline{\nabla}\psi\cdot\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle\underbrace{-\sigma\varepsilon^{16}\int_{\Sigma}\frac{\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\partial}{}_{i}\psi}{|N|}\cdot\partial_{t}(\overline{\partial}{}_{i}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\partial_{t}^{7}\psi)\,\mathrm{d}x^{\prime}-\sigma\varepsilon^{16}\int_{\Sigma}\frac{\overline{\nabla}\psi\cdot\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\partial}{}_{i}\psi}{|N|^{3}}\overline{\nabla}\psi\,\partial_{t}(\overline{\partial}{}_{i}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\partial_{t}^{7}\psi)\,\mathrm{d}x^{\prime}}_{=:\text{
ST}_{1}^{*,R}}$
$\displaystyle\underbrace{-\sigma\varepsilon^{16}\int_{\Sigma}\left(\left[\overline{D_{t}^{-}}\partial_{t}^{6},\frac{1}{|N|}\right]\partial_{t}\overline{\nabla}\psi+\left[\overline{D_{t}^{-}}\partial_{t}^{6},\frac{1}{|N|^{3}}\right]((\overline{\nabla}\psi\cdot\partial_{t}\overline{\nabla}\psi)\overline{\nabla}\psi)-\frac{1}{|N|^{3}}[\overline{D_{t}^{-}}\partial_{t}^{6},\overline{\nabla}\psi]\partial_{t}\overline{\nabla}\psi\right)\cdot\partial_{t}\overline{\nabla}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}}_{=:\text{
ST}_{2}^{*,R}}$

where the right side is further equal to

$\displaystyle-\frac{\sigma\varepsilon^{16}}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\frac{|\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi|^{2}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}-\frac{|\overline{\nabla}\psi\cdot\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\,\mathrm{d}x^{\prime}$
(3.74)
$\displaystyle+\underbrace{\frac{\sigma\varepsilon^{16}}{2}\int_{\Sigma}\partial_{t}\left(\frac{1}{|N|}\right)\left|\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi\right|^{2}-\partial_{t}\left(\frac{1}{|N|^{3}}\right)\left|\overline{\nabla}\psi\cdot\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi\right|^{2}\,\mathrm{d}x^{\prime}}_{=:\text{
ST}_{3}^{*,R}}+\text{ ST}_{1}^{*,R}+\text{ ST}_{2}^{*,R}.$

The first line above together with the inequality (3.37) gives the
$\sqrt{\sigma}$-weighted boundary regularity as in step 2 in Section 3.3.1.
The term$\text{ ST}_{1}^{*,R}$ is generated by commuting
$\overline{D_{t}^{-}}$ with $\overline{\nabla}$ (the one falling on
$\partial_{t}^{7}\psi$) and is directly controlled by the energy. The
term$\text{ ST}_{3}^{*,R}$ is controlled in the same way as$\text{
ST}_{2}^{R}$ in step 2 of Section 3.3. The term$\text{ ST}_{2}^{*,R}$ is
controlled by integrating $\partial_{t}$ by parts under time integral, which
was also analyzed in [55, Section 4.6]. The term$\text{ ST}_{0}^{*,R}$ is
controlled by integrating by parts in $\partial_{t}$ and then in
$\overline{\partial}{}_{j}$ under time integral (which is similar to$\text{
ST}_{2}^{*,R}$). Thus, we conclude their estimates by

$\int_{0}^{t}\text{ ST}_{0}^{*,R}+\text{ ST}_{1}^{*,R}+\text{
ST}_{2}^{*,R}+\text{ ST}_{3}^{*,R}\,\mathrm{d}\tau\lesssim
P(E^{\kappa}(0))+\int_{0}^{t}P(E^{\kappa}(\tau))\,\mathrm{d}\tau.$ (3.75)

Next we analyze the terms$\text{ ST}^{*}_{1,\kappa},\text{ ST}^{*}_{2,\kappa}$
involving the $\kappa$-regularization terms. Note that we have to commute
$\overline{D_{t}^{-}}$ with $1-\overline{\Delta}{}$ or
$\langle\overline{\partial}{}\rangle=\sqrt{1-\overline{\Delta}{}}$ when
deriving the $\sqrt{\kappa}$-weighte terms. Integrating
$1-\overline{\Delta}{}$ by parts in$\text{ ST}^{*}_{1,\kappa}$

$\displaystyle\text{
ST}_{1,\kappa}^{*}=-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})^{2}\psi\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
(3.76) $\displaystyle=$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}D_{t}^{\varphi,-}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\,\partial_{t}\left(\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}[\overline{D_{t}^{-}},1-\overline{\Delta}{}]\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\,\partial_{t}\left([1-\overline{\Delta}{},\overline{D_{t}^{-}}]\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle=:$
$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right|_{0}^{2}+\text{
ST}^{*,R}_{11,\kappa}+\text{ ST}^{*,R}_{12,\kappa}$

On $\Sigma$, the material derivative
$D_{t}^{\varphi,-}=\overline{D_{t}^{-}}=\partial_{t}+\bar{v}^{-}\cdot\overline{\nabla}$,
so the commutator is

$[\overline{D_{t}^{-}},1-\overline{\Delta}{}]f=[\overline{\Delta}{},\bar{v}^{-}\cdot\overline{\nabla}]f=\overline{\Delta}{}\bar{v}^{-}\cdot\overline{\nabla}f+2\overline{\partial}{}_{i}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\overline{\partial}{}_{i}f.$

Then$\text{ ST}^{*,R}_{11,\kappa}$ is controlled under time integral by
integrating $\overline{\partial}{}_{j}$ by parts in the second term

$\displaystyle\int_{0}^{t}\text{ ST}^{*,R}_{11,\kappa}\,\mathrm{d}\tau=$
$\displaystyle-\kappa\varepsilon^{16}\int_{0}^{t}\int_{\Sigma}\overline{\Delta}{}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle+2\kappa\varepsilon^{16}\int_{0}^{t}\int_{\Sigma}\overline{\partial}{}_{i}\bar{v}^{-}_{j}\overline{\partial}{}_{i}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\overline{\partial}{}_{j}\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau+\text{
lower order terms}$ $\displaystyle\lesssim$
$\displaystyle~{}\delta\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{8}\psi\right|_{1}^{2}+\frac{1}{4\delta}\int_{0}^{t}\left|\sqrt{\kappa}\varepsilon^{8}\overline{\partial}{}\partial_{t}^{7}\psi\right|_{2}^{2}|\overline{\partial}{}\bar{v}^{-}|_{W^{1,\infty}}^{2}\,\mathrm{d}\tau\leq\delta
E_{8}^{\kappa}(t)+\int_{0}^{t}P(E_{8}^{\kappa}(\tau),E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau.$
(3.77)

The control of$\text{ ST}^{*,R}_{12,\kappa}$ is easier because there is no
term containing 9 time derivatives of $\psi$. It is directly controlled by
using the $\sqrt{\kappa}$-weighted boundary energy obtained above.

$\text{
ST}^{*,R}_{12,\kappa}\lesssim\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right|_{0}\left(\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{8}\psi\right|_{2}+\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{2}\right)\sqrt{E_{4}^{\kappa}(t)}\leq
E_{8}^{\kappa}(t)\sqrt{E_{4}^{\kappa}(t)}.$

The control of$\text{ ST}_{2,\kappa}^{*}$ is similar to$\text{
ST}_{1,\kappa}^{*}$. Using
$\langle\overline{\partial}{}\rangle^{2}=1-\overline{\Delta}{}$, we have

$\displaystyle\text{ ST}_{2,\kappa}^{*}=$
$\displaystyle-\int_{\Sigma}\left|\sqrt{\kappa}\varepsilon^{8}D_{t}^{\varphi,-}\partial_{t}^{8}\langle\overline{\partial}{}\rangle\psi\right|^{2}\,\mathrm{d}x^{\prime}$
(3.78)
$\displaystyle+\kappa\varepsilon^{16}\int_{\Sigma}[\overline{D_{t}^{-}},\overline{\partial}{}_{i}]\left(\partial_{t}^{8}\overline{\partial}{}_{i}\psi\right)\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}+\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{8}\overline{\partial}{}_{i}\psi\,\left([\overline{\partial}{}_{i}\partial_{t},\overline{D_{t}^{-}}]\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle=:$
$\displaystyle-\int_{\Sigma}\left|\sqrt{\kappa}\varepsilon^{8}D_{t}^{\varphi,-}\partial_{t}^{8}\langle\overline{\partial}{}\rangle\psi\right|^{2}\,\mathrm{d}x^{\prime}+\text{
ST}^{*,R}_{21,\kappa}+\text{ ST}^{*,R}_{22,\kappa},$

where we use the concrete form of the commutators

$[\overline{D_{t}^{-}},\overline{\partial}{}_{i}]=-\overline{\partial}{}_{i}\bar{v}_{j}^{-}\overline{\partial}{}_{j}f,~{}~{}[\overline{\partial}{}_{i}\partial_{t},\overline{D_{t}^{-}}]f=\partial_{t}\left(\overline{\partial}{}_{i}\bar{v}_{j}^{-}\overline{\partial}{}_{j}\overline{D_{t}^{-}}f\right)+\partial_{t}\bar{v}_{j}^{-}\overline{\partial}{}_{j}\overline{\partial}{}_{i}f$

to get estimates similar to$\text{ ST}^{*,R}_{11,\kappa}$ and$\text{
ST}^{*,R}_{12,\kappa}$

$\int_{0}^{t}\text{ ST}^{*,R}_{21,\kappa}+\text{
ST}^{*,R}_{22,\kappa}\,\mathrm{d}\tau\lesssim~{}\delta
E_{8}^{\kappa}(t)+\int_{0}^{t}P(E_{8}^{\kappa}(\tau),E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau.$

Hence, the control of$\text{ ST}^{*}$ in (3.61) is concluded by

$\displaystyle\int_{0}^{t}\text{
ST}\,\mathrm{d}\tau+\frac{1}{2}\int_{\Sigma}\frac{|\sqrt{\sigma}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{7}\overline{\nabla}\psi|^{2}}{{\sqrt{1+|\overline{\nabla}\psi|^{2}}}^{3}}\,\mathrm{d}x^{\prime}$
(3.79)
$\displaystyle+\int_{\Sigma}\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right|^{2}\,\mathrm{d}x^{\prime}+\int_{0}^{t}\int_{\Sigma}\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{8}\langle\overline{\partial}{}\rangle\psi\right|^{2}\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle~{}\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(E^{\kappa}(\tau))\,\mathrm{d}\tau.$

The term${\text{ ST}^{*}}^{\prime}$ is controlled in the same way as$\text{
ST}^{*}$ by replacing $\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi$
with
$(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi$.
We no longer get energy terms, but we can integrate
$(\bar{v}^{+}\cdot\overline{\nabla})$ by parts and use symmetry and the above
boundary regularity to control them. Invoking the jump condition, we have

$\displaystyle{\text{ ST}^{*}}^{\prime}=$
$\displaystyle~{}\sigma\varepsilon^{16}\overline{D_{t}^{-}}\partial_{t}^{7}\mathcal{H}\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})^{2}\psi\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\partial_{t}\psi\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle{\text{ ST}_{0}^{*}}^{\prime}+{\text{
ST}_{1,\kappa}^{*}}^{\prime}+{\text{ ST}_{2,\kappa}^{*}}^{\prime}.$ (3.80)

Following the analysis (3.72)-(3.75), the first term is controlled thanks to
the boundary regularity and symmetric structure after integrating
$(\bar{v}^{+}\cdot\overline{\nabla})$ by parts.

$\displaystyle{\text{ ST}_{0}^{*}}^{\prime}\stackrel{{\scriptstyle
L}}{{=}}\frac{1}{2}\sigma\varepsilon^{16}\int_{\Sigma}(\overline{\nabla}\cdot\bar{v}^{+})\left(\frac{|\overline{D_{t}^{-}}\partial_{t}^{7}\psi|^{2}}{|N|}-\frac{|\overline{\nabla}\psi\cdot\overline{\nabla}\overline{D_{t}^{-}}\partial_{t}^{7}\psi|^{2}}{|N|^{3}}\right)\,\mathrm{d}x^{\prime}\leq
P(|\overline{\nabla}\psi|_{L^{\infty}})|\bar{v}^{+}|_{W^{1,\infty}}\left|\sqrt{\sigma}\varepsilon^{8}\overline{\nabla}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right|_{0}^{2}.$
(3.81)

Similarly, we can use the symmetric structure to control${\text{
ST}_{1,\kappa}^{*}}^{\prime}+{\text{ ST}_{2,\kappa}^{*}}^{\prime}$. We only
check the commutators arising in the control of${\text{
ST}_{1,\kappa}^{*}}^{\prime}$ as an example.

$\displaystyle{\text{ ST}_{1,\kappa}^{*,R}}^{\prime}:=$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}[\overline{D_{t}^{-}},1-\overline{\Delta}{}]\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,(\bar{v}^{+}\cdot\overline{\nabla})\left(\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}\left(\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,(\bar{v}^{+}\cdot\overline{\nabla})\left([1-\overline{\Delta}{},\overline{D_{t}^{-}}]\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,[1-\overline{\Delta}{},(\bar{v}^{+}\cdot\overline{\nabla})]\left(\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle{\text{
ST}_{11,\kappa}^{*,R}}^{\prime}+{\text{
ST}_{12,\kappa}^{*,R}}^{\prime}+{\text{ ST}_{13,\kappa}^{*,R}}^{\prime}.$
(3.82)

The control of ${\text{ ST}_{11,\kappa}^{*,R}}^{\prime}+{\text{
ST}_{12,\kappa}^{*,R}}^{\prime}$ is similar to ${\text{
ST}_{11,\kappa}^{*,R}}+{\text{ ST}_{12,\kappa}^{*,R}}$. We have

$\displaystyle{\text{ ST}_{11,\kappa}^{*,R}}^{\prime}\stackrel{{\scriptstyle
L}}{{=}}$
$\displaystyle-\kappa\varepsilon^{16}\int_{\Sigma}\overline{\Delta}{}\bar{v}^{-}_{j}\overline{\partial}{}_{j}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\,(\bar{v}^{+}\cdot\overline{\nabla})\left(\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle+2\kappa\varepsilon^{16}\int_{\Sigma}\overline{\partial}{}_{j}\bar{v}^{-}_{j}\overline{\partial}{}_{i}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\overline{\partial}{}_{j}(\bar{v}^{+}\cdot\overline{\nabla})\left(\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\lesssim$
$\displaystyle~{}|\bar{v}^{-}|_{W^{2,\infty}}|\bar{v}^{+}|_{L^{\infty}}\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{3}\left(\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{3}+\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{8}\psi\right|_{2}\right)\lesssim
E_{4}^{\kappa}(t)E_{8}^{\kappa}(t),$ (3.83)

and

$\displaystyle{\text{
ST}_{12,\kappa}^{*,R}}^{\prime}\lesssim~{}|\bar{v}^{-}|_{W^{2,\infty}}|\bar{v}^{+}|_{L^{\infty}}\left(\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{8}\psi\right|_{2}+\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{3}\right)^{2}\lesssim
E_{4}^{\kappa}(t)E_{8}^{\kappa}(t).$ (3.84)

The extra term${\text{ ST}_{13,\kappa}^{*,R}}^{\prime}$ is also directly
controlled

$\displaystyle{\text{ ST}_{13,\kappa}^{*,R}}^{\prime}=$
$\displaystyle~{}\kappa\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\left(\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right)\,\left(\overline{\Delta}{}\bar{v}_{j}^{+}\overline{\partial}{}_{j}+2\overline{\partial}{}_{i}\bar{v}_{j}^{+}\overline{\partial}{}_{j}\overline{\partial}{}_{i}\right)\left(\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\lesssim$
$\displaystyle~{}|\bar{v}^{+}|_{W^{2,\infty}}|\bar{v}^{-}|_{L^{\infty}}\left(\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{8}\psi\right|_{2}+\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{3}\right)^{2}\lesssim
E_{4}^{\kappa}(t)E_{8}^{\kappa}(t).$ (3.85)

Thus we have

$\displaystyle{\text{
ST}_{1,\kappa}^{*}}^{\prime}\lesssim\frac{1}{2}\int_{\Sigma}(\overline{\nabla}\cdot\bar{v}^{+})\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{7}(1-\overline{\Delta}{})\psi\right|^{2}+E_{4}^{\kappa}(t)E_{8}^{\kappa}(t).$
(3.86)

Similarly, we have

$\displaystyle\int_{0}^{t}{\text{
ST}_{2,\kappa}^{*}}^{\prime}\,\mathrm{d}\tau=$
$\displaystyle~{}\kappa\varepsilon^{16}\int_{0}^{t}\int_{\Sigma}[\overline{D_{t}^{-}},\overline{\partial}{}_{i}](\partial_{t}^{8}\overline{\partial}{}_{i}\psi)\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\varepsilon^{16}\int_{0}^{t}\int_{\Sigma}(\overline{D_{t}^{-}}\partial_{t}^{8}\overline{\partial}{}_{i}\psi)\,\overline{\partial}{}_{i}\left((\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right)\,\mathrm{d}x^{\prime}\,\mathrm{d}\tau$
$\displaystyle-\kappa\varepsilon^{16}\int_{0}^{t}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}\partial_{t}\psi\,(\bar{v}^{+}\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle\lesssim$
$\displaystyle~{}\delta\left|\sqrt{\kappa}\varepsilon^{8}\overline{D_{t}^{-}}\partial_{t}^{8}\langle\overline{\partial}{}\rangle\psi\right|_{0}^{2}+\int_{0}^{t}|\bar{v}^{\pm}|_{W^{1,\infty}}^{2}\left(\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{7}\psi\right|_{3}^{2}+\left|\sqrt{\kappa}\varepsilon^{8}\partial_{t}^{8}\psi\right|_{2}^{2}\right)\,\mathrm{d}\tau.$
(3.87)

Hence, we have the estimate of ${\text{ ST}^{*}}^{\prime}$:

$\displaystyle\int_{0}^{t}{\text{
ST}^{*}}^{\prime}\,\mathrm{d}\tau\lesssim\delta
E_{8}^{\kappa}(t)+\int_{0}^{t}E_{8}^{\kappa}(\tau)E_{4}^{\kappa}(\tau)\,\mathrm{d}\tau.$
(3.88)

What’s more, we can also control the remainder term
$R_{\Sigma}^{*,\pm}:=\pm\varepsilon^{16}\int_{\Sigma}\mathbf{Q}^{*,\pm}\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}.$
Indeed, we use Gauss-Green formula to write it to be an interior intergral.

$\displaystyle R_{\Sigma}^{*,\pm}\stackrel{{\scriptstyle
L}}{{=}}-\varepsilon^{16}\int_{\Omega^{\pm}}\partial_{3}\mathbf{Q}^{*,\pm}\,\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\varphi\,\mathrm{d}x$
(3.89)

Recall that
$\mathbf{Q}^{*,\pm}=D_{t}^{\varphi,-}\partial_{t}^{7}q^{\pm}-D_{t}^{\varphi,-}\partial_{t}^{7}\psi\partial_{3}q^{\pm}$.
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 $\partial q$ to a tangential derivative of $v,b$. We
now integrate by parts $\overline{D_{t}^{-}}$ to get

$\displaystyle\int_{0}^{t}R_{\Sigma}^{*,\pm}\,\mathrm{d}\tau\stackrel{{\scriptstyle
L}}{{=}}$
$\displaystyle~{}\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{\pm}}\partial_{3}(\partial_{t}^{7}q^{\pm}-\partial_{t}^{7}\varphi\partial_{3}q^{\pm})\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot
D_{t}^{\varphi,-}\overline{\nabla}\partial_{t}^{7}\varphi\,\mathrm{d}x\,\mathrm{d}\tau$
$\displaystyle-\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{\pm}}\partial_{3}(\partial_{t}^{7}q^{\pm}-\partial_{t}^{7}\varphi\partial_{3}q^{\pm})\overline{D_{t}^{\pm}}\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\varphi\,\mathrm{d}x.$
(3.90)

Using the reduction for $\partial_{3}q$ again, we can control the above
integral by

$\displaystyle\int_{0}^{t}R_{\Sigma}^{*,\pm}\,\mathrm{d}\tau\lesssim\delta\|\varepsilon^{8}\partial_{3}\partial_{t}^{7}q^{\pm}\|_{0,\pm}^{2}+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau\lesssim\delta
E_{8}^{\kappa}(t)+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau.$
(3.91)

#### Step 2: Control of VS term.

Now we start to analyze the most difficult boundary term

$\displaystyle\text{
VS}^{*}:=~{}\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}.$
(3.92)

Note that there is no spatial derivative $\overline{\partial}{}$ in$\text{
VS}^{*}$, so we cannot integrate $\overline{\partial}{}^{1/2}$ by parts as in
step 3 in Section 3.3.1. To overcome this difficulty, we try to rewrite the
term $\overline{D_{t}^{-}}\partial_{t}^{7}\psi$ by invoking the kinematic
boundary condition

$\displaystyle\overline{D_{t}^{-}}\partial_{t}^{7}\psi=$
$\displaystyle~{}\partial_{t}^{8}\psi+\bar{v}^{-}\cdot\overline{\nabla}\partial_{t}^{7}\psi=\partial_{t}^{7}(v^{-}\cdot
N)-v^{-}\cdot\partial_{t}^{7}N$ $\displaystyle=$
$\displaystyle~{}\partial_{t}^{7}v^{-}\cdot
N+[\partial_{t}^{7},N_{i},v_{i}^{-}],$

and thus

$\displaystyle\text{ VS}^{*}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}\cdot
N\,\mathrm{d}x^{\prime}$
$\displaystyle+\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{-}\,\partial_{t}^{7}v^{-}\cdot(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})N\,\mathrm{d}x^{\prime}+\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})[\partial_{t}^{7},N_{i},v_{i}^{-}]\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle\text{ VS}^{*}_{0}+\text{
VS}^{*,ZB}_{1}+\text{ VS}^{*,ZB}_{2}$ (3.93)

Using divergence theorem, we convert$\text{ VS}^{*}_{0}$ to an interior
integral in $\Omega^{-}$

$\displaystyle\text{ VS}_{0}^{*}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,\nabla^{\varphi}\cdot\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}\right)\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$ $\displaystyle:\text{ VS}^{*}_{01}+\text{ VS}^{*}_{02},$
(3.94)

where $\left\llbracket\bar{v}\right\rrbracket=\bar{v}^{+}-\bar{v}^{-}$ is
defined via Sobolev extension in $\Omega^{-}$. In$\text{ VS}^{*}_{01}$, we
want to commute $\nabla^{\varphi}\cdot$ with
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ in order to
get a similar cancellation structure as in $ZB+Z$. The commutator is

$\displaystyle[\partial^{\varphi}_{i},\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}]f=$
$\displaystyle~{}\overline{\partial}{}_{i}(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}f)-\frac{\overline{\partial}{}_{i}\varphi}{\partial_{3}\varphi}\partial_{3}(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}f)-(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left(\overline{\partial}{}_{i}f-\frac{\overline{\partial}{}_{i}\varphi}{\partial_{3}\varphi}\partial_{3}f\right)$
$\displaystyle=$
$\displaystyle~{}(\overline{\partial}{}_{i}\left\llbracket\bar{v}\right\rrbracket_{j})(\overline{\partial}{}_{j}f)-\frac{\overline{\partial}{}_{i}\varphi}{\partial_{3}\varphi}(\partial_{3}\left\llbracket\bar{v}\right\rrbracket_{j})(\overline{\partial}{}_{j}f)+\left\llbracket\bar{v}\right\rrbracket_{j}\overline{\partial}{}_{j}\left(\frac{\overline{\partial}{}_{i}\varphi}{\partial_{3}\varphi}\right)\partial_{3}f$
$\displaystyle=$
$\displaystyle~{}\partial^{\varphi}_{i}\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}f-(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}_{i}\partial^{\varphi}_{3}f+\mathbf{N}_{i}\frac{(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{3}\varphi}{\partial_{3}\varphi}\partial^{\varphi}_{3}f,~{}~{}i=1,2,$
$\displaystyle[\partial^{\varphi}_{3},\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}]f=$
$\displaystyle~{}\partial^{\varphi}_{3}\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}f+\frac{(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{3}\varphi}{\partial_{3}\varphi}\partial^{\varphi}_{3}f.$

Commuting $\nabla^{\varphi}\cdot$ with
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$, we get

$\displaystyle\text{ VS}^{*}_{01}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,\nabla^{\varphi}\cdot\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}\right)\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left(\nabla^{\varphi}\cdot\partial_{t}^{7}v^{-}\right)\,\mathrm{d}\mathcal{V}_{t}-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,\partial_{3}\partial_{t}^{7}v^{-}\cdot(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\,\mathrm{d}x$
$\displaystyle+\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,\partial^{\varphi}_{i}\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{3}\varphi\,\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N}\,\mathrm{d}x$
$\displaystyle=:$ $\displaystyle\text{ VS}^{*}_{011}+\text{
VS}^{*,Z}_{011}+\text{ VS}^{*,R}_{011}+\text{ VS}^{*,R}_{012}$ (3.95)

Next we introduce
$\mathbf{F}^{\sharp}:=\partial_{t}^{7}f-\partial_{t}^{7}\psi\partial^{\varphi}_{3}f$
to be the Alinhac good unknown of $f$ with respect to $\partial_{t}^{7}$ in
order to commute $\nabla^{\varphi}$ with $\partial_{t}^{7}$. Namely, we have

$\partial_{t}^{7}\partial^{\varphi}_{i}f=\partial^{\varphi}_{i}\mathbf{F}^{\sharp}+\mathfrak{C}_{i}^{\sharp}(f),~{}~{}\partial_{t}^{7}D_{t}^{\varphi}f=D_{t}^{\varphi}\mathbf{F}^{\sharp}+\mathfrak{D}^{\sharp}(f),$

where $\mathfrak{C}^{\sharp},\mathfrak{D}^{\sharp}$ are defined in the same
way as (3.2)-(3.2) with $\mathcal{T}^{\gamma}=\partial_{t}^{7}$. With this
formulation, we have

$\displaystyle\nabla^{\varphi}\cdot\partial_{t}^{7}v^{-}=\nabla^{\varphi}\cdot\mathbf{V}^{\sharp,-}+\partial^{\varphi}_{i}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}^{-})=\partial_{t}^{7}(\nabla^{\varphi}\cdot
v^{-})-\mathfrak{C}_{i}^{\sharp}(v_{i}^{-})+\partial^{\varphi}_{i}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}^{-}).$

Now we insert the good unknowns in$\text{ VS}^{*}_{011}$ to get

$\displaystyle\text{ VS}_{011}^{*}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}\underbrace{-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left(\mathfrak{C}_{i}^{\sharp}(v_{i}^{-})-\partial^{\varphi}_{i}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}^{-})\right)\,\mathrm{d}\mathcal{V}_{t}}_{\text{
VS}^{*,Z}_{012}}$ $\displaystyle=$
$\displaystyle-\varepsilon^{16}\int_{\Omega^{-}}\mathcal{F}_{p}^{\pm}D_{t}^{\varphi,-}\partial_{t}^{7}p^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}D_{t}^{\varphi,-}p^{-}\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}(\frac{1}{2}|b^{-}|^{2})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}+\text{ VS}^{*,Z}_{012}$ $\displaystyle=:$
$\displaystyle\text{ VS}^{*}_{0111}+\text{ VS}^{*,B}_{0111}+\text{
VS}^{*,Z}_{012}.$ (3.96)

By the definition of $\mathbf{P}^{\sharp,-}$

$D_{t}^{\varphi,-}\partial_{t}^{7}p^{-}=D_{t}^{\varphi,-}\mathbf{P}^{\sharp,-}+D_{t}^{\varphi,-}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}p^{-}),~{}~{}\partial_{t}^{7}D_{t}^{\varphi,-}p^{-}=D_{t}^{\varphi,-}\mathbf{P}^{\sharp}+\mathfrak{D}^{\sharp}(p^{-}).$

Then we integrate
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ by parts and
use symmetry to find

$\displaystyle\text{ VS}_{0111}^{*}=$
$\displaystyle-\varepsilon^{16}\int_{\Omega^{-}}\mathcal{F}_{p}^{\pm}D_{t}^{\varphi,-}\mathbf{P}^{\sharp,-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{P}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle\underbrace{+\varepsilon^{16}\int_{\Omega^{-}}\mathcal{F}_{p}^{\pm}D_{t}^{\varphi,-}\mathbf{P}^{\sharp,-}\left((\overline{\nabla}\cdot\left\llbracket\bar{v}\right\rrbracket)D_{t}^{\varphi,-}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}p^{-})-(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left(\mathfrak{D}^{\sharp}(p^{-})-D_{t}^{\varphi,-}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}p^{-})\right)\right)\,\mathrm{d}\mathcal{V}_{t}}_{\text{
VS}^{*,R}_{0111}}$
$\displaystyle\underbrace{-\varepsilon^{16}\int_{\Omega^{-}}\mathcal{F}_{p}^{\pm}D_{t}^{\varphi,-}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}p^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathfrak{D}^{\sharp}(p^{-})\,\mathrm{d}\mathcal{V}_{t}}_{\text{
VS}^{*,R}_{0112}}$
$\displaystyle\overset{\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla}}{===}$
$\displaystyle-\frac{1}{2}\int_{\Omega^{-}}(\overline{\nabla}\cdot\left\llbracket\bar{v}\right\rrbracket)(\sqrt{\mathcal{F}_{p}^{\pm}}\varepsilon^{8}D_{t}^{\varphi,-}\mathbf{P}^{\sharp,-})^{2}\,\mathrm{d}\mathcal{V}_{t}+\text{
VS}^{*,R}_{0111}+\text{ VS}^{*,R}_{0112},$ (3.97)

where the first term on the right side is controlled by
$\left\|(\mathcal{F}_{p}^{\pm})^{\frac{1}{2}}\varepsilon^{8}\mathbf{P}^{*,-}\right\|_{0}^{2}\|\overline{\nabla}\left\llbracket\bar{v}\right\rrbracket\|_{L^{\infty}}^{2}$.
Next we adapt the analysis for $Z^{\pm}+ZB^{\pm}$ term to the control of
$\text{ VS}_{1}^{*,ZB}+\text{ VS}_{011}^{*,Z}$ and $\text{
VS}_{2}^{*,ZB}+\text{ VS}_{012}^{*,Z}$. Using Gauss-Green formula, we have

$\displaystyle\text{ VS}_{1}^{*,ZB}+\text{ VS}_{011}^{*,Z}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}(D_{t}^{\varphi,-}\partial^{\varphi}_{3}\partial_{t}^{7}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\varepsilon^{16}\int_{\Omega^{-}}[\partial^{\varphi}_{3},D_{t}^{\varphi,-}](\partial_{t}^{7}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}+D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,\partial^{\varphi}_{3}\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\right)\cdot\partial_{t}^{7}v^{-}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=:$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}(D_{t}^{\varphi,-}\partial^{\varphi}_{3}\partial_{t}^{7}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}\,\mathrm{d}\mathcal{V}_{t}+\text{
VS}_{1}^{*,ZR}.$ (3.98)

The main term is controlled by integrating $D_{t}^{\varphi,-}$ by parts under
time integral and invoking the momentum equation to replace
$\partial^{\varphi}_{3}q^{-}$ by tangential derivatives of $v^{-},b^{-}$:

$\displaystyle\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}(D_{t}^{\varphi,-}\partial^{\varphi}_{3}\partial_{t}^{7}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}(\partial^{\varphi}_{3}\partial_{t}^{7}q^{-})\,D_{t}^{\varphi,-}\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}\right)\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau+\varepsilon^{16}\int_{\Omega^{-}}(\partial^{\varphi}_{3}\partial_{t}^{7}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{N}\cdot\partial_{t}^{7}v^{-}\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}$
$\displaystyle\lesssim$
$\displaystyle~{}\delta\|\varepsilon^{8}\partial^{\varphi}_{3}\partial_{t}^{7}q^{-}\|_{0,-}^{2}+P(E_{4}^{\kappa}(0))E_{8}^{\kappa}(0)+\int_{0}^{t}\|\varepsilon^{8}\partial^{\varphi}_{3}\partial_{t}^{7}q^{-}\|_{0,-}\|\varepsilon^{8}(\overline{\partial}{}\partial_{t}^{7}v^{-},\partial_{t}^{8}v^{-})\|_{0}P(E_{4}^{\kappa}(\tau))\,\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle~{}\delta
E_{8}^{\kappa}(t)+P(E_{4}^{\kappa}(0))E_{8}^{\kappa}(0)+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau,~{}~{}~{}\forall\delta\in(0,1).$
(3.99)

For$\text{ VS}_{2}^{*,ZB}+\text{ VS}_{012}^{*,Z}$, we recall that the term
$\mathfrak{C}_{i}^{\sharp}(v_{i}^{-})$ in$\text{ VS}_{012}^{*,Z}$ includes a
term $[\partial_{t}^{7},\mathbf{N}_{i}/\partial_{3}\varphi,v_{i}^{-}]$ which
also appears in$\text{ VS}_{2}^{*,ZB}$. Thus we can again use the Gauss-Green
formula to analyze this term. Let us first compute the commutator in$\text{
VS}_{012}^{*,Z}$:

$\displaystyle\mathfrak{C}_{i}^{\sharp}(v_{i}^{-})-\partial^{\varphi}_{i}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}^{-})=$
$\displaystyle-\partial^{\varphi}_{i}\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}+\left[\partial_{t}^{7},\frac{\mathbf{N}_{i}}{\partial_{3}\varphi},\partial_{3}v_{i}^{-}\right]-\partial_{3}v_{i}^{-}\left[\partial_{t}^{7}\mathbf{N}_{i},\frac{1}{\partial_{3}\varphi}\right]-\mathbf{N}_{i}\partial_{3}v_{i}^{-}\left[\partial_{t}^{6},\frac{1}{(\partial_{3}\varphi)^{2}}\right]\partial_{t}\partial_{3}\varphi$
$\displaystyle=$
$\displaystyle-\partial^{\varphi}_{i}\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i}+\left[\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}^{-}\right]+\sum_{k=1}^{6}\binom{7}{k}[\partial_{t}^{k},(\partial_{3}\varphi)^{-1}]\mathbf{N}_{i}\,\partial_{t}^{7-k}\partial_{3}v_{i}^{-}$
$\displaystyle-\partial_{3}v_{i}^{-}\left[\partial_{t}^{7},\mathbf{N}_{i},\frac{1}{\partial_{3}\varphi}\right]-\mathbf{N}_{i}\partial_{3}v_{i}^{-}\left[\partial_{t}^{6},\frac{1}{(\partial_{3}\varphi)^{2}}\right]\partial_{t}\partial_{3}\varphi$
$\displaystyle=:$
$\displaystyle~{}\frac{1}{\partial_{3}\varphi}\left[\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}^{-}\right]+\mathfrak{C}^{\sharp,R}_{i}(v_{i}^{-})$

Then

$\displaystyle\text{ VS}_{2}^{*,ZB}+\text{ VS}_{012}^{*,Z}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Sigma}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})[\partial_{t}^{7},N_{i},v_{i}^{-}]\,\mathrm{d}x^{\prime}-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left(\frac{1}{\partial_{3}\varphi}\left[\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}^{-}\right]\right)\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle\underbrace{-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathfrak{C}^{\sharp,R}_{i}(v_{i}^{-})\,\mathrm{d}\mathcal{V}_{t}}_{\text{
VS}_{2}^{*,ZR}}$ $\displaystyle=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{3}\left(D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\right)(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})[\partial_{t}^{7},\mathbf{N}_{i},v_{i}^{-}]\,\mathrm{d}x+\text{
VS}_{2}^{*,ZR}+\text{ lower order terms},$ (3.100)

where the first term on the right side is again controlled by integrating
$D_{t}^{\varphi,-}$ by parts under time integral. We omit the details and just
list the result

$\int_{0}^{t}\int_{\Omega^{-}}\varepsilon^{16}\partial^{\varphi}_{3}\left(D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\right)(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})[\partial_{t}^{7},\mathbf{N}_{i},v_{i}^{-}]\,\mathrm{d}x\,\mathrm{d}\tau\leq\delta
E_{8}^{\kappa}(t)+P(E_{4}^{\kappa}(0))E_{8}^{\kappa}(0)+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau,~{}~{}~{}\forall\delta\in(0,1).$

Now the term$\text{ VS}_{1}^{*}$ is controlled except for those remainder
terms$\text{ VS}^{*,R}_{011}$,$\text{ VS}^{*,R}_{012}$,$\text{
VS}^{*,B}_{0111}$,$\text{ VS}^{*,R}_{0111}$,$\text{ VS}^{*,R}_{0112}$, $\text{
VS}_{1}^{*,ZR}$ and$\text{ VS}_{2}^{*,ZR}$. In fact, apart from$\text{
VS}^{*,B}_{0111}$, the other remainder terms can be directly controlled by
counting the number of derivatives and invoking the reduction for
$\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N}$ and
$\partial^{\varphi}_{3}\partial_{t}^{7}q^{-}$. There is no loss of Mach number
in these remainder terms. In fact, when $\partial_{t}^{8}p^{-}$ appears in the
remainder terms, either we have $\varepsilon^{16}\mathcal{F}_{p}^{\pm}$-weight
to control it directly, or we can integrate by parts $D_{t}^{\varphi,-}$ and
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ under time
integral to move one time derivative to $v_{i}^{-}$. Besides, the control of
$\partial_{t}^{7}\varphi,\partial_{t}^{8}\varphi$ depends on the boundary
regularity contributed by surface tension and so depends on $\sigma^{-1}$.
Therefore, we can conclude the estimates of $\text{ VS}_{1}^{*}$ by

$\text{ VS}_{01}^{*}+\text{ VS}^{*,ZB}_{1}+\text{ VS}^{*,ZB}_{2}\leq\text{
VS}^{*,B}_{0111}+\delta
E_{8}^{\kappa}(t)+P(E_{4}^{\kappa}(0))E_{8}^{\kappa}(0)+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau~{}~{}~{}\forall\delta\in(0,1).$
(3.101)

Next we control$\text{
VS}_{02}^{*}=\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}D_{t}^{\varphi,-}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}$.
First, we commute $D_{t}^{\varphi,-}$ with $\partial^{\varphi}_{i}$ to get

$\displaystyle\text{ VS}_{02}^{*}=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial^{\varphi}_{i}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}v_{j}^{-}\partial^{\varphi}_{j}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=:$ $\displaystyle\text{ VS}_{021}^{*}+\text{ VS}_{021}^{*,R}.$
(3.102)

In the first term, we integrate by parts $D_{t}^{\varphi,-}$ under time
integral and commute $D_{t}^{\varphi,-}$ with
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ to get

$\displaystyle\int_{0}^{t}\text{
VS}_{021}^{*}\,\mathrm{d}\tau\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}\partial^{\varphi}_{i}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau+\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}\partial_{t}^{7}q^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}$
$\displaystyle-\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}\partial^{\varphi}_{i}\partial_{t}^{7}q^{-}\,[D_{t}^{\varphi,-},(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})]\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle=:$ $\displaystyle\int_{0}^{t}\text{
VS}_{0211}^{*}\,\mathrm{d}\tau+\text{ VS}_{022}^{*,R}+\int_{0}^{t}\text{
VS}_{023}^{*,R}\,\mathrm{d}\tau.$ (3.103)

Next we insert the good unknowns $\mathbf{Q}^{\sharp,-}$ and
$\mathbf{V}^{\sharp,-}$ and invoke again the momentum equation
$\rho^{-}D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}-(b^{-}\cdot\nabla^{\varphi})\mathbf{B}^{\sharp,-}=-\nabla^{\varphi}\mathbf{Q}^{\sharp,-}+\mathcal{R}_{v}^{\sharp,-}-\mathfrak{C}^{\sharp}(q^{-})$
to get

$\displaystyle\text{ VS}_{0211}^{*}=$
$\displaystyle~{}-\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}(\mathbf{Q}^{\sharp,-}-\partial_{t}^{7}\varphi\partial^{\varphi}_{3}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}(\mathbf{V}^{\sharp,-}-\partial_{t}^{7}\varphi\partial^{\varphi}_{3}v_{i})\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}\rho^{-}D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\cdot(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}-\varepsilon^{16}\int_{\Omega^{-}}(b^{-}\cdot\nabla^{\varphi})\mathbf{B}^{\sharp,-}\cdot(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle+\varepsilon^{16}\int_{\Omega^{-}}\left(\mathfrak{C}^{\sharp}(q^{-})-\mathcal{R}_{v}^{\sharp,-}\right)\cdot(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}\partial^{\varphi}_{i}(\partial_{t}^{7}\varphi\partial^{\varphi}_{3}q^{-})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\partial_{t}^{7}v^{-}_{i}\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle=:$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}\rho^{-}D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}+\text{
VS}_{0211}^{*,B}+\text{ VS}_{0211}^{*,R}+\text{ VS}_{0212}^{*,R},$ (3.104)

where the first term is again controlled by integrating by parts in
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ and using
symmetry

$\displaystyle\varepsilon^{16}\int_{\Omega^{-}}\rho^{-}D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}=\frac{\varepsilon^{16}}{2}\int_{\Omega^{-}}\left(\overline{\nabla}\cdot(\rho^{-}\left\llbracket\bar{v}\right\rrbracket)\right)\left|D_{t}^{\varphi,-}\mathbf{V}^{\sharp,-}\right|^{2}\,\mathrm{d}\mathcal{V}_{t}\leq
P(E_{4}^{\kappa}(t))E_{8}^{\kappa}(t).$ (3.105)

Next we wish to combine$\text{ VS}^{*,B}_{0211}$ with$\text{
VS}^{*,B}_{0111}:=\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\partial_{t}^{7}(\frac{1}{2}|b^{-}|^{2})\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}$ to get a cancellation structure. In$\text{
VS}^{*,B}_{0111}$, we invoke the evolution equation
$D_{t}^{\varphi,-}b_{j}^{-}=(b^{-}\cdot\nabla^{\varphi})v^{-}-b^{-}(\nabla^{\varphi}\cdot
v^{-})$ to get

$\displaystyle\text{ VS}_{0111}^{*,B}\overset{L}{=}$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}b_{j}^{-}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}(\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}$ $\displaystyle=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}\left(b_{j}^{-}(\nabla^{\varphi}\cdot
v^{-})\right)\,\mathrm{d}\mathcal{V}_{t}+\underbrace{\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,\left[b_{j},(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}\right](\nabla^{\varphi}\cdot
v^{-})\,\mathrm{d}\mathcal{V}_{t}}_{\text{ VS}^{*,BR}_{0111}}$
$\displaystyle=$
$\displaystyle-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{B}_{j}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}((b^{-}\cdot\nabla^{\varphi})v_{j}^{-})\,\mathrm{d}\mathcal{V}_{t}$
$\displaystyle-\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathfrak{D}^{\sharp}(b_{j}^{-})\,\mathrm{d}\mathcal{V}_{t}+\text{
VS}^{*,BR}_{0111},$ (3.106)

where the first term on the right side is again controlled by integrating by
parts in $(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ and
using symmetry, and the third term on the right side is controlled directly
after inserting the expression of $\mathfrak{D}^{\sharp}(b)$. We denote

$\text{
VS}_{0112}^{*,B}:=\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{j}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\partial_{t}^{7}((b^{-}\cdot\nabla^{\varphi})v_{j}^{-})\,\mathrm{d}\mathcal{V}_{t}$

to be the second term on the right side above. Inserting the good unknown
$\mathbf{V}^{\sharp,-}$, the term$\text{ VS}_{0112}^{*,B}$ is equal to

$\displaystyle\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{i}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left((b^{-}\cdot\nabla^{\varphi})\mathbf{V}_{i}^{\sharp,-}\right)\,\mathrm{d}\mathcal{V}_{t}+\underbrace{\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{i}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\left([\partial_{t}^{7},b_{j}^{-}]\partial^{\varphi}_{j}v_{i}^{-}+b_{j}^{-}\mathfrak{C}_{j}^{\sharp}(v_{i}^{-})\right)\,\mathrm{d}\mathcal{V}_{t}}_{\text{
VS}^{*,BR}_{0112}}$ $\displaystyle=$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{i}\,(b^{-}\cdot\nabla^{\varphi})\left((\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{V}_{i}^{\sharp,-}\right)\,\mathrm{d}\mathcal{V}_{t}+\varepsilon^{16}\int_{\Omega^{-}}D_{t}^{\varphi,-}\mathbf{B}^{\sharp,-}_{i}\,\left[(b^{-}\cdot\nabla^{\varphi}),(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\right]\mathbf{V}_{i}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}+\text{
VS}^{*,BR}_{0112}$ $\displaystyle=:$ $\displaystyle\text{
VS}_{0113}^{*,B}+\text{ VS}^{*,BR}_{0113}+\text{ VS}^{*,BR}_{0112}.$ (3.107)

Now we can integrate by parts $(b^{-}\cdot\nabla^{\varphi})$ and then
$D_{t}^{\varphi,-}$ in$\text{ VS}_{0113}^{*,B}$ in order to produce the
cancellation with$\text{ VS}_{0211}^{*,B}$. Under time integral,
$\int_{0}^{t}\text{ VS}_{0113}^{*,B}\,\mathrm{d}\tau$ is equal to

$\displaystyle\int_{0}^{t}\underbrace{\varepsilon^{16}\int_{\Omega^{-}}(b^{-}\cdot\nabla^{\varphi})\mathbf{B}^{\sharp,-}_{i}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})D_{t}^{\varphi,-}\mathbf{V}_{i}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}}_{=-\text{
VS}_{0211}^{*,B}}\,\mathrm{d}\tau+\varepsilon^{16}\int_{\Omega^{-}}(b^{-}\cdot\nabla^{\varphi})\mathbf{B}^{\sharp,-}_{i}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{V}_{i}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}$
$\displaystyle+\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}[(b^{-}\cdot\nabla^{\varphi}),D_{t}^{\varphi,-}]\mathbf{B}^{\sharp,-}_{i}\,(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})\mathbf{V}_{i}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau+\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}(b^{-}\cdot\nabla^{\varphi})\mathbf{B}^{\sharp,-}_{i}\,[D_{t}^{\varphi,-},(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})]\mathbf{V}_{i}^{\sharp,-}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle=:$ $\displaystyle-\int_{0}^{t}\text{
VS}_{0211}^{*,B}\,\mathrm{d}\tau+\text{ VS}^{*,BR}_{0211}+\int_{0}^{t}\text{
VS}^{*,BR}_{0212}+\text{ VS}^{*,BR}_{0213}\,\mathrm{d}\tau.$ (3.108)

Note that
$[D_{t}^{\varphi,-},(b^{-}\cdot\nabla^{\varphi})]=-(\nabla^{\varphi}\cdot
v^{-})(b^{-}\cdot\nabla^{\varphi})f$ and when we commute
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ with either
$D_{t}^{\varphi,-}$ or $(b^{-}\cdot\nabla^{\varphi}),$, no normal derivative
will be generated because the weight functions in front of $\partial_{3}$
(namely, $b^{-}\cdot\mathbf{N}$ and
$(v^{-}\cdot\mathbf{N}-\partial_{t}\varphi)$) are still vanishing on the
interface $\Sigma$ after taking
$(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$. Therefore,
the commutators above are all controllable in $\|\cdot\|_{8,*,-}$ norm and no
loss of Mach number occurs. The following remainder terms are controlled
directly

$\displaystyle\text{ VS}_{022}^{*,R}+\text{
VS}^{*,BR}_{0211}+\int_{0}^{t}\text{ VS}_{021}^{*,R}+\text{
VS}_{023}^{*,R}+\text{ VS}^{*,BR}_{0111}+\text{ VS}^{*,BR}_{0112}+\text{
VS}^{*,BR}_{0212}+\text{ VS}^{*,BR}_{0213}\,\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle~{}\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(E^{\kappa}(\tau))\,\mathrm{d}\tau.$
(3.109)

In the terms$\text{ VS}_{0211}^{*,R}+\text{ VS}_{0212}^{*,R}$, we can
integrate $(\left\llbracket\bar{v}\right\rrbracket\cdot\overline{\nabla})$ by
parts to get to get the desired control thanks to the $\sqrt{\sigma}$-weighted
boundary regularity of $\psi$

$\displaystyle\text{ VS}_{0211}^{*,R}+\text{
VS}_{0212}^{*,R}\lesssim_{\sigma^{-1}}\left(|\varepsilon^{8}\partial_{t}^{7}\psi|_{2}+|\varepsilon^{8}\partial_{t}^{8}\psi|_{1}\right)\|\varepsilon^{8}D_{t}^{\varphi,-}\partial_{t}^{7}v\|_{0,-}P(E_{4}^{\kappa}(t)).$
(3.110)

Thus, the control of $\text{ VS}_{02}^{*}$ term is concluded by

$\displaystyle\int_{0}^{t}\text{ VS}_{02}^{*}+\text{
VS}_{0111}^{*,B}\,\mathrm{d}\tau\leq\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau.$
(3.111)

Finally, combining (3.92), (3.93), (3.101) and (3.111), we get the estimate
of$\text{ VS}^{*}$ term

$\displaystyle\int_{0}^{t}\text{ VS}^{*}\,\mathrm{d}\tau\leq\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(\sigma^{-1},E^{\kappa}(\tau))\,\mathrm{d}\tau.$
(3.112)

#### Step 3: Control of RT term.

In step 3, we control the terms$\text{ RT}^{*}$ and $\text{ RT}^{*,\pm}$
defined in (3.65)-(3.66), The latter one can be directly controlled by using
symmetry

$\displaystyle\text{
RT}^{*,\pm}=\mp\frac{1}{2}\int_{\Sigma}(\overline{\nabla}\cdot(\partial_{3}q^{\pm}\,v^{\pm}))\left|\overline{D_{t}^{-}}\partial_{t}^{7}\psi\right|^{2}\,\mathrm{d}x^{\prime}\leq\,\sigma^{-1}E_{4}^{\kappa}(t)E_{8}^{\kappa}(t).$
(3.113)

The term$\text{
RT}^{*}=-\varepsilon^{16}\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
cannot be controlled in the same way as in the estimates of spatial
derivatives because we do not have $L^{2}(\Sigma)$-control for
$\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi$ without $\kappa$-weight
nor can we integrate by parts $\partial_{t}^{1/2}$. To overcome this
difficulty, we need to invoke the kinematic boundary condition to reduce the
number of time derivatives. We have

$\overline{D_{t}^{-}}\partial_{t}^{7}\psi=\partial_{t}^{7}v^{-}\cdot
N+[\partial_{t}^{7},v^{-},N],~{}~{}\partial_{t}\overline{D_{t}^{-}}\partial_{t}^{7}\psi=\partial_{t}^{8}v^{-}\cdot
N+8\partial_{t}^{7}v^{-}\cdot\partial_{t}N+\text{ lower order terms}.$

Plugging it to$\text{ RT}^{*}$, we find

$\displaystyle\text{ RT}^{*}\stackrel{{\scriptstyle
L}}{{=}}-\varepsilon^{16}\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\partial_{t}^{7}v^{-}\cdot
N\,\partial_{t}^{8}v^{-}\cdot
N\,\mathrm{d}x^{\prime}-8\varepsilon^{16}\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\partial_{t}^{7}v^{-}\cdot
N\,\partial_{t}^{7}v^{-}\cdot\partial_{t}N\,\mathrm{d}x^{\prime}=:\text{
RT}^{*}_{1}+\text{ RT}^{*}_{2}.$ (3.114)

The term$\text{ RT}^{*}_{2}$ can be controlled by using Gauss-Green formula

$\displaystyle\text{ RT}^{*}_{2}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}-8\varepsilon^{16}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial_{t}^{7}v^{-}\cdot\partial_{t}\mathbf{N})\,\mathrm{d}\mathcal{V}_{t}-8\varepsilon^{16}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial_{3}\partial_{t}^{7}v^{-}\cdot\partial_{t}\mathbf{N})\,\mathrm{d}x,$
(3.115)

where $\left\llbracket\partial_{3}q\right\rrbracket$ is defined via Sobolev
extension. The first term above is directly controlled after invoking the
reduction
$\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N}\stackrel{{\scriptstyle
L}}{{=}}-\partial_{t}^{7}(\varepsilon^{2}D_{t}^{\varphi,-}p^{-}+\overline{\nabla}\cdot\bar{v}^{-})$.
For the second term, it suffices to integrate $\partial_{t}$ by parts under
time integral

$\displaystyle-8\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial_{t}^{7}v^{\pm}\cdot\mathbf{N})(\partial_{3}\partial_{t}^{7}v^{-}\cdot\partial_{t}\mathbf{N})\,\mathrm{d}x\,\mathrm{d}\tau$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-8\varepsilon^{16}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial_{3}\partial_{t}^{6}v^{-}\cdot\partial_{t}\mathbf{N})\,\mathrm{d}x\bigg{|}^{t}_{0}+8\int_{0}^{t}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial_{t}^{8}v^{-}\cdot\mathbf{N})(\partial_{3}\partial_{t}^{6}v^{-}\cdot\partial_{t}\mathbf{N})\,\mathrm{d}x$
$\displaystyle\lesssim$
$\displaystyle~{}\delta\|\varepsilon^{8}\partial_{3}\partial_{t}^{6}v^{-}\|_{0,-}^{2}+P(E^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau$
(3.116)

Using the same trick as above, the term$\text{ RT}^{*}_{1}$ is directly
controlled by repeated invoking
$\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N}\stackrel{{\scriptstyle
L}}{{=}}-\partial_{t}^{7}(\varepsilon^{2}D_{t}^{\varphi,-}p^{-}+\overline{\nabla}\cdot\bar{v}^{-})$

$\displaystyle\int_{0}^{t}\text{
RT}^{*}_{1}\,\mathrm{d}\tau\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket\left((\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial_{t}^{8}v^{-}\cdot\mathbf{N})+(\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial^{\varphi}_{3}\partial_{t}^{8}v^{-}\cdot\mathbf{N})\right)\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle\overset{\partial_{t},~{}L}{===}$
$\displaystyle\int_{0}^{t}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket\left((\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial_{t}^{8}v^{-}\cdot\mathbf{N})-\partial_{t}(\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N})\right)\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle-\varepsilon^{16}\int_{\Omega^{-}}\left\llbracket\partial_{3}q\right\rrbracket(\partial_{t}^{7}v^{-}\cdot\mathbf{N})(\partial^{\varphi}_{3}\partial_{t}^{7}v^{-}\cdot\mathbf{N})\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}$
$\displaystyle\lesssim$ $\displaystyle~{}\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau.$
(3.117)

Hence, we conclude the estimate of$\text{ RT}^{*}$ by

$\displaystyle\int_{0}^{t}\text{ RT}^{*}\,\mathrm{d}\tau\lesssim\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(E_{4}^{\kappa}(\tau))E_{8}^{\kappa}(\tau)\,\mathrm{d}\tau.$
(3.118)

#### Step 4: The cancellation structure between $ZB^{*}$ and $Z^{*}$.

Now we control the term $ZB^{*,\pm}+Z^{*,\pm}$. Note that we cannot integrate
by parts $\overline{\partial}{}^{1/2}$ due to the lack of spatial derivatives.
First, $ZB^{*,\pm}$ can be written as

$\displaystyle ZB^{*,\pm}=$
$\displaystyle\mp\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}q^{\pm}(\partial_{3}v^{\pm}\cdot
N)\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}\pm\varepsilon^{16}\int_{\Sigma}\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\partial_{3}q^{\pm}(\partial_{3}v^{\pm}\cdot
N)\overline{D_{t}^{-}}\partial_{t}^{7}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle\mp\varepsilon^{16}\int_{\Sigma}\mathbf{Q}^{*,\pm}\,\left[D_{t}^{\varphi,-}\partial_{t}^{7},N_{i},v_{i}^{\pm}\right]\,\mathrm{d}x^{\prime}$
$\displaystyle=:$
$\displaystyle~{}ZB^{*,R,\pm}_{1}+ZB^{*,R,\pm}_{2}+ZB^{*,\pm}_{0}.$ (3.119)

The second term on the right side can be directly controlled. We have

$\displaystyle
ZB^{*,R,\pm}_{2}\leq\left|D_{t}^{\varphi,-}\partial_{t}^{7}\psi\right|_{0}^{2}\left|\partial_{3}q^{\pm}\,(\partial_{3}v^{\pm}\cdot
N)\right|_{L^{\infty}}\leq P(\sigma^{-1},E_{4}^{\kappa}(t))E_{8}^{\kappa}(t).$
(3.120)

For the first term, using again
$\overline{D_{t}^{-}}\partial_{t}^{7}\psi=\partial_{t}^{7}v\cdot N+$lower
order terms, we can convert it to an interior integral.

$\displaystyle\int_{0}^{t}ZB^{*,R,\pm}_{1}\,\mathrm{d}\tau\stackrel{{\scriptstyle
L}}{{=}}$
$\displaystyle~{}\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{\pm}}(\partial_{3}v^{\pm}\cdot\mathbf{N})\left(\partial^{\varphi}_{3}D_{t}^{\varphi,-}\partial_{t}^{7}q^{\pm}\,\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}+D_{t}^{\varphi,-}\partial_{t}^{7}q^{\pm}\,\partial^{\varphi}_{3}\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}\right)\,\mathrm{d}\mathcal{V}_{t}$
(3.121) $\displaystyle\overset{D_{t}^{\varphi,-}}{==}$
$\displaystyle~{}\varepsilon^{16}\int_{\Omega^{\pm}}(\partial_{3}v^{\pm}\cdot\mathbf{N})\partial^{\varphi}_{3}\partial_{t}^{7}q^{\pm}\,\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}\,\mathrm{d}\mathcal{V}_{t}\bigg{|}^{t}_{0}+\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{\pm}}(\partial_{3}v^{\pm}\cdot\mathbf{N})\left([\partial^{\varphi}_{3},D_{t}^{\varphi,-}]\partial_{t}^{7}q^{\pm}\right)\,\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}\,\mathrm{d}\mathcal{V}_{t}\,\mathrm{d}\tau$
$\displaystyle+\varepsilon^{16}\int_{0}^{t}\int_{\Omega^{\pm}}(\partial_{3}v^{\pm}\cdot\mathbf{N})\left(\partial^{\varphi}_{3}\partial_{t}^{7}q^{\pm}\,D_{t}^{\varphi,-}\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}+D_{t}^{\varphi,-}\partial_{t}^{7}q^{\pm}\,\partial^{\varphi}_{3}\partial_{t}^{7}v^{\pm}\cdot\mathbf{N}\right)\,\mathrm{d}\mathcal{V}_{t}+\text{l.o.t}$

Now we can invoke the reduction for $\partial^{\varphi}_{3}q$ and
$\partial^{\varphi}_{3}v\cdot\mathbf{N}$ to convert $\partial^{\varphi}_{3}$
to a tangential derivative

$\partial^{\varphi}_{3}\partial_{t}^{7}q\stackrel{{\scriptstyle
L}}{{=}}\partial_{t}^{7}(\rho
D_{t}^{\varphi}v-(b\cdot\nabla^{\varphi})b),~{}~{}\partial^{\varphi}_{3}\partial_{t}^{7}v\cdot\mathbf{N}\stackrel{{\scriptstyle
L}}{{=}}-\partial_{t}^{7}(\mathcal{F}_{p}D_{t}^{\varphi}p+\overline{\nabla}\cdot\bar{v}).$

Note that the second equation above produces an extra
$\mathcal{F}_{p}=O(\varepsilon^{2})$ weight, so there is no loss of Mach
number when $D_{t}^{\varphi,-}\partial_{t}^{7}q$ appears. When
$D_{t}^{\varphi,-}\partial_{t}^{7}q$ is multiplied by
$\partial_{t}^{7}\overline{\nabla}\cdot\bar{v}$, we can further integrate by
parts in $\partial_{t}$ and then in $\overline{\nabla}\cdot$ to move one time
derivative to $v$. Hence, $ZB^{*,R,\pm}_{1}$ is controlled in
$\|\cdot\|_{8,*}$ norm without loss of $\varepsilon$-weights

$\displaystyle\int_{0}^{t}ZB^{*,R,\pm}_{1}\,\mathrm{d}\tau\lesssim~{}\delta
E_{8}^{\kappa}(t)+P(E^{\kappa}(0))+\int_{0}^{t}P(E^{\kappa}(\tau))\,\mathrm{d}\tau.$
(3.122)

Next we will see again the cancellation structure in
$ZB_{0}^{*,\pm}+Z^{*,\pm}$. From (3.2), we have

$\displaystyle\sum_{i=1}^{3}\mathfrak{C}_{i}^{*}(v_{i})=$
$\displaystyle~{}\partial^{\varphi}_{3}(\nabla^{\varphi}\cdot
v)D_{t}^{\varphi,-}\partial_{t}^{7}\varphi+\left[D_{t}^{\varphi,-}\partial_{t}^{7},\frac{\mathbf{N}_{i}}{\partial_{3}\varphi},\partial_{3}v_{i}\right]+\partial_{3}v_{i}\left[D_{t}^{\varphi,-}\partial_{t}^{7},\mathbf{N}_{i},(\partial_{3}\varphi)^{-1}\right]$
$\displaystyle+(\partial_{3}v\cdot\mathbf{N})\left[D_{t}^{\varphi,-}\partial_{t}^{6},\frac{1}{(\partial_{3}\varphi)^{2}}\right]\partial_{t}\partial_{3}\varphi+(\partial_{3}\varphi)^{-1}\mathbf{N}\cdot[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]v+(\partial_{3}\varphi)^{-1}(\partial^{\varphi}_{3}v\cdot\mathbf{N})[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]\varphi,$

where we have further analysis on the second term and the fifth term

$\displaystyle\left[D_{t}^{\varphi,-}\partial_{t}^{7},\frac{\mathbf{N}_{i}}{\partial_{3}\varphi},\partial_{3}v_{i}\right]\stackrel{{\scriptstyle
L}}{{=}}$
$\displaystyle~{}(\partial_{3}\varphi)^{-1}\left[D_{t}^{\varphi,-}\partial_{t}^{7},\mathbf{N}_{i},\partial_{3}v_{i}\right]-\partial^{\varphi}_{3}\partial_{t}\varphi\partial^{\varphi}_{3}D_{t}^{\varphi,-}\partial_{t}^{6}v\cdot\mathbf{N},$
(3.123)
$\displaystyle(\partial_{3}\varphi)^{-1}\mathbf{N}\cdot[D_{t}^{\varphi,-}\partial_{t}^{7},\partial_{3}]v=$