# Local Well-Posedness of the Free-Boundary Incompressible
Magnetohydrodynamics
with Surface Tension

Xumin Gu, Chenyun Luo, and Junyan Zhang School of Mathematics, Shanghai
University of Finance and Ecomonics, Shanghai 200433, China. Email:
<EMAIL_ADDRESS>XG was supported in part by NSFC Grant
12031006Department of Mathematics, Chinese University of Hong Kong, Shatin,
NT, Hong Kong. Email<EMAIL_ADDRESS>CL was supported in part by the
Direct Grant for Research 2020/2021, Project Code: 4053457Department of
Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA. Email:
<EMAIL_ADDRESS>
###### Abstract

We prove the local well-posedness of the 3D free-boundary incompressible ideal
magnetohydrodynamics (MHD) equations with surface tension, which describe the
motion of a perfect conducting fluid in an electromagnetic field. We adapt the
ideas developed in the remarkable paper [coutand2007LWP] by Coutand and
Shkoller to generate an approximate problem with artificial viscosity indexed
by $\kappa>0$ whose solution converges to that of the MHD equations as
$\kappa\to 0$. However, the local well-posedness of the MHD equations is no
easy consequence of Euler equations thanks to the strong coupling between the
velocity and magnetic fields. This paper is the continuation of the second and
third authors’ previous work [luozhangMHDST3.5] in which the a priori energy
estimate for incompressible free-boundary MHD with surface tension is
established. But the existence is not a trivial consequence of the a priori
estimate as it cannot be adapted directly to the approximate problem due to
the loss of the symmetric structure.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary lemmas

##  1 Introduction

We consider the following 3D incompressible ideal MHD system which describes
the motion of a conducting fluid with free surface boundary in an electro-
magnetic field under the influence of surface tension

$\begin{cases}(\partial_{t}+u\cdot\nabla)u-B\cdot\nabla B+\nabla
P=0,~{}~{}P:=p+\frac{1}{2}|B|^{2}~{}~{}~{}&\text{in}~{}\mathcal{D};\\\
(\partial_{t}+u\cdot\nabla)B-B\cdot\nabla
u=0,~{}~{}~{}&\text{in}~{}\mathcal{D};\\\ \text{div }u=0,~{}~{}\text{div
}B=0,~{}~{}~{}&\text{in}~{}\mathcal{D},\end{cases}$ (1.1)

with boundary conditions

$\begin{cases}(\partial_{t}+u\cdot\nabla)|_{\partial\mathcal{D}}\in\mathcal{T}(\partial\mathcal{D}),\\\
P=\sigma\mathcal{H}~{}~{}~{}&\text{on}~{}\partial\mathcal{D},\\\ B\cdot
n=0~{}~{}~{}&\text{on}~{}\partial\mathcal{D}.\end{cases}$ (1.2)

Here $\mathcal{D}:=\cup_{0\leq t\leq T}\\{t\\}\times\mathcal{D}_{t}$ and
$\mathcal{D}_{t}\subseteq\mathbb{R}^{3}$ is the bounded domain occupied by the
conducting fluid (plasma) whose boundary $\partial\mathcal{D}_{t}$ moves with
the velocity of the fluid. Here $u=(u_{1},u_{2},u_{3})$ is the fluid velocity,
$B=(B_{1},B_{2},B_{3})$ is the magnetic field, $p$ is the fluid pressure and
$P:=p+\frac{1}{2}|B|^{2}$ is the total pressure. The quantity $\mathcal{H}$ is
the mean curvature of the free surface $\partial\mathcal{D}_{t}$, $\sigma>0$
is a given constant, called surface tension coefficient and $n$ denotes the
exterior unit normal to $\partial\mathcal{D}_{t}$. Throughout the manuscript,
we will use the notation $D_{t}:=\partial_{t}+u\cdot\nabla$ to denote the
material derivative.

The first boundary condition shows that the boundary of the plasma moves with
the velocity of the fluid. It can be equivalently expressed as the velocity of
$(\partial\mathcal{D}_{t})$ is equal to $u\cdot n$. The second boundary
condition shows that the motion of the plasma is under the influence of
surface tension. Here we note that $\mathcal{H}$ is determined by the unknown
moving domain and thus not known a priori. The third boundary condition
implies that the plasma liquid is a perfect conductor. We also note that
$\text{div }B=0$ and $B\cdot n|_{\partial\mathcal{D}_{t}}=0$ are both required
only for initial data and they automatically propagate to any positive time.
Therefore, the system (1.1)-(1.2) is not over-determined.

Under the conditions above, we have the following conservation of physical
energy [luozhangMHDST3.5, Section 1].

$\frac{d}{dt}\left(\frac{1}{2}\int_{\mathcal{D}_{t}}|u|^{2}+|B|^{2}\,dx+\sigma\int_{\partial\mathcal{D}(t)}dS(\partial\mathcal{D}(t))\right)=0$
(1.3)

Given a simply connected domain $\mathcal{D}_{0}\subseteq\mathbb{R}^{3}$ and
initial data $u_{0}$ and $B_{0}$ satisfying $\text{div }u_{0}=0$ and
$\text{div }B_{0}=0,B_{0}\cdot n|_{\\{t=0\\}\times\partial\mathcal{D}_{0}}=0$,
we want to find a set $\mathcal{D}$ and vector fields $u$ and $B$ solving
(1.1)-(1.2) with initial data

$\mathcal{D}_{0}=\\{x:(0,x)\in\mathcal{D}\\},\quad(u,B)=(u_{0},B_{0}),\quad\text{in}\,\,\\{t=0\\}\times\Omega_{0}.$
(1.4)

######  Remark.

When the surface tension is neglected, the classical Rayleigh-Taylor sign
condition $-\nabla_{n}P\geq c_{0}>0$ is necessary for the well-posedness. Ebin
[ebin1987] and Hao-Luo [haoluo2018] constructed the counterexamples for Euler
equations and MHD equations respectively to show that the free-boundary
problems can be ill-posed when the Rayleigh-Taylor sign condition is violated.

###  1.1 History and Background

####  1.1.1 Physical background: Plasma-Vacuum model

The free-boundary problem (1.1)-(1.2) originates from the plasma-vacuum free-
interface model, which is an important theoretic model both in laboratory and
in astro-physical magnetohydrodynamics. The plasma is confined in a vacuum
with another magnetic field $\hat{B}$, and there is a free interface
$\Gamma(t)$, moving with the motion of plasma, between the plasma region
$\Omega_{+}(t)$ and the vacuum region $\Omega_{-}(t)$. Such model requires
that (1.1) holds in the plasma region $\Omega_{+}(t)$ and the pre-Maxwell
system holds in vacuum $\Omega_{-}(t)$:

$\text{curl }\hat{B}=\mathbf{0},~{}~{}~{}\text{div }\hat{B}=0.$ (1.5)

On the interface $\Gamma(t)$, the perfect conducting condition is required
that there is no jump in the the normal component:

$B\cdot n=\hat{B}\cdot
n=0,~{}~{}~{}[P]:=p+\frac{1}{2}|B|^{2}-\frac{1}{2}|\hat{B}|^{2}=\sigma\mathcal{H}$
(1.6)

where $n$ is the exterior unit normal to $\Gamma(t)$. Finally, there is a
rigid wall $W$ wrapping the vacuum region on which the following boundary
condition holds

$\hat{B}\times n_{w}=\mathcal{J}$

where $\mathcal{J}$ is the given outer surface current density (as an external
input of energy) and $n_{w}$ is the exterior unit normal of $W$. Note that for
the ideal MHD equations, $B\cdot n=0$ on $\partial\mathcal{D}_{t}$ and
$\text{div}\,B=0$ in $\mathcal{D}_{t}$ should be the constraints on initial
data and these constraints propagate. See more details in [MHDphy, Chapter 4,
6].

######  Remark.

We also note that, in the study of MHD surface waves in the free-interface
models, the effect of surface tension is crucially important for modelling MHD
flows in liquid metals, e.g., film flow, liquid-metal jets and drops, etc. See
also [MHDSTphy1, MHDSTphy2, MHDSTphy3] and references therein. Even if we
consider the MHD flows in astrophysical plasmas, where the surface tension
effect and magnetic diffusion are usually neglected, it is still useful to
keep surface tension as a stabilization effect in numerical simulations of the
magnetic Rayleigh-Taylor instability [MHDSTphy4, MHDSTphy5].

In this manuscript we consider the case that $\hat{B}=\mathbf{0}$, i.e., we
neglect the magnetic field in the vacuum. It characterize the free-surface
motion of an isolated liquid plasma under the influence of surface tension.

####  1.1.2 Review of previous results

In the absence of magnetic field $B$, the system (1.1)-(1.2) is reduced to the
free-boundary incompressible Euler equations. The study of free-surface
incompressible Euler equations has blossomed in the past several decades. In
the case of no surface tension ($\sigma=0$), the first breakthrough is Wu
[wu1997LWPww, wu1999LWPww] in which she proved the local well-posedness (LWP)
for the irrotational case without surface tension. See also [ambrose05,
lannes2005ww, mz2009, alazard2011st, alazard2014cauchy] for the LWP with or
without surface tension. In the case of nonzero vorticity, Christodoulou-
Lindblad [christodoulou2000motion] first proved the a priori estimates and
then Lindblad [lindblad2003, lindblad2005well] proved the LWP by using Nash-
Moser iteration. Later Coutand-Shkoller [coutand2007LWP, coutand2010LWP]
proved the LWP by using tangential smoothing and the energy estimates without
loss of regularity in the case of both $\sigma=0$ and $\sigma>0$. See also
Zhang-Zhang [zhangzhang08Euler] for the study of incompressible water wave. In
the case of nonzero surface tension, we refer to [Schweizer05, coutand2007LWP,
coutand2008VX, shatah2008a, shatah2008b, shatah2011] for LWP, and
[ignatova2016, DK2017, DKT] for low regularity estimates.

However, the study of free-boundary MHD equations is far less developed as
opposed to Euler equations. The strong coupling between the magnetic field and
the motion of fluid destroys some good properties of Euler equations such as
the propagation of the irrotational assumption. Most of the known results
focus on the case of zero surface tension. When the surface tension is
neglected, extra stabilization such as the Rayleigh-Taylor sign condition is
required. Lee [leeMHD1, leeMHD2] proved the LWP for viscous-resistive MHD and
the vanishing viscosity-resistivity limit. For the free-boundary problem of
ideal incompressible MHD under Rayleigh-Taylor sign condition, Hao-Luo
[haoluo2014] proved the a priori estimates and [haoluo2019] proved the
linearized LWP. Then the first author and Wang [gu2016construction] proved the
LWP. The second and the third authors [luozhangMHD2.5] proved the minimal
regularity $H^{\frac{5}{2}+\varepsilon}$ estimates for a small fluid domain.
For the plasma-vacuum model under Rayleigh-Taylor sign condition, Hao
[hao2017] proved the a priori estimates when $\mathcal{J}=\mathbf{0}$ and the
first author [guaxi1, guaxi2] proved the LWP for axi-symmetric case with a
non-zero vacuum magnetic field in a non-simply connected domain. We note that
there is another non-collinearity condition111Such condition comes from the
study of stability of current-vortex sheet which is a two-fluid (plasma-
plasma) model in free-boundary MHD. $|B\times\hat{B}|\geq c_{0}>0$ which gives
extra $1/2$-order regularity of the free interface than Rayleigh-Taylor sign
condition for the plasma-vacuum model. Under this condition, Morando-
Trebeschi-Trakhinin [iMHDlinear] proved the LWP for linearized plasma-vacuum
system and Sun-Wang-Zhang [sun2017well] proved the nonlinear LWP. Coulombo-
Morando-Secchi-Trebeschi [iMHDVX] proved the a priori estimates for 3D
incompressible current-vortex sheets and Sun-Wang-Zhang [sun2015well] proved
the LWP. So far, the energy estimates and well-posedness of the plasma-vacuum
model in general cases (i.e., $\mathcal{J}\neq 0$ and without axi-symmetric
assumption) under Rayleigh-Taylor sign condition are still open problems.

In the case of nonzero surface tension, there are very few results for the
free-boundary MHD system and most previous works focus on the resistive or
viscous MHD. To the best of our knowledge, The second and the third authors’
previous work [luozhangMHDST3.5] which proved the $H^{7/2}$ a priori estimates
is the only avaliable result for incompressible ideal MHD with surface
tension. We also refer to Chen-Ding [chendingMHDlimit] for inviscid limit,
Wang-Xin [wangxinMHD] for GWP of incompressible resistive MHD around a
transversal uniform magnetic field, and Padula-Solonnikov [Solonnikov], Guo-
Zeng-Ni [GuoMHDSTviscous] for incompressible viscous-resistive MHD.

Finally, for compressible MHD, we refer to Secchi-Trakhinin [secchi2013well]
for the LWP of plasma-vacuum model under non-collinearity condition, and Chen-
Wang [chengqMHD], Trakhinin [trakhininMHD2009] and Wang-Yu [wangyuMHD2d] for
compressible current-vortex sheets in 3D and 2D. Very recently, Trakhinin-Wang
proved the LWP of free-boundary compressible ideal MHD under Rayleigh-Taylor
sign condition [trakhininMHD2020] or with surface tension [trakhininMHD2021].
All these results are proved by Nash-Moser iteration and thus there is no
energy estimate without regularity loss. The third author proved the LWP
[ZhangCRMHD2] and the incompressible limit [ZhangCRMHD1] of compressible
resistive MHD under Rayleigh-Taylor sign condition with energy estimates of no
regularity loss. Finding suitable energy estimates without regularity loss for
compressible ideal MHD with or without surface tension is also a widely open
problem. The plasma-vacuum model in compressible MHD under Rayleigh-Taylor
sign condition is also unsolved. See Trakhinin [trakhininMHD2016] for detailed
discussion.

In this manuscript, we prove the local well-posedness with energy estimates of
no regularity loss for the free-boundary problem in incompressible ideal MHD
with surface tension. Our result is a necessary step to study the plasma-
vacuum model under the influence of surface tension, which is an original
theoretical model in the study of confined plasma in both laboratory and
astro-physical MHD.

###  1.2 Reformulation in Lagrangian coordinates

We reformulate the MHD equations in Lagrangian coordinates and thus the free-
surface domain becomes fixed. Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded
domain. Denoting coordinates on $\Omega$ by $y=(y_{1},y_{2},y_{3})$, we define
$\eta:[0,T]\times\Omega\to\mathcal{D}$ to be the flow map of the velocity $u$,
i.e.,

$\partial_{t}\eta(t,y)=u(t,\eta(t,y)),\quad\eta(0,y)=y.$ (1.7)

We introduce the Lagrangian velocity, magnetic field and pressure respectively
by

$v(t,y)=u(t,\eta(t,y)),\quad b(t,y)=B(t,\eta(t,y)),\quad
q(t,y)=P(t,\eta(t,y)).$ (1.8)

Let $\partial$ be the spatial derivative with respect to $y$ variable. We
introduce the cofactor matrix $a=[\partial\eta]^{-1}$. Specifically,
$a^{\mu\alpha}=\frac{\partial y^{\mu}}{\partial x^{\alpha}}$, where
$x^{\alpha}(t,y)=\eta^{\alpha}(t,y)$. Also, we define $J:=\det[\partial\eta]$,
which is well-defined since $\eta(t,\cdot)$ is almost the identity map when
$t$ is sufficiently small. It’s worth noting that $a$ verifies the Piola’s
identity and $J=1$ in the incompressible case, i.e.,

$\partial_{\mu}a^{\mu\alpha}=\partial_{\mu}(Ja^{\mu\alpha})=0\text{ and }J=1.$
(1.9)

Here, the Einstein summation convention is used for repeated upper and lower
indices. In above and throughout, all Greek indices range over 1, 2, 3, and
the Latin indices range over 1, 2.

For the sake of simplicity and clean notation, here we consider the model case
when

$\Omega=\mathbb{T}^{2}\times(0,1),$ (1.10)

where $\partial\Omega=\Gamma_{0}\cup\Gamma$ and
$\Gamma=\mathbb{T}^{2}\times\\{1\\}$ is the top (moving) boundary,
$\Gamma_{0}=\mathbb{T}^{2}\times\\{0\\}$ is the fixed bottom. We mention here
that $\Omega$ is known as the reference domain, which allows us to work in one
coordinate patch. We refer the interested readers to [coutand2007LWP] for
detailed discussions. Using a partition of unity, e.g., [DKT], a general
domain can also be treated with the same tools we shall present. However,
choosing $\Omega$ as above allows us to focus on the real issues of the
problem without being distracted by the cumbersomeness of the partition of
unity. Let $N$ stands for the outward unit normal of $\partial\Omega$. In
particular, we have $N=(0,0,-1)$ on $\Gamma_{0}$ and $N=(0,0,1)$ on $\Gamma$.

Under this setting, the system (1.1)-(1.2) can be reformulated as:

$\begin{cases}\partial_{t}v_{\alpha}-b_{\beta}a^{\mu\beta}\partial_{\mu}b_{\alpha}+a^{\mu}_{\alpha}\partial_{\mu}q=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
\partial_{t}b_{\alpha}-b_{\beta}a^{\mu\beta}\partial_{\mu}v_{\alpha}=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
a^{\mu\alpha}\partial_{\mu}v_{\alpha}=0,~{}~{}a^{\mu\alpha}\partial_{\mu}b_{\alpha}=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
v^{3}=b^{3}=0~{}~{}~{}&\text{on}~{}\Gamma_{0};\\\
a^{\mu\alpha}N_{\mu}q+\sigma(\sqrt{g}\Delta_{g}\eta^{\alpha})=0~{}~{}~{}&\text{on}~{}\Gamma;\\\
a^{\mu\nu}b_{\nu}N_{\mu}=0~{}~{}~{}&\text{on}~{}\Gamma,\\\ \end{cases}$ (1.11)

where $N$ is the unit outer normal vector to $\partial\Omega$, and
$\Delta_{g}$ is the Laplacian of the metric $g_{ij}$ induced on
$\Gamma(t)=\eta(t,\Gamma)$ by the embedding $\eta$. Specifically, we have:

$g_{ij}=\overline{\partial}{}_{i}\eta^{\mu}\overline{\partial}{}_{j}\eta_{\mu},~{}\Delta_{g}(\cdot)=\frac{1}{\sqrt{g}}\overline{\partial}{}_{i}(\sqrt{g}g^{ij}\overline{\partial}{}_{j}(\cdot)),\text{
where }g:=\det(g_{ij}).$ (1.12)

Here, we use $\overline{\partial}{}$ to emphasis that the derivative is
tangential to $\Gamma$. In particular,
$\overline{\partial}{}=(\overline{\partial}{}_{1},\overline{\partial}{}_{2})=(\partial_{1},\partial_{2})$.

By the second equation of (1.11) and the divergence-free condition on $b$, we
get $\partial_{t}(a^{\mu\alpha}b_{\mu})=0$ which implies
$a^{\mu\alpha}b_{\mu}=b_{0}^{\alpha}$ and thus
$b^{\alpha}=b_{0}^{\mu}\partial_{\mu}\eta^{\alpha}=(b_{0}\cdot\partial)\eta^{\alpha}$.
See Gu-Wang [gu2016construction, (1.13)-(1.15)] for the proof. Therefore, the
system (1.11) can be equivalently written as the following system of
$(\eta,v,q)$

$\begin{cases}\partial_{t}\eta=v~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
\partial_{t}v-(b_{0}\cdot\partial)^{2}\eta+\nabla_{a}q=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
\text{div }_{a}v=0,&\text{in}~{}[0,T]\times\Omega;\\\ \text{div
}b_{0}=0~{}~{}~{}&\text{in}~{}\\{t=0\\}\times\Omega;\\\
v^{3}=b_{0}^{3}=0~{}~{}~{}&\text{on}~{}\Gamma_{0};\\\
a^{3\alpha}q+\sigma(\sqrt{g}\Delta_{g}\eta^{\alpha})=0~{}~{}~{}&\text{on}~{}\Gamma;\\\
b_{0}^{3}=0~{}~{}~{}&\text{on}~{}\Gamma,\\\
(\eta,v)=(\operatorname{Id},v_{0})~{}~{}~{}&\text{on}~{}\\{t=0\\}{\times}\overline{\Omega}.\end{cases}$
(1.13)

######  Notation 1.1.

The differential operator
$\nabla_{a}:=(\nabla_{a}^{1},\nabla_{a}^{2},\nabla_{a}^{3})$ with
$\nabla_{a}^{\alpha}:=a^{\mu\alpha}\partial_{\mu}$, and for a smooth vector
field $X$, we denote by $\text{div }_{a}X:=\nabla_{a}\cdot
X=a^{\mu\alpha}\partial_{\mu}X_{\alpha}$ the Eulerian divergence of $X$ and by
$\text{div}\,X:=\partial\cdot X=\delta^{\mu\alpha}\partial_{\mu}X_{\alpha}$
the flat divergence.

######  Remark.

The initial data of $q$ is determined by $v_{0}$ and $b_{0}$. In particular,
$q_{0}$ satisfies an elliptic equation

$\displaystyle\begin{cases}-\Delta q_{0}=(\partial v_{0})(\partial
v_{0})-(\partial b_{0})(\partial b_{0}),~{}~{}\text{in}\,\,\Omega,\\\
q_{0}=0,~{}~{}\text{on}\,\,\Gamma,\\\ \frac{\partial q_{0}}{\partial
N}=0,~{}~{}\text{on}\,\,\Gamma_{0}.\end{cases}$ (1.14)

The boundary condition on $\Gamma$ is derived by restricting the boundary
condition $a^{3\alpha}q+\sigma(\sqrt{g}\Delta_{g}\eta^{\alpha})=0$ at $t=0$.
Then it becomes
$\delta^{3\alpha}q_{0}+\sigma\overline{\Delta}{}\eta_{0}^{\alpha}=0,$ where
$\overline{\Delta}{}:=\overline{\partial}{}_{1}^{2}+\overline{\partial}{}_{2}^{2}$,
and this yields $q_{0}=0$ on $\Gamma$ since
$\overline{\partial}{}^{2}\eta_{0}^{3}=0$. On the other hand, the boundary
condition on $\Gamma_{0}$ is derived from by restricting
$\partial_{t}v-(b_{0}\cdot\partial)^{2}\eta+\nabla_{a}q=0$ on
$\\{t=0\\}\times\Gamma_{0}$ and then taking the normal component, where we
have used the fact that $a^{13}=a^{23}=0$ and $a^{33}=1$ on $\Gamma_{0}$.

###  1.3 Main result

We prove the local well-posedness of (1.13) in the presenting manuscript. We
denote $\|f\|_{s}:=\|f(t,\cdot)\|_{H^{s}(\Omega)}$ for any function
$f(t,y)\text{ on }[0,T]\times\Omega$ and
$|f|_{s}:=|f(t,\cdot)|_{H^{s}(\Gamma)}$ for any function $f(t,y)\text{ on
}[0,T]\times\Gamma$. Let $\Pi$ be the canonical normal project defined on the
tangent bundle of the moving interface. Our main result is:

######  Theorem 1.2.

Let $v_{0}\in H^{4.5}(\Omega)\cap H^{5}(\Gamma)$ and $b_{0}\in
H^{4.5}(\Omega)$ be divergence-free vector fields with $(b_{0}\cdot
N)|_{\Gamma}=0$. Then there exists some $T>0$, only depending on
$\sigma,v_{0},b_{0}$, such that the system (1.13) with initial data
$(v_{0},b_{0},q_{0})$ has a unique strong solution $(\eta,v,q)$ with the
energy estimates

$\sup_{0\leq t\leq T}E(t)\leq\mathcal{C},$ (1.15)

where $\mathcal{C}$ is a constant depends on
$\|v_{0}\|_{4.5},\|b_{0}\|_{4.5}$, $|v_{0}|_{5}$, and

$\displaystyle E(t)$
$\displaystyle:=\left\|\eta(t)\right\|_{4.5}^{2}+\left\|v(t)\right\|_{4.5}^{2}+\left\|\partial_{t}v(t)\right\|_{3.5}^{2}+\left\|\partial_{t}^{2}v(t)\right\|_{2.5}^{2}+\left\|\partial_{t}^{3}v(t)\right\|_{1.5}^{2}+\left\|\partial_{t}^{4}v(t)\right\|_{0}^{2}$
(1.16)
$\displaystyle+\left\|(b_{0}\cdot\partial)\eta(t)\right\|_{4.5}^{2}+\left\|\partial_{t}(b_{0}\cdot\partial)\eta(t)\right\|_{3.5}^{2}+\left\|\partial_{t}^{2}(b_{0}\cdot\partial)\eta(t)\right\|_{2.5}^{2}+\left\|\partial_{t}^{3}(b_{0}\cdot\partial)\eta(t)\right\|_{1.5}^{2}+\left\|\partial_{t}^{4}(b_{0}\cdot\partial)\eta(t)\right\|_{0}^{2}$
$\displaystyle+\left|\overline{\partial}{}\left(\Pi\partial_{t}^{3}v(t)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\left(\Pi\overline{\partial}{}\partial_{t}^{2}v(t)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\left(\Pi\overline{\partial}{}^{2}\partial_{t}v(t)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\big{(}\Pi\overline{\partial}{}^{3}v(t)\big{)}\right|_{0}^{2}+\left|\overline{\partial}{}\big{(}\Pi\overline{\partial}{}^{3}(b_{0}\cdot\partial)\eta(t)\big{)}\right|_{0}^{2}.$

Moreover, the $H^{5}(\Gamma)$-regularity of $v$ on the free-surface can also
be recovered, in the sense that there exists some $0<T_{1}<T$, depending only
on $\sigma^{-1},v_{0},b_{0}$, such that

$\sup_{0\leq t\leq T_{1}}|\eta(t)|_{5}^{2}+|v(t)|_{5}^{2}\leq
C(\sigma^{-1},\|v_{0}\|_{4.5},\|b_{0}\|_{4.5}).$ (1.17)

######  Remark (Smoothing effect of $b_{0}\cdot\partial$).

It can be seen that in (1.16) $v$ and $(b_{0}\cdot\partial)\eta$ are of the
same interior regularity (i.e., $H^{4.5}(\Omega)$). This suggests that
$(b_{0}\cdot\partial)$ and $\partial_{t}$ behave the same when falling on the
flow map $\eta$. This observation turns out to be very important when studying
the energy of the approximate equations (1.19) defined below.

###  1.4 Strategy of the proof

####  1.4.1 Necessity of the tangential smoothing

In [luozhangMHDST3.5], the second and third authors proved the a priori
estimates of (1.13). However, it is often highly nontrivial to prove the local
well-posedness for a free-boundary problem of inviscid fluid, especially when
equipped with the Young-Laplace boundary condition, by a simple iteration
scheme and fixed-point argument for the linearized equations. The reason is
that the linearization breaks the subtle cancellation structure on the free
surface and thus causes the loss of tangential derivatives of the flow map
$\eta$, which also occurs for incompressible Euler equations with surface
tension.

In their remarkable work [coutand2007LWP], Coutand and Shkoller introduced an
approximate system in the Lagrangian coordinates by smoothing the nonlinear
coefficients in the tangential direction. This can be adapted to study the MHD
equations and the tangential smoothing preserves the essential transport-type
structure of the original equations. Specifically, we define
$\Lambda_{\kappa}$ to be the standard mollifier with parameter $\kappa>0$ on
$\mathbb{R}^{2}$ as in (LABEL:lkk0). Let
$\tilde{\eta}:=\Lambda_{\kappa}^{2}\eta$ and
$\tilde{a}=[\partial\tilde{\eta}]^{-1}$. Then we set nonlinear
$\kappa$-approximation problem by replacing $a$ with $\tilde{a}$. However,
such construction is not applicable to MHD because we also need to control
$\|[\Lambda_{\kappa}^{2},(b_{0}\cdot\partial)]\eta\|_{4.5}$ in which there is
a normal derivative $b_{0}^{3}\partial_{3}$ that is not compatible with the
tangential mollification in the interior. Motivated by Gu-Wang
[gu2016construction], we first mollify the flow map on the boundary, then
extend it into the interior by the harmonic extension, i.e.,

$\begin{cases}-\Delta\tilde{\eta}=-\Delta\eta~{}~{}~{}&\text{ in }\Omega,\\\
\tilde{\eta}=\Lambda_{\kappa}^{2}\eta~{}~{}~{}&\text{ on }\Gamma.\end{cases}$
(1.18)

Define
$\tilde{a}:=[\partial\tilde{\eta}]^{-1},~{}\tilde{J}:=\det[\partial\tilde{\eta}]$
and $\tilde{A}:=\tilde{J}\tilde{a}$, then we have the Piola’s identity
$\partial_{\mu}\tilde{A}^{\mu\alpha}=0$. The nonlinear approximate system is
defined to be

$\begin{cases}\partial_{t}\eta=v~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
\partial_{t}v-(b_{0}\cdot\partial)^{2}\eta+\nabla_{\tilde{A}}q=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega;\\\
\text{div}_{\tilde{a}}v=0,&\text{in}~{}[0,T]\times\Omega;\\\ \text{div
}b_{0}=0~{}~{}~{}&\text{in}~{}\\{t=0\\}\times\Omega;\\\
v^{3}=b_{0}^{3}=0~{}~{}~{}&\text{on}~{}\Gamma_{0};\\\
\tilde{A}^{3\alpha}q=-\sigma\sqrt{g}(\Delta_{g}\eta\cdot\tilde{n})\tilde{n}^{\alpha}+\kappa\left((1-\overline{\Delta}{})(v\cdot\tilde{n})\right)\tilde{n}^{\alpha}~{}~{}~{}&\text{on}~{}\Gamma;\\\
b_{0}^{3}=0~{}~{}~{}&\text{on}~{}\Gamma,\\\
(\eta,v)=(\operatorname{Id},v_{0})~{}~{}~{}&\text{in}~{}\\{t=0\\}{\times}\overline{\Omega}.\end{cases}$
(1.19)

In this paper, we will (i). derive the uniform-in-$\kappa$ a priori estimates
of the system (1.19), and then (ii). solve the nonlinear
$\kappa$-approximation system (1.19).

####  1.4.2 Necessity of the artificial viscosity

There is an artificial viscosity term
$\kappa\left((1-\overline{\Delta}{})(v\cdot\tilde{n})\right)\tilde{n}^{\alpha}$
in the smoothed surface tension equation on the boundary. This was first
introduced by Coutand-Shkoller in [coutand2007LWP] where the authors mentioned
that the artificial viscosity term appears to be necessary in order to prove
the existence of an inviscid fluid with non-trivial vorticity and surface
tension. This term also appears in the subsequent work that studies the free-
surface fluid with surface tension, e.g., Cheng-Coutand-Shkoller
[coutand2008VX] for the vortex sheets, Coutand-Hole-Shkoller [coutand2013LWP]
for the compressible Euler, and very recently Trakhinin-Wang
[trakhininMHD2021] for the compressible MHD.

######  Remark.

Very recently, the first author and Lei [GuLeielastoST] proved the LWP of
incompressible elastodynamics with surface tension by proving the inviscid
limit of visco-elastodynamics system in standard Sobolev spaces. We also note
that the inviscid limit of free-boundary MHD was recently proved by Chen-Ding
[chendingMHDlimit] in co-normal Sobolev spaces. However, analogous inviscid
limit in standard Sobolev space is not applicable to MHD due to the existence
of MHD boundary layers.

An essential reason for introducing such artificial viscosity term is that the
presence of surface tension forces us to control all of the time derivatives.
In particular, the pressure $q$ satisfies an elliptic equation and it appears
that one can only get control of it by considering the Neumann boundary
condition instead of Dirichlet boundary condition due to the presence of
surface tension. The Neumann boundary condition contains the time derivative
of $v$, and thus we have to include the time derivatives in our energy.

However, the full time derivatives of $v$ and $(b_{0}\cdot\partial)\eta$ only
has $L^{2}(\Omega)$ regularity and we cannot get estimates of the full time
derivatives of $q$ due to the low spatial regularity. Therefore, we do not
have any control for the terms containing full time derivatives on the
boundary due to the failure of Sobolev trace lemma. For the original system,
one can use the subtle cancellation structure developed in [DK2017,
luozhangMHDST3.5] to resolve this difficulty. But such cancellation structure
no longer holds for the nonlinear $\kappa$-approximate problem due to the
presence of tangential smoothing. Therefore, introducing the artificial
viscosity term could produce $\kappa$-weighted higher order terms on the
boundary, which enables us to finish the energy control.

######  Remark.

The Young-Laplace boundary condition only gives us the information in the
Eulerian normal direction. Therefore, the artificial viscosity can only be
imposed in the smoothed Eulerian normal direction
$\kappa\left((1-\overline{\Delta}{})(v\cdot\tilde{n})\right)\tilde{n}^{\alpha}$
instead of all the components, otherwise the system would be over-determined.

####  1.4.3 Difference from the case without surface tension

The first author and Wang [gu2016construction] proved the LWP of
incompressible MHD without surface tension, in which the pressure $q$ can be
controlled by the elliptic equation with Dirichlet (zero) boundary condition
and thus one can avoid the estimates of all time derivatives which turn out to
be very complicated in the presenting manuscript. This tells an essential
difference from the case without the surface tension.

On the other hand, as mentioned in [coutand2007LWP, DK2017, luozhangMHDST3.5],
surface tension has a stronger stabilization effect than the Rayleigh-Taylor
sign condition in the case without surface tension. In fact, the presence of
surface tension allows us to control the boundary norms of the normal
component of $v$ and $(b_{0}\cdot\partial)\eta$ by comparing with the
corresponding Eulerian normal projections instead of using normal trace
theorem to reduce to interior tangential estimates. We refer Section
LABEL:sect_bdy_estimate_3.3 for details. This property allows us to gain extra
$1/2$ derivatives in the interior, and there is no need to introduce the
Alinhac good unknowns and correction terms as in [gu2016construction].

####  1.4.4 Illustration on the energy functional

Let $\Pi$ be the canonical normal project defined on the tangent bundle of the
moving interface and $\tilde{n}$ be the (Eulerian) unit normal (We refer Lemma
2.1 for the precise definition). The energy functional of the nonlinear
approximate problem (1.19) is defined to be

$E_{\kappa}=E_{\kappa}^{(1)}+E_{\kappa}^{(2)}+E_{\kappa}^{(3)},$

where

$\displaystyle E_{\kappa}^{(1)}:=$
$\displaystyle\left\|\eta(\kappa)\right\|_{4.5}^{2}+\left\|v(\kappa)\right\|_{4.5}^{2}+\left\|\partial_{t}v(\kappa)\right\|_{3.5}^{2}+\left\|\partial_{t}^{2}v(\kappa)\right\|_{2.5}^{2}+\left\|\partial_{t}^{3}v(\kappa)\right\|_{1.5}^{2}+\left\|\partial_{t}^{4}v(\kappa)\right\|_{0}^{2}$
$\displaystyle+\left\|(b_{0}\cdot\partial)\eta(\kappa)\right\|_{4.5}^{2}+\left\|\partial_{t}(b_{0}\cdot\partial)\eta(\kappa)\right\|_{3.5}^{2}+\left\|\partial_{t}^{2}(b_{0}\cdot\partial)\eta(\kappa)\right\|_{2.5}^{2}+\left\|\partial_{t}^{3}(b_{0}\cdot\partial)\eta(\kappa)\right\|_{1.5}^{2}+\left\|\partial_{t}^{4}(b_{0}\cdot\partial)\eta(\kappa)\right\|_{0}^{2}$
$\displaystyle+\left|\overline{\partial}{}\left(\Pi\partial_{t}^{3}v(\kappa)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\left(\Pi\overline{\partial}{}\partial_{t}^{2}v(\kappa)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\left(\Pi\overline{\partial}{}^{2}\partial_{t}v(\kappa)\right)\right|_{0}^{2}+\left|\overline{\partial}{}\big{(}\Pi\overline{\partial}{}^{3}v(\kappa)\big{)}\right|_{0}^{2}+\left|\overline{\partial}{}\big{(}\Pi\overline{\partial}{}^{3}(b_{0}\cdot\partial)\eta(\kappa)\big{)}\right|_{0}^{2},$
$\displaystyle E_{\kappa}^{(2)}:=$
$\displaystyle\sigma\int_{0}^{T}\bigg{(}\left|\sqrt{\kappa}\partial_{t}^{4}v(\kappa)\cdot\tilde{n}(\kappa)\right|_{1}^{2}+\left|\sqrt{\kappa}\partial_{t}^{3}v(\kappa)\cdot\tilde{n}(\kappa)\right|_{2}^{2}+\left|\sqrt{\kappa}\partial_{t}^{2}v(\kappa)\cdot\tilde{n}(\kappa)\right|_{3}^{2}$
$\displaystyle+\left|\sqrt{\kappa}\partial_{t}v(\kappa)\cdot\tilde{n}(\kappa)\right|_{4}^{2}+\left|\sqrt{\kappa}(b_{0}\cdot\partial)v(\kappa)\cdot\tilde{n}(\kappa)\right|_{4}^{2}\bigg{)}\,dt,$
$\displaystyle E_{\kappa}^{(3)}:=$
$\displaystyle\int_{0}^{T}\Big{(}\big{\|}\sqrt{\kappa}\partial_{t}^{4}v(\kappa)\big{\|}_{1.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}^{4}(b_{0}\cdot\partial)\eta\big{\|}_{1.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}^{3}v(\kappa)\big{\|}_{2.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}^{3}(b_{0}\cdot\partial)\eta\big{\|}_{2.5}^{2}\Big{)}$
$\displaystyle+\big{\|}\sqrt{\kappa}\partial_{t}^{2}v(\kappa)\big{\|}_{3.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}^{2}(b_{0}\cdot\partial)\eta\big{\|}_{3.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}v(\kappa)\big{\|}_{4.5}^{2}+\big{\|}\sqrt{\kappa}\partial_{t}(b_{0}\cdot\partial)\eta\big{\|}_{4.5}^{2}\Big{)}\,dt.$

The energy constructed above looks much more complicated than (1.16), but it
is in fact quite natural. First, $E_{\kappa}^{(1)}$ constitutes the non-
weighted energies which are needed in order to close the a priori estimate for
the MHD equations without the artificial viscosity (cf. Luo-Zhang
[luozhangMHDST3.5]). Then $E_{\kappa}^{(2)}$ consists of the $\kappa$-weighted
higher order energy terms produced by the artificial viscosity when dealing
with the tangential estimates.

Besides, extra error terms are generated when all the derivatives fall on the
smoothed Eulerian normal $\tilde{n}$ in the construction of
$E_{\kappa}^{(2)}$. Since $E_{\kappa}^{(2)}$ only gives us higher order
control of the normal component instead of all components. Most of the top
order error terms should be treated by moving them to the interior with the
help of Sobolev trace lemma, and we use $E_{\kappa}^{(3)}$ to record all of
them. Nevertheless, due to the strong coupling structure (see Subsection 1.4.5
for more details) between the velocity and the magnetic field, the terms in
$E_{\kappa}^{(3)}$ must be controlled together via the Hodge-type div-curl
estimate and thus we have to include the associated magnetic terms in
$E_{\kappa}^{(3)}$ as well.

When closing the energy estimates of $E_{\kappa}^{(2)}$ and
$E_{\kappa}^{(3)}$, and $\overline{\partial}{}^{3}\partial_{t}$-tangential
estimates, one needs the control of $\sqrt{\kappa}$-weighted
$H^{5}(\Gamma)$-norms of $\eta,v$ and $(b_{0}\cdot\partial)\eta$ recorded in
Lemma LABEL:super_control. These $\sqrt{\kappa}$-weighted bounds can be
established by considering
$\overline{\partial}{}^{4},\overline{\partial}{}^{4}\partial_{t},\overline{\partial}{}^{4}(b_{0}\cdot\partial)$-differentiated
smoothed Young-Laplace boundary condition. See also Coutand-Shkoller
[coutand2007LWP, Lemma 12.6].

######  Remark.

In the proof of Lemma LABEL:super_control, the self-adjointness of
$\Lambda_{\kappa}$ is used to keep the structure and close the energy
estimates. This is the reason that we need to mollify $\eta$ twice in (1.18).

####  1.4.5 Difference between Euler equations and MHD with surface tension

As mentioned in [luozhangMHD2.5, luozhangMHDST3.5], the Cauchy invariance for
the Euler equations no longer holds for MHD equations, which makes it
impossible to get a higher regularity of the flow map $\eta$ than that of the
velocity $v$. Without such property, one cannot control the
$\overline{\partial}{}^{4}$-tangential energy estimate directly as what
Coutand-Shkoller did in [coutand2007LWP] for the incompressible Euler
equations.

In addition to this, the strong coupling between $v$ and
$b=(b_{0}\cdot\partial)\eta$ yields that the Sobolev norm of their vorticities
have to be controlled together (see equation (LABEL:curlbeq)). As a
consequence, the full Sobolev norms of $v$ and $b$ (and their time
derivatives) have to be studied simultaneously.

Finally, the $|\sqrt{\kappa}\eta|_{6}$ regularity for 3D incompressible Euler
equations in [coutand2007LWP] cannot be achieved either. But this does not
affect the proof for MHD system unless one wants to get a
$H^{6}(\Gamma)$-posteriori estimates for the flow map $\eta$.

####  1.4.6 Penalization method to solve the linearized problem

Finally, it remains to solve the nonlinear approximation problem. With the
help of tangential smoothing, it is not difficult for us to finish the
iteration from the linearized approximate problem to the nonlinear one. But it
is still difficult to solve the linearized approximate problem by the fixed-
point argument even if one can get the a priori estimates without the loss of
regularity. The reason is that we do not have any suitable equation for $q$
and thus the structure of the linearized system is no longer preserved in the
verification of fixed-point argument. Motivated by [coutand2007LWP], we use
the penalization method to solve the linearized system. We introduce a
penalized pressure defined by
$q_{\lambda}:=-\lambda^{-1}\text{div}_{\mathring{\tilde{A}}}w_{\lambda}$ and
prove the existence of $L^{2}$-weak solution to the penalized problem by
Galerkin’s method. Then we take the weak limit by passing $\lambda\to 0$ to
get the weak solution of the linearized approximate problem. Finally, one can
prove the weak solution is strong by $H^{1}$-estimates together with the
inverse theorem of div-curl decomposition (cf. Lemma LABEL:hodge (2)).

######  Remark.

The penalization method is not needed in the compressible case because the
free-boundary compressible MHD is a first-order symmetric hyperbolic system
with characteristic boundary conditions and the corresponding linearized
problem can be solved by the duality argument in Lax-Phillips [lax60]. We
refer to Trakhinin-Wang [trakhininMHD2020, trakhininMHD2021] for details.

######  Remark.

We cannot directly prove the weak solution of the penalized problem is strong
as in [coutand2007LWP] as the divergence part cannot be controlled because of
the presence of the magnetic field. That is why we first take the weak limit
and then verify the $H^{1}$-estimates for the linearized system.

######  Remark.

In the a priori estimates and iteration process of the linearized approximate
problem, the energy control is much simpler than the uniform-in-$\kappa$
estimates of the nonlinear approximate problem (LABEL:MHDLkk) because we no
longer require the energy is $\kappa$-independent. Therefore, one can use the
elliptic estimates for equations with merely BMO-coefficients proved by Dong-
Kim [DKell] (see also Disconzi-Kukavica [DK2017, Proposition 3.4]) to get the
boundary control. See Section LABEL:sect_exist_kk-prob for details.

###  1.5 Organization of the paper

The presenting manuscript is organized as follows. In Section 2 we record the
lemmas that are repeatedly used in the proof. Then we introduce the nonlinear
$\kappa$-approximation problem and do the div-curl-boundary estimates in
Section LABEL:sect_nonlinearkk. The non-weighted energy $E_{\kappa}^{(1)}$ and
$\sqrt{\kappa}$-weighted boundary norms are treated in Section LABEL:tgkk and
$\sqrt{\kappa}$-weighted interior norms are treated in Section
LABEL:sect_E3kk. Then the uniform-in-$\kappa$ estimates for the nonlinear
$\kappa$-approximate problem are closed in Section LABEL:close. In Section
LABEL:linearlwpkk we solve the linearized approximate system by penalization
method. In Section LABEL:sect_exist_kk-prob we use Picard iteration to solve
the nonlinear $\kappa$-approximate problem. Finally, the local well-posedness
and energy estimates of the original system are established in Section
LABEL:lwp1.

The following notations will be frequently used in the rest of this
manuscript.

List of Notations:

  * •

$\Omega:=\mathbb{T}^{2}\times(0,1)$. $\Gamma:=\mathbb{T}^{2}\times\\{1\\}$ is
the free boundary and $\Gamma_{0}:=\mathbb{T}^{2}\times\\{0\\}$ is the fixed
bottom.

  * •

$\|\cdot\|_{s}$: We denote $\|f\|_{s}:=\|f(t,\cdot)\|_{H^{s}(\Omega)}$ for any
function $f(t,y)\text{ on }[0,T]\times\Omega$.

  * •

$|\cdot|_{s}$: We denote $|f|_{s}:=|f(t,\cdot)|_{H^{s}(\Gamma)}$ for any
function $f(t,y)\text{ on }[0,T]\times\Gamma$.

  * •

$P(\cdot)$: A generic non-decreasing continuous function in its arguments;

  * •

$\overline{\partial}{},\overline{\Delta}{}$:
$\overline{\partial}{}=\partial_{1},\partial_{2}$ denotes the tangential
derivative and $\overline{\Delta}{}:=\partial_{1}^{2}+\partial_{2}^{2}$
denotes the tangential Laplacian.

  * •

For a smooth scalar function $f$,
$\nabla^{\alpha}_{a}f:=a^{\mu\alpha}\partial_{\mu}f$. Also, for a smooth
vector field $X$, $\text{div }_{a}X:=a^{\mu\alpha}\partial_{\mu}X_{\alpha}$
and $(\text{curl
}_{a}X)_{\lambda}:=\epsilon_{\lambda\tau\alpha}a^{\mu\tau}\partial_{\mu}X^{\alpha}$,
where $\epsilon_{\lambda\tau\alpha}$ is the sign of the 3-permutation
$(\lambda\tau\alpha)\in S_{3}.$

  * •

Let $X$ be given as above. The flat divergence and curl are given as
$\text{div}\,X:=\delta^{\mu\alpha}\partial_{\mu}X_{\alpha}$, and $(\text{curl
}X)^{\lambda}:=\epsilon^{\lambda\tau\alpha}\partial_{\tau}X_{\alpha}$.

##  2 Preliminary lemmas

###  2.1 Geometric identities

The following geometric identities will be used repeatedly (and silently)
throughout this manuscript.

######  Lemma 2.1.

Let $\hat{n}$ the unit outer normal to $\eta(\Gamma)$ and
$\mathcal{T},\mathcal{N}$ be the tangential and normal bundle of
$\eta(\Gamma)$ respectively. Denote
$\Pi:\mathcal{T}|_{\eta(\Gamma)}\to\mathcal{N}$ to be the canonical normal
projection. Denote $\overline{\partial}{}_{A}$ be $\partial_{t}$ or
$\overline{\partial}{}_{1},\overline{\partial}{}_{2}$. Then we have the
identities

$\displaystyle\hat{n}:=n\circ\eta=$ $\displaystyle\frac{a^{T}N}{|a^{T}N|},$
(2.1) $\displaystyle|a^{T}N|=$
$\displaystyle|(a^{31},a^{32},a^{33})|=\sqrt{g},$ (2.2)
$\displaystyle\Pi^{\alpha}_{\lambda}=$
$\displaystyle\hat{n}^{\alpha}\hat{n}_{\lambda}=\delta^{\alpha}_{\lambda}-g^{kl}\overline{\partial}{}_{k}\eta_{\alpha}\overline{\partial}{}_{l}\eta_{\lambda},$
(2.3) $\displaystyle\Pi^{\alpha}_{\lambda}=$
$\displaystyle\Pi^{\alpha}_{\mu}\Pi^{\mu}_{\lambda},$ (2.4)
$\displaystyle-\Delta_{g}(\eta^{\alpha}|_{\Gamma})=$
$\displaystyle\mathcal{H}\circ\eta\hat{n}^{\alpha},$ (2.5)
$\displaystyle\sqrt{g}\Delta_{g}\eta^{\alpha}=$
$\displaystyle\sqrt{g}g^{ij}\Pi^{\alpha}_{\lambda}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\eta^{\lambda}=\sqrt{g}g^{ij}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\eta^{\alpha}-\sqrt{g}g^{ij}g^{kl}\overline{\partial}{}_{k}\eta^{\alpha}\overline{\partial}{}_{l}\eta^{\mu}\overline{\partial}{}_{i}\overline{\partial}{}_{j}\eta_{\mu},$
(2.6)
$\displaystyle\overline{\partial}{}_{A}(\sqrt{g}\Delta_{g}\eta^{\alpha})=$
$\displaystyle\overline{\partial}{}_{i}\bigg{(}\sqrt{g}g^{ij}\Pi^{\alpha}_{\lambda}\overline{\partial}{}_{A}\overline{\partial}{}_{j}\eta^{\lambda}+\sqrt{g}(g^{ij}g^{kl}-g^{ik}g^{lj})\overline{\partial}{}_{j}\eta^{\alpha}\overline{\partial}{}_{k}\eta_{\lambda}\overline{\partial}{}_{A}\overline{\partial}{}_{l}\eta^{\lambda}\bigg{)},$
(2.7) $\displaystyle\overline{\partial}{}_{A}\hat{n}_{\mu}=$
$\displaystyle-g^{kl}\overline{\partial}{}_{k}\overline{\partial}{}_{A}\eta^{\tau}\hat{n}_{\tau}\overline{\partial}{}_{l}\eta_{\mu},$
(2.8) $\displaystyle\partial_{t}(\sqrt{g}g^{ij})=$
$\displaystyle\sqrt{g}(g^{ij}g^{kl}-2g^{lj}g^{ik})\overline{\partial}{}_{k}v^{\lambda}\overline{\partial}{}_{l}\eta_{\lambda}.$
(2.9)

###### Proof.

See Lemma 2.5 in Disconzi-Kukavica [DK2017]. ∎

######  Remark.

Recall that
$g_{ij}=\overline{\partial}{}_{i}\eta_{\mu}\overline{\partial}{}_{j}\eta^{\mu}$
and $g=\det[g_{ij}]$ and $[g^{ij}]=[g_{ij}]^{-1}$. This means that
$g_{ij},~{}g$ and $g^{ij}$ are rational functions of
$\overline{\partial}{}\eta$ and so is $\Pi$.