# Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly
Conducting Wall

Jiawei WANG Junyan ZHANG Institute of Applied Physics and Computational
Mathematics, Beijing, P.R. China. Email<EMAIL_ADDRESS>of Mathematics, National University of Singapore, Singapore. Email:
<EMAIL_ADDRESS>
###### Abstract

We prove the incompressible limit of compressible ideal
magnetohydrodynamic(MHD) flows in a reference domain where the magnetic field
is tangential to the boundary. Unlike the case of transversal magnetic fields,
the linearized problem of our case is not well-posed in standard Sobolev space
$H^{m}~{}(m\geq 2)$, while the incompressible problem is still well-posed in
$H^{m}$. The key observation to overcome the difficulty is a hidden structure
contributed by Lorentz force in the vorticity analysis, which reveals that one
should trade one normal derivative for two tangential derivatives together
with a gain of Mach number weight $\varepsilon^{2}$. Thus, the energy
functional should be defined by using suitable anisotropic Sobolev spaces. The
weights of Mach number should be carefully chosen according to the number of
tangential derivatives, such that the energy estimates are uniform in Mach
number. Besides, part of the proof is similar to the study of compressible
water waves, so our result opens the possibility to study the incompressible
limit of free-boundary problems in ideal MHD.

Keywords: Compressible ideal MHD, Incompressible limit, Perfectly conducting
wall.

MSC(2020) codes: 35L60, 35Q35, 76M45, 76W05.

###### Contents

  1. 1 Introduction
  2. 2 Difficulties and strategies
  3. 3 Uniform energy estimates
  4. 4 Incompressible limit
  5. References

##  1 Introduction

In this paper, we consider the compressible ideal magnetohydrodynamics(MHD)
equations

$\begin{cases}D_{t}\rho+\rho(\nabla\cdot
u)=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \rho D_{t}u=B\cdot\nabla
B-\nabla
P,~{}~{}P:=p+\frac{1}{2}|B|^{2}~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}B=B\cdot\nabla u-B(\nabla\cdot
u)~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \nabla\cdot
B=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}S=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\end{cases}$ (1.1)

describing the motion of a compressible conducting fluid in an electro-
magnetic field. Here $\Omega:=\mathbb{T}^{d-1}\times(-1,1)$ is the reference
domain in $\mathbb{R}^{d}~{}(d=2,3)$ with boundary
$\Sigma:=\Sigma_{+}\cup\Sigma_{-}$ and $\Sigma_{\pm}:=\\{x_{d}=\pm 1\\}$.
$\nabla:=(\partial_{x_{1}},\cdots,\partial_{x_{d}})$ is the standard spatial
derivative. $D_{t}:=\partial_{t}+u\cdot\nabla$ is the material derivative. The
fluid velocity, the magnetic field, the fluid density, the fluid pressure and
the entropy are denoted by $u=(u_{1},\cdots,u_{d})$, $B=(B_{1},\cdots,B_{d})$,
$\rho$, $p$ and $S$ respectively. Note that the last equation of (1.1) is
derived from the equation of total energy and Gibbs relation. See [6, Ch. 4.3]
for more details. Throughout this paper, we will use follow the convention of
Einstein’s summation, that is, repeated indices represent taking sum over the
indices.

We assume the fluid pressure $p=p(\rho,S)$ to be a given smooth function of
$\rho,S$ which satisfies

$\displaystyle\rho>0,~{}~{}~{}\frac{\partial
p}{\partial\rho}>0,\quad\text{in}\,\,\bar{\Omega}.$ (1.2)

These two conditions also guarantee the hyperbolicity of system (1.1).

The initial and boundary conditions of system (1.1) are

$\displaystyle(u,B,\rho,S)|_{t=0}=(u_{0},B_{0},\rho_{0},S_{0})~{}~{}~{}$
$\displaystyle\text{in }[0,T]\times\Omega,$ (1.3) $\displaystyle
u_{d}=B_{d}=0~{}~{}~{}$ $\displaystyle\text{on }[0,T]\times\Sigma,$ (1.4)

where the boundary condition for $u_{d}$ is the slip boundary condition, and
the boundary condition for $B_{d}$ shows that $\Sigma_{\pm}$ are perfectly
conducting walls.

######  Remark 1.1.

The conditions $\nabla\cdot B=0$ in $\Omega$ and $B_{d}=0$ on $\Sigma$ are
both constraints for initial data so that the MHD system is not over-
determined. One can show that they propagate within the lifespan of the
solution. Using the theory of hyperbolic system with charateristic boundary
condition [22], one can show that the correct number of boundary conditions
when assuming $u_{d}|_{\Sigma}=0$ is 1 (see [31]). So, $B_{d}|_{\Sigma}=0$ has
to be an initial constraint.

The initial-boundary value problem (1.1)-(1.4) is used to characterize the
motion of a plasma confined in a rigid perfectly conducting wall, which is an
important model in the study of laboratory plasma confinement problems. See
[6, Ch. 4.6] for more details. To make the initial-boundary value problem
(1.1)-(1.4) solvable, we need to require the initial data satisfying the
compatibility conditions up to certain order. For $m\in\mathbb{N}$, we define
the $m$-th order compatibility conditions to be

$\partial_{t}^{j}u_{d}|_{t=0}=0,~{}~{}0\leq j\leq m.$ (1.5)

Let $\mathcal{F}:=\log\rho$. Since $\frac{\partial p}{\partial\rho}>0$
indicates $\frac{\partial\mathcal{F}}{\partial p}>0$, using $D_{t}S=0$, the
first equation of (1.1) is equivalent to

$\frac{\partial\mathcal{F}}{\partial p}D_{t}p+\nabla\cdot u=0.$ (1.6)

Thus the compressible MHD system is now reformulated as follows

$\begin{cases}\frac{\partial\mathcal{F}}{\partial p}D_{t}p+\nabla\cdot
u=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \rho D_{t}u=B\cdot\nabla
B-\nabla
P,~{}~{}P:=p+\frac{1}{2}|B|^{2}~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}B=B\cdot\nabla u-B(\nabla\cdot
u)~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \nabla\cdot
B=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}S=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ p=p(\rho,S),\frac{\partial
p}{\partial\rho}>0~{}~{}~{}&\text{in}~{}[0,T]\times\bar{\Omega},\\\
u_{d}=B_{d}=0~{}~{}~{}&\text{on}~{}[0,T]\times\Sigma,\\\
(u,B,\rho,S)|_{t=0}=(u_{0},B_{0},\rho_{0},S_{0})~{}~{}~{}&\text{on}~{}\\{t=0\\}\times\Omega.\end{cases}$
(1.7)

###  1.1 The equation of states and sound speed

This paper is devoted to studying the behavior of the solution of (1.7) as the
sound speed goes to infinity, which is known to be the incompressible limit.
Physically, the sound speed is defined by $c_{s}:=\sqrt{\frac{\partial
p}{\partial\rho}}$. Mathematically, Mathematically, it is convenient to view
the sound speed as a parameter $\lambda$. This being said, a typical choice of
the equation of states $p_{\lambda}(\rho,S)$ would be the polytropic gas

$\displaystyle
p_{\lambda}(\rho,S)=\lambda^{2}\left(\rho^{\gamma}\exp(S/C_{V})-1\right),~{}~{}~{}\gamma\geq
1,~{}~{}C_{V}>0.$ (1.8)

When viewing the density as a function of the pressure and the entropy, this
indicates

$\displaystyle\rho_{\lambda}(p/\lambda^{2},S)=\left(\left(1+\frac{p}{\lambda^{2}}\right)e^{-\frac{S}{C_{V}}}\right)^{\frac{1}{\gamma}},\quad\text{and}\,\,\log\left(\rho_{\lambda}(p/\lambda^{2},S)\right)=\gamma^{-1}\log\left(\left(1+\frac{p}{\lambda^{2}}\right)e^{-\frac{S}{C_{V}}}\right).$
(1.9)

Hence, we can view $\mathcal{F}=\log\rho$ as a parametrized family
$\\{\mathcal{F}_{\varepsilon}(p,S)\\}$ as well, where
$\varepsilon=\frac{1}{\lambda}$. Indeed, we have

$\frac{\partial\mathcal{F}_{\varepsilon}}{\partial
p}=\gamma^{-1}\log\left((1+\varepsilon^{2}p)e^{-\frac{S}{C_{V}}}\right).$
(1.10)

Since we work on the case when the entropy and velocity are both bounded
(later we will assume $u,S\in H^{4}(\Omega)$), we again slightly abuse the
terminology, and call $\lambda$ the sound speed and call $\varepsilon$ the
Mach number and thus $M=O(\varepsilon)$. Furthermore, there exists $C>0$ such
that

$\displaystyle
C^{-1}\varepsilon^{2}\leq\frac{\partial\mathcal{F}_{\varepsilon}}{\partial
p}(p,S)=\frac{1}{\rho}\frac{\partial\rho}{\partial p}(p,S)\leq
C\varepsilon^{2}.$ (1.11)

Also, we assume

$\displaystyle|\partial_{p}^{s}\mathcal{F}_{\varepsilon}(p,S)|\leq
C,\quad|\partial_{p}^{s}\mathcal{F}_{\varepsilon}(p,S)|\leq
C|\partial_{p}\mathcal{F}_{\varepsilon}(p,S)|^{s}\leq
C\mathcal{F}^{\prime}_{\varepsilon}(p)$ (1.12)

holds for $0\leq s\leq 8$. Thus, when considering the incompressible limit,
that is, when $\varepsilon>0$ is sufficiently small, it is more convenient to
reformulate the compressible MHD system by replacing
$\frac{\partial\mathcal{F}}{\partial p}$ with $\varepsilon^{2}$ as follows

$\begin{cases}\varepsilon^{2}D_{t}p+\nabla\cdot
u=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \rho D_{t}u=B\cdot\nabla
B-\nabla
P,~{}~{}P:=p+\frac{1}{2}|B|^{2}~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}B=B\cdot\nabla u-B(\nabla\cdot
u)~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ \nabla\cdot
B=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\
D_{t}S=0~{}~{}~{}&\text{in}~{}[0,T]\times\Omega,\\\ p=p(\rho,S),\frac{\partial
p}{\partial\rho}>0~{}~{}~{}&\text{in}~{}[0,T]\times\bar{\Omega},\\\
u_{d}=B_{d}=0~{}~{}~{}&\text{on}~{}[0,T]\times\Sigma,\\\
(u,B,\rho,S)|_{t=0}=(u_{0},B_{0},\rho_{0},S_{0})~{}~{}~{}&\text{on}~{}\\{t=0\\}\times\Omega.\\\
\end{cases}$ (1.13)

###  1.2 An overview of previous results

The incompressible limit of compressible inviscid fluids is considered to be a
type of singular limit of hyperbolic system: the pressure for compressible
fluids is a variable of hyperbolic system whereas the pressure for
incompressible fluids is a Lagrangian multiplier and the equation of state is
no longer valid. Early works about compressible Euler equations can be dated
back to Klainerman-Majda [14, 15] when the domain is the whole space
$\mathbb{R}^{d}$ or the periodic domain $\mathbb{T}^{d}$, Ebin [5] and
Schochet [23, 24] when the domain is bounded, and Isozaki [9] when considering
an exterior domain. The abovementioned papers consider the case of “well-
prepared” initial data, which means the compressible initial data is exactly a
small perturbation of a given incompressible initial data. When the initial
data is “ill-prepared”, that is, the compressible initial data is the small
perturbation of incompressible initial data plus a highly oscillatory part, we
refer to [34, 2, 9, 25, 8, 29, 18, 1] and references therein. The precise
definitions of “well-prepared data” and “ill-prepared data” can be found in
[29, 18, 1].

For compressible ideal MHD, the incompressible limit in the whole space was
studied by Jiang-Ju-Li [10]. However, when the domain has a boundary, the
study of incompressible limit for ideal compressible MHD is much more subtle.
The first difficulty is that the vorticity estimate for ideal compressible MHD
cannot be closed when using standard Sobolev spaces, and thus the method
presented in [23, 24] is no longer valid. When the magnetic field is not
tangential to the boundary, one can use such transversality to compensate the
loss of normal derivative arising in the vorticity analysis. See Yanagisawa
[35] for the well-posedness result under the condition $B\times
N|_{\partial\Omega}=\mathbf{0}$. As for the singular limits, we refer to Ju-
Schochet-Xu [12] for the singular limits when both Mach number and Alfvén
number converge to zero and Jiang-Ju-Xu [11] for the low Alfvén number limit
of incompressible ideal MHD whose local existence was proved via a low-Mach-
number regime.

Unfortunately, when the magnetic field is tangential to the boundary, that is,
$B\cdot N|_{\partial\Omega}=0$, Ohno-Shirota [20] proved that the linearized
problem near a non-zero magnetic field is ill-posed in standard Sobolev
spaces. In this case, one has to use suitable anisotropic Sobolev spaces
introduced by Chen [3] to proved the well-posedness. See Yanagisawa-Matsumura
[36] and Secchi [26, 28].

Therefore, an essential difficulty in establishing the incompressible limit
for ideal MHD with $B\cdot N|_{\partial\Omega}=0$ is the “incompatibility of
function spaces for the existence”: compressible ideal MHD flow with the
perfectly conducting wall condition may not be well-posed in standard Sobolev
spaces, while the corresponding incompressible problem is well-posed (see, for
example, Gu-Wang [7]). So far, there is only a partial answer for this
incompressible limit problem. Ju and the first author [13] proved the
incompressible limit in a suitable closed subspace of standard Sobolev space,
introduced by Secchi [27], by adding more restrictive constraints for the
boundary value of initial data. Specifically, [27, 13] require the initial
data to satisfy

$\partial_{3}^{2k}(u_{3},B_{3})|_{\Sigma}=0,~{}\partial_{3}^{2k+1}(p,u_{1},u_{2},B_{1},B_{2},S)|_{\Sigma}=0,~{}~{}k=0,1,\cdots,\lfloor\frac{m-1}{2}\rfloor.$

However, the physical interpretation of these “extra constraints” is still
unclear. In other words, it is still unknown how to thoroughly overcome the
difficulty caused by the abovementioned “incompatibility”.

The aim of this paper is to give a definitive answer to the incompressible
limit problem of ideal MHD with the perfectly conducting wall condition when
the initial data is “well-prepared”. Indeed, in the vorticity analysis, there
is a hidden structure brought by the Lorentz force term. To the best of our
knowledge, such an observation never appears in any previous works, but it can
really help us easily prove the uniform estimates in Mach number and also
illustrates why the anisotropic Sobolev spaces are naturally introduced and
necessary for compressible MHD but are not needed for incompressible MHD.

###  1.3 The main theorems

Before stating our results, we should first define the anisotropic Sobolev
space $H_{*}^{m}(\Omega)$ for $m\in\mathbb{N}$ and
$\Omega=\mathbb{T}^{d-1}\times[-1,1]$. Let $\omega=\omega(x_{d})$ be a cutoff
function111The choice of $\omega(x_{d})$ is not unique. We just need
$\omega(x_{d})$ vanishes on $\Sigma$ and is comparable to the distance
function near $\Sigma$. on $[-1,1]$ defined by
$\omega(x_{d})=(1-x_{d})(1+x_{d})$. Then we define $H_{*}^{m}(\Omega)$ for
$m\in\mathbb{N}^{*}$ as follows

$H_{*}^{m}(\Omega):=\left\\{f\in
L^{2}(\Omega)\bigg{|}(\omega\partial_{d})^{\alpha_{d+1}}\partial_{1}^{\alpha_{1}}\cdots\partial_{d}^{\alpha_{d}}f\in
L^{2}(\Omega),~{}~{}\forall\alpha\text{ with
}\sum_{j=1}^{d-1}\alpha_{j}+2\alpha_{d}+\alpha_{d+1}\leq m\right\\},$

equipped with the norm

$\|f\|_{H_{*}^{m}(\Omega)}^{2}:=\sum_{\sum_{j=1}^{d-1}\alpha_{j}+2\alpha_{d}+\alpha_{d+1}\leq
m}\|(\omega\partial_{d})^{\alpha_{d+1}}\partial_{1}^{\alpha_{1}}\cdots\partial_{d}^{\alpha_{d}}f\|_{L^{2}(\Omega)}^{2}.$
(1.14)

For any multi-index
$\alpha:=(\alpha_{0},\alpha_{1},\cdots,\alpha_{d},\alpha_{d+1})\in\mathbb{N}^{d+2}$,
we define

$\partial_{*}^{\alpha}:=\partial_{t}^{\alpha_{0}}(\omega\partial_{3})^{\alpha_{d+1}}\partial_{1}^{\alpha_{1}}\cdots\partial_{d}^{\alpha_{d}},~{}~{}\langle\alpha\rangle:=\sum_{j=0}^{d-1}\alpha_{j}+2\alpha_{d}+\alpha_{d+1},$

and define the space-time anisotropic Sobolev norm $\|\cdot\|_{m,*}$ to be

$\|f\|_{m,*}^{2}:=\sum_{\langle\alpha\rangle\leq
m}\|\partial_{*}^{\alpha}f\|_{L^{2}(\Omega)}^{2}=\sum_{\alpha_{0}\leq
m}\|\partial_{t}^{\alpha_{0}}f\|_{H_{*}^{m-\alpha_{0}}(\Omega)}^{2}.$ (1.15)

We also denote the interior Sobolev norm to be
$\|f\|_{s}:=\|f(t,\cdot)\|_{H^{s}(\Omega)}$ for any function $f(t,x)\text{ on
}[0,T]\times\Omega$ and denote the boundary Sobolev norm to be
$|f|_{s}:=|f(t,\cdot)|_{H^{s}(\Sigma)}$ for any function $f(t,x)\text{ on
}[0,T]\times\Sigma$. From now on, we assume the dimension to be $d=3$, that
is, $\Omega=\mathbb{T}^{2}\times(-1,1)$ and $\Sigma_{\pm}=\\{x_{3}=\pm 1\\}$.
In the proof, we will see the 2D case follow in the same manner as the 3D case
up to a slight modification in the vorticity analysis.

First, we establish a local-in-time estimate that is uniform in Mach number
$\varepsilon$.

######  Theorem 1.1 (Uniform estimate in $\varepsilon$).

Let $\varepsilon\in(0,1)$ be fixed. Let $(u_{0},B_{0},\rho_{0},S_{0})\in
H_{*}^{8}(\Omega)\times H_{*}^{8}(\Omega)\times H_{*}^{8}(\Omega)\times
H_{*}^{8}(\Omega)$ be the initial data of (1.13) satisfying the compatibility
conditions (1.5) up to 7-th order and

$E(0)\leq M$ (1.16)

for some $M>0$ independent of $\varepsilon$. Then there exists $T>0$ depending
only on $M$, such that (1.13) admits a unique solution $(p(t),u(t),B(t),S(t))$
verifies the energy estimate

$\sup_{t\in[0,T]}E(t)\leq P(E(0)),$ (1.17)

where $P(\cdots)$ is a generic polynomial in its arguments, and the energy
$E(t)$ is defined to be

$\displaystyle E(t)=$
$\displaystyle~{}E_{4}(t)+E_{5}(t)+E_{6}(t)+E_{7}(t)+E_{8}(t),$ (1.18)
$\displaystyle E_{4}(t)=$
$\displaystyle\sum_{k=0}^{4}\left\|\left(\varepsilon^{(k-1)_{+}}\partial_{t}^{k}u,\varepsilon^{(k-1)_{+}}\partial_{t}^{k}B,\varepsilon^{(k-1)_{+}}\partial_{t}^{k}S,\varepsilon^{k}\partial_{t}^{k}p\right)\right\|^{2}_{4-k}$
$\displaystyle E_{4+l}(t)=$
$\displaystyle\sum_{\begin{subarray}{c}\langle\alpha\rangle=2l,\\\
\alpha_{0}<2l\end{subarray}}\sum_{k=0}^{4-l}\left\|\varepsilon^{(k-1)_{+}+2l}\mathcal{T}^{\alpha}\partial_{t}^{k}(u,B,S,p)\right\|_{4-k-l}^{2}$
$\displaystyle+\sum_{k=0}^{4-l}\left\|\varepsilon^{(k-1)_{+}+2l}\partial_{t}^{k+2l}(u,B,S),\varepsilon^{k+2l}\partial_{t}^{k+2l}p\right\|_{4-k-l}^{2},~{}1\leq
l\leq 4,$

where $K_{+}:=\max\\{K,0\\}$ and we denote
$\mathcal{T}^{\alpha}:=(\omega(x_{3})\partial_{3})^{\alpha_{4}}\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
to be a high-order tangential derivative for the multi-index
$\alpha=(\alpha_{0},\alpha_{1},\alpha_{2},0,\alpha_{4})$ with length (for the
anisotropic Sobolev spaces)
$\langle\alpha\rangle=\alpha_{0}+\alpha_{1}+\alpha_{2}+2\times 0+\alpha_{4}$.
In the rest of this paper, we sometimes write $\mathcal{T}^{k}$ to represent a
tangential derivative $\mathcal{T}^{\alpha}$ with order
$\langle\alpha\rangle=k$ when we do not need to specify what the derivative
$\mathcal{T}^{\alpha}$ contains.

######  Remark 1.2 (Correction of $E_{4}(t)$).

We note that the norm $\|p\|_{4}^{2}$ in $E_{4}(t)$ defined by (1.18) should
be replaced by $\|\varepsilon p\|_{0}^{2}+\|\nabla p\|_{3}^{2}$ because we do
not have $L^{2}$ estimates of $p$ without $\varepsilon$ weight. We still write
$\|p\|_{4}^{2}$ as above for simplicity of notations.

######  Remark 1.3 (“Prepared” initial data).

The above estimate only requires $\nabla\cdot u_{0}=O(\varepsilon)$ and
$\partial_{t}u|_{t=0}=O(1)$. In this case, the compressible data $u_{0}$ is a
small perturbation of an incompressible data $u_{0}^{0}$, and this
perturbation is completely contributed by the compressibility. Such
compressible data are usually called “well-prepared initial data”222One can
find the definitions of “well-prepared” and “ill-prepared” in [18, 1] for
rescaled Euler system, which is equivalent to the statement in our paper..

######  Remark 1.4 (Relations with anisotropic Sobolev space).

The energy functional $E(t)$ above is considered as a variant of
$\|\cdot\|_{8,*}$ norm at time $t>0$. For different multi-index $\alpha$, we
impose different Mach number weights according to the number of tangential
derivatives that appear in $\partial_{*}^{\alpha}$, such that the energy
estimate for the slightly modified $\|\cdot\|_{8,*}$ norm is uniform in
$\varepsilon>0$.

The next main theorem concerns the incompressible limit. We consider the
incompressible MHD equations together with a transport equation satisfied by
$(u^{0},B^{0},\pi,S^{0})$ with incompressible initial data
$(u_{0}^{0},B_{0}^{0})$ and $S_{0}^{0}$:

$\begin{cases}\varrho(\partial_{t}u^{0}+u^{0}\cdot\nabla
u^{0})-B^{0}\cdot\nabla
B^{0}+\nabla(\pi+\frac{1}{2}|B^{0}|^{2})=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
\partial_{t}B^{0}+u^{0}\cdot\nabla B^{0}-B^{0}\cdot\nabla
u^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
\partial_{t}S^{0}+u^{0}\cdot\nabla
S^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\ \nabla\cdot
u^{0}=\nabla\cdot B^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
u_{3}^{0}=B_{3}^{0}=0&~{}~{}~{}\text{on}~{}[0,T]\times\Sigma,\\\
(u_{0},B_{0},S_{0})|_{t=0}=(u_{0}^{0},B_{0}^{0},S_{0}^{0})&~{}~{}~{}\text{on}~{}\\{t=0\\}\times\Omega.\end{cases}$
(1.19)

######  Theorem 1.2 (Incompressible limit).

Under the hypothesis of Theorem 1.1, we assume that
$(u_{0},B_{0},S_{0})\to(u_{0}^{0},B_{0}^{0},S_{0}^{0})$ in $H^{4}(\Omega)$ as
$\varepsilon\to 0$ for $\nabla\cdot u^{0}=\nabla\cdot B^{0}=0$ in $\Omega$
with $u_{03}^{0}=B_{03}^{0}=0$ on $\Sigma$. Then it holds that

$\displaystyle(u,B,S)\to$ $\displaystyle(u^{0},B^{0},S^{0})\quad\text{weakly-*
in }L^{\infty}([0,T];H^{4}(\Omega))\text{ and strongly in
}C([0,T];H^{4-\delta}(\Omega))$

for $\delta>0$. $(u^{0},B^{0},S^{0})$ solves (1.19), that is, the
incompressible MHD equations together with a transport equation of $S^{0}$.
Here $\varrho$ satisfies the transport equation

$\partial_{t}\varrho+u^{0}\cdot\nabla\varrho=0,~{}~{}\varrho|_{t=0}=\rho(0,S_{0}^{0}),$

where we consider the equation of state (1.9) as
$\rho=\rho(\varepsilon^{2}p,S)$, that is, a function of $\varepsilon^{2}p$ and
$S$. The function $\pi$ satisfying $\nabla\pi\in C([0,T];H^{3}(\Omega))$
represents the fluid pressure for incompressible MHD system (1.19).

######  Remark 1.5 (Choice of regularity).

We propose $H^{4}$ regularity for the energy functional because lots of
commutator estimates require the bound for
$\|\overline{\partial}{}^{2}(u,B)\|_{L^{\infty}}$. Recall that
$H^{\frac{d}{2}+\delta}\hookrightarrow L^{\infty}$, so it would be convenient
to choose $H^{2+\lceil\frac{d}{2}+\delta\rceil}$ regularity, that is, $H^{4}$
for $d=2,3$. The initial data are set in $H^{8}(\Omega)$ in order to guarantee
they satisfy the compatibility conditions up to 7-th order and the uniform-
in-$\varepsilon$ bound $E(0)\leq M$. Our results can be easily generalized to
the case when $H^{4}$ and $H_{*}^{8}$ are replaced by $H^{m}$ and
$H_{*}^{2m}~{}(m\geq 4)$ respectively.

######  Remark 1.6 (The space for the convergence of initial data).

The compressible initial data converges to the incompressible data in
$H^{4}(\Omega)$ instead of $H_{*}^{8}(\Omega)$ because the higher-order energy
$E_{5}(0)\sim E_{8}(0)$ automatically vanishes as $\varepsilon\to 0$. Thus, it
suffices to require the convergence in $H^{4}(\Omega)$.

######  Remark 1.7 (Boundedness of the domain).

In the proof of the main theorem, we do not use the boundedness of the domain,
such as Poincaré’s inequality, to close the energy estimate. Thus, after
making necessary modifications (including setting $\rho_{0}-1\in
H_{*}^{8}(\Omega)$ instead of $\rho_{0}$ itself and assuming the initial data
to be “localized”), our proof is still valid for the case of unbounded
reference domains such as $\mathbb{R}^{d-1}\times(-1,1)$ and the half space.
Indeed, using a partition of unity as in [4], the proof can be directly
generalized to the domains which are diffeomorphic to the reference domains
via $H_{*}^{8}(\Omega)$-diffeomorphisms.

###  1.4 Organization of the paper

This paper is organized as follows. In section 2, we discuss the main
difficulties and briefly introduce our strategies to tackle the problem. Then
section 3 is devoted to the proof of uniform estimates in Mach number.
Combined with the compactness argument, we conclude the incompressible limit
in section 4.

##### Acknowledgment.

The research of Jiawei Wang is supported by the NSFC (Grants 12071044 and
12131007). Jiawei Wang would like to thank his Ph.D. advisor Prof. Qiangchang
Ju for helpful suggestions. Junyan Zhang would like to thank Chenyun Luo for
the kind hospitality during his visit at The Chinese University of Hong Kong.

### List of Notations

  * •

(Tangential derivatives) $\mathcal{T}_{0}=\partial_{t}$ denotes the time
derivative, $\mathcal{T}_{j}=\overline{\partial}{}_{j}~{}(1\leq j\leq d-1)$
denotes the tangential spatial derivatives and
$\mathcal{T}_{d+1}:=\omega(x_{d})\partial_{d}$ with
$\omega(x_{d})=(1-x_{d})(1+x_{d})$.

  * •

(Tangential components) $\bar{u}:=(u_{1},\cdots,u_{d-1})$ and
$\bar{B}:=(B_{1},\cdots,B_{d-1})$. We also write
$\overline{\nabla}:=(\overline{\partial}{}_{1},\overline{\partial}{}_{2})$ as
the tangential components of the operator $\nabla$.

  * •

($L^{\infty}$-norm) $\|\cdot\|_{\infty}:=\|\cdot\|_{L^{\infty}(\Omega)}$,
$|\cdot|_{\infty}:=\|\cdot\|_{L^{\infty}(\Sigma)}$.

  * •

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

  * •

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

  * •

(Anisotropic Sobolev norms) $\|\cdot\|_{m,*}$: For any function $f(t,x)\text{
on }[0,T]\times\Omega$, $\|f\|_{m,*}^{2}:=\sum_{\langle\alpha\rangle\leq
m}\|\partial_{*}^{\alpha}f(t,\cdot)\|_{0}^{2}$ denotes the $m$-th order space-
time anisotropic Sobolev norm of $f$.

  * •

(Polynomials) $P(\cdots)$ denotes a generic polynomial in its arguments.

  * •

(Commutators) $[T,f]g=T(fg)-f(Tg)$, $[f,T]g=-[T,f]g$,
$[T,f,g]:=T(fg)-T(f)g-fT(g)$ where $T$ is a differential operator and $f,g$
are functions.

  * •

(Equality modulo lower order terms) $A\stackrel{{\scriptstyle L}}{{=}}B$ means
$A=B$ modulo lower order terms.

##  2 Difficulties and strategies

Before going to the detailed proofs, we will discuss the main difficulties in
this incompressible limit problem and briefly introduce our strategies to
derive energy estimates that are uniform in the Mach number. Note that our
initial data is assumed to be well-prepared, so the uniform estimates together
with compactness argument are enough to derive the incompressible limit.

###  2.1 Choice of function spaces

Compressible ideal MHD with perfectly conducting wall conditions is a first-
order quasilinear hyperbolic system with characteristic boundary conditions of
constant multiplicity, for which there is a potential of normal derivative
loss. In fact, inside a rigid wall with the slip boundary condition, Euler
equations and elastodynamic equations with degenerate deformation tensor on
the boundary (see [38]) also have characteristic boundary conditions, but
their vorticity can be controlled in the setting of standard Sobolev spaces
because there are no other quantities involved in the pressure part. The loss
of normal derivatives is then compensated via the control of vorticity and
divergence. However, the vorticity estimates for compressible ideal MHD cannot
be closed in the setting of standard Sobolev spaces, which was explicitly
presented in [37].

On the one hand, people observed that such normal derivative loss can be
compensated if the vertical component of the magnetic field does not vanish on
the boundary. See Yanagisawa [35] for the well-posedness and [19, 32, 33] for
the study of corresponding free-boundary problems (MHD contact
discontinuities). But the transversality of magnetic fields violates the
perfectly conducting wall condition. On the other hand, Chen [3] first
introduced the anisotropic Sobolev spaces defined in (1.14)-(1.15) in the
study of compressible gas dynamics inside a rigid wall. Under this setting, a
normal derivative is considered as a second-order derivative, which exactly
compensates the derivative loss. Detailed analysis for MHD in such setting is
referred to the second author’s paper [16, Sect. 2.5]. Using such function
spaces, Yanagisawa-Matsumura [36] and Secchi [26, 28] proved the local well-
posedness and see also [30, 31, 16] for the study of free-boundary problems,
but the estimates obtained in these previous works are not uniform in Mach
number.

As stated in Section 1.2, the essential difficulty in proving the
incompressible limit is that the function spaces for well-posedness of
compressible problem are not “compatible” with the ones for the incompressible
problems. Since our initial data is well-prepared, that is, a given
incompressible data plus a slight perturbation, we shall still start with the
incompressible counterpart and try to find out the relationships between the
incompressible problem and the compressible problem.

###  2.2 Key observation: hidden structure of Lorentz force in the vorticity
analysis

First, the entropy is easy to control thanks to $D_{t}S=0$, so it suffices to
analyze the relations between $(u,B)$ and $p$. We start with the control of
$\|(u,B)\|_{4}$. Using div-curl decomposition, the $H^{4}$ Sobolev norms are
bounded by $\|\nabla\times u,\nabla\times B\|_{3}$, $\|\nabla\cdot
u,\nabla\cdot B\|_{3}$ and the normal traces $|u_{3},B_{3}|_{3.5}$. The
boundary conditions (1.4) eliminate the normal traces, and the divergence part
is reduced to tangential derivatives $\|\varepsilon^{2}D_{t}p\|_{3}$ thanks to
the continuity equation and the divergence constraint $\nabla\cdot B=0$. As
for the vorticity, taking curl in the momentum equation yields the evolution
equation of vorticity

$\rho D_{t}(\nabla\times u)-(B\cdot\nabla)(\nabla\times B)=\cdots,$ (2.1)

where we find that the term $\nabla(p+\frac{1}{2}|B|^{2})$ is already
eliminated. In the $H^{3}$-control of $\nabla\times u$, we invoke the
evolution equation of $B$ to get

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|\partial^{3}(\nabla\times
u)|^{2}+|\partial^{3}(\nabla\times
B)|^{2}\,\mathrm{d}x=-\int_{\Omega}\partial^{3}\nabla\times(B(\nabla\cdot
u))\cdot\partial^{3}(\nabla\times B)\,\mathrm{d}x+\text{ controllable terms},$
(2.2)

where we find that there are 5 derivatives falling on $u$ and thus the
vorticity estimates cannot be closed in the setting of 4-th order standard
Sobolev spaces. Such derivative loss in the vorticity analysis must appear
because taking curl operator eliminates the term $\nabla(\frac{1}{2}|B|^{2})$
in the momentum equation but does not eliminate the term $B(\nabla\cdot u)$.
However, in the proof of $L^{2}$ estimate, the contribution of
$\nabla(\frac{1}{2}|B|^{2})$ should be cancelled with the contribution of
$B(\nabla\cdot u)$.

If we further analyze this problematic term by using the structure of MHD
system, we will find that the anisotropic Sobolev space is naturally
introduced in order to close the energy estimates. Using the continuity
equation $\varepsilon^{2}D_{t}p=-\nabla\cdot u$, we find that the highest
order term in (2.2) is $\varepsilon^{2}B\times(\partial^{3}\nabla D_{t}p)$.
Then commuting $\nabla$ with $D_{t}$ and invoking the momentum equation $\rho
D_{t}u+B\times(\nabla\times B)=-\nabla p$, we get

$\varepsilon^{2}B\times(\partial^{3}\nabla D_{t}p)\stackrel{{\scriptstyle
L}}{{=}}\varepsilon^{2}B\times(\partial^{3}D_{t}\nabla
p)\stackrel{{\scriptstyle L}}{{=}}-\varepsilon^{2}\rho
B\times(\partial^{3}D_{t}^{2}u)-\varepsilon^{2}B\times
D_{t}(B\times\partial^{3}(\nabla\times B)).$ (2.3)

On the right side of the above identity, the second term still contains 4
normal derivatives $\partial^{3}\nabla\times$ and 1 tangential derivative
$D_{t}$ which exhibits a loss of derivative. The key observation here is that
the contribution of this term in (2.2) gives an energy term and thus avoid the
loss of derivative. Specifically, using $\nabla\times(B(\nabla\cdot
u))=(\nabla\times B)(\nabla\cdot u)-B\times\nabla(\nabla\cdot u)$ and omitting
the contribution of $(\nabla\times B)(\nabla\cdot u)$ in (2.2) (this part has
no loss of derivative), we have

$\displaystyle-\int_{\Omega}\partial^{3}\nabla\times(B(\nabla\cdot
u))\cdot\partial^{3}(\nabla\times B)\,\mathrm{d}x\stackrel{{\scriptstyle
L}}{{=}}\int_{\Omega}B\times\partial^{3}\nabla\underbrace{(\nabla\cdot
u)}_{=-\varepsilon^{2}D_{t}p}\cdot\partial^{3}(\nabla\times B)\,\mathrm{d}x$
(2.4) $\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle\int_{\Omega}\varepsilon^{2}\left(\rho
B\times(\partial^{3}D_{t}^{2}u)\right)\cdot\partial^{3}(\nabla\times
B)\,\mathrm{d}x+\int_{\Omega}\varepsilon^{2}\left(B\times
D_{t}\left(B\times\partial^{3}(\nabla\times
B)\right)\right)\cdot\partial^{3}(\nabla\times B)\,\mathrm{d}x,$

where we insert (2.3) in the second equality. Then setting $\mathbf{u}=B$,
$\mathbf{v}=D_{t}\left(B\times\partial^{3}(\nabla\times B)\right)$ and
$\mathbf{w}=\partial^{3}(\nabla\times B)$ in the vector identity
$(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}=-(\mathbf{u}\times\mathbf{w})\cdot\mathbf{v}$,
we find that the last term in (2.6) contributes to an energy term

$\displaystyle\int_{\Omega}\varepsilon^{2}\left(B\times
D_{t}\left(B\times\partial^{3}(\nabla\times
B)\right)\right)\cdot\partial^{3}(\nabla\times B)\,\mathrm{d}x$ (2.5)
$\displaystyle=$
$\displaystyle-\int_{\Omega}\varepsilon^{2}\left(B\times\partial^{3}(\nabla\times
B)\right)\cdot D_{t}\left(B\times\partial^{3}(\nabla\times
B)\right)\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}\left|B\times\partial^{3}(\nabla\times
B)\right|^{2}\,\mathrm{d}x+\text{ controllable terms}.$

The above analysis for compressible ideal MHD shows that the vorticity
estimates cannot be closed in standard Sobolev spaces, but in fact we trade
one normal derivative (in $\nabla\times$) for two tangential derivatives
$\varepsilon^{2}D_{t}^{2}$ thanks to the special structure of Lorentz force
$B\times(\nabla\times B)$ whose contribution in the vorticity analysis is an
$\varepsilon$-weighted energy term instead of introducing loss of derivatives.
This exactly illustrates why the anisotropic Sobolev space is naturally
introduced to study compressible ideal MHD.

It should also be noted that this difficulty never occurs for incompressible
ideal MHD because the divergence-free condition automatically eliminates the
bad term above. Indeed, taking the limit $\varepsilon\to 0_{+}$, the right
side of (2.2) automatically vanishes and the vorticity estimates for the
incompressible problem can be established in standard Sobolev spaces. Besides,
to prove the incompressible limit of ideal MHD with (1.4), an extra Mach
number weight $\varepsilon^{2}$ must be added together with the two tangential
derivatives. So, we find that the “anisotropic part” of the energy, namely the
terms $E_{5}\sim E_{8}$ in (1.18), will automatically vanish when we take the
incompressible limit $\varepsilon\to 0_{+}$. The remaining term $E_{4}(t)$
exactly consists of the standard Sobolev norms, which coincides with the
energy for incompressible ideal MHD.

######  Remark 2.1 (Slight modifications in 2D).

When dimension $d=2$, we have to replace the curl operator $\nabla\times$ by
$\nabla^{\perp}\cdot$ where $\nabla^{\perp}:=(-\partial_{2},\partial_{1})$.
Then (2.2) reads

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|\partial^{3}(\nabla^{\perp}\cdot
u)|^{2}+|\partial^{3}(\nabla^{\perp}\cdot
B)|^{2}\,\mathrm{d}x=-\int_{\Omega}\left((B\cdot\nabla^{\perp})\partial^{3}\nabla\cdot
u\right)(\partial^{3}\nabla^{\perp}\cdot B)\,\mathrm{d}x+\text{ controllable
terms},$ (2.6)

where the highest order term can be reduced as follows after invoking the
first two equations of (1.13)

$\displaystyle-(B\cdot\nabla^{\perp})\partial^{3}\nabla\cdot
u\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}\varepsilon^{2}B\cdot\partial^{3}D_{t}\nabla^{\perp}p\stackrel{{\scriptstyle
L}}{{=}}\varepsilon^{2}(B_{2}\partial^{3}D_{t}\partial_{1}p-B_{1}\partial^{3}D_{t}\partial_{2}p)$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}-\varepsilon^{2}\rho(B_{2}\partial^{3}D_{t}^{2}u_{1}-B_{1}\partial^{3}D_{t}^{2}u_{2})-\varepsilon^{2}(B_{2}^{2}+B_{1}^{2})D_{t}\partial^{3}(\nabla^{\perp}\cdot
B),$

where we note that the momentum equations can be written as
$\partial_{1}p=-\rho D_{t}u_{1}-B_{2}(\nabla^{\perp}\cdot B)$ and
$\partial_{2}p=-\rho D_{t}u_{2}+B_{1}(\nabla^{\perp}\cdot B)$. The first term
can be treated as in the 3D case, and the contribution of the second term
gives an energy term
$-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}|B|^{2}|\partial^{3}(\nabla^{\perp}\cdot
B)|^{2}\,\mathrm{d}x$ which also automatically vanishes when we take the
incompressible limit $\varepsilon\to 0_{+}$.

It should be emphasized that our analysis is very different from the previous
works about well-posedness [36, 26, 16]. Indeed, these previous works [36, 26,
16] avoided using div-curl analysis to reduce the normal derivatives of $u$
and $B$. Instead, they controlled the normal derivatives by directly computing
the corresponding $L^{2}$-type estimates which yields non-zero boundary
integrals such as
$-\int_{\Sigma}\partial_{3}^{4}P~{}\partial_{3}^{4}v_{3}\,\mathrm{d}x^{\prime}$
and only gives the estimates for $(u,B,\varepsilon p)$, not $(u,B,p)$. Since
taking normal derivatives does not preserve the boundary conditions, one has
to use the MHD equations to replace $\partial_{3}P$ and $\partial_{3}u_{3}$ by
tangential derivatives of the other variables, and then uses Gauss-Green
formula to rewrite the boundary integrals into the interior. During this
process, more time derivatives might be produced without extra weights of Mach
number (e.g., the product of the underlined terms in $-\nabla
P\to\underline{D_{t}v}-(\bar{B}\cdot\overline{\nabla})B$ and
$\partial_{3}v_{3}\to-\varepsilon^{2}D_{t}p-\underline{\overline{\nabla}\cdot\bar{v}}$)
. However, the continuity equation indicates that one may have to add more
Mach number weights to higher-order time derivatives. Thus, following the
methods in [36, 26, 16] might cause a potential of loss of Mach number weight,
and one can find related details in [16, Sect. 2.5]. In other words, these
previous works [36, 26, 16] only revealed that one should trade one normal
derivative for two derivatives but ignored the necessity of adding extra
weight of Mach number $\varepsilon^{2}$. One of the advantanges of our method
is that all normal derivatives are reduced via div-curl analysis which does
not involve any boundary estimates; and every time when we reduce a normal
derivative, we never lose weights of Mach number because the continuity
equation must be used in the reduction.

###  2.3 Reduction of pressure and design of energy functional

The main idea of proving the uniform-in-$\varepsilon$ estimates is to
repeatedly reduce normal derivatives to tangential derivatives until all
derivatives are tangential (with suitable Mach number weights); and reduce
spatial derivatives of $p$ to tangential derivatives of $u,B$ until all
derivatives on $p$ are time derivatives. Once we only have tangential
derivatives, it suffices to mimic the $L^{2}$ estimates to close the
tangential estimates because taking tangential derivatives preserves the
boundary conditions.

From the definition of $D_{t}$ and $u_{3}|_{\Sigma}=B_{3}|_{\Sigma}=0$, we
know both $D_{t}$ and $B\cdot\nabla$ are tangential derivatives. Based on this
fact and the analysis in section 2.2, we have the following reductions:

  * a.

Vorticity: $\nabla\times(u,B)\to\varepsilon^{2}\mathcal{T}^{2}u$.

  * b.

Divergence: $\nabla\cdot(u,B)\to\varepsilon^{2}\mathcal{T}p$.

  * c.

Reduction of pressure: The momentum equation reads $-\nabla(p+|B|^{2}/2)=\rho
D_{t}u-B\cdot\nabla B$ gives the relation $\nabla P\to\mathcal{T}(u,B)$ and
thus $\nabla p\to\mathcal{T}(u,B)$ plus $\nabla B$ where $\nabla B$ is further
reduced by using (a) and (b). This indicates that $\partial_{t}^{k}\nabla p$
should have the same Mach number weight as $\partial_{t}^{k+1}(u,B,S)$.

  * d.

Tangential estimates: When estimating $E_{4+l}(t)$ (defined in (1.18)),
$\mathcal{T}^{\alpha}(u,B)$ is controlled together with
$\varepsilon\mathcal{T}^{\alpha}p$ in the estimates of full tangential
derivatives, i.e., when $\langle\alpha\rangle=4+l$.

In the above subsection we start with $E_{4}$ and reduce $\|u,B\|_{4}$ (part
of $E_{4}$) to $\|\varepsilon^{2}\mathcal{T}p\|_{3}$ (still a part of $E_{4}$)
and $\|\varepsilon^{2}\mathcal{T}^{2}u\|_{3}$ (part of $E_{5}$) via the div-
curl analysis. The divergence part introduces time derivative, but the number
of derivatives does not exceed 4, so we can repeatedly use (c) to reduce the
spatial derivatives of $p$ to tangential derivatives of $u,B$ until there is
no spatial derivative falling on $p$. Finally, it suffices to control 4-th
order time derivatives of $u,B$ and $p$ with suitable Mach number weights. The
Mach number weights of these quantities are determined by the relation (d) and
the “preparedness” of initial data. See also diagram 2 below.

For the control of terms that have 5 derivatives, for example the term
$\|\varepsilon^{2}\mathcal{T}^{2}u\|_{3}$, we repeat the div-curl analysis to
get $\|\varepsilon^{4}\mathcal{T}^{3}p\|_{2}$ (still a part of $E_{5}$) from
the divergence part and $\|\varepsilon^{4}\mathcal{T}^{4}u\|_{2}$ (part of
$E_{6}$) from the curl part. Again, the divergence part is reduced to
$E_{5}(t)$ which is controlled in a similar way as $E_{4}(t)$; and then we do
further div-curl decomposition for $\|\varepsilon^{4}\mathcal{T}^{4}u\|_{2}$
which then requires the control of $E_{7}(t)$. Repeatedly, we finally need to
control the $L^{2}$ norms of $\varepsilon^{8}\mathcal{T}^{8}(u,B)$, which is
controlled together with $\varepsilon^{8}\mathcal{T}^{8}p$ when
$\mathcal{T}^{8}$ are not purely time derivatives, and
$\varepsilon^{9}\partial_{t}^{8}p$ when $\mathcal{T}^{8}$ are purely time
derivatives. Then the control of $\mathcal{T}^{8}$ is completely parallel to
the proof of $L^{2}$ energy conservation because there isn’t any normal
derivative.

We present the above complicated reduction procedure in the following
diagrams. For more details of the reduction procedures, we refer to section
3.2.3. We also list an example of reduction of $E_{4+l}(t)$ by repeatedly
using (c) and (d) when $l=0$. The other cases follow in the same manner.

${\|(u,B)\|_{4}}$${\|\varepsilon^{2}\mathcal{T}^{2}(u,B)\|_{3}}$${\|\varepsilon^{4}\mathcal{T}^{4}(u,B)\|_{2}}$${\|\varepsilon^{6}\mathcal{T}^{6}(u,B)\|_{1}}$${\|\varepsilon^{8}\mathcal{T}^{8}(u,B)\|_{0}}$${\|\varepsilon^{2}\mathcal{T}p\|_{3}}$${\|\varepsilon^{4}\mathcal{T}^{3}p\|_{2}}$${\|\varepsilon^{6}\mathcal{T}^{5}p\|_{1}}$${\|\varepsilon^{8}\mathcal{T}^{7}p\|_{0}}$${\text{reduction
of }E_{4}}$${\text{reduction of }E_{5}}$${\text{reduction of
}E_{6}}$${\text{reduction of }E_{7}}$${\text{reduction of }E_{8}}$closed div
curl div curl div curl div curl $\scriptstyle{\partial_{t}\text{ appears}}$
$\scriptstyle{\partial_{t}\text{ appears}}$ $\scriptstyle{\partial_{t}\text{
appears}}$ $\scriptstyle{\partial_{t}\text{ appears}}$
$\scriptstyle{\partial_{t}\text{ appears}}$ tangential tangential tangential
tangential tangential (2.7)

Diagram (2.7): From standard Sobolev space to anisotropic Sobolev space.

${E_{5}(t)}$${\|(u,B)\|_{4}}$${\|\partial_{t}(u,B)\|_{3}}$${\|\varepsilon\partial_{t}^{2}(u,B)\|_{2}}$${\|\varepsilon^{2}\partial_{t}^{3}(u,B)\|_{1}}$${\|\varepsilon^{3}\partial_{t}^{4}(u,B)\|_{0}}$${\|p\|_{4}}$${\|\varepsilon\partial_{t}p\|_{3}}$${\|\varepsilon^{2}\partial_{t}^{2}p\|_{2}}$${\|\varepsilon^{3}\partial_{t}^{3}p\|_{1}}$${\|\varepsilon^{4}\partial_{t}^{4}p\|_{0}}$closed
curl curl (d) curl (d) curl (d) (c) (c) (c) (c) tangential (2.8)

Diagram (2.8): Reduction of $E_{4}(t)$ via (c) and (d).

Recall that the vorticity estimates for compressible Euler equations can be
closed under the setting of standard Sobolev spaces. As presented in [17], the
reduction scheme for Euler equations is merely the control of $E_{4}(t)$ but
does not involve any higher-order energies. In other words, the reduction
scheme is finished within the first column of diagram (2.7), or equivalently
diagram (2.8) with $E_{5}(t)$ replaced by $E_{4}(t)$. Compared with Euler
equations, the extra thing for compressible ideal MHD is that the vorticity
analysis for $E_{4+l}(t)$ requires the control of $E_{4+l+1}(t)$ until there
is no normal derivative on $u,B$. That’s why we have the first row in diagram
(2.7). Then, as presented in each column of diagram (2.7), we need to mimic
the scheme in [17], that is, run the scheme presented in diagram (2.8), for
each $E_{4+l}(t)$ to close the energy estimates.

###  2.4 Further applications of our method

Based on the above analysis, we give a complete and definitive answer to the
incompressible limit problem of ideal MHD under the perfectly conducting wall
condition when the initial data is well-prepared. Besides, compared with the
previous work [13] by Ju and the first author, we give a clear illustration on
the “incompatibility” of the function spaces of well-posedness and thoroughly
resolve this issue. It should be noted that, in the uniform-in-$\varepsilon$
estimates, the “preparedness” of initial data is only used to guarantee
$E(0)<\infty$. If the initial data is not “prepared”, that is, $\nabla\cdot
u_{0}=O(1)$ is not small and $\partial_{t}u|_{t=0}=O(1/\varepsilon)$ is
unbounded, we believe that the uniform-in-$\varepsilon$ estimates can be
proven in a similar way after doing suitable technical modifications. Hence,
one can expect to generalize our result to the case when the initial data is
not “prepared”.

It should also be noted that the reduction procedure presented in diagram
(2.8) is parallel to the case of compressible water wave, which is presented
in the paper [17, Sect. 2.1.3] by Luo and the second author. Indeed, it is
exactly the appearance of the magnetic field and the perfectly conducting wall
condition that force us to do further vorticity analysis as stated in section
2.2 and presented in diagram (2.7). Therefore, one can expect to extend our
results to free-boundary problems. In fact, we believe that our method
together with the techniques about free-boundary problems presented in [17]
can be generalized to study the current-vortex sheets in ideal compressible
MHD, which will be presented in a forthcoming paper by the second author.

##  3 Uniform energy estimates

###  3.1 $L^{2}$ estimate

First, we establish $L^{2}$-energy estimate for (1.13). Invoking the momentum
equation and integrate by parts, we have:

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}p^{2}+\rho|u|^{2}+|B|^{2}+S^{2}\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\Omega}\nabla\cdot
u(\varepsilon^{2}p^{2}+|B|^{2}+S^{2})\,\mathrm{d}x-\int_{\Omega}\nabla\cdot(pu)\,\mathrm{d}x$
$\displaystyle+\int_{\Omega}(B\cdot\nabla B)\cdot u+(B\cdot\nabla u)\cdot
B\,\mathrm{d}x-\int_{\Omega}\frac{1}{2}\nabla|B|^{2}\cdot u+(B\cdot
B)\nabla\cdot u\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\Omega}\nabla\cdot
u(\varepsilon^{2}p^{2}+|B|^{2}+S^{2})\,\mathrm{d}x,$ (3.1)

which give the energy estimate:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|(\varepsilon
p,u,B,S)\|_{0}^{2}\leq P(E_{4}(t)).$ (3.2)

###  3.2 Reduction of normal derivatives

####  3.2.1 Reduction of pressure

We show how to reduce the control of the pressure to that of the velocity and
magnetic field when there is at least one spatial derivative on $P$. This
follows from using the momentum equation

$\displaystyle-\nabla(p+|B|^{2}/2)=\rho D_{t}u-B\cdot\nabla B.$ (3.3)

Then

$\|\nabla(p+|B|^{2}/2)\|_{0}\lesssim\|\rho\|_{\infty}\|D_{t}u\|_{0}+\|B\cdot\nabla
B\|_{0},$ (3.4)

where $D_{t}u=\partial_{t}u+\bar{u}\cdot\overline{\nabla}u+u_{3}\partial_{3}u$
and $B\cdot\nabla B=\bar{B}\cdot\overline{\nabla}B+B_{3}\partial_{3}B$.
Similarly, using (3.3), we also have the reduction for $L^{\infty}$ norm

$\displaystyle\|\nabla(p+|B|^{2}/2)\|_{\infty}\lesssim\|\rho\|_{\infty}\|D_{t}u\|_{\infty}+\|B\cdot\nabla
B\|_{\infty}.$ (3.5)

Recall that
$D_{t}=\partial_{t}+\bar{u}\cdot\overline{\nabla}+u_{3}\partial_{3}$ and
$B\cdot\nabla=\bar{B}\cdot\overline{\nabla}+B_{3}\partial_{3}$ are both
tangential derivatives thanks to $u_{3}=B_{3}=0$ on $\Sigma$. The above
inequalities (3.4)-(3.5) show that $\nabla P$ is reduced to $\partial_{t}u$,
$\overline{\partial}{}u$, $\omega(x_{3})\partial_{3}u$,
$\overline{\partial}{}B$ and $\omega(x_{3})\partial_{3}B$ for some weight
function $\omega(x_{3})$ vanishing on $\Sigma$. Thus, the momentum equation
reduces a normal derivative of $P$ to a tangential derivative of $u,B$.

Also, one can reduce the Sobolev norm of $\nabla P$ in the same way. Let
$\mathcal{T}=\partial_{t}$ or $\overline{\partial}{}$ or
$\omega(x_{3})\partial_{3}$ and $D=\partial$ or $\partial_{t}$. The above
estimate yields the control of $\|\mathcal{T}^{\alpha}D^{k}\nabla p\|_{0}$
after taking $\mathcal{T}^{\alpha}D^{k}$, $k+\langle\alpha\rangle\geq 1$, to
(3.3). Specifically, at the leading order,
$\|\mathcal{T}^{\alpha}D^{k}\nabla(p+|B|^{2}/2)\|_{0}$ is controlled by

$\displaystyle\|\rho\|_{\infty}\|\mathcal{T}^{\alpha}D^{k}\mathcal{T}u\|_{0}+\|\mathcal{T}^{\alpha}D^{k}\mathcal{T}B\|_{0}.$
(3.6)

As for the fluid pressure $p$, by definition we have
$\partial_{i}p=\partial_{i}P-\partial_{i}B\cdot B$. Thus, to control the
Sobolev norms of $\nabla p$, we still need to do further analysis for $\nabla
B$, which will be controlled together with $\nabla u$.

####  3.2.2 Div-Curl analysis

From the analysis of section 3.2.1, in order to control the Sobolev norms
defined in (1.18), we shall reduce the normal derivatives of $u$ and $B$ and
then analyze the tangential derivatives. We use the following lemma to reduce
the normal derivatives of $u,B$ to their divergence and curl.

######  Lemma 3.1 (Hodge elliptic estimates).

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

$\|X\|_{s}^{2}\lesssim\|X\|_{0}^{2}+\|\nabla\cdot X\|_{s-1}^{2}+\|\nabla\times
X\|_{s-1}^{2}+|\overline{\partial}{}X\cdot N|^{2}_{s-\frac{3}{2}}.$ (3.7)

We start from the control of $E_{4}(t)$. We apply Lemma 3.1 to
$\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}(u,B)\|_{4-k}$ for $0\leq k\leq 3$
to get

$\displaystyle\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}u\|_{4-k}^{2}\lesssim$
$\displaystyle\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}u\|_{0}^{2}+\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}u\|_{3-k}^{2}+\|\varepsilon^{(k-1)_{+}}\nabla\cdot\partial_{t}^{k}u\|_{3-k}^{2},$
(3.8)
$\displaystyle\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}B\|_{4-k}^{2}\lesssim$
$\displaystyle\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}B\|_{0}^{2}+\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}B\|_{3-k}^{2},$
(3.9)

where we already use $\nabla\cdot B=0$ and the boundary condition
$u_{3}=B_{3}=0$ on $\Sigma$.

The $L^{2}$ norm of $u$ and $B$ has been controlled in (3.2), while the
control of $\|\varepsilon^{(k-1)_{+}}\partial_{t}^{k}(u,B)\|_{0}~{}(1\leq
k\leq 3)$ is parallel to the case of $k=0$ and will be postponed to later
sections. The divergence part is reduced to the estimates of $p$ by using the
continuity equation

$\|\varepsilon^{(k-1)_{+}}\nabla\cdot\partial_{t}^{k}u\|_{3-k}^{2}=\left\|\varepsilon^{(k-1)_{+}+2}\partial_{t}^{k}(D_{t}p)\right\|_{3-k}^{2},$
(3.10)

which will be further reduced to the tangenital estimates of $(u,B)$ by using
the argument in Section 3.2.1.

According to the reduction procedure in section 3.2.1 and (b)-(d) in section
2.3, the control of $E_{4}(t)$ will be reduced to the control of
$\mathcal{T}^{\alpha}(u,B)$ and $\partial_{t}^{4}p$ with suitable Mach number
weights, plus the curl part. Next, we analyze the curl part via the evolution
equation of vorticity.

######  Lemma 3.2.

$\displaystyle\sum_{k=0}^{3}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}u\right\|_{3-k}^{2}+\left\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}B\right\|_{3-k}^{2}+\left\|\varepsilon^{(k-1)_{+}+1}B\times(\nabla\times\partial_{t}^{k}B)\right\|_{3-k}^{2}\right)\leq
P(E_{4}(t),E_{5}(t)).$ (3.11)

###### Proof.

First, we take $\nabla\times$ in the momentum equation $\rho
D_{t}u-(B\cdot\nabla)B=-\nabla P$ to get the evolution equation of vorticity

$\displaystyle\rho D_{t}(\nabla\times u)-(B\cdot\nabla)\nabla\times
B=\rho[D_{t},\nabla\times]u-[B\cdot\nabla,\nabla\times]B-(\nabla\rho)\times(D_{t}u),$
(3.12)

where we notice that the right side only contains the first-order derivatives
and does not lose Mach number weight. Note that the equation of state is
smooth, so $\partial_{S}\rho$ is bounded.

$[D_{t},\partial_{i}](\cdot)=-\partial_{i}u_{j}\partial_{j}(\cdot),~{}~{}[B\cdot\nabla,\partial_{i}](\cdot)=-\partial_{i}B_{k}\partial_{k}(\cdot),~{}~{}\nabla\rho=\frac{\partial\rho}{\partial
p}\nabla p+\partial_{S}\rho\nabla S=O(\varepsilon^{2})\nabla
p+\partial_{S}\rho\nabla S.$

We first prove the case when $k=0$, and the cases for $1\leq k\leq 3$ follow
in the same manner. Considering the structure of the evolution equation (3.12)
and $\rho$ is bounded below by a positive constant, it suffices to prove that

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho\left|\partial^{3}\nabla\times
u\right|^{2}+\left|\partial^{3}\nabla\times B\right|^{2}+\left|\varepsilon
B\times(\partial^{3}\nabla\times B)\right|^{2}\,\mathrm{d}x\leq
P(E_{4}(t),E_{5}(t)).$

We take $\partial^{3}$ in (3.12) to get

$\displaystyle\rho D_{t}\partial^{3}(\nabla\times
u)-(B\cdot\nabla)\partial^{3}\nabla\times B=\underbrace{\partial^{3}(\text{RHS
of }\eqref{curlv eq})+[\rho D_{t},\partial^{3}](\nabla\times
u)-[B\cdot\nabla,\partial^{3}](\nabla\times B)}_{\mathcal{R}_{1}},$ (3.13)

where the order of derivatives on the right side must be $\leq 4$. Now,
standard $L^{2}$-type estimate yields that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|\partial^{3}\nabla\times
u|^{2}\,\mathrm{d}x=\int_{\Omega}\rho(\partial_{t}\partial^{3}\nabla\times
u)\cdot\partial^{3}\nabla\times
u\,\mathrm{d}x+\frac{1}{2}\int_{\Omega}\partial_{t}\rho|\partial^{3}\nabla\times
u|^{2}\,\mathrm{d}x$ $\displaystyle=$ $\displaystyle\int_{\Omega}\rho
D_{t}\partial^{3}\nabla\times u\cdot\partial^{3}\nabla\times u\,\mathrm{d}x$
$\displaystyle\underbrace{-\int_{\Omega}\rho(u\cdot\nabla)\partial^{3}\nabla\times
u\cdot\partial^{3}\nabla\times
u\,\mathrm{d}x-\frac{1}{2}\int_{\Omega}(\frac{\partial\rho}{\partial
p}\partial_{t}p+\partial_{S}\rho\partial_{t}S)|\partial^{3}\nabla\times
u|^{2}\,\mathrm{d}x}_{:=\mathcal{I}_{1}}.$ (3.14)

Then invoking (3.13) gives us the following terms

$\displaystyle\int_{\Omega}(\rho D_{t}\partial^{3}\nabla\times
u)\cdot(\partial^{3}\nabla\times u)\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left((B\cdot\nabla)\partial^{3}\nabla\times
B\right)\cdot(\partial^{3}\nabla\times
u)\,\mathrm{d}x\underbrace{+\int_{\Omega}\mathcal{R}_{1}\cdot(\partial^{3}\nabla\times
u)\,\mathrm{d}x}_{:=\mathcal{I}_{2}}$ $\displaystyle\overset{B\cdot\nabla}{=}$
$\displaystyle-\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot\left(\partial^{3}\nabla\times(B\cdot\nabla
u)\right)\,\mathrm{d}x\underbrace{-\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot\left([B\cdot\nabla,\partial^{3}\nabla\times]u\right)\,\mathrm{d}x}_{:=\mathcal{I}_{3}}+\mathcal{I}_{2}.$
(3.15)

Next, we insert the evolution equation of $B$, that is,
$D_{t}B=(B\cdot\nabla)u-B(\nabla\cdot u)$ to get

$\displaystyle-\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot\left(\partial^{3}\nabla\times(B\cdot\nabla u)\right)\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle-\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot(\partial^{3}\nabla\times
D_{t}B)\,\mathrm{d}x-\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot\left(\partial^{3}\nabla\times(B\nabla\cdot u)\right)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}|\partial^{3}\nabla\times
B|^{2}\,\mathrm{d}x\underbrace{-\int_{\Omega}\partial^{3}\nabla\times
B\cdot\left([\partial^{3}\nabla\times,D_{t}]B+(u\cdot\nabla)\partial^{3}\nabla\times
B\right)\,\mathrm{d}x}_{:=\mathcal{I}_{4}}$
$\displaystyle\underbrace{+\int_{\Omega}(\partial^{3}\nabla\times
B)\cdot\partial^{3}\nabla\times(\varepsilon^{2}BD_{t}p)\,\mathrm{d}x}_{:=\mathcal{K}}.$
(3.16)

Based on the concrete form of the commutators, a straightforward product
estimate for $\mathcal{I}_{j}$ $(1\leq j\leq 4)$ gives us

$\displaystyle\sum_{j=1}^{4}\mathcal{I}_{j}\leq P(E_{4}(t)).$ (3.17)

The most difficult term is $\mathcal{K}$ because the highest-order term has
5-th order derivative and thus cannot be controlled by $E_{4}(t)$:

$\displaystyle\varepsilon^{2}\partial^{3}\nabla\times(BD_{t}p)=$
$\displaystyle~{}\varepsilon^{2}\partial^{3}((\nabla D_{t}p)\times
B+D_{t}p(\nabla\times B))$ $\displaystyle=$
$\displaystyle-\varepsilon^{2}B\times(\partial^{3}\nabla
D_{t}p)\underbrace{-\varepsilon^{2}[\partial^{3},B\times](\nabla
D_{t}p)+\partial^{3}(\varepsilon^{2}D_{t}p(\nabla\times
B))}_{\mathcal{R}_{2}},$

where straightforward calculation shows that $\|\mathcal{R}_{2}\|_{0}\leq
P(E_{4}(t))$. We then further analyze the problematic term
$-\varepsilon^{2}B\times(\partial^{3}\nabla D_{t}p)$. We invoke a simple
vector identity (in 3D)
$(B\cdot\nabla)B-\frac{1}{2}\nabla|B|^{2}=-B\times(\nabla\times B)$ to rewrite
the momentum equation to be

$\rho D_{t}u+B\times(\nabla\times B)=-\nabla p.$

So, we commute $D_{t}$ with $\nabla$ and insert this equation into
$\varepsilon^{2}B\times(\partial^{3}\nabla D_{t}p)$ to get

$\displaystyle-\varepsilon^{2}B\times(\partial^{3}\nabla
D_{t}p)=-\varepsilon^{2}B\times(\partial^{3}D_{t}\nabla
p)-\varepsilon^{2}B\times(\partial^{3}(\nabla u_{j}\partial_{j}p))$
$\displaystyle=$ $\displaystyle~{}\varepsilon^{2}B\times\partial^{3}D_{t}(\rho
D_{t}u)+\varepsilon^{2}B\times\partial^{3}D_{t}(B\times(\nabla\times
B))-\varepsilon^{2}B\times(\partial^{3}(\nabla u_{j}\partial_{j}p))$
$\displaystyle=$ $\displaystyle~{}\varepsilon^{2}\rho
B\times(\partial^{3}D_{t}^{2}u)+\underbrace{\varepsilon^{2}B\times
D_{t}(B\times\partial^{3}(\nabla\times B))}_{=\mathcal{K}_{1}}$
$\displaystyle\underbrace{+\varepsilon^{2}B\times([\partial^{3}D_{t},\rho]D_{t}u)+\varepsilon^{2}B\times\left([\partial^{3},D_{t}](\nabla\times
B)+D_{t}\left([\partial^{3},B\times](\nabla\times
B)\right)\right)-\varepsilon^{2}B\times(\partial^{3}(\nabla
u_{j}\partial_{j}p))}_{\mathcal{R}_{3}},$ (3.18)

where $\|\mathcal{R}_{3}\|_{0}\leq P(E_{4}(t))$ can be proved by using again
the concrete forms of these commutators as the order of derivatives does not
exceed 4. In particular, in the commutator
$D_{t}\left([\partial^{3},B\times](\nabla\times B)\right)$, the highest-order
terms have the form $D_{t}\left(\partial B\times\partial^{2}(\nabla\times
B)\right)$ or $D_{t}\left(\partial^{3}B\times(\nabla\times B)\right)$ and
contain a time derivative, in which there is no loss of either derivative or
$\varepsilon$-weights because there is already an $\varepsilon^{2}$ factor in
$\mathcal{R}_{3}$. Now, we analyze the contribution of $\mathcal{K}_{1}$ in
the integral $\mathcal{K}$, which reads

$\displaystyle\int_{\Omega}\varepsilon^{2}(\partial^{3}\nabla\times
B)\cdot\left(B\times D_{t}\left(B\times\partial^{3}(\nabla\times
B)\right)\right)\,\mathrm{d}x.$ (3.19)

Then we let $\mathbf{u}=B$,
$\mathbf{v}=D_{t}\left(B\times\partial^{3}(\nabla\times B)\right)$ and
$\mathbf{w}=\partial^{3}(\nabla\times B)$ in the vector identity
$(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}=-(\mathbf{u}\times\mathbf{w})\cdot\mathbf{v}$
to get

$\displaystyle\int_{\Omega}\varepsilon^{2}(\partial^{3}\nabla\times
B)\cdot\left(B\times D_{t}\left(B\times\partial^{3}(\nabla\times
B)\right)\right)\,\mathrm{d}x=-\int_{\Omega}\varepsilon^{2}D_{t}\left(B\times(\partial^{3}\nabla\times
B)\right)\cdot\left(B\times(\partial^{3}\nabla\times B)\right)\,\mathrm{d}x,$
(3.20)

in which the right side contributes to an $\varepsilon$-weighted energy terms
instead of introducing a loss of derivative

$\displaystyle-\int_{\Omega}\varepsilon^{2}D_{t}\left(B\times(\partial^{3}\nabla\times
B)\right)\cdot\left(B\times(\partial^{3}\nabla\times B)\right)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}\left|B\times(\partial^{3}\nabla\times
B)\right|^{2}\,\mathrm{d}x+\underbrace{\frac{1}{2}\int_{\Omega}(\nabla\cdot
u)\left|B\times(\partial^{3}\nabla\times
B)\right|^{2}\,\mathrm{d}x}_{\mathcal{R}_{4}}.$ (3.21)

The term $\mathcal{R}_{4}$ is directly controlled by
$\|u\|_{W^{1,\infty}}\|B\|_{L^{\infty}}^{2}\|B\|_{4}^{2}\leq P(E_{4}(t))$.
Therefore, we control the term $\mathcal{K}$ with both $E_{4}(t)$ and
$E_{5}(t)$ as follows:

$\displaystyle\mathcal{K}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}\left|B\times(\partial^{3}\nabla\times
B)\right|^{2}\,\mathrm{d}x\leq\|B\|_{4}\left(P(E_{4}(t))+\|\rho
B\|_{\infty}\|\varepsilon^{2}D_{t}^{2}u\|_{3}\right)\leq
P(E_{4}(t),E_{5}(t)),$ (3.22)

which gives us the energy estimate

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\|\nabla\times
u\|_{3}^{2}+\|\nabla\times B\|_{3}^{2}+\left\|\varepsilon B\times(\nabla\times
B)\right\|_{3}^{2}\right)\leq P(E_{4}(t),E_{5}(t)).$ (3.23)

Recall that
$D_{t}=\partial_{t}+u\cdot\nabla=\partial_{t}+\bar{u}\cdot\overline{\nabla}+u_{3}\partial_{3}$
and $u_{3}|_{\Sigma}=0$ imply that $D_{t}$ is a tangential derivative. So, the
above analysis shows that the curl estimate for compressible ideal MHD cannot
be closed in standard Sobolev space, but in fact we can trade one normal
derivative $(\nabla\times)$ for two tangential derivatives together with
square Mach number weight, that is, $\varepsilon^{2}\mathcal{T}^{2}$. This
step naturally introduces the so-called anisotropic Sobolev space and also
explains why we add $\varepsilon^{2}$weight to $E_{5}(t)$.

Similarly, we can prove the same conclusion for
$\partial^{\alpha}\partial_{t}^{k}$ with $k+|\alpha|=3$ by replacing
$\partial^{3}$ with
$\varepsilon^{(k-1)_{+}}\partial^{\alpha}\partial_{t}^{k}$. Indeed, the
highest order derivatives in the above commutators do not exceed 4-th order,
and there is no loss of Mach number weight because none of the above steps
creates negative power of Mach number. Besides, the problematic term can still
be analyzed in the same manner:

$\varepsilon^{(k-1)_{+}+2}B\times(\partial^{\alpha}\partial_{t}^{k}\nabla
D_{t}p)\stackrel{{\scriptstyle L}}{{=}}-\varepsilon^{(k-1)_{+}}\rho
B\times\partial^{\alpha}\partial_{t}^{k}(\varepsilon^{2}D_{t}^{2}u)-\varepsilon^{(k-1)_{+}}\varepsilon^{2}B\times
D_{t}\left(B\times(\partial^{\alpha}\partial_{t}^{k}(\nabla\times B))\right),$

where the $L^{2}$ norms of the omitted lower-order terms are controlled by
$P(E_{4}(t))$. The contribution of the last term in the above equality is
again the energy term
$-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\varepsilon^{2}\left|B\times(\partial^{\alpha}\partial_{t}^{k}\nabla\times
B)\right|^{2}\,\mathrm{d}x$. The proof of this follows in the same way if we
replace $\partial^{3}$ above by
$\varepsilon^{(k-1)_{+}}\partial^{\alpha}\partial_{t}^{k}$ (that is, using the
vector identity
$(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}=-(\mathbf{u}\times\mathbf{w})\cdot\mathbf{v}$.
Hence, we can conclude the vorticity analysis for $E_{4}(t)$ with the
following inequality

$\displaystyle\sum_{k=0}^{3}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}u\right\|_{3-k}^{2}+\left\|\varepsilon^{(k-1)_{+}}\nabla\times\partial_{t}^{k}B\right\|_{3-k}^{2}+\left\|\varepsilon^{(k-1)_{+}+1}B\times\left(\partial_{t}^{k}\nabla\times
B\right)\right\|_{3-k}^{2}\right)\leq P(E_{4}(t),E_{5}(t)).$ (3.24)

∎

The above div-curl analysis shows that it is necessary to control
$\|\varepsilon^{2+(k-1)_{+}}\mathcal{T}^{2}\partial_{t}^{k}u\|_{3-k}$, which
should be analyzed together with
$\|\varepsilon^{2+(k-1)_{+}}\mathcal{T}^{2}\partial_{t}^{k}B\|_{3-k}$. When
$k=3$, $\mathcal{T}^{2}\partial_{t}^{3}$ is a purely tangential derivative and
we would like to postpone the tangential estimates to the next subsection.
When $0\leq k\leq 2$, we again apply Lemma 3.1 to
$\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}(u,B)\|_{3-k}$ to
obtain

$\displaystyle\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}u\|_{3-k}^{2}\lesssim$
$\displaystyle~{}\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}u\|_{0}^{2}+\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\nabla\cdot\mathcal{T}^{2}u\|_{2-k}^{2}+\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\nabla\times\mathcal{T}^{2}u\|_{2-k}^{2},$
(3.25)
$\displaystyle\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}B\|_{3-k}^{2}\lesssim$
$\displaystyle~{}\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}B\|_{0}^{2}+\|\varepsilon^{2+(k-1)_{+}}[\mathcal{T}^{2},\nabla\cdot]\partial_{t}^{k}B\|_{2-k}^{2},$
(3.26)

where we already use the boundary conditions and the divergence constraint to
eliminate the boundary normal traces and $\nabla\cdot B$. The divergence part
is reduced to the estimates of $p$ by using the continuity equation

$\displaystyle\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\nabla\cdot\mathcal{T}^{2}u\|_{2-k}^{2}\leq$
$\displaystyle~{}\|\varepsilon^{2+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}\nabla\cdot
u\|_{2-k}^{2}+\|\varepsilon^{2+(k-1)_{+}}[\mathcal{T}^{2},\nabla\cdot]\partial_{t}^{k}u\|_{2-k}^{2}$
$\displaystyle=$
$\displaystyle~{}\left\|\varepsilon^{4+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}(D_{t}p)\right\|_{2-k}^{2}+\left\|\varepsilon^{2+(k-1)_{+}}[\mathcal{T}^{2},\nabla\cdot]\partial_{t}^{k}u\right\|_{2-k}^{2},$
(3.27)

where
$\|\varepsilon^{2+(k-1)_{+}}[\mathcal{T}^{2},\nabla\cdot]\partial_{t}^{k}u\|_{2-k}^{2}$
can be controlled by
$\varepsilon^{4}E_{4}(t)\leq\varepsilon^{4}E_{4}(0)+\int_{0}^{t}E_{5}(\tau)\mathrm{d}\tau$.
Indeed, such commutators only appear when we commute the third component of
$\nabla\cdot$ with $\omega\partial_{3}$. We have the following identity which
can be shown by induction:

$[(\omega\partial_{3})^{k},\partial_{3}](\cdot)=\sum_{\ell\leq
k-1}c_{k,\ell}(\omega\partial_{3})^{\ell}\partial_{3}(\cdot)=\sum_{\ell\leq
k-1}d_{k,\ell}\partial_{3}(\omega\partial_{3})^{\ell}(\cdot),$ (3.28)

where $c_{k,\ell}$ and $d_{k,\ell}$ are smooth functions that depend on $k$,
$\ell$ and the derivatives (up to order $k$) of $\omega$. For example, we
assume $k=0$ and $\mathcal{T}^{2}=(\omega\partial_{3})\overline{\partial}{}$,
for which the highest-order term in the commutator is
$\|\varepsilon^{2}(\partial_{3}\omega(x_{3}))\partial_{3}\overline{\partial}{}u_{3}\|_{2}\leq\varepsilon^{4}E_{4}(t).$
This term can be either controlled by
$\varepsilon^{4}E_{4}(0)+\int_{0}^{t}E_{5}(\tau)\mathrm{d}\tau$ or absorbed by
$E_{4}(t)$ when $\varepsilon$ is sufficiently small. Similar estimates apply
to the commutator
$\|\varepsilon^{2+(k-1)_{+}}[\mathcal{T}^{2},\nabla\cdot]\partial_{t}^{k}B\|_{2-k}^{2}$
so we omit the details.

The term
$\|\varepsilon^{4+(k-1)_{+}}\partial_{t}^{k}\mathcal{T}^{2}(D_{t}p)\|_{2-k}^{2}$
will be further reduced to the tangenital estimates of $(u,B)$ by using the
argument in Section 3.2.1. Finally, we need to control the tangential
derivatives of $u$ and $B$ (including time derivatives) in $E_{5}(t)$ and the
$L^{2}$ norm of $\partial_{t}^{5}p$ with suitable Mach number weights.

Next, we analyze the vorticity term. The proof is parallel to Lemma 3.2.

######  Lemma 3.3.

$\displaystyle\sum_{k=0}^{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{2+(k-1)_{+}}\nabla\times\mathcal{T}^{2}\partial_{t}^{k}u\right\|_{2-k}^{2}+\left\|\varepsilon^{2+(k-1)_{+}}\nabla\times\mathcal{T}^{2}\partial_{t}^{k}B\right\|_{2-k}^{2}+\left\|\varepsilon^{2+(k-1)_{+}+1}B\times\left(\nabla\times\mathcal{T}^{2}\partial_{t}^{k}B\right)\right\|_{2-k}^{2}\right)$
(3.29) $\displaystyle\leq$ $\displaystyle~{}P(E_{4}(t),E_{5}(t),E_{6}(t)).$

###### Proof.

For the case of $k=0$, $\alpha_{3}=\alpha_{4}=0$,
$\mathcal{T}^{\alpha}=\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
with $\alpha_{0}+\alpha_{1}+\alpha_{2}=2$, we write $\mathcal{T}^{\alpha}$ to
be $\mathcal{T}^{2}$ for convenience and take
$\partial^{2}\nabla\times\mathcal{T}^{2}$ in the momentum equation $\rho
D_{t}u-(B\cdot\nabla)B=-\nabla(p+\frac{1}{2}|B|^{2})$ to get

$\rho
D_{t}(\partial^{2}\nabla\times\mathcal{T}^{2}u)-(B\cdot\nabla)(\partial^{2}\nabla\times\mathcal{T}^{2}B)=[\rho
D_{t},\partial^{2}\nabla\times\mathcal{T}^{2}]u-[B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]B.$
(3.30)

Again, it suffices to prove that

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho\left|\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u\right|^{2}+\left|\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B\right|^{2}+\left|\varepsilon^{3}B\times(\partial^{2}\nabla\times\mathcal{T}^{2}B)\right|^{2}\,\mathrm{d}x\leq
P(E_{4}(t),E_{5}(t),E_{6}(t)).$

We compute that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u|^{2}\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle\int_{\Omega}\rho
D_{t}(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\cdot(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\,\mathrm{d}x$
$\displaystyle\underbrace{-\int_{\Omega}\rho(u\cdot\nabla)(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\cdot(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\,\mathrm{d}x-\int_{\Omega}(\frac{\partial\rho}{\partial
p}\partial_{t}p+\partial_{S}\rho\partial_{t}S)|\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u|^{2}\,\mathrm{d}x}_{:=\mathcal{I}_{1}^{\prime}}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\varepsilon^{2}(B\cdot\nabla)(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\,\mathrm{d}x$
$\displaystyle\underbrace{+\int_{\Omega}\varepsilon^{2}\left([\rho
D_{t},\partial^{2}\nabla\times\mathcal{T}^{2}]u-[B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]B\right)\cdot(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\,\mathrm{d}x}_{:=\mathcal{I}_{2}^{\prime}}+\mathcal{I}_{1}^{\prime}.$
(3.31)

Then we integrate $B\cdot\nabla$ by parts and invoke the evolution equation of
$B$, namely $D_{t}B=(B\cdot\nabla)u-B(\nabla\cdot u)$ to get

$\displaystyle\int_{\Omega}\varepsilon^{2}(B\cdot\nabla)(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}u)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\int_{\Omega}\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}(B\cdot\nabla
u))\,\mathrm{d}x-\int_{\Omega}\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot\varepsilon^{2}([B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\int_{\Omega}\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}(D_{t}B))\,\mathrm{d}x-\int_{\Omega}\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}(B\nabla\cdot
u))\,\mathrm{d}x$
$\displaystyle-\int_{\Omega}\varepsilon^{2}(\partial^{2}\nabla\times\mathcal{T}^{2}B)\cdot\varepsilon^{2}([B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}|\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B|^{2}\,\mathrm{d}x\underbrace{+\int_{\Omega}\left(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B\right)\cdot\left(\varepsilon^{4}\partial^{2}\nabla\times\mathcal{T}^{2}(BD_{t}p)\right)\,\mathrm{d}x}_{:=\mathcal{K}^{\prime}}$
$\displaystyle\underbrace{-\int_{\Omega}\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B\cdot\varepsilon^{2}\left([\partial^{2}\nabla\times\mathcal{T}^{2},D_{t}]B+(u\cdot\nabla)(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B)+[B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u\right)\,\mathrm{d}x}_{:=\mathcal{I}_{3}^{\prime}}$
(3.32)

A straightforward product estimate for $\mathcal{I}_{1}^{\prime}$ gives us

$\mathcal{I}_{1}^{\prime}\leq P(E_{4}(t),E_{5}(t)).$ (3.33)

For $\mathcal{I}_{2}^{\prime}$, we first consider the commutator
$\varepsilon^{2}[\rho u\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u$.
Since,

$\displaystyle[\rho
u\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u=\partial(\rho
u_{j})\partial^{2}\mathcal{T}^{2}\partial_{j}u_{i}+\mathcal{T}(\rho
u_{j})\partial^{3}\mathcal{T}\partial_{j}u_{i}+\text{ lower order terms},$
(3.34)

we obtain

$\displaystyle\varepsilon^{2}\|\partial(\rho
u_{j})\partial^{2}\mathcal{T}^{2}\partial_{j}u_{i}\|_{0}\leq$
$\displaystyle~{}\|\partial(\rho
u)\|_{\infty}\|\varepsilon^{2}\mathcal{T}^{2}u\|_{3}\leq
P(E_{4}(t),E_{5}(t)),$ (3.35) $\displaystyle\varepsilon^{2}\|\mathcal{T}(\rho
u_{j})\partial^{3}\mathcal{T}\partial_{j}u_{i}\|_{0}\leq$
$\displaystyle~{}\|\mathcal{T}(\rho
u)\|_{\infty}\|\varepsilon^{2}\mathcal{T}\partial_{j}u\|_{3}\leq
P(E_{4}(t),E_{5}(t)),~{}j=1,2,$ (3.36)

Since $u_{3}|_{\Sigma}=0$, using fundamental theorem of calculus, we have
$u_{3}(x_{3})=u_{3}(-1)+\int_{-1}^{x_{3}}\partial_{3}u_{3}(\xi_{3})\mathrm{d}\xi_{3}$
(W.L.O.G. $x_{3}<0$), so the length of the interval $[-1,x_{3}]$ is comparable
to the weight function $\omega(x_{3})=(1-x_{3})(1+x_{3})$. We have

$|\mathcal{T}(\rho u_{3})(x_{3})|\leq
0+\int_{-1}^{x_{3}}\|\partial_{3}\mathcal{T}(\rho
u_{3})(\xi_{3})\|_{\infty}\mathrm{d}\xi_{3}\lesssim\omega(x_{3})\|\partial_{3}\mathcal{T}(\rho
u_{3})\|_{\infty}.$

Then we can combine this $\omega(x_{3})$ with $\partial_{3}$ to convert a
normal derivative to a tangential derivative.

$\displaystyle\varepsilon^{2}\|\mathcal{T}(\rho
u_{3})(x_{3})\partial^{3}\mathcal{T}\partial_{3}u_{i}\|_{0}\lesssim$
$\displaystyle~{}\|\partial_{3}\mathcal{T}(\rho
u_{3})\|_{\infty}\|\varepsilon^{2}\omega(x_{3})\partial^{3}\mathcal{T}\partial_{3}u\|_{0}$
$\displaystyle\leq$ $\displaystyle~{}\|\partial_{3}\mathcal{T}(\rho
u_{3})\|_{\infty}\|\varepsilon^{2}\partial^{3}\mathcal{T}\omega(x_{3})\partial_{3}u\|_{0}+P(E_{4}(t),E_{5}(t))$
$\displaystyle\leq$ $\displaystyle~{}P(E_{4}(t),E_{5}(t)).$ (3.37)

Thus,

$\|\varepsilon^{2}[\rho
u\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]u\|_{0}\leq
P(E_{4}(t),E_{5}(t)).$ (3.38)

Similarly, since $B_{3}|_{\Sigma}=0$,

$\|\varepsilon^{2}[\rho\partial_{t},\partial^{2}\nabla\times\mathcal{T}^{2}]u\|_{0}+\|\varepsilon^{2}[B\cdot\nabla,\partial^{2}\nabla\times\mathcal{T}^{2}]B\|_{0}\leq
P(E_{4}(t),E_{5}(t)),$ (3.39)

then

$\mathcal{I}_{2}^{\prime}\leq P(E_{4}(t),E_{5}(t)).$ (3.40)

We can also get the estimate of $\mathcal{I}_{3}^{\prime}$ in a similar way:

$\mathcal{I}_{3}^{\prime}\leq P(E_{4}(t),E_{5}(t)).$ (3.41)

For $\mathcal{K}^{\prime}$, we consider the estimate of
$\varepsilon^{4}\partial^{2}\nabla\times\mathcal{T}^{2}(BD_{t}p)$. Since,

$\displaystyle\varepsilon^{4}\partial^{2}\nabla\times\mathcal{T}^{2}(BD_{t}p)$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{4}\partial^{2}\mathcal{T}^{2}(D_{t}\nabla
p\times B)\underbrace{+\varepsilon^{4}\partial^{2}\mathcal{T}^{2}(\nabla
u\cdot\nabla p\times
B)+\varepsilon^{4}\partial^{2}\mathcal{T}^{2}(D_{t}p\nabla\times
B)}_{:=\mathcal{R}_{1}^{\prime}}$ $\displaystyle=$
$\displaystyle-B\times\varepsilon^{4}\partial^{2}\mathcal{T}^{2}D_{t}\nabla
p\underbrace{-\varepsilon^{4}[\partial^{2}\mathcal{T}^{2},B\times](D_{t}\nabla
p)}_{\mathcal{R}_{2}^{\prime}}+\mathcal{R}_{1}^{\prime}$ $\displaystyle=$
$\displaystyle~{}B\times(\varepsilon^{4}\partial^{2}\mathcal{T}^{2}D_{t}(\rho
D_{t}u))+B\times\varepsilon^{4}\partial^{2}\mathcal{T}^{2}D_{t}(B\times(\nabla\times
B))+\mathcal{R}_{1}^{\prime}+\mathcal{R}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle~{}B\times(\varepsilon^{4}\rho\partial^{2}\mathcal{T}^{2}D_{t}^{2}u)+\underbrace{\varepsilon^{4}B\times
D_{t}\left(B\times\partial^{2}\mathcal{T}^{2}(\nabla\times
B)\right)}_{\mathcal{K}_{1}^{\prime}}+\mathcal{R}_{1}^{\prime}+\mathcal{R}_{2}^{\prime}$
$\displaystyle\underbrace{+B\times\varepsilon^{4}([\partial^{2}\mathcal{T}^{2}D_{t},\rho]D_{t}u)+\varepsilon^{2}B\times\left([\partial^{2}\mathcal{T}^{2},D_{t}](\nabla\times
B)+D_{t}\left([\partial^{2}\mathcal{T}^{2},B\times](\nabla\times
B)\right)\right)}_{\mathcal{R}_{3}^{\prime}},$ (3.42)

we get that

$\displaystyle\|\mathcal{R}_{1}^{\prime}+\mathcal{R}_{2}^{\prime}+\mathcal{R}_{3}^{\prime}\|_{0}\lesssim$
$\displaystyle
P(E_{4}(t))(\|\varepsilon^{2}\mathcal{T}^{2}B\|_{3}+\|\varepsilon^{2}\mathcal{T}^{3}B\|_{2}+\|\varepsilon^{4}\mathcal{T}^{2}D_{t}p\|_{2}+\|\varepsilon^{4}\mathcal{T}^{2}u\|_{3}).$
(3.43)

It should be noted that when commuting $D_{t}$ with $\overline{\partial}{}$ or
$\partial_{t}$, no extra normal derivative is generated as $D_{t}$ itself is
also a tangential derivative. Indeed, when $j=3$ in
$[D_{t},\overline{\partial}{}_{i}](\cdot)=-\overline{\partial}{}_{i}u_{j}\partial_{j}(\cdot)$
for $i=t,1,2$, we have $\overline{\partial}{}_{i}u_{3}|_{\Sigma}=0$ and so
this is still a tangential derivative.

The analysis of $\mathcal{K}_{1}^{\prime}$ is also similar to the analysis of
$\mathcal{K}_{1}$. That is, the contribution of $\mathcal{K}^{\prime}_{1}$ in
$\mathcal{K}^{\prime}$ is an energy term plus controllable error terms

$\displaystyle\int_{\Omega}\left(\varepsilon^{2}\partial^{2}\nabla\times\mathcal{T}^{2}B\right)\cdot\left(\varepsilon^{4}B\times
D_{t}\left(B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right)\right)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\left|\varepsilon^{3}B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right|^{2}\,\mathrm{d}x+\frac{1}{2}\int_{\Omega}(\nabla\cdot
u)\left|\varepsilon^{3}B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right|^{2}\,\mathrm{d}x$
$\displaystyle\lesssim$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\left|\varepsilon^{3}B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right|^{2}\,\mathrm{d}x+P(E_{4}(t))E_{5}(t),$
(3.44)

and thus

$\mathcal{K}^{\prime}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\left|\varepsilon^{3}B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right|^{2}\,\mathrm{d}x\leq
P(E_{4}(t))\|\varepsilon^{4}\mathcal{T}^{2}D_{t}^{2}u\|_{2}+P(E_{4}(t),E_{5}(t)).$
(3.45)

Here we use the fact that $\mathcal{T}^{2}$ commutes with $\nabla$ when
$\alpha_{4}=0$. So we obtain the vorticity estimates for $k=0$ and
$\alpha_{4}=0$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{2}\nabla\times\mathcal{T}^{2}u\right\|_{2}^{2}+\left\|\varepsilon^{2}\nabla\times\mathcal{T}^{2}B\right\|_{2}^{2}+\left\|\varepsilon^{3}B\times(\nabla\times\mathcal{T}^{2}B)\right\|_{2}^{2}\right)$
(3.46) $\displaystyle\leq$
$\displaystyle~{}P(E_{4}(t),E_{5}(t))\|\varepsilon^{4}\mathcal{T}^{2}D_{t}^{2}u\|_{2}\leq
P(E_{4}(t),E_{5}(t),E_{6}(t)).$

For the case of $\alpha_{4}\geq 1$, since
$[\partial_{3},\omega\partial_{3}]\neq 0$, we need to reconsider the estimates
of commutators. First, we consider commutators:
$[\mathcal{T}^{2},\nabla\times](\cdot)$ and
$[\mathcal{T}^{2},\nabla\cdot](\cdot)$. According to the identity (3.28), such
commutators contain at most two derivatives and at most one of them is
$\partial_{3}$. Thus, we have

$\|\varepsilon^{2}[\mathcal{T}^{2},\nabla\times]u\|_{2}+\|\varepsilon^{2}[\mathcal{T}^{2},\nabla\times]B\|_{2}+\|\varepsilon^{2}[\mathcal{T}^{2},\nabla\cdot]u\|_{2}\leq\varepsilon^{2}\left(E_{4}(0)+\int_{0}^{t}P(E_{4}(\tau))\mathrm{d}\tau\right).$
(3.47)

Next, for example, we consider the commutator
$\varepsilon^{2}[B\cdot\nabla,\partial^{2}\nabla\times(\omega\partial_{3})\mathcal{T}]B$
in $\mathcal{I}_{2}^{\prime}$ (we take $\alpha_{4}=1$,
$\mathcal{T}^{\alpha}=(\omega\partial_{3})\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$
with $\alpha_{0}+\alpha_{1}+\alpha_{2}=1$ without loss of generality),

$\displaystyle[B\cdot\nabla,\partial^{2}\nabla\times(\omega\partial_{3})\mathcal{T}]B=$
$\displaystyle~{}[B_{3}\partial_{3},\omega\partial_{3}^{4}\mathcal{T}]B+\text{
lower order terms}$ $\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}B_{3}\partial_{3}(\omega\partial_{3}^{4}\mathcal{T}B)-\omega\partial_{3}^{4}\mathcal{T}(B_{3}\partial_{3}B)$
$\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle~{}B_{3}(\partial_{3}\omega)\partial_{3}^{3}\mathcal{T}B,$
(3.48)

where the “lower order terms” represent the terms that do not have the leading
order derivative and the terms that $\partial^{2}\nabla\times$ contributes to
tangential derivatives instead of $\partial_{3}^{3}$ as above.

Since $B_{3}|_{\Sigma}=0$, we again use the fundamental theorem of calculus to
get

$\displaystyle\varepsilon^{2}\|B_{3}(\partial_{3}\omega)\partial_{3}^{3}\mathcal{T}B\|_{0}=$
$\displaystyle~{}\|\varepsilon^{2}|B_{3}(x_{3})|\partial_{3}^{3}\mathcal{T}B\|_{0}$
$\displaystyle\lesssim$
$\displaystyle~{}\|\partial_{3}B_{3}\|_{\infty}\|\varepsilon^{2}\omega(x_{3})\partial_{3}^{3}\mathcal{T}B\|_{0}$
$\displaystyle\leq$ $\displaystyle~{}P(E_{4}(t)).$ (3.49)

All the other terms in $\mathcal{I}_{2}^{\prime}$ and
$\mathcal{I}_{3}^{\prime}$ can be treated similarly. Finally, we need to
consider the term $\mathcal{K}_{1}^{\prime}$ because we shall control an extra
commutator arising from

$D_{t}\left(B\times\partial^{2}\mathcal{T}^{2}(\nabla\times
B)\right)=D_{t}\left(B\times\partial^{2}(\nabla\times\mathcal{T}^{2}B)\right)+D_{t}\left(B\times\partial^{2}\left([\mathcal{T}^{2},\nabla\times]B\right)\right).$

Again, using (3.28), $[\mathcal{T}^{2},\nabla\times]B$ contains at most two
derivatives and at most one of these two derivatives is $\partial_{3}$. Thus,
the highest-order part in the term
$D_{t}\left(B\times\partial^{2}\left([\mathcal{T}^{2},\nabla\times]B\right)\right)$
contains at most 3 normal derivatives and two tangential derivatives, whose
$L^{2}(\Omega)$ can still be bounded by $E_{5}(t)$ aftering multiplying the
weight $\varepsilon^{4}$.

At this point, we obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{2}\nabla\times\mathcal{T}^{2}u\right\|_{2}^{2}+\left\|\varepsilon^{2}\nabla\times\mathcal{T}^{2}B\right\|_{2}^{2}+\left\|\varepsilon^{3}B\times(\nabla\times\mathcal{T}^{2}B)\right\|_{2}^{2}\right)$
$\displaystyle\leq$
$\displaystyle~{}P(E_{4}(t),E_{5}(t))\|\varepsilon^{4}\mathcal{T}^{4}u\|_{2}\leq
P(E_{4}(t),E_{5}(t),E_{6}(t)).$ (3.50)

For $k=1,2$, we obtain the following energy estimates in similar way as in the
proof of Lemma 3.2

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{2+(k-1)_{+}}\nabla\times\mathcal{T}^{2}\partial_{t}^{k}u\right\|_{2-k}^{2}+\left\|\varepsilon^{2+(k-1)_{+}}\nabla\times\mathcal{T}^{2}\partial_{t}^{k}B\right\|_{2-k}^{2}+\left\|\varepsilon^{2+(k-1)_{+}+1}B\times(\nabla\times\mathcal{T}^{2}\partial_{t}^{k}B)\right\|_{2-k}^{2}\right)$
$\displaystyle\leq$ $\displaystyle~{}P(E_{4}(t),E_{5}(t),E_{6}(t)).$ (3.51)

The proof of Lemma 3.3 is completed.∎

The proof of Lemma 3.3 shows that we need to trade one normal derivative for
two more tangentials with Mach number weight $\varepsilon^{2}$, that is, we
need to control
$\|\varepsilon^{4}\mathcal{T}^{4}(\partial_{t}^{k}u,\partial_{t}^{k}B)\|_{2-k}~{}(k=0,1,2)$.
We again do the analogous div-curl analysis for the case $k=0,1$ and reduce it
to
$\|\varepsilon^{6}\mathcal{T}^{6}(\partial_{t}^{k}u,\partial_{t}^{k}B)\|_{1-k}$,
and then repeat once more such that we finally need to control
$\|\varepsilon^{8}\mathcal{T}^{8}(u,B)\|_{0}$ which is a purely tangential
estimate. To sum up, we can obtain the following estimates by mimicing the
proof of Lemma 3.2 and Lemma 3.3.

######  Lemma 3.4.

The div-curl analysis and vorticity estimates for
$\|\varepsilon^{4}\mathcal{T}^{4}(u,B)\|_{2}$,
$\|\varepsilon^{4}\partial_{t}\mathcal{T}^{4}(u,B)\|_{1}$ and
$\|\varepsilon^{6}\mathcal{T}^{6}(u,B)\|_{1}$ are listed below:

$\displaystyle\|\varepsilon^{4}\mathcal{T}^{4}\partial_{t}^{k}u\|_{2-k}^{2}\lesssim~{}\|\varepsilon^{4}\mathcal{T}^{4}\partial_{t}^{k}u\|_{0}^{2}+\|\varepsilon^{4}\nabla\times\mathcal{T}^{4}\partial_{t}^{k}u\|_{1-k}^{2}+\|\varepsilon^{4}\nabla\cdot\mathcal{T}^{4}\partial_{t}^{k}u\|_{1-k}^{2},~{}k=0,1,$
(3.52)
$\displaystyle\|\varepsilon^{4}\mathcal{T}^{4}\partial_{t}^{k}B\|_{2-k}^{2}\lesssim~{}\|\varepsilon^{4}\mathcal{T}^{4}\partial_{t}^{k}B\|_{0}^{2}+\|\varepsilon^{4}\nabla\times\mathcal{T}^{4}\partial_{t}^{k}B\|_{1-k}^{2}+\|\varepsilon^{4+(k-1)_{+}}[\mathcal{T}^{4},\nabla\cdot]\partial_{t}^{k}B\|_{1-k}^{2},~{}k=0,1,$
(3.53)
$\displaystyle\sum_{k=0,1}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\varepsilon^{4}\nabla\times\partial_{t}^{k}\mathcal{T}^{4}u\right\|_{1-k}^{2}+\left\|\varepsilon^{4}\nabla\times\partial_{t}^{k}\mathcal{T}^{4}B\right\|_{1-k}^{2}+\left\|\varepsilon^{5}B\times(\nabla\times\partial_{t}^{k}\mathcal{T}^{4}B)\right\|_{1-k}^{2}\right)$
$\displaystyle\leq$
$\displaystyle~{}P(E_{4}(t),E_{5}(t),E_{6}(t))\|\varepsilon^{6}\mathcal{T}^{6}\partial_{t}^{k}u\|_{1-k},$
(3.54)
$\displaystyle\|\varepsilon^{6}\mathcal{T}^{6}u\|_{1}^{2}\lesssim~{}\|\varepsilon^{6}\mathcal{T}^{6}u\|_{0}^{2}+\|\varepsilon^{6}\nabla\times\mathcal{T}^{6}u\|_{0}^{2}+\|\varepsilon^{6}\nabla\cdot\mathcal{T}^{6}u\|_{0}^{2},$
(3.55)
$\displaystyle\|\varepsilon^{6}\mathcal{T}^{6}B\|_{1}^{2}\lesssim~{}\|\varepsilon^{6}\mathcal{T}^{6}B\|_{0}^{2}+\|\varepsilon^{6}\nabla\times\mathcal{T}^{6}B\|_{0}^{2}+\|\varepsilon^{6}[\mathcal{T}^{6},\nabla\cdot]B\|_{0}^{2},$
(3.56) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle\left(\left\|\varepsilon^{6}\nabla\times\mathcal{T}^{6}u\right\|_{0}^{2}+\left\|\varepsilon^{6}\nabla\times\mathcal{T}^{6}B\right\|_{0}^{2}+\left\|\varepsilon^{7}B\times(\nabla\times\mathcal{T}^{6}B)\right\|_{0}^{2}\right)\leq
P(E_{4}(t)E_{5}(t),E_{6}(t),E_{7}(t))\|\varepsilon^{8}\mathcal{T}^{8}u\|_{0}.$
(3.57)

Besides, the divergence part is reduced to the estimates of $p$ by using the
continuity equation

$\displaystyle\|\varepsilon^{4}\partial_{t}^{k}\nabla\cdot\mathcal{T}^{4}u\|_{1-k}^{2}=$
$\displaystyle\left\|\varepsilon^{6}\partial_{t}^{k}\mathcal{T}^{4}(D_{t}p)\right\|_{1-k}^{2}+\|\varepsilon^{4}[\mathcal{T}^{4},\nabla\cdot]\partial_{t}^{k}u\|_{1-k}^{2},~{}~{}~{}k=0,1,$
$\displaystyle\|\varepsilon^{6}\nabla\cdot\mathcal{T}^{6}u\|_{0}^{2}=$
$\displaystyle\left\|\varepsilon^{8}\mathcal{T}^{6}(D_{t}p)\right\|_{0}^{2}+\|\varepsilon^{6}[\mathcal{T}^{6},\nabla\cdot]u\|_{0}^{2},$
(3.58)

where the commutators can be controlled

$\displaystyle\sum_{k=0,1}\|\varepsilon^{4}[\mathcal{T}^{4},\nabla\cdot]\partial_{t}^{k}u\|_{1-k}^{2}\leq$
$\displaystyle~{}\varepsilon^{4}E_{5}(t)\lesssim\varepsilon^{4}E_{5}(0)+\int_{0}^{t}E_{6}(\tau)\mathrm{d}\tau,$
(3.59)
$\displaystyle\|\varepsilon^{6}[\mathcal{T}^{6},\nabla\cdot]u\|_{0}^{2}\leq$
$\displaystyle~{}\varepsilon^{4}E_{6}(t)\lesssim\varepsilon^{4}E_{6}(0)+\int_{0}^{t}E_{7}(\tau)\mathrm{d}\tau.$
(3.60)

by using the identity (3.28). Moreover,
$\left\|\varepsilon^{6}\partial_{t}^{k}\mathcal{T}^{4}(D_{t}p)\right\|_{1-k}^{2}$
and $\left\|\varepsilon^{8}\mathcal{T}^{6}(D_{t}p)\right\|_{0}^{2}$ will be
further reduced to the tangenital estimates of $(u,B)$ by using the argument
in Section 3.2.1.

####  3.2.3 The remaining tangential estimates

Recall that the entropy $S$ is directly controlled via $D_{t}S=0$, so we can
temporarily ignore that. Let us summarize what kind of tangential derivatives
we shall control in order to close the estimates that are uniform in Mach
number $\varepsilon$. Again we start with
$\|\varepsilon^{(k-1)_{+}}(\partial_{t}^{k}u,\partial_{t}^{k}B)\|_{4-k}$ in
$E_{4}(t)$. We set such Mach number weights based on the following facts:

  * a.

$H^{4}$ regularity: Several commutator estimates require the bound for
$\|\overline{\partial}{}^{2}(u,B)\|_{L^{\infty}}$. Recall that
$H^{\frac{d}{2}+\delta}\hookrightarrow L^{\infty}$, so it would be convenient
to choose $H^{2+\lceil\frac{d}{2}+\delta\rceil}$ regularity, that is, $H^{4}$
for $d=2,3$.

  * b.

The initial data is well-prepared, so $\|\nabla p(0)\|_{3}$ should be bounded
with respect to $\varepsilon$ and we shall add Mach number weights to
$\partial_{t}^{k}p$ when $k\geq 1$.

  * c.

The $L^{2}$ estimate in section 3.1 suggests that $u,B,\varepsilon p$ should
share the same weights of Mach number $\varepsilon$ because taking tagential
derivatives preserves the boundary conditions.

  * d.

The reduction procedure in section 3.2.1 shows that $\nabla p$ is converted to
$\mathcal{T}(u,B)$ and $\nabla B$. When $\mathcal{T}$ is a spatial derivative,
$\mathcal{T}(u,B)$ and $\nabla B$ should be controlled via div-curl analysis
as shown in Lemma 3.2. When $\mathcal{T}=\partial_{t}$, we find that
$\partial_{t}^{k}p,~{}\partial_{t}^{k+1}(u,B)$ should share the same weights
of Mach number.

Thus, $E_{4}(t)$ consists of the following quantities.

$\displaystyle\|u,B,S,p\|_{4},~{}\|\partial_{t}u,\partial_{t}B,\partial_{t}S,\varepsilon\partial_{t}p\|_{3},~{}\|\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B,\varepsilon\partial_{t}^{2}S,\varepsilon^{2}\partial_{t}^{2}p\|_{2},$
$\displaystyle\|\varepsilon^{2}\partial_{t}^{3}u,\varepsilon^{2}\partial_{t}^{3}B,\varepsilon^{2}\partial_{t}^{3}S,\varepsilon^{3}\partial_{t}^{3}p\|_{1},~{}\|\varepsilon^{3}\partial_{t}^{4}u,\varepsilon^{3}\partial_{t}^{4}B,\varepsilon^{3}\partial_{t}^{4}S,\varepsilon^{4}\partial_{t}^{4}p\|_{0}.$

The Sobolev norms of
$\|\varepsilon^{(k-1)_{+}}(\partial_{t}^{k}u,\partial_{t}^{k}B)\|_{4-k}$ for
$0\leq k\leq 3$ are controlled via div-curl analysis. The divergence part can
be absorbed by $E_{4}(t)$ itself because the continuity equation produces
extra $\varepsilon^{2}$ weight and $\varepsilon^{2}<\varepsilon$ since we
assume the Mach number is small. Even if we do not assume $\varepsilon$ is
small, we can alternatively control the divergence by repeatedly using the
reduction of pressure such that there are only tangential derivatives. The
curl part, thanks to the special structure of Lorentz force, is reduced to
$\|\varepsilon^{(k-1)_{+}+2}D_{t}^{2}\partial_{t}^{k}u\|_{3-k}$ which is a
part of $E_{5}(t)$. So, it remains to control $E_{5}(t)$ and the full time
derivatives, namely
$\|\varepsilon^{3}\partial_{t}^{4}u,\varepsilon^{3}\partial_{t}^{4}B,\varepsilon^{4}\partial_{t}^{4}p\|_{0}$.
See also diagram (2.8).

Then $E_{5}(t)$ is designed in the same manner as $E_{4}(t)$:

$\displaystyle\|\varepsilon^{2}\mathcal{T}^{2}(u,B,S,p)\|_{3}~{}(\text{produced
from the vorticity analysis of }\|u,B\|_{4}),$
$\displaystyle\|\varepsilon^{2}\mathcal{T}^{2}(\partial_{t}u,\partial_{t}B,\partial_{t}S,\varepsilon\partial_{t}p)\|_{2}~{}(\text{produced
from the vorticity analysis of }\|\partial_{t}u,\partial_{t}B\|_{3}),$
$\displaystyle\|\varepsilon^{2}\mathcal{T}^{2}(\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B,\varepsilon\partial_{t}^{2}S,\varepsilon^{2}\partial_{t}^{2}p)\|_{1}~{}(\text{produced
from the vorticity analysis of
}\|\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B\|_{2}),$
$\displaystyle\|\varepsilon^{2}\mathcal{T}^{2}(\varepsilon^{2}\partial_{t}^{3}u,\varepsilon^{2}\partial_{t}^{3}B,\varepsilon^{2}\partial_{t}^{3}S,\varepsilon^{3}\partial_{t}^{3}p)\|_{0}~{}(\text{produced
from the vorticity analysis of
}\|\varepsilon^{2}\partial_{t}^{3}u,\varepsilon^{2}\partial_{t}^{3}B\|_{1}).$

The div-curl analysis of $E_{5}(t)$ produces
$\|\varepsilon^{(k-1)_{+}+4}\mathcal{T}^{2}D_{t}^{2}\partial_{t}^{k}u\|_{2-k}$
which is part of $E_{6}(t)$. In order to close the energy estimates, we now
need to control $E_{6}(t)$ and also the full time derivatives
$\|\varepsilon^{2}\mathcal{T}^{2}(\varepsilon\partial_{t}^{3}u,\varepsilon\partial_{t}^{3}B,\varepsilon^{2}\partial_{t}^{3}p)\|_{0}$.

Repeat the above procedure once more, we can see $E_{6}(t)$ should be designed
as

$\displaystyle\|\varepsilon^{4}\mathcal{T}^{4}(u,B,S,p)\|_{2}~{}(\text{produced
from the vorticity analysis of }\|\varepsilon^{2}\mathcal{T}^{2}(u,B)\|_{3}),$
$\displaystyle\|\varepsilon^{4}\mathcal{T}^{4}(\partial_{t}u,\partial_{t}B,\partial_{t}S,\varepsilon\partial_{t}p)\|_{1}~{}(\text{produced
from the vorticity analysis of
}\|\varepsilon^{2}\mathcal{T}^{2}(\partial_{t}u,\partial_{t}B)\|_{2},$
$\displaystyle\|\varepsilon^{4}\mathcal{T}^{4}(\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B,\varepsilon\partial_{t}^{2}S,\varepsilon^{2}\partial_{t}^{2}p)\|_{0}~{}(\text{produced
from the vorticity analysis of
}\|\varepsilon^{2}\mathcal{T}^{2}(\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B)\|_{1}).$

Combining the result of vorticity estimate, it remains to control
$\|\varepsilon^{6}\mathcal{T}^{4}D_{t}^{2}\partial_{t}^{k}u\|_{1-k}$ (part of
$E_{7}(t)$) and also the full time derivatives
$\|\varepsilon^{4}\mathcal{T}^{4}(\varepsilon\partial_{t}^{2}u,\varepsilon\partial_{t}^{2}B,\varepsilon\partial_{t}^{2}S,\varepsilon^{2}\partial_{t}^{2}p)\|_{0}$.

Repeat the above procedure once more, we can see $E_{7}(t)$ should be designed
as

$\displaystyle\|\varepsilon^{6}\mathcal{T}^{6}(u,B,S,p)\|_{1}~{}(\text{produced
from the vorticity analysis of }\|\varepsilon^{4}\mathcal{T}^{4}(u,B)\|_{2}),$
$\displaystyle\|\varepsilon^{6}\mathcal{T}^{6}(\partial_{t}u,\partial_{t}B,\partial_{t}S,\varepsilon\partial_{t}p)\|_{0}~{}(\text{produced
from the vorticity analysis of
}\|\varepsilon^{4}\mathcal{T}^{4}(\partial_{t}u,\partial_{t}B)\|_{1}\text{ and
{(c)}}.$

Discussion about the weights of $p$. For the pressure $p$, we only have the
control of
$\|\varepsilon^{k+2l}\mathcal{T}^{\alpha}\partial_{t}^{k}p\|_{4-k-l}$ at this
step, which has one more $\varepsilon$-weight than what we defined in (1.18)
when $k\geq 1,~{}l\geq 1$ and $\alpha_{0}<2l$. To replace $\varepsilon^{k+2l}$
by $\varepsilon^{(k-1)_{+}+2l}$ when $l\geq 1$ and $\alpha_{0}<2l$, we just
need to notice that $\mathcal{T}^{\alpha}$ now must contain at least one
tangential spatial derivative, and then we can use the momentum equation

$-\partial_{i}p=B_{k}\partial_{i}B_{k}+\rho
D_{t}u_{i}-(B\cdot\nabla)B_{i},~{}-\omega\partial_{3}p=B_{k}\underbrace{\omega\partial_{3}B_{k}}_{\text{tangential}}+\omega\rho
D_{t}u_{3}-(\omega B\cdot\nabla)B_{3}$

to reduce the control of
$\|\varepsilon^{(k-1)_{+}+2l}\mathcal{T}^{\alpha}\partial_{t}^{k}p\|_{4-k-l}$
($\langle\alpha\rangle=2l,~{}\alpha_{0}<2l$) to
$\|\varepsilon^{(k-1)_{+}+2l}\mathcal{T}^{\beta}\partial_{t}^{k}(u,B)\|_{4-k-l}$
for some $\beta$ satisfying $\langle\beta\rangle=2l$ and
$\alpha_{0}\leq\beta_{0}\leq\alpha_{0}+1$, which has been included in
$E_{4+l}(t)$.

Combining the result of vorticity estimate, it remains to control
$\|\varepsilon^{6}\mathcal{T}^{\alpha}(\partial_{t}u,\partial_{t}B,\partial_{t}p)\|_{0}$
for $\langle\alpha\rangle=6,~{}\alpha_{0}<6$,
$\|\varepsilon^{6}(\partial_{t}^{6}u,\partial_{t}^{6}B,\varepsilon\partial_{t}^{7}p)\|_{0}$
and $\|\varepsilon^{8}\mathcal{T}^{\alpha}D_{t}^{2}u\|_{0}$ for
$\langle\alpha\rangle=6$. This indicates us to define $E_{8}(t)$ to be

$E_{8}(t)=\sum_{\langle\alpha\rangle=8,\alpha_{0}<8}\left\|\varepsilon^{8}\mathcal{T}^{\alpha}(u,B,S,p)\right\|_{0}^{2}+\left\|\varepsilon^{8}\partial_{t}^{8}(u,B,S,\varepsilon
p)\right\|_{0}^{2}.$

To sum up, the remaining estimates are all tangential estimates, which can be
proved by analyzing the $L^{2}$ estimates after taking the following
tangential derivatives to MHD system (1.13)

$\displaystyle\varepsilon^{3}\partial_{t}^{4},~{}\varepsilon^{4}\mathcal{T}^{2}\partial_{t}^{3},~{}\varepsilon^{5}\mathcal{T}^{4}\partial_{t}^{2},~{}\varepsilon^{6}\mathcal{T}^{6}\partial_{t},~{}\varepsilon^{8}\mathcal{T}^{8}.$
(3.61)

###  3.3 Tangential estimates

####  3.3.1 The $\mathcal{T}^{\alpha}$-differentiated equations

By the div-curl analysis, the crucial step is to study the higher order
tangential energy estimate of (1.13). In particular, we define the following
tangential derivatives

$\mathcal{T}_{0}=\partial_{t},\quad\mathcal{T}_{1}=\partial_{1},\quad\mathcal{T}_{2}=\partial_{2},\quad\mathcal{T}_{4}=\omega(x_{3})\partial_{3}.$
(3.62)

We take $\mathcal{T}^{\alpha}$ in the momentum equation $\rho D_{t}u=-\nabla
p-B\times(\nabla\times B)$ to get

$\rho
D_{t}(\mathcal{T}^{\alpha}u)-(B\cdot\nabla)(\mathcal{T}^{\alpha}B)+\mathcal{T}^{\alpha}\nabla(p+|B|^{2}/2)=[\rho
D_{t},\mathcal{T}^{\alpha}]u-[B\cdot\nabla,\mathcal{T}^{\alpha}]B,$ (3.63)

where
$\mathcal{T}^{\alpha}:=\mathcal{T}_{0}^{\alpha_{0}}\mathcal{T}_{1}^{\alpha_{1}}\mathcal{T}_{2}^{\alpha_{2}}\mathcal{T}_{4}^{\alpha_{4}}$
and $\langle\alpha\rangle=\alpha_{0}+\alpha_{1}+\alpha_{2}+0\times
2+\alpha_{4}\leq 8$. Accoding to section 3.2.3, it suffices to consider the
following cases

  * •

$\langle\alpha\rangle=\alpha_{0}=4$ with $\varepsilon^{3}$ weight,

  * •

$\langle\alpha\rangle=5,~{}\alpha_{0}\geq 3$ with $\varepsilon^{4}$ weight,

  * •

$\langle\alpha\rangle=6,~{}\alpha_{0}\geq 2$ with $\varepsilon^{5}$ weight,

  * •

$\langle\alpha\rangle=7,~{}\alpha_{0}\geq 1$ with $\varepsilon^{6}$ weight,

  * •

$\langle\alpha\rangle=8$ with $\varepsilon^{8}$ weight.

Since these derivatives are all tangential, taking either of them still
preserves the boundary conditions. For simplicity, we will only show the
details of the following cases (either full spatial derivatives or full time
derivatives)

$\varepsilon^{8}\mathcal{T}^{\alpha}\text{ with }\langle\alpha\rangle=8\text{
and
}\alpha_{0}=0,~{}~{}\varepsilon^{8}\partial_{t}^{8},~{}~{}\varepsilon^{k-1}\partial_{t}^{k}~{}(4\leq
k\leq 7).$

and the other cases (space-time mixed derivatives) can be proven in the same
manner.

####  3.3.2 Tangential energy estimate with full spatial derivatives

In this subsection we study the spatially-differentiated equations, i.e., the
equations obtained by commuting $\mathcal{T}^{\alpha}$, $\alpha_{0}=0$, and
$\langle\alpha\rangle=8$, with (1.13). We aim to prove the following estimate

######  Proposition 3.5.

For $\mathcal{T}^{\alpha}$ with multi-index $\alpha$ satisfying $\alpha_{0}=0$
and $\langle\alpha\rangle=8$, we have the energy inequality:

$\sum_{\langle\alpha\rangle=8,\alpha_{0}=0}\frac{\mathrm{d}}{\mathrm{d}t}\left\|\varepsilon^{8}\mathcal{T}^{\alpha}(u,B,p)\right\|_{0}^{2}\leq
P(E(t)).$ (3.64)

###### Proof.

We first consider the case of $\alpha_{4}=0$, that is,
$\mathcal{T}^{\alpha}=\overline{\partial}{}_{1}^{\alpha_{1}}\overline{\partial}{}_{2}^{\alpha_{2}}$
with $\alpha_{1}+\alpha_{2}=8$. To simplify our notation, we will write
$\overline{\partial}{}^{8}$ to represent such $\mathcal{T}^{\alpha}$.

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho|\varepsilon^{8}\overline{\partial}{}^{8}u|^{2}\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}\varepsilon^{16}\rho\overline{\partial}{}^{8}\partial_{t}u\cdot\overline{\partial}{}^{8}u\,\mathrm{d}x+\frac{1}{2}\int_{\Omega}\partial_{t}\rho|\varepsilon^{8}\overline{\partial}{}^{8}u|^{2}\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(\rho
D_{t}u)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}u)\,\mathrm{d}x\underbrace{-\int_{\Omega}\varepsilon^{16}\left(\rho\overline{\partial}{}^{8}(u\cdot\nabla
u)+[\overline{\partial}{}^{8},\rho]D_{t}u+(\frac{\partial\rho}{\partial
p}\partial_{t}p+\partial_{S}\rho\partial_{t}S)\overline{\partial}{}^{8}u\right)\cdot\overline{\partial}{}^{8}u\,\mathrm{d}x}_{:=\mathcal{J}_{1}}.$
(3.65)

The first integral above, after invoking the momentum equation and integrating
$B\cdot\nabla$ by parts, is equal to

$\displaystyle\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(B\cdot\nabla
B)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}u)\,\mathrm{d}x-\int_{\Omega}\varepsilon^{2k}\overline{\partial}{}^{8}(\nabla(p+|B|^{2}/2))\cdot(\varepsilon^{2k}\overline{\partial}{}^{4+k}u)\,\mathrm{d}x.$
(3.66)

For the first term, we integrate $B\cdot\nabla$ by parts

$\displaystyle\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(B\cdot\nabla
B)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}u)\,\mathrm{d}x$
$\displaystyle\overset{B\cdot\nabla}{=}$
$\displaystyle-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B\cdot(\varepsilon^{8}\overline{\partial}{}^{8}(B\cdot\nabla
u))\,\mathrm{d}x\underbrace{+\int_{\Omega}\varepsilon^{8}[\overline{\partial}{}^{8},B\cdot\nabla]B\cdot(\varepsilon^{8}\overline{\partial}{}^{8}u)\,\mathrm{d}x-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B\cdot(\varepsilon^{8}[B\cdot\nabla,\overline{\partial}{}^{8}]u)\,\mathrm{d}x}_{:=\mathcal{J}_{2}}.$
(3.67)

For the second term, we integrate by parts and invoke the continuity equation
to get

$\displaystyle-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(\nabla(p+|B|^{2}/2))\cdot(\varepsilon^{8}\overline{\partial}{}^{8}u)\,\mathrm{d}x=\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(p+|B|^{2}/2)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(|B|^{2}/2)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}p\cdot(\varepsilon^{8}\overline{\partial}{}^{8}(\varepsilon^{2}D_{t}p))\,\mathrm{d}x,$
(3.68)

where the boundary intergral on $\Sigma$ vanishes thanks to the boundary
condition $u_{3}=B_{3}=0$ on $\Sigma$.

The first integral in (3.3.2), after inserting the evolution equation of $B$,
is equal to

$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}|\varepsilon^{8}\overline{\partial}{}^{8}B|^{2}\,\mathrm{d}x\underbrace{-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B\cdot((u\cdot\nabla)\overline{\partial}{}^{8}B+[\overline{\partial}{}^{8},u\cdot\nabla]B)\,\mathrm{d}x}_{:=\mathcal{J}_{3}}$
$\displaystyle\underbrace{-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B\cdot(\varepsilon^{8}\overline{\partial}{}^{8}(B\nabla\cdot
u))\,\mathrm{d}x}_{:=\mathcal{K}_{1}},$ (3.69)

which gives the energy of $B$. The last term on the right side of (3.3.2)
gives the energy of $\varepsilon p$

$\displaystyle-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}p\cdot(\varepsilon^{8}\overline{\partial}{}^{8}(\varepsilon^{2}D_{t}p))\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\left|\varepsilon^{8}\overline{\partial}{}^{8}(\varepsilon
p)\right|^{2}\,\mathrm{d}x\underbrace{-\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}p\cdot(\varepsilon^{10}[\overline{\partial}{}^{8},u\cdot\nabla]p)\,\mathrm{d}x}_{:=\mathcal{J}_{4}}.$
(3.70)

Then we analyze the first term on the right side of (3.3.2). Since
$\overline{\partial}{}(\frac{1}{2}|B|^{2})=\overline{\partial}{}B\cdot B$, we
have

$\displaystyle\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}(|B|^{2}/2)\cdot(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\underbrace{\int_{\Omega}(\varepsilon^{8}B)\cdot(\overline{\partial}{}^{8}B)(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x}_{:=\mathcal{K}_{2}}+\underbrace{\sum_{0<\langle\alpha\rangle<8}\int_{\Omega}\varepsilon^{8}C_{\alpha}(\overline{\partial}{}^{\alpha}B)\cdot(\overline{\partial}{}^{8-\alpha}B)(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x}_{:=\mathcal{J}_{5}}$ (3.71)

where $C_{\alpha}$ are some positive constants. Now let us control the
commutators $\mathcal{J}_{1}\sim\mathcal{J}_{5}$. For
$\mathcal{J}_{1}\sim\mathcal{J}_{4}$, since $u_{3}=B_{3}=0$ on $\Sigma$, it
suffices to analyze one of the following two types of commutators

$\varepsilon^{8}[\overline{\partial}{}^{8},u\cdot\nabla]f,~{}~{}\varepsilon^{8}[\overline{\partial}{}^{8},B\cdot\nabla]f.$

For example. we expand the first one to find that

$\displaystyle\varepsilon^{8}[\overline{\partial}{}^{8},u\cdot\nabla]f=\varepsilon^{8}\overline{\partial}{}^{8}u\cdot\nabla
f+\sum_{j=1,2}\varepsilon^{8}(\overline{\partial}{}\bar{u}_{j})(\overline{\nabla}_{j}\overline{\partial}{}^{7}f)+\varepsilon^{8}(\overline{\partial}{}u_{3})(\partial_{3}\overline{\partial}{}^{7}f)+\sum_{k=2}^{7}\binom{8}{k}\varepsilon^{8}\overline{\partial}{}^{k}u_{j}\partial_{j}\overline{\partial}{}^{8-j}f,$

and it is easy to see that the last term is directly controlled by $E(t)$. For
the first term, we have

$\|\varepsilon^{8}\overline{\partial}{}^{8}u\cdot\nabla
f+\varepsilon^{8}(\overline{\partial}{}\bar{u})\cdot(\overline{\nabla}\overline{\partial}{}^{7}f)\|_{0}\leq\|\varepsilon^{8}\overline{\partial}{}^{8}u\|_{0}\|\nabla
f\|_{\infty}+\|\overline{\partial}{}u\|_{L^{\infty}}\|\varepsilon^{8}\overline{\partial}{}^{8}f\|_{0},$

which is directly controlled by $E(t)$. For the third term, note that
$\overline{\partial}{}u_{3}|_{\Sigma}=0$, using fundamental theorem of
calculus, we have

$|\overline{\partial}{}u_{3}(x_{3})|\lesssim\omega(x_{3})\|\overline{\partial}{}\partial_{3}u_{3}\|_{L^{\infty}},$

and thus

$\|\varepsilon^{8}(\overline{\partial}{}u_{3})(\partial_{3}\overline{\partial}{}^{7}f)\|_{0}\lesssim\|\overline{\partial}{}\partial_{3}u_{3}\|_{L^{\infty}}\|\varepsilon^{8}(\omega\partial_{3})\overline{\partial}{}^{7}f\|_{0}$

which is again controlled by $E(t)$. As for $\mathcal{J}_{5}$, we integrate
$\overline{\partial}{}$ by parts to get

$\displaystyle\mathcal{J}_{5}\lesssim\sum_{0<\langle\alpha\rangle<8}\|\varepsilon^{8}\overline{\partial}{}^{\alpha+1}B\cdot\overline{\partial}{}^{8-\alpha}B)\|_{0}\|\varepsilon^{8}\overline{\partial}{}^{7}(\nabla\cdot
u)\|_{0}\lesssim
E(t)\|\varepsilon^{10}\overline{\partial}{}^{7}D_{t}p\|_{0}\leq P(E(t)).$

Thus, all the commutators $\mathcal{J}_{i}~{}(i=1,\cdots,5)$ are directly
controlled

$\displaystyle\mathcal{J}_{1}+\mathcal{J}_{2}+\mathcal{J}_{3}+\mathcal{J}_{4}+\mathcal{J}_{5}\leq
P(E(t)).$ (3.72)

Next, we show that $\mathcal{K}_{1}$ has cancellation with $\mathcal{K}_{2}$.
We have

$\displaystyle\mathcal{K}_{1}=$
$\displaystyle-\sum_{i,j=1}^{3}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot(\varepsilon^{8}\overline{\partial}{}^{8}(B_{i}\partial_{j}u_{j}))\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\underbrace{-\int_{\Omega}\varepsilon^{8}B\cdot\overline{\partial}{}^{8}B(\varepsilon^{8}\overline{\partial}{}^{8}\nabla\cdot
u)\,\mathrm{d}x}_{=-\mathcal{K}_{2}}-\sum_{i,j=1}^{3}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot(\varepsilon^{8}\overline{\partial}{}B_{i}\overline{\partial}{}^{7}\partial_{j}u_{j})\,\mathrm{d}x$
$\displaystyle\underbrace{-\sum_{i,j=1}^{3}\sum_{2\leq\langle\alpha\rangle\leq
8}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot(\varepsilon^{8}C_{\alpha}\overline{\partial}{}^{\alpha}B_{i}\overline{\partial}{}^{8-\alpha}\partial_{j}u_{j})\,\mathrm{d}x}_{\mathcal{J}_{6}}.$
(3.73)

The last term $\mathcal{J}_{6}$ is again directly controlled:
$\mathcal{J}_{6}\leq P(E(t))$. The first term cancels with $\mathcal{K}_{2}$.
For the second term, we need to further analyze the case when $j=3$. Thanks to
the continuity equation, we have
$-\partial_{3}u_{3}=\varepsilon^{2}D_{t}p+\overline{\nabla}\cdot\bar{u}$ and
thus

$\displaystyle-\sum_{i,j=1}^{3}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot(\varepsilon^{8}\overline{\partial}{}B_{i}\overline{\partial}{}^{7}\partial_{3}u_{3})\,\mathrm{d}x=\sum_{i=1}^{3}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot\left(\varepsilon^{8}\overline{\partial}{}B_{i}\overline{\partial}{}^{7}(\varepsilon^{2}D_{t}p+\overline{\nabla}\cdot\bar{u})\right)\,\mathrm{d}x\lesssim
P(E(t)).$ (3.74)

When $j=1,2$, the second term is again directly controlled by $P(E(t))$
because all the derivatives are tangential.

$\displaystyle-\sum_{i=1}^{3}\sum_{j=1,2}\int_{\Omega}\varepsilon^{8}\overline{\partial}{}^{8}B_{i}\cdot(\varepsilon^{8}\overline{\partial}{}B_{i}\overline{\partial}{}^{7}\overline{\partial}{}_{j}\bar{u}_{j})\,\mathrm{d}x\leq
P(E(t)).$ (3.75)

Thus, we conclude that

$\displaystyle\mathcal{K}_{1}+\mathcal{K}_{2}\leq P(E(t)).$ (3.76)

Combining (3.66)-(3.76), we conclude that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|\varepsilon^{8}\overline{\partial}{}^{8}(\varepsilon
p,u,B)\right\|_{0}^{2}\leq P(E(t)).$ (3.77)

To control $\|\varepsilon^{8}\overline{\partial}{}^{8}p\|$, it suffices to
invoke the momentum equation
$-\overline{\partial}{}_{i}p=B\cdot(\overline{\partial}{}_{i}B)+\rho
D_{t}\bar{u}_{i}-(B\cdot\nabla)B$ and thus we convert
$\varepsilon^{8}\overline{\partial}{}^{8}p$ to
$\varepsilon^{8}\overline{\partial}{}^{7}\mathcal{T}(u,B)$ which is part of
$E_{8}(t)$. We then conclude that

$\displaystyle\left\|\varepsilon^{8}\overline{\partial}{}^{8}(p,u,B)(t)\right\|_{0}^{2}\leq
E_{8}(0)+\int_{0}^{t}P(E(\tau))\mathrm{d}{\tau}.$ (3.78)

For the case of $\alpha_{4}\geq 1$, since
$[\partial_{3},\omega\partial_{3}]\neq 0$, we need to reconsider the estimates
of commutators. For the commutators of type
$[\mathcal{T}^{\alpha},\partial_{3}](\cdot)$, we can use identity (3.28). For
example, we need to control the extra terms when commuting
$\mathcal{T}^{\alpha}$ with $\nabla$:

$-\int_{\Omega}(\varepsilon^{8}\mathcal{T}^{\alpha}u_{i})(\varepsilon^{8}[\mathcal{T}^{\alpha},\partial_{i}]P)\,\mathrm{d}x\text{
and
}\int_{\Omega}(\varepsilon^{8}[\partial_{i},\mathcal{T}^{\alpha}]u_{i})(\varepsilon^{8}\mathcal{T}^{\alpha}P)\,\mathrm{d}x.$

Without loss of generality, we assume
$\mathcal{T}^{\alpha}=(\omega(x_{3})\partial_{3})\overline{\partial}{}^{7}$.
In the first integral above, the highest-order term should be

$-\int_{\Omega}(\varepsilon^{8}\mathcal{T}^{\alpha}u_{3})(\varepsilon^{8}~{}\partial_{3}\omega~{}\partial_{3}\overline{\partial}{}^{7}P)\,\mathrm{d}x.$

Now we invoke the momentum equation to replace $-\partial_{3}P$ with $\rho
D_{t}u_{3}-(B\cdot\nabla)B_{3}$. Recall that $D_{t}$ and $B\cdot\nabla$ are
both tangential derivatives thanks to $u_{3}|_{\Sigma}=B_{3}|_{\Sigma}=0$. So
the above integral is controlled by
$\|\varepsilon^{8}\mathcal{T}^{\alpha}u\|_{0}(\|\varepsilon^{8}\mathcal{T}^{\alpha^{\prime}}u\|_{0}+\|\varepsilon^{8}\mathcal{T}^{\alpha^{\prime\prime}}B\|_{0})$
plus lower-order terms where
$\langle\alpha^{\prime}\rangle=\langle\alpha^{\prime\prime}\rangle=8$.
Similarly, for the second integral above, one can invoke the continuity
equation, which reads
$\partial_{3}u_{3}=-\overline{\nabla}\cdot\bar{u}-\varepsilon^{2}D_{t}p$, to
finish the control. A slight difference is that we do not have the energy for
$\|\varepsilon^{8}\mathcal{T}^{\alpha}p\|_{0}$. Luckily, at this step,
$\mathcal{T}^{\alpha}$ must contain a spatial derivative
$\omega(x_{3})\partial_{3}$. So, it suffices to again invoke the momentum
equation

$-\omega\partial_{3}p=\underbrace{\omega\partial_{3}}_{\text{tangential}}(\frac{1}{2}|B|^{2})+\omega\rho
D_{t}u_{3}-\omega(B\cdot\nabla B_{3})$

such that all terms that contain the fluid pressure are replaced by the
velocity and the magnetic field.

As for the other commutators, for example, we re-consider
$\varepsilon^{8}[\omega\partial_{3}\overline{\partial}{}^{7},B\cdot\nabla]B$
in the analogue of $\mathcal{J}_{2}$ (W.L.O.G. $\alpha_{4}=1$). We have

$\displaystyle[\omega\partial_{3}\overline{\partial}{}^{7},B\cdot\nabla]B=$
$\displaystyle~{}[\omega\partial_{3}\overline{\partial}{}^{7},B_{3}\partial_{3}]B+[\omega\partial_{3}\overline{\partial}{}^{7},\bar{B}\cdot\overline{\nabla}]B$
$\displaystyle=$
$\displaystyle~{}\omega\partial_{3}\overline{\partial}{}^{7}(B_{3}\partial_{3}B)-B_{3}\partial_{3}(\omega\partial_{3}\overline{\partial}{}^{7}B)+[\omega\partial_{3}\overline{\partial}{}^{7},\bar{B}\cdot\overline{\nabla}]B.$
(3.79)

Compared with the case of $\alpha_{4}=0$, the potential risk is that
$\partial_{3}$ may fall on $\omega(x_{3})$ and then converts the tangential
derivative $\omega\partial_{3}$ to be a normal derivative $\partial_{3}$. So,
it suffices to analyze the term
$-B_{3}(\partial_{3}\omega)(\partial_{3}\overline{\partial}{}^{7}B)$. Luckily,
there is a $B_{3}$ (replaced by $u_{3}$ if we commute $\mathcal{T}^{\alpha}$
with $u\cdot\nabla$). Since $B_{3}|_{\Sigma}=0$, we again use the fundamental
theorem of calculus to obtain

$|B_{3}(x_{3})|\leq\omega(x_{3})\|\partial_{3}B_{3}\|_{\infty},$

and thus

$\displaystyle\varepsilon^{8}\|B_{3}(\partial_{3}\omega)(\partial_{3}\overline{\partial}{}^{7}B)\|_{0}\lesssim$
$\displaystyle\|\partial_{3}B_{3}\|_{\infty}\|\varepsilon^{8}(\omega\partial_{3})\overline{\partial}{}^{7}B\|_{0}\leq
E(t).$ (3.80)

Other terms arise from commutators can be treated similarly. Therefore, we
obtain the energy estimate

$\displaystyle\sum_{\langle\alpha\rangle=8,\alpha_{0}=0}\frac{\mathrm{d}}{\mathrm{d}t}\left\|\varepsilon^{8}\mathcal{T}^{\alpha}(u,B,p)\right\|_{0}^{2}\leq
P(E(t)).$ (3.81)

The proof is completed.∎

####  3.3.3 Tangential energy estimate with full time derivatives

In this subsection we study the time-differentiated equations, i.e., the
equations obtained by commuting $\partial_{t}^{k}$, with (1.13). We aim to
prove the following proposition which gives stronger estimates for full time
derivatives than what we need in (3.61).

######  Proposition 3.6.

The following tangential estimate for full time derivatives holds:

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\sum_{k=4}^{7}\left\|\varepsilon^{k-1}\partial_{t}^{k}(u,B,\varepsilon
p)\right\|_{0}^{2}+\left\|\varepsilon^{8}\partial_{t}^{8}(u,B,\varepsilon
p)\right\|_{0}^{2}\right)\leq P(E(t)).$ (3.82)

###### Proof.

Let us first consider the case when $k=8$, that is,
$\varepsilon^{8}\partial_{t}^{8}$-estimates. We just need to replace
$\overline{\partial}{}^{8}$ by $\partial_{t}^{8}$ in the proof of Proposition
3.5 and then check if the analogues of the commutators
$\mathcal{J}_{1}\sim\mathcal{J}_{6}$ can be controlled by $P(E(t))$ or not.
From (3.3.2) and (3.3.2), we know that the analogue of $\mathcal{J}_{5}$ and
$\mathcal{J}_{6}$ can be controlled in the same way because
$\partial_{t}^{8}(u,B)$ and $\overline{\partial}{}^{8}(u,B)$ have the same
weights of Mach number. Next, let us analyze the commutators in the analogues
of $\mathcal{J}_{1}\sim\mathcal{J}_{3}$. For example, we consider

$\varepsilon^{8}[\partial_{t}^{8},B\cdot\nabla]f,~{}~{}f=u\text{ or }B.$

This commutator is equal to

$\displaystyle\varepsilon^{8}(\partial_{t}^{8}B_{j}\partial_{j}f+8\partial_{t}B_{j}\partial_{j}\partial_{t}^{7}f)+\sum_{k=2}^{7}\varepsilon^{8}\binom{8}{k}\partial_{t}^{k}B_{j}\partial_{j}\partial_{t}^{8-k}f$
$\displaystyle=$
$\displaystyle~{}(\varepsilon^{8}\partial_{t}^{8}B_{j})(\partial_{j}f)+8(\partial_{t}B_{j})\partial_{j}(\varepsilon^{8}\partial_{t}^{7}f)+\sum_{k=2}^{7}\binom{8}{k}(\varepsilon^{k-1}\partial_{t}^{k}B_{j})(\varepsilon^{9-k}\partial_{j}\partial_{t}^{8-k}f),$

where the $L^{2}(\Omega)$ norms of the first term and the third term can be
directly controlled by $E(t)$. For the second term, when $j=1,2$, it can be
directly controlled; when $j=3$, we again use $\partial_{t}B_{3}|_{\Sigma}=0$
and fundamental theorem of calculus to create a weight function
$\omega(x_{3})$:

$\|(\partial_{t}B_{j})\partial_{j}(\varepsilon^{8}\partial_{t}^{7}f)\|\leq\|\partial_{t}\bar{B}\|_{\infty}\|\varepsilon^{8}\overline{\partial}{}B\partial_{t}^{7}f\|_{0}+\|\partial_{3}\partial_{t}B_{3}\|_{\infty}\|\varepsilon^{8}(\omega\partial_{3})\partial_{t}^{7}f\|_{0}\lesssim\sqrt{E_{4}(t)E_{8}(t)}\leq
E(t).$

It remains to analyze the commutator $\mathcal{J}_{4}$ arising from (3.3.2).
The difference is that $\partial_{t}^{8}p$ needs one more $\varepsilon$
weight. Luckily, this term is produced because we invoke the continuity
equation which automatically gives us $\varepsilon^{2}$ weight. So, the
analogue of $\mathcal{J}_{4}$ is controlled in the following way:

$\displaystyle-\int_{\Omega}\varepsilon^{8}\partial_{t}^{8}p\left(\varepsilon^{10}[\partial_{t}^{8},u\cdot\nabla]p\right)\,\mathrm{d}x=-\int_{\Omega}\varepsilon^{9}\partial_{t}^{8}p\left(\varepsilon^{9}[\partial_{t}^{8},u\cdot\nabla]p\right)\,\mathrm{d}x$
(3.83) $\displaystyle=$
$\displaystyle-\int_{\Omega}\varepsilon^{9}\partial_{t}^{8}p\left(\varepsilon^{9}(\partial_{t}^{8}u_{j}\partial_{j}p+8\partial_{t}u_{j}\partial_{j}\partial_{t}^{8}p)\right)\,\mathrm{d}x$
$\displaystyle-\int_{\Omega}(\varepsilon^{9}\partial_{t}^{8}p)\left(\sum_{k=2}^{7}\binom{8}{k}(\varepsilon^{k-1}\partial_{t}^{k}u_{j})(\varepsilon^{10-k}u_{j}\partial_{j}\partial_{t}^{8-k}p)\right)\,\mathrm{d}x\lesssim
E(t).$

Therefore, we conclude the $\varepsilon^{8}\partial_{t}^{8}$-estimates as
follows

$\frac{\mathrm{d}}{\mathrm{d}t}\left\|\varepsilon^{8}\partial_{t}^{8}(u,B,\varepsilon
p)\right\|_{0}^{2}\leq P(E(t)).$ (3.84)

For $4\leq k\leq 7$, the proof is still parallel to Proposition 3.5 if we
replace $\varepsilon^{8}\partial_{t}^{8}$ by
$\varepsilon^{k-1}\partial_{t}^{k}~{}(4\leq k\leq 7)$. We only need to re-
consider the estimates of the commutators arising in the analogues of
$\mathcal{J}_{1}\sim\mathcal{J}_{6}$. For example, we look at the case $k=7$.
Similarly as above, we first consider th $L^{2}$ estimates of the following
commuator

$\varepsilon^{6}[\partial_{t}^{7},B\cdot\nabla]f,~{}~{}f=u\text{ or }B,$

which is equal to

$\displaystyle\varepsilon^{6}(\partial_{t}^{7}B_{j}\partial_{j}f+7\partial_{t}B_{j}\partial_{j}\partial_{t}^{6}f)+\sum_{k=2}^{6}\varepsilon^{6}\binom{7}{k}\partial_{t}^{k}B_{j}\partial_{j}\partial_{t}^{7-k}f$
$\displaystyle=$
$\displaystyle~{}(\varepsilon^{6}\partial_{t}^{7}B_{j})(\partial_{j}f)+7(\partial_{t}B_{j})\partial_{j}(\varepsilon^{6}\partial_{t}^{7}f)+\sum_{k=2}^{6}\binom{7}{k}(\varepsilon^{k-1}\partial_{t}^{k}B_{j})(\varepsilon^{7-k}\partial_{j}\partial_{t}^{7-k}f).$

Again, the first term and the third term are directly controlled by $E(t)$.
For the second term, using $\partial_{t}B_{3}|_{\Sigma}=0$ and fundamental
theorem of calculus, we then obtain

$\|(\partial_{t}B_{j})\partial_{j}(\varepsilon^{6}\partial_{t}^{7}f)\|_{0}\leq\|\partial_{t}\bar{B}\|_{\infty}\|\varepsilon^{6}\overline{\partial}{}\partial_{t}^{6}f\|_{0}+\|\partial_{3}\partial_{t}B_{3}\|_{\infty}\|\varepsilon^{6}(\omega\partial_{3})\partial_{t}^{6}f\|_{0}\lesssim\sqrt{E_{4}(t)E_{7}(t)}\lesssim
E(t).$

As for the analogue of $\mathcal{J}_{4}$, the continuity equation gives us
extra $\varepsilon^{2}$ weight such that the estimate is uniform in Mach
number:

$\displaystyle-\int_{\Omega}\varepsilon^{6}\partial_{t}^{7}p\left(\varepsilon^{8}[\partial_{t}^{7},u\cdot\nabla]p\right)\,\mathrm{d}x=-\int_{\Omega}\varepsilon^{7}\partial_{t}^{7}p\left(\varepsilon^{7}[\partial_{t}^{7},u\cdot\nabla]p\right)\,\mathrm{d}x$
(3.85) $\displaystyle=$
$\displaystyle-\int_{\Omega}\varepsilon^{7}\partial_{t}^{7}p\left(\varepsilon^{7}(\partial_{t}^{7}u_{j}\partial_{j}p+7\partial_{t}u_{j}\partial_{j}\partial_{t}^{6}p)\right)\,\mathrm{d}x$
$\displaystyle-\int_{\Omega}(\varepsilon^{7}\partial_{t}^{7}p)\left(\sum_{k=2}^{6}\binom{7}{k}(\varepsilon^{k-1}\partial_{t}^{k}u_{j})(\varepsilon^{8-k}u_{j}\partial_{j}\partial_{t}^{8-k}p)\right)\,\mathrm{d}x\lesssim
E(t).$

Therefore, we obtain the energy estimate

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|\varepsilon^{6}\partial_{t}^{7}(\varepsilon
p,u,B)\right\|_{0}^{2}\leq P(E(t)).$ (3.86)

The case when $4\leq k\leq 6$ can be proved in the same way, so we conclude
the following energy estimate for full time derivatives

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\sum_{k=4}^{7}\left\|\varepsilon^{k-1}\partial_{t}^{k}(u,B,\varepsilon
p)\right\|_{0}^{2}\leq P(E(t)).$ (3.87)

The proof is completed. ∎

####  3.3.4 Space-time mixed tangential derivatives

We still need to prove the tangential estimates for
$\varepsilon^{8}\mathcal{T}^{\alpha}~{}(0<\alpha_{0}<8)$. Indeed, they can be
proved in the same way as Proposition 3.5 and Proposition 3.6 because of the
following reasons

  * •

The derivatives $\varepsilon^{8}\mathcal{T}^{\alpha}~{}(0<\alpha_{0}<8)$ are
tangential, so they preserve the boundary conditions and thus all boundary
integrals must vanish.

  * •

The commutators arising in the proof of Proposition 3.5 and Proposition 3.6 do
not produce extra time derivatives, so the weight $\varepsilon^{8}$ is enough
to close the estimate.

  * •

When $\alpha_{4}\neq 0$, that is, $(\omega\partial_{3})^{\alpha_{4}}$ appears
in $\mathcal{T}^{\alpha}$, it suffices to use the same strategy as presented
in Proposition 3.5. Indeed, we only commute $(\omega\partial_{3})$’s with
either $u\cdot\nabla$ or $B\cdot\nabla$, so there must a $u_{3}$ or $B_{3}$
appearing when $\partial_{3}$ falls on $\omega$. Thus, we can use the boundary
conditions for $u_{3},~{}B_{3}$ to reproduce the weight function
$\omega(x_{3})$.

Based on the above analysis, we conclude the tangential estimates in following
proposition.

######  Proposition 3.7.

We have the energy inequality:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bigg{(}$
$\displaystyle\sum_{\langle\alpha\rangle=8,\alpha_{0}<8}\left\|\varepsilon^{8}\mathcal{T}^{\alpha}(u,B,p)\right\|_{0}^{2}+\left\|\varepsilon^{8}\partial_{t}^{8}(u,B,\varepsilon
p)\right\|_{0}^{2}$ (3.88)
$\displaystyle+\sum_{l=0}^{3}\sum_{\langle\alpha\rangle=2l,\alpha_{0}<2l}\left\|\varepsilon^{3+l}\mathcal{T}^{\alpha}\partial_{t}^{4-l}(u,B,p)\right\|_{0}^{2}+\sum_{l=0}^{3}\left\|\varepsilon^{3+l}\partial_{t}^{4+l}(u,B,\varepsilon
p)\right\|_{0}^{2}\bigg{)}\leq P(E(t)).$

###  3.4 Uniform estimates

Since $D_{t}S=0$, we can easily prove the estimates for $S$ by directly
commuting $D_{t}$ with $\partial_{*}^{\alpha}$ and corresponding weights of
Mach number. The proof does not involve any boundary term because we do not
integrate by parts, so we omit the details.

$\frac{\mathrm{d}}{\mathrm{d}t}\sum_{l=0}^{4}\sum_{\langle\alpha\rangle=2l}\left\|\varepsilon^{(k-1)_{+}+2l}\mathcal{T}^{\alpha}\partial_{t}^{k}S\right\|_{4-k-l}^{2}\leq
P(E(t)).$ (3.89)

Combining the tangential estimates presented in Propositions 3.5-3.7 with the
div-curl analysis in section 3.2.2, the reduction of pressure in section
3.2.1, the $L^{2}$ estimates and the summary in section 3.2.3, we can get the
following energy inequality

$\frac{\mathrm{d}}{\mathrm{d}t}E(t)\leq P(E(t)),$ (3.90)

where $E(t)$ is defined by (1.18). Since the right side of the energy
inequality does not rely on $\varepsilon$, we can use Grönwall’s inequality to
prove that there exists some $T>0$ independent of $\varepsilon$ such that

$\sup_{t\in[0,T]}E(t)\leq P(E(0)).$ (3.91)

Theorem 1.1 is proven.

###  3.5 The case of 2D MHD

The previous sections are devoted to the incompressible limit of 3D MHD. Now
we explain how to modify the proof such that it is valid for 2D MHD flows. The
proof for the 2D case is essentially the same as the 3D case: we only need to
do a slight modification in the vorticity analysis because the vorticity in 2D
is a scalar function, not a vector function. First, lemma 3.1 should be
modified as below

######  Lemma 3.8 (Hodge elliptic estimates).

For any sufficiently smooth vector field $X\in\mathbb{R}^{2}$ and any real
number $s\geq 1$, one has

$\|X\|_{s}^{2}\lesssim\|X\|_{0}^{2}+\|\nabla\cdot
X\|_{s-1}^{2}+\|\nabla^{\perp}\cdot
X\|_{s-1}^{2}+|\overline{\partial}{}_{1}X\cdot N|^{2}_{s-\frac{3}{2}},$ (3.92)

where $\nabla=(\partial_{1},\partial_{2})$ and
$\nabla^{\perp}=(-\partial_{2},\partial_{1})$.

Taking $\nabla^{\perp}\cdot$ in the momentum equation, we obtain the analogue
of (3.12) as

$\rho D_{t}(\nabla^{\perp}\cdot u)-(B\cdot\nabla)(\nabla^{\perp}\cdot
B)=\rho[D_{t},\nabla^{\perp}\cdot]u-[B\cdot\nabla,~{}\nabla^{\perp}\cdot]B-(\nabla^{\perp}\rho)\cdot(D_{t}u),$
(3.93)

which has the same structure as (3.12). Thus, one can follow the analysis in
section 3.2.2. The only slight difference is the treatment of Lorentz force
term. We take $\partial^{3}$-estimate of $\nabla^{\perp}\cdot(u,B)$ for an
example. Following the proof of lemma 3.2, we need to control the following
integral

$\mathcal{K}^{\prime\prime}:=\int_{\Omega}\partial^{3}(\nabla^{\perp}\cdot
B)~{}\partial^{3}\nabla^{\perp}\cdot(\varepsilon^{2}BD_{t}p)\,\mathrm{d}x,$
(3.94)

where the problematic term is

$\varepsilon^{2}(B\cdot\nabla^{\perp})\partial^{3}D_{t}p=\varepsilon^{2}(-B_{1}\partial^{3}\partial_{2}D_{t}p+B_{2}\partial^{3}\partial_{1}D_{t}p)$

after inserting the continuity equation. Commuting $\nabla^{\perp}$ with
$D_{t}$, we have

$\displaystyle\varepsilon^{2}(-B_{1}\partial^{3}\partial_{2}D_{t}p+B_{2}\partial^{3}\partial_{1}D_{t}p)$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{2}\left(-B_{1}\partial^{3}D_{t}(\partial_{2}p)+B_{2}\partial^{3}D_{t}(\partial_{1}p)\right)-\varepsilon^{2}\left(B_{1}\partial^{3}(\partial_{2}u_{j}\partial_{j}p)-B_{2}\partial^{3}(\partial_{1}u_{j}\partial_{j}p)\right),$

where the $L^{2}(\Omega)$ norm of the second term is directly controlled by
$E_{4}(t)$. For the first term, we again invoke the momentum equation, which
in 2D reads

$\displaystyle-\partial_{1}p=\rho
D_{t}u_{1}-B_{1}\partial_{1}B_{1}-B_{2}\partial_{2}B_{1}+\partial_{1}(|B|^{2}/2)=\rho
D_{t}u_{1}+B_{2}(\partial_{1}B_{2}-\partial_{2}B_{1})=\rho
D_{t}u_{1}+B_{2}(\nabla^{\perp}\cdot B),$ $\displaystyle-\partial_{2}p=\rho
D_{t}u_{2}-B_{1}\partial_{1}B_{2}-B_{2}\partial_{2}B_{2}+\partial_{2}(|B|^{2}/2)=\rho
D_{t}u_{1}-B_{1}(\partial_{1}B_{2}-\partial_{2}B_{1})=\rho
D_{t}u_{2}-B_{1}(\nabla^{\perp}\cdot B).$

Now, we have

$\displaystyle\varepsilon^{2}(-B_{1}\partial^{3}\partial_{2}D_{t}p+B_{2}\partial^{3}\partial_{1}D_{t}p)$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{2}\rho(B_{1}\partial^{3}D_{t}^{2}u_{2}-B_{2}\partial^{3}D_{t}^{2}u_{1})\underbrace{-\varepsilon^{2}(B_{1}^{2}+B_{2}^{2})\partial^{3}D_{t}(\nabla^{\perp}\cdot
B)}_{\varepsilon^{2}|B|^{2}\text{ has definite
sign}}+~{}L^{2}(\Omega)\text{-controllable terms},$

and thus

$\displaystyle\mathcal{K}^{\prime\prime}\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle\int_{\Omega}\partial^{3}(\nabla^{\perp}\cdot
B)\varepsilon^{2}\rho(B_{1}\partial^{3}D_{t}^{2}u_{2}-B_{2}\partial^{3}D_{t}^{2}u_{1})\,\mathrm{d}x-\int_{\Omega}\partial^{3}(\nabla^{\perp}\cdot
B)~{}\varepsilon^{2}|B|^{2}\partial^{3}D_{t}(\nabla^{\perp}\cdot
B)\,\mathrm{d}x$ (3.95) $\displaystyle\stackrel{{\scriptstyle L}}{{=}}$
$\displaystyle\int_{\Omega}\partial^{3}(\nabla^{\perp}\cdot
B)\varepsilon^{2}\rho(B_{1}\partial^{3}D_{t}^{2}u_{2}-B_{2}\partial^{3}D_{t}^{2}u_{1})\,\mathrm{d}x$
$\displaystyle-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\left|\varepsilon|B|\partial^{3}(\nabla^{\perp}\cdot
B)\right|^{2}\,\mathrm{d}x+\frac{1}{2}\int_{\Omega}(\nabla\cdot
u)\left(\left|\varepsilon|B|\partial^{3}(\nabla^{\perp}\cdot
B)\right|^{2}\right)\,\mathrm{d}x,$

where the last term on the right side can be directly controlled, and the
first term requires the control of $\|\varepsilon^{2}D_{t}^{2}u\|_{3}$. The
conclusion of Lemma 3.2 is then modified to be

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\rho\left|\partial^{3}(\nabla^{\perp}\cdot
u)\right|^{2}+(1+\varepsilon^{2}|B|^{2})\left|\partial^{3}(\nabla^{\perp}\cdot
B)\right|^{2}\,\mathrm{d}x\leq P(E_{4}(t))+E_{5}(t).$ (3.96)

##  4 Incompressible limit

This section is devoted to showing that we can pass the solution of (1.13) to
the incompressible limit (1.19) provided that we have the convergence for
initial data. In other words, we study the behavior of the solution of (1.13)
as the Mach number $\varepsilon$ tends to $0$. Recall that the Mach number
$\varepsilon$ is defined in Section 1.1.

The energy estimate presented in Theorem 1.1 implies the boundedness of
$\sum\limits_{k=0,1}\|\partial_{t}^{k}(u,B,S)\|_{4-k}$ uniformly in
$\varepsilon$ within the time interval $[0,T]$. Thus, by Aubin-Lions Lemma, up
to a subsequence, we have

$\displaystyle(u,B,S)\to(u^{0},B^{0},S^{0})$ $\displaystyle\text{weakly-* in
}L^{\infty}([0,T];H^{4}(\Omega))\text{ and strongly in
}C([0,T];H^{4-\delta}(\Omega)),$ (4.1)

for $\delta>0$. Moreover, by using the continuity equation
$\varepsilon^{2}D_{t}p+\nabla\cdot u=0$, we have

$\nabla\cdot u\to\nabla\cdot u^{0}=0\quad\text{in
}L^{\infty}([0,T];H^{3}(\Omega))\text{ and strongly in
}C([0,T];H^{3-\delta}(\Omega))$ (4.2)

for $\delta>0$ because $\|\varepsilon\partial_{t,x}p\|_{3}$ and $\|u\|_{4}$
are uniformly bounded in $[0,T]$.

Now, $(u^{0},B^{0},S^{0})\in L^{\infty}([0,T];H^{4}(\Omega))$ solves the
incompressible MHD equations together with a transport equation

$\begin{cases}\varrho(\partial_{t}u^{0}+u^{0}\cdot\nabla
u^{0})-B^{0}\cdot\nabla
B^{0}+\nabla(\pi+\frac{1}{2}|B^{0}|^{2})=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
\partial_{t}B^{0}+u^{0}\cdot\nabla B^{0}-B^{0}\cdot\nabla
u^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\ \nabla\cdot
u^{0}=\nabla\cdot B^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
\partial_{t}S^{0}+u^{0}\cdot\nabla
S^{0}=0&~{}~{}~{}\text{in}~{}[0,T]\times\Omega,\\\
u_{3}^{0}=B_{3}^{0}=0&~{}~{}~{}\text{on}~{}[0,T]\times\Sigma,\\\
(u^{0},B^{0},S^{0})|_{t=0}=(u_{0}^{0},B_{0}^{0},S_{0}^{0}),\end{cases}$ (4.3)

for a suitable fluid pressure function $\pi$ satisfying $\nabla\pi\in
C([0,T];H^{3}(\Omega))$. Here $\varrho$ satisfies
$\partial_{t}\varrho+u^{0}\cdot\nabla\varrho=0,$ with initial data
$\varrho_{0}:=\rho(0,S_{0}^{0})$ . Moreover, the uniqueness of the limit