\section{Introduction}\label{introud}
\numberwithin{equation}{section}

In this paper, we investigate
the global(-in-time) well-posdedness of  the following \emph{three-dimensional} initial-boundary value problem for the incompressible, inviscid, non-resistive magnetohydrodynamic (MHD) fluid  with velocity damping in a horizontally periodic domain $\Omega$ with finite height:
\begin{equation}\label{0101}
\begin{cases}
\rho v_{t}+ \rho v\cdot\nabla v
+\nabla \left(P+\lambda|M|^2/2\right)+a \rho v =\lambda M\cdot\nabla M, \\
 {M}_{t}+ v\cdot\nabla {M}=M\cdot\nabla v, \\
\div  v =\mathrm{div}M=0,\\
(v,M)|_{t=0}=(v^0,M^0),\\
v_3|_{\partial\Omega } =0.
\end{cases}
\end{equation}
Next we shall explain the mathematical notations appearing in the problem above.

The unknowns ${v}:={v}(x,t)$, ${M}:={M}(x,t)$ and ${P}:={P}(x,t)$ denotes the velocity and magnetic field and  kinetic pressure of incompressible MHD fluids, resp., where $(x,t)\in \mathbb{R}^3\times \mathbb{R}_0^+$ and $\mathbb{R}_0^+:=[0,\infty)$.
The positive constants $\rho$, $a$ and $\lambda$ stands for the density of fluids,
  damping coefficient and  permeability of the vacuum, resp.
The horizontally periodic domain $\Omega$ with finite height is defined as follows:
\begin{align}\label{0101a}
\Omega:=\{(x_{\mathrm{h}}, x_3)^{\mathrm{T}}~|~x_{\mathrm{h}}:=(x_1, x_2)^{\mathrm{T}}
\in\mathbb{T},\;\;x_3\in(0,h)\}\subset\mathbb{R}^3,
\end{align}
where $\mathbb{T}=(l_1\mathbb{R}/\mathbb{Z})\times (l_2\mathbb{R}/\mathbb{Z})$,
 $\mathbb{R}/\mathbb{Z}$ represents the usual $1$-torus, and the superscript ${\mathrm{T}}$ denotes the transposition. Since $\Omega$ is horizontally periodic,
the set $\mathbb{T}\times \{0,h\}$, denoted by $\partial\Omega$, customarily represents the boundary of $\Omega$.
  Without loss of generality, we assume that $l_1=l_2=h=1$ for the sake of simplicity.

For the the viscous case, i.e. the equations \eqref{0101}$_1$--\eqref{0101}$_3$ with the viscous term  $-\nu \Delta v$  in place of the velocity damping term $a \rho v$,  the global  well-posedness, in which the initial data is a small perturbation around a non-zero trivial stationary state
{ (i.e., $(v,M)=(0,\bar{M})$, where $\bar{M}$ is a non-zero constant vector and $\bar{M}$ often called the impressive magnetic field)},
has been widely investigated, see \cite{lin2015global,zhang2014elementary,ZTGS} and \cite{ABIHZPOTG,xu2015global} for the 2D and 3D
 Cauchy problems resp., and see \cite{ren2016global} and \cite{TZWYJGw} for the 2D and 3D initial-boundary value problems resp.. The existence of global solutions to the 2D Cauchy problem with large initial perturbations was obtained by Zhang under a strong impressive magnetic field \cite{ZTGS}. {As for the well-poseness  of the 3D Cauchy and initial-boundary value problems  with large initial perturbations, to our best knowledge,  all available results are about the local(-in-time) existence, see \cite{fefferman2017local,HOCELETFCMDSFA,chemin2016local} for examples.
 We mention here that the corresponding compressible case has been also widely studied, see \cite{LYSYZG,LXLSNWDH,WJHWYF} and the references
 cited therein.}

 For the the inviscid case, Bardos--Sulem--Sulem used hyperbolicity of \eqref{0101}$_1$--\eqref{0101}$_3$ with $\mu=0$  to establish an interesting global existence result of classical
solutions of the Cauchy problem with small initial data in the H\"older space $H^s(\mathbb{R}^3)$ \cite{BCSCSPLL}; also see \cite{CYLZGW,HLBXLYP} for the case
of Sobolev spaces). We remark that such a result is not known
in the 3D incompressible Euler equations (i.e. the case of the absence of magnetic field).

For the inviscid case with a velocity damping term, Wu--Wu--Xu first given the existence of a unique global solution with algebraic decay-in-time for the 2D Cauchy problem of  \eqref{0101}$_1$--\eqref{0101}$_4$ with small initial perturbation  \cite{WJHWYFXXJG}. Recently, Du--Yang--Zhou also obtained the existence of a unique global solution with exponential decay-in-time for the initial-boundary value problem in a 2D slab domain with small initial perturbation around some non-trivial equilibrium \cite{DYYWZYOSJMA}.
To our best knowledge, there are not any available result for the 3D case,
 thought  the 3D well-posedness problem of inviscid fluids with velocity damping had been widely investigated, see \cite{TZWY2013JDE,PRHZK2009JDE,WWKYT2001JDE,STCTBWDH2012CPDE,HFMPRHARMA2003} for examples.  \emph{The goal of this paper is devoted to further providing the first 3D result by developing  a new type of two-layer energy structure to overcome the difficulty arising from  three-dimensional nonlinear terms in the MHD equations.}
It should be noted that the mathematical methods adopted in \cite{WJHWYFXXJG,DYYWZYOSJMA} for the well-posedness problem in
Eulerian coordinates cannot be applied to our 3D problem, therefore  next we shall reformulate the initial-boundary value problem \eqref{0101} in Lagrangian coordinates as in \cite{WYJ2019ARMA,JFJSSETEFP,JFJSOUI}.

\subsection{Reformulation in Lagrangian coordinates.}\label{subsec:02}

Let the flow map $\zeta$ to be the solution to the initial-value problem
\begin{equation}
\nonumber
\begin{cases}
  \zeta_t(y,t)=v(\zeta(y,t),t)&\mbox{in }\Omega,
\\
\zeta(y,0)=\zeta^0(y)&\mbox{in }\Omega,
\end{cases}
\end{equation}
where the
invertible mapping $\zeta^0:=\zeta^0(y):\Omega\to \Omega$ satisfies
\begin{align}
&\label{zeta0inta}
J^0:=\det(\nabla \zeta^0)=1  \mbox{ in } \Omega,\\
&\label{zeta0inta0}
\zeta^0_3 =y_3  \mbox{ on } \partial\Omega .
\end{align}
Here and in what follows  $\det$ denotes a determinant of matrix, and $f^0$ with the superscript $0$ represents the initial data of $f$. Sometimes we also use
$f_0$ with the subscript $0$ to represent the initial data of $f$.

We denote the Eulerian coordinates by $(x,t)$ with
$x =\zeta(y,t)$ and the Lagrangian coordinates by $(y, t)\in\Omega\times \mathbb{R}_0^+$.
We further assume that, for each fixed $t>0$,
\begin{align}
& \zeta|_{y_3=i}   : \mathbb{R}\to \mathbb{R}\mbox{ is a } C^1(\mathbb{R})\mbox{-diffeomorphism mapping for }i=0,\ 1,\label{20210301715x}\\
&\zeta   :  \overline{\Omega}\to \overline{\Omega} \mbox{ is a } C^1(\overline{\Omega})\mbox{-diffeomorphism mapping},\label{20210301715}
\end{align}
where $\overline{\Omega}:=\mathbb{R}^2\times[0,1]$.

Since $v$ satisfies the divergence-free condition,
and slip boundary condition \eqref{0101}$_5$, we can deduce from \eqref{zeta0inta}--\eqref{zeta0inta0} that
\begin{align}
&\nonumber
J:=\det(\nabla \zeta)=1  \mbox{ in } \Omega,\\
&\nonumber
 \zeta_3 =y_3  \mbox{ on } \partial\Omega .
\end{align}

We introduce the matrix $\mathcal{A}=(\mathcal{A}_{ij})_{3\times 3}$, which is defined via
\begin{align}\nonumber
\mathcal{A}^{\mathrm{T}}=(\nabla\zeta)^{-1}:=
(\partial_j \zeta_i)^{-1}_{3\times 3}.
\end{align}
Then we further define the differential operators $\nabla_{\mathcal{A}}$, $\mathrm{div}_{\mathcal{A}}$ and $\mathrm{curl}_{\mathcal{A}}$
as follows: for a given scalar function $f$ and a given vector function $X:=(X_1,X_2, X_3)^{\mathrm{T}}$,
\begin{align}
&\nabla_{\mathcal{A}}f:=(\mathcal{A}_{1k}\partial_kf,
\mathcal{A}_{2k}\partial_kf, \mathcal{A}_{3k}\partial_kf)^{\mathrm{T}},\ \nabla_{\mathcal{A}}X:=(\nabla_{\mathcal{A}}X_1,\nabla_{\mathcal{A}}X_2,\nabla_{\mathcal{A}}X_3)^{\mathrm{T}},   \nonumber  \\
&\mathrm{curl}_{\mathcal{A}}X:=(\mathcal{A}_{2k}\partial_{k}X_3-\mathcal{A}_{3k}\partial_{k}X_2,
\mathcal{A}_{3k}\partial_{k}X_1-\mathcal{A}_{1k}\partial_{k}X_3,
\mathcal{A}_{1k}\partial_{k}X_2-\mathcal{A}_{2k}\partial_{k}X_1
)^{\mathrm{T}},\nonumber \\
&\mathrm{div}_{\mathcal{A}}(X_1,X_2, X_3)^{\mathrm{T}}:=\mathcal{A}_{lk}\partial_k X_l, \nonumber
\end{align}
where we have used the Einstein convention of summation over repeated indices, and $\partial_k=\partial_{y_k}$.
In particular, $\mathrm{curl}X:=\mathrm{curl}_IX$, where $I$ is a $3\times 3$ identity matrix. In addition, we will denote $(\mathrm{curl}_{\mathcal{A}} X^1,\ldots, \mathrm{curl}_{\mathcal{A}}X^n) $ by $\mathrm{curl}_{\mathcal{A}}(X^1,\ldots X^n) $ for simplicity, where $X^i=(X^i_1,X^i_2,X^i_3)^{\mathrm{T}}$ is a vector function for $1\leqslant i \leqslant  n$.

Defining the Lagrangian unknowns
\begin{equation*}
(u ,Q, B)(y,t)=(v,P+\lambda|M|^2/2,M)(\zeta(y,t),t) \mbox{ for } (y,t)\in \Omega \times\mathbb{R}^+_0,
\end{equation*}
then in Lagrangian coordinates, the initial-boundary value problem \eqref{0101}  reads as follows:
\begin{equation}\label{01dsaf16asdfasf00}
\begin{cases}
\zeta_t=u,\ \div_\mathcal{A}u=0 &\mbox{in } \Omega,\\
\rho u_t+\nabla_{\mathcal{A}}Q+a\rho u=\lambda B\cdot\nabla_{\mathcal{A}}B &\mbox{in } \Omega,\\
B_t=B\cdot \nabla_{\mathcal{A}}u  &\mbox{in } \Omega, \\
\div_\mathcal{A}B=0  &\mbox{in } \Omega, \\
(u,\zeta, B)|_{t=0}=(u^0,\zeta^0, B^0) &\mbox{in } \Omega, \\
(\zeta_3-y_3)=u_3=0  &\mbox{on }\partial\Omega .
\end{cases}
\end{equation}

We can  derive from \eqref{01dsaf16asdfasf00}$_3$ the differential version of magnetic flux conservation \cite{JFJSARMA2019}:
\begin{equation} \nonumber
\mathcal{A}^{\mathrm{T}}B=\mathcal{A}_0^{\mathrm{T}}B^0,
 \end{equation}
 which yields
\begin{eqnarray}
 \label{0124}  B=\nabla\zeta \mathcal{A}^{\mathrm{T}}_0 B^0.
\end{eqnarray}

Let $\bar{M}=(\bar{M}_1,0,0)^{\mathrm{T}}$ and $\bar{M}_1$
is a nonzero constant.
If we assume
\begin{equation}
\label{201903081437}
\mathcal{A}^{\mathrm{T}}_0 B^0=\bar{M}  \mbox{ (i.e.  }B^0=\nabla\zeta^0\bar{M}  = \partial_{\bar{M}}\zeta^0\mbox{)},
 \end{equation}
then \eqref{0124} reduces to
 \begin{eqnarray}
\label{0124xx}   B=\partial_{\bar{M}}\zeta  .
\end{eqnarray}
It should be noted that $B$ given by \eqref{0124xx} automatically satisfies \eqref{01dsaf16asdfasf00}$_3$
and \eqref{01dsaf16asdfasf00}$_4$. Moreover, we see from \eqref{0124xx} that the magnetic tension in Lagrangian coordinates
has the relation
\begin{equation}\nonumber
 \lambda B\cdot \nabla_{\mathcal{A}} B=  \lambda\bar{M}_1^2\partial_{1}^2\zeta.
 \end{equation}

Let
$$e_1=(1,0,0)^{\mathrm{T}},\ m= \lambda\bar{M}_1^2 /\rho ,\
q= Q /\rho ,\ \eta=\zeta-y,\ \mathcal{A}=(\nabla \eta+I)^{\mathrm{T}} \mbox{ and }   \eta^0=\zeta^0-y.$$
Consequently, under the assumption \eqref{201903081437}, the problem \eqref{01dsaf16asdfasf00} is equivalent to
the following initial-boundary value problem:
\begin{equation}\label{01dsaf16asdfasf}
                              \begin{cases}
\eta_t=u &\mbox{ in } \Omega,\\
u_t+\nabla_{\mathcal{A}}q+a u=m  \partial_1^2\eta  &\mbox{ in } \Omega,\\
\div_\mathcal{A}u=0  &\mbox{ in } \Omega, \\
(u,\eta)|_{t=0}=(u^0,\eta^0) &\mbox{ in } \Omega, \\
(u_3,\eta_3)=0  &\mbox{on } \partial\Omega ,
\end{cases}
\end{equation}
with the expression of magnetic field
\begin{align}\label{202103190614}
B=\bar{M}_1(\partial_1\eta+e_1).
\end{align}

\subsection{Main results.}\label{subsec:03}

 Before stating our main result, we introduce some notations which will be frequently used throughout this paper.

(1) Simplified basic notations:  $I_a:=(0,a)$, in particular, $I_\infty=\mathbb{R}^+$. $\overline{S}$ denotes the closure of the set $S\subset \mathbb{R}^n$ with $n\geqslant 1$, in particular $\overline{I_T}:=[0,T]$.  $\Omega_a:=\Omega\times I_a$.  $\vec{\mathbf{n}}:=({\mathbf{n}}_1,{\mathbf{n}}_2, {\mathbf{n}}_3)^{\mathrm{T}}$
denotes the outward unit normal vector on  $\partial\Omega$. $\int:=\int_\Omega=\int_{(0,1)^3}$. $(u)_{\Omega}=\int  u\mathrm{d}y $.


 $a\lesssim b$ means that $a\leqslant cb$.
If not stated explicitly,  the positive constant $c$ may depend on $a$,  $m $ and $\Omega$, and may vary from one place to other place.
Sometimes we use $c_i$ for $i\geqslant 1$ to replace $c$ in order to emphasize that $c_i$ is fixed value.

 $\alpha$ always denotes the multiindex with respect to the variable $y$,
$|\alpha|=\alpha_1+\alpha_2+\alpha_3$ is called the order of multiindex and  $\partial^{\alpha}:=\partial_{1}^{\alpha_1} \partial_{2}^{\alpha_2} \partial_{3}^{\alpha_3}$.
 ``$\nabla^{i}f\in \mathbb{X}$" represents that $\partial^\alpha f\in \mathbb{X}$ for any multiindex $\alpha$ satisfying $|\alpha|=i$, where $\mathbb{X}$ denotes some set of functions.

(2) Simplified Banach spaces:
\begin{align}
& L^p:=L^p (\Omega)=W^{0,p}(\Omega),\;\;H^{i}:=W^{i,2}(\Omega),
\ L^p_TH^i:=L^p(I_T,H^i),\nonumber \\
&{H}^{l,i}:=\{w\in H^{i}~|~ { \partial_1^{k}w}\in H^{i},\;0\leqslant k\leqslant l\},
\  H^{l,j}_{\mathrm{s}}:=\{w\in {H}^{l,j}~|~w_3|_{\partial\Omega }=0\}, \nonumber \\
&H^{j}_{1}:=\{w \in H^{j} ~|~\det(\nabla w +I )=1\},\
{
H^{l,j}_{1,\mathrm{s}}:= H^{l,j}_{\mathrm{s}}\cap H^{j}_{1}},\nonumber \\
& {{C}}^0_{B,\mathrm{weak}}(\overline{I_T} ,L^2):= L^\infty_TL^2\cap  C^0(\overline{I_T}, L^2_{\mathrm{weak}}),  \  \underline{H}^i:=\{w\in H^i~|~(w)_{\Omega}=0\}, \nonumber
\end{align}
where $1\leqslant p\leqslant \infty$,
and  $i$, $l\geqslant 0$ and  $j\geqslant 1$  are integers. In addition,  $\|(f^1,\cdot,f^n)\|_B:=\sqrt{\sum_{1\leqslant i\leqslant n}\|f^i\|_B^2}$, where $f^i$ may be a  scalar function, a vector or a matrix function for $1\leqslant i\leqslant n$.

 (3) Simplified function classes: for any given integer $j\geqslant 1$,
\begin{align}
& H^j_{*}:=\{\xi\in H^j~|~ \xi(y)+y : \overline{\Omega}\to \overline{\Omega}\mbox{ satisfies the properties }\nonumber \\
&\qquad \quad \ \mbox{of  diffeomorphism \eqref{20210301715x} and
\eqref{20210301715} as }\zeta\},\nonumber\\
&\mathfrak{C}^0( \overline{I_T},{H}^{2,3}_{\mathrm{s}}):=\{\eta\in C^0(\overline{I_T} ,{H}^{1,3}_{\mathrm{s}}) ~|~  \nabla^3 \partial_1^2\eta \in   {C}^0_{B,\mathrm{weak}}(\overline{I_T} ,L^2) \},\nonumber \\
& \underline{\mathfrak{H}}^{2,3}_{1,*,T}:=\{\eta\in \mathfrak{C}^0(\overline{I_T}, H^{2,3}_{\mathrm{s}} )
~|~ \eta(t) \in    H^3_1\cap H^{3}_{*}\mbox{ for each }t\in \overline{I_T}\},\nonumber\\
&    \mathfrak{U}_{T}^{1,3}:=  \{u\in C^0(\overline{I_T}, H^3_{\mathrm{s}}) ~|~\nabla^3 \partial_1 u \in {C}^0_{B,\mathrm{weak}}( \overline{I_T} ,L^2) , \nonumber   \\
&\qquad \qquad\qquad \qquad\qquad \quad  \
u_t\in C^0(\overline{I_T}, H^2_{\mathrm{s}}),\  u_t\in L^\infty_TH^3  \}\nonumber .
\end{align}

(4) Simplified norms and semi-norms: for integers $i\geqslant 0$ and $j\geqslant 0$,
\begin{align}
\|\cdot \|_{i} :=\|\cdot \|_{H^i},\;\;\;\;
\|\cdot\|_{j,i}:= \|\partial_{1}^{j}\cdot\|_{i},\;\;\;
\ \|\cdot\|_{\underline{j},i}:=\sqrt{\sum_{0\leqslant k\leqslant j}\|\cdot\|_{k,i}^2}.\nonumber
\end{align}

 (5) Simplified functionals:
 \begin{align}&  \mathcal{E}^L:=\|(u,\eta,\partial_1\eta)\|_{3}^2+\|( u_t, \nabla q)\|_2^2,\;\; \mathcal{D}^L:= \|(u,\partial_1\eta)\|_3^2+\|( u_t,\nabla q)\|_2^2\nonumber,\\[2mm]
& \mathcal{E}^{H}:= \|(u, \partial_1\eta )\|_{\underline{1},3}^2
+\|( u_t, \nabla q)\|_2^2,\;\;  \mathcal{D}^{H}:=\|(u,\partial_1u, \partial_1^2\eta)\|_{3}^2+\|( u_t,\nabla q)\|_2^2,\nonumber\\
& I^0_L:= \|(u^0,\eta^0,\partial_1\eta^0)\|_3^2 , \ I^0_H:= \|(u^0,\partial_1 \eta^0)\|_{\underline{1},3}.  \nonumber
               \end{align}
We call $\mathcal{E}^L$, resp. $\mathcal{D}^L$ the lower energy, resp. dissipation functionals, and $\mathcal{E}^{H}$, resp. $\mathcal{D}^H$ the higher energy, resp. dissipation  functionals.

Now we state the existence result of unique global solutions for the initial-boundary value problem \eqref{01dsaf16asdfasf} with   small initial data.

\begin{thm}\label{thm2}
Let
$(u^0,\eta^0)\in H^{1,3}_{\mathrm{s}} \times
 (H^{2,3}_{1,\mathrm{s}}\cap H^3_*)$
satisfy the incompressible condition
$$\mathrm{div}_{\mathcal{A}^0}u^0=0\mbox{ in }\Omega,$$ where
 $\mathcal{A}^0:=(\nabla \eta^0+I)^{-\mathrm{T}}$. Then there exist  a sufficiently small constant $\delta >0$ and a positive constant $c$, such that, for any $(u^0,\eta^0)$ satisfying
$$ e^{c \sqrt{I_H^0}}\|(u^0,\eta^0,\partial_1\eta^0)\|_3^2\leqslant\delta^2, $$
 the initial-boundary problem \eqref{01dsaf16asdfasf} admits  a unique global classical solution
$(u,\eta,q)\in \mathfrak{U}_{\mathrm{s},\infty}^{1,3}\times \underline{\mathfrak{H}}^{2,3,*}_{1,\mathrm{s},\infty}\times C(\mathbb{R}^+_0 ,\underline{H}^3)$. Moreover the solution enjoys the following stability estimates:
\begin{align}\label{1.200}
&\mathcal{E}^L(t)+\int_0^t\mathcal{D}^L(s)\mathrm{d}s\lesssim e^{c\sqrt{I^0_H}  }I^0_L,\\
&\label{202101241112}
 {e}^{c_1 t}\mathcal{E}^H(t) + \int_0^t e^{c_1 \tau}\mathcal{D}^H(s)\mathrm{d}s
\lesssim I^0_H,\\
&e^{c_1 t }\|\eta(t)-\eta^\infty\|_3^2\lesssim {I^0_H},\label{1.200xx}
\end{align}
where $\eta^\infty$ only depends on $y_2$ and $y_3$. In addition, $\nabla_{\mathcal{A}}q\in L^\infty_\infty H^3$.
\end{thm}
\begin{rem}
In this paper, we only consider the impressive horizontal magnetic field $\bar{M}$.
\emph{However it is not clear to the authors that whether there exists a global well-posdness result for non-horizontal magnetic field.}
\end{rem}
\begin{rem}
For the case $\Omega=\mathbb{R}^2\times (0,1)$, we also obtain similar result, where the exponential stability estimates \eqref{202101241112}  and \eqref{1.200xx} should be replaced by  algebraic stability estimates. \emph{We will verify this assertion in a forthcoming paper.}
\end{rem}
\begin{rem}
For each fixed $t\in \mathbb{R}_0^+$, the solution $\eta(y,t)$ in Theorem \ref{thm2} belongs to $  H^3_{*}$. Let $\zeta=\eta+y$, then $\zeta $  satisfies \eqref{20210301715x} and \eqref{20210301715} for each fixed $t\in \mathbb{R}_0^+$. We denote the inverse transformation of $\zeta$ by $\zeta^{-1}$, and then define that
$$
\begin{aligned}
&( v,N,Q)(x,t):=( u(y,t), \bar{M}_1\partial_1\eta(y,t),\rho q(y,t))|_{y=\zeta^{-1}(x,t)},\\
& M:=N+\bar{M}\mbox{ and } P:=Q-\lambda|M|^2/2.
\end{aligned}$$
Consequently  $(  v,M,P)$  is a global solution for \eqref{0101}; moreover, making use of \eqref{1.200}, \eqref{202101241112} and the fifth conclusion in Lemma \ref{20021032019018}, $(v,N,Q)$ enjoys stability estimates, which are similar to \eqref{1.200} and \eqref{202101241112}.
\end{rem}

Next we briefly sketch the proof of Theorem \ref{thm2}, and the details will be presented in Section \ref{sec:global}.

The key proof for the existence of global  small solutions is to derive an  \emph{a priori} energy inequalities \eqref{1.200} and \eqref{202101241112}.
To this purpose,
  let $(\eta,u)$ be a solution to \eqref{01dsaf16asdfasf} and satisfy
\begin{align}
\label{aprpiosesnew}
&\sup\nolimits_{  t\in \overline{I_T}}\|(u,\eta,\partial_1\eta)(t)\|_3\leqslant  {\delta} \in (0,1] \mbox{ for some } T,\\
& \det(\nabla \eta+I)=1,\label{aprpiosasfesnew}
\end{align}
where $\delta$ is sufficiently small.

Under the  \emph{a priori}  assumption  \eqref{aprpiosesnew} of lower-order energy,  we first derive the both energy inequalities of horizontal-type and vorticity-type, see \eqref{202008250856}  and \eqref{202005021632}.
Summing up the two energy inequalities, we can arrive at the lower-order energy inequality  \begin{align}\label{for:0202n0319}
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^L+ \mathcal{D}^L\lesssim (\|u\|_3+\|\nabla q\|_0)\|\eta\|_3^2
\end{align}
for some  energy functional $\tilde{\mathcal{E}}^L$, which is equivalent to $\mathcal{E}^L$.

Obviously we can not directly close the lower-order energy by using \eqref{for:0202n0319}, unless the term ``$\|u\|_3+\|\nabla q\|_0$" decays in time. It should be noted that the term ``$ (\|u\|_3+\|\nabla q\|_0)\|\eta\|_3^2$" (arising from nonlinear terms, see \eqref{20202105112053} and \eqref{2020202105112055}) does not appear in \eqref{for:0202n0319} for the two dimensional cases.  Motivated by the multi-layers energy method introduced in \cite{GYTIAE2}, in which Guo--Tice investigated the global well-posdedness of surface wave problem, we naturally look for another energy inequality, which includes the decay-in-time of $\|u\|_3$.

By careful analysis of energy estimates, we find that, if
 $(u^0,\eta^0)$ additionally satisfies the regularity
 $$\partial_1(u^0,\partial_1\eta^0) \in H^3,$$ the following higher-order energy inequality can be established:
\begin{align}\label{2020122621050319}
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^H + \mathcal{D}^H\leqslant 0,
\end{align}
where $\tilde{\mathcal{E}}^H  $ is equivalent to $\mathcal{E}^H$.
Moreover, thanks to the estimate
$$\|\partial_1\eta\|_3\lesssim\|\partial_1^2\eta\|_3,$$
we have \begin{align}
\nonumber
\mathcal{E}^H \mbox{ is equivalent to }\mathcal{D}^H.
\end{align}
Thus the  higher-order  energy inequality  \eqref{2020122621050319}, together with the above equivalence, immediately implies the exponential stability of higher-order energy \eqref{202101241112},
which particularly presents the exponential decay-in-timely of $ \|u\|_3$ and $\|\nabla q\|_0$. Hence the \emph{a priori }stability estimate \eqref{1.200} further follows  from \eqref{for:0202n0319}.

Thanks to the \emph{a priori} stability estimates \eqref{1.200}--\eqref{202101241112} and the unique local solvability  of the  initial-boundary problem \eqref{01dsaf16asdfasf}, we immediately get the unique global solvability  of the  initial-boundary problem \eqref{01dsaf16asdfasf}.   Finally, \eqref{1.200xx} can be deduced from \eqref{202101241112} by an asymptotic analysis method.

The rest of this paper is organized as follows.
In Sections \ref{sec:global}, we will provide the proof of  Theorem \ref{thm2}. In Section \ref{202102241211}, we will establish the local well-posdedness result for the initial-boundary problem \eqref{01dsaf16asdfasf}.  Finally, in \ref{sec:09}, we will list some well-known mathematical results, which will be used in Sections \ref{sec:global}--\ref{202102241211}.
\section{Proof of Theorem \ref{thm2}}\label{sec:global}
This section is devoted to the proof of Theorem \ref{thm2}.
The key step is to derive the a \emph{priori} stability estimates \eqref{1.200}--\eqref{202101241112} for the initial-boundary value  problem \eqref{01dsaf16asdfasf}.
To this end, let $(u,\eta,q)$ be a solution to the problem \eqref{01dsaf16asdfasf},  and satisfy \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew} with sufficiently small
  $ {\delta}$. Next we  proceed the derivation of \emph{a priori}  estimates.

\subsection{Preliminary estimates}
To begin with, we shall establish some preliminary estimates.
\begin{lem}
\label{201805141072}
Let $0\leqslant i\leqslant 2$ and $1\leqslant j\leqslant 2$.
Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, we have
\begin{enumerate}[(1)]
  \item  the estimates for $\mathcal{A}$ and $\tilde{\mathcal{A}}$:
\begin{align}
&\label{aimdse}
\|\mathcal{A}\|_{C^0(\overline{\Omega})}+ \|\mathcal{A}\|_2 \lesssim  1,\\
&  \label{06041533fwqg}
\|\tilde{\mathcal{A}}\|_{i}\lesssim   \| \eta\|_{i+1}\\
&  \label{06041533fwqgn}
\| \tilde{\mathcal{A}}\|_{j,i}\lesssim   \|  \eta\|_{j,i+1},\\
& \|\mathcal{A}_t\|_i \lesssim   \|   u\|_{i+1} , \label{06142100x}\\
&  \label{06041safsa533fwqgn}
\| \tilde{\mathcal{A}}_t\|_{1,i}\lesssim   \|  (u,\eta)\|_{1,i+1}.
\end{align}
Here and in what follows $\tilde{\mathcal{A}}:=\mathcal{A}-I$.
\item  the estimates of $\mathrm{div}{ {u}}$:
\begin{align}
&\label{201808181500} \|\mathrm{div} u\|_{i}\lesssim \|\eta\|_{3}\|u\|_{i+1},\\
&\label{201808181500n001} \|\mathrm{div} u\|_{1,i}\lesssim \|(u,\eta )\|_{3}\|(u,\eta )\|_{1,i+1}.
\end{align}
\item  the estimates of $\mathrm{div} { {\eta}}$:
\begin{align}
&\label{improtian1}
\|\mathrm{div} { {\eta}}\|_{i} \lesssim \|\eta\|_{3} \|\eta\|_{i+1} ,\\
&\label{improtian1002}
\|\mathrm{div} { {\eta}}\|_{1,i} \lesssim \|\eta\|_{3} \|\eta\|_{1,i+1} ,\\
&\label{improtian1003}
\|\mathrm{div} { {\eta}}\|_{2,2} \lesssim \|\eta\|_{3} \|\eta\|_{2,3}+\|\eta\|_{1,3}^2.
\end{align}
\end{enumerate}
\end{lem}
\begin{pf}
 Recalling  the condition \eqref{aprpiosasfesnew} and the definition of $\mathcal{A}$, we have
\begin{equation}
\label{06131422}
\mathcal{A}=(A^*_{ij})_{3\times 3},
\end{equation}
where
$A^{*}_{ij}$ is the algebraic complement minor of $(i,j)$-th entry of matrix $(\partial_j \zeta_i)_{3\times 3}$.  Exploiting \eqref{01dsaf16asdfasf}$_1$, \eqref{aprpiosesnew}, the produce estimate \eqref{fgestims} and Poinc\'are's inequality \eqref{202012241002}, we easily derive \eqref{aimdse}--\eqref{06041safsa533fwqgn} from the expression \eqref{06131422}.
In addition, we easily derive \eqref{201808181500}  and \eqref{201808181500n001} from the incompressible condition $\mathrm{div}_{\mathcal{A}}u=0$ by  \eqref{fgestims}.
{ Similarly, we easily get \eqref{improtian1}--\eqref{improtian1003} from the relation  \eqref{aprpiosasfesnew}.}
Interesting readers can refer to \cite{JFJSJMFMOSERT} for the detailed derivation.   \hfill $\Box$
\end{pf}
\begin{lem}\label{lem:07291427nnmmm}
Let $i=0$, $1$, and  the multindex $\alpha$ satisfy $|\alpha|\leqslant2$, $w(t)\in H^3$ for each $t\in \overline{I_T}$
and
\begin{align}
\label{2021011281929}
{W}^{i,\alpha}:= \partial_{1}^{i}\partial^{\alpha}\left(\mathrm{curl}_{\mathcal{A}_{t}}u
-m \mathrm{curl}_{\partial_1\mathcal{A}} \partial_1\eta\right).
\end{align}
Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, we have
\begin{align}
&\label{202101231432}
\left\|\partial^{\alpha}\left(\operatorname{curl}_{\mathcal{A}} w-\operatorname{curl}w\right)\right\|_0
\lesssim\|\eta\|_3\|w\|_3,\\
&\label{202101231245}
\left\|\partial^{\alpha} \left(\operatorname{curl}_{\mathcal{A}} w-\operatorname{curl}w\right)\right\|_{1,0}\lesssim
 \|(\eta,w)\|_3 \|(\eta,w)\|_{1,3},\\
&\label{201907291200000}
\left\|W^{0,\alpha}\right\|_0\lesssim\|(u,\partial_1\eta)\|_3^2 ,\\
&\label{201907291200000nm}
\left\|W^{1,\alpha}\right\|_0\lesssim\|u\|_3\|(u,\eta)\|_{1,3}
+\| \eta\|_{1,3}\| \eta\|_{2,3} .
\end{align}
\end{lem}
\begin{pf}
By  \eqref{06041533fwqg}, \eqref{06041533fwqgn} and the product estimate \eqref{fgestims}, we easily get \eqref{202101231432} and \eqref{202101231245}.
Similarly, making use of \eqref{06041533fwqgn}--\eqref{06041safsa533fwqgn} and \eqref{fgestims}, we easily derive \eqref{201907291200000}--\eqref{201907291200000nm}  from the definition of $W^{i,\alpha}$.
The proof is complete.
\hfill$\Box$
\end{pf}
\subsection{Horizontal spatial estimates}
Now we derive the estimates of horizontally spatial derivatives of $\left(u,\eta\right)$.
\begin{lem}\label{lem:08241445}
Let $0\leqslant k\leqslant1$. Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew},
we have
\begin{align}
&\label{202008241448}
\frac{\mathrm{d}}{\mathrm{d}t}\left(\|\partial_1^{k} u\|^2_0
+m \|\partial_1^{k+1} \eta\|_{0}^2 \right)+ a\|\partial_1^{k} u\|_{0}^2
\lesssim
\begin{cases}
0
&\hbox{for }k=0; \\
\sqrt{\mathcal{E}^L}\mathcal{D}^H
&\hbox{for }k=1,
\end{cases}
\\
&
\frac{\mathrm{d}}{\mathrm{d}t}\left(\int\partial_1^{k}\eta\cdot\partial_1^{k} u\mathrm{d}y
+\frac{a}{2}\|\partial_1^{k} \eta\|_{0}^2\right)
+m \| \partial_1^{k+1} \eta\|_{0}^2
\nonumber \\
&\lesssim\|\partial_1^{k}u\|_{0}^2
+
\begin{cases}
{ \|\eta\|_2^2\|\nabla q\|_0}
&\hbox{for }k=0; \\
\sqrt{\mathcal{E}^L}\mathcal{D}^H
&\hbox{for } k=1\label{202008241446}.
\end{cases}
\end{align}
\end{lem}
\begin{pf}
Applying $\partial_1^{k}$ to \eqref{01dsaf16asdfasf} yields
\begin{equation}\label{01dsaf16asdfasf0321}
                              \begin{cases}
\partial_1^{k}\eta_t=\partial_1^{k}u &\mbox{in } \Omega,\\
\partial_1^{k}u_t+\partial_1^{k}\nabla_{\mathcal{A}}q+a \partial_1^{k}u=m  \partial_1^2\partial_1^{k}\eta  &\mbox{in } \Omega,\\
\partial_1^{k}\div_\mathcal{A}u=0  &\mbox{in } \Omega, \\
\partial_1^{k}(u_3, \eta_3)=0  &\mbox{on } \partial\Omega ,
\end{cases}
\end{equation}
Multiplying \eqref{01dsaf16asdfasf0321}$_2$ by $\partial_1^{k}u$ and $\partial_1^{k}\eta$ in $L^2$, resp., and then using
the integration by parts and \eqref{01dsaf16asdfasf}$_1$,
we have
\begin{align}
&\label{202008241510n}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\|\partial_1^{k} u\|_0^2
+m \|\partial_1\partial_1^{k}\eta\|_0^2\right)+a\|\partial_1^{k} u\|_{0}^2
=-\int \partial_1^{k}\nabla_{\mathcal{A}}q \cdot\partial_1^{k}u\mathrm{d}y
\end{align} and
\begin{align}
&\label{202008241510}
\frac{\mathrm{d}}{\mathrm{d}t}\left(\int\partial_1^{k}\eta\cdot\partial_1^{k} u\mathrm{d}y
+\frac{a}{2}\|\partial_1^{k} \eta\|_{0}^2\right)
+m \|\partial_1\partial_1^{k}\eta\|_0^2
=\|\partial_1^{k}u\|_{0}^2
-\int \partial_1^{k}\nabla_{\mathcal{A}}q\cdot\partial_1^{k} \eta\mathrm{d}y.
\end{align}
Next we estimate for the integrals involving the pressure in \eqref{202008241510n} and \eqref{202008241510} by two cases.

(1) \emph{Case $k=0$.}

Noting that $\eta_3|_{\partial\Omega}=0$,
 thus
\begin{align}
\mathcal{A}_{13}|_{\partial\Omega}=\mathcal{A}_{23}|_{\partial\Omega}=0.
\label{202010230522238}
\end{align}
Exploiting the integration by parts, \eqref{01dsaf16asdfasf}$_3$  and the  boundary-value conditions of $u_3|_{\partial\Omega}=0$ and \eqref{202010230522238}, we have
\begin{align}\nonumber
-\int\nabla_{\mathcal{A}}q\cdot u\mathrm{d}y=\int q\mathrm{div}_{\mathcal{A}}u\mathrm{d}y=0.
\end{align}
Putting the above identity into \eqref{202008241510n} with $k=0$ yields \eqref{202008241448} with $k=0$.

Now we turn to estimating for the last integral with $k=0$ in \eqref{202008241510}.
Since $\det(\nabla \eta+I)=1$,  we have
\begin{align}
\label{202103051850}
\mathrm{div}\eta=\mathrm{div}\Psi(\eta),
\end{align} where
\begin{align}\nonumber
  \Psi(\eta):=-\left(\begin{array}{c}
          \eta_1(\partial_2\eta_2+\partial_3\eta_3 )-
           \eta_1(\partial_2\eta_3 \partial_3\eta_2
- \partial_2\eta_2\partial_3\eta_3) \\
         \eta_2\partial_3\eta_3-\eta_1\partial_1\eta_2-
        \eta_1( \partial_1\eta_2 \partial_3\eta_3
-\partial_1\eta_3\partial_3\eta_2)\\
         -\eta_1\partial_1\eta_3
-\eta_2\partial_2\eta_3-\eta_1(\partial_1\eta_3 \partial_2\eta_2
-\partial_1\eta_2\partial_2\eta_3)
        \end{array}\right).
\end{align}
It is easy to estimate that
\begin{equation}\label{201910040902n00}
\|\Psi(\eta)\|_0\lesssim  \|\eta\|_2^2.
\end{equation}

Thanks to \eqref{202103051850}, \eqref{201910040902n00} and the boundary-value conditions $$\eta_3|_{\partial\Omega}=0,\ \Psi_3(\eta)|_{\partial\Omega}=0,$$
we have
\begin{align}
\int q\mathrm{div} \eta\mathrm{d}y
\leqslant\|\Psi(\eta)\|_0\|\nabla q\|_0
\lesssim \|\eta\|_2^2\|\nabla q\|_0. \nonumber
\end{align}
In addition, we can estimate that
\begin{equation}\nonumber
\left|\int \nabla_{\tilde{\mathcal{A}}}q\cdot\eta\mathrm{d}y\right|
\lesssim \|\eta\|_2^2\|\nabla q\|_0.
\end{equation}
Exploiting  the above two estimates and integration by parts, we get
\begin{align}
-\int\nabla_{\mathcal{A}}q\cdot\eta\mathrm{d}y
 =\int ( q\mathrm{div}\eta- \nabla_{\tilde{\mathcal{A}}}q\cdot\eta)\mathrm{d}y\lesssim \|\eta\|_2^2\|\nabla q\|_0. \label{20202105112053} \end{align}
 Putting the above estimate into \eqref{202008241510} yields  \eqref{202008241446} with $k=0$.

(2) \emph{Case $k=1$.}

Thanks to \eqref{06041533fwqg}, \eqref{201808181500n001} and \eqref{202012241002},  we can estimate that
\begin{align}
&-\int\partial_1\nabla_{\mathcal{A}}q\cdot\partial_1u\mathrm{d}y
 =\int(\partial_1q\partial_1 \mathrm{div}u
+ \nabla_{\tilde{\mathcal{A}}}q\cdot\partial_1^2u)\mathrm{d}y\nonumber\\
&\lesssim
\|(u, \eta)\|_3\|(u,\eta)\|_{1,1}\|\partial_1 q\|_0
+\|\partial_1^2u\|_0\| \eta\|_3 \|\nabla q\|_0\lesssim
\sqrt{\mathcal{E}^L}\mathcal{D}^H. \nonumber
\end{align}
Putting the above estimate into \eqref{202008241510n} yields \eqref{202008241448} with $k=1$.

Thanks to \eqref{06041533fwqg}, \eqref{improtian1002}   and \eqref{202012241002}, we have
\begin{align}
&-\int\partial_1\nabla_{\mathcal{A}}q \cdot\partial_1 \eta\mathrm{d}y
=\int( \nabla_{\tilde{\mathcal{A}}}q \cdot\partial_1^2 \eta
+ \partial_1q \mathrm{div}\partial_1 \eta)\mathrm{d}y\nonumber \\
&\lesssim  \| \eta\|_{2,0} \|\eta\|_3\|\nabla q\|_0
+\| \eta\|_{1,1}\|\eta\|_3\|  q\|_{1}
\lesssim \sqrt{\mathcal{E}^L}\mathcal{D}^H. \nonumber
\end{align}
Plugging the above estimate into \eqref{202008241510} yields \eqref{202008241446} with $k=1$.
The proof is complete.
\hfill $\Box$
\end{pf}

Lemma \ref{lem:08241445} immediately implies the following two horizontal-type energy  inequalities.
\begin{pro}\label{pro0902}
Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, there exist two energy functionals $\mathcal{E}^H_{h}$ and $\mathcal{E}^L_{h}$ of $(\eta,u)$
 such that
\begin{align}
&\label{202008250856n0}
\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^L_{h}
+  \|(u,\partial_1\eta)\|_{0}^2
\lesssim \|\eta\|_2^2 \|\nabla q\|_0 ,\\
&\label{202008250856}
\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^H_{h}
+ \|(u,\partial_1u, \partial_1^2\eta)\|_{0}^2 \lesssim\sqrt{\mathcal{E}^L}\mathcal{D}^H,
\end{align}
where, for each $t>0$,
$$
 \mathcal{E}^L_{h}\mbox{ and } \mathcal{E}^H_{h}
\mbox{ are equivalent to }
\|(u,\eta,\partial_1\eta )\|_{0}^2\mbox{ and }\|(u,\partial_1\eta)\|_{\underline{1},0}^2 \mbox{, resp.}.
$$
\end{pro}

\subsection{Vorticity estimates}
Let the multindex $\alpha $ satisfy $|\alpha|\leqslant 2$.
Applying $\partial^{\alpha  } \partial_1^{i}\mathrm{curl}_{\mathcal{A}} $
to \eqref{01dsaf16asdfasf}$_2$ yields
\begin{align}\label{202005021542000}
\partial_t\partial^{\alpha}{ \partial_1^{i}}\mathrm{curl}_{\mathcal{A}}u
+a\partial^{\alpha}{ \partial_1^{i}}\mathrm{curl}_{\mathcal{A}}u
=m \partial_1(\partial^{\alpha}{ \partial_1^{i}}\mathrm{curl}_{\mathcal{A}}\partial_1\eta)
+W^{i,\alpha},
\end{align}
where $W^{i,\alpha}$ is defined by \eqref{2021011281929}.

Now we derive the following vorticity estimates for $u$ and $ \eta$.
\begin{lem}\label{2019100216355nnn00}
Let $|\alpha|\leqslant 2$. Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, we have
\begin{align}
&\label{202005021610nnmm}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\int\left(|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u |^2+m |\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\partial_1\eta|^2\right)\mathrm{d}y
 +a\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u\|^2_{0}\nonumber\\
&\lesssim{
\begin{cases}
 \|u\|_3 \|(u,{ \partial_1\eta})\|_3^2
&\hbox{for }   i=0; \\
  \| \eta\|_{2,3}(\|u\|_3\| \eta\|_{2,3}+ \| ( u,\eta)\|_{1,3}\|\eta\|_{1,3}
)
  +  \| u\|_3  \| (u,\eta)\|_{1,3}^2
&\hbox{for }  i=1,
\end{cases}}\\
&\label{202005021600nnmm}
\frac{\mathrm{d}}{\mathrm{d}t}
\int\left(\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\eta \cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} u
+\frac{a}{2}|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \eta |^2\right)\mathrm{d}y
+{m }\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\partial_1\eta\|^2_{0}\nonumber\\
&\lesssim\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u\|^2_{0}\nonumber \\
&\quad +
\begin{cases}
\|\eta\|_3 \|(u,\partial_1\eta)\|_3^2
+{\|u\|_3  \|\eta\|_3^2}
&\hbox{for }   i=0; \\
  \|(u,\eta)\|_3   \| (u,\eta)\|_{1,3}^2
+ \| \eta\|_{2,3}(\|\eta\|_3\| \eta\|_{2,3} +\|\eta\|_{1,3}^2)
&\hbox{for }  i=1.
\end{cases}
\end{align}
\end{lem}
\begin{pf}
(1) Multiplying \eqref{202005021542000} by $\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u$ in $L^2$ and then using the integration by parts, we get
\begin{align}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
\int\left(|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u |^2+m |\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\partial_1\eta|^2\right)\mathrm{d}y
 +a\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u\|^2_{0}\nonumber \\
&= m \int\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \partial_1\eta\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}_t} \partial_1\eta\mathrm{d}y-m \int\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\partial_1\eta
\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\partial_1\mathcal{A}} u\mathrm{d}y
\nonumber
\\
&\quad +\int{W^{i,\alpha}}
\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} u \mathrm{d}y=:\sum_{j=1}^{3} I_j^{i,\alpha}.
\label{202005031742nnmm} \end{align}

Making use of
\eqref{aimdse}, \eqref{06041533fwqgn}, \eqref{06142100x}, and \eqref{201907291200000},
the three integrals $I_1^{0,\alpha} $--$I_3^{0,\alpha} $ can be estimated as follows:
\begin{align}
I_1^{0,\alpha}
\lesssim&\|\mathrm{curl}_{\mathcal{A}}\partial_1\eta\|_2 \|\mathrm{curl}_{\mathcal{A}_t} \partial_1\eta\|_2
\lesssim   \|u\|_3\| \eta\|_{1,3}^2 ,\label{202105021842}\\
 I_2^{0,\alpha} \lesssim& \|\mathrm{curl}_{\mathcal{A}}\partial_1\eta\|_2\| \mathrm{curl}_{\partial_1\mathcal{A}} u \|_2\lesssim  \|u\|_3\| \eta\|_{1,3}^2,  \\
 I_3^{0,\alpha}\lesssim&\|W^{0,\alpha}\|_0\|\mathrm{curl}_{ \mathcal{A}} u\|_2
\lesssim \|u\|_3 \|(u,\partial_1\eta)\|_3^2.
\end{align}

Exploiting \eqref{aprpiosesnew},
 \eqref{aimdse}, \eqref{06041533fwqgn}--\eqref{06041safsa533fwqgn}, \eqref{201907291200000nm} and \eqref{202012241002},
the three integrals $I_1^{1,\alpha}$--$I_3^{1,\alpha}$ can be estimated as follows:
\begin{align}
I_1^{1,\alpha}&\lesssim \| \eta\|_{2,3} (\|u\|_3\|\eta\|_{2,3}+\|(u,\eta)\|_{1,3}
\|\eta\|_{1,3}) , \\
I_2^{1,\alpha}
& \lesssim \|\eta\|_{2,3}(\|u\|_3\|\eta\|_{2,3}+\| u\|_{1,3}\|\eta\|_{1,3} ) ,\\
I_3 ^{1,\alpha}
&\lesssim  \|(u,\eta)\|_{1,3}(\|u\|_3 \|(u,\eta)\|_{1,3}+\| \eta\|_{1,3}\|\eta\|_{2,3}) .\label{202105021842xx}
\end{align}
Thanks to the above six estimates and  \eqref{202012241002}, we immediately get \eqref{202005021610nnmm} from \eqref{202005031742nnmm}.

(2) Multiplying \eqref{202005021542000}
by $\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\eta$ in $L^2$, and then using
the integration by parts, we obtain
\begin{align}
&\frac{\mathrm{d}}{\mathrm{d}t}
\int\left(\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\eta \cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} u
+\frac{a}{2}|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \eta |^2\right)\mathrm{d}y
 +{m }\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}\partial_1\eta\|^2_{0}\nonumber \\
&
=\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u\|^2_{0}+
\int \partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}_t}\eta \cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} u\mathrm{d}y
+{a}\int\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}_t} \eta
\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \eta  \mathrm{d}y\nonumber \\
&\quad -m \int (\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \partial_1\eta)
\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\partial_1\mathcal{A}} \eta \mathrm{d}y+\int
{ W^{i,\alpha}}\cdot\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}} \eta\mathrm{d}y
\nonumber \\
&=:\|\partial^{\alpha}{ \partial_1^i}\mathrm{curl}_{\mathcal{A}}u\|^2_{0}{ + \sum_{j=4}^7 I_j^{i,\alpha}}.
\label{202005031720nnmm}
\end{align}


Similarly to \eqref{202105021842}--\eqref{202105021842xx},
the   integrals $I_4^{i,\alpha} $--$I_7^{i,\alpha} $  can be estimated as follows:
\begin{align}
& I_4^{0,\alpha} \lesssim \|u\|_3^2\|\eta\|_3,\
 I_5^{0,\alpha}
\lesssim \|u\|_3\|\eta\|_3^2 , \label{2020202105112055} \\
&  I_6^{0,\alpha} \lesssim
 \|\eta\|_3\| \eta\|_{1,3}^2 ,\ I_7^{0,\alpha}
 \lesssim  \|\eta\|_3 \|(u,\partial_1\eta)\|_3^2, \nonumber \\
 &I_4^{1,\alpha},\ I_5^{1,\alpha}
\lesssim   \|(u,\eta)\|_3  \|(u,\eta)\|_{1,3}^2  , \nonumber
\\
&I_6^{1,\alpha} \lesssim   \| \eta\|_{2,3} ( \|\eta\|_3\|\eta\|_{2,3} +\|\eta\|_{1,3}^2)
 , \nonumber  \\
&I_7^{1,\alpha} \lesssim   \|\eta\|_{1,3} (\|u\|_3\|(u,\eta)\|_{1,3}
+\|\eta\|_{1,3}\|\eta\|_{2,3}). \nonumber
\end{align}
Thanks to the above eight estimates,
we immediately
get \eqref{202005021600nnmm} from \eqref{202005031720nnmm}.
This completes the proof of Lemma \ref{2019100216355nnn00}.
\hfill$\Box$
\end{pf}

Next we further establish the vorticity-type energy inequalities.
\begin{pro}\label{pro1632nn1}
Under the assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, there exist two energy functionals $\mathcal{E}^H_{\mathrm{cul}}$ and $\mathcal{E}^L_{\mathrm{cul}}$ of $(\eta,u)$
 such that
\begin{align}
\label{202005021632nn1}
&\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^L_{\mathrm{cul}}
+  \|\mathrm{curl}( u,\partial_1 \eta)\|_2^2
\lesssim \|(u,\eta)\|_3\|(u,\partial_1\eta)\|_3^2
+ \|u\|_3\|\eta\|_3^2
\end{align}
and
\begin{align}\label{202005021632}
&\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^H_{\mathrm{cul}}
+ \|\mathrm{curl}(u, \partial_1 u,\partial_1^2  \eta)\|_{2 }^2
\lesssim \|(u,\eta,\partial_1\eta)\|_3 \|(u, \partial_1\eta )\|_{\underline{1},3}^2,
\end{align}
where, for each $t\in \overline{I_T}$,
\begin{align}
& \mathcal{E}^L_{\mathrm{cul}}
\mbox{ is equivalent to }
 \|\mathrm{curl}_{\mathcal{A}}(u,\eta, \partial_1\eta )\|_{2}^2,\label{2022100520202157} \\
&\mathcal{E}^H_{\mathrm{cul}}
\mbox{ is equivalent to }
{  \|  \mathrm{curl}_{\mathcal{A}}(u,\partial_1\eta)
\|_{\underline{1},2}^2+\|\mathrm{curl}_{\mathcal{A}}\eta\|_{1,2}^2}.\label{2022100520202157x}
\end{align}
\end{pro}
\begin{pf}
We easily derive from \eqref{202005021610nnmm} and \eqref{202005021600nnmm} with $i=0$ that
\begin{align}
\label{202101242005}
&\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^L_{\mathrm{cul}}
+  \| \mathrm{curl}_{\mathcal{A}}(u,\partial_1\eta)\|_2^2
\lesssim
\|(u,\eta)\|_3\|(u,\partial_1\eta)\|_3^2
+ \|u\|_3\|\eta\|_3^2,
\end{align}
where  $\mathcal{E}^L_{\mathrm{cul}}$ satisfies  \eqref{2022100520202157}.
Employing \eqref{202101231432} with $w=u$ and $ \partial_1\eta$, we further get \eqref{202005021632nn1}
from \eqref{202101242005}.

 We derive from \eqref{202005021610nnmm} and \eqref{202005021600nnmm} with $i=1$ that
\begin{align}
\label{202005021632nx2}
&\frac{\mathrm{d}}{\mathrm{d}t}
\mathcal{E}^{H}_{\mathrm{cul}}
+\|\mathrm{curl}_{\mathcal{A}}u\|^2_{2} +  \|\mathrm{curl}_{\mathcal{A}}\left(u
, \partial_1\eta\right)\|^2_{1,2}
\lesssim
\|(u,\eta,\partial_1\eta)\|_3(\|u\|_3^2+\| (u, \eta,\partial_1 \eta)\|_{1,3}^2),
\end{align}
where  $\mathcal{E}^{H}_{\mathrm{cul}}$ satisfies \eqref{2022100520202157x}.
Consequently, making use of \eqref{202101231432} and \eqref{202101231245}, we derive \eqref{202005021632nn1} and \eqref{202005021632}
from \eqref{202101242005} and \eqref{202005021632nx2}.
This completes the proof of Proposition \ref{pro1632nn1}.
\hfill$\Box$
\end{pf}

\subsection{Equivalent estimates }\label{subsec:equivalent}

This section is devoted to establishing the equivalent estimates for
 $\mathcal{E}^L$, $\mathcal{D}^L$, $\mathcal{E}^H$ and $\mathcal{D}^H$.
\begin{lem}\label{201612132242nx}
Under assumptions \eqref{aprpiosesnew} and \eqref{aprpiosasfesnew}, we have
\begin{align}
&\label{201903161543}
\mathcal{E}^L\mbox{ is equivalent to }\|(u,\eta,\partial_1 \eta )\|_3^2,\\
&\label{202103192137}
\mathcal{D}^L\mbox{ is equivalent to }\|(u,\partial_1 \eta )\|_3^2,\\
&\mathcal{E}^H,\ \mathcal{D}^H\mbox{ and }\|(u,\partial_1u,\partial_1^2\eta)\|_{3}^2\mbox{ are equivalent}.
\label{2019031615431}
\end{align}
\end{lem}
\begin{pf}To obtain \eqref{201903161543}--\eqref{2019031615431}, the key step is to establish the estimates of $\nabla q$ and $u_t$.
By  \eqref{01dsaf16asdfasf}$_2$, we have
\begin{equation}
\begin{cases}
-\Delta q= f^1  &\mbox{in } \Omega,\\
\partial_3 q =f^2  \vec{\mathbf{n}}_3 &\mbox{on } \partial\Omega,
\end{cases}
\label{2022010502022007}
\end{equation}
where $f^1:=\mathrm{div}\left(u_t+a u-m \partial_1^2\eta+\nabla_{\tilde{\mathcal{A}}}q\right)$ and
$f^2:=- {\tilde{\mathcal{A}}}_{3j}\partial_j q$.

Note that
\begin{align*}
\int f^1\mathrm{d}y
+\int_{\partial\Omega} f^2  { \vec{\mathbf{n}}_3}\mathbf{d}y_{\mathrm{h}}=0,
\end{align*}
  where $\vec{\mathbf{n}}_3$ is the third component of the outward unit normal vector $\vec{\mathbf{n}} $ on  $\partial\Omega$, thus applying  the elliptic estimate \eqref{neumaasdfann1n} to \eqref{2022010502022007}, and then using \eqref{01dsaf16asdfasf}$_3$, we have
\begin{align}
\|\nabla q\|_2&\lesssim\|f^1\|_1+\|f^2\|_2 \nonumber \\
&\lesssim \|(\mathrm{div}_{\mathcal{A}_t}u,\mathrm{div}_{\tilde{\mathcal{A}}}u_t,\mathrm{div}_{\tilde{\mathcal{A}}}u)\|_1
+\|(\partial_1^2\eta, \nabla_{\tilde{\mathcal{A}}}q)\|_2 \nonumber \\
&\lesssim\| \eta\|_{2,2}
+ \|(u,\eta)\|_3\|(u,\nabla q,u_t)\|_2. \label{202008241115}
\end{align}

In addition, by \eqref{01dsaf16asdfasf}$_2$, we have
\begin{align}\nonumber
\|u_t\|_2
\lesssim\|(u,\partial_1^2\eta,\nabla q)\|_2,
\end{align}
which, together with \eqref{202008241115}, yields, for sufficiently small $\delta$,
\begin{align}\label{202008241115nmq}
\|(u_t,\nabla q)\|_2
\lesssim\|(u,\partial_1^2\eta )\|_2.
\end{align}
Consequently, we conclude the desired results by   \eqref{202008241115nmq}
and  \eqref{202012241002}.
The proof is complete.
\hfill$\Box$
\end{pf}

\subsection{Stability estimates}

Now we are in the position to  derive  \emph{a priori} stability estimates \eqref{1.200} and \eqref{202101241112}.


 Making use of \eqref{201808181500}--\eqref{improtian1003},   and Hodge-type elliptic estimate \eqref{08171537}, we can deduce  that, for  $0\leqslant i\leqslant 1$ and $0\leqslant j\leqslant 2$,
\begin{align}
\| u \|_{i,3}\lesssim &\|  u \|_{i,0}
+\| \mathrm{curl} u  \|_{i,2} +\begin{cases}
\|\eta\|_{3} \|u\|_{3}&\mbox{for }  i=0 ,
\\
\|(u,\eta)\|_3\|( u,\eta)\|_{1,3} &\mbox{for }i=1   \end{cases} \label{202008240900} \end{align}
and
\begin{align}
\|  \eta \|_{j,3}\lesssim &\|  \eta \|_{j,0}
+\| \mathrm{curl}  \eta \|_{j,2}  +  \begin{cases}
\|\eta\|_{3}^2&\mbox{for }  j=0 ;
\\
\|\eta\|_{3} \|\eta\|_{1,3} &\mbox{for }  j=1; \\
\|\eta\|_{3} \|\eta\|_{2,3}+ \|\eta\|_{1,3}^2 &\mbox{for }  j=2. \end{cases}\label{20200824sfas0900} \end{align}

By \eqref{202008250856}, \eqref{202005021632} and \eqref{202012241002}, we have
\begin{align}\label{201910050940}
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^H+c\tilde{\mathcal{D}}^H\lesssim \sqrt{\mathcal{E}^L}\mathcal{D}^H,
\end{align}
where
$$\begin{aligned}
&\tilde{\mathcal{E}}^H:=
\mathcal{E}^H_h +\mathcal{E}^H_{\mathrm{cul}} \mbox{ and }
 \tilde{\mathcal{D}}^H:=
\|(u,\partial_1u,\partial_1^2\eta)\|_{0}^2
+\|\mathrm{curl}(u,\partial_1 u,\partial_1^2 \eta )\|_2^2.
\end{aligned}$$
In addition, it follows from \eqref{202101231432}, \eqref{202101231245}, \eqref{2019031615431},
\eqref{202008240900} and \eqref{20200824sfas0900}
that for sufficiently small ${\delta}$,
\begin{align}
&\label{201908081232}
\tilde{\mathcal{E}}^H,\ {\mathcal{E}^H},\ \tilde{\mathcal{D}}^H,\ {\mathcal{D}^H}\mbox{ and }\|(u,\partial_1u, \partial_1^2\eta)\|^2_{3}
\mbox{ are equivalent}.
\end{align}
Thus  we deduce from \eqref{201910050940}--\eqref{201908081232} that,
\emph{for any sufficiently small $\delta$,}
\begin{align}
\nonumber
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^H+\mathcal{D}^H\leqslant 0.
\end{align}

Exploiting \eqref{201908081232}, we further deduce from the above inequality that  there exists a constant  $c_1>0$ such that
\begin{align}    \label{estemalas}
 & {e}^{c_1 t}\mathcal{E}^H(t) + \int_0^t e^{c_1 s}\mathcal{D}^H(s)\mathrm{d}{s}
\lesssim I_H^0.
\end{align}

In addition, by \eqref{202008250856n0} and \eqref{202005021632nn1}, we have
\begin{align}\label{201910050940nnm}
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^L+c\tilde{\mathcal{D}}^L\lesssim  \|(u,\eta)\|_3 \|(u, \partial_1\eta )\|_{3}^2 + (\|u\|_3+\|\nabla q\|_0)\|\eta\|_3^2,
\end{align}
where
$$\begin{aligned}
 \tilde{\mathcal{E}}^L:=
\mathcal{E}^L_h+\mathcal{E}^L_{\mathrm{cul}}\mbox{ and }\tilde{\mathcal{D}}^L:=
 \|( u,\partial_1\eta )\|_{0}^2
 +\|\mathrm{curl}(u,\partial_1\eta)\|_{2}^2.
\end{aligned}$$
In addition, thanks to \eqref{202101231432},
  \eqref{201903161543}, \eqref{202103192137}, \eqref{202008240900} and \eqref{20200824sfas0900}, we have, for sufficiently small $\delta$,
\begin{align}
&\label{2019080sdaf81232nnm}
\tilde{\mathcal{E}}^L,\ {\mathcal{E}}^L\mbox{ and }\|( u,\eta,\partial_1\eta)\|_{3}^2
\mbox{ are equivalent}
\end{align}
 and
\begin{align}
 \label{201908081232nnm}
\tilde{\mathcal{D}}^L\mbox{ is equivalent to }{\mathcal{D}^L}.
\end{align}

Thanks to \eqref{201908081232nnm}, we get from \eqref{201910050940nnm} that,  \emph{for sufficiently small $\delta$},
\begin{align}\nonumber
\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathcal{E}}^L+c\tilde{\mathcal{D}}^L\lesssim   (\|u\|_3+\|\nabla q\|_0)\|\eta\|_3^2.
\end{align}
Exploiting \eqref{estemalas}, \eqref{2019080sdaf81232nnm}   and Gronwall's lemma, we deduce from the above estimate  that
\begin{align} \label{202101242212}
 \mathcal{E}^L(t) +\int_0^t \mathcal{D}^L(s)\mathrm{d}{s} \lesssim
e^{c  \sqrt{I_H^0}}I_L^0.
\end{align}
In particular, by \eqref{estemalas} and \eqref{202101242212}, there exist constants $c_2\geqslant 1$ and $\delta_1\in (0,1]$  such that, for any $\delta\leqslant \delta_1$,
\begin{align}
& \mathcal{E}^L(t) +\int_0^t \mathcal{D}^L(s)\mathrm{d}{s} \leqslant c_2^2
e^{c_2 \sqrt{I_H^0}}I_L^0,\label{20210asfdsa1242212}\\
&{e}^{c_1 t}\mathcal{E}^H(t) + \int_0^t e^{c_1 \tau}\mathcal{D}^H(s)\mathrm{d}{s}
\leqslant c_2^2 I_H^0 .\label{estemsadfaalas}
\end{align}
\subsection{Proof of Theorem \ref{thm2} }\label{subsec:08}
Now we state the local well-posedness result for the initial-boundary value problem \eqref{01dsaf16asdfasf}.
\begin{pro}\label{202102182115}
Let $b >0$ be a constant and $\iota >0$ be the  constant in Lemma \ref{pro:1221}.
We assume that   $(  u^0,\eta^0)\in H^{1,3}_{\mathrm{s}}\times (H^{2,3}_{ \mathrm{s}}\cap H^3_*) $,
${  \|(u^0,\partial_1u^0,\partial_1^2\eta^0)\|_{3}\leqslant b }$  and $\mathrm{div}_{\mathcal{A}^0}u^0=0$, where $\mathcal{A}^0$ is defined by $\zeta^0$ and $ \zeta^0 = \eta^0+y$.
Then there exist a constant $\delta_2 \in (0,\iota/2]$  such that  if $\eta^0$   satisfies
\begin{align}
\| \eta^0\|_{\underline{1},3}\leqslant \delta_2,
\nonumber
\end{align}
the initial-boundary value problem  \eqref{01dsaf16asdfasf} admits a unique local classical solution
$(u,\eta,q)\in \mathfrak{U}_{T }^{1,3}\times   \mathfrak{C}^0 (\overline{I_{T }},{H}^{2,3}_{\mathrm{s}} )\times   C(\overline{I_T},\underline{H}^3)$  for some local existence time $T>0$.
Moreover, $(u,\eta,q) $ satisfies
\begin{align}& \sup\nolimits_{t\in \overline{I_T}}
\| \eta\|_{ 3}\leqslant 2\delta_2
,\nonumber \\
&\label{202012212151}
  \sup\nolimits_{t\in \overline{I_T}}\| (u,\eta, \partial_1 \eta )\|_3
  \leqslant c_3 \sqrt{I^0_L}  \mbox{ and }
  \sup\nolimits_{t\in \overline{I_T}}\| ( u , \partial_1 \eta)\|_{\underline{1},3}
  \leqslant c_3 \sqrt{ I^0_H}
\end{align}
for some positive constant $c_3\geqslant 1$.
It should be noted that $\delta_2$ and $T$ depend on $a$, $m $ and $\Omega$; moreover $T$ further depends on $b$.
\end{pro}
\begin{rem}
 {Here the uniqueness means that if there is another solution
$(\tilde{u}, \tilde{\eta},\tilde{q})\in\mathfrak{U}_{T }^{1,3}\times   \mathfrak{C}^0 (\overline{I_{T }},{H}^{2,3}_{\mathrm{s}} )\times   C(\overline{I_T},\underline{H}^3)$
 satisfying $0<\inf_{(y,t)\in \mathbb{R}^2\times \overline{I_T}} \det(\nabla \tilde{\eta}+I)$ and $\nabla_{(\nabla \tilde{\eta}+I)^{-\mathrm{T}}}\tilde{q}\in L^\infty_TH^3$, then
 $(\tilde{\eta},\tilde{u},\tilde{q})=(\eta,u,q)$ by virtue of the smallness condition
 ``$\sup_{t\in \overline{I_T}}\|\eta\|_{3}\leqslant 2\delta_2$''. In addition,
  we have, by the fact ``$\sup\nolimits_{t\in \overline{I_T}}\| \eta\|_3\leqslant  \iota$" and  Lemma \ref{pro:1221},
$$\inf\nolimits_{(y,t)\in \mathbb{R}^2\times  {I_T}} \det(\nabla \eta+I)\geqslant 1/4.$$}
\end{rem}
\begin{pf} The proof of Proposition \ref{202102182115} will be provided in Section \ref{202102241211}.
\hfill $\Box$
\end{pf}

Thanks to the \emph{priori} estimates \eqref{20210asfdsa1242212}, \eqref{estemsadfaalas}
and Proposition  \ref{202102182115}, we can easily establish
Theorem \ref{thm2}. Next we briefly give the proof.

Let  $(u^0,\eta^0)$ satisfies the assumptions in Theorem \ref{thm2},
\begin{align}
 e^{c_2 c_3\sqrt{I_H^0}}I_L^0  \leqslant\delta^2 \mbox{ and }
\delta =\min\left\{ \delta_1, \delta_2\right\}/ {{c}_2}c_3^2.\nonumber
\end{align}  By Proposition  \ref{202102182115}, there exists a unique local solution $( u,\eta,q)$ to the initial-boundary value  problem \eqref{01dsaf16asdfasf} with a maximal existence time $T^{\max}$, which satisfies that
\begin{itemize}
  \item for any $\tau\in  I_{T^{\max}}$,
the solution $(u,\eta,q)$ belongs to $\mathfrak{U}_{\tau}^{1,3}  \times{\mathfrak{H}}^{2,3}_{1,*,\tau}\times  C^0(\overline{I_\tau},\underline{H}^3)$, $$
 \sup\nolimits_{t\in {I_{\tau}}}\| \eta\|_{ 3}\leqslant 2\delta_2;$$
\item $\limsup_{t\to T^{\max} }\|\eta ( t)\|_{\underline{1},3} > \delta_2$
or $\limsup_{t\to T^{\max} }\|(u,\partial_1u,\partial_1\eta)( t)\|_{ 3}=\infty$, if $T^{\max}<\infty$.
\end{itemize}

Let
\begin{equation}
\nonumber
T^{*}:=\sup\left\{ \tau \in I_{T^{\max}}~\left|~
\|( u,\eta,\partial_1\eta)(t)\|_{3} \leqslant {{c}_2}c_3^2\delta  \;\mbox{ for any }t\leqslant \tau\right.\right\}.
\end{equation}
We easily see that the definition of $T^*$ makes sense and $T^*>0$.
Thus, to obtain the existence of global solutions,  it suffices to verify $T^*=\infty$. Next we shall prove this fact by contradiction.

We assume that $T^*<\infty$. Then, for any given $T^{**}\in I_{T^*}$,
\begin{equation}
\label{201911262sadf202}
\sup\nolimits_{t\in \overline{I_T^{**}}}\|(u,\eta,\partial_1\eta)(t) \|_3   \leqslant {{c}_2}c_3^2\delta \leqslant \delta_1.
\end{equation}
Thanks to the estimate  \eqref{201911262sadf202}, we can
follow the arguments of  \eqref{20210asfdsa1242212}--\eqref{estemsadfaalas} and use a regularity method and \eqref{202012212151} to derive that
\begin{align}
&\label{202104191546}
\|e^{c_1 t}\mathcal{E}^H(t)\|_{L^{\infty}(I_{T^{**}})} + \int_0^{{T^{**}}} e^{c_1 \tau}\mathcal{D}^H(s)\mathrm{d}s
\leqslant {c}_2^2
\lim_{t\to 0}\sup\nolimits_{\tau\in (0,t)}\|(u,\partial_1\eta)(\tau)\|_{\underline{1},3}^2
\leqslant c_2^2 c_3^2{ I^0_H},\\
&\label{202104191547}
\|\mathcal{E}^L(t)\|_{L^{\infty}(I_{T^{**}})} + \int_0^{T^{**}} \mathcal{D}^L(s)\mathrm{d}s
\leqslant {c}_2^2e^{c_2c_3\sqrt{I_{H}^0}}\lim_{t\to 0}\sup\nolimits_{\tau\in (0,t)}
 \|(u,\eta,\partial_1\eta)\|_3^2
\leqslant c_2^2c_3^2\delta^2.
\end{align}
By the weak continuity of $(\eta,u)$ and the fact
\begin{align}\sup\nolimits_{t\in \overline{I_{\tau}}} \|f\|_0
= \|f\|_{L^\infty(I_{\tau},L^2)}
\mbox{ for any }f\in C(\overline{I_{\tau}}, L^2_{\mathrm{weak}})\ \mbox{with}\ \tau>0, \nonumber
\end{align}
we further derive from \eqref{202104191546} and \eqref{202104191547} that
\begin{align}
& \sup\nolimits_{t\in \overline{I_T^{**}}}\|(u,\partial_1\eta)(t)\|_{\underline{1},3}
\leqslant c_2c_3 \sqrt{ I^0_H} \nonumber ,\\
&\label{202104191542}\sup\nolimits_{ t\in \overline{I_T^{**}}}\|(u,\eta,\partial_1\eta)(t)\|_{3}
\leqslant c_2c_3\delta.
\end{align}

We take $(u(T^{**},\eta(T^{**}) ))$ as a initial data. Noting that
$$\begin{aligned}
&\|(u,\partial_1u, \partial_1^2\eta)(T^{**})\|_{ 3}
\leqslant b := c_2c_3\sqrt{ I^0_H}
\mbox{ and }\| \eta(T^{**})\|_{\underline{1},3}\leqslant c_2c_3\delta\leqslant \delta_2,
\end{aligned}$$ then,
by Proposition \ref{202102182115} and \eqref{202104191542}, there exists a unique local  classical solution, denoted by $( u^*, \eta^*$, $q^*)$, to the initial-boundary value problem \eqref{01dsaf16asdfasf} with $(u(T^{**}),\eta(T^{**}) )$ in place of $(u^0,\eta^0 )$; moreover,
\begin{align}
&\sup\nolimits_{t\in [T^{**},T]}\| (u^* ,\eta^*, \partial_1 \eta^*  )\|_3 \leqslant {{c}_2}c_3^2\delta
\mbox{ and } \sup\nolimits_{t\in [T^{**},T]}\| \eta^*\|_{ 3}\leqslant 2\delta_2  , \nonumber
\end{align}
where the local existence time $T>0$ depends on $ b$, $a$, $m $ and $\Omega$.

In view of the existence result of $(u^*, \eta^*, q^*)$, the uniqueness conclusion in Proposition  \ref{202102182115}  and the fact that $T^{\max}$ denotes the maximal existence time, we immediately see that $T^{\max}>T^*+T/2$ and
$ \sup\nolimits_{t\in  {[0,T^*+T/2]}}\| (u,\eta, \partial_1 \eta)\|_3 \leqslant  {{c}_2}c_3^2\delta$.
This contradicts with the definition of $T^*$. Hence $T^*=\infty$ and thus $T^{\max}=\infty$.
This completes the proof of the existence of global solutions.
The uniqueness of global solution is obvious due to the uniqueness of local solutions in
Proposition \ref{202102182115} and the fact $\sup\nolimits_{t\geqslant0}\| \eta\|_{\underline{1},3}\leqslant 2\delta_2 $.
In addition,  it is obvious that $(\eta,u)$ indeed enjoys
\eqref{1.200} and \eqref{202101241112} by
 the derivation of \eqref{202104191546} and \eqref{202104191547}.

Finally, we derive \eqref{1.200xx}.
By \eqref{202101241112}, we easily see that
\begin{align}
\label{202101242112}
&\partial_1\eta(t)\to 0\mbox{ in }H^3\mbox{ as }t\to \infty
\end{align}
and
\begin{align}
\label{20safda2101242112}
\left\|\int_0^t u\mathrm{d}\tau\right\|_3\lesssim\int_0^t \|u  \|_3\mathrm{d}\tau\lesssim \|(u^0,\partial_1\eta^0)\|_{\underline{1},3}   \mbox{ for any }t>0\end{align}
Thanks to \eqref{20safda2101242112}, there exist a subsequence $\{t_n\}_{n=1}^\infty$ and some function $\eta^\infty \in H^3$ such that
$$\eta(t_n)-\eta^0= \int_0^{t_n} u\mathrm{d}\tau \to\eta^\infty -\eta^0\mbox{ weakly in }H^3.$$
Exploiting $\eqref{01dsaf16asdfasf}_1$ and \eqref{202101241112}, we have
$$
\begin{aligned}
\|\eta(t)-\eta^\infty\|_3\leqslant &\liminf_{t_n\to \infty} \left\|\int_t^{t_n } u\mathrm{d}\tau\right\|_3
\lesssim \sqrt{ I^0_H} \liminf_{t_n \to \infty}\int_t^{t_n} e^{-c_1 t/2}\mathrm{d}\tau
 \lesssim   \sqrt{ I^0_H}e^{-c_1 t/2}  ,
\end{aligned}$$
which, together with \eqref{202101242112}, yields that \eqref{1.200xx} holds and $\eta^\infty $ only depends on $y_2$ and $y_3$. This completes the proof of Theorem  \ref{thm2}.

\section{Local well-posedness}\label{202102241211}
 This section is devoted to the proof of local well-posedness of the initial-boundary value problem \eqref{01dsaf16asdfasf}.   Next we roughly sketch the proof frame to establish the  local well-posedness result.

Similarly to \cite{GXMWYJJMPA2019}, in which the authors investigated the well-posdeness problem of  incompressible inviscid MHD fluid equations with free boundary, we first  alternatively  use an iteration method to establish the existence result of the unique local solution $(u,\eta,{q})$ of the linear $\kappa$-approximate problem
\begin{equation} \label{01dsaf16asdfasf0000xx}
                              \begin{cases}
\eta_t-\kappa\partial_1^2\eta=u  ,\\
u_t+\nabla_{{\mathcal{B}}} {q}+a u=m   \partial_1^2\eta ,\\
\div_{{\mathcal{B}}} u=0  , \\
(u,\eta)|_{t=0}=(u^0,\eta^0) , \\
(u_3,\eta_3)|_{\partial\Omega}=0,\\
  \mathcal{B}_{3j}\partial_j{q}|_{\partial\Omega}=0,
\end{cases}
\end{equation}
where $\kappa>0$, $\mathcal{B}:=(\nabla \varsigma +I)^{-\mathrm{T}}$, and the given function $\varsigma$ belongs to
some proper function space,
see Proposition \ref{202102151544} in Section \ref{subsec:local01} for details. It should be noted that the Neumann boundary-value condition \eqref{01dsaf16asdfasf0000xx}$_6$ makes sure the solvability of $q$.

 Then we derive the $\kappa$-independent estimates of the solutions, denoted by $(u^\kappa,\eta^\kappa, {q}^\kappa)$ for $\kappa\in \mathbb{R}^+$, of the $\kappa$-approximate problem above,
and then  take the limit of $(u^\kappa,\eta^\kappa, {q}^\kappa)$ with respect to $\kappa\to 0$ in a common definition domain. In particular, the limit function, denoted by $(u,\eta,{q})$, is the unique local solution to the linearized problem
\begin{equation}\label{01dsaf16asdfsafasf0000}
                              \begin{cases}
\eta_t =u,\\
u_t+\nabla_{{\mathcal{B}}} {q}+a u=m   \partial_1^2\eta,\\
\div_{{\mathcal{B}}} u=0, \\
(u,\eta)|_{t=0}=(u^0,\eta^0), \\
(u_3,\eta_3)|_{\partial\Omega}=0,
\end{cases}
\end{equation}
see Proposition \ref{thm08} in Section \ref{subsubsce:03} for details. It should be noted that the solution $q$ of \eqref{01dsaf16asdfsafasf0000} automatically satisfies \eqref{01dsaf16asdfasf0000xx}$_6$ due to $u_3$ satisfying $\partial_t u_3 |_{\partial\Omega}=0$.

Finally, since the linearized problem \eqref{01dsaf16asdfsafasf0000} admits a unique local solution for any given $\varsigma$, thus we easily arrive at Proposition \ref{202102182115}  by a standard iteration method as in \cite{Coutand1,Coutand2}, in which the authors investigated the well-posdeness problem of  incompressible Euler equations with free boundary.

Now we turn to introducing some new notations appearing in this section.
$P(x_1,\ldots, x_n )$
and $\dot{P}(x_1,\ldots, x_n )$ represent the generic polynomials with respect to the parameters $x_1$, $\ldots$, $x_n$, where all the coefficients in $P$ and $\dot{P}$ are equal one, and $\dot{P}$ satisfies $\dot{P}(0,\ldots,0)=0$;
$a\lesssim_\kappa b$ means that $a\leqslant c^\kappa b$, where $c^\kappa$ denotes a generic positive constant, which may depend on $\kappa$, $a$, $m $ and $\Omega$. It should be noted that $P(x_1,\ldots, x_n )$, $\dot{P}(x_1,\ldots, x_n )$ and $c^\kappa$ may vary from line to line.


We define the difference quotient with respect variable $y_1$ as follows:
\begin{align}
\label{202220105041846}
D_1^{\tau}f (y_1,y_2,y_3)=(f(y_1+\tau,y_2 ,y_3)-f(y_1,y_2,y_3 ))/\tau\mbox{ for }
|\tau|\in  (0,1).
\end{align}
 The notation $\mathcal{X}$ represents $\mathrm{Id}$\mbox{ or }$D^\tau_1$, where $\mathrm{Id}$ denotes and identity mapping, and $D^\tau_1$ the difference quotient with respect to the variable $y_1$.

We always use the notations $\mathcal{B}$, resp. $\mathcal{J}$ to represent $(\nabla \varsigma +I)^{-\mathrm{T}}$, resp. $\det(\nabla \varsigma+I)$, where $\varsigma$ at least satisfies
\begin{align}
\varsigma \in {C}^0(\overline{I_T},{H}^{3})\mbox{ and }\inf\nolimits_{(y,t)\in {\Omega_T}}\det(\nabla \varsigma +I)\geqslant 1/4 \mbox{ for some }T>0 .  \label{202safas2103250842}
\end{align}
In addition, we define that  \begin{align}
\mathbb{A}^{3,1/4}_{T,\iota }:=\{\psi\in  {C}^0(\overline{I_T},{H}^{3}_{\mathrm{s}})~|~  \|\psi\|_3\leqslant \iota\}  \label{2022104161633}
  \end{align}
and
\begin{align}
\mathbb{S}_T:=\{&(w,\xi,\beta) \in C^0(\overline{I_{T}} ,{H}^3_{ \mathrm{s}})
\times C^0( \overline{I_{T}},{H}^{3}_{\mathrm{s}})\times (C ^0(\overline{I_{T}},\underline{H}^3 )  )~|~   \partial_1^2\xi  \in C^0(\overline{I_T},H^2_{\mathrm{s}}),\nonumber \\
& \partial_1^2\xi \in  L^2_{T}H^{3} ,\
\nabla^3  \partial_1\xi
\in C^0_{ B, \mathrm{weak}}( \overline{I_T},L^2), \  \nabla_{{\mathcal{B}}} \beta\in {L_{T}^2H^{ 3}}, \;
 {{\mathcal{B}}}_{3j}\partial_j \beta
|_{\partial\Omega}=0  \}  \label{20210413asfda21251},
\end{align}
 where $T>0$ and $\iota$ is the positive constant provided in Lemma \ref{pro:1221}. \emph{It should be noted that the function $\varsigma$, which belongs to $\mathbb{A}^{3,1/4}_{T,\iota }$, automatically satisfies \eqref{202safas2103250842} by Lemma \ref{pro:1221}. }


Finally, some preliminary estimates for $\mathcal{B}$ and $\mathcal{J}$ are collected as follows.
\begin{lem}\label{lem:0817}
Let $\varsigma $ satisfy \eqref{202safas2103250842} and $i=0$, $1$. Then
\begin{align}
&\label{202008121505}
\| \mathcal{B} \|_{C^0(\bar{\Omega})}
+\| (\mathcal{B},\mathcal{J}^{-1}) \|_2
\lesssim  P(\|{\varsigma}\|_3),\\
&\label{202008121535nnn}
\|{\mathcal{B}}-I\|_{ {i},2}
\lesssim  \|{\varsigma}\|_{{i},3}{P}(( \|{\varsigma}\|_{3} ) \mbox{ for any }t\in \overline{I_T}.
\end{align} If additionally  $\varsigma$ further satisfies
$(\partial_1\varsigma,\varsigma_t)\in  L^\infty_TH^{i,3}$ , then, for a.e. $t\in {I_T}$,
\begin{align}
&\label{202008121535}
\| \partial_t(\mathcal{B},\mathcal{J})\|_{i, 2}
\lesssim
          \dot{P}( \|( \varsigma ,\varsigma _t)\|_{\underline{i}, 3})  ,
\\
&\label{202008121535n}
\|  \mathcal{B}\|_{ 1+i, 3}
\lesssim
      \dot{P}( \| {\varsigma}\|_{\underline{1+i},3}).
\end{align}
\end{lem}
\begin{pf} Since the derivation of the estimates
\eqref{202008121505}--\eqref{202008121535n} are very elementary, we omit it. \hfill$\Box$
\end{pf}

\subsection{Solvability of the linear $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx}}\label{subsec:local01}

In this section, we construct the unique local solution of the  linear $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx}. To this purpose, we shall rewrite  the linear $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx} as the following equivalent problem (in the sense of classical solutions):
\begin{equation} \label{01dsaf16asdfasf0safd000}
                              \begin{cases}
\eta_t-\kappa\partial_1^2\eta=u ,\\
u_t+\nabla_{{\mathcal{B}}} {q}+a u=m  \partial_1^2\eta  ,\\
-\Delta_{{\mathcal{B}}}{q}=K^1  , \\
(u,\eta)|_{t=0}=(u^0,\eta^0)  , \\
(u_3,\eta_3)|_{\partial\Omega}=0 ,\ \mathcal{B}_{3j}\partial_j{q}|_{\partial\Omega}=0,
\end{cases}
\end{equation}
where $\Delta_{{\mathcal{B}}}:=\mathrm{div}_{{\mathcal{B}}}\nabla_{{\mathcal{B}}}$, $\varsigma|_{t=0}=\eta^0$, $(u^0,\eta^0)$ satisfies
$\mathrm{div}_{(\nabla \eta^0+I)^{-\mathrm{T}}}u^0=0$, $\inf\nolimits_{y \in \overline{\Omega}} \det{(\nabla \eta^0+I)^{-\mathrm{T}}} >0$, and
\begin{align}
& K^1:=a\mathrm{div}_{{\mathcal{B}}}{u} -\mathrm{div}_{\mathcal{B}_t}u-\mathcal{J}_t\mathrm{div}_{{\mathcal{B}}}u/\mathcal{J}
  -m   \mathrm{div}_{{\mathcal{B}}} \partial_1^2{\eta}.  \label{202102141605}
\end{align}
Then the solvability of the linear $\kappa$-approximate problem  \eqref{01dsaf16asdfasf0000xx}
reduces to the solvability of the initial-boundary value problem  \eqref{01dsaf16asdfasf0safd000}.

We want to establish the local well-posdeness result for \eqref{01dsaf16asdfasf0safd000}
by an iteration method.
To begin with, we shall investigate the solvability of the linear problem
\begin{equation} \label{01dsa000}
                              \begin{cases}
\eta_t-\kappa\partial_1^2\eta=w , \\
u_t+a u= K^2, \\
-\Delta_{{\mathcal{B}}}{q}=K^1 ,  \\
(u,\eta)|_{t=0}=(u^0,\eta^0) ,   \\
(u_3,\eta_3)|_{\partial\Omega}=0 ,\ \mathcal{B}_{3j}\partial_j{q}|_{\partial\Omega}=0,
\end{cases}
\end{equation}
where $(\xi,w,\theta)\in \mathbb{S}_T$ is given, and $K^2:=m    \partial_1^2\xi-\nabla_{{\mathcal{B}}}\theta  $.


It is easy to see that the above linear problem is  equivalent to the following three sub-problems: the  initial-boundary value problem of partly dissipative equation for $\eta$
\begin{equation}\label{parabolic}
                              \begin{cases}
\eta_t-\kappa\partial_1^2\eta=w ,\\
\eta|_{t=0}=\eta^0  ,\\
\eta_3|_{\partial\Omega} =0  ,
\end{cases}
\end{equation}
the initial-value problem of ODE  for $u$
\begin{equation}\label{transport}
                              \begin{cases}
 u_t+a u=K^2 ,\\
u|_{t=0}=u^0  , \\
u_3|_{\partial\Omega}=0
\end{cases}
\end{equation}
and the Neumann boundary-value problem of elliptic equation for $q$
\begin{equation}\label{elliptic}
                              \begin{cases}
-\Delta_{{\mathcal{B}}}{q}=K^1  &\mbox{in } \Omega,\\
\mathcal{B}_{3j}\partial_j{q} =0 &\mbox{on } \partial\Omega.
\end{cases}
\end{equation}
Thus the solvability of the linear problem \eqref{01dsa000} reduces to the solvability of the  three sub-problems above. Next we establish the { global} well-posedness results for the above  three sub-problems  in sequence.
 \begin{pro}\label{pro:parabolic}
Let $\kappa>0$, $T>0$, $w\in L^2_{T}H^{3}_{\mathrm{s}}$
and $\eta^0\in H^{1,3}_{\mathrm{s}}$.
Then the initial-boundary value problem \eqref{parabolic} admits a unique solution $\eta\in {C}^0( \overline{I_T},{H}^{3}_{\mathrm{s}} )$, which satisfies $ \partial_1^2 \eta \in C^0(\overline{I_T},H^2_{\mathrm{s}})$,
$\nabla^{3}\partial_1 \eta \in C^0_{B,\mathrm{weak}}(\overline{I_T},{L}^2)$,
and enjoys the estimates
\begin{align}
&
\sup\nolimits_{t\in \overline{I_T}}(\| \eta \|_{ 3}+\sqrt{\kappa}\| \eta \|_{ 1,3})
+ \|(\kappa\partial_1^{2 }\eta,\eta_t)\|_{L^2_TH^{3}}\nonumber \\
&\lesssim \|\eta^0 \|_{ 3}+\sqrt{\kappa}\|\eta^0 \|_{ 1,3}+(1+\sqrt{T})\|  w\|_{L^2_TH^{3}}, \label{20200805100sadfsa5}\\
&\label{20200805DSAFS100sadfsa5}\ {\kappa}\sup\nolimits_{t\in\overline{I_s}} \|   \eta^\varepsilon  \|_{1,3}^2
 \lesssim  \kappa\| \eta^0 \|_{1 ,3}^2+  \| w\|_{{L^2_sH^{3}}}^2\mbox{ for any }s\in (0,T].
\end{align}
\end{pro}
\begin{pf}
Let $\varepsilon\in (0,1)$,
$\chi$ be a one-dimensional standard mollifier (see \cite[pp. 38]{NASII04} for the definition), and $\chi^{\varepsilon}(s):=\chi(s/\varepsilon)/\varepsilon$.
Let $\tilde{w}=w$ in $\Omega_T $ and $\tilde{w}=0$ outside $\Omega\times (\mathbb{R}\backslash I_T)$. We define the mollification of $\tilde{w}$ with respect to $t$:
\begin{align}
\nonumber
S^{t}_\varepsilon (\tilde{w}): =\chi^{\varepsilon} * \tilde{w}.
\end{align}
Then
$S^{t}_\varepsilon  (\tilde{w}) \in C^\infty(\mathbb{R},H^{1,3}_{\mathrm{s}})$. We can check that
\begin{align}
&\|  S^{t}_\varepsilon (\tilde{w} ) \|_{L^2_TH^{3}}  \lesssim \| w\|_{L^2_TH^{3}}
\label{202210208asdfsa2248},\\
&  S^{t}_\varepsilon  (\tilde{w}) \to  w
\mbox{ strongly in }{L^2_TH^{3}},
\nonumber
\end{align}

Now we consider the  $\tau$-approximate problem for \eqref{parabolic}:
\begin{equation}\label{parabsalic}
                              \begin{cases}
 \eta^\tau_t= {L}(\eta^\tau)+ S^{t}_\varepsilon  (\tilde{w} )
 ,\\
\eta^\tau|_{t=0}=\eta^0 ,\\
\eta ^\tau_3|_{\partial\Omega}=0 ,
\end{cases}
\end{equation}
where $L:$ $H^{1,3}_{\mathrm{s}}\to H^{1,3}_{\mathrm{s}}$
by the rule $L(f)= \kappa D^{-\tau}_1 (D^\tau_1f)$ for $f\in  H^{1,3}_{\mathrm{s}}$, see  \eqref{202220105041846} for the definition of difference quotient $D^\tau_1$.

It is easy to see  that  $L(\varpi)\in  C^0(\overline{I_T},H^{1,3}_{\mathrm{s}})$ for $\varpi\in  C^0(\overline{I_T},H^{1,3}_{\mathrm{s}})$ and $L\in \mathcal{L}(H^{1,3}_{\mathrm{s}})$,
where $\mathcal{L}(H^{1,3}_{\mathrm{s}})$ is a set of all linear bounded operators of $H^{1,3}_{\mathrm{s}}$.
In particular, $ L\in  C^0(\overline{I_T},\mathcal{L}(H^{1,3}_{\mathrm{s}}))$.
By existence theory of the initial-value problem of a abstract ODE equation
(see \cite[Proposition 2.17]{NASII04}),
there exists a unique solution
$\eta^\tau\in C^0(\overline{I_T},H^{1,3}_{\mathrm{s}})\cap C^1( \overline{I_T},H^{1,3}_{\mathrm{s}})$ to \eqref{parabsalic}.
Obviously
$\eta^\tau$,  $\eta^\tau_t$ automatically belong to $L^2_TH^{1,3}_{\mathrm{s}}$
by the second conclusion in Lemma \ref{20021032019018} and the separability of $H^{1,3}_{\mathrm{s}}$.

Let $\alpha$ satisfy $0\leqslant |\alpha|\leqslant  3$.
Applying $\partial^\alpha $
to \eqref{parabsalic}$_1$, and then multiplying the resulting identity by $ \partial^\alpha  \eta^\tau$  in $L^2 $, we have
\begin{align}
 \frac{\mathrm{d}}{\mathrm{d}t} \int |\partial^{\alpha} \eta^\tau|^2\mathrm{d}y
+ \kappa\int |D_1^{\tau}\partial^{\alpha} \eta^\tau |^2\mathrm{d}y
  =\int S^{t}_\varepsilon ( \partial^{\alpha} \tilde{w}) \cdot
\partial^{\alpha} \eta^\tau \mathrm{d}y . \label{202008051043}
\end{align}
Making use of  \eqref{202210208asdfsa2248},
we easily deduce from \eqref{202008051043} that,
\begin{align}
\| \eta^\tau\|_{C^0(\overline{I_T},H^{3})}
\lesssim\| \eta^0 \|_{ 3}+  \sqrt{T}\| w\|_{L^2_TH^{3}}.  \label{202102082127}
\end{align}

In addition, we easily deduce from  \eqref{parabsalic}$_1$ that
\begin{align}\label{2020safda08051043}
& \frac{\kappa}{2}\frac{\mathrm{d}}{\mathrm{d}t}\| D_1^{\tau}\partial^{\alpha} \eta^\tau\|^2_0
+\int| \partial^{\alpha} \eta^\tau_t |^2\mathrm{d}y
=\int (\partial^{\alpha} S^{t}_\varepsilon (\tilde{w}))
\cdot  \partial^{\alpha} \eta^\tau_t \mathrm{d}y.
\end{align}
Noting that
\begin{align}
\label{2021032101049}
\| D_1^{\tau}\partial^{\alpha} \eta^\tau|_{t=0}\| _0
\lesssim \| \partial^{\alpha}\eta^0\| _{1 ,0},
\end{align}
thus, making use of  Young's inequality, \eqref{202210208asdfsa2248}, \eqref{parabsalic}$_1$ and \eqref{2021032101049}, we deduce from \eqref{2020safda08051043} that, for any $s\in (0,T] $
\begin{align}
 & {\kappa}\sup\nolimits_{t\in\overline{I_s}} \| D_1^\tau \eta^\tau \|_{3}^2
+ \| ({L}( \eta^\tau),  \eta^\tau_t)\|_{L^2_sH^{3}}^2
\lesssim  \kappa\| \eta^0 \|_{1 ,3}^2+  \| w\|_{{L^2_sH^{3}}}^2.  \label{202008051056}
\end{align}

Thanks to the regularity of $\eta^\tau$, we can we easily derive \eqref{parabsalic}$_1$ that, for any $\varphi\in H^1$ and for any $r$, $s\in I_T$,
\begin{align}
& \int D_1^\tau\partial^{\alpha}
(\eta^\tau(y,t) -  \eta^\tau(y,r))\cdot\varphi\mathrm{d}y \nonumber \\
&=
- \int_r^t\int ( S^{t}_\varepsilon ( \partial^{\alpha}  \tilde{w})   +
L(\partial^{\alpha}  \eta^\tau)) \cdot D_1^{-\tau}\varphi  \mathrm{d}y \mathrm{d}s.
\label{202210041111912}
\end{align}
Exploiting  \eqref{202210208asdfsa2248},
the uniform estimate of $L( \eta^\tau)$ in \eqref{202008051056} and the fact
$$\|D_1^{-\tau}\varphi\|_0\lesssim \| \varphi\|_{1,0} ,$$
we easily derive from the identity \eqref{202210041111912} that
\begin{align}
&\label{20202108021asdf27}
 D_1^{\tau}\nabla^3  \eta^\tau\mbox{ is uniformly continuous in }H^{-1} .
\end{align}
Here and in what follows, $H^{-1}$ denotes the dual space of $H^1_0:= \{w\in {H}^1~|~w|_{\partial\Omega} =0\} $.

Making use of \eqref{202102082127}, \eqref{202008051056}, \eqref{20202108021asdf27} and \eqref{202104141653asfda}--\eqref{202104141653} there exists a sequence of $\{\eta^\tau\}_{|\tau|\in (0,1)}$ (still denoted by $\eta^\tau$ for simplicity) such that, as $\tau\to 0$,
\begin{align}
&( \eta_t^\tau , D_1^{-\tau}(D_1^\tau \eta^\tau) )\rightharpoonup  (\eta_t^\varepsilon ,\partial_1^{2 } \eta^\varepsilon )\mbox{ weakly in }L^2_TH^{3}_{\mathrm{s}}, \
 \eta^\tau \to  \eta^\varepsilon  \mbox{ strongly in }
C^0(\overline{I_T},H^{2}), \nonumber  \\
& (\eta^\tau,D_1^\tau \eta^\tau ) \rightharpoonup (\eta^\varepsilon,\partial_1 \eta^\varepsilon) \mbox{ weakly-* in }L^\infty_TH^{3}_{\mathrm{s}} ,\
  D_1^\tau\nabla^3 \eta^\tau \to\nabla^3\partial_1 \eta^\varepsilon
 \mbox{ in } C^0(\overline{I_T},L^2_{\mathrm{weak}}) . \nonumber
\end{align}
Moreover, the limit function $\eta^\varepsilon$ is just the unique solution to the problem \begin{equation}
\nonumber
                              \begin{cases}
 \eta^\varepsilon_t=  \kappa\partial_1^2  \eta^\varepsilon + S^{t}_\varepsilon  (\tilde{w} )
  ,\\
\eta^\varepsilon|_{t=0}=\eta^0  ,\\
\eta ^\varepsilon_3|_{\partial\Omega}=0
\end{cases}
\end{equation} and satisfies the following estimates:
\begin{align}
&\sup\nolimits_{t\in \overline{I_T}} \| \eta^\varepsilon  \|_{ 3}
 \lesssim \|\eta^0 \|_{ 3} +  \sqrt{T} \|  w\|_{L^2_TH^{3}} , \label{20200805100sasfsdafadfsa5} \\
& {\kappa}\sup\nolimits_{t\in\overline{I_s}} \|  \eta^\varepsilon  \|_{1,3}^2
+ \|(\kappa\partial_1^2\eta^\varepsilon,\eta^\varepsilon_t )\|_{L^2_sH^{3}}^2
\lesssim  \kappa\| \eta^0 \|_{1 ,3}^2+  \| w\|_{{L^2_sH^{3}}}^2\mbox{ for any }s\in (0,T].\label{202fsa5}
\end{align}
Thus we further have $ \partial_1^2 \eta^\varepsilon \in C^0(\overline{I_T},H^2_{\mathrm{s}})$ and  $\eta^\varepsilon \in C^0(\overline{I_T}, H^{ 3}_{\mathrm{s}})$.

Noting that $\eta^{\varepsilon}$ enjoys the estimates \eqref{20200805100sasfsdafadfsa5} and \eqref{202fsa5}, thus we  get a limit function $\eta$, which satisfies the desired conclusion in Proposition \ref{pro:parabolic}, by taking limit of some sequence of $\{\eta^\varepsilon\}_{\varepsilon>0}$. We omit such limit process, since it is very similar to the argument of obtaining $\eta^\varepsilon$.
\hfill $\Box$
\end{pf}


\begin{pro}\label{pro:transport}
Let $a\in \mathbb{R}$, $K^2\in L^1_{T}H^{3}_{\mathrm{s}}$ and $u^0\in H^{3}_{\mathrm{s}} $, then the initial-boundary value problem \eqref{transport} admits a unique solution $u\in C^0(\overline{I_T},H^{3}_{\mathrm{s}})$, which satisfies
$u_t\in L^1_{T}H^{3}_{\mathrm{s}}$ and
\begin{align}
&\label{202104101021}
\|u(t)\|_{ 3}\lesssim\|u^0\|_{ 3}
+\int_0^t\|K^2(s)\|_{ 3}\mathrm{d}s.
\end{align}
\end{pro}
\begin{pf}
Let
\begin{align}\label{202008051415}
u=u^0e^{-at}+\int_0^tK^2(s)e^{-a(t-\tau)}\mathrm{d}s.
\end{align}
Since $K^2\in L^1_{T}H^{ 3}_{\mathrm{s}}$ and $u^0\in H^{  3}_{\mathrm{s}}$, it is easy to check that $u$ given by \eqref{202008051415} is the unique solution of \eqref{transport};
moreover, $u$ belongs to $C^0(\overline{I_T},H^{ 3}_{\mathrm{s}})$ and satisfies \eqref{202104101021}.
\hfill $\Box$
\end{pf}

\begin{pro}\label{pro:elliptic}
Let $T>0$,  $\iota$ be the positive constant provided in Lemma \ref{pro:1221}
and $\varsigma\in \mathbb{A}^{3,1/4}_{T,\iota }$  defined by \eqref{2022104161633}.
If $K^1\in C^0(\overline{I_T},H^1)$ and satisfies $\int  K^1\mathcal{J}\mathrm{d}y=0$ for $t\in \overline{I_T}$, then there exist a unique solution ${q}\in C^0(\overline{I_T},\underline{H}^3)  $, which satisfies the boundary-value problem \eqref{elliptic} for each $t\in I_T$ and enjoys the estimate
\begin{align}
\label{202008051safa442}
&\| q\|_{C^0(\overline{I_T}, {H}^3) }\lesssim P( \|{\varsigma}\|_{L^\infty_TH^{3}}) \|K^1\|_{L^\infty_TH^{1}} .
\end{align}

If we additionally assume that $K^1\in   L^r_TH^2$ for $r=2$ or $\infty$, then
\begin{align}\label{202008051442}
&\|\nabla_{{\mathcal{B}}}q\|_{L^r_TH^{3}}\lesssim P( \|{\varsigma}\|_{L^\infty_TH^{ 3}}) \|K^1\|_{L^r_TH^2} .
\end{align}
\end{pro}
\begin{pf}
 Let $\varphi=\varsigma+y $, then $\varphi$ is a $C^1$-diffeomorphism mapping by Lemma \ref{pro:1221}.
We denote $\varphi^{-1}$ the inverse mapping of $\varphi$ with respect to variable $y$, and then define $\tilde{K}^1:=K^1 (\varphi^{-1},t)$. Obviously $\tilde{K}^1\in  C^0(\overline{I_T}, H^{1})$ by \eqref{202104031901}.

Now we consider the following Neumann boundary-value problem of elliptic equation
\begin{align}\label{elliptic1}
\begin{cases}
-\Delta p=\tilde{K}^1&\mbox{in } \Omega,\\
\partial_3 p =0 &\mbox{on } \partial\Omega.
\end{cases}
\end{align}
It is easy to see that
\begin{align}
\int \tilde{K}^1\mathrm{d}x=\int K^1\mathcal{J}\mathrm{d}y=0 .
\nonumber
\end{align}

By Lemma \ref{lem:08181945}, there exists
a unique solution $p\in C^0(\overline{I_T},\underline{H}^3) $ of the boundary-value problem \eqref{elliptic1} such that
\begin{align}\label{2020080516sasafddfa42}
\|p \|_{ C^0(\overline{I_T}, {H}^3)}\lesssim \|\tilde{K}^1\|_{L^\infty_TH^{1}}.
\end{align}

Let $\tilde{q}=p(\varphi)$, then $\tilde{q}\in C^0(\overline{I_T}, {H}^3) $
by \eqref{2020103250855}. We further define that $ {q}:=\tilde{q}- (\tilde{q})_{\Omega} $, then $ {q} \in C^0(\overline{I_T}, \underline{H}^3)$ and satisfies  \eqref{202008051safa442} by \eqref{2020080516sasafddfa42} and \eqref{2022104101908}. Since $p$ satisfies \eqref{elliptic1}, we see that $q$ is the solution of the boundary-value problem \eqref{elliptic}. In addition, the uniqueness of the solution of the problem \eqref{elliptic} is obvious in the class $C^0(\overline{I_T}, \underline{H}^3)$.

If we additionally assume that $K^1\in   L^r_TH^{2}$, then  $\tilde{K}^1\in  L^r_TH^{2}$, where $r=2$ and $\infty$. By the second assertion in Lemmas \ref{20021032019018}--\ref{lem:08181945}, \eqref{202104032134} and the separability of $\underline{H}^3$,   we have $p\in L^r_TH^{3}$ and \begin{align}\label{2020080516sadfa42}
\|p \|_{L^r_TH^{4}}\lesssim P(\|{\varsigma}\|_{L^\infty_TH^{3}}) \|\tilde{K}^1\|_{L^r_TH^{2}}.
\end{align}
By \eqref{2020080516sadfa42}, \eqref{2021sfa04031901} and \eqref{2022104101908},
\begin{align}
&\|\nabla_{\mathcal{B}}q\|_{L^r_TH^{3}}=\|\nabla p(x,t)|_{x=\varphi(y,t)}\|_{L^r_TH^{3}}\nonumber \\
&\lesssim P(\|\varsigma\|_{L^\infty_TH^{3}}) \|\nabla p\|_{L^r_TH^{3}}\lesssim P(\|{\varsigma}\|_{L^\infty_TH^{3}}) \|{K}^1\|_{L^r_TH^{2}}, \nonumber
\end{align}
which yields \eqref{202008051442}.  \hfill $\Box$
\end{pf}

With Propositions \ref{pro:parabolic}--\ref{pro:elliptic} in hand, next we will use an
iterative method  to establish the existence result of a unique local solution to the linear $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx}.

\begin{pro}
\label{202102151544}
Let $\mathbb{A}^{3,1/4}_{\alpha ,\iota }$, resp.  $\mathbb{S}_{ \alpha  }$ be defined by \eqref{2022104161633} resp. \eqref{20210413asfda21251} with some positive constant $\alpha  $ in place of $T$.
 We assume that  $\kappa>0$,  $a\geqslant 0$,  $(u^0,\eta^0)\in  {H}^{3}_{\mathrm{s}}\times{H}^{1,3}_{\mathrm{s}} $  and $\varsigma\in\mathbb{A}^{3,1/4}_{\alpha ,\iota }$  satisfies  $ \varsigma_t \in C^0(\overline{I_{\alpha }},H^{2})\cap {L^{\infty}_{\alpha }H^{3}}$, then  the $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx} defined on $\Omega_\alpha$  admits a unique solution, denoted by $(u^\kappa,\eta^\kappa,q^\kappa)$, which belongs to $\mathbb{S}_{ \alpha  }$.
\end{pro}
\begin{pf}
Let $T  \leqslant \alpha $. In view of Propositions \ref{pro:parabolic}--\ref{pro:elliptic}, we easily see that there exists  a solution sequence $\{ (u^n,\eta^n,q^n)\}_{n=1}^\infty$ defined on $\overline{I_T}$. Moreover,
\begin{enumerate}
\item[(1)]  $(u^1,\eta^1, q^1)=0$;
\item[(2)]
$(u^n,\eta^n,Q^n)\in \mathbb{S}_{T  }$ for $n\geqslant 1$;
\item[(3)]
$(u^n,\eta^n,q^n)$ for $n\geqslant2$ enjoys the following relations:
\begin{equation}
\nonumber
                              \begin{cases}
{\eta}_t^{n}-\kappa\partial_1^2{\eta}^{n}={u}^{n-1},\\
{u}^{n}_t+a {u}^{n}= m   \partial_1^2{\eta}^{n-1}
-\nabla_{{\mathcal{B}}}{q}^{n-1}=:K^{2,n},\\
-\Delta_{{\mathcal{B}}}{q}^{n}
=a\mathrm{div}_{{\mathcal{B}}}{u}^{n}-\mathrm{div}_{\mathcal{B}_t}{u}^{n}
-\mathcal{J}_t\mathrm{div}_{{\mathcal{B}}}{u}^{n}/\mathcal{J}
-m  \mathrm{div}_{{\mathcal{B}}} \partial_1^2{\eta}^{n} =:K^{1,n}  , \\
({u}^{n},{\eta}^{n})|_{t=0}=(u^0,\eta^0) ,\\
( {u}^{n}_3,{\eta}^{n}_3)|_{\partial\Omega}=0 ,\
{\mathcal{B}}_{3j}\partial_j{q}^{n} |_{\partial\Omega} =0,
\end{cases}
\end{equation}
where $K^{1,n} \in  C^0(\overline{I_T  },H^1)\cap L^2_T  {H}^{2} $,
$K^{2,n} \in  L^2_{T  }H^{ 3}_{\mathrm{s}}$ and
\begin{align}
 \int  K^{1,n}\mathcal{J}\mathrm{d}y=0\mbox{ for each }t\in \overline{I_T} . \nonumber
\end{align}
It should be noted that we have used the relation $\partial_{j}(\mathcal{J} \mathcal{B}_{ij})=0$ for $i=1$, $2$, and the regularity $ \varsigma_t \in C^0(\overline{I_{\alpha }},H^1)$ in the derivation of the above identity.
\end{enumerate}

 We  further define
$(\bar{u}^{n+1},\bar{\eta}^{n+1}, \bar{q}^{n+1}):=(u^{n+1}-u^{n},\eta^{n+1}-\eta^{n},q^{n+1}-q^{n})$
and $(\bar{u}^{n},\bar{\eta}^{n}, \bar{q}^{n}):=(u^{n}-u^{n-1},\eta^{n}-\eta^{n-1},q^{n}-q^{n-1})$ for $n\geqslant2$.
Then we have
\begin{equation}\label{01dsaf16asdfasf00001}
                              \begin{cases}
\bar{\eta}_t^{n+1}-\kappa\partial_1^2\bar{\eta}^{n+1}=\bar{u}^{n},\\
\bar{u}^{n+1}_t+a \bar{u}^{n+1}= m   \partial_1^2\bar{\eta}^{n}
-\nabla_{{\mathcal{B}}}\bar{q}^{n},\\
-\Delta_{{\mathcal{B}}}\bar{q}^{n+1}
=a\mathrm{div}_{{\mathcal{B}}}\bar{u}^{n+1}-\mathrm{div}_{\mathcal{B}_t}\bar{u}^{n+1}
-\mathcal{J}_t\mathrm{div}_{{\mathcal{B}}}\bar{u}^{n+1}/\mathcal{J}
-m  \mathrm{div}_{{\mathcal{B}}}(\partial_1^2\bar{\eta}^{n+1})  , \\
(\bar{u}^{n+1},\bar{\eta}^{n+1})|_{t=0}=(0,0) ,\\
(\bar{u}^{n+1}_3,\bar{\eta}^{n+1}_3)|_{\partial\Omega}=0 ,\\
{\mathcal{B}}_{3j}\partial_{i}\bar{q}^{n+1} |_{\partial\Omega} =0
 .
\end{cases}
\end{equation}

Applying   Propositions \ref{pro:parabolic}--\ref{pro:elliptic} to \eqref{01dsaf16asdfasf00001}, and then using \eqref{202008121505} and \eqref{202008121535}, we get
\begin{align}
& \nonumber
\| \bar{\eta}^{n+1} \|_{C^0(\overline{I_T}, H^{3} )}
+\|\partial_1  \bar{\eta}^{n+1} \|_{L_{T}^{\infty}H^{3}}
+\| \partial_1^2\bar{\eta}^{n+1} \|_{L_{T}^2H^3}
\lesssim_{\kappa}T(1+\sqrt{T})\|\bar{u}^{n}\|_{L^\infty_TH^{3} },\\
&  \nonumber
\| \bar{u}^{n+1}  \|_{C^0(\overline{I_T}, H^{3} )}
\lesssim \sqrt{T} \|(\partial_1^2\bar{\eta}^{n},\nabla_{{\mathcal{B}}}\bar{q}^{n} )\|_{L_{T}^2H^{ 3}}, \\
&\|\bar{q}^{n+1}\|_{C^0(\overline{I_T}, H^{3} )}\lesssim_\kappa
P ( \|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})
\|(\bar{u}^{n+1},\partial_1^2\bar{\eta}^{n+1})\|_{L_{T}^\infty H^2}, \nonumber  \\
&\|\nabla_{{\mathcal{B}}}\bar{q}^{n+1}\|_{L_{T}^2H^{ 3}}\lesssim_\kappa
P ( \|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{ 3}})
\left( \sqrt{T}\|\bar{u}^{n+1}\|_{L_{T}^{\infty}H^{ 3}}+\|\partial_1^2\bar{\eta}^{n+1}\|_{L_{T}^2H^{ 3}}\right). \nonumber
\end{align}
In addition, by \eqref{01dsaf16asdfasf00001}$_1$  and \eqref{01dsaf16asdfasf00001}$_2$,
$$\|(\bar{\eta}_t^{n+1},\bar{u}_t^{n+1})\|_{C^0(\overline{I_T}, H^{2} )}\lesssim \|( \bar{u}^{n+1},\bar{u}^{n},\partial_1^2 \bar{\eta}^{n+1} , \partial_1^2\bar{\eta}^{n} ,\nabla  \bar{q}^{n} )\|_{L^\infty_TH^2} . $$

Thus we immediately see from the above five estimates that
$$\{ ( u^n, \eta^n,
q^n, \eta^n_t,u^{n}_t)\}_{n=1}^\infty$$ is a Cauchy sequence in
$ C^0(\overline{I_T}, H^{ 3}_{\mathrm{s}}) \times C^0(\overline{I_T}, H^{ 3}_{\mathrm{s}})\times
  C^0(\overline{I_T},\underline{H}^3)\times C^0(\overline{I_T}, H^{2}_{\mathrm{s}} )\times C^0(\overline{I_T}, H^{2}_{\mathrm{s}} )$.
Hence there exists one limit function $(u^\kappa,\eta^\kappa,q^\kappa)$, which is the unique local solution of  the linear  $\kappa$-approximate problem \eqref{01dsaf16asdfasf0safd000} with $(u^\kappa,\eta^\kappa,q^\kappa)$ in place of $(\eta ,u ,q)$, and  also the unique local solution of  the linear  $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx}.
In addition, it is easy to see that the solution $(u^\kappa,\eta^\kappa,q^\kappa)\in \mathbb{S}_{T  }$ by further using  Propositions \ref{pro:parabolic} and \ref{pro:elliptic}.

Noting that  the local time $T  _1$ is independent of the initial data and  the local solution constructed above satisfies $(u,\eta)|_{t=T  }\in {H}^{ 3}_{\mathrm{s}}\times {H}^{1,3}_{\mathrm{s}} $, thus, if $ T <\alpha $, we can further extend the local solution to be a global solution defined on $\overline{I_\alpha }$ by finite steps; moreover the obtained global solution is the unique solution of of  the linear  $\kappa$-approximate problem \eqref{01dsaf16asdfasf0000xx} and  belongs to $\mathbb{S}_{ \alpha  }$.
This completes the proof  of Proposition \ref{202102151544}.
\hfill$\Box$
\end{pf}

\subsection{Solvability of linearized  problem \eqref{01dsaf16asdfasf0000xx}}\label{subsubsce:03}

To investigate the solvability of linearized  problem \eqref{01dsaf16asdfsafasf0000}, we shall first $\kappa$-independent estimates of the solutions of the $\kappa$-approximate problem.

\begin{lem}\label{pro:0812}
Under the assumptions of Proposition \ref{202102151544},
we further assume that $\kappa \in (0,1)$, $\|\eta^0\|_{\underline{1},3}\leqslant \delta \in \mathbb{R}^+$, $(\partial_1 u^0, \partial_1^2 \eta^0) \in H^3$ and $\varsigma\in \mathbb{A}_{\alpha,\varsigma}^{3,1/4} $ satisfies
\begin{align}
& \partial_1\varsigma \in L^\infty_\alpha H^{1,3} ,\  \varsigma_t \in L^\infty_\alpha H^{1,3}\cap C^0(\overline{I_T}, H^2),\
\varsigma|_{t=0}=\eta^0 .   \nonumber
\end{align}
Then there exist polynomials
$P  ( I^0_H,\|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{\alpha}H^{3}} )$,
$P  ( \|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{\alpha}H^{1,3}} )$, positive constants $c$, $c_3\geqslant 1$,
and  $\delta_0>0$ (may depending on  $a$,  $m $ and $\Omega$),
such that, for any $\delta\leqslant \delta_0 $,
the local solution  $(\eta^\kappa,u^\kappa,q^\kappa)$ provided by Proposition \ref{202102151544} belongs to $\mathfrak{U}_{T_1}^{1,3}  \times \mathfrak{C}^0( \overline{I_{T_1}},{H}^{2,3}_{\mathrm{s}}) \times  C^0(\overline{I_{T_1}},\underline{H}^3)$
 and
enjoys the following estimates:
\begin{align}
&\label{2020081safdasaf21000}
\sup\nolimits_{t\in \overline{I_{T_1}}}\mathcal{E}_{\kappa}^L(t) +\kappa \| \partial_1^2\eta \|^2_{L^2_{T}H^{3}} \leqslant 4c_3^2{I^0_L}
\\
&\label{2020081safdasaf210000415}
{ \sup\nolimits_{t\in \overline{I_{T_1}}}\mathcal{E}^H_{\kappa}(t) \leqslant 4c_3^2{I^0_H}  }  ,\\
&\label{2020081609300416}
   \| q^\kappa\|_{C^0(\overline{I_{T_1}}, {H}^{3} ) } +\|(u_t^\kappa, \eta_t^\kappa,\nabla_{\mathcal{B}}q^\kappa )\|_{L^\infty_{T_1}H^3}\lesssim
P\left( I^0_H ,\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T_1}H^{3}}^2\right),
\end{align}
where $\mathcal{E}_{\kappa}^L(t):=\|(\eta^\kappa,\partial_1\eta^\kappa,u^\kappa)(t)\|_{3}^2$, $\mathcal{E}^H_{\kappa}(t):=\|(u^\kappa,\partial_1\eta^\kappa)(t)\|_{\underline{1},3}^2$ and
\begin{align}
T_1:=\min\{ 1/3c_5 P  ( \|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{\alpha}H^{1,3}} ) ,\alpha, 1\}   .
\label{20220103021516}
\end{align}
Moreover, for $(\eta^\kappa,u^\kappa)$   restricted on $[0,T_1]$,
\begin{align}
&\partial_1 \nabla^3(u^\kappa,  \partial_1\eta^\kappa )
\mbox{ is uniformly continuous in }H^{-1}  . \label{202103021540}
\end{align}
\end{lem}
\begin{pf}Let $T\leqslant \min\{\alpha,1\}$,
\begin{align}
\|\eta^0\|_{\underline{1},3}\leqslant \delta\in (0,1].
\label{20220010504202237}
 \end{align}
 We define that
 $$\|\cdot \|_{*,i}^2:=\|(\cdot ,D^\tau_1\cdot) \|_{ i}^2 \mbox{ and }  \mathcal{E}^\tau_{\kappa}(t):=\|(u^\kappa,\partial_1\eta^\kappa)(t)\|_{*, 3}^2.$$ In addition, we denote  $(u^{\kappa},\eta^{\kappa},{q}^{\kappa})$
  by $(u, \eta,{q})$ for simplicity.
Next we establish the desired uniform estimates for $(u,\eta,q)$  by seven steps.

(1) \emph{Estimates of $\eta$}

Recalling \eqref{20200805100sadfsa5}, it is easy to see that
\begin{align}
&\sup\nolimits_{t\in\overline{I_T}}\| \eta (t)\|_{3}^2+\kappa^2\| \partial_1^2\eta \|^2_{L^2_{T}H^{3}} \lesssim
I^0_L +T\sup\nolimits_{t\in\overline{I_T}}\|u(t)\|_3^2.\label{202104101103}
\end{align}


By \eqref{20200805DSAFS100sadfsa5}, we have
\begin{align}
&\limsup_{s\to 0^+}\sup_{t\in\overline{I_s}}\| \eta (t)\|_{1,3}^2  \lesssim
 \| \eta^0\|_{1,3}^2.\label{20210410sdaf1103}
\end{align}
Since
\begin{equation}\label{parabsadfolic}
                              \begin{cases}
D^\tau_1\eta_t-\kappa D^\tau_1\partial_1^2\eta=D^\tau_1 w ,\\
D^\tau_1\eta|_{t=0}=\eta^0  ,\\
D^\tau_1 \eta_3|_{\partial\Omega} =0,
\end{cases}
\end{equation}
applying Proposition \ref{pro:parabolic} to \eqref{parabsadfolic} yields that
\begin{align}
&\limsup_{s\to 0^+}\sup_{t\in\overline{I_s}}\| \partial_1  D^\tau_1 \eta (t)\|_{3}^2  \lesssim
 \| \eta^0\|_{2,3}^2.\label{2021041sdaf0sdaf1103}
\end{align}

(2) \emph{Estimates of $q$}

Noting that $q$ satisfies
\begin{equation}\label{202104152037}
                              \begin{cases}
-\Delta_{{\mathcal{B}}}q=K^1&\mbox{in } \Omega,\\
{\mathcal{B}}_{3j}\partial_{i}q=0  &\mbox{on } \partial\Omega,
\end{cases}
\end{equation}
where $K^1$ is defined in \eqref{202102141605}. Applying the estimate \eqref{202008051safa442} to the above boundary-value problem, and then using \eqref{202008121505} and \eqref{202008121535}, we have
\begin{align}
\label{202008121055}
\|{q}\|_{C^0(\overline{I_T}H^3)}
&\lesssim P\left( \|{\varsigma}\|_{L^{\infty}_{T}H^3}\right)\|K^1\|_{L^{\infty}_{T}H^1} \lesssim P\left( \|({\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{T}H^3}\right)\sup\nolimits_{t\in\overline{I_T}}
\|(u,\partial_1^2\eta)(t)\|_2.
\end{align}

(3) \emph{$L^2$-norm estimates of $(u,\partial_1\eta)$}.

Applying $\mathcal{X}$ to  \eqref{01dsaf16asdfasf0000xx}$_2$ yields that
\begin{align}\label{202104091638}
\partial_t\mathcal{X} u+a\mathcal{X} u=m \mathcal{X}\partial_1^{ 2}\eta-\mathcal{X}\nabla_{\mathcal{B}}q
\end{align}
Multiplying \eqref{202104091638} by $\mathcal{X} u$ in $L^2$   yields that (for a.e. $t\in I_T$)
\begin{align}\label{202008121145}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|\mathcal{X} u|^2\mathrm{d}y
+\int(a |\mathcal{X}u|^2\mathrm{d}y-m (\mathcal{X} \partial_1^{ 2}\eta)\cdot \mathcal{X} u)\mathrm{d}y
=-\int (\mathcal{X}\nabla_{{\mathcal{B}}}{q})\cdot \mathcal{X}u\mathrm{d}y.
\end{align}
By \eqref{202008121505}, \eqref{202008121535n} and \eqref{202008121055}, we can estimate that
\begin{align}\label{202008121207}
-\int \mathcal{X}\nabla_{{\mathcal{B}}}{q}\cdot \mathcal{X} u\mathrm{d}y
 \lesssim
\|\mathcal{B}\|_{\underline{1},2}\|\nabla {q}\|_{\underline{1},0}\| u\|_{1,0}
 \lesssim P\left(  \|({\varsigma}, \partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3}\right)
 \sup\nolimits_{t\in\overline{I_T}}\|(u, \partial_1^2\eta) \|_2^2.
\end{align}

In addition, by \eqref{01dsaf16asdfasf0000xx}$_1$ and the integration by parts, we get
\begin{align}\label{202008121210}
\begin{aligned}
-m   \int (\mathcal{X}\partial_1^{ 2}\eta)\cdot  \mathcal{X} u\mathrm{d}y
&=\frac{m   }{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|\mathcal{X}\partial_1\eta|^2\mathrm{d}y
+m   \kappa\int|\mathcal{X}\partial_1^{2}\eta|^2\mathrm{d}y.
\end{aligned}
\end{align}
Plugging \eqref{202008121207}--\eqref{202008121210} into \eqref{202008121145} and then integrating the resulting inequality over $(0,t)$, we arrive at
\begin{align}
&
\|\mathcal{X} \left(u, \partial_1\eta\right)(t)\|_{0}^2
+{  \kappa}\int_0^t\|\mathcal{X} \partial_1^2\eta(s)\|_{0}^2\mathrm{d}s\nonumber \\
&\lesssim \| \left(u^0, \partial_1\eta^0\right)\|_{\underline{1},0}^2 + TP\left( \|({\varsigma}, \partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3}\right)
\sup\nolimits_{t\in\overline{I_T}}  \|(u, \partial_1^2\eta)(t)\|_2^2.\label{202008121220}
\end{align}

(4) \emph{Curl estimates of $(u,\partial_1\eta)$.}

Applying $\mathrm{curl}_{{\mathcal{B}}}$ to \eqref{01dsaf16asdfasf0000xx}$_2$  yields that
\begin{align}\nonumber
\partial_t\mathrm{curl}_{{\mathcal{B}}}u+a\mathrm{curl}_{{\mathcal{B}}}u
=m  \mathrm{curl}_{{\mathcal{B}}} \partial_1^2\eta
+\mathrm{curl}_{ {{\mathcal{B}}}_t}u.
\end{align}
Let the multiindex $\alpha$ satisfy $|\alpha|\leqslant2$. Applying $\partial^{\alpha}\mathcal{X}$
to the above identity yields
\begin{align}\label{202005021542nnbm}
\begin{aligned}
\partial_t\partial^{\alpha} \mathcal{X} \mathrm{curl}_{{\mathcal{B}}}u
+a\partial^{\alpha}\mathcal{X} \mathrm{curl}_{{\mathcal{B}}}u
=&{m  }\partial_1\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}} \partial_1\eta
+K^{\mathcal{X} ,\alpha}_3 ,
\end{aligned}
\end{align}
where $K_3^{\mathcal{X},\alpha}:=
\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{{\mathcal{B}}}_{t}} u-m  \partial^{\alpha}\mathcal{X}
\mathrm{curl}_{\partial_1{{\mathcal{B}}}} \partial_1\eta
$.



Multiplying \eqref{202005021542nnbm} by
$\partial^{\alpha}\mathcal{X}\mathrm{curl}_{ {\mathcal{B}}}u$ in $L^2$, we obtain
\begin{align}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int|\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}u|^2\mathrm{d}y
+a\int|\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}u|^2\mathrm{d}y
\nonumber\\
&-m\int {  }\left(\partial_1\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}\partial_1\eta\right)
\cdot\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}u\mathrm{d}y =\int K_3^{\mathcal{X},\alpha}\cdot \partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}} u\mathrm{d}y
=:I_8 .\label{2020081214246}
\end{align}
Using the integration by parts, \eqref{01dsaf16asdfasf0000xx}$_1$ and a regularity method, we obtain
\begin{align}
&-m  \int { }\left(\partial_1\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}\partial_1\eta\right)
\cdot\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}u\mathrm{d}y \nonumber \\
&=\frac{m }{2}\frac{\mathrm{d}}{\mathrm{d}t}
\int|\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}} \partial_1\eta |^2\mathrm{d}y
+{m  \kappa} \int|\partial^{\alpha}\mathcal{X}\mathrm{curl} \partial_1^2\eta |^2\mathrm{d}y
+ I_{9}+I_{10}, \label{202008121536}
\end{align}
where we have defined that
$$
\begin{aligned}
&I_{9}:=m \int(\partial^{\alpha} \mathcal{X}\mathrm{curl}_{{\mathcal{B}}} \partial_1\eta )\cdot
\partial^{\alpha}\mathcal{X}\left(\mathrm{curl}_{\partial_1{\mathcal{B}}}u
-\mathrm{curl}_{\partial_t{\mathcal{B}}} \partial_1\eta \right)\mathrm{d}y,\\
&I_{10}:=
 m  \kappa
\int\bigg(\left(\partial^{\alpha} \mathcal{X}\mathrm{curl}_{\partial_1{\mathcal{B}}}\partial_1\eta\right)
 \cdot\partial^{\alpha} \mathcal{X}\mathrm{curl}_{{\mathcal{B}}} \partial_1^2\eta
+ |\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}-I} \partial_1^2\eta |^2
 \\
&\qquad\quad + 2 \left(\partial^{\alpha}\mathcal{X}\mathrm{curl}_{{\mathcal{B}}-I}\partial_1^2\eta\right)
\cdot\partial^{\alpha}\mathcal{X}\mathrm{curl} \partial_1^2\eta \bigg)\mathrm{d}y.
\end{aligned}
$$


By the estimates \eqref{202008121505}, \eqref{202008121535} and \eqref{202008121535n}, we can estimate that,
\begin{itemize}
  \item for the case  $\mathcal{X}=\mathrm{Id}$,
\begin{align}
&\label{202104152110}
I_{8}+I_{9}
 \lesssim P(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})
 \|(u,\partial_1\eta)\|_3^2,\\
&\label{202104152111}
I_{10}\lesssim \kappa \big( P(\| {\varsigma} \|_{L^{\infty}_{T}H^{1,3}})
\| \eta\|_{1,3}(\|\mathrm{curl}  \eta \|_{2,2}+  \|{\mathcal{B}}-I\|_2\| \eta\|_{2,3})
\nonumber \\
&\qquad \quad+
\|{\mathcal{B}}-I\|_2\left(\|{\mathcal{B}}-I\|_2\| \eta \|_{2,3}^2
+  \|  \eta \|_{2,3}\|\mathrm{curl}  \eta \|_{2,2}\right)\big).
\end{align}
  \item for the case  $\mathcal{X}=D^\tau_1$,
\begin{align}
&\label{202104152112}
I_{8}+I_{9}
 \lesssim P(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})
 \|(u,\partial_1\eta)\|_{*,3}^2,\\
&\label{202104152114}
 I_{10} \lesssim  \kappa \big( P(\| {\varsigma} \|_{L^{\infty}_{T}H^{2,3}})
\|\partial_1\eta\|_{*,3} (\|\mathrm{curl}  \partial_1^2\eta \|_{*,2}+\|{\mathcal{B}}-I\|_{\underline{1},2}\| \partial_1^2\eta\|_{*,3})
\nonumber \\
&\qquad \quad+
\|{\mathcal{B}}-I\|_{\underline{1},2}(\|{\mathcal{B}}-I\|_{\underline{1},2}\|\partial_1^2 \eta \|_{*,3}^2
+  \|\partial_1^2 \eta \|_{*,3}\|\mathrm{curl} \partial_1^2\eta  \|_{*,2})\big).
\end{align}
\end{itemize}

Noting that $\varsigma|_{t=0}=\eta^0$, by Newton--Leibniz formula,   \eqref{202008121535nnn} with $t=0$, \eqref{202008121535}  and \eqref{20220010504202237}, we have
\begin{align}
\| ({\mathcal{B}}(t)-I)\|_{\underline{i},2}
&\lesssim
\left\| ({\mathcal{B}}^0-I)\right\|_{\underline{i},2}+
\int_0^t\left\| {\mathcal{B}}_{\tau}(s)\right\|_{\underline{i},2}\mathrm{d}s\nonumber\\
&\lesssim
\begin{cases}
 \|\eta^0\|_{3}
+T   P(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{3}})
& \mbox{for } i=0; \\
  \|\eta^0\|_{ {1},3}
+T   P(\|({\varsigma},   {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})
 & \mbox{for }i=1.
\end{cases}
\label{202103020944}
\end{align}
Making use of the above estimates \eqref{202104152110}--\eqref{202103020944}, Young's inequality and the fact ``$\kappa\in(0,1)$",   we deduce from \eqref{2020081214246}--\eqref{202008121536}  that
\begin{align}
& \frac{\mathrm{d}}{\mathrm{d}t}
 \|\mathrm{curl}_{ \mathcal{B}} ( u ,  \partial_1\eta)\|_{2}^2
+{c \kappa}\| \mathrm{curl} \eta \|_{2,2}^2\nonumber \\
&\lesssim P( \|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})
 \|(u,\partial_1\eta)\|_3^2\nonumber
\\
&\quad + \kappa \left(\|\eta^0\|_{3}
+T {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{3}})\right)
\| \eta\|_{2,3}^2 \label{202008121724}
\end{align}
and
\begin{align}\label{202104101500}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}
 \|\mathrm{curl}_{{\mathcal{B}}} (u ,  \partial_1\eta )\|_{*,2}^2
+{c \kappa}\| \mathrm{curl} \partial_1^2\eta \|_{*,2}^2\nonumber \\
&\lesssim P( \|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})
\|(u,\partial_1\eta)\|_{*,3}^2
\nonumber \\
&\quad +  \kappa  \left(\|\eta^0\|_{\underline{1},3}
+T  {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\right)
\|\partial_1^2\eta\|_{*,3}^2.
\end{align}

In addition, similarly to \eqref{202103020944}, we have
\begin{align}
\|\mathcal{X}\mathrm{curl}f(t)\|_{ 2}&\lesssim\left \|\mathcal{X}
\left(\mathrm{curl}_{{\mathcal{B}}^0-I}f(t),
\mathrm{curl}_{{\mathcal{B}}}f(t),\mathrm{curl}_{\int_0^t {\mathcal{B}}_{s}\mathrm{d}s}f(t)\right)\right\|_{ 2}
\nonumber \\
&\lesssim \|\mathcal{X}\mathrm{curl}_{{\mathcal{B}}}f(t)\|_{ 2}\nonumber \\
&\quad +
\begin{cases}
 (\|\eta^0\|_{3}
+ T {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{3}}))\|f(t)\|_{3}
 & \mbox{for } \mathcal{X}=\mathrm{Id}; \\
 (\|\eta^0\|_{\underline{1},3}
+T {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\|f(t)\|_{*,3}
 &  \mbox{for }\mathcal{X}=D^\tau_1.
\end{cases}
\nonumber
\end{align}
Integrating \eqref{202008121724} and \eqref{202104101500} over $(0,t)$, resp.,
and then using \eqref{20220010504202237}, \eqref{20210410sdaf1103} and the above estimate, we conclude that
\begin{align}
&\|(\mathrm{curl}u,\mathrm{curl}\partial_1\eta)(t)\|_{ 2}^2
+{  \kappa}\int_0^t\| \mathrm{curl}  \eta (s)\|_{2,2}^2\mathrm{d}s\nonumber \\
&\lesssim    I^0_L +(\|\eta^0\|_3^2+T
P(  \|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{T}H^3}))
\sup\nolimits_{t\in\overline{I_T}} \|(u,\partial_1\eta)(t)\|_3^2\nonumber \\
&\quad+ \kappa  \left(\|\eta^0\|_{3}
+T  {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})\right)
 \int_0^t\| \eta(s)\|_{2,3}^2\mathrm{d}s
\label{202008121745}
\end{align}
and
\begin{align}\label{202104101434}
&\|(\mathrm{curl}u,\mathrm{curl}\partial_1\eta)(t)\|_{*,2}^2
+{   \kappa}\int_0^t\| \mathrm{curl} \partial_1^2\eta (s)\|_{*,2}^2\mathrm{d}s\nonumber \\
&\lesssim  I^0_H
+( \|\eta^0\|_{\underline{1},3}+TP(\|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}}))
\sup\nolimits_{t\in\overline{I_T}}\ \|(u,\partial_1\eta)(t)\|_{*,3}^2\nonumber \\
&\quad+ \kappa P\left(\|\eta^0\|_{\underline{1},3}
+T   (\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\right)
 \int_0^t\|\partial_1^2\eta(s)\|_{*,3}^2\mathrm{d}s.
\end{align}

(5) \emph{Divergence estimates of $(u,\partial_1\eta)$.}

Similarly to \eqref{202103020944},   we have
\begin{align}
\|\mathcal{X}\mathrm{div}f(t)\|_{ 2}\lesssim \left\|\mathcal{X}\left(\mathrm{div}_{{\mathcal{B}}^0-I}f(t),\mathrm{div}_{{\mathcal{B}}}f(t),\mathrm{div}_{\int_0^t  {\mathcal{B}}_{\tau} \mathrm{d}\tau } f(t)\right) \right\|_{ 2} . \nonumber
\end{align}
Noting that $\mathrm{div}_{{\mathcal{B}}}u=0$, thus taking $f=u$ in the above estimate yields
\begin{align}
\|\mathcal{X}\mathrm{div}u \|_{ 2}
\lesssim &
\begin{cases}
(\|\eta^0\|_{ 3}
+T {P}(\|({\varsigma},{\varsigma}_t)\|_{L^{\infty}_{T}H^3}))
\|u\|_{3}
& \mbox{for } \mathcal{X}=\mathrm{Id} ; \\
(\|\eta^0\|_{\underline{1},3}
+T {P}(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}}))
\|u\|_{*,3}
 & \mbox{for }\mathcal{X}=D^\tau_1.
\end{cases}\label{202008121802}
\end{align}

Applying $\mathrm{div}_{{\mathcal{B}}}\partial_1$ to \eqref{01dsaf16asdfasf0000xx}$_1$
and then use \eqref{01dsaf16asdfasf0000xx}$_3$ yields
\begin{align}
 \partial_t \mathrm{div}_{{\mathcal{B}}}\partial_1\eta
-\kappa\partial_1\mathrm{div}_{{\mathcal{B}}}\partial_1^2\eta =\mathrm{div}_{ {\mathcal{B}_t}}\partial_1\eta-\mathrm{div}_{\partial_1{\mathcal{B}}}u
-\kappa\mathrm{div}_{\partial_1{\mathcal{B}}}\partial_1^2\eta=:K^{ 4}. \nonumber
\end{align}
We further derive from the above identity that
\begin{align}
& \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\| \partial^{\alpha }\mathcal{X}\mathrm{div}_{\mathcal{B}}
\partial_1\eta   \|^2_0 +
\kappa  \| \partial^\alpha\mathcal{X} \mathrm{div}_{{\mathcal{B}}}\partial_1^2\eta  \|_2^2  \nonumber \\
&= \int \big( \partial^{\alpha } \mathcal{X}K^4 \partial^\alpha\mathcal{X} \mathrm{div}_{{ \mathcal{B}}}\partial_1\eta
-\kappa \partial^\alpha\mathcal{X} \mathrm{div}_{{\mathcal{B}}}(\partial_1^2\eta \partial^\alpha\mathcal{X} \mathrm{div}_{{\partial_1 \mathcal{B}}}\partial_1\eta )\big)\mathrm{d}y .  \label{202104112141}
\end{align}
Following the arguments of \eqref{202008121745} and \eqref{202104101434} by further using \eqref{20210410sdaf1103} and \eqref{2021041sdaf0sdaf1103},  we can derive from
\eqref{202104112141} that
\begin{align}
&\|\mathrm{div} \eta(t)\|_{1,2}^2
+ {\kappa} \int_0^t\|\mathrm{div} \eta(s)\|_{2,2}^2\mathrm{d}s\nonumber \\
&\lesssim \| \eta^0\|_{1,3}
+(\|\eta^0\|_3+TP(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})\sup\nolimits_{t\in\overline{I_T}} \|(u,\partial_1\eta)(t)\|_3^2
\nonumber \\
&\quad
+\kappa \left(\|\eta^0\|_3+T P(\|({\varsigma},\varsigma_t)\|_{L^{\infty}_{T}H^3})\right)
\int_0^t\| \eta(s)\|_{2,3}^2\mathrm{d}s\nonumber \\
&\quad
+\kappa   P(\| {\varsigma} \|_{L^{\infty}_{T}H^{1,3}})
\int_0^t \|  \eta(s)\|_{1,3} \|  \eta(s)\|_{2,3}\mathrm{d}s  \label{202008122100}
\end{align} and
\begin{align}
&\| \mathrm{div}\partial_1\eta(t)\|_{*,2}^2
+ {\kappa} \int_0^t\|\mathrm{div}\partial_1^2 \eta(s)\|_{*,2}^2\mathrm{d}s\nonumber \\
&\lesssim \| \partial_1\eta^0\|_{\underline{1},3} +(\|\eta^0\|_{\underline{1},3}+T
P(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\sup\nolimits_{t\in\overline{I_T}} \|(u,\partial_1\eta)(t)\|_{*,3}^2
\nonumber \\
&\quad
+\kappa  \left(\|\eta^0\|_{\underline{1},3}
+T P(\|({\varsigma}, \varsigma_t)\|_{L^{\infty}_{T}H^{1,3}})\right)
\int_0^t\|\partial_1^2\eta(s)\|_{*,3}^2\mathrm{d}s\nonumber \\
&\quad
+\kappa   P(\| {\varsigma} \|_{L^{\infty}_{T}H^{2,3}})
\int_0^t\|\partial_1 \eta(s)\|_{*,3}\|\partial_1^2\eta(s)\|_{*,3} \mathrm{d}s .\label{202104101601}
\end{align}

Consequently,  we immediately deduce from \eqref{202008121745}--\eqref{202008121802}, \eqref{202008122100} and \eqref{202104101601} that
\begin{align}
&\|(\mathrm{div}u,\mathrm{curl}u,\mathrm{div}\partial_1\eta, \mathrm{curl}\partial_1\eta)(t)\|_{2}^2
+ {\kappa}\int_0^t\|(\mathrm{div} \eta,
\mathrm{curl} \eta)(s)\|_{2,2}^2\mathrm{d}s\nonumber \\
&\lesssim  I^0_L + (\|\eta^0\|_3+TP(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})\sup\nolimits_{t\in\overline{I_T}}
\|(u,\partial_1\eta)(t)\|_{3}^2\nonumber \\
&\quad+   \kappa \left(\|\eta^0\|_{3}+T P(\|({\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^3})\right)
\int_0^t\|\eta(s)\|_{2,3}^2
\mathrm{d}s\nonumber   \\ &\quad+\kappa   P(\| {\varsigma} \|_{L^{\infty}_{T}H^{1,3}})
\int_0^t \|  \eta(s)\|_{1,3} \|  \eta(s)\|_{2,3}\mathrm{d}s \label{202008122105}
\end{align}
and
\begin{align}\label{202104101622}
&\|(\mathrm{div}u,\mathrm{curl}u,\mathrm{div}\partial_1\eta, \mathrm{curl}\partial_1\eta)(t)\|_{*,2}^2
+ {\kappa}\int_0^t\|(\mathrm{div}\partial_1^2\eta,
 \mathrm{curl}\partial_1^2\eta)(s)\|_{*,2}^2\mathrm{d}s \nonumber \\
&\lesssim  I^0_H +(\|\eta^0\|_{\underline{1},3}+TP(\|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\sup\nolimits_{t\in\overline{I_T}
}\|(u,\partial_1\eta)(t)\|_{*,3}^2\nonumber \\
&\quad+ \kappa \left(\|\eta^0\|_{\underline{1},3}+T P(\|({\varsigma} , {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})\right)
\int_0^t\| \partial_1^2\eta (s)\|_{*,3}^2
\mathrm{d}s\nonumber\\
&\quad
+\kappa   P(\| {\varsigma} \|_{L^{\infty}_{T}H^{2,3}})
\int_0^t\|\partial_1 \eta(s)\|_{*,3}\|\partial_1^2\eta(s)\|_{*,3} \mathrm{d}s .
\end{align}


(6) \emph{Summing up the estimates of $(u,\eta,{q})$.}

Thanks to the estimates \eqref{202104101103},
\eqref{202008121220},
\eqref{202008122105}, {\eqref{202104101622}},
 the Hodge-type elliptic estimate \eqref{08171537} and Young's inequality, we have, \emph{for sufficiently small $\delta$},
\begin{align}
&\sup\nolimits_{t\in\overline{I_T}}\mathcal{E}_{\kappa}^L(t)
+  \kappa
\|\partial_1^2\eta^\kappa \|_{L^2_{{T}}H^{3}}^2\nonumber \\
&\leqslant c_5I^0_L
+c_5TP( \|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{3}})
\bigg(\sup\nolimits_{t\in\overline{I_T}}\mathcal{E}_{\kappa}^L(t)
+  \kappa \|\partial_1^2\eta^\kappa \|_{L^2_{{T}}H^{3}}^2\bigg) \label{202008122137}
\end{align}
and
\begin{align}\label{202104101640}
&\sup\nolimits_{t\in\overline{I_T}}\mathcal{E}^{\tau}_{\kappa}(t)
+ \kappa
\int_0^t \|\partial_1^2\eta^\kappa (s )\|_{*,3}^2\mathrm{d}s\nonumber \\
&\leqslant c_5I^0_H +c_5TP ( \|({\varsigma},\partial_1{\varsigma}, {\varsigma}_t)\|_{L^{\infty}_{T}H^{1,3}})
\bigg(\sup\nolimits_{t\in\overline{I_T}}\mathcal{E}^{\tau}_{\kappa}(t)
+\ \kappa \int_0^t \| \partial_1^2\eta^\kappa (s )\|_{*,3}^2\mathrm{d}s\bigg).
\end{align}

Obviously, we can choose a new  polynomial, still denoted by $P  ( \|({\varsigma},\partial_1{\varsigma},  {\varsigma}_t)\|_{L^{\infty}_{\alpha}H^{1,3}} )$, which is not less than the two polynomials in \eqref{202008122137} and \eqref{202104101640}.
Now we use $c_5$ and the new polynomial $P$ to define $T_1$ by \eqref{20220103021516}. Taking $T={T}_1$ in \eqref{202008122137} and \eqref{202104101640},  thus we get \eqref{2020081safdasaf21000} and
\begin{align}
&\sup\nolimits_{t\in\overline{I_T}}\mathcal{E}^{\tau}_{\kappa}(t)+ \kappa
\int_0^t \|\partial_1^2\eta^\kappa (s )\|_{*,3}^2\mathrm{d}s \leqslant  4c_3^2{I^0_H}   , \nonumber
\end{align}
which, together with \eqref{202104141653asfda}, yields that
\begin{align}
&\mathrm{ess}\sup\nolimits_{t\in {I_T}}\mathcal{E}_{\kappa}^H(t)  +    \kappa
\|\partial_1^2\eta^\kappa \|_{L^2_{{T}}H^{1,3}}^2\leqslant  4c_3^2{I^0_H}   .
\label{202105072000}
\end{align}
We further derive from \eqref{01dsaf16asdfasf0safd000}$_1$, \eqref{01dsaf16asdfasf0safd000}$_2$,  \eqref{202105072000} and the fact $(u_3,\partial_1^2\eta_3, \mathcal{B}_{3j}\partial_j q)|_{\partial\Omega}=0$ that
\begin{align}
\eta\in  C^0(\overline{I_T}, H^{1,3})\mbox{ and }u_t\in  C^0(\overline{I_T}, H^{2}_{\mathrm{s}}).
\label{2022010500720046}
\end{align}

By \eqref{2020081safdasaf21000} and \eqref{202008121055}, we get
\begin{align}
&\label{202008160942}
\|q\|_{C^0(\overline{I_T},H^3)}^2\lesssim
P\left(I^0_H , \|({\varsigma},   {\varsigma}_t {)}\|_{L^{\infty}_{T}H^3}^2\right).
\end{align}
Applying   \eqref{202008051442} to \eqref{202104152037}, and then using \eqref{202105072000}, we have
\begin{align}
\label{2020081safas21055}
 \|\nabla_{{\mathcal{B}}}q\|_{L^\infty_TH^{3}}
&\lesssim P\left( \|{\varsigma}\|_{L^{\infty}_{T}H^3}\right)\|K^1\|_{L^{\infty}_{T}H^2} \lesssim P\left(I^0_H , \|({\varsigma},   {\varsigma}_t {)}\|_{L^{\infty}_{T}H^3}^2\right).
\end{align}

In addition, exploiting \eqref{202105072000} and \eqref{2020081safas21055}, we deduce from \eqref{01dsaf16asdfasf0000xx}$_1$ and \eqref{01dsaf16asdfasf0000xx}$_2$  that
\begin{align}\label{202008asf160946}
\|(u_t,\eta_t)\|_{L^\infty_TH^3}
\lesssim
P\left(I^0_H , \|({\varsigma}, {\varsigma}_t {)}\|_{L^{\infty}_{T}H^3}^2\right).
\end{align}
Putting the estimates \eqref{202008160942}--\eqref{202008asf160946} together yields \eqref{2020081609300416} immediately.

(7)  \emph{Estimates of uniform  continuity of $\partial_1 \nabla^3(u^\kappa,  \partial_1\eta^\kappa )$ in $H^{-1}$. }

Let the multiindex $\beta$ satisfy $ |\beta|=  3$.
Similarly to \eqref{202210041111912}, we easily deduce from \eqref{01dsaf16asdfasf0000xx}$_1$ and \eqref{01dsaf16asdfasf0000xx}$_2$ that, for any $\varphi\in H^1 $ and for any $t$, $r\in I_T$,
\begin{align}
&\int\partial^\beta D_1^\tau\partial_1 (\eta(y,t)- \eta(y,r))
\cdot\varphi\mathrm{d}y=-\int_r^t\int\partial^\beta D_1^\tau(  u + \kappa \partial_1^2 \eta) \cdot \partial_1\varphi\mathrm{d}y\mathrm{d}s,  \label{2022104191625}\\
&  \label{2022104191625xxx}
\int \partial^\beta D_1^\tau( u (y,t) - u(y,r))  \cdot \varphi\mathrm{dy}\nonumber \\
&= \int_r^t\int
 ( \partial^{\beta }\nabla_{\mathcal{B}}q \cdot D_1^{-\tau} \varphi
- m  \partial^\beta D_1^\tau\partial_1 \eta \cdot \partial_1 \varphi
-a\partial^\beta D_1^\tau u  \cdot  \varphi )
\mathrm{d}y\mathrm{d}s.
\end{align}


Exploiting the uniform estimates \eqref{202105072000} and  \eqref{2020081safas21055}, we easily deduce that
\begin{align}
& D_1^\tau\nabla^3(u ,  \partial_1\eta  )
\mbox{ is uniformly continuous in }H^{-1} . \label{20210302asfd1540}
\end{align}
In addition, it is easy to check that
\begin{align}
\|D_1^\tau\nabla^3(u ,  \partial_1\eta  )\|_{L^\infty_TL^2}\lesssim\| \partial_1\nabla^3(u ,  \partial_1\eta  )\|_{L^\infty_TL^2} . \label{2021030safda2asfd1540}
\end{align}
Making use of \eqref{20210302asfd1540}, \eqref{2021030safda2asfd1540}, \eqref{2022205050220333} and the regularity $(u,\partial_1\eta )\in  C^0(\overline{I_T}, H^{3})$, we arrive at
\begin{align}
& \partial_1\nabla^3(u ,  \partial_1\eta )\in C^0(\overline{I_T}, L^2_{ \mathrm{weak}}) , \label{202103safafsda02asfd1540}
\end{align}
which, together with \eqref{202105072000}, yields \eqref{2020081safdasaf210000415}.
Noting that \eqref{2022104191625} and \eqref{2022104191625xxx} also hold with $\partial_1$  in place of $D_1^\tau$, thus we further derive
 the assertion in \eqref{202103021540}.

Thanks to \eqref{2022010500720046}, \eqref{202008asf160946}, \eqref{202103safafsda02asfd1540} and  the fact $(u ,\eta ,q)  \in \mathbb{S}_{T_1 }$, we easily see that
\begin{align}
(u ,\eta ,q )\in \mathfrak{U}_{T_1}^{1,3}  \times \mathfrak{C}^0( \overline{I_{T_1}},{H}^{2,3}_{\mathrm{s}}) \times  C^0(\overline{I_{T_1}},\underline{H}^3) . \nonumber
\end{align}
This completes the proof.
\hfill $\Box$
\end{pf}


Thanks to Lemma \ref{pro:0812}, we easily get a unique local solution  to the linearized problem   \eqref{01dsaf16asdfsafasf0000}  by a compactness argument and a regularity argument.
\begin{pro}\label{thm08}
Let the assumptions of Lemma \ref{pro:0812} be satisfied, and $\delta_0$, $T_1$ be provided by Lemma \ref{pro:0812}. Then, for any $\delta\leqslant \delta_0$,
the linearized problem \eqref{01dsaf16asdfsafasf0000}
defined on $\Omega_{T_1}$  admits a unique solution
$(u^{{L}},\eta^{{L}}, {q}^{L} )\in\mathfrak{U}_{T_1}^{1,3}\times   \mathfrak{C}^0( {\overline{I_{T_1}}},{H}^{2,3}_{\mathrm{s}})\times C^0(\overline{I_{T_1}},\underline{H}^3)$; moreover the solution satisfies
\begin{align}
&\label{202104191930}
 \|\eta^{{L}}\|_{C^0(\overline{I_{T_1}},H^3)}
\leqslant\|\eta^0\|_3+2 c_3\sqrt{ I^0_L }T,\\
&\label{202008131221}
 \|( u^{{L}},\eta^{{L}}, \partial_1 \eta^{{L}})\|_{C^0(\overline{I_{T_1}},H^3)}\leqslant 2 c_3
\sqrt{I^0_L},\\
&\label{2020081312210416}
\sup\nolimits_{t\in\overline{I_{T_1}}}  \|(u^{{L}}, \partial_1 \eta^{{L}})\|_{\underline{1},3}\leqslant 2 c_3 \sqrt{I^0_H},\\
&\label{2020081609540416}
  \|{q}^{L}  \|_{C^0(\overline{I_{T_1}},H^3)}+\| (u_t^{{L}}  ,\nabla_{\mathcal{B}}q^{{L}}  )\|_{L^\infty_{T_1}H^3 }
\leqslant P\left(  I^0_H ,\|({\varsigma},  \varsigma_t)\|_{L^{\infty}_{T_1}H^{3}}\right).
\end{align}
\end{pro}
\begin{pf}
Let $\kappa\in (0,1)$ and $(u^\kappa,\eta^\kappa,q^\kappa)\in\mathbb{S}_{{T_1} }$ be the local solution stated as in Lemma \ref{pro:0812}. Thanks to the $\kappa$-independent estimates \eqref{2020081safdasaf21000}--\eqref{2020081609300416} and \eqref{202103021540}
satisfied  by $(u^\kappa,\eta^\kappa,q^\kappa)$, we can easily follow the compactness argument as in the proof of Proposition \ref{pro:parabolic} to obtain a limit function $(u^{{L}},\eta^{{L}},q^{{L}})$, which is the solution of the linearized problem \eqref{01dsaf16asdfsafasf0000} and satisfies\begin{itemize}
\item the estimates
\begin{align}
&\label{20200813122fdas1}
 \|( u^{{L}},\eta^{{L}}, \partial_1 \eta^{{L}})\|_{L^\infty_{T_1}H^3}\leqslant 2 c_3
\sqrt{I^0_L},\\
&\label{202008131221041sdfa6}
 \|(u^{{L}}, \partial_1 \eta^{{L}})\|_{L^\infty_{T_1}H^{1,3}}\leqslant 2 c_3 \sqrt{I^0_H},\\
&\label{2020081609540fdsa416}
   \| (u_t^{{L}}  ,q,\nabla_{\mathcal{B}}q^{{L}}  )\|_{L^\infty_{T_1}H^3 }
\leqslant P\left(  I^0_H ,\|({\varsigma},  \varsigma_t)\|_{L^{\infty}_{T_1}H^{3}}\right).
\end{align}  \item the additional regularity
 \begin{align}
&\label{20200813sdafa122fdas1}
u^{{L}}\in {C^0(I_{T_1}H^2_{\mathrm{s}})}  ,\ \nabla^3 \partial_1 u \in {C}^0( \overline{I_T} ,L^2_{ \mathrm{weak}}),\  \eta^{{L}}\in \mathfrak{C}^0( \overline{I_{T_1}},{H}^{2,3}_{\mathrm{s}} ).
\end{align}
                                             \end{itemize}
In addition,
$$(q)_{\Omega}=0\mbox{ and }u_t|_{\partial\Omega}=0\mbox{ for a.e. }t\in I_T.$$

Thanks to Proportion \ref{pro:transport} and the regularity of $(u,\eta,q)$, we further have
\begin{align}
u^{{L}}\in {C^0(I_{T_1}H^3_{\mathrm{s}})}.
\end{align}

By \eqref{01dsaf16asdfsafasf0000}$_2$ and \eqref{01dsaf16asdfsafasf0000}$_3$, $q$ satisfies
\begin{equation}\nonumber
                              \begin{cases}
 \Delta_{{\mathcal{B}}}q=K^4:=m\mathrm{div}_{{\mathcal{B}}}  \partial_1^2\eta +\mathrm{div}_{{\mathcal{B}}_t}  u&\mbox{in } \Omega,\\
{\mathcal{B}}_{3j}\partial_{i}q=0  &\mbox{on } \partial\Omega,
\end{cases}
\end{equation}
where $K^4\in {C^0(I_{T_1}H^1 )}$. Applying Proposition \ref{pro:elliptic} to the above boundary-value problem, we easily obtain
\begin{align}
\label{202008121055xx}
  q \in {C^0(\overline{I_{T_1}},\underline{H}^3)} .
\end{align}
Thus we further derive from \eqref{01dsaf16asdfsafasf0000}$_2$ that
\begin{align}
\label{2020081sdfa21055}
  u_t \in C^0(\overline{I_{T_1}},H^2_{\mathrm{s}}).
\end{align}
Collecting the regularity of $(u,\eta,q)$ in \eqref{202008131221041sdfa6}--\eqref{2020081sdfa21055}, we easily see  that
 $(u^{{L}},\eta^{{L}},q^{{L}})\in \mathfrak{U}_{T_1}^{1,3}\times \mathfrak{C}^0( \overline{I_{T_1}},{H}^{2,3}_{\mathrm{s}} )\times  C(\overline{I_{T_1}},\underline{H}^3)$. Thus we further get \eqref{202008131221}--\eqref{2020081609540416} from \eqref{20200813122fdas1}--\eqref{2020081609540fdsa416}.

The estimate \eqref{202104191930} is obvious by \eqref{01dsaf16asdfsafasf0000}$_1$
satisfied by $(\eta^{{L}},u^{{L}})$ and the estimate \eqref{202008131221}.
In addition, the uniqueness can be easily verified by a standard energy method.
The proof of Proposition \ref{thm08} is complete.
\hfill$\Box$
\end{pf}

\subsection{Proof of Proposition \ref{202102182115}}\label{subsce:03}
Now we are in the position  to the proof of Proposition  \ref{202102182115}.
Let   $(u^0,\eta^0)$ satisfy the assumptions in Proposition \ref{202102182115}, $\|\eta^0\|_{\underline{1},3}\leqslant \delta\leqslant \{\delta_0, \iota/2\}$ and
\begin{align}
\nonumber
{T_2}=  \min\{1/3c_3 P(4c_3(b+1) , \delta/2c_3(b+1),1 \},
\end{align}
where the positive constants $c_3\geqslant 1$  and $\delta_0$ are provided by Proposition \ref{thm08} with $\alpha={ T_2}$.
By Proposition \ref{thm08}
and Lemma \ref{pro:1221}, for any $T\leqslant T_2$, we can construct a solution sequence
$$\{( u^{n},\eta^{n}, Q^{n})\}_{n=1}^\infty\subset\mathfrak{U}_{T }^{1,3}\times   \mathfrak{C}^0 (\overline{I_{T }},{H}^{2,3}_{\mathrm{s}} )\times {C^0(\overline{I_{T_1}},\underline{H}^3)} $$  such that
\begin{enumerate}[(1)]
\item $(u^{1}, \eta^1,{q}^{1})=(\eta^0, u^0,0)$;
  \item $(u^{n+1},\eta^{n+1}, {q}^{n+1})$ satisfies
\begin{equation}\label{01dsaf16asdfasf00002}
                              \begin{cases}
 \eta^{n+1}_t=u^{n+1} ,\\
u^{n+1}_t+\nabla_{\mathcal{A}^{n}}{q}^{n+1}+ au^{n+1}=m  \partial_1^2\eta^{n+1} ,\\
\div_{{\mathcal{A}}^{n}}u^{n+1}=0  , \\
(u^{n+1},\eta^{n+1})|_{t=0}=(u^0,\eta^0)  , \\
(u^{n+1}_3,\eta^{n+1}_3)|_{\partial\Omega}=\mathcal{A}^{n}_{3j}\partial_jq^{n+1}|_{\partial\Omega} =0
\end{cases}
\end{equation}
for $n\geqslant 1$, where $\mathcal{A}^{n}=(\nabla(\eta^{n}+I))^{-\mathrm{T}}$.
\item $(u^{n},\eta^{n}, q^{n})$ enjoys the estimates, for $n\geqslant 1$,
\begin{align}
&
  \| \eta^{n}  \|_{C^0(\overline{I_{T_1}},H^3)}^2 \leqslant 2\delta\leqslant \iota\leqslant 1,
\label{20dsfa2008131747} \\
&\label{202008131747} \|(u^{n},\eta^{n},\partial_1\eta^{n}) \|_{C^0(\overline{I_{T_1}},H^3)}  \leqslant 2c_3\sqrt{I^0_L}\lesssim b+1 , \\
&\label{202104101412}{
\sup\nolimits_{t\in\overline{I_T}} \|(u^{n}, \partial_1\eta^{n})(t)\|_{\underline{1},3} \leqslant 2c_3\sqrt{I^0_H } \lesssim b+1,} \\
&\label{202008131740n0416}
 \|q^n \|_{C^0(\overline{I_T},H^3)} + \|(u_t^{n} ,\nabla_{\mathcal{A}^{n-1}}q^n)\|_{L^\infty_{T}H^3}
\lesssim P ( b ) .
\end{align}
\end{enumerate}


Similarly to \eqref{202008121505} and \eqref{202008121535},
by using \eqref{20dsfa2008131747}  and \eqref{202008131747}, we have
\begin{align}
&\label{202008141600}
\sup\nolimits_{t\in  {I_{T}}} \|(\mathcal{A}^{n}_t,\mathcal{A}^{n},\partial_1\mathcal{A}^{n},1/J^{n})\|_2
\lesssim P(b ),\\
& \sup\nolimits_{t\in  {I_{T}}} \| \mathcal{A}^{n}-I\|_2
\lesssim \|\eta^n\|_3.\label{202008141dsafa600}
\end{align}

Similarly to \eqref{2022104191625} and \eqref{2022104191625xxx}, by a regularity method, we can easily verify that, for any $\varphi\in H^1 $ and for any $t$, $r\in \overline{I_T}$, $(\eta^{n+1}, u^{n+1})$  satisfies \begin{align}
&\int\partial^\beta \partial_1^2 (\eta^{n+1}(t)- \eta^{n+1}(r))
\cdot\varphi\mathrm{d}y=-\int_r^t\int\partial^\beta\partial_1  u^{n+1}  \cdot \partial_1\varphi\mathrm{d}y\mathrm{d}s  ,  \nonumber\\
&  \int \partial^\beta \partial_1(u^{n+1} (t) -u^{n+1}(r))  \cdot \varphi\mathrm{dy}\nonumber \\
& = \int_r^t\int
 ( \partial^{ \beta}   \nabla_{\mathcal{A}^{n}}q^{n+1}
-m \partial^\beta\partial_1^2 \eta^{n+1}+a\partial^\beta  u^{n+1}   )\cdot \partial_1\varphi
\mathrm{d}y\mathrm{d}s. \nonumber
\end{align}
Thus we further derive from the above two identities  that, for $n\geqslant 1$,
\begin{align}
\partial_1\nabla^3\left(u^{n}, \partial_1 \eta^{n}
\right)
\mbox{ are uniformly continuous in }H^{-1}.
\label{20210302154safdsadf0}
\end{align}
Next we shall prove that
$\{(\eta^{n}, u^{n}, {q}^{n})\}_{m=1}^{\infty}$
is a Cauchy sequence.

From now on, we always assume $n\geqslant 2$. We define that
$$\bar{\eta}^{n+1}:={\eta}^{n+1}-{\eta}^{n},\quad
\bar{u}^{n+1}:={u}^{n+1}-{u}^{n},\quad
\bar{{q}}^{n+1}:={{q}}^{n+1}-{{q}}^{n}\mbox{ and }\bar{\mathcal{A}}^{n}:={\mathcal{A}}^{n}-{\mathcal{A}}^{n-1}.$$
Then it follows from \eqref{01dsaf16asdfasf00002} that
\begin{equation}\label{01dsaf16asdfasf00003}
                              \begin{cases}
\bar{\eta}^{n+1}_t=\bar{u}^{n+1} ,\\
\bar{u}^{n+1} _t+\nabla_{\mathcal{A}^{n}}\bar{{q}}^{n+1}+a \bar{u}^{n+1}=
m    \partial_1^2\bar{\eta}^{n+1}
-\nabla_{\bar{\mathcal{A}}^{n}}{{q}}^{n} ,\\
\div_{{\mathcal{A}}^{n}}\bar{u}^{n+1}=-\div_{{\bar{\mathcal{A}}}^{n}}{u}^{n}   , \\
(\bar{u}^{n+1},\bar{\eta}^{n+1} )|_{t=0}=(0,0) , \\
(\bar{u}^{n+1}_3,\bar{\eta}^{n+1}_3)|_{\partial\Omega}=0   .
\end{cases}
\end{equation}


By \eqref{01dsaf16asdfasf00003}$_1$ and \eqref{01dsaf16asdfasf00003}$_4$, we have
\begin{align}\label{202008140740}
\begin{aligned}
 \|\bar{\eta}^{n+1}(t)\|_{C^0(\overline{I_{T}},H^3)}  \leqslant \int_0^t\|\bar{u}^{n+1}(s)\|_3\mathrm{d}s
\leqslant  {T}\|\bar{u}^{n+1}\|_{L^{\infty}_{T}H^3}.
\end{aligned}
\end{align}
Similarly to \eqref{202008141600}, we can estimate that
\begin{align}\label{202008140824}
& \sup\nolimits_{t\in  {I_T}}\|\bar{\mathcal{A}}^{n} \|_2
\lesssim
 \|\bar{\eta}^{n}\|_{{L^{\infty}_{T}H^3}} ,\\
& \sup\nolimits_{t\in  {I_T}}\|\bar{\mathcal{A}}^{n} \|_{1,2}
\lesssim
 \| \bar{\eta}^{n}\|_{{L^{\infty}_{T}H^{1,3}}} , \label{20200814sd0824}\\
\label{2020081608}
&\sup\nolimits_{t\in  {I_T}}
\| \bar{\mathcal{A}}^{n}_t \|_2
\lesssim  P(b)
 \|(\bar{u}^{n},\bar{\eta}^{n})\|_{{L^{\infty}_{T}H^3}} .
\end{align}

Thanks to \eqref{01dsaf16asdfasf00003}$_2$, $\bar{{q}}^{n+1}$ satisfies
\begin{equation}\label{01dsaf16asdfasf00004}
  \begin{cases}
 \Delta_{\mathcal{A}^{n}}\bar{{q}}^{n+1}
= \mathrm{div}_{\mathcal{A}^{n}}\left( m \partial_1^2\bar{\eta}^{n+1}
-\nabla_{\bar{\mathcal{A}}^{n}}{{q}}^{n}
-\bar{u}^{n+1}_t-a\bar{u}^{n+1}\right)
& \hbox{in }\Omega, \\
\nabla_{\mathcal{A}^{n}}\bar{{q}}^{n+1}\cdot\vec{\mathbf{n}} =
- \nabla_{\bar{\mathcal{A}}^{n}} q^{n}\cdot\vec{\mathbf{n}}  & \hbox{on }\partial\Omega,
\end{cases}
\end{equation}
where $\Delta_{\mathcal{A}^{n}}:=\mathrm{div}_{\mathcal{A}^{n}}\nabla_{\mathcal{A}^{n}}$.
Let $\zeta^n =\eta^n+y$. $(\zeta^n)^{-1}$ denotes the inverse function of $\zeta^n$ with respect to the variable $y$. We define that \begin{align}
&K^5:=
-\nabla_{\bar{\mathcal{A}}^{n}}{q}^{n} ,\
K^6:=  m  \partial_1^2\bar{\eta}^{n+1}
- \bar{u}^{n+1}_t- a  \bar{u}^{n+1}  -\nabla_{\bar{\mathcal{A}}^{n}}{q}^{n} ,\nonumber \\
&(\beta,\tilde{K}^5,\tilde{K}^6):=(\bar{q}^{n+1}, {K}^5,K^6)|_{y=(\zeta^n)^{-1}(x,t)},\nonumber
 \end{align}then, by \eqref{01dsaf16asdfasf00004},
\begin{equation}\nonumber
  \begin{cases}
 \Delta\beta=\mathrm{div}\tilde{K}^6 &\mbox{in }\Omega,\\
 \nabla \beta \cdot\vec{\mathbf{n}}= \tilde{K}^5 \cdot\vec{\mathbf{n}}  &\hbox{on }\partial\Omega.
\end{cases}
\end{equation}


Applying the elliptic estimate \eqref{neumaasdfann1n} to the above boundary-value problem and then making use of  \eqref{2021sfa04031901} and \eqref{2022104101908}, we have, for a.e. $t\in I_T$,
\begin{align}
\|\bar{q}^{n+1} \|_3\lesssim & P(b) \|\beta \|_3
 \lesssim P(b)\left(\|\mathrm{div}\tilde{K}^6\|_1+ \| \tilde{K}^5\|_2\right)\nonumber \\
\lesssim & P(b)\left(\| \mathrm{div}_{\mathcal{A}^{n}}{K}^6\|_1+\|{K}^5\|_2\right).
\label{2sfa02008141000}
\end{align}
Noting that
$$\mathrm{div}_{\mathcal{A}^{n}}{K}^6=\mathrm{div}_{\mathcal{A}^{n}}\left(
m  \partial_1^2\bar{\eta}^{n+1}-\nabla_{\bar{\mathcal{A}}^{n}}{q}^{n}-a \bar{u}^{n+1}\right)
 +\div_{{\mathcal{A}}^{n}_t}\bar{u}^{n+1}+
\partial_t \div_{{\bar{\mathcal{A}}}^{n}}{u}^{n} ,$$
thus, making use of \eqref{202008131740n0416}, \eqref{202008141600}, \eqref{202008140824} and \eqref{2020081608}, we easily estimate that
$$ \| \mathrm{div}_{\mathcal{A}^{n}}{K}^6\|_1+\|{K}^5\|_2
\lesssim P(b) \|\left(\bar{u}^{n}, \bar{u}^{n+1}, \bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}} .$$
Putting the above estimate into \eqref{2sfa02008141000} yields
\begin{align}\label{202008141000}
\| \bar{q}^{n+1}\|_{C^0(\overline{I_{T}},H^3)}
 \lesssim P(b) \|\left(\bar{u}^{n}, \bar{u}^{n+1}, \bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}} .
\end{align}


Similarly to the derivation of \eqref{202008121220}, we  have that
\begin{align}
\|(\bar{u}^{n+1},\partial_1\bar{\eta}^{n+1})(t)\|_{C^0(\overline{I_{T}},L^2)}  \lesssim  P(b)T\|\left(\bar{u}^{n}, \bar{u}^{n+1}, \bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}}.
 \label{202008141635}
\end{align}

By \eqref{01dsaf16asdfasf00003}$_1$--\eqref{01dsaf16asdfasf00003}$_2$, we have
\begin{align}
&
\partial_t \mathrm{div}_{{\mathcal{A}}^{n}}\partial_1\bar{\eta}^{n+1}
 =\mathrm{div}_{ {\mathcal{A}}^{n}_t}\partial_1\bar{\eta}^{n+1}
-\partial_1 \mathrm{div}_{\bar{{\mathcal{A}}}^{n}}{u}^{n} -\mathrm{div}_{\partial_1{{\mathcal{A}}}^{n}}\bar{u}^{n+1},\nonumber \\
&
\partial_t \mathrm{curl}_{{\mathcal{A}}^{n}}\bar{u}^{n+1}
+a\mathrm{curl}_{{\mathcal{A}}^{n}}   \bar{u}^{n+1}
 - {m  }
\partial_1 \mathrm{curl}_{\mathcal{A}^{n}} \partial_1\bar{\eta}^{n+1}
\nonumber \\
&=\mathrm{curl}_{ {\mathcal{A}}_t^{n}}\bar{u}^{n+1}
- {m }\mathrm{curl}_{\partial_1\mathcal{A}^{n}}\partial_1\bar{\eta}^{n+1}  +  \mathrm{curl}_{\bar{\mathcal{A}}^{n}}(\nabla_{\mathcal{A}^{n-1}}q^{n}). \nonumber
\end{align}
Following the argument  of \eqref{202008122105} by further making use of \eqref{202104101412}, \eqref{202008141600}, \eqref{202008141dsafa600}, \eqref{202008140824} and \eqref{20200814sd0824}, we derive can from \eqref{01dsaf16asdfasf00003}$_3$ and the above identities that
\begin{align}
& \|(\mathrm{div}\bar{u}^{n+1},\mathrm{curl}\bar{u}^{n+1},\mathrm{div}\partial_1\bar{\eta}^{n+1},
  \mathrm{curl}\partial_1\bar{\eta}^{n+1})\|_{C^0(\overline{I_{T}},H^3)} \nonumber \\
&\lesssim  P(b)T
\|\left(  \bar{u}^{n+1},\bar{\eta}^{n}, \partial_1\bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}}+  \|\eta^0\|_3\|\left( \bar{u}^{n+1}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}}, \label{202008142035}
\end{align}
where we have used \eqref{202008140824} in the last inequality of \eqref{202008142035}.

Summing up the estimates \eqref{202008140740}, \eqref{202008141635} and \eqref{202008142035} and then using Hodge-type elliptic estimate \eqref{08171537}, we obtain,\emph{ for sufficiently small $\delta$,}
\begin{align}\nonumber
 \|\left(\bar{u}^{n+1},\bar{\eta}^{n+1}, \partial_1\bar{\eta}^{n+1} \right)\|_{C^0(\overline{I_{T}},H^3)}
 \lesssim P(b)T \|\left( \bar{u}^{n},\bar{u}^{n+1},\bar{\eta}^{n}, \partial_1\bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}}
.
\end{align}
In particular
\begin{align}\label{202008142102}
 \|\left(\bar{u}^{n+1},\bar{\eta}^{n+1}, \partial_1\bar{\eta}^{n+1} \right)\|_{C^0(\overline{I_{T}},H^3)}
 \leqslant \|\left( \bar{u}^{n}, \bar{\eta}^{n}, \partial_1\bar{\eta}^{n} \right)\|_{{L^{\infty}_{T}H^3}}/2
\end{align}  \emph{for sufficiently small $T\in (0,T_2]$,} where the smallness depends  on $b$, $a$, $m $ and $\Omega$.

In addition, by \eqref{01dsaf16asdfasf00003}$_2$  and \eqref{202008141600}, we get that
\begin{align}\label{202008151041}
\|  \bar{u}^{n+1}_t\|_{C^0(\overline{I_{T}},H^2)}  \lesssim P(b)
 \|\left(\bar{u}^{n+1}, \bar{\eta}^{n}, \partial_1\bar{\eta}^{n+1},\bar{{q}}^{n+1}\right)\|_{{L^{\infty}_{T}H^3}}.
\end{align}

We immediately see from \eqref{202008141000}, \eqref{202008142102} and \eqref{202008151041} that the sequence $\{( u^{n},\eta^{n},{q}^{n},u_t^n)\}_{n=1}^{\infty}$ is a Cauchy sequence in $C^0(\overline{I_{T}},  H^3_{\mathrm{s}}\times H^{1,3}_{\mathrm{s}}\times \underline{H}^3\times H^{2}_{\mathrm{s}} )$. Thus
$$(u^{n}, \eta^{n}, {q}^{n},u_t^n)\to (u,\eta,{q},u_t)
\mbox{ strongly in }C^0(\overline{I_{T}}, H^3_{\mathrm{s}}\times H^{1,3}_{\mathrm{s}}\times \underline{H}^3\times H^{2}_{\mathrm{s}}).$$
Thanks to the strong convergence of $(u^{n},\eta^{n},{q}^{n},u^n_t)$, we can pass to limits in \eqref{01dsaf16asdfasf00002}, and then get a  solution $(u,\eta,{q},u_t)$ to the problem \eqref{01dsaf16asdfasf}.

Finally, thanks to \eqref{202008131747}--\eqref{202008131740n0416} and \eqref{20210302154safdsadf0}, we easily follow the  compactness argument and the regularity argument in Proposition \ref{thm08} to further derive that
\begin{align}
&(u,\eta,{q})\in  \mathfrak{U}_{T }^{1,3}\times \mathfrak{C}^0 (\overline{I_{T }},{H}^{2,3} )\times  C^0(\overline{I_{T}},\underline{H}^3),\  \nabla_{\mathcal{A} }q \in {L^\infty_TH^3}, \nonumber \\
&\label{2020081508001n}
 \|({u}, {\eta} , \partial_1{\eta})\|_{C^0(\overline{I_{T_1}},H^3)} \leqslant 2 c_3 \sqrt{I^0_L } ,\\
&\label{2020081508001nn}
\sup\nolimits_{t\in \overline{I_T}}\|({u} , \partial_1{\eta} )\|_{\underline{1},3} \leqslant 2 c_3 \sqrt{I^0_H}.
\end{align}
Finally it is easy to check that  $(\eta,u,q)$ is the unique solution to the problem
\eqref{01dsaf16asdfasf}, provided that $\delta$ is sufficiently small.
This completes the proof of Proposition \ref{202102182115}.
\hfill $\Box$

