after possibly passing to a subsequence. Theorem 1.2 is proven.

###  4.4 Double limits in 2D: a subsonic zone

When the space dimension $d=2$, the substantial part of the proof for well-
posedness, uniform estimates and limit process remains unchanged. In fact, we
shall only re-consider the following aspects

  * •

The curl operator now becomes
$\nabla^{\varphi,\perp}\cdot:=(-\partial^{\varphi}_{2},\partial^{\varphi}_{1})\cdot$,
so we need to check the special structure given by Lorentz force in the
vorticity analysis. This has been analyzed in [63, Section 3.6.4].

  * •

The interface is now a 1D curve instead of a 2D surface, thus it is impossible
to have “non-parallel” magnetic fields $b^{\pm}$ on $\Sigma$. The functions
$\mu^{\pm}$ are no longer uniquely determined by $b_{1}^{\pm}$.

We prove the following lemma about the choice of Friedrichs symmetrizer.

######  Lemma 4.5.

There exist functions $\bar{\mu}^{\pm}(t,x_{1})$ satisfying

$\left\llbracket v_{1}-\bar{\mu}b_{1}\right\rrbracket=0~{}\text{ and
}~{}|\bar{\mu}^{\pm}|<1/a^{\pm}~{}~{}~{}\text{ on
}[0,T]\times\Sigma,~{}~{}~{}a^{\pm}:=\sqrt{\rho^{\pm}\left(1+(c_{A}^{\pm}/c_{s}^{\pm})^{2}\right)}$

if and only if the following inequality holds

$\displaystyle|\left\llbracket
v_{1}\right\rrbracket|<\frac{|b_{1}^{+}|}{a^{+}}+\frac{|b_{1}^{-}|}{a^{-}}~{}~{}~{}\text{
on }[0,T]\times\Sigma.$ (4.57)

Under (4.57), the functions $\bar{\mu}^{\pm}$ are chosen to be

$\displaystyle\bar{\mu}^{\pm}=\pm\frac{\text{sgn}(b_{1}^{\pm})\,a^{\mp}\left\llbracket
v_{1}\right\rrbracket}{a^{-}|b_{1}^{+}|+a^{+}|b_{1}^{-}|}.$ (4.58)

###### Proof.

First, we shall exclude the possibility for $b_{1}^{\pm}=0$ at some point
$(x_{1},0)\in\Sigma$ because of $\left\llbracket v_{1}\right\rrbracket\neq 0$
everywhere on $\Sigma$.

Case 1: One of the two magnetic fields is vanishing on $\Sigma$, e.g., we
assume $|b_{1}^{+}|>0=|b_{1}^{-}|$ on $\Sigma$, then $\left\llbracket
v_{1}-\bar{\mu}b_{1}\right\rrbracket=0$ directly gives us
$\bar{\mu}^{+}=\left\llbracket v_{1}\right\rrbracket/b_{1}^{+}$ and
$\bar{\mu}^{-}$ can be any function satisfying the hyperbolicity constraint
$|\bar{\mu}^{-}|<1/a^{-}$. For simplicity, we may choose $\bar{\mu}^{-}=0$.
Solving the constraint $|\bar{\mu}^{+}|<1/a^{+}$ gives us $|\left\llbracket
v_{1}\right\rrbracket|<|b_{1}^{+}|/a^{+}$ as a special case of (4.57).
Similarly, when $|b_{1}^{-}|>0=|b_{1}^{+}|$ on $\Sigma$, we can choose
$\bar{\mu}^{-}=-\left\llbracket v_{1}\right\rrbracket/b_{1}^{-}$ and
$\bar{\mu}^{+}$ can be any function satisfying the hyperbolicity constraint
$|\bar{\mu}^{+}|<1/a^{+}$ and we may choose $\bar{\mu}^{+}=0$ for simplicity.

######  Remark 4.1.

One can verify that if $b_{0}=0$ on $\Sigma$, then $b$ must be identically
zero on $\Sigma$. In fact, restricting the equation of $b$ onto $\Sigma$ and
doing $L^{2}$ estimate shows that
$\frac{\mathrm{d}}{\mathrm{d}t}|b^{\pm}|_{0}^{2}\leq C|\partial
v|_{L^{\infty}}|b^{\pm}|_{0}^{2}$. Using Gronwall’s inequality and
$b_{0}|_{\Sigma}=0$ yields the result.

Case 2: $b_{1}^{\pm}$ are not identically zero on $\Sigma$. In this case, we
may assume $|b_{1}^{\pm}|>0$ on $\Sigma$ as well. In fact, if $b_{1}^{-}$
vanishes at some point $(x_{1},0)\in\Sigma$, then we can follow the choice of
$\bar{\mu}^{\pm}$ as in case 1 to determine the function $\bar{\mu}^{\pm}$ at
this point. Note that the functions $\bar{\mu}^{+}=\left\llbracket
v_{1}\right\rrbracket/b_{1}^{+},~{}\bar{\mu}^{-}=0$ still satisfies (4.58), so
they do not break the continuity and differentiability of (4.58) at the points
where one of $b_{1}^{\pm}$ vanishes. Thus, making the assumption
$|b_{1}^{\pm}|>0$ on $\Sigma$ is reasonable.

The “if” part is easy to prove. Indeed, when the stability condition (4.57)
holds on $\Sigma$, we can set $\bar{\mu}^{\pm}$ as in (4.58). The direct
computation shows that such $\bar{\mu}^{\pm}$ satisfy $\left\llbracket
v_{1}-\bar{\mu}b_{1}\right\rrbracket=0$ on $\Sigma$ and
$|\bar{\mu}^{\pm}|<1/a^{\pm}$. Let us prove the “only if” part. When
$|b_{1}^{\pm}|>0$ on $\Sigma$, we can write

$\bar{\mu}^{+}=\frac{\left\llbracket
v_{1}\right\rrbracket+\bar{\mu}^{-}b_{1}^{-}}{b_{1}^{+}}.$

Using $|\bar{\mu}^{+}|<1/a^{+}$, we can solve the inequality by

$-\frac{1}{a^{+}}-\frac{\left\llbracket
v_{1}\right\rrbracket}{b_{1}^{+}}<\frac{b_{1}^{-}}{b_{1}^{+}}\bar{\mu}^{-}<\frac{1}{a^{+}}-\frac{\left\llbracket
v_{1}\right\rrbracket}{b_{1}^{+}},$

where $\bar{\mu}^{-}$ should also satisfy $|\bar{\mu}^{-}|<1/a^{-}$. Assume
$b_{1}^{\pm}<0$ for simplicity (that is, the horizontal directions of
$b^{\pm}$ on $\Sigma$ are the same). Combining these two requirements, we find
that the following inequality is necessary

$\displaystyle|\left\llbracket
v_{1}\right\rrbracket|<-\frac{b_{1}^{+}}{a^{+}}-\frac{b_{1}^{-}}{a^{-}}=\frac{|b_{1}^{+}|}{a^{+}}+\frac{|b_{1}^{-}|}{a^{-}}.$
(4.59)

Similar calculation for the case $b_{1}^{\pm}>0$ and the case
$b_{1}^{+}b_{1}^{-}<0$ also leads to the same inequality as above. ∎

##  5 Improved incompressible limit for well-prepared initial data

In this final section, we aim to drop the redundant assumptions on the well-
prepared initial data, namely $\partial_{t}^{k}v|_{t=0}=O(1)$ for $2\leq k\leq
4$, when taking the incompressible limit. Compared with the energy $E(t)$ that
we use to prove the local existence in [63], there is a new difficulty in the
control of the “weaker” energy $\mathfrak{E}(t)$: There exhibits a loss of
weight of Mach number in $\overline{\partial}{}^{3}\partial_{t}$-tangential
estimates when analyzing $\mathfrak{E}_{4}(t)$. In particular, we have to
control the following quantity in the cancellation structure in
$Z^{\pm}+ZB^{\pm}$,

$\int_{\Omega}(\overline{\partial}{}^{2}\partial_{3}\partial_{t}v_{i})(\overline{\partial}{}\mathbf{N}_{i})(\overline{\partial}{}^{2}\partial_{3}\partial_{t}q)\,\mathrm{d}x,$

in which $\partial_{t}q$ has to be uniformly bounded with respect to Mach
number. However, now we only have $\nabla^{\varphi}\cdot v=O(\varepsilon)$ and
$\partial_{t}q=O(1/\varepsilon)$, which leads to a loss of
$\varepsilon$-weight. Besides, similar difficulty also appears in the control
of
$-\int_{\Omega^{\pm}}\mathbf{V}^{\pm}\cdot\mathfrak{C}(q^{\pm})\,\mathrm{d}\mathcal{V}_{t}$.
Indeed, such loss of $\varepsilon$-weight necessarily happens in
$\overline{\partial}{}^{3}\partial_{t}$-tangential estimates because of the
following two reasons

  1. 1.

$\overline{\partial}{}^{3}\partial_{t}q$ needs one more $\varepsilon$-weight
than $\overline{\partial}{}^{3}\partial_{t}v$;

  2. 2.

The (extension of) normal vector $\mathbf{N}$, which arises from the
commutator
$[\overline{\partial}{}^{3}\partial_{t},\mathbf{N}_{i}/\partial_{3}\varphi,\partial_{3}f]$
in $\mathfrak{C}_{i}(f)$, may NOT absorb a time derivative.

As we can see, this type of difficulty never appears in the fixed-domain
setting because the commutator terms in $\mathfrak{C}(f)$ are contributed by
the free-interface motion. To get rid of the loss of Mach number, we have to
find a new way to control $v_{t}$, $b_{t}$ and also avoid the appearance of
$|\sqrt{\sigma}\overline{\nabla}\partial_{t}\psi|_{3}$ without
$\varepsilon$-weight.

###  5.1 The weaker energy for the improved incompressible limit

As in (1.40)-(1.41), we consider the new energy functional

$\displaystyle\mathfrak{E}(t):=\mathfrak{E}_{4}(t)+E_{5}(t)+E_{6}(t)+E_{7}(t)+E_{8}(t)$
$\displaystyle\widetilde{\mathfrak{E}}(t):=\widetilde{\mathfrak{E}}_{4}(t)+\widetilde{E}_{5}(t)+\widetilde{E}_{6}(t)+\widetilde{E}_{7}(t)+\widetilde{E}_{8}(t)$

where

$\displaystyle\mathfrak{E}_{4}(t)=\sum_{\pm}$
$\displaystyle\left\|(v^{\pm},b^{\pm},S^{\pm},p^{\pm})\right\|_{4,\pm}^{2}+\left|\sqrt{\sigma}\psi\right|_{5}^{2}+\left\|\partial_{t}(v^{\pm},b^{\pm},S^{\pm},(\mathcal{F}_{p}^{\pm})^{\frac{1}{2}}p^{\pm})\right\|_{3,\pm}^{2}+\left|\sqrt{\sigma}\partial_{t}\psi\right|_{4}^{2}$
$\displaystyle+\sum_{k=2}^{4}\left\|\varepsilon\partial_{t}^{k}(v^{\pm},b^{\pm},S^{\pm},(\mathcal{F}_{p}^{\pm})^{\frac{(k-3)_{+}}{2}}p^{\pm})\right\|_{4-k,\pm}^{2}+\left|\sqrt{\sigma}\varepsilon\partial_{t}^{k}\psi\right|_{5-k}^{2}$
(5.1)

and

$\displaystyle\widetilde{\mathfrak{E}}_{4}(t)=\mathfrak{E}_{4}(t)+\left|\psi\right|_{4.5}^{2}+\left|\partial_{t}\psi\right|_{3.5}^{2}+\left|\partial_{t}^{2}\psi\right|_{2.5}^{2}+\left|\varepsilon\partial_{t}^{3}\psi\right|_{1.5}^{2}+\left|\varepsilon\partial_{t}^{4}\psi\right|_{0.5}^{2}.$
(5.2)

We aim to prove uniform-in-$\varepsilon$ estimates for $\mathfrak{E}(t)$ for
each $\sigma>0$ and prove uniform-in-$({\varepsilon,\sigma})$ estimates for
$\widetilde{\mathfrak{E}}(t)$ under the stability condition (1.29) (replaced
with (1.37) in the 2D case). Let us analyze the estimates for different $k$ in
$\mathfrak{E}_{4}(t)$ and $\widetilde{\mathfrak{E}}_{4}(t)$.

#### The case $k=0$

When $k=0$, the reduction of $v,b$ is the same as in Section 3. That is, we
use the div-curl analysis to convert normal derivatives to tangential
derivatives

$\displaystyle\|(v^{\pm},b^{\pm})\|_{4,\pm}^{2}\leq$
$\displaystyle~{}C(|\psi|_{4},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|v^{\pm},b^{\pm}\|_{0,\pm}^{2}+\left\|\nabla^{\varphi}\cdot
v^{\pm},\nabla^{\varphi}\times(v^{\pm},b^{\pm})\right\|_{3,\pm}^{2}+\left\|\overline{\partial}{}^{4}(v^{\pm},b^{\pm})\right\|_{0,\pm}^{2}\right),$
(5.3) $\displaystyle\left\|\nabla^{\varphi}\cdot
v^{\pm}\right\|_{3,\pm}^{2}\lesssim$
$\displaystyle~{}\left\|\mathcal{F}_{p}^{\pm}D_{t}^{\varphi\pm}p^{\pm}\right\|_{3,\pm}^{2},\quad\left\|\nabla^{\varphi}\times(v^{\pm},b^{\pm})\right\|_{3,\pm}^{2}\lesssim~{}\delta\mathfrak{E}_{4}(t)+\int_{0}^{t}P(\mathfrak{E}_{4}(\tau))+\mathfrak{E}_{5}(\tau)\,\mathrm{d}\tau,$
(5.4) $\displaystyle\left\|\nabla q\right\|_{3,\pm}^{2}\lesssim$
$\displaystyle~{}\left\|\rho
D_{t}^{\varphi\pm}v^{\pm}\right\|_{3,\pm}^{2}+\left\|(b^{\pm}\cdot\nabla^{\varphi})b^{\pm}\right\|_{3,\pm}^{2}.$
(5.5)

The $\overline{\partial}{}^{4}$-control is proved in almost the same way as in
[63, Section 3.3] which gives the control of
$\left|\sqrt{\sigma}\overline{\nabla}\psi\right|_{4}^{2}$. The only difference
is the treatment of RT because we need to avoid using $\sqrt{\sigma}$-weight
enenrgy when taking the limit $\sigma\to 0$. Using Kato-Ponce type porduct
estimate (B.6) in Lemma B.6, we have

$\displaystyle\text{
RT}=\int_{\Sigma}\left\llbracket\partial_{3}q\right\rrbracket\overline{\partial}{}^{4}\psi\overline{\partial}{}^{4}\psi_{t}\leq\left|\left\llbracket\partial_{3}q\right\rrbracket\overline{\partial}{}^{4}\psi\right|_{\frac{1}{2}}|\psi_{t}|_{3.5}\leq\|q^{\pm}\|_{3,\pm}|\psi|_{4.5}|\psi_{t}|_{3.5}.$
(5.6)

So, we need to find $({\varepsilon,\sigma})$-independent control of
$\left|\psi\right|_{4.5}^{2}$ and $\left|\partial_{t}\psi\right|_{3.5}^{2}$.

#### The case $k=1$

When $k=1$, we cannot use the above div-curl inequality because we must avoid
$\overline{\partial}{}^{3}\partial_{t}$-estimate. Instead, we use the div-curl
inequality (B.2) to get

$\displaystyle\|(\partial_{t}v^{\pm},\partial_{t}b^{\pm})\|_{3,\pm}^{2}\leq
C(|\psi|_{3.5},|\overline{\nabla}\psi|_{W^{1,\infty}})\bigg{(}$
$\displaystyle\|\partial_{t}v^{\pm},\partial_{t}b^{\pm}\|_{0,\pm}^{2}+\left\|\nabla^{\varphi}\cdot(\partial_{t}v^{\pm},\partial_{t}b^{\pm}),\nabla^{\varphi}\times(\partial_{t}v^{\pm},\partial_{t}b^{\pm})\right\|_{2,\pm}^{2}$
$\displaystyle+\left|\partial_{t}v^{\pm}\cdot N,\partial_{t}b^{\pm}\cdot
N\right|_{2.5}^{2}\bigg{)}.$ (5.7)

The divergence part and the curl part are controlled in the same way as [63,
Section 3.6.1], so we do not repeat the analysis here. The boundary normal
trace for $b_{t}$ is easy to control. Using $b^{\pm}\cdot N=0$, we have
$b_{t}\cdot N=\bar{b}\cdot\overline{\nabla}\psi_{t}$ and thus

$\displaystyle\left|\partial_{t}b^{\pm}\cdot
N\right|_{2.5}^{2}=\left|\bar{b}^{\pm}\cdot\overline{\nabla}\psi_{t}\right|_{2.5}^{2}\lesssim\|b^{\pm}\|_{3,\pm}^{2}|\psi_{t}|_{3.5}^{2}.$
(5.8)

For the normal trace $\left|\partial_{t}v^{\pm}\cdot N\right|_{2.5}^{2}$, we
invoke the kinematic boundary condition $\partial_{t}\psi=v^{\pm}\cdot N$ to
get

$\displaystyle\left|\partial_{t}v^{\pm}\cdot
N\right|_{2.5}^{2}\leq\left|\partial_{t}^{2}\psi\right|_{2.5}^{2}+\left|\bar{v}^{\pm}\cdot\overline{\nabla}\psi_{t}\right|_{2.5}^{2}\lesssim\left|\partial_{t}^{2}\psi\right|_{2.5}^{2}+\|v^{\pm}\|_{3,\pm}^{2}|\psi_{t}|_{3.5}^{2}.$
(5.9)

Since we avoid $\overline{\partial}{}^{3}\partial_{t}$-tangential estimates,
we must seek for another way to find $\varepsilon$-independent estimates for
$\left|\partial_{t}^{2}\psi\right|_{2.5}^{2}$ and
$\left|\sqrt{\sigma}\partial_{t}\psi\right|_{4}^{2}$. Also, under the
stability condition (1.29), we need to find
$({\varepsilon,\sigma})$-independent control of
$\left|\psi\right|_{4.5}^{2},\left|\partial_{t}\psi\right|_{3.5}^{2},\left|\partial_{t}^{2}\psi\right|_{2.5}^{2}$
and $\left|\sqrt{\sigma}\partial_{t}\psi\right|_{4}^{2}$.

#### The case $2\leq k\leq 4$

When $k=2,3,4$, the reduction stays the same as in [63, Section 3.3-3.5] (for
the incompressible limit for fixed $\sigma>0$) or the analysis in Section 4.2.
The reason is that $\partial_{t}^{k}q$ share the same weight of Mach number as
$\partial_{t}^{k}v$ which helps us avoid the loss of $\varepsilon$-weight in
$\mathfrak{C}(q)$.

###  5.2 The evolution equation of the free interface and its
paralinearization

To prove the uniform-in-$\varepsilon$ estimates for $\mathfrak{E}(t)$ and the
uniform-in-$({\varepsilon,\sigma})$ estimates for
$\widetilde{\mathfrak{E}}(t)$ under the stability condition (1.29), it remains
to prove the $\varepsilon$-indenpedent control of
$|\psi|_{4.5},~{}|\psi_{t}|_{3.5},~{}|\psi_{tt}|_{2.5}$ and
$|\sqrt{\sigma}\psi_{t}|_{4}$ by
$P(\mathfrak{E}(0))+P(\mathfrak{E}(t))\int_{0}^{t}P(\mathfrak{E}(\tau))\,\mathrm{d}\tau$
and $({\varepsilon,\sigma})$-indenpedent control of them by
$P(\widetilde{\mathfrak{E}}(0))+P(\widetilde{\mathfrak{E}}(t))\int_{0}^{t}P(\widetilde{\mathfrak{E}}(\tau))\,\mathrm{d}\tau$.
Since we already avoid $\overline{\partial}{}^{3}\partial_{t}$-tangential
estimates, we shall further analyze the evolution equation of the free
interface.

####  5.2.1 Derivation of the equation

We take $\partial_{t}$ in the kinematic boundary condition to get
$\partial_{t}^{2}\psi=\partial_{t}v^{\pm}\cdot
N-\bar{v}^{\pm}\cdot\overline{\nabla}\partial_{t}\psi$. Plugging the momentum
equation of (1.17) into the term $\partial_{t}v^{\pm}\cdot N$, we get

$\displaystyle\partial_{t}v^{\pm}\cdot
N=-\frac{1}{\rho^{\pm}}N\cdot\nabla^{\varphi}q^{\pm}-(\bar{v}^{\pm}\cdot\overline{\nabla})v^{\pm}\cdot
N+\frac{1}{\rho^{\pm}}(\bar{b}^{\pm}\cdot\overline{\nabla})b^{\pm}\cdot
N\quad\text{ on }\Sigma.$

Using $\partial_{t}\psi=v^{\pm}\cdot N$ and $b^{\pm}\cdot N=0$ on $\Sigma$, we
have

$\displaystyle-(\bar{v}^{\pm}\cdot\overline{\nabla})v^{\pm}\cdot N=$
$\displaystyle-(\bar{v}^{\pm}\cdot\overline{\nabla})\partial_{t}\psi+v^{\pm}\cdot(\bar{v}^{\pm}\cdot\overline{\nabla})N=-\bar{v}_{j}^{\pm}\overline{\partial}{}_{j}\partial_{t}\psi-\bar{v}_{i}^{\pm}\bar{v}_{j}^{\pm}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi,$
$\displaystyle(\bar{b}^{\pm}\cdot\overline{\nabla})b^{\pm}\cdot N=$
$\displaystyle~{}\bar{b}_{i}^{\pm}\bar{b}_{j}^{\pm}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi,$

and thus

$\displaystyle\partial_{t}^{2}\psi=-\frac{1}{\rho^{\pm}}N\cdot\nabla^{\varphi}q^{\pm}+\left(\frac{1}{\rho^{\pm}}\bar{b}_{i}^{\pm}\bar{b}_{j}^{\pm}-\bar{v}_{i}^{\pm}\bar{v}_{j}^{\pm}\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\bar{v}^{\pm}\cdot\overline{\nabla})\partial_{t}\psi.$
(5.10)

Next we want to separate the boundary value of $q^{\pm}$ from its interior
contribution in order to create an energy term involving the surface tension.
First, taking $\nabla^{\varphi}\cdot$ in the momentum equation and invoking
the continuity equation in (1.17), we derive a wave-type equation

$\mathcal{F}_{p}^{\pm}(D_{t}^{\varphi\pm})^{2}p^{\pm}-\Delta^{\varphi}q^{\pm}=(\partial^{\varphi}_{i}v_{j}^{\pm})(\partial^{\varphi}_{j}v_{i}^{\pm})-(\partial^{\varphi}_{i}b_{j}^{\pm})(\partial^{\varphi}_{j}b_{i}^{\pm}),$

which can be written as a wave equation of $q^{\pm}$ thanks to
$q^{\pm}=p^{\pm}+\frac{1}{2}|b^{\pm}|^{2}$

$\displaystyle\mathcal{F}_{p}^{\pm}(D_{t}^{\varphi\pm})^{2}q^{\pm}-\Delta^{\varphi}q^{\pm}=\varepsilon^{2}(D_{t}^{\varphi\pm})^{2}\left(\frac{1}{2}|b^{\pm}|^{2}\right)+(\partial^{\varphi}_{i}v_{j}^{\pm})(\partial^{\varphi}_{j}v_{i}^{\pm})-(\partial^{\varphi}_{i}b_{j}^{\pm})(\partial^{\varphi}_{j}b_{i}^{\pm})\quad\text{
in }[0,T]\times\Omega^{\pm},$ (5.11)

with a jump condition $\left\llbracket
q\right\rrbracket=\sigma\mathcal{H}(\psi)$ on $\Sigma$ and a Neumann-type
boundary condition $\partial_{3}q^{\pm}=0$ on $\Sigma^{\pm}$ (got by
restricting the momentum equations on $\Sigma^{\pm}$), where we omit the terms
in which $D_{t}^{\varphi\pm}$ falls on $\mathcal{F}_{p}^{\pm}$.

######  Definition 5.1 (Dirichlet-to-Neumann operator).

For a function $f:\Sigma\to\mathbb{R}$, we now define the Dirichlet-to-Neumann
operator with respect to $(\psi,\Omega^{\pm})$ by

$\displaystyle\mathfrak{N}_{\psi}^{\pm}f:=\mp
N\cdot\nabla^{\varphi}(\mathcal{E}_{\psi}^{\pm}f),$ (5.12)

where $\mathcal{E}_{\psi}^{\pm}f$ is defined to be the harmonic extension of
$f$ into $\Omega^{\pm}$, namely

$-\Delta^{\varphi}(\mathcal{E}_{\psi}^{\pm}f)=0\quad\text{ in
}\Omega^{\pm},\quad\mathcal{E}_{\psi}^{\pm}f=f\quad\text{ on
}\Sigma,\quad\partial_{3}(\mathcal{E}_{\psi}^{\pm}f)=0\quad\text{ on
}\Sigma^{\pm}.$ (5.13)

Thus, we can define a decomposition $q^{\pm}=q_{\psi}^{\pm}+q_{w}^{\pm}$
satisfying

$q_{\psi}^{\pm}:=\mathcal{E}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})\text{ in
}\Omega^{\pm}$ (5.14)

and

$-\Delta^{\varphi}q_{w}^{\pm}=-\mathcal{F}_{p}^{\pm}(D_{t}^{\varphi\pm})^{2}q^{\pm}+\mathcal{F}_{p}^{\pm}(D_{t}^{\varphi\pm})^{2}\left(\frac{1}{2}|b^{\pm}|^{2}\right)+(\partial^{\varphi}_{i}v_{j}^{\pm})(\partial^{\varphi}_{j}v_{i}^{\pm})-(\partial^{\varphi}_{i}b_{j}^{\pm})(\partial^{\varphi}_{j}b_{i}^{\pm})\text{
in }\Omega^{\pm}$ (5.15)

with boundary conditions $q_{w}^{\pm}=0\text{ on }\Sigma$ and
$N\cdot\nabla^{\varphi}q_{w}^{\pm}=\partial_{3}q_{w}^{\pm}=0$ on
$\Sigma^{\pm}$. The second boundary conditions holds thanks to the slip
condition for $v_{3}^{\pm},b_{3}^{\pm}$ on $\Sigma^{\pm}$. Thus, the evolution
equation of $\psi$ can be written as

$\displaystyle\rho^{\pm}\partial_{t}^{2}\psi=$
$\displaystyle\pm\mathfrak{N}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})-N\cdot\nabla^{\varphi}q_{w}^{\pm}+\left(\bar{b}_{i}^{\pm}\bar{b}_{j}^{\pm}-\rho^{\pm}\bar{v}_{i}^{\pm}\bar{v}_{j}^{\pm}\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\rho^{\pm}\bar{v}^{\pm}\cdot\overline{\nabla})\partial_{t}\psi.$
(5.16)

We now want to resolve $q^{\pm}|_{\Sigma}$ in terms of $\rho^{\pm}$ and
$F_{\psi}^{\pm}$ by inverting the Dirichlet-to-Neumann operators
$\mathfrak{N}_{\psi}^{\pm}$. However, we no longer have
$\int_{\Sigma}\rho\partial_{t}\psi\,\mathrm{d}x^{\prime}=\int_{\Sigma}\rho\partial_{t}^{2}\psi\,\mathrm{d}x^{\prime}=0$
due to the compressibility of fluids. Thus, we have to eliminate the zero-
frequency part in $\rho^{\pm}\partial_{t}^{2}\psi$ before inverting the
Dirichlet-to-Neumann operators. In other words, we cannot directly apply
$(\mathfrak{N}_{\psi}^{\pm})^{-1}$ to
$(1/\rho)\mathfrak{N}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})$.

For a function $f:\Sigma=\mathbb{T}^{2}\to\mathbb{R}$, we define the
Littlewood-Paley projection

$\mathcal{P}_{\neq
0}f:=f-(f)_{\Sigma},~{}~{}(f)_{\Sigma}:=\int_{\mathbb{T}^{2}}f\,\mathrm{d}x^{\prime}.$

Under this setting, we have

$\rho^{\pm}\partial_{t}^{2}\psi=\mathcal{P}_{\neq
0}(\rho^{\pm}\partial_{t}^{2}\psi)+(\rho^{\pm}\partial_{t}^{2}\psi)_{\Sigma}$
(5.17)

and we insert it back to the evolution equation to get

$\displaystyle\mathcal{P}_{\neq 0}(\rho^{\pm}\partial_{t}^{2}\psi)=$
$\displaystyle~{}\pm\mathfrak{N}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})-(\rho^{\pm}\partial_{t}^{2}\psi)_{\Sigma}-N\cdot\nabla^{\varphi}q_{w}^{\pm}+\left(\bar{b}_{i}^{\pm}\bar{b}_{j}^{\pm}-\rho^{\pm}\bar{v}_{i}^{\pm}\bar{v}_{j}^{\pm}\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\rho^{\pm}\bar{v}^{\pm}\cdot\overline{\nabla})\partial_{t}\psi$
(5.18) $\displaystyle=$
$\displaystyle:\pm\mathfrak{N}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})+F_{\psi}^{\pm}.$
(5.19)

Note that the zero-frequency modes of both $\mathcal{P}_{\neq
0}(\rho^{\pm}\partial_{t}^{2}\psi)$ and
$\pm\mathfrak{N}_{\psi}^{\pm}(q^{\pm}|_{\Sigma})$ are vanishing on the
interface $\Sigma$, so we deduce that $\int_{\Sigma}F_{\psi}^{\pm}=0$ and then
$(\mathfrak{N}_{\psi}^{\pm})^{-1}(F_{\psi}^{\pm})$ is well-defined. Now we can
resolve the traces $q^{\pm}|_{\Sigma}$ from the evolution equations of $\psi$.
We have

$\displaystyle\mathfrak{N}_{\psi}^{+}(q^{+}|_{\Sigma})+\mathfrak{N}_{\psi}^{-}(q^{-}|_{\Sigma})=\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)-F_{\psi}^{+}+F_{\psi}^{-}$
$\displaystyle\Rightarrow$
$\displaystyle\mp\mathfrak{N}_{\psi}^{\mp}(\left\llbracket
q\right\rrbracket|_{\Sigma})+\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(q^{\pm}|_{\Sigma})=\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)-F_{\psi}^{+}+F_{\psi}^{-}$
$\displaystyle\Rightarrow$
$\displaystyle~{}q^{\pm}|_{\Sigma}=\widetilde{\mathfrak{N}}^{-1}\left(\pm\mathfrak{N}_{\psi}^{\mp}(\sigma\mathcal{H}(\psi))+\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)-\left\llbracket
F_{\psi}\right\rrbracket\right),$ (5.20)

where
$\widetilde{\mathfrak{N}}:=\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}$
(and equivalently we have
$\mathfrak{N}_{\psi}^{\pm}=\frac{1}{2}(\widetilde{\mathfrak{N}}\pm(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-}))$
represents the mixed Dirichlet-to-Neumann operator and $\left\llbracket
F_{\psi}\right\rrbracket:=F_{\psi}^{+}-F_{\psi}^{-}$.

Plugging (5.20) back into the evolution equation of the free interface, we get

$\displaystyle\mathcal{P}_{\neq 0}(\rho^{+}\partial_{t}^{2}\psi)=$
$\displaystyle~{}\mathfrak{N}_{\psi}^{+}(q^{+}|_{\Sigma})+F_{\psi}^{+}=\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\left(\mathfrak{N}_{\psi}^{-}(\sigma\mathcal{H}(\psi))+\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)-F_{\psi}^{+}+F_{\psi}^{-}\right)+F_{\psi}^{+}$
$\displaystyle=$
$\displaystyle~{}\sigma\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\mathfrak{N}_{\psi}^{-}(\mathcal{H}(\psi))+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{-}-\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{+}+F_{\psi}^{+}+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\left(\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)\right)$
$\displaystyle=$
$\displaystyle~{}\sigma\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\mathfrak{N}_{\psi}^{-}(\mathcal{H}(\psi))+\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{+}+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{-}+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\left(\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)\right).$ (5.21)

Similarly, we have

$\displaystyle\mathcal{P}_{\neq
0}(\rho^{-}\partial_{t}^{2}\psi)=\sigma\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}\mathfrak{N}_{\psi}^{+}(\mathcal{H}(\psi))+\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{+}+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}F_{\psi}^{-}-\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}\left(\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)\right).$ (5.22)

Now, using the expressions of $\mathfrak{N}_{\psi}^{\pm}$ in terms of
$\widetilde{\mathfrak{N}}$ and
$\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-}$, we have

$\displaystyle\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}\mathfrak{N}_{\psi}^{-}f=$
$\displaystyle~{}\frac{1}{2}\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}(\widetilde{\mathfrak{N}}f+(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})f)=~{}\frac{1}{2}\mathfrak{N}_{\psi}^{+}f+\frac{1}{2}\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})f,$
(5.23)
$\displaystyle\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}\mathfrak{N}_{\psi}^{+}f=$
$\displaystyle~{}\frac{1}{2}\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}(\widetilde{\mathfrak{N}}f-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})f)=~{}\frac{1}{2}\mathfrak{N}_{\psi}^{-}f-\frac{1}{2}\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})f,$
(5.24)

and also for $g^{\pm}:\Sigma\to\mathbb{R}$ with
$\int_{\Sigma}g^{\pm}\,\mathrm{d}x^{\prime}=0$, we have

$\displaystyle\mathfrak{N}_{\psi}^{-}\widetilde{\mathfrak{N}}^{-1}g^{+}+\mathfrak{N}_{\psi}^{+}\widetilde{\mathfrak{N}}^{-1}g^{-}=$
$\displaystyle\frac{1}{2}(\widetilde{\mathfrak{N}}-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-}))\widetilde{\mathfrak{N}}^{-1}g^{+}+\frac{1}{2}(\widetilde{\mathfrak{N}}+(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-}))\widetilde{\mathfrak{N}}^{-1}g^{-}$
$\displaystyle=$
$\displaystyle~{}\frac{g^{+}+g^{-}}{2}-\frac{1}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left\llbracket
g\right\rrbracket.$ (5.25)

Let $f=\mathcal{H}(\psi)$ and $g^{\pm}=F_{\psi}^{\pm}$ in (5.23)-(5.2.1). We
find that $\eqref{psi equ +}+\eqref{psi equ -}$ can be written as

$\displaystyle\mathcal{P}_{\neq
0}(\rho^{+}\partial_{t}^{2}\psi)+\mathcal{P}_{\neq
0}(\rho^{-}\partial_{t}^{2}\psi)=$
$\displaystyle~{}\frac{\sigma}{2}\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))+F_{\psi}^{+}+F_{\psi}^{-}$
$\displaystyle+\frac{\sigma}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})(\mathcal{H}(\psi))$
$\displaystyle-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(F_{\psi}^{+}-F_{\psi}^{-})+(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left(\mathcal{P}_{\neq
0}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)\right).$ (5.26)

Recall that
$F_{\psi}^{\pm}=\mathfrak{F}_{\psi}^{\pm}-(\rho^{\pm}\partial_{t}^{2}\psi)_{\Sigma}$
and $\rho^{\pm}\partial_{t}^{2}\psi=\mathcal{P}_{\neq
0}(\rho^{\pm}\partial_{t}^{2}\psi)+(\rho^{\pm}\partial_{t}^{2}\psi)_{\Sigma}$
where

$\mathfrak{F}_{\psi}^{\pm}:=-N\cdot\nabla^{\varphi}q_{w}^{\pm}+\left(\bar{b}_{i}^{\pm}\bar{b}_{j}^{\pm}-\rho^{\pm}\bar{v}_{i}^{\pm}\bar{v}_{j}^{\pm}\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\rho^{\pm}\bar{v}^{\pm}\cdot\overline{\nabla})\partial_{t}\psi.$

Thus, the evolution equation of the free interface becomes

$\displaystyle(\rho^{+}+\rho^{-})\partial_{t}^{2}\psi=$
$\displaystyle~{}\frac{\sigma}{2}\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))+\left(\bar{b}_{i}^{+}\bar{b}_{j}^{+}-\rho^{+}\bar{v}_{i}^{+}\bar{v}_{j}^{+}+\bar{b}_{i}^{-}\bar{b}_{j}^{-}-\rho^{-}\bar{v}_{i}^{-}\bar{v}_{j}^{-}\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\rho^{+}\bar{v}_{i}^{+}+\rho^{-}\bar{v}_{i}^{-})\overline{\partial}{}_{i}\partial_{t}\psi$
$\displaystyle-N\cdot\nabla^{\varphi}q_{w}^{+}-N\cdot\nabla^{\varphi}q_{w}^{-}$
$\displaystyle+\frac{\sigma}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})(\mathcal{H}(\psi))-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left(\left\llbracket\mathfrak{F}_{\psi}-\rho\partial_{t}^{2}\psi\right\rrbracket\right),$
(5.27)

where the first line is expected to give the $\sqrt{\sigma}$-weighted
regularity (contributed by surface tension) and the non-weighted regularity
(provided that stability condition (1.29)) for the free interface, the second
line will be converted to the interior estimate of the right side of (5.15),
and the last line consists of remainder terms that can be directly controlled
by using paradifferential calculus.

####  5.2.2 Preliminaries on paradifferential calculus

In the equation (5.27), the term
$\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))$
is a fully nonlinear term. Although it is well-known that the Dirichlet-to-
Neumann operator is a first-order elliptic operator and the mean-curvature
operator is a second-order elliptic operator, it is still necessary for us to
find out their concrete forms and “symmetrize” the paradifferential
formulations in order for an explicit energy estimate. In the remaining part
of this paper, we will introduce several preliminary lemmas about
paradifferential calculus that have been proven in Alazard-Burq-Zuily [2].
Following the notations in Métivier [38], we first introduce the basic
definition of a paradifferential operator. Note that the dimension $d$ below
is not the same as the one in Section 1.

######  Definition 5.2 (Symbols).

Given $r\geq 0,~{}m\in\mathbb{R}$, we denote $\Gamma_{r}^{m}(\mathbb{T}^{d})$
to be the space of locally bounded functions $a(x^{\prime},\xi)$ on
$\mathbb{T}^{d}\times(\mathbb{R}^{d}\backslash\\{0\\})$, which are
$C^{\infty}$ with respect to $\xi(\xi\neq\mathbf{0})$, such that for any
$\alpha\in\mathbb{N}^{d},\xi\neq\mathbf{0}$, the function
$x^{\prime}\mapsto\partial_{\xi}^{\alpha}a(x^{\prime},\xi)$ belongs to
$W^{r,\infty}(\mathbb{T}^{d})$ and there exists a constant $C_{\alpha}$ such
that

$\left|\partial_{\xi}^{\alpha}a(\cdot,\xi)\right|_{W^{r,\infty}(\mathbb{T}^{d})}\leq
C_{\alpha}(1+|\xi|)^{m-|\alpha|},~{}~{}\forall|\xi|\geq 1/2.$

######  Definition 5.3 (Paradifferential operator).

Given a symbol $a$, we shall define the paradifferential operator $T_{a}$ by

$\displaystyle\widehat{T_{a}u}(\xi):=(2\pi)^{-d}\int_{\mathbb{R}^{d}}\tilde{\chi}(\xi-\eta,\eta)\hat{a}(\xi-\eta,\eta)\phi(\eta)\hat{u}(\eta)\,\mathrm{d}\eta$
(5.28)

where
$\hat{a}(\theta,\xi)=\int_{\mathbb{T}^{d}}\exp(-ix^{\prime}\cdot\theta)a(x^{\prime},\xi)\,\mathrm{d}x^{\prime}$
is the Fourier transform of $a$ in variable $x^{\prime}$. Here $\tilde{\chi}$
and $\phi$ are two given cut-off functions such that

$\phi(\eta)=0~{}~{}\text{for }|\eta|\leq 1,\quad\phi(\eta)=1~{}~{}\text{for
}|\eta|\geq 2,$

and $\tilde{\chi}(\theta,\eta)$ is homogeneous of degree 0 and satisfies that
for $0<\varepsilon_{1}<\varepsilon_{2}\ll 1$, $\tilde{\chi}(\theta,\eta)=1$ if
$|\theta|\leq\varepsilon_{1}|\eta|$ and $\tilde{\chi}(\theta,\eta)=0$ if
$|\theta|\geq\varepsilon_{2}|\eta|$. We also introduce the semi-norm

$\displaystyle M_{r}^{a}(a):=\sup_{|\alpha|\leq\frac{d}{2}+1+r}\sup_{|\xi|\geq
1/2}\left|(1+|\xi|)^{|\alpha|-m}\partial_{\xi}^{\alpha}a(\cdot,\xi)\right|_{W^{r,\infty}(\mathbb{T}^{d})}.$
(5.29)

For $m\in\mathbb{R}$, we say $T$ is of order $m$ if for all $s\in\mathbb{R}$,
$T$ is bounded from $H^{s}$ to $H^{s-m}$.

######  Proposition 5.1.

Let $m\in\mathbb{R}$. If $a\in\Gamma_{0}^{m}(\mathbb{T}^{d})$, then $T_{a}$ is
of order $m$. Moreover, for any $s\in\mathbb{R}$, there exists a constant $K$
such that $\|T_{a}\|_{H^{s}\to H^{s-m}}\leq KM_{0}^{m}(a)$.

######  Proposition 5.2 (Composition, [2, Theorem 3.7]).

Let $m\in\mathbb{R}$ and $r>0$. If
$a\in\Gamma_{r}^{m}(\mathbb{T}^{d}),~{}b\in\Gamma_{r}^{m^{\prime}}(\mathbb{T}^{d})$,
then $T_{a}T_{b}-T_{a\\#b}$ is of order $m+m^{\prime}-r$ where

$a\\#b:=\sum_{|\alpha|<r}\frac{1}{i^{|\alpha|}\alpha!}\partial_{\xi}^{\alpha}a\partial_{x^{\prime}}^{\alpha}b.$

Moreover, for all $s\in\mathbb{R}$, there exists a constant $K$ such that

$\displaystyle\|T_{a}T_{b}-T_{a\\#b}\|_{H^{s}\to H^{s-m-m^{\prime}+r}}\leq
KM_{r}^{m}(a)M_{r}^{m^{\prime}}(b).$ (5.30)

######  Proposition 5.3 (Adjoint, [2, Theorem 3.10]).

Let $m\in\mathbb{R}$, $r>0$ and $a\in\Gamma_{r}^{m}(\mathbb{T}^{d})$. We
denote by $(T_{a})^{*}$ the adjoint operator of $T_{a}$. Then
$(T_{a})^{*}-T_{a^{*}}$ is of order $m-r$ where

$a^{*}:=\sum_{|\alpha|<r}\frac{1}{i^{\alpha}\alpha!}\partial_{\xi}^{\alpha}\partial_{x^{\prime}}^{\alpha}\bar{a}.$

Moreover, for any $s\in\mathbb{R}$, there exists a constant $K$ such that
$\|(T_{a})^{*}-T_{a^{*}}\|_{H^{s}\to H^{s-m+r}}\leq KM_{r}^{m}(a)$.

The symbolic calculus adopted in this paper is not of $C^{\infty}$-regularity.
We shall introduce the following class of symbols. Here and thereafter in this
section, $\psi\in C([0,T];H^{s+\frac{1}{2}}(\mathbb{T}^{d}))$ is a given
function with $s>2+\frac{d}{2}$.

######  Definition 5.4.

Given $m\in\mathbb{R}$, we denote $\Sigma^{m}$ to be the class of symbols $a$
of the form $a=a^{(m)}+a^{(m-1)}$ with

$a^{(m)}(t,x^{\prime},\xi)=F(\overline{\nabla}_{x^{\prime}}\psi(t,x^{\prime}),\xi),\quad
a^{(m-1)}(t,x^{\prime},\xi)=\sum_{|\alpha|_{2}}G_{\alpha}(\overline{\nabla}_{x^{\prime}}\psi(t,x^{\prime}),\xi)\partial_{x^{\prime}}^{\alpha}\psi(t,x^{\prime})$

such that

  * i.

$T_{a}$ maps real-valued functions to real-valued functions;

  * ii.

$F$ is a $C^{\infty}$ real-valued functions of
$(\zeta,\xi)\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\backslash\\{0\\})$,
homogeneous of degree $m$ in $\xi$, such that there exists a continuous
function $K=K(\zeta)>0$ such that $F(\zeta,\xi)\geq K(\zeta)|\xi|^{m}$ for all
$(\zeta,\xi)\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\backslash\\{0\\})$;

  * iii.

$G_{\alpha}$ is a $C^{\infty}$ complex-valued function of
$(\zeta,\xi)\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\backslash\\{0\\})$,
homogeneous of degree $m-1$ in $\xi$.

######  Definition 5.5 (“Equivalence” of operators).

Given $m\in\mathbb{R}$ and consider two families of operators of order $m$:
$\\{A(t):t\in[0,T]\\}$ and $\\{B(t):t\in[0,T]\\}$, We say $A\sim B$ if $A-B$
is of order $m-1.5$ and satisfies the estimate: for all $r\in\mathbb{R}$ there
exists a continuous function $C(\cdot)$ such that

$\forall t\in[0,T],~{}~{}\|A(t)-B(t)\|_{H^{r}\to H^{r-(m-1.5)}}\leq
C(|\psi(t)|_{s+\frac{1}{2}}).$

From now on, we use the notation $|\cdot|_{s_{1}\to s_{2}}$ to represent the
operator norm $\|\cdot\|_{H^{s_{1}}\to H^{s_{2}}}$, and use the notation
$|\cdot|_{s}$ to represent $\|\cdot\|_{H^{s}(\mathbb{T}^{d})}$, as we only
apply paradifferential calculus on the free interface $\Sigma$. With this
definition, we have

######  Proposition 5.4 ([2, Prop. 4.3]).

Let $m,m^{\prime}\in\mathbb{R}$. Then

  1. 1.

If $a\in\Sigma^{m},~{}b\in\Sigma^{m^{\prime}}$, then $T_{a}T_{b}\sim
T_{a\\#b}$ where $a\\#b$ is given by

$a\\#b=a^{(m)}b^{(m^{\prime})}+a^{(m-1)}b^{(m^{\prime})}+a^{(m)}b^{(m^{\prime}-1)}+\frac{1}{i}\partial_{\xi}a^{(m)}\cdot\partial_{x^{\prime}}b^{(m^{\prime})}.$

  2. 2.

If $a\in\Sigma^{m}$, then $(T_{a})^{*}\sim T_{b}$ where $b\in\Sigma^{m}$ is
given by

$b=a^{(m)}+\overline{a^{(m-1)}}+\frac{1}{i}(\partial_{x^{\prime}}\cdot\partial_{\xi})a^{(m)}.$

As a corollary, we have

######  Corollary 5.5 ([2, Prop. 4.3(2)]).

If $a\in\Sigma^{m}$ satisfies $\textbf{Im
}a^{(m-1)}=-0.5(\partial_{\xi}\cdot\partial_{x^{\prime}})a^{(m)}$, then
$(T_{a})^{*}\sim T_{a}$.

The next proposition is significant for the estimate of Sobolev norms via
paradifferential calculus.

######  Proposition 5.6 ([2, Prop. 4.4 and 4.6]).

Let $m\in\mathbb{R},~{}r\in\mathbb{R}$. Then for all symbol $a\in\Sigma^{m}$
and $t\in[0,T]$, the following estimate holds.

$\displaystyle|T_{a(t)}u|_{r-m}\leq$
$\displaystyle~{}C(|\psi(t)|_{s-1})|u|_{r},$ (5.31)
$\displaystyle|u|_{r+m}\leq$
$\displaystyle~{}C(|\psi(t)|_{s-1})\left(|T_{a(t)}u|_{r}+|u|_{0}\right).$
(5.32)

####  5.2.3 Paralinearization of the nonlinear terms

Now we can start to paralinearize the term
$\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))$
in (5.27).

######  Lemma 5.7 (Paralinearization of the Dirichlet-to-Neumann operator, [4,
Section 4.4]).

For $f,\psi\in H^{s+\frac{1}{2}}(\mathbb{T}^{d})$, we have

$\displaystyle\mathfrak{N}_{\psi}^{\pm}f=T_{\Lambda^{\pm}}\psi+R_{\Lambda,1}^{\pm}(\psi,f)+R_{\Lambda,2}^{\pm}(\psi,f),$
(5.33)

with the symbols $\lambda^{\pm}=\lambda^{(1),\pm}+\lambda^{(0),\pm}$ give by

$\displaystyle\Lambda^{(1),\pm}=$
$\displaystyle\sqrt{(1+|\overline{\nabla}\psi|^{2})|\xi|^{2}-(\overline{\nabla}\psi\cdot\xi)^{2}},$
(5.34) $\displaystyle\Lambda^{(0),-}=-\overline{\Lambda^{(0),+}}=$
$\displaystyle\frac{1+|\overline{\nabla}\psi|^{2}|}{2\Lambda^{(1),-}}\left(\overline{\nabla}\cdot(\alpha^{(1)}\overline{\nabla}\psi)+i\partial_{\xi}\Lambda^{(1),-}\cdot\overline{\nabla}\alpha^{(1)}\right),$
(5.35)

and
$\alpha^{(1)}:=(\Lambda^{(1),-}+i\overline{\nabla}\psi\cdot\xi)/(1+|\overline{\nabla}\psi|^{2})$.
The remainder terms satisfy the following estimates

$\displaystyle|R_{\Lambda,1}^{\pm}(\psi,f)|_{s-\frac{1}{2}}\leq
C(|\psi|_{C^{2}},|f|_{3})|f|_{s+\frac{1}{2}},\quad|R_{\Lambda,2}^{\pm}(\psi,f)|_{s-\frac{1}{2}}\leq
C(|\psi|_{s-\frac{1}{2}})|\overline{\partial}{}f|_{s-2}.$ (5.36)

######  Lemma 5.8 (Paralinearization of the mean curvature operator, [2, Lemma
3.25]).

There holds $\mathcal{H}(\psi)=-T_{\mathfrak{H}}f+R_{\mathfrak{H}}$ where
$\mathfrak{H}=\mathfrak{H}^{(2)}+\mathfrak{H}^{(1)}$ is defined by

$\displaystyle\mathfrak{H}^{(2)}=$
$\displaystyle\frac{1}{\sqrt{1+|\overline{\nabla}\psi|^{2}}}\left(|\xi|^{2}-\frac{(\overline{\nabla}\psi\cdot\xi)^{2}}{1+|\overline{\nabla}\psi|^{2}}\right),$
(5.37) $\displaystyle\mathfrak{H}^{(1)}=$
$\displaystyle-\frac{i}{2}(\overline{\nabla}_{x^{\prime}}\cdot\partial_{\xi})\mathfrak{H}^{(2)},$
(5.38)

and the remainder term $R_{\mathfrak{H}}$ satisfies

$\displaystyle|R_{\mathfrak{H}}|_{2s-3}\leq C(|\psi|_{s+\frac{1}{2}}).$ (5.39)

With the paralinearization of operators $\mathfrak{N}_{\psi}^{\pm}$ and
$\mathcal{H}(\psi)$, the term
$\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))$
in (5.27) becomes

$\displaystyle\sigma\left(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-}\right)(\mathcal{H}(\psi))=-\sigma
T_{\Lambda}T_{\mathfrak{H}}\psi+\sigma\mathcal{R}_{\psi}^{\sigma},$ (5.40)

where $-\overline{\Lambda^{(0),+}}=\Lambda^{(0),-}$ shows that $\textbf{Re
}(\Lambda^{(0),+})+\textbf{Re }(\Lambda^{(0),-})=0$, $\textbf{Im
}(\Lambda^{(0),+})=\textbf{Im }(\Lambda^{(0),+})$ and thus

$\displaystyle\Lambda:=$
$\displaystyle~{}\underbrace{(\Lambda^{(1),+}+\Lambda^{(1),-})}_{=:\Lambda^{(1)}}+\underbrace{(\Lambda^{(0),+}+\Lambda^{(0),-})}_{=:\Lambda^{(0)}}=2\Lambda^{(1),-}+2i\textbf{Im
}(\Lambda^{(0),-})$ (5.41) $\displaystyle\mathcal{R}_{\psi}^{\sigma}:=$
$\displaystyle\sum_{\pm}T_{\Lambda^{\pm}}R_{\mathfrak{H}}+R_{\Lambda,1}^{\pm}(\psi,\mathcal{H}(\psi))+R_{\Lambda,2}^{\pm}(\psi,\mathcal{H}(\psi))\text{
and }|\mathcal{R}_{\psi}^{\sigma}|_{s-\frac{1}{2}}\leq
C(|\psi|_{s+\frac{1}{2}})|\psi|_{s+1}.$ (5.42)

In order for an explicit energy estimate for $\psi$ and $\psi_{t}$, we shall
symmetrize the 3-rd order paradifferential operator
$T_{\Lambda}T_{\mathfrak{H}}$. That is, find suitable symbols
${\mathfrak{m}}\in\Sigma^{1.5}$ and ${\mathfrak{n}}\in\Sigma^{0}$ such that
$T_{\mathfrak{n}}T_{\Lambda}T_{\mathfrak{H}}\sim
T_{\mathfrak{m}}T_{\mathfrak{m}}T_{\mathfrak{n}}$ and
$T_{\mathfrak{m}}\sim(T_{\mathfrak{m}})^{*}$.

######  Proposition 5.9 (Symmetrisation of the composition).

Let ${\mathfrak{n}}\in\Sigma^{0}$ and ${\mathfrak{m}}\in\Sigma^{1.5}$ be
defined by

$\displaystyle{\mathfrak{n}}:=$
$\displaystyle~{}\frac{1}{\sqrt[3]{2}\sqrt[4]{1+|\overline{\nabla}\psi|^{2}}}=2^{-\frac{1}{3}}|N|^{-\frac{1}{2}},$
(5.43) $\displaystyle{\mathfrak{m}}:=$
$\displaystyle~{}\underbrace{\sqrt{\mathfrak{H}^{(2)}\Lambda^{(1)}}}_{=:{\mathfrak{m}}^{(1.5)}}+\underbrace{\frac{1}{2i}(\partial_{\xi}\cdot\partial_{x^{\prime}})\sqrt{\mathfrak{H}^{(2)}\Lambda^{(1)}}}_{=:{\mathfrak{m}}^{(0.5)}}.$
(5.44)

Then $T_{\mathfrak{n}}T_{\lambda}T_{\mathfrak{H}}\sim
T_{\mathfrak{m}}T_{\mathfrak{m}}T_{\mathfrak{n}}$ and
$T_{\mathfrak{m}}\sim(T_{\mathfrak{m}})^{*}$ are both fulfilled.

###### Proof.

Given the symbol $\Lambda$ and $\mathfrak{H}$, we shall find suitable symbols
${\mathfrak{n}}\in\Sigma^{0},{\mathfrak{m}}\in\Sigma^{1.5}$ such that
${\mathfrak{n}}(x^{\prime},\xi)$ is independent of $\xi$ and
${\mathfrak{n}}\\#(\Lambda\\#\mathfrak{H})=({\mathfrak{m}}\\#{\mathfrak{m}})\\#{\mathfrak{n}}$,
i.e.,

$\displaystyle~{}{\mathfrak{n}}^{(0)}(\Lambda\\#\mathfrak{H})+{\mathfrak{n}}^{(-1)}\Lambda^{(1)}\mathfrak{H}^{(2)}+\frac{1}{i}\partial_{\xi}{\mathfrak{n}}^{(0)}\cdot\partial_{x^{\prime}}(\Lambda^{(1)}\mathfrak{H}^{(2)})$
$\displaystyle=$
$\displaystyle~{}({\mathfrak{m}}\\#{\mathfrak{m}}){\mathfrak{n}}^{(0)}+({\mathfrak{m}}^{(1.5)})^{2}{\mathfrak{n}}^{(-1)}+\frac{1}{i}\partial_{\xi}(({\mathfrak{m}}^{(1.5)})^{2})\cdot\partial_{x^{\prime}}{\mathfrak{n}}^{(0)}.$

Recall that

$\displaystyle(\Lambda\\#\mathfrak{H})=$
$\displaystyle~{}\Lambda^{(1)}\mathfrak{H}^{(2)}+\Lambda^{(0)}\mathfrak{H}^{(2)}+\Lambda^{(1)}\mathfrak{H}^{(1)}+\frac{1}{i}\partial_{\xi}\mathfrak{H}^{(2)}\cdot\partial_{x^{\prime}}\Lambda^{(1)},$
$\displaystyle({\mathfrak{m}}\\#{\mathfrak{m}})=$
$\displaystyle~{}({\mathfrak{m}}^{(1.5)})^{2}+2({\mathfrak{m}}^{(1.5)})({\mathfrak{m}}^{(0.5)})+\frac{1}{i}\partial_{\xi}({\mathfrak{m}}^{(1.5)})\cdot\partial_{x^{\prime}}({\mathfrak{m}}^{(1.5)}).$

We choose the principal symbol
${\mathfrak{m}}^{(1.5)}:=\sqrt{\Lambda^{(1)}\mathfrak{H}^{(2)}}$ in order for
cancelling the leading-order symbols. Since we require
$(T_{{\mathfrak{m}}})^{*}\sim T_{{\mathfrak{m}}^{*}}$, we must have
$\textbf{Im
}({\mathfrak{m}}^{(0.5)})=-0.5(\partial_{x^{\prime}}\cdot\partial_{\xi}){\mathfrak{m}}^{(1.5)}$
(cf. [2, Prop. 4.3]). With this choice for ${\mathfrak{m}}$, it remains to
solve the symbolic equation

${\mathfrak{n}}^{(0)}(\Lambda\\#\mathfrak{H}-{\mathfrak{m}}\\#{\mathfrak{m}})=\frac{1}{i}\partial_{\xi}((\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{x^{\prime}}{\mathfrak{n}}^{(0)}-\frac{1}{i}\partial_{x^{\prime}}(\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{\xi}{\mathfrak{n}}^{(0)},$
(5.45)

with

$\displaystyle\Lambda\\#\mathfrak{H}-{\mathfrak{m}}\\#{\mathfrak{m}}=\Lambda^{(0)}\mathfrak{H}^{(2)}+\Lambda^{(1)}\mathfrak{H}^{(1)}-2({\mathfrak{m}}^{(1.5)})({\mathfrak{m}}^{(0.5)})+\frac{1}{i}\partial_{\xi}\mathfrak{H}^{(2)}\cdot\partial_{x^{\prime}}\Lambda^{(1)}-\frac{1}{i}\partial_{\xi}({\mathfrak{m}}^{(1.5)})\cdot\partial_{x^{\prime}}({\mathfrak{m}}^{(1.5)}).$

The sub-principle ${\mathfrak{n}}^{(-1)}$ does not appear, so we can choose
${\mathfrak{n}}^{(-1)}=0$. Since the principal symbols of $\Lambda$ and
$\mathfrak{H}$ are real-valued, we now just need to solve

$\textbf{Re
}(\Lambda\\#\mathfrak{H}-{\mathfrak{m}}\\#{\mathfrak{m}})=0,~{}~{}{\mathfrak{n}}^{(0)}\textbf{Im
}(\Lambda\\#\mathfrak{H}-{\mathfrak{m}}\\#{\mathfrak{m}})=-\partial_{\xi}((\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{x^{\prime}}{\mathfrak{n}}^{(0)}+\partial_{x^{\prime}}(\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{\xi}{\mathfrak{n}}^{(0)}.$

The condition for the real part is fulfilled if we have

$\underbrace{\textbf{Re
}(\Lambda^{(0)})}_{=0}\mathfrak{H}^{(2)}=2{\mathfrak{m}}^{(1.5)}\textbf{Re
}({\mathfrak{m}}^{(0.5)})\Rightarrow\textbf{Re }({\mathfrak{m}}^{(0.5)})=0.$

For the imaginary part, inserting the symbols
$\mathfrak{H}^{(1)},~{}\textbf{Im }(\Lambda^{(0)}),~{}{\mathfrak{m}}^{(1.5)}$
and $\textbf{Im }({\mathfrak{m}}^{(0.5)})$, we get

$\displaystyle\textbf{Im
}(\Lambda\\#\mathfrak{H}-{\mathfrak{m}}\\#{\mathfrak{m}})=\frac{1}{2}\partial_{\xi}\mathfrak{H}^{(2)}\cdot\partial_{x^{\prime}}\Lambda^{(1)}-\frac{1}{2}\partial_{x^{\prime}}\mathfrak{H}^{(2)}\cdot\partial_{\xi}\Lambda^{(1)},$

and thus we need to solve

$\displaystyle{\mathfrak{n}}^{(0)}\left(\frac{1}{2}\partial_{\xi}\mathfrak{H}^{(2)}\cdot\partial_{x^{\prime}}\Lambda^{(1)}-\frac{1}{2}\partial_{x^{\prime}}\mathfrak{H}^{(2)}\cdot\partial_{\xi}\Lambda^{(1)}\right)=-\partial_{\xi}((\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{x^{\prime}}{\mathfrak{n}}^{(0)}+\partial_{x^{\prime}}(\Lambda^{(1)}\mathfrak{H}^{(2)})\cdot\partial_{\xi}{\mathfrak{n}}^{(0)}.$
(5.46)

Notice that $\mathfrak{H}^{(2)}=(c\Lambda^{(1)})^{2}$ with
$c=\frac{1}{2}(1+|\overline{\nabla}\psi|^{2})^{-\frac{3}{4}}$. Plugging it to
the above equation, after a long and tedious calculation, we get the following
relation

${\mathfrak{n}}^{(0)}\left(c^{2}\partial_{x^{\prime}}\Lambda^{(1)}(\Lambda^{(1)})-(\partial_{x^{\prime}}c)c(\Lambda^{(1)})^{2}-c^{2}\partial_{x^{\prime}}(\Lambda^{(1)})\Lambda^{(1)}\right)=-3c^{2}(\Lambda^{(1)})^{2}\partial_{x^{\prime}}{\mathfrak{n}}^{(0)},$

that is,

$\frac{\partial_{x^{\prime}}{\mathfrak{n}}^{(0)}}{{\mathfrak{n}}^{(0)}}=\frac{-(\partial_{x^{\prime}}c)c(\Lambda^{(1)})^{2}}{-3c^{2}(\Lambda^{(1)})^{2}}=\frac{1}{3}\frac{\partial_{x^{\prime}}c}{c}\Rightarrow{\mathfrak{n}}^{(0)}=c^{\frac{1}{3}}=2^{-\frac{1}{3}}(1+|\overline{\nabla}\psi|^{2})^{-\frac{1}{4}}.$

∎

We expect to take $(s-\frac{1}{2})$-th order derivatives in (5.27). In view of
the paradifferential formulation, we shall alternatively take
$T_{{\mathfrak{M}}}$ with

$\displaystyle{\mathfrak{M}}:=({\mathfrak{m}}^{(1.5)})^{\frac{2s-1}{3}}=2^{\frac{2s-1}{6}}|\xi|^{s-\frac{1}{2}}\left(1-\left|\frac{N}{|N|}\cdot\frac{\xi}{|\xi|}\right|^{2}\right)^{\frac{s}{2}-\frac{1}{4}}\in\Sigma^{s-\frac{1}{2}}$
(5.47)

for sake of simplicity. Below, we list several commutator estimates for the
paradifferential operators.

######  Lemma 5.10.

For any $r\in\mathbb{R},~{}s>2+\frac{d}{2}$, any functions $a$ and $f$, the
following commutator estimates hold

$\displaystyle\left|\left[T_{\mathfrak{M}},T_{\mathfrak{m}}\right]\right|_{r+s-1\to
r}\leq$ $\displaystyle~{}C\left(|\psi|_{s+0.5}\right),$
$\displaystyle\left|[T_{\mathfrak{n}},T_{a}]f\right|_{s-\frac{1}{2}}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}})|a|_{W^{1,\infty}}|f|_{s-1.5},$
$\displaystyle\left|\left[T_{\mathfrak{M}},a\right]T_{\mathfrak{n}}f\right|_{0}+\left|T_{\mathfrak{M}}\left[T_{\mathfrak{n}},a\right]f\right|_{0}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}})|a|_{s-0.5}|f|_{s-1.5},$
$\displaystyle\left|[T_{\mathfrak{m}},a]f\right|_{0}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}})|a|_{s-0.5}|f|_{0.5}.$

We also need to commute $\partial_{t}$ with paradifferential operators. These
steps will generate paradifferential operators whose symbols are spatial or
time derivatives.

######  Lemma 5.11.

For any $r\in\mathbb{R}$, the following estimates hold

$\displaystyle|T_{\partial_{t}{\mathfrak{n}}}|_{r\to
r}+|T_{\partial_{t}{\mathfrak{M}}}|_{r\to r-(s-0.5)}\leq$
$\displaystyle~{}C(|\psi|_{W^{1,\infty}},|\partial_{t}\psi|_{W^{1,\infty}}),$
$\displaystyle|T_{\partial_{t}^{2}{\mathfrak{n}}}|_{r\to
r}+|T_{\partial_{t}^{2}{\mathfrak{M}}}|_{r\to r-(s-0.5)}\leq$
$\displaystyle~{}C\left(\left|\overline{\nabla}\psi,\partial_{t}\psi,\partial_{t}^{2}\psi\right|_{W^{1,\infty}}\right),$
$\displaystyle|T_{\overline{\partial}{}{\mathfrak{n}}}|_{r\to
r}+|T_{\overline{\partial}{}{\mathfrak{M}}}|_{r\to r-(s-0.5)}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}}),$
$\displaystyle|T_{\partial_{t}{\mathfrak{m}}}|_{r\to
r-1.5}+|T_{\overline{\partial}{}{\mathfrak{m}}}|_{r\to r-1.5}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}},|\overline{\nabla}\partial_{t}\psi|_{W^{1,\infty}}).$

###  5.3 Uniform estimates for the free interface

With the symmetrized paralinearization of
$(\mathfrak{N}_{\psi}^{+}+\mathfrak{N}_{\psi}^{-})\mathcal{H}(\psi)$ derived
in Section 5.2.2, we can now prove the uniform-in-$({\varepsilon,\sigma})$
estimates of $\psi$ under the stability condition (1.29). The equation (5.27)
can be written as

$\displaystyle(\rho^{+}+\rho^{-})\partial_{t}^{2}\psi=$
$\displaystyle~{}-\frac{\sigma}{2}T_{\Lambda}T_{\mathfrak{H}}\psi-(\rho^{+}+\rho^{-})(\bar{\mathbf{w}}_{i}\bar{\mathbf{w}}_{j})\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi-2(\rho^{+}+\rho^{-})\bar{\mathbf{w}}_{i}\overline{\partial}{}_{i}\partial_{t}\psi$
$\displaystyle+\left(\rho^{+}(\bar{\mathbf{b}}_{i}^{+}\bar{\mathbf{b}}_{j}^{+}-\bar{\mathbf{u}}_{i}\bar{\mathbf{u}}_{j})+\rho^{-}(\bar{\mathbf{b}}_{i}^{-}\bar{\mathbf{b}}_{j}^{-}-\bar{\mathbf{u}}_{i}\bar{\mathbf{u}}_{j})\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi$
$\displaystyle-(N\cdot\nabla^{\varphi}q_{w}^{+}+N\cdot\nabla^{\varphi}q_{w}^{-})+\Psi^{R},$
(5.48)

where $T_{\Lambda},T_{\mathfrak{H}}$ are the paradifferential operators
defined in Proposition 5.7 and Proposition 5.8, the quantities
$\mathbf{w},\mathbf{u},\mathbf{b}$ are defined by

$\displaystyle\mathbf{w}:=\frac{\rho^{+}v^{+}+\rho^{-}v^{-}}{\rho^{+}+\rho^{-}},\quad\mathbf{u}:=\frac{\sqrt{\rho^{+}\rho^{-}}}{\rho^{+}+\rho^{-}}\left\llbracket
v\right\rrbracket,\quad\mathbf{b}^{\pm}:=\frac{b^{\pm}}{\sqrt{\rho^{\pm}}},$
(5.49)

and $\Psi^{R}$ is defined by

$\displaystyle\Psi^{R}:=\frac{\sigma}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})(\mathcal{H}(\psi))-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left(\left\llbracket\mathfrak{F}_{\psi}-\rho\partial_{t}^{2}\psi\right\rrbracket\right)+\frac{\sigma}{2}\mathcal{R}_{\psi}^{\sigma}.$
(5.50)

We pick $s=4$ in the paradifferential operator $T_{\mathfrak{M}}$, that is,
${\mathfrak{M}}=({\mathfrak{m}}^{(1.5)})^{\frac{7}{3}}\in\Sigma^{3.5}$ and
then consider the energy functionals

$\displaystyle\mathcal{E}(t):=$
$\displaystyle~{}\frac{1}{2}\int_{\Sigma}(\rho^{+}+\rho^{-})\left|(\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x^{\prime}+\frac{1}{4}\int_{\Sigma}\left|\sqrt{\sigma}T_{\mathfrak{m}}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|_{0}^{2}\,\mathrm{d}x^{\prime},$
(5.51) $\displaystyle\widetilde{\mathcal{E}}(t):=$
$\displaystyle~{}\frac{1}{2}\int_{\Sigma}\rho^{+}\left(\left|\bar{\mathbf{b}}^{+}\cdot\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}-\left|\bar{\mathbf{u}}\cdot\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\right)+\rho^{-}\left(\left|\bar{\mathbf{b}}^{-}\cdot\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}-\left|\bar{\mathbf{u}}\cdot\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\right)\,\mathrm{d}x^{\prime}.$
(5.52)

######  Lemma 5.12 (Comparison between $\mathcal{E},\widetilde{\mathcal{E}}$
and Sobolev norms).

For any fixed $\sigma>0$, we have the following relations between
$\mathcal{E},\widetilde{\mathcal{E}}$ and standard Sobolev norms.

$\displaystyle\left|\sqrt{\sigma}\psi\right|_{5}^{2}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\mathcal{E}(t)+|\sqrt{\sigma}\psi|_{0}^{2}\right),$
$\displaystyle|\psi|_{4.5}^{2}\lesssim$
$\displaystyle~{}|\sqrt{\sigma}\psi|_{5}^{2}+\sigma^{-1}|\psi|_{0}^{2},$
$\displaystyle|\psi_{t}|_{3.5}^{2}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi,\overline{\nabla}\psi_{t},\bar{v}^{\pm},\rho^{\pm}|_{W^{1,\infty}})\left(\mathcal{E}(t)+|\psi|_{4.5}^{2}+|\psi_{t}|_{0}^{2}\right)$

where $C(\cdot)$ represents a generic positive continuous function in its
arguments. Moreover, when the stability condition (1.29) holds, there exist
positive continuous functions $C_{1},C_{1}^{\prime},C_{1}^{\prime\prime}$
depending on
$|\overline{\nabla}\psi,\bar{v}^{\pm},\bar{b}^{\pm},\rho^{\pm}|_{W^{1,\infty}}$
and independent of $\sigma$, such that

$\displaystyle
C_{1}\left(|\overline{\nabla}\psi,\bar{v}^{\pm},\bar{b}^{\pm},\rho^{\pm}|_{W^{1,\infty}}\right)\left|\psi\right|_{4.5}^{2}\leq\widetilde{\mathcal{E}}(t)+C_{1}^{\prime}|\psi|_{0}^{2},~{}~{}\widetilde{\mathcal{E}}(t)\leq
C_{1}^{\prime\prime}\left(|\overline{\nabla}\psi,\bar{v}^{\pm},\bar{b}^{\pm},\rho^{\pm}|_{W^{1,\infty}}\right)\left|\psi\right|_{4.5}^{2}.$

###### Proof.

Recall that $T_{\mathfrak{M}}$ and $T_{\mathfrak{m}}$ are paradifferential
operators of order 3.5 and 1.5 respectively, thus the first inequality is a
direct consequence of Proposition 5.6. The second inequality is a directly
consequence of Sobolev interpolation and Young’s inequality

$|\psi|_{4.5}^{2}\leq|\sqrt{\sigma}\psi|_{5}^{1.8}|\sigma^{-\frac{1}{2}}\psi|_{0}^{0.2}\leq\frac{|\sqrt{\sigma}\psi|_{5}^{2}}{10/9}+\frac{|\psi|_{0}^{2}}{10\sigma}.$

To prove the third inequality, we again use Proposition 5.6 to get

$\displaystyle|\psi_{t}|_{3.5}^{2}\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi|_{W^{1,\infty}})\left(|T_{\mathfrak{M}}T_{\mathfrak{n}}\psi_{t}|_{0}^{2}+|\psi_{t}|_{0}^{2}\right)$
$\displaystyle\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi,\bar{v}^{\pm},\rho^{\pm}|_{W^{1,\infty}})\left(\mathcal{E}(t)+|T_{\mathfrak{M}}T_{\partial_{t}{\mathfrak{n}}}\psi|_{0}^{2}+|T_{\partial_{t}{\mathfrak{M}}}T_{\mathfrak{n}}\psi|_{0}^{2}+|\psi_{t}|_{0}^{2}\right)$
$\displaystyle\leq$
$\displaystyle~{}C(|\overline{\nabla}\psi,\overline{\nabla}\psi_{t},\bar{v}^{\pm},\rho^{\pm}|_{W^{1,\infty}})\left(\mathcal{E}(t)+|\psi|_{3.5}^{2}+|\psi_{t}|_{0}^{2}\right).$

For the last inequality in this lemma, the right side is trivial. When the
stability condition (1.29) holds, it suffices to prove that
$\widetilde{\mathcal{E}}(t)$ is a positive-definite energy, then the left side
automatically holds. Multiplying $(\rho^{+}\rho^{-})^{-\frac{1}{2}}$ in
(1.29), the stability condition becomes

$\displaystyle\exists\delta_{0}\in(0,\frac{1}{8}),\quad|\bar{\mathbf{b}}^{+}\times\bar{\mathbf{b}}^{-}|\geq(1-\delta_{0})^{-1}|\bar{\mathbf{b}}^{\pm}\times\left\llbracket\bar{v}\right\rrbracket|\sqrt{1+(c_{A}^{\pm}/c_{s}^{\pm})^{2}}>(1-\delta_{0})^{-1}|\bar{\mathbf{b}}^{\pm}\times\left\llbracket\bar{v}\right\rrbracket|.$
(5.53)

Since $\left\llbracket v\right\rrbracket\cdot N=0$ and $\mathbf{b}^{\pm}$ are
nonzero and not collinear, we may assume $\left\llbracket
v\right\rrbracket=c_{1}\mathbf{b}^{+}+c_{2}\mathbf{b}^{-}$. Plugging this into
the stability condition, we get $c_{1},c_{2}\leq 1-\delta_{0}$. Using Cauchy-
Schwarz inequality, we derive that

$\displaystyle\inf_{\begin{subarray}{c}\mathbf{z}\in\mathbb{R}^{2}\\\
|\mathbf{z}|=1\end{subarray}}(1-\delta_{0})^{2}\left((\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+2(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})+(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}\right)-\left(\left\llbracket\bar{v}\right\rrbracket\cdot\mathbf{z}\right)^{2}\geq
0$

Invoking
$\bar{\mathbf{u}}=\frac{\sqrt{\rho^{+}\rho^{-}}}{\rho^{+}+\rho^{-}}\left\llbracket\bar{v}\right\rrbracket$
and using the non-collinearity, the above inequality implies that

$\displaystyle~{}\inf_{\begin{subarray}{c}\mathbf{z}\in\mathbb{R}^{2}\\\
|\mathbf{z}|=1\end{subarray}}\left(\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+2\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})+\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}-(\rho^{+}+\rho^{-})\left(\bar{\mathbf{u}}\cdot\mathbf{z}\right)^{2}\right)>0.$

Notice that

$\displaystyle\rho^{+}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+\rho^{-}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}-\left(\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+2\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})+\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}\right)$
$\displaystyle=$
$\displaystyle~{}\frac{1}{\rho^{+}+\rho^{-}}\left((\rho^{+})^{2}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}-2\rho^{+}\rho^{-}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})+(\rho^{-})^{2}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}\right)\geq
0.$

Thus, it implies that

$\displaystyle~{}\inf_{\begin{subarray}{c}\mathbf{z}\in\mathbb{R}^{2}\\\
|\mathbf{z}|=1\end{subarray}}\left(\rho^{+}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+\rho^{-}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}-(\rho^{+}+\rho^{-})\left(\bar{\mathbf{u}}\cdot\mathbf{z}\right)^{2}\right)>0,$
(5.54)

or equivalently, there exists some $\delta_{0}^{\prime}>0$ such that

$\displaystyle~{}\inf_{\mathbf{z}\in\mathbb{R}^{2}}\left(\rho^{+}(\bar{\mathbf{b}}^{+}\cdot\mathbf{z})^{2}+\rho^{-}(\bar{\mathbf{b}}^{-}\cdot\mathbf{z})^{2}-(\rho^{+}+\rho^{-})\left(\bar{\mathbf{u}}\cdot\mathbf{z}\right)^{2}\right)\geq
2\delta_{0}^{\prime}|\mathbf{z}|^{2}.$ (5.55)

Now let $\mathbf{z}=\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi$,
the above inequality shows that
$\widetilde{\mathcal{E}}(t)\geq\delta_{0}^{\prime}|\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi|_{0}^{2}\gtrsim|\psi|_{4.5}^{2}-|\psi|_{0}^{2}$.
∎

######  Remark 5.1 (The 2D case).

When the space dimension $d=2$, we no longer have the non-collinearity, but
the stability condition (1.37) still guarantees the ellipticity of the
corresponding second-order differential operator, i.e.,

$\exists\delta_{0}^{\prime}>0,\quad\rho^{+}\left((\mathbf{b}_{1}^{+})^{2}-\mathbf{u}_{1}^{2}\right)+\rho^{-}\left((\mathbf{b}_{1}^{-})^{2}-\mathbf{u}_{1}^{2}\right)\geq\delta_{0}^{\prime}.$

In fact, the stability condition (1.37) implies
$|\mathbf{b}_{1}^{+}|+|\mathbf{b}_{1}^{-}|\geq(1+\delta_{0})|\left\llbracket
v_{1}\right\rrbracket|$. Taking square and invoking
$\mathbf{u}:=\frac{\sqrt{\rho^{+}\rho^{-}}}{\rho^{+}+\rho^{-}}\left\llbracket
v\right\rrbracket$, we get

$\frac{\rho^{+}\rho^{-}}{\rho^{+}+\rho^{-}}\left((\mathbf{b}_{1}^{+})^{2}+2\mathbf{b}_{1}^{+}\mathbf{b}_{1}^{-}+(\mathbf{b}_{1}^{-})^{2}\right)\geq(1+\delta_{0})(\rho^{+}+\rho^{-})\mathbf{u}_{1}^{2},$

in which we find that the left side does not exceed
$(\rho^{+}+\rho^{-})^{-1}\left(\rho^{+}(\mathbf{b}_{1}^{+})^{2}+\rho^{-}(\mathbf{b}_{1}^{-})^{2}\right)$
by direct calculation. The desired result immediately follows thanks to
$|\mathbf{u}_{1}|>0$ (otherwise the interface is not a vortex sheet).

In view of Lemma 5.12, it suffices to prove energy estimates for
$\mathcal{E}(t)$ and $\widetilde{\mathcal{E}}(t)$ under the stability
condition (1.29). We start with the estimate of $|\psi_{t}|_{3.5}$.

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\int_{\Sigma}(\rho^{+}+\rho^{-})\left|(\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle\int_{\Sigma}(\rho^{+}+\rho^{-})(\partial_{t}^{2}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}+\int_{\Sigma}(\rho^{+}+\rho^{-})(\bar{\mathbf{w}}\cdot\overline{\nabla}\partial_{t}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle+\int_{\Sigma}(\rho^{+}+\rho^{-})(\partial_{t}\bar{\mathbf{w}}\cdot\overline{\nabla}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}+\frac{1}{2}\int_{\Sigma}\partial_{t}(\rho^{+}+\rho^{-})\left|(\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x^{\prime}$
$\displaystyle=:$ $\displaystyle~{}I_{0}+I_{1}+I_{1}^{R}+I_{2}^{R}.$ (5.56)

The remainder terms are easy to control. Using Proposition 5.6, we have

$\displaystyle I_{1}^{R}+I_{2}^{R}\leq
C\left(\left|\rho^{\pm},\partial_{t}\rho^{\pm},\bar{v}^{\pm},\partial_{t}\bar{v}^{\pm}\right|_{L^{\infty}},|\psi|_{3}\right)\left(|\psi_{t}|_{3.5}^{2}+|\psi|_{4.5}^{2}\right).$
(5.57)

For the main term $I_{0}$, we first commute
$(\rho^{+}+\rho^{-})\partial_{t}^{2}$ with $T_{\mathfrak{M}}T_{\mathfrak{n}}$

$\displaystyle(\rho^{+}+\rho^{-})\partial_{t}^{2}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi=$
$\displaystyle~{}(\rho^{+}+\rho^{-})\left(T_{\mathfrak{M}}T_{\mathfrak{n}}\partial_{t}^{2}\psi+(T_{\partial_{t}^{2}{\mathfrak{M}}}T_{\mathfrak{n}}+T_{{\mathfrak{M}}}T_{\partial_{t}^{2}{\mathfrak{n}}})\psi+2(T_{\partial_{t}{\mathfrak{M}}}T_{{\mathfrak{n}}}+T_{\mathfrak{M}}T_{\partial_{t}{\mathfrak{n}}})\partial_{t}\psi\right)$
$\displaystyle=$
$\displaystyle~{}T_{\mathfrak{M}}T_{\mathfrak{n}}\left((\rho^{+}+\rho^{-})\partial_{t}^{2}\psi\right)-\left([T_{\mathfrak{M}},\rho^{+}+\rho^{-}]T_{\mathfrak{n}}\partial_{t}^{2}\psi+T_{\mathfrak{M}}([T_{\mathfrak{n}},\rho^{+}+\rho^{-}]\partial_{t}^{2}\psi)\right)$
$\displaystyle+(\rho^{+}+\rho^{-})\left((T_{\partial_{t}^{2}{\mathfrak{M}}}T_{\mathfrak{n}}+T_{{\mathfrak{M}}}T_{\partial_{t}^{2}{\mathfrak{n}}})\psi+2(T_{\partial_{t}{\mathfrak{M}}}T_{{\mathfrak{n}}}+T_{\mathfrak{M}}T_{\partial_{t}{\mathfrak{n}}})\partial_{t}\psi\right).$

The commutators can be controlled straightforwardly thanks to Lemma 5.10 and
Lemma 5.11:

$\displaystyle\left|[T_{\mathfrak{M}},\rho^{+}+\rho^{-}]T_{\mathfrak{n}}\partial_{t}^{2}\psi\right|_{0}+\left|T_{\mathfrak{M}}([T_{\mathfrak{n}},\rho^{+}+\rho^{-}]\partial_{t}^{2}\psi)\right|_{0}\leq
C(|\overline{\nabla}\psi|_{W^{1,\infty}})\left(|\rho^{+}+\rho^{-}|_{3.5}|\partial_{t}^{2}\psi|_{2.5}\right),$
$\displaystyle\left|(T_{\partial_{t}^{2}{\mathfrak{M}}}T_{\mathfrak{n}}+T_{{\mathfrak{M}}}T_{\partial_{t}^{2}{\mathfrak{n}}})\psi\right|_{0}+\left|(T_{\partial_{t}{\mathfrak{M}}}T_{{\mathfrak{n}}}+T_{\mathfrak{M}}T_{\partial_{t}{\mathfrak{n}}})\partial_{t}\psi\right|_{0}\leq
C(|\psi_{tt},\psi_{t},\overline{\nabla}\psi|_{W^{1,\infty}})\left(|\psi|_{3.5}+|\psi_{t}|_{3.5}\right).$

In the remaining of this section, we no longer explicitly write the
commutators between the paradifferential operators and functions or
$\partial_{t},\overline{\partial}{}_{i}$, as they can be controlled in the
same way as above. Instead, we will again use the notation
$\stackrel{{\scriptstyle L}}{{=}}$ to skip these terms and analyze the main
terms.

Then we can plug the equation (5.48) into the integral to get

$\displaystyle I_{00}:=$
$\displaystyle\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}((\rho^{+}+\rho^{-})\partial_{t}^{2}\psi)\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{\sigma}{2}\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}T_{\Lambda}T_{\mathfrak{H}}\psi\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-2\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}((\rho^{+}+\rho^{-})\bar{\mathbf{w}}_{i}\overline{\partial}{}_{i}\partial_{t}\psi)\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}((\rho^{+}+\rho^{-})\bar{\mathbf{w}}_{i}\bar{\mathbf{w}}_{j}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi)\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle+\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}\left(\rho^{+}(\bar{\mathbf{b}}_{i}^{+}\bar{\mathbf{b}}_{j}^{+}-\bar{\mathbf{u}}_{i}\bar{\mathbf{u}}_{j})+\rho^{-}(\bar{\mathbf{b}}_{i}^{-}\bar{\mathbf{b}}_{j}^{-}-\bar{\mathbf{u}}_{i}\bar{\mathbf{u}}_{j})\right)\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle-\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}\left(N\cdot\nabla^{\varphi}q_{w}^{+}+N\cdot
q_{w}^{-}\right)\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle+\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}\Psi^{R}\right)\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}=:I_{000}+I_{001}+I_{002}+I_{003}+I_{0}^{w}+I_{0}^{R}.$
(5.58)

Since Proposition 5.9 indicates that
$T_{\mathfrak{n}}T_{\Lambda}T_{\mathfrak{H}}\sim
T_{\mathfrak{m}}T_{\mathfrak{m}}T_{\mathfrak{n}}$ and
$(T_{\mathfrak{m}})^{*}\sim T_{\mathfrak{m}}$, we have

$\displaystyle~{}T_{\mathfrak{M}}T_{\mathfrak{n}}T_{\Lambda}T_{\mathfrak{H}}\psi\stackrel{{\scriptstyle
L}}{{=}}T_{\mathfrak{M}}T_{\mathfrak{m}}T_{\mathfrak{m}}T_{\mathfrak{n}}\psi$
$\displaystyle=$
$\displaystyle~{}(T_{\mathfrak{m}})^{*}T_{\mathfrak{m}}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi+\big{(}((T_{\mathfrak{m}})^{*}-T_{\mathfrak{m}})T_{\mathfrak{M}}+T_{\mathfrak{m}}[T_{\mathfrak{M}},T_{\mathfrak{m}}]+[T_{\mathfrak{M}},T_{\mathfrak{m}}]T_{\mathfrak{m}}\big{)}T_{\mathfrak{n}}\psi\stackrel{{\scriptstyle
L}}{{=}}(T_{\mathfrak{m}})^{*}T_{\mathfrak{m}}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi$

and using the duality, we get

$\displaystyle I_{000}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\frac{\sigma}{2}\int_{\Sigma}(T_{\mathfrak{m}}T_{{\mathfrak{M}}}T_{\mathfrak{n}}\psi)\,T_{\mathfrak{m}}(\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\,\mathrm{d}x^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{\sigma}{4}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\left|T_{\mathfrak{m}}T_{{\mathfrak{M}}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x^{\prime}$
$\displaystyle+\frac{\sigma}{2}\int_{\Sigma}(T_{\mathfrak{m}}T_{{\mathfrak{M}}}T_{\mathfrak{n}}\psi)\,\left(T_{\partial_{t}{\mathfrak{m}}}+\bar{\mathbf{w}}_{i}T_{\overline{\partial}{}_{i}{\mathfrak{m}}}+\frac{1}{2}(\overline{\nabla}\cdot\bar{\mathbf{w}})-[T_{{\mathfrak{m}}},\bar{\mathbf{w}}_{i}]\overline{\partial}{}_{i}\right)T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\,\mathrm{d}x^{\prime},$
(5.59)

where the second term can be directly controlled (uniformly in $\sigma$) by
$\mathcal{E}(t)C\left(|\overline{\nabla}\psi,\psi_{t},\bar{v}^{\pm},\rho^{\pm}|_{W^{1,\infty}}\right)$
thanks to Lemma 5.10. Next we analyze $I_{1}+I_{001}$ and $I_{002},I_{003}$.
For a generic function $\mathbf{a}\in H^{3.5}(\Sigma\to\mathbb{R}^{2})$ and a
generic $\bm{\rho}\in H^{3.5}(\Sigma\to\mathbb{R}_{+})$, we have

$\displaystyle\int_{\Sigma}\left(T_{\mathfrak{M}}T_{\mathfrak{n}}(\bm{\rho}\mathbf{a}_{i}\mathbf{a}_{j}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\psi)\right)\,\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle\int_{\Sigma}\bm{\rho}\mathbf{a}_{i}\mathbf{a}_{j}\overline{\partial}{}_{i}\overline{\partial}{}_{j}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\,\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}\stackrel{{\scriptstyle
L}}{{=}}-\int_{\Sigma}\bm{\rho}(\mathbf{a}_{i}\overline{\partial}{}_{i}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi)\,(\mathbf{a}_{j}\overline{\partial}{}_{j}(\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi)\,\mathrm{d}x^{\prime}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}\bm{\rho}\left|\mathbf{a}_{i}\overline{\partial}{}_{i}T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x^{\prime}.$

Setting $\mathbf{a}=\bar{\mathbf{w}},\bar{\mathbf{u}},\bar{\mathbf{b}}$ and
$\bm{\rho}=\rho^{\pm}$ or $\rho^{+}+\rho^{-}$, we immediately get

$I_{002}\stackrel{{\scriptstyle
L}}{{=}}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}(\rho^{+}+\rho^{-})\left|(\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x,\quad
I_{003}\stackrel{{\scriptstyle
L}}{{=}}-\frac{\mathrm{d}}{\mathrm{d}t}\widetilde{\mathcal{E}}(t).$

For $I_{1}+I_{001}$, we have

$\displaystyle I_{001}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-2\int_{\Sigma}(\rho^{+}+\rho^{-})(\bar{\mathbf{w}}\cdot\overline{\nabla})\partial_{t}T_{{\mathfrak{m}}}T_{\mathfrak{n}}\psi\,\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\Rightarrow I_{1}+I_{001}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\int_{\Sigma}(\rho^{+}+\rho^{-})(\bar{\mathbf{w}}\cdot\overline{\nabla})\partial_{t}T_{{\mathfrak{m}}}T_{\mathfrak{n}}\psi\,\left((\partial_{t}+\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right)\,\mathrm{d}x^{\prime}$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}(\rho^{+}+\rho^{-})\left|(\bar{\mathbf{w}}\cdot\overline{\nabla})T_{\mathfrak{M}}T_{\mathfrak{n}}\psi\right|^{2}\,\mathrm{d}x,$
(5.60)

which cancels with the main term in $I_{002}$. When $\sigma>0$ is given and
the stability condition (1.29) is not assumed, the quantity
$\widetilde{\mathcal{E}}(t)$ is not necessarily positive, but its
contribution, namely the term $I_{003}$, can be controlled by integrating by
parts for 1/2-derivative

$I_{003}\lesssim
P(\|v^{\pm},\rho^{\pm},b^{\pm}\|_{4,\pm})(|\psi|_{5}^{2}+|\psi|_{5}|\psi_{t}|_{4})\lesssim\sigma^{-1}P(\mathfrak{E}_{4}(t)).$

Now, it remains to control $I_{0}^{w}$ and $I_{0}^{R}$. In view of Proposition
5.6, it suffices to control the $H^{3.5}(\Sigma)$ norms of
$N\cdot\nabla^{\varphi}q_{w}^{\pm}$ and $\Psi^{R}$. For the term
$q_{w}^{\pm}$, we use trace lemma and div-curl inequality with tangential
trace (see (B.3)) to get

$\displaystyle|N\cdot\nabla^{\varphi}q_{w}^{\pm}|_{3.5}^{2}\leq|\psi|_{4.5}^{2}\|\nabla^{\varphi}q_{w}^{\pm}\|_{4,\pm}^{2}$
$\displaystyle\leq$
$\displaystyle~{}C(|\psi|_{4.5})\left(\|\nabla^{\varphi}q_{w}^{\pm}\|_{0,\pm}^{2}+\|\Delta^{\varphi}q_{w}^{\pm}\|_{3,\pm}^{2}+\|\nabla^{\varphi}\times\nabla^{\varphi}q_{w}^{\pm}\|_{3,\pm}^{2}+\left|N\times\nabla^{\varphi}q_{w}^{\pm}\right|_{3.5}^{2}+\left|N\cdot\nabla^{\varphi}q_{w}^{\pm}\right|_{H^{3,5}(\Sigma^{\pm})}^{2}\right)$
(5.61)

where the last three terms are zero because of

$\nabla^{\varphi}\times\nabla^{\varphi}q_{w}^{\pm}=\vec{0},\quad
N\cdot\nabla^{\varphi}q_{w}^{\pm}|_{\Sigma^{\pm}}=0,\quad
q_{w}^{\pm}|_{\Sigma}=0\Rightarrow
N\times\nabla^{\varphi}q_{w}^{\pm}|_{\Sigma}=(-\overline{\partial}{}_{2}q_{w}^{\pm},\overline{\partial}{}_{1}q_{w}^{\pm},\overline{\partial}{}_{2}\psi\overline{\partial}{}_{1}q_{w}^{\pm}-\overline{\partial}{}_{1}\psi\overline{\partial}{}_{2}q_{w}^{+})^{\top}|_{\Sigma}=\vec{0}.$

Then invoking the definition of $q_{w}^{\pm}$ and using
$\mathcal{F}_{p}^{\pm}\lesssim\varepsilon^{2}$, we find

$\displaystyle\|\Delta^{\varphi}q_{w}^{\pm}\|_{3,\pm}^{2}\leq
C(|\psi|_{4},|\psi_{t}|_{3})\left(\left\|\varepsilon^{2}\mathcal{T}^{2}(b^{\pm},p^{\pm})\right\|_{3,\pm}^{2}+P\left(\left\|v^{\pm},b^{\pm}\right\|_{4,\pm}\right)\right)\leq
P(\widetilde{\mathfrak{E}}_{4}(t))\widetilde{\mathfrak{E}}_{5}(t).$ (5.62)

######  Remark 5.2 (Necessity of anisotropic Sobolev spaces).

The term bounded by $\widetilde{\mathfrak{E}}_{5}(t)$is contributed exactly by
the extra $\frac{1}{2}|b^{\pm}|^{2}$ in the total pressure. From this, we can
also see the necessity of the anisotropic Sobolev spaces when studying ideal
compressible MHD. For Euler equations, the source terms for the wave equation
only contain quadratic first-order terms, and one can use the trick in [32,
62] to close the energy bound by $\widetilde{\mathfrak{E}}_{4}(t)$. For
incompressible MHD, the second-order time derivative term vanishes because
$q^{\pm}$ satisfies an elliptic equation.

The term $|\Psi^{R}|_{3.5}$ can also be directly controlled. Recall that

$\Psi^{R}:=\frac{\sigma}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})(\mathcal{H}(\psi))-(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left(\left\llbracket\mathfrak{F}_{\psi}-\rho\partial_{t}^{2}\psi\right\rrbracket\right)+\frac{\sigma}{2}\mathcal{R}_{\psi}^{\sigma}.$

Using the Sobolev estimates for the Dirichlet-to-Neumann operators, we have

$\displaystyle\left|\frac{\sigma}{2}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})(\mathcal{H}(\psi))\right|_{3.5}+\left|(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}\left(\left\llbracket\mathfrak{F}_{\psi}-\rho\partial_{t}^{2}\psi\right\rrbracket\right)\right|_{3.5}$
$\displaystyle\lesssim$
$\displaystyle~{}|\sigma\mathfrak{H}(\psi)|_{2.5}+\left|\mathfrak{F}_{\psi}^{\pm}\right|_{2.5}+\left|\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi\right|_{2.5}$
$\displaystyle\lesssim$
$\displaystyle~{}\left((1+\sigma)|\psi|_{4.5}+|\psi_{t}|_{3.5}\right)\left(\left\|\varepsilon^{2}\partial_{t}^{2}p^{\pm}\right\|_{2,\pm}+P(\|v^{\pm},b^{\pm}\|_{3,\pm},|\psi|_{3})\right)+\left|\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi\right|_{2.5}\lesssim
P(\widetilde{\mathfrak{E}}_{4}(t)).$ (5.63)

Setting $s=4$ in the remainder estimate (5.42), we have
$|\sigma\mathcal{R}_{\psi}^{\sigma}|_{3.5}\leq
C(|\psi|_{4.5})|\sigma\psi|_{5}\leq\sqrt{\sigma}C(\widetilde{\mathcal{E}}(t))\sqrt{\mathcal{E}(t)}\leq\sqrt{\sigma}C(\widetilde{\mathfrak{E}}_{4}(t)).$

Summarizing the estimate above, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(t)\lesssim\sigma^{-1}P(\mathfrak{E}_{4}(t))\mathfrak{E}_{5}(t),$
(5.64)

and under the stability condition (1.29), we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\mathcal{E}(t)+\widetilde{\mathcal{E}}(t))\lesssim
P(\widetilde{\mathfrak{E}}_{4}(t))\widetilde{\mathfrak{E}}_{5}(t).$ (5.65)

Invoking Lemma 5.12, we actually prove the following uniform-in-$\varepsilon$
estimate for fixed $\sigma>0$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\sqrt{\sigma}\psi\right|_{5}^{2}+\left|\psi_{t}\right|_{3.5}^{2}\right)\lesssim\sigma^{-1}P(\mathfrak{E}_{4}(t))\mathfrak{E}_{5}(t),$
(5.66)

and the following uniform-in-$({\varepsilon,\sigma})$ estimate under the
stability condition (1.29)

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\sqrt{\sigma}\psi\right|_{5}^{2}+\left|\psi\right|_{4.5}^{2}+\left|\psi_{t}\right|_{3.5}^{2}\right)\lesssim
P(\widetilde{\mathfrak{E}}_{4}(t))\widetilde{\mathfrak{E}}_{5}(t).$ (5.67)

It remains to prove the uniform estimates for $|\psi_{tt}|_{2.5}^{2}$. We just
need to take one more $\partial_{t}$ in the paralinearized evolution equation
(5.48). The proof follows in the same way as the above analysis for the
$T_{\mathfrak{M}}T_{\mathfrak{n}}$-differentiated version of (5.48), so we
skip the details and only list the differences. The first difference is that
we should replace $T_{\mathfrak{M}}$ by $T_{{\mathfrak{M}}^{\prime}}$ where
${\mathfrak{M}}^{\prime}\in\Sigma^{\frac{5}{2}}$ is defined by

${\mathfrak{M}}^{\prime}:=(\mathfrak{m}^{(1.5)})^{\frac{5}{3}}=2^{\frac{5}{6}}|\xi|^{\frac{5}{2}}\left(1-\left|\frac{N}{|N|}\cdot\frac{\xi}{|\xi|}\right|^{2}\right)^{\frac{5}{4}}\in\Sigma^{\frac{5}{2}}.$

The second difference is essential: we notice that the remainder term
$\Psi^{R}$ already contains second-order time derivative
$(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\left\llbracket\rho\right\rrbracket\partial_{t}^{2}\psi)$.
Taking one more time derivative, we are required to control
$(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\left\llbracket\rho\right\rrbracket\partial_{t}^{3}\psi)$.
Using Lemma C.3 and Lemma C.4, we have

$\left|(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})\widetilde{\mathfrak{N}}^{-1}(\left\llbracket\rho\right\rrbracket\partial_{t}^{3}\psi)\right|_{2.5}\lesssim|\left\llbracket\rho\right\rrbracket|_{1.5}|\partial_{t}^{3}\psi|_{1.5}.$

Since we require
$|\left\llbracket\rho\right\rrbracket|_{1.5}\lesssim\varepsilon$, we can
control the right side by $|\varepsilon\partial_{t}^{3}\psi|_{1.5}$. Under the
stability condition (1.29), this term is already a part of
$\widetilde{\mathfrak{E}}_{4}(t)$. For fixed $\sigma>0$, we again invoke the
kinematic boundary condition to get

$|\varepsilon\partial_{t}^{3}\psi|_{1.5}\lesssim\|\varepsilon\partial_{t}^{2}v^{\pm}\|_{2,\pm}|\psi|_{2.5}+\|\varepsilon\partial_{t}v^{\pm}\|_{2,\pm}|\psi_{t}|_{2.5}+\|\varepsilon
v^{\pm}\|_{2,\pm}|\psi_{tt}|_{2.5},$

where the right side is already controlled by $P(\mathfrak{E}_{4}(t))$. As for
the fifth-order terms arising from $N\cdot\nabla^{\varphi}q_{w}^{\pm}$, they
now contribute to $\|\varepsilon^{2}\partial_{t}^{3}q^{\pm}\|_{2,\pm}$ which
is still a part of $E_{5}(t)$. Thus, we can conclude the following uniform-
in-$\varepsilon$ estimate for fixed $\sigma>0$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\sqrt{\sigma}\psi_{t}\right|_{4}^{2}+\left|\psi_{tt}\right|_{2.5}^{2}\right)\lesssim\sigma^{-1}P(\mathfrak{E}_{4}(t))\mathfrak{E}_{5}(t),$
(5.68)

and the following uniform-in-$({\varepsilon,\sigma})$ estimate under the
stability condition (1.29)

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\sqrt{\sigma}\psi_{t}\right|_{4}^{2}+\left|\psi_{t}\right|_{3.5}^{2}+\left|\psi_{tt}\right|_{2.5}^{2}\right)\lesssim
P(\widetilde{\mathfrak{E}}_{4}(t))\widetilde{\mathfrak{E}}_{5}(t).$ (5.69)

###  5.4 Improved double limits without the boundedness of $\geq 2$ time
derivatives

####  5.4.1 Improved incompressible limit for fixed $\sigma>0$

From the analysis in Section 5.1-Section 5.3, we can prove the uniform-
in-$\varepsilon$ estimates for $\mathfrak{E}_{4}(t)$. For any $\delta\in(0,1)$

$\displaystyle\mathfrak{E}_{4}(t)\lesssim\delta\mathfrak{E}_{4}(t)+P(\mathfrak{E}_{4}(0))+P(\mathfrak{E}_{4}(t))\int_{0}^{t}P(\sigma^{-1},\mathfrak{E}_{4}(\tau))+\mathfrak{E}_{5}(\tau)\,\mathrm{d}\tau.$
(5.70)

For $1\leq l\leq 4$, since we do not change anything $\mathfrak{E}_{4+l}(t)$,
we still have

$\displaystyle l=1,2,3:$
$\displaystyle~{}~{}\mathfrak{E}_{4+l}(t)\lesssim\delta\mathfrak{E}_{4+l}(t)+P(\mathfrak{E}_{4+l}(0))+P(\mathfrak{E}_{4}(t))\int_{0}^{t}P\left(\sigma^{-1},\sum_{j=0}^{l}\mathfrak{E}_{4+j}(\tau)\right)+\mathfrak{E}_{4+l+1}(\tau)\,\mathrm{d}\tau;$
(5.71) $\displaystyle l=4:$
$\displaystyle~{}~{}\mathfrak{E}_{8}(t)\lesssim\delta\mathfrak{E}_{8}(t)+P(\mathfrak{E}_{8}(0))+P(\mathfrak{E}_{4}(t))\int_{0}^{t}P(\sigma^{-1},\mathfrak{E}_{8}(\tau))\,\mathrm{d}\tau.$
(5.72)

Therefore, we get the Gronwall-type energy inequality for $\mathfrak{E}(t)$

$\displaystyle\mathfrak{E}(t)\lesssim\delta\mathfrak{E}(t)+P(\mathfrak{E}(0))+P(\mathfrak{E}(t))\int_{0}^{t}P(\sigma^{-1},\mathfrak{E}(\tau))\,\mathrm{d}\tau.$
(5.73)

Choosing $\delta>0$ suitably small, the term $\delta\mathfrak{E}(t)$ can be
absorbed by the left side. Thus, there exists a time $T_{\sigma}^{\prime}>0$
depending on $\sigma^{-1}$ and the initial data, but independent of
$\varepsilon$, such that

$\displaystyle\sup_{0\leq t\leq T_{\sigma}^{\prime}}\mathfrak{E}(t)\lesssim
P(\sigma^{-1},\mathfrak{E}(0)).$ (5.74)

With the uniform-in-$\varepsilon$ estimates for $\mathfrak{E}(t)$, we now take
the incompressible limit. Again, since $\|\partial_{t}(v,b)\|_{3}$ is
uniformly bounded with respect to $\varepsilon$ and we still have
$\varepsilon$-independent bound $|\psi_{t}|_{3.5}$, the Aubin-Lions
compactness lemma gives the same strong convergence result as in Section 4.3.

####  5.4.2 Improved double limits under the stability conditions

With the estimates (5.67) and (5.69) in Section 5.3, we can get

$|v_{t}\cdot N|_{2.5}^{2}+|b_{t}\cdot N|_{2.5}^{2}\lesssim
P(\widetilde{\mathfrak{E}}_{4}(0))+P(\widetilde{\mathfrak{E}}_{4}(t))\int_{0}^{t}P(\widetilde{\mathfrak{E}}_{4}(\tau))\widetilde{\mathfrak{E}}_{5}(\tau)\,\mathrm{d}\tau.$

This finishes the control of $\widetilde{\mathfrak{E}}_{4}(t)$. Since
$\widetilde{\mathfrak{E}}_{4+l}(t)=\widetilde{E}_{4+l}(t)$ when $1\leq l\leq
4$ and the strategies to control them remain unchanged, we can now close the
energy estimates for $\widetilde{\mathfrak{E}}(t)$, uniformly in $\varepsilon$
and $\sigma$, under the stability condition (1.29) ($d=3$) or (1.37) ($d=2$).

$\displaystyle\widetilde{\mathfrak{E}}(t)\leq
P(\widetilde{\mathfrak{E}}(0))+P(\widetilde{\mathfrak{E}}(t))\int_{0}^{t}P(\widetilde{\mathfrak{E}}(\tau))\,\mathrm{d}\tau.$
(5.75)

By Grönwall’s inequality, there exists $T^{\prime}>0$ independent of $\sigma$
and $\varepsilon$ such that

$\displaystyle\sup_{t\in[0,T]}\widetilde{\mathfrak{E}}(t)\leq
P(\widetilde{\mathfrak{E}}(0)).$ (5.76)

Thus, by Aubin-Lions compactness lemma, we can prove the same convergence
result as in Theorem 1.2.

##### Data avaliability.

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### Ethics Declarations

##### Conflict of interest.

The author declares that there is no conflict of interest.

##  Appendix A Reynolds transport theorems

We record the Reynolds transport theorems used in this paper. For the proof,
we refer to Luo-Zhang [36, Appendix A]

######  Lemma A.1.

Let $f,g$ be smooth functions defined on $[0,T]\times\Omega$. Then:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}fg\partial_{3}\varphi\,\mathrm{d}x=\int_{\Omega}(\partial^{\varphi}_{t}f)g\partial_{3}\varphi\,\mathrm{d}x+\int_{\Omega}f(\widetilde{\varphi}_{t}g)\partial_{3}\varphi\,\mathrm{d}x+\int_{x_{3}=0}fg\partial_{t}\psi\,\mathrm{d}x^{\prime},$
(A.1)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}fg\partial_{3}\mathring{\varphi}\,\mathrm{d}x=\int_{\Omega}(\partial_{t}^{\mathring{\varphi}}f)g\partial_{3}\mathring{\varphi}\,\mathrm{d}x+\int_{\Omega}f(\partial_{t}^{\mathring{\varphi}}g)\partial_{3}\mathring{\varphi}\,\mathrm{d}x+\int_{x_{3}=0}fg\partial_{t}\mathring{\psi}\,\mathrm{d}x^{\prime}.$
(A.2)

######  Lemma A.2 (Integration by parts for covariant derivatives).

Let $f,g$ be defined as in Lemma A.1. Then:

$\displaystyle\int_{\Omega}(\partial^{\varphi}_{i}f)g\partial_{3}\varphi\,\mathrm{d}x=-\int_{\Omega}f(\partial^{\varphi}_{i}g)\partial_{3}\varphi\,\mathrm{d}x+\int_{x_{3}=0}fgN_{i}\,\mathrm{d}x^{\prime},$
(A.3)
$\displaystyle\int_{\Omega}(\partial^{\mathring{\varphi}}_{i}f)g\partial_{3}\mathring{\varphi}\,\mathrm{d}x=-\int_{\Omega}f(\partial^{\mathring{\varphi}}_{i}g)\partial_{3}\mathring{\varphi}\,\mathrm{d}x+\int_{x_{3}=0}fg\mathring{N}_{i}\,\mathrm{d}x^{\prime}.$
(A.4)

The following theorem holds.

######  Theorem A.3 (Reynolds transport theorem).

Let $f$ be a smooth function defined on $[0,T]\times\Omega$. Then:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|f|^{2}\partial_{3}\varphi\,\mathrm{d}x=\int_{\Omega}\rho(D_{t}^{\varphi}f)f\partial_{3}\varphi\,\mathrm{d}x.$
(A.5)

Theorem A.3 leads to the following corollary. The first one records the
integration by parts formula for $D_{t}^{\varphi}$.

######  Corollary A.4 (Reynolds transport theorem - a variant).

It holds that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}fg\partial_{3}\varphi\,\mathrm{d}x=\int_{\Omega}(D_{t}^{\varphi}f)g\partial_{3}\varphi\,\mathrm{d}x+\int_{\Omega}f(D_{t}^{\varphi}g)\partial_{3}\varphi\,\mathrm{d}x+\int_{\Omega}(\nabla^{\varphi}\cdot
v)fg\partial_{3}\varphi\,\mathrm{d}x.$ (A.6)

##  Appendix B Preliminary lemmas about Sobolev inequalities

######  Lemma B.1 (Hodge-type elliptic estimates).

For any sufficiently smooth vector field $X$ and $s\geq 1$, one has

$\displaystyle\|X\|_{s}^{2}\leq
C(|\psi|_{s},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|X\|_{0}^{2}+\|\nabla^{\varphi}\cdot
X\|_{s-1}^{2}+\|\nabla^{\varphi}\times
X\|_{s-1}^{2}+\|\overline{\partial}{}^{\alpha}X\|_{0}^{2}\right),$ (B.1)
$\displaystyle\|X\|_{s}^{2}\leq
C^{\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|X\|_{0}^{2}+\|\nabla^{\varphi}\cdot
X\|_{s-1}^{2}+\|\nabla^{\varphi}\times X\|_{s-1}^{2}+|X\cdot
N|_{s-\frac{1}{2}}^{2}\right),$ (B.2) $\displaystyle\|X\|_{s}^{2}\leq
C^{\prime\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|X\|_{0}^{2}+\|\nabla^{\varphi}\cdot
X\|_{s-1}^{2}+\|\nabla^{\varphi}\times X\|_{s-1}^{2}+|X\times
N|_{s-\frac{1}{2}}^{2}\right),$ (B.3)

for any multi-index $\alpha$ with $|\alpha|=s$. The constant
$C(|\psi|_{s},|\overline{\nabla}\psi|_{W^{1,\infty}})>0$ depends linearly on
$|\psi|_{s}^{2}$ and the constants
$C^{\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})>0$
and
$C^{\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})>0$
depend linearly on $|\psi|_{s+\frac{1}{2}}^{2}$.

######  Lemma B.2 (Normal trace lemma).

For any sufficiently smooth vector field $X$ and $s\geq 0$, one has

$\displaystyle\left|X\cdot N\right|_{s-\frac{1}{2}}^{2}\lesssim
C^{\prime\prime\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})\left(\|\langle\overline{\partial}{}\rangle^{s}X\|_{0}^{2}+\|\nabla^{\varphi}\cdot
X\|_{s-1}^{2}\right)$ (B.4)

where the constant
$C^{\prime\prime\prime}(|\psi|_{s+\frac{1}{2}},|\overline{\nabla}\psi|_{W^{1,\infty}})>0$
depends linearly on $|\psi|_{s+\frac{1}{2}}^{2}$.

We list two lemmas for the estimates of traces in the anisotropic Sobolev
spaces. Define

$L_{T}^{2}(H_{*}^{m}(\Omega^{\pm}))=\bigcap_{k=0}^{m}H^{k}((-\infty,T];H_{*}^{m-k}(\Omega^{\pm}))$

with the norm
$\|u\|_{m,*,T,\pm}:=\int_{-\infty}^{T}\|u(t)\|_{m,*,\pm}^{2}\,\mathrm{d}t$.
Similarly, we define

$L_{T}^{2}(H^{m}(\Sigma))=\bigcap_{k=0}^{m}H^{k}((-\infty,T];H^{m-k}(\Sigma))$

with the norm $|u|_{m,T}:=\int_{-\infty}^{T}|u(t)|_{m}^{2}\,\mathrm{d}t$.

######  Lemma B.3 (Trace lemma for anisotropic Sobolev spaces).

Let $m\geq 1,~{}m\in\mathbb{N}^{*}$, then we have the following trace lemma
for the anisotropic Sobolev space.

  1. 1.

If $f\in L_{T}^{2}(H_{*}^{m+1}(\Omega^{\pm}))$, then its trace $f|_{\Sigma}$
belongs to $L_{T}^{2}(H^{m}(\Omega^{\pm}))$ and satisfies

$|f|_{m,T}\lesssim\|f\|_{m+1,*,T,\pm}.$

  2. 2.

There exists a linear continuous operator
$\mathfrak{R}_{T}^{\pm}:L_{T}^{2}(H^{m}(\Sigma))\to
L_{T}^{2}(H_{*}^{m+1}(\Omega^{\pm}))$ such that
$(\mathfrak{R}_{T}^{\pm}g)|_{\Sigma}=g$ and

$\|\mathfrak{R}_{T}^{\pm}g\|_{m+1,*,T,\pm}\lesssim|g|_{m,T}.$

###### Proof.

See [63, Appendix B]. ∎

There is one derivative loss in the above trace lemma, which is 1/2-order more
than the trace lemma for standard Sobolev spaces. Indeed, for $\Omega^{\pm}$
defined in this paper, we have the following estimate that will be applied to
control the non-characteristic variables $q,v\cdot\mathbf{N}$ and
$b\cdot\mathbf{N}$.

######  Lemma B.4 (An estimate for traces of non-characteristic variables).

Let $\Omega^{\pm}:=\mathbb{T}^{d-1}\times\\{0\lessgtr x_{d}\lessgtr\pm H\\}$,
$\Sigma=\mathbb{T}^{d-1}\times\\{x_{d}=0\\}$ and
$\Sigma^{\pm}=\mathbb{T}^{d-1}\times\\{\pm H\\}$. Let
$\mathcal{T}^{\alpha}=(\omega(x_{d})\partial_{d})^{\alpha_{d+1}}\partial_{t}^{\alpha_{0}}\overline{\partial}{}_{1}^{\alpha_{1}}\cdots\overline{\partial}{}_{d-1}^{\alpha_{d-1}}\partial_{d}^{\alpha_{d}}$
with
$\langle\alpha\rangle:=\alpha_{0}+\cdots+\alpha_{d-1}+2\alpha_{d}+\alpha_{d+1}=m-1,~{}m\in\mathbb{N}^{*}$.
Let $q^{\pm}(t,x)\in H_{*}^{m}(\Omega)$ satisfy
$\|q^{\pm}(t)\|_{m,*,\pm}+\|\partial_{d}q^{\pm}(t)\|_{m-1,*,\pm}<\infty$ for
any $0\leq t\leq T$ and let $f^{\pm}\in H_{*}^{2}(\Omega^{\pm})\cap
H^{\frac{3}{2}}(\Omega^{\pm})$ be a function vanishing on $\Sigma^{\pm}$. Then
we have

$\int_{\Sigma}(\langle\overline{\partial}{}\rangle^{\frac{1}{2}}\mathcal{T}^{\gamma}q^{\pm})\,(\langle\overline{\partial}{}\rangle
f^{\pm})\,\mathrm{d}x^{\prime}\leq(\|\partial_{d}q^{\pm}\|_{m-1,*,\pm}+\|q^{\pm}\|_{m,*,\pm})\|\langle\overline{\partial}{}\rangle^{\frac{1}{2}}f^{\pm}\|_{1,\pm}$
(B.5)

In particular, for $s\geq 1$, we have the following inequality for any
$g^{\pm}\in H_{*}^{s}(\Omega^{\pm})$ with $g^{\pm}|_{\Sigma^{\pm}}=0$.

$|g^{\pm}|_{s-1/2}^{2}\leq\|\langle\overline{\partial}{}\rangle^{s}g^{\pm}\|_{0,\pm}\|\langle\overline{\partial}{}\rangle^{s-1}\partial_{d}g^{\pm}\|_{0,\pm}\leq\|g^{\pm}\|_{s,*,\pm}\|\partial_{d}g^{\pm}\|_{s-1,*,\pm}.$

###### Proof.

See [63, Appendix B]. ∎

The following lemma concerns the Sobolev embeddings.

######  Lemma B.5.

We have the following inequalities

$\displaystyle H^{m}(\Omega^{\pm})\hookrightarrow
H_{*}^{m}(\Omega^{\pm})\hookrightarrow$ $\displaystyle H^{\lfloor
m/2\rfloor}(\Omega^{\pm}),~{}~{}\forall m\in\mathbb{N}^{*}$
$\displaystyle\|u\|_{L^{\infty}(\Omega^{\pm})}\lesssim\|u\|_{H_{*}^{3}(\Omega^{\pm})},~{}~{}$
$\displaystyle\|u\|_{W^{1,\infty}(\Omega^{\pm})}\lesssim\|u\|_{H_{*}^{5}(\Omega^{\pm})},~{}~{}|u|_{W^{1,\infty}(\Omega^{\pm})}\lesssim\|u\|_{H_{*}^{5}(\Omega^{\pm})}.$

We also need the following Kato-Ponce type multiplicative Sobolev inequality.

######  Lemma B.6 ([27]).

Let $J=(1-\Delta)^{1/2}$, $s\geq 0$. Then the following estimates hold:

$\|J^{s}(fg)\|_{L^{2}}\lesssim\|f\|_{W^{s,p_{1}}}\|g\|_{L^{p_{2}}}+\|f\|_{L^{q_{1}}}\|g\|_{W^{s,q_{2}}},$
(B.6)

where $1/2=1/p_{1}+1/p_{2}=1/q_{1}+1/q_{2}$ and $2\leq p_{1},q_{2}<\infty$.

$\|[J^{s},f]g\|_{L^{p}}\lesssim\|\partial
f\|_{L^{\infty}}\|J^{s-1}g\|_{L^{p}}+\|J^{s}f\|_{L^{p}}\|g\|_{L^{\infty}}$
(B.7)

where $s\geq 0$ and $1<p<\infty$.

We also need the following transport-type estimate in order to close the
uniform estimates for the nonlinear approximate system.

######  Lemma B.7.

Let $f(t)\in W^{1,1}(0,T)$ and $g\in L^{1}(0,T)$ and $\kappa>0$. Assume that

$f(t)+\kappa f^{\prime}(t)\leq g(t)~{}~{}\text{a.e. }t\in(0,T).$

Then for any $t\in(0,T)$,

$\sup_{\tau\in[0,t]}f(\tau)\leq
f(0)+\mathop{\mathrm{ess~{}sup}}\limits_{\tau\in(0,t)}|g(\tau)|.$

##  Appendix C Paraproducts and the Dirichlet-to-Neumann operator

###  C.1 Bony’s paraproduct decomposition

We already introduce the paradifferential operator in Section 5.2.2. Here we
present the relations between paradifferential operators and paraproducts. The
cutoff function $\tilde{\chi}(\xi,\eta)$ in the definition of $T_{a}u$ is

$\tilde{\chi}(\xi,\eta)=\sum_{k=0}^{\infty}\Theta_{k-3}(\xi)\vartheta(\eta),$

where $\Theta(\xi)=1$ when $|\xi|\leq 1$ and $\Theta(\xi)=0$ when $|\xi|\geq
2$ and

$\Theta_{k}(\xi):=\Theta(\frac{\xi}{2}),~{}~{}k\in\mathbb{Z},\quad\vartheta_{0}=\Theta,\quad\vartheta_{k}:=\Theta_{k}-\Theta_{k-1},~{}~{}k\geq
1.$

Based on this, we can introduce the Littlewood-Paley projections
$\mathcal{P}_{k}$ and $\mathcal{P}_{\leq k}$ as follows

$\displaystyle\widehat{\mathcal{P}_{k}u}(\xi):=\vartheta_{k}(\xi)\hat{u}(\xi),~{}~{}\forall
k\geq 0,\quad\mathcal{P}_{k}u:=0~{}~{}\forall k<0,\quad\mathcal{P}_{\leq
k}u:=\sum_{l\leq k}\mathcal{P}_{l}u.$

When the symbol $a(x,\xi)$ (in the paradifferential operator $T_{a}$) does not
depend on $\xi$, we can take $\psi(\eta)\equiv 1$ and then we have

$T_{a}u=\sum_{k}\mathcal{P}_{\leq k-3}a(\mathcal{P}_{k}u)$

which is the usual Bony’s paraproduct. In general, the well-known Bony’s
paraproduct decomposition is

$au=T_{a}u+T_{u}a+R(u,a),\quad R(u,a)=\sum_{|k-l|\leq
2}(\mathcal{P}_{k}a)(\mathcal{P}_{l}u).$

We have the following estimates for the remainder $R(u,a)$

######  Lemma C.1 ([3, Section 2.3]).

For $s\in\mathbb{R},~{}r<d/2,~{}\delta>0$, we have

$|T_{a}u|_{H^{s}}\lesssim\min\\{|a|_{L^{\infty}}|u|_{H^{s}},|a|_{H^{r}}|u|_{H^{s+\frac{d}{2}-r}},|a|_{H^{\frac{d}{2}}}|u|_{H^{s+\delta}}\\}$

and for any $s>0,s_{1},s_{2}\in\mathbb{R}$ satisfying
$s_{1}+s_{2}=s+\frac{d}{2}$, we have

$|R(u,a)|_{H^{s}}\lesssim|a|_{H^{s_{1}}}|u|_{H^{s_{2}}}.$

###  C.2 Basic properties of the Dirichlet-to-Neumann operator

Let the space dimension $d=3$ for simplicity. Given a function
$f:\Sigma=\mathbb{T}^{2}\to\mathbb{R}$, we define the Dirichlet-to-Neumann
(DtN) operator (with respect to $\psi$ and region $\Omega^{\pm}$) by

$\mathfrak{N}_{\psi}^{\pm}f:=\mp
N\cdot\nabla^{\varphi}(\mathcal{E}_{\psi}^{\pm}f)|_{\Sigma},\quad-\Delta^{\varphi}(\mathcal{E}_{\psi}^{\pm}f)=0\text{
in
}\Omega^{\pm},~{}~{}\mathcal{E}_{\psi}^{\pm}f|_{\Sigma}=f,~{}~{}\partial_{3}(\mathcal{E}_{\psi}^{\pm}f)|_{\Sigma^{\pm}}=0.$

Here the Laplacian operator is defined by
$\Delta^{\varphi}:=\nabla^{\varphi}\cdot\nabla^{\varphi}=\partial_{i}(\bm{E}^{ij}\partial_{j})$
with

$\bm{E}=\frac{1}{\partial_{3}\varphi}\begin{bmatrix}\partial_{3}\varphi&0&-\overline{\partial}{}_{1}\varphi\\\
0&\partial_{3}\varphi&-\overline{\partial}{}_{2}\varphi\\\
-\overline{\partial}{}_{1}\varphi&-\overline{\partial}{}_{2}\varphi&\frac{1+|\overline{\nabla}\varphi|^{2}}{\partial_{3}\varphi}\end{bmatrix}=\frac{1}{\partial_{3}\varphi}\bm{P}\bm{P}^{\top},~{}~{}\bm{P}:=\begin{bmatrix}\partial_{3}\varphi&0&0\\\
0&\partial_{3}\varphi&0\\\
-\overline{\partial}{}_{1}\varphi&-\overline{\partial}{}_{2}\varphi&1\end{bmatrix},$

and $\varphi(t,x):=x_{3}+\chi(x_{3})\psi(t,x^{\prime})$ is defined in (1.5) as
the extension of $\psi$ into $\Omega^{\pm}$. The choice of $\chi(x_{3})$ is
slightly different from [2, 3, 4], but it does not introduce any substantial
difference because the expression of $\Delta^{\varphi}$ is still written to be
$\Delta^{\varphi}:=\nabla^{\varphi}\cdot\nabla^{\varphi}=\partial_{i}(\bm{E}^{ij}\partial_{j})$
and we have $\Delta^{\varphi}\varphi=0$ in $\Omega^{\pm}$. The DtN operators
satisfy the following estimates and we refer to [48, Appendix A.4] for the
proof.

######  Lemma C.2 (Sobolev estimates for DtN operators).

For $s>2+\frac{d}{2},~{}-\frac{1}{2}\leq r\leq s-1$ and $\psi\in
H^{s}(\mathbb{T}^{d})$, we have

$|\mathfrak{N}_{\psi}^{\pm}f|_{r}\leq C(|\psi|_{s})|f|_{r+1}.$

######  Lemma C.3 (Remainder estimates for DtN operators).

For $s>2+\frac{d}{2}$ and $\psi\in H^{s}(\mathbb{T}^{d})$, we have

$\mathfrak{N}_{\psi}^{\pm}f=T_{\Lambda^{(1),\pm}}f+R_{1}^{\pm}(f)$

with $\Lambda^{(1),+}=\Lambda^{(1),\pm}$ defined in Proposition 5.7 and

$\forall
r\in[\frac{1}{2},s-1],\quad|R_{1}^{\pm}(f)|_{r}+|(\mathfrak{N}_{\psi}^{+}-\mathfrak{N}_{\psi}^{-})f|_{r}\leq
C(|\psi|_{s})|f|_{r}.$

######  Lemma C.4 (Sobolev estimates for the inverse of DtN operators).

For $s>2+\frac{d}{2},~{}-\frac{1}{2}\leq r\leq s-1$ and $\psi\in
H^{s}(\mathbb{T}^{d})$, we have

$|(\mathfrak{N}_{\psi}^{\pm})^{-1}f|_{r+1}\leq C(|\psi|_{s})|f|_{r}.$

## References

  * [1] Alazard, T.  Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions.  Adv. Differ. Equ., 10(1):19–44, 2005.
  * [2] Alazard, T., Burq, N., Zuily, C.  On the water-wave equations with surface tension.  Duke Math. J., 158(3), 413-499, 2011.
  * [3] Alazard, T., Burq, N., Zuily, C.  On the Cauchy problem for gravity water waves.  Invent. Math., 198(1), 71-163, 2014.
  * [4] Alazard, T., Métivier, G. Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves.  Commun. Partial Differ. Equ., 34(12), 1632-1704, 2009.
  * [5] Alinhac, S.  Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels.(French. English summary) [Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems].  Commun. Partial Differ. Equ., 14(2), 173-230, 1989.
  * [6] Ambrose, D., Masmoudi, N.  Well-posedness of 3D vortex sheets with surface tension.  Commun. Math. Sci., 5(2), 391-430, 2007.
  * [7] Axford, W. I.  Note on a problem of magnetohydrodynamic stability.  Canad. J. Phys., 40(5), 654-655, 1962.
  * [8] Baranov, V. B., Krasnobaev, K. V., Kulikovsky, A. G.  A model of interaction of the solar wind with the interstellar medium.  Sov. Phys. Dokl., Vol. 15, 791–793 (1971).
  * [9] Bony, J. M.  Calcul symbolique et propagation des singularités pour les équations aux derivees partielles non linéaires.  Ann. Sci. École Norm. Sup. (4), 14(2), 209-246, 1981.
  * [10] Chen, G.-Q., Wang, Y.-G.  Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics.  Arch. Rational Mech. Anal., 187(3), 369-408, 2008.
  * [11] Chen, S.-X.  Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary.  Translated from Chin. Ann. Math. 3(2), 222–232 (1982). Front. Math. China, 2(1), 87–102 (2007).
  * [12] Cheng, C.-H. A., Coutand, D., Shkoller, S.  On the Motion of Vortex Sheets with Surface Tension in Three-Dimensional Euler Equations with Vorticity.  Commun. Pure Appl. Math., 61(12), 1715-1752, 2007.
  * [13] Coulombel, J.-F., Morando, A., Secchi, P., Trbeschi, P.  A priori Estimates for 3D Incompressible Current-Vortex Sheets.  Commun. Math. Phys., 311(1), 247-275, 2012.
  * [14] Coulombel, J.-F., Secchi, P.  The stability of compressible vortex sheets in two space dimensions.  Indiana Univ. Math. J., 53(4), 941-1012, 2004.
  * [15] Coulombel, J.-F., Secchi, P.  Nonlinear compressible vortex sheets in two space dimensions.  Ann. Sci. de l’École Norm.(4), 41(1), 85-139, 2008.
  * [16] Disconzi, M. M., Luo, C.  On the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid.  Arch. Rational Mech. Anal., 237(2), 829-897, 2020.
  * [17] Ebin, D. G.  Motion of slightly compressible fluids in a bounded domain. I.  Commun. Pure. Appl. Math., 35(4), 51–485, 1982.
  * [18] Ebin, D. G.  Ill-posedness of the Rayleigh–Taylor and Helmholtz problems for incompressible fluids.  Commum. Partial Differ. Equ., 13(10), 1265-1295, 1988.
  * [19] Fejer, J. A., Miles, J. W.  On the stability of a plane vortex sheet with respect to threedimensional disturbances.  J. Fluid Mech., 15, 335–336 (1963).
  * [20] Friedrichs, K. O.  Symmetric positive linear differential equations.  Commun. Pure Math. Anal., 11(3), 333-418, 1958.
  * [21] Goedbloed, H., Keppens, R., Poedts, S.  Magnetohydrodynamics of Laboratory and Astrophysical plasmas. Cambridge University Press, 2020.
  * [22] Gu, X., Wang, Y.  On the construction of solutions to the free-surface incompressible ideal magnetohydrodynamic equations.  J. Math. Pures Appl., Vol. 128: 1-41, 2019.
  * [23] Gu, X., Wang, Y.  Well-posedness and Low Mach Number Limit of the Free Boundary Problem for the Euler-Fourier System.  arXiv:2308.15332, preprint.
  * [24] Guo, Y., Tice, I.  Compressible, Inviscid Rayleigh-Taylor Instability.  Indiana Univ. Math. J., 60(2), 677-711, 2011.
  * [25] Hao, C., Luo, T.  A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows.  Arch. Rational Mech. Anal., 212(3), 805–847, 2014.
  * [26] Hao, C., Luo, T.  Ill-posedness of free boundary problem of the incompressible ideal MHD.  Commun. Math. Phys., 376(1), 259-286, 2020.
  * [27] Kato, T., Ponce, G.  Commutator estimates and the Euler and Navier-Stokes equations.  Commun. Pure Appl. Math., 41(7), 891-907, 1988.
  * [28] Klainerman, S., Majda, A.  Compressible and incompressible fluids.  Commun. Pure. Appl. Math., 35(5):629–651, 1982.
  * [29] Kreiss, H.-O.  Initial boundary value problems for hyperbolic systems.  Commun. Pure Appl. Math., 23(3), 277-298, 1970.
  * [30] Landau, L. D., Lifshitz, E. M., Pitaevskiĭ, L. P.  Electrodynamics of Continuous Media. Vol. 8 of Course of Theoretical Physics. Pergamon Press, 1984.
  * [31] Li, C., Li, H.  Well-posedness of the free-boundary problem in incompressible MHD with surface tension.  Calc. Var. & PDE, 61: 191, 2022.
  * [32] Lindblad, H., Luo, C.  A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit.  Commun. Pure Appl. Math., 71(7), 1273-1333, 2018.
  * [33] Lindblad, H., Zhang, J.  Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics.  Arch. Rational Mech. Anal., 247(5), no. 89: 94 pp., 2023.
  * [34] Liu, S., Xin, Z.  Local Well-posedness of the Incompressible Current-Vortex Sheet Problems.  arXiv:2309.03534, preprint, 2023.
  * [35] Luo, C.  On the Motion of a Compressible Gravity Water Wave with Vorticity.  Ann. PDE, 4(2), 2506-2576, 2018.
  * [36] Luo, C., Zhang, J.  Compressible Gravity-Capillary Water Waves: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits.  arXiv: 2211.03600, preprint.
  * [37] Majda, A.  Compressible Fluids Flow and Systems of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, Vol. 53, Springer-Verlag New York, 1984.
  * [38] Métivier, G.  Para-differential calculus and applications to the Cauchy problem for nonlinear systems. Edizioni della Normale, Pisa, (5) (2008).
  * [39] Métivier, G., Schochet, S.  The incompressible limit of the non-isentropic Euler equations.  Arch. Rational Mech. Anal., 158(1):61–90, 2001.
  * [40] Miles, J. W.  On the disturbed motion of a plane vortex sheet.  J. Fluid Mech., 4, 538-552. (1958)
  * [41] Molokov, S., Reed, C. B.  Review of Free-Surface MHD Experiments and Modeling. United States: N. p., 2000.
  * [42] Morando, A., Secchi, P., Trebeschi, P., Yuan, D.  Nonlinear Stability and Existence of Two-Dimensional Compressible Current-Vortex Sheets.  Arch. Rational Mech. Anal., 247(3), No. 50, 1-83, 2023.
  * [43] Ohno, M., Shirota, T.  On the initial-boundary-value problem for the linearized equations of magnetohydrodynamics.  Arch. Rational Mech. Anal., 144(3), 259-299, 1998.
  * [44] Schochet, S.  The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit.  Commun. Math. Phys., 104(1), 49-75, 1986.
  * [45] Schochet, S.  Fast Singular Limits of Hyperbolic PDEs.  J. Differ. Equ., 114(2), 476-512, 1994.
  * [46] Shatah, J., Zeng, C.  Local well-posedness for fluid interface problems.  Arch. Rational Mech. Anal., 199(2), 653-705, 2011.
  * [47] Stevens, B.  Short-time structural stability of compressible vortex sheets with surface tension.  Arch. Rational Mech. Anal., 222(2), 603-730, 2016.
  * [48] Sun, Y., Wang, W., Zhang, Z.  Nonlinear Stability of the Current-Vortex Sheet to the Incompressible MHD Equations.  Commun. Pure Appl. Math., 71(2), 356-403, 2018.
  * [49] Sun, Y., Wang, W., Zhang, Z.  Well-posedness of the Plasma-Vacuum Interface Problem for Ideal Incompressible MHD.  Arch. Rational Mech. Anal., 234(1), 81-113, 2019.
  * [50] Syrovatskiĭ, S. I.  The stability of tangential discontinuities in a magnetohydrodynamic medium. (In Russian)  Z. Eksperim. Teoret. Fiz., 24, 622-629. (1953)
  * [51] Syrovatskiĭ, S. I.  Instability of tangential discontinuities in a compressible medium. (In Russian)  Z. Eksperim. Teoret. Fiz., 27, 121-123. (1954)
  * [52] Trakhinin, Y.  Existence of compressible current-vortex sheets: variable coefficients linear analysis.  Arch. Rational Mech. Anal., 177(3), 331-366, 2005.
  * [53] Trakhinin, Y.  The existence of current-vortex sheets in ideal compressible magnetohydrodynamics.  Arch. Rational Mech. Anal., 191(2), 245-310, 2009.
  * [54] Trakhinin, Y., Wang, T.  Nonlinear Stability of MHD Contact Discontinuities with Surface Tension.  Arch. Rational Mech. Anal., 243(2), 1091-1149, 2022.
  * [55] Wang, J., Zhang, J.  Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall.  arXiv:2308.01142, preprint.
  * [56] Wang, Y.-G., Yu, F.  Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets.  Arch. Rational Mech. Anal., 208(2), 341-389, 2013.
  * [57] Wang, Y., Xin, Z.  Existence of Multi-dimensional Contact Discontinuities for the Ideal Compressible Magnetohydrodynamics.  Commun. Pure Appl. Math., 77(1), 583-629, 2024.
  * [58] Wu, S.  Recent progress in mathematical analysis of vortex sheets. ICM 2002, Vol. III, 233-242.
  * [59] Wu, S.  Mathematical analysis of vortex sheets.  Commun. Pure Appl. Math., 59(8), 1065-1206, 2005.
  * [60] Yanagisawa, T., Matsumura, A.  The fixed boundary value problems for the equations of ideal magnetohydrodynamics with a perfectly conducting wall condition.  Commun. Math. Phys., 136(1), 119-140, 1991.
  * [61] Zhang, J.  Local Well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics.  Calc. Var. Partial Differ. Equ., 62(4), no.124: 60 pp., 2023.
  * [62] Zhang, J.  Local Well-posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics.  Arch. Rational Mech. Anal., 244(3), 599-697, 2022.
  * [63] Zhang, J.  Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD.  arXiv:2312.11254v3, preprint.