# On two conserved quantities in the inviscid electron and Hall
magnetohydrodynamic equations

Yanqing Wang111 College of Mathematics and Information Science, Zhengzhou
University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email:
<EMAIL_ADDRESS>Jing Yang222College of Mathematics and Information
Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, P.
R. China Email<EMAIL_ADDRESS>and Yulin Ye333Corresponding author.
School of Mathematics and Statistics, Henan University, Kaifeng, 475004, P. R.
China. Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we are concerned with the energy and magnetic helicity
conservation of weak solutions for both the electron and Hall
magnetohydrodynamic equations. Various sufficient criteria to ensure the
energy and magnetic helicity conservation in Onsager’s critical spaces
$\underline{B}^{\alpha}_{p,VMO}$ and $B^{\alpha}_{p,c(\mathbb{N})}$ in these
systems are established. Moreover, for the E-MHD equations, we observe that
the conservation criteria of energy and magnetic helicity to the E-MHD
equations correspond to the helicity and energy to the ideal incompressible
Euler equations, respectively.

MSC(2020): 76W05, 76X05, 35D30, 76B03, 35Q35
Keywords: electron magnetohydrodynamic equations; Hall magnetohydrodynamic
equations; energy; magnetic helicity; weak solutions

##  1 Introduction

The magnetohydrodynamic (MHD) equations, the electron (E-MHD) and Hall (H-MHD)
magnetohydrodynamic equations play an central role in the theory of plasma
physical (see e.g. [22, 19, 8]). The inviscid H-MHD equations may be written
as follows

$\left\\{\begin{aligned} &u_{t}+u\cdot\nabla u-h\cdot\nabla h+\nabla\Pi=0,\\\
&h_{t}+u\cdot\nabla h-h\cdot\nabla
u+\text{d}_{\text{I}}\nabla\times\Big{[}(\nabla\times h)\times h\Big{]}=0,\\\
&\mathrm{div}\,u=\mathrm{div}\,h=0.\end{aligned}\right.$ (1.1)

Here $u$ represents the velocity field, $h$ stands for the magnetic field and
$\Pi=\pi+\frac{1}{2}|h|^{2}$ denotes the magnetic pressure with $\pi$ being
the fluid pressure, respectively. We hereafter denote $\vec{j}=\text{curl}h$
as the electric current. Without loss of generality, we set the iron inertial
length $\text{d}_{\text{I}}=1$. In the present paper, we consider the case of
a bounded domain with periodic boundary condition in $\mathbb{R}^{d}$, namely
$\Omega=\mathbb{T}^{d}$ with $d=2,3$. Formally, when the velocity $u$ in the
H-MHD equations (LABEL:hallMHD) is zero, this system becomes the following
E-MHD equations

$\left\\{\begin{aligned} &h_{t}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}=0,\\\ &\mathrm{div}\,h=0.\end{aligned}\right.$ (1.2)

For the derivation of the E-MHD system and its background, we refer the
readers to [22]. Moreover, if ignoring the Hall effect, namely letting
$\text{d}_{I}=0$, then the Hall-MHD equations (LABEL:hallMHD) reduce to the
standard incompressible MHD equations

$\left\\{\begin{aligned} &u_{t}+u\cdot\nabla u-h\cdot\nabla h+\nabla\Pi=0,\\\
&h_{t}+u\cdot\nabla h-h\cdot\nabla u=0,\\\
&\mathrm{div}\,u=\mathrm{div}\,h=0.\end{aligned}\right.$ (1.3)

As known to all, in physics, the common conserved quantities in these three
systems are the total energy and magnetic helicity (see [19, 8]). Precisely,
the energy in the E-MHD system (LABEL:E-MHD) and H-MHD (MHD) equations
(LABEL:hallMHD) for smooth solutions are
$\frac{1}{2}\int_{\mathbb{T}^{d}}|h|^{2}dx$ and
$\frac{1}{2}\int_{\mathbb{T}^{d}}|v|^{2}+|h|^{2}dx.$ The magnetic helicity in
these system is $\int_{\mathbb{T}^{d}}H\cdot{\rm curl\,}H\ dx,$ where $H={\rm
curl}^{-1}h$ represents the magnetic vector potential.

The mathematical study of conserved quantities is very close to the Onsager’s
conjecture on the energy conservation of weak solutions for the ideal Euler
equations in [27]. The initial Onsager’s conjecture is that the critical
regularity for weak solutions conserving the energy in the Euler equations is
1/3 in Hölder spaces. As the initial Onsager’s conjecture for the Euler
equations, two parts of the analogous Onsager’s conjecture for the usual MHD
system (LABEL:MHD) are well studied. For the positive part, the sufficient
conditions of the weak solutions implying the energy and magnetic helicity
conservation for the standard MHD system (LABEL:MHD) can be found in [35, 20,
17, 4, 29]. Precisely, a weak solution $(u,h)$ of the ideal MHD equations
(LABEL:MHD) satisfies the total energy conservation if one of the following
conditions holds

  * •

Caflisch-Klapper-Steele [4]: $u\in C(0,T;B_{3,\infty}^{\alpha_{1}}),h\in
C(0,T;B_{3,\infty}^{\alpha_{2}}),\alpha_{1}>1/3,\alpha_{1}+2\alpha_{2}>1;$

  * •

Kang and Lee [20]: $u\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\alpha_{1}}),h\in
L^{3}(0,T;B_{3,c(\mathbb{N})}^{\alpha_{2}}),\alpha_{1}\geq
1/3,\alpha_{1}+2\alpha_{2}\geq 1;$

  * •

Yu [35]: $u\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\alpha_{1}}),h\in
L^{3}(0,T;B_{3,\infty}^{\alpha_{2}}),\alpha_{1}\geq
1/3,\alpha_{1}+2\alpha_{2}\geq 1;$

  * •

Wang-Huang-Ye-Liu [29]: $u\in L^{3}(0,T;\underline{B}^{\alpha}_{3,VMO})\cap
L^{p_{1}}(0,T;\underline{B}^{\alpha}_{q_{1},VMO}),h\in
L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty})$, $\alpha\geq
1/3,\alpha+2\beta\geq
1,1/q_{1}+2/q_{2}=1,1/p_{1}+2/p_{2}=1,1/p_{2}+1/q_{1}\leq 1.$

The magnetic helicity of weak solutions for the ideal MHD equations
(LABEL:MHD) is invariant if

  * •

Caflisch-Klapper-Steele [4]: $u\in C(0,T;B_{3,\infty}^{\alpha_{1}}),h\in
C(0,T;B_{3,\infty}^{\alpha_{2}}),\alpha_{1}+2\alpha_{2}>0;$

  * •

Kang and Lee [20]: $u,h\in L^{3}([0,T];L^{3})$;

  * •

Faraco-Lindberg-Székelyhidi [17]: $h\in
L^{p_{1}}((0,T)\times\mathbb{T}^{d}),h\times u\in
L^{p_{2}}((0,T)\times\mathbb{T}^{d}),$ $1/p_{1}+1/p_{2}=1$;

  * •

Wang-Huang-Ye-Liu [29]: $u\in L^{p_{1}}(0,T;L^{q_{1}}(\mathbb{T}^{d}))$ and
$h\in L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))$, $1\leq
p_{1},p_{2},q_{1},q_{2}<\infty$ and $1/p_{1}+2/p_{2}=1$,$1/q_{1}+2/q_{2}=1$.

It is worth pointing out that, for $\alpha>1/3$, the inclusion relations of
the aforementioned Besov spaces are that

$C^{\alpha}\subseteq B^{\alpha}_{3,\infty}\subseteq
B^{\frac{1}{3}}_{3,c(\mathbb{N})}\subseteq\underline{B}^{\frac{1}{3}}_{3,VMO}\subseteq
B^{\frac{1}{3}}_{3,\infty},$ (1.4)

where $B^{\frac{1}{3}}_{3,c(\mathbb{N})}$ and
$\underline{B}^{\frac{1}{3}}_{3,VMO}$ are Onsager’s critical spaces which were
initially introduced by Cheskidov-Constantin-Friedlander-Shvydkoy in [7] and
Fjordholm-Wiedemann in [18], respectively. Regarding the negative part of
Onsager’s conjecture, the readers can be referred to [23, 24, 25, 2, 4, 17,
11] for the non-uniqueness theory of weak solutions for both the
incompressible ideal and viscous MHD equations.

Now, we turn our attentions back to the ideal E-MHD and H-MHD equations.
Compared with nonlinear term in terms of convection type of both the ideal
Euler and the standard MHD equations, the Hall term
$\nabla\times\Big{[}(\nabla\times h)\times h\Big{]}$ in the E-MHD and H-MHD
equations involves the second order derivative rather than the first order
derivative. To the knowledge of authors, there are only two works for positive
part of Onsager type results in the inviscid E-MHD and H-MHD equations.
Precisely, in [21], Kang-Deng-Bie showed that if

$u,h\in L^{p}\cap
B^{\alpha}_{p,\infty}((0,T)\times\mathbb{T}^{3}),~{}\vec{j}=\text{curl}h\in
B^{\beta}_{p,\infty}((0,T)\times\mathbb{T}^{3}),\frac{2}{p}+\frac{1}{q}=1,2\alpha+\beta>1,$
(1.5)

then the total energy of weak solutions for the ideal H-MHD equations
(LABEL:hallMHD) is conserved. On the other hand, Dai-Krol-Liu in [12] showed
that the weak solutions for the ideal E-MHD equations (LABEL:E-MHD) preserve
the energy if $h\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{2}{3}})$ and magnetic
helicity if

$h\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{1}{3}}),$ (1.6)

respectively. The intention of this paper is to establish new energy and
magnetic helicity conservation criterion of weak solutions for both the
inviscid E-MHD and H-MHD equations.

######  Theorem 1.1.

Let $d=2,3$ and $h\in C([0,T];L^{2}(\mathbb{T}^{d}))$ be a weak solution of
the incompressible E-MHD equations (LABEL:E-MHD). Then for any $t\in[0,T]$,
the energy $\frac{1}{2}\int_{\mathbb{T}^{d}}|h|^{2}dx$ is invariant provided
that one of the following four conditions is satisfied

  1. (1)

$h\in L^{3}(0,T;\underline{B}^{\frac{2}{3}}_{3,VMO}(\mathbb{T}^{d}))$;

  2. (2)

$\vec{j}\in L^{r_{1}}(0,T;\underline{B}^{\alpha}_{p,VMO}(\mathbb{T}^{d})),h\in
L^{r_{2}}(0,T;B^{\beta}_{q,\infty}(\mathbb{T}^{d})),$$\alpha+2\beta\geq
1,\frac{1}{r_{1}}+\frac{2}{r_{2}}=1,\frac{1}{p}+\frac{2}{q}=1$;

  3. (3)

$\vec{j}\in L^{r_{1}}(0,T;B^{\alpha}_{p,\infty}(\mathbb{T}^{d})),h\in
L^{r_{2}}(0,T;\underline{B}^{\beta}_{q,VMO}(\mathbb{T}^{d})),$$\alpha+2\beta\geq
1,\frac{1}{r_{1}}+\frac{2}{r_{2}}=1,\frac{1}{p}+\frac{2}{q}=1$;

  4. (4)

$\vec{j}\in
L^{3}(0,T;B^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}(\mathbb{T}^{d})).$

######  Remark 1.1.

According to inclusion relations (1.4) for Besov spaces, we know that this
theorem improves the corresponding results in [12].

######  Remark 1.2.

Since there holds
$B^{\gamma}_{\ell,c(\mathbb{N})}\subseteq\underline{B}^{\gamma}_{\ell,VMO}$,
for any $\gamma\in(0,1)$ and $\ell\in[1,\infty]$, the results in (2) and (3)
are still valid by replacing $\underline{B}^{\gamma}_{\ell,VMO}$ by
$B^{\gamma}_{\ell,c(\mathbb{N})}$.

######  Remark 1.3.

It seems that the energy conservation criteria for the E-MHD equations is by
the analogy of the helicity conservation roles of the ideal Euler equations
due to Cheskidov-Constantin-Friedlander-Shvydkoy [7]. And the results (2)-(4)
are partial motived for the investigation of the helicity of the ideal Euler
equations by Chae in [6, 5] and by De Rosa [14].

Before we state the result concerning the magnetic helicity for E-MHD
equations (LABEL:E-MHD), we recall that the magnetic vector potential
equations take the form below

$H_{t}+(\nabla\times h)\times h+\nabla p=0,~{}\text{div}H=0,$ (1.7)

where $H={\rm curl}^{-1}h$.

######  Theorem 1.2.

Let $d=3$ and $\vec{j}\in C(0,T;L^{\frac{3}{2}}(\mathbb{T}^{3}))$ be a weak
solution of the inviscid E-MHD equations (LABEL:E-MHD). If $h\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{3})),$ then for any
$t\in[0,T]$, the magnetic helicity is invariant, i.e.

$\int_{\mathbb{T}^{3}}H\cdot{\rm curl\,}H\
dx=\int_{\mathbb{T}^{3}}H_{0}\cdot{\rm curl\,}H_{0}\ dx,$

where $H={\rm curl}^{-1}h$.

######  Remark 1.4.

Owing to inclusion relations (1.4), we see that this theorem generalizes the
sufficient condition (1.6) for the conservation of magnetic helicity.

######  Remark 1.5.

It seems that the magnetic helicity conservation criteria for the E-MHD
equations corresponds to the energy conservation roles of the ideal Euler
equations Cheskidov-Constantin-Friedlander-Shvydkoy [7].

Next, we formulate our results on inviscid Hall-MHD equations (LABEL:hallMHD)
as follows.

######  Theorem 1.3.

Let $d=2,3$ and the pair $(u,h)\in C([0,T];L^{2}(\mathbb{T}^{d}))$ be a weak
solution of the inviscid Hall-MHD equations (LABEL:hallMHD). Then the total
energy $\frac{1}{2}\int_{\mathbb{T}^{d}}(|u|^{2}+|h|^{2})dx$ is invariant
provided that

$u\in L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{d}))\
\text{and}\ h\in
L^{3}(0,T;\underline{B}^{\frac{2}{3}}_{3,VMO}(\mathbb{T}^{d})).$

Similar as (1.7), an analogue of the magnetic vector potential equations for
(LABEL:hallMHD) can be given by

$H_{t}+h\times u+(\nabla\times h)\times h+\nabla p=0,~{}\text{div}H=0,$ (1.8)

where $H={\rm curl}^{-1}h$. Then the sufficient conditions for the
conservation of magnetic helicity involving (LABEL:hallMHD) and (1.8) are
stated as follows.

######  Theorem 1.4.

Let $d=3$ and $h\in C([0,T];L^{3}(\mathbb{T}^{3}))$ be a weak solution of the
inviscid Hall-MHD equations (LABEL:hallMHD). If $u\in
L^{3}(0,T;L^{3}(\mathbb{T}^{3}))$ and $h\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{3})),$ then for any
$t\in[0,T]$, the magnetic helicity is constant, i.e.

$\int_{\mathbb{T}^{3}}H\cdot{\rm curl\,}H\
dx=\int_{\mathbb{T}^{3}}H_{0}\cdot{\rm curl\,}H_{0}\ dx,$

where $H={\rm curl}^{-1}h$.

######  Remark 1.6.

It seems that this is the first result for weak solutions keeping magnetic
helicity for the inviscid Hall-MHD equations (LABEL:hallMHD).

Next, we are concerned with the energy and the generalized magnetic helicity
in the viscous E-MHD equations and the viscous H-MHD equations. Indeed, for
the following viscous E-MHD equations

$\left\\{\begin{aligned} &h_{t}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}-\Delta h=0,\\\ &\mathrm{div}\,h=0,\end{aligned}\right.$ (1.9)

in [12], Dai-Krol-Liu proved that for any
$\phi(x,t)\in\mathcal{D}([0,T]\times\mathbb{R}^{3})$ and $t\in(0,T]$, the
following generalized magnetic helicity equation holds

$\displaystyle\int_{\mathbb{R}^{3}\times\\{t\\}}h\cdot H\phi
dx+2\int^{t}_{0}\int_{\mathbb{R}^{3}}\nabla H:\nabla h\phi dxds$ (1.10)
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{3}\times\\{0\\}}h\cdot H\phi
dx+\int^{t}_{0}\int_{\mathbb{R}^{3}}h\cdot(\phi_{t}+\Delta\phi)dxds+\int^{t}_{0}\int_{\mathbb{R}^{3}}((\nabla\times
h)\times h)\cdot(\nabla\phi\times H)dxds,$

if

$h\in
L^{3}(0,T;L^{\frac{9}{2}}(\mathbb{R}^{3}))~{}~{}\text{and}~{}~{}\vec{j}\in
L^{3}(0,T;L^{\frac{9}{5}}(\mathbb{R}^{3}))\cap
L^{3}(0,T;L^{\frac{18}{5}}(\mathbb{R}^{3}))\ \text{ outside a}\
C^{\frac{1}{2}}\ \text{curve}.$ (1.11)

The main difference between the inviscid and viscous equations is that the
second one obeys the additional regularity $L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$.
Partially inspired by the recent work [31], as a consequence result of Theorem
1.1 and Theorem 1.2, one can get the energy and magnetic helicity equality
criterion for the viscous E-MHD equations (LABEL:visE-MHD) as follows.

######  Corollary 1.5.

Let $h\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ be a weak solutions of (LABEL:visE-MHD).

  1. 1.

Then the energy $\frac{1}{2}\int_{\mathbb{T}^{d}}|h|^{2}dx$ is invariant
provided that

$\vec{j}\in L^{r_{1}}(0,T;L^{p}(\mathbb{T}^{d}))\ \text{and}\ h\in
L^{r_{2}}(0,T;L^{q}(\mathbb{T}^{d})),\frac{2}{r_{1}}+\frac{1}{r_{2}}=1,\frac{2}{p}+\frac{1}{q}=1.$

  2. 2.

Let $d=3$, then the following magnetic helicity equality holds that

$\int_{\mathbb{T}^{3}}h\cdot H(x,T)dx-\int_{\mathbb{T}^{3}}h\cdot
H(x,0)dx+2\int_{0}^{T}\int_{\mathbb{T}^{3}}(\nabla\times h)\cdot hdxdt=0,$

provided that one of the following conditions is satisfied

    1. (1)

$\vec{j}\in L^{p}(0,T;L^{q}(\mathbb{T}^{3})),h\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3})),$

    2. (2)

$h\in L^{4}(0,T;L^{4}(\mathbb{T}^{3}));$

    3. (3)

$h\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))\ \text{with}\
\frac{1}{p}+\frac{3}{q}=1,\ \text{if}\ 3<q<4;$ or $\frac{2}{p}+\frac{2}{q}=1,\
\text{if}\ q\geq 4;$

    4. (4)

$\vec{j}\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))\ \text{with}\
\frac{1}{p}+\frac{3}{q}=2,\ \text{if}\ \frac{3}{2}<q<\frac{9}{5};$ or
$\frac{1}{p}+\frac{6}{5q}=1,\ \text{if}\ q\geq\frac{9}{5}.$

######  Remark 1.7.

Notice that (1.11) is a special case of this theorem.

######  Remark 1.8.

This corollary extends the famous Lion-Shinbrot type energy balance criteria
of the 3D Navier-Stokes equations to the magnetic helicity conservation of the
3D viscous E-MHD equations (LABEL:visE-MHD).

######  Remark 1.9.

Inspired by [30], it should be pointed out that this corollary still holds for
the whole spaces $\mathbb{R}^{3}$.

Moreover, Dumas and Sueur in [16] showed that the weak solutions of the
viscous H-MHD equations below

$\left\\{\begin{aligned} &u_{t}+u\cdot\nabla u-h\cdot\nabla h+\nabla\Pi-\Delta
u=0,\\\ &h_{t}+u\cdot\nabla h-h\cdot\nabla u+\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}-\Delta h=0,\\\
&\mathrm{div}\,u=\mathrm{div}\,h=0,\end{aligned}\right.$ (1.12)

conserve the energy if

$u\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{1}{3}})~{}~{}\text{and}~{}~{}h\in
L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{2}{3}}),$ (1.13)

and preserve the magnetic helicity if

$h\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{1}{3}}).$ (1.14)

In the spirt of the above theorems and the classical work [7], one can refine
Dumas and Sueur’s result as follows.

######  Corollary 1.6.

Let $(u,h)\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{3})))$ be a weak solution of the viscous Hall-MHD
equations (LABEL:vhallMHD).

  1. (1)

Then the total energy $\frac{1}{2}\int_{\mathbb{T}^{3}}(|u|^{2}+|h|^{2})dx$ is
invariant provided that

$u\in L^{3}(0,T;B^{\frac{1}{3}}_{3,\infty}(\mathbb{T}^{3}))\ \text{and}\ h\in
L^{3}(0,T;\underline{B}^{\frac{2}{3}}_{3,VMO}(\mathbb{T}^{3}))$

  2. (2)

The following magnetic helicity equality holds that

$\int_{\mathbb{T}^{3}}h\cdot H(x,T)dx-\int_{\mathbb{T}^{3}}h\cdot
H(x,0)dx+2\int_{0}^{T}\int_{\mathbb{T}^{3}}(\nabla\times h)\cdot hdxdt=0,$

provided $h\in L^{3}(0,T;B^{\frac{1}{3}}_{3,\infty}(\mathbb{T}^{3}))$.

Finally, we would like to mention that four-thirds law of conserved quantities
in the inviscid electron and Hall magnetohydrodynamic systems can be found in
[28]. Moreover, Dai and Liu recently constructed the weak solutions to the
nonresistive E-MHD system (LABEL:E-MHD) which do not conserve energy and
magnetic helicity in [13]. For the non-unique weak solutions in Leray-Hopf
class for the three-dimensional Hall-MHD system, the reader may refer to [10].

The paper is organized as follows. In section 2, we present some notations
used in this paper and the key lemma concerning the Constantin-E-Titi type
commutator estimates for the functions in Besov-VMO space
$\underline{B}^{\alpha}_{p,VMO}$ and homogeneous Besov space
$\dot{B}^{\beta}_{p,\infty}$. Section 3 and section 4 are devoted to the proof
of the energy and magnetic helicity conservation for the inviscid E-MHD
equations (LABEL:E-MHD) and H-MHD equations (LABEL:hallMHD), respectively.
Finally, in Section 5, we consider the conserved quantities in the viscous
E-MHD equations (LABEL:visE-MHD) and Hall-MHD equations (LABEL:vhallMHD).

##  2 Notations and some auxiliary lemmas

First, we introduce some notations used in this paper. For $p\in[1,\,\infty]$,
the notation $L^{p}(0,\,T;X)$ stands for the set of measurable functions $f$
on the interval $(0,\,T)$ with values in $X$ and $\|f\|_{X}$ belonging to
$L^{p}(0,\,T)$.

Second, for $s\in\mathbb{R}$ and $1\leq p\leq\infty$, we define the Besov
semi-norm $\left\|f\right\|_{\dot{B}^{s}_{p,\infty}(\mathbb{T}^{d})}$ and
Besov norm $\left\|f\right\|_{B^{s}_{p,\infty}(\mathbb{T}^{d})}$ of
$f\in\mathcal{S}^{{}^{\prime}}$ as

$\displaystyle\|f\|_{\dot{B}_{p,\infty}^{s}(\mathbb{T}^{d})}=\||y|^{-s}\|f(x-y)-f(x)\|_{L_{x}^{p}(\mathbb{T}^{d})}\|_{L_{y}^{\infty}(\mathbb{T}^{d})},$
$\displaystyle\left\|f\right\|_{B^{s}_{p,\infty}(\mathbb{T}^{d})}=\left\|f\right\|_{{L^{p}}(\mathbb{T}^{d})}+\left\|f\right\|_{\dot{B}^{s}_{p,\infty}(\mathbb{T}^{d})}.$

In the spirit of [18], we recall the Besov-VMO space
$L^{p}(0,T;\underline{B}^{\alpha}_{q,VMO}(\mathbb{T}^{d}))$ of function $f$ if
it satisfies

$\|f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}<\infty,$

and

$\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon^{\alpha}}\left(\int_{0}^{T}\Big{[}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(x)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,\,~{}\,~{}\,|f(x)-f(y)|^{q}dydx\Big{]}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}$
$\displaystyle=$ $\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon^{\alpha}}\left(\int_{0}^{T}\Big{[}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x)-f(x-y)|^{q}dydx\Big{]}^{\frac{p}{q}}dt\right)^{\frac{1}{p}}=0.$

Eventually, we let $\eta_{\varepsilon}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be
a standard mollifier, i.e. $\eta(x)=C_{0}e^{-\frac{1}{1-|x|^{2}}}$ for $|x|<1$
and $\eta(x)=0$ for $|x|\geq 1$, where $C_{0}$ is a constant such that
$\int_{\mathbb{R}^{d}}\eta(x)dx=1$. For $\varepsilon>0$, we define the
rescaled mollifier by
$\eta_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\eta(\frac{x}{\varepsilon})$,
and for any function $f\in L^{1}_{\rm loc}(\mathbb{R}^{d})$, its mollified
version is defined by

$f^{\varepsilon}(x)=(f*\eta_{\varepsilon})(x)=\int_{\mathbb{R}^{d}}f(x-y)\eta_{\varepsilon}(y)dy,\
\ x\in\mathbb{R}^{d}.$ (2.1)

Next, we state several useful lemmas which will be frequently used in the
proof of present paper.

######  Lemma 2.1.

([33, 1]) Let $\alpha\in(0,1)$, $p,q\in[1,\infty]$, and $k\in\mathbb{N}^{+}$.
Assume that $f\in L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty}(\mathbb{T}^{d}))$ and
$g\in L^{p}(0,T;\underline{B}^{\alpha}_{q,VMO}(\mathbb{T}^{d}))$, then letting
$\varepsilon\to 0$, there hold that

  1. (1)

$\|f^{\varepsilon}-f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{O}(\varepsilon^{\alpha})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty}(\mathbb{T}^{d}))}$;

  2. (2)

$\|\nabla^{k}f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{O}(\varepsilon^{\alpha-k})\|f\|_{L^{p}(0,T;\dot{B}^{\alpha}_{q,\infty}(\mathbb{T}^{d}))}$;

  3. (3)

$\|g^{\varepsilon}-g\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{o}(\varepsilon^{\alpha})$;

  4. (4)

$\|\nabla^{k}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{o}(\varepsilon^{\alpha-k}).$

In the following, we will state a generalized Constantin-E-Titi type
commutator estimates involving the Besov-VMO spaces. The special case $p=q=3$
and $\alpha=\beta=\frac{1}{3}$ in (1) of the following lemma was presented in
[1] and the following version is due to [33]. For the convenience of readers,
we outline the proof to make the paper more self-contained.

######  Lemma 2.2.

([33]) Assume that $0<\alpha,\beta<1$, $1\leq p,q,p_{1},p_{2}\leq\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Then, there holds

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{o}(\varepsilon^{\alpha+\beta}),\
\text{as}\ \varepsilon\to 0,$ (2.2)

provided that one of the following three conditions is satisfied,

  1. (1)

$f\in L^{p_{1}}(0,T;\underline{B}^{\alpha}_{q_{1},VMO}(\mathbb{T}^{d}))$,
$g\in L^{p_{2}}(0,T;\underline{B}^{\beta}_{q_{2},VMO}(\mathbb{T}^{d}))$,
$1\leq q_{1},q_{2}\leq\infty$, $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$;

  2. (2)

$f\in L^{p_{1}}(0,T;\underline{B}^{\alpha}_{q_{1},VMO}(\mathbb{T}^{d}))$,
$g\in L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty}(\mathbb{T}^{d}))$, $1\leq
q_{1},q_{2}\leq\infty$,
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}},q_{2}\geq\frac{q_{1}}{q_{1}-1}$
and $p_{2}\geq\frac{q_{1}}{q_{1}-1}$.

###### Proof.

The key point of this lemma is that the following Constantin-E-Titi identity
in [9]

$\displaystyle(fg)^{\varepsilon}(x)-f^{\varepsilon}g^{\varepsilon}(x)$ (2.3)
$\displaystyle=$
$\displaystyle\int_{B_{\varepsilon}(0)}\eta_{\varepsilon}(y)\Big{[}f(x-y)-f(x)\Big{]}\Big{[}g(x-y)-g(x)\Big{]}dy-(f-f^{\varepsilon})(g-g^{\varepsilon})(x)$
$\displaystyle=$ $\displaystyle\,I+II.$

(1) It follows from Hölder’s inequality that

$\displaystyle|I|$ $\displaystyle\leq$ $\displaystyle
C\Big{[}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{q_{1}}dy\Big{]}^{\frac{1}{q_{1}}}\Big{[}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|g(x-y)-g(x)|^{q_{2}}dy\Big{]}^{\frac{1}{q_{2}}}.$

Performing a space-time integration and using the definition of Besov-VMO
space, as $\varepsilon\to 0$, we note that

$\displaystyle\|I\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}$ (2.4)
$\displaystyle\leq$ $\displaystyle
C\Big{\|}\left(\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{q_{1}}dy\right)^{\frac{1}{q_{1}}}\Big{\|}_{L^{p_{1}}(0,T)}\Big{\|}\left(\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|g(x-y)-g(x)|^{q_{1}}dy\right)^{\frac{1}{q_{2}}}\Big{\|}_{L^{p_{2}}(0,T)}$
$\displaystyle\leq$ $\displaystyle o(\varepsilon^{\alpha+\beta}),$

where we need to require
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.

We deduce from the Hölder inequality and Lemma 2.1 that, as $\varepsilon\to
0$,

$\displaystyle\|II\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq$ $\displaystyle
C\|f-f^{\varepsilon}\|_{L^{p_{1}}(0,T;L^{q_{1}}(\mathbb{T}^{d}))}\|g-g^{\varepsilon}\|_{L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))}\leq$
$\displaystyle o(\varepsilon^{\alpha+\beta}).$ (2.5)

Then substituting (2.5) and (2.4) into (2.3), we finish the proof of this
part.

(2) The Hölder inequality implies that

$|I|\leq\frac{1}{\varepsilon^{d}}\Big{[}\int_{B_{\varepsilon}(0)}\big{|}\eta\big{(}\frac{y}{\varepsilon}\big{)}[g(x-y)-g(x)]\big{|}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}\Big{[}\int_{B_{\varepsilon}(0)}|f(x-y)-f(x)|^{q_{1}}dy\Big{]}^{\frac{1}{q_{1}}}.$

Combining this, the Hölder inequality again and Minkowski inequality, we find

$\displaystyle\|I\|_{L^{q}(\mathbb{T}^{d})}$ $\displaystyle\leq$
$\displaystyle
C\Big{[}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(0)}\big{\|}g(x-y)-g(x)\|_{L^{q_{2}}(\mathbb{T}^{d})}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}\Big{[}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{q_{1}}dydx\Big{]}^{\frac{1}{q_{1}}},$

where we used $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$ and
$q_{2}\geq\frac{q_{1}}{q_{1}-1}$. Thereby, by utilizing Hölder inequality and
Minkowski inequality once again, as $\varepsilon\to 0$, we observe that

$\displaystyle\|I\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}$ (2.6)
$\displaystyle\leq$ $\displaystyle
C\Big{[}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(0)}\|g(x-y)-g(x)\|_{L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))}^{\frac{q_{1}}{q_{1}-1}}dy\Big{]}^{1-\frac{1}{q_{1}}}\Big{\|}\int_{\mathbb{T}^{d}}\int_{B_{\varepsilon}(0)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-~{}\,~{}\,~{}\,~{}\,|f(x-y)-f(x)|^{q_{1}}dydx\Big{\|}_{L^{p_{1}}(0,T)}$
$\displaystyle\leq$ $\displaystyle o(\varepsilon^{\alpha+\beta}).$

Plugging (2.6) and (2.5) into (2.3), we get the desired estimate. The proof of
this lemma is completed. ∎

######  Lemma 2.3.

([33]) Assume that $0<\alpha,\beta<1$, $1\leq p,q,p_{1},p_{2}\leq\infty$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Let $f^{\varepsilon}$ and
$g^{\varepsilon}$ be defined by (2.1) Then as $\varepsilon\to 0$, there holds

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq\text{o}(\varepsilon^{\alpha+\beta}),$
(2.7)

provided that one of the following three conditions is satisfied,

  1. (1)

$f\in L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})}(\mathbb{T}^{d}))$,
$g\in L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty}(\mathbb{T}^{d}))$, $1\leq
q_{1},q_{2}\leq\infty$, $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$;

  2. (2)

$\nabla f\in
L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})}(\mathbb{T}^{d}))$,
$\nabla g\in L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty}(\mathbb{T}^{d}))$,
$\frac{2}{d}+\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, $1\leq
q_{1},q_{2}<d$;

  3. (3)

$f\in L^{p_{1}}(0,T;\dot{B}^{\alpha}_{q_{1},c(\mathbb{N})}(\mathbb{T}^{d}))$,
$\nabla g\in L^{p_{2}}(0,T;\dot{B}^{\beta}_{q_{2},\infty}(\mathbb{T}^{d}))$,
$\frac{1}{d}+\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, $1\leq q_{2}<d$,
$1\leq q_{1}\leq\infty$.

######  Lemma 2.4.

([34, 33]) Let $p,q,p_{1},q_{1},p_{2},q_{2}\in[1,+\infty)$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}},\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$.
Assume $f\in L^{p_{1}}(0,T;L^{q_{1}}(\mathbb{T}^{d}))$ and $g\in
L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))$, then as $\varepsilon\to 0$, it
holds

$\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\rightarrow
0,$ (2.8)

and

$\|(f\times g)^{\varepsilon}-f^{\varepsilon}\times
g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\rightarrow 0.$ (2.9)

######  Lemma 2.5.

([26]) Suppose that $f\in L^{p}(0,T;L^{q}(\mathbb{T}^{d}))$. Then for any
$\varepsilon>0$, there holds

$\|\nabla f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
C\varepsilon^{-1}\|f\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))},$ (2.10)

and, if $p,q<\infty$

$\limsup_{\varepsilon\rightarrow 0}\varepsilon\|\nabla
f^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}=0.$

######  Lemma 2.6.

([26, 31]) Let $1\leq p,q,p_{1},p_{2},q_{1},q_{2}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$. Assume $f\in
L^{p_{1}}(0,T;W^{1,q_{1}}(\mathbb{T}^{d}))$ and $g\in
L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))$. Then for any $\varepsilon>0$, there
holds

$\displaystyle\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
C\varepsilon\|f\|_{L^{p_{1}}(0,T;W^{1,q_{1}}(\mathbb{T}^{d}))}\|g\|_{L^{p_{2}}(0,T;L^{q_{2}}(\mathbb{T}^{d}))}.$
(2.11)

Moreover, if $p_{2},q_{2}<\infty$ then

$\displaystyle\limsup_{\varepsilon\to
0}\varepsilon^{-1}\|(fg)^{\varepsilon}-f^{\varepsilon}g^{\varepsilon}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{d}))}=0.$
(2.12)

Finally, for the convenience of readers, we state the definitions of weak
solutions for both the inviscid E-MHD equations (LABEL:E-MHD) and H-MHD
equations (LABEL:hallMHD), respectively. To do this, we list some identities
first. For any vectors $\vec{A}$ and $\vec{B}$, there hold that

$\displaystyle\nabla(\vec{A}\cdot\vec{B})=\vec{A}\cdot\nabla\vec{B}+\vec{B}\cdot\nabla\vec{A}+\vec{A}\times(\nabla\times\vec{B})+\vec{B}\times(\nabla\times\vec{A}),$
(2.13)
$\displaystyle\nabla\times(\vec{A}\times\vec{B})=\vec{A}\text{div}\vec{B}-\vec{B}\text{div}\vec{A}+\vec{B}\cdot\nabla\vec{A}-\vec{A}\cdot\nabla\vec{B},$
$\displaystyle\mathrm{div}\,(\nabla\times\vec{A})=0,\
\nabla\times(\nabla\vec{B})=0,$

which together with the divergence-free condition
$\mathrm{div}\,u=\mathrm{div}\,h=0$ allow us to get

$\displaystyle h\cdot\nabla h=\frac{1}{2}\nabla|h|^{2}+\vec{j}\times h,\
\text{with}\ \vec{j}=\nabla\times h,$ (2.14)
$\displaystyle\nabla\times(h\times u)=u\cdot\nabla h-h\cdot\nabla u,$

which means that

$\displaystyle\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}=\nabla\times[\vec{j}\times h]=\nabla\times\Big{[}\text{div}(h\otimes
h)-\nabla\frac{1}{2}|h|^{2}\Big{]}=\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}.$ (2.15)

Hence, we get a equivalent form of the inviscid E-MHD equation (LABEL:E-MHD)
as

$\displaystyle h_{t}+\nabla\times\Big{[}\text{div}(h\otimes h)\Big{]}=0.$
(2.16)

Moreover, we rewrite magnetic potential equation (1.7) as

$H_{t}+\text{div}(h\otimes h)+\nabla p=0.$ (2.17)

Thanks to $\eqref{VI}_{2}$ and $\eqref{non2}$, we reformulate
$\eqref{hallMHD}_{2}$ in the Hall-MHD equations as

$h_{t}+\nabla\times(h\times u)+\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}=0,$ (2.18)

which implies its magnetic potential equations

$H_{t}+h\times u+\text{div}(h\otimes h)+\nabla p=0.$ (2.19)

Based on this, we present the definitions of weak solutions for the
incompressible inviscid H-MHD equations and its magnetic potential equations
(2.19). The definitions of weak solutions for incompressible inviscid E-MHD
equations and its magnetic potential equations can be given in a similar way,
hence we omit the details.

######  Definition 2.1 (Weak solutions of incompressible inviscid H-MHD
equations).

The pair $(u,b)\in C_{{\rm weak}}(0,T;L^{2}(\mathbb{T}^{d}))$ with initial
data $u_{0},b_{0}\in L^{2}(\mathbb{T}^{d})$ is a weak solution to the
incompressible inviscid H-MHD equations (LABEL:hallMHD) if

  1. (1)

For all $t\in[0,T]$, $(u(t,x),h(t,x))$ is divergence-free in the sense of
distributions, namely,

$\int_{0}^{T}\int_{\mathbb{T}^{d}}u\cdot\nabla\phi
dx=0,\quad\int_{0}^{T}\int_{\mathbb{T}^{d}}h\cdot\nabla\phi dx=0,$

for any smooth function $\phi(x,t)\in
C_{0}^{\infty}([0,T]\times\mathbb{T}^{d})$

  2. (2)

Equations hold in the sense of distributions, i.e., for any divergence-free
test vector field $\varphi\in C_{0}^{\infty}((0,T)\times\mathbb{T}^{d})$,

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\partial_{t}\varphi\cdot
u+\nabla\varphi:(u\otimes u-h\otimes h)dxdt=0,$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\partial_{t}\varphi\cdot
h+\nabla\varphi:(h\otimes u-u\otimes h)+(h\otimes h):\nabla\nabla\times\varphi
dxdt=0.$

Then we can define the weak solution of magnetic potential equations (2.19) as
follows.

######  Definition 2.2 (Weak solutions of the magnetic potential equations for
inviscid H-MHD equations).

$H\in C_{{\rm weak}}(0,T;L^{2}(\mathbb{T}^{d}))$ with initial data in
$L^{2}(\mathbb{T}^{d})$ is called a weak solution of equation (2.19), if for
any divergence-free test vector field $\varphi\in
C_{0}^{\infty}((0,T)\times\mathbb{T}^{d})$, there holds

$\int_{0}^{T}\int_{\mathbb{T}^{d}}H\cdot\partial_{t}\varphi+(u\times
h)\varphi+(h\otimes h):\nabla\varphi dxdt=0.$

##  3  Two conserved quantities in the inviscid E-MHD equations

In this section, we commence the proof of the energy and maganetic helicity
conservation for the inviscid E-MHD equations (LABEL:E-MHD). First, we show
the energy conservation of weak solutions for the E-MHD equations.

###### Proof of Theorem 1.1.

In what follows, for the sake of simplicity, we assume that the weak solution
$h$ of the inviscid E-MHD equations (LABEL:E-MHD) is differential in time and
only space mollification is applied. This ensures that the weak solution $h$
satisfies

$h_{t}^{\varepsilon}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}=0,$ (3.1)

which holds pointwise in space and time derivative is a classical derivative
at Lebesgue almost all times. For the rigorous argument, we refer the reader
to [15, Section 2. p.741-p.744] by Drivas and Eyink. Multiplying $\eqref{p2}$
by $h^{\varepsilon}$, then integrating it over $(0,T)\times\mathbb{T}^{3}$, we
have

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\frac{1}{2}\frac{d}{dt}|h^{\varepsilon}|^{2}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt=0.$ (3.2)

To obtain the energy balance of weak solutions, we need to take the limits on
both sides of equation (3.2) as $\varepsilon\to 0$. For the convenience of
proof, we denote the second term on the left-hand side of (3.2) by $L$. First,
in light of (2.15) and the integration by parts, we know that

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$ (3.3) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}\nabla(\nabla\times
h^{\varepsilon})dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}(h^{\varepsilon}\otimes
h^{\varepsilon})\nabla(\nabla\times h^{\varepsilon})dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}\nabla(\nabla\times h^{\varepsilon})dxdt,$

where we have used the divergence-free condition and the following facts

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}(h^{\varepsilon}\otimes
h^{\varepsilon})\nabla(\nabla\times h^{\varepsilon})dxdt=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}\text{div}(h^{\varepsilon}\otimes
h^{\varepsilon})-\nabla\frac{1}{2}|h^{\varepsilon}|^{2}\Big{]}\nabla\times
h^{\varepsilon}dxdt$ (3.4) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(\nabla\times
h^{\varepsilon})\times h^{\varepsilon}\Big{]}\nabla\times h^{\varepsilon}dxdt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}\vec{j}^{\varepsilon}\times
h^{\varepsilon}\Big{]}\cdot\vec{j}^{\varepsilon}dxdt=0,$

and

$\vec{A}\times\vec{B}\cdot\vec{A}=\vec{A}\times\vec{A}\cdot\vec{B}=0,\
\text{for\ any\ vectors}\ \vec{A}\ \text{and}\ \vec{B}.$ (3.5)

Then, combining (3.3) and (3.2), we have

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\frac{1}{2}\frac{d}{dt}|h^{\varepsilon}|^{2}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}\nabla(\nabla\times h^{\varepsilon})dxdt=0.$ (3.6)

Now, we are in a position to show that the term $L$ tends to zero as
$\varepsilon\to 0$. Indeed, thanks to the Hölder inequality, we see that

$|L|\leq C\|(h^{\varepsilon}\otimes h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\|\nabla(\nabla\times
h^{\varepsilon})\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}$ (3.7)

Since $h\in L^{3}(0,T;\underline{B}^{\frac{2}{3}}_{3,VMO}(\mathbb{T}^{d}))$,
then applying Lemma 2.2, we observe that, as $\varepsilon\rightarrow 0$,

$\|(h^{\varepsilon}\otimes h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\frac{4}{3}}).$ (3.8)

Furthermore, it follows from Lemma 2.1 that, as $\varepsilon\rightarrow 0$,

$\|\nabla(\nabla\times
h^{\varepsilon})\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{-\frac{4}{3}}).$ (3.9)

As a consequence, combining (3.7), (3.8) and (3.9), we have

$|L|=\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\Big{|}\leq o(1).$ (3.10)

Thus, passing to the limit of $\varepsilon$ in (3.2), we conclude the desired
energy conservation.

(2) Recalling (3.2), the second term $L$ can be rewriten as

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times(\vec{j}\times
h)^{\varepsilon}h^{\varepsilon}dxdt$ (3.11) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}(\vec{j}\times
h)^{\varepsilon}\nabla\times h^{\varepsilon}dxdt.$

Notice that

$\int_{0}^{T}\int_{\mathbb{T}^{d}}(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\nabla\times
h^{\varepsilon}dxdt=\int_{0}^{T}\int_{\mathbb{T}^{d}}(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\cdot\vec{j}^{\varepsilon}dxdt=0.$ (3.12)

Hence, combining (3.11) and (3.12), we get

$\displaystyle
L=\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt=\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(\vec{j}\times
h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\Big{]}\nabla\times h^{\varepsilon}dxdt.$ (3.13)

Now, we need to show the term $L$ tends to zero as $\varepsilon\to 0$. First,
by virtue of the Hölder inequality, we discover that

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\Big{|}$ (3.14)
$\displaystyle\leq$ $\displaystyle\|(\vec{j}\times
h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\|_{L^{\frac{r_{1}r_{2}}{r_{1}+r_{2}}}(0,T;L^{\frac{pq}{p+q}}(\mathbb{T}^{d}))}\|\nabla\times
h^{\varepsilon}\|_{L^{r_{2}}(0,T;L^{q}(\mathbb{T}^{d}))},$

where we need to require that $\frac{1}{r_{1}}+\frac{2}{r_{2}}=1$ and
$\frac{1}{p}+\frac{2}{q}=1$. On the other hand, by means of Lemma 2.2, we
deduce from $\vec{j}\in
L^{r_{1}}(0,T;\underline{B}^{\alpha}_{p,VMO}(\mathbb{T}^{d})),h\in
L^{r_{2}}(0,T;B^{\beta}_{q,\infty}(\mathbb{T}^{d}))$ that

$\|(\vec{j}\times h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\|_{L^{\frac{r_{1}r_{2}}{r_{1}+r_{2}}}(0,T;L^{\frac{pq}{p+q}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\alpha+\beta}),\ \text{as}\ \varepsilon\to 0.$ (3.15)

Moreover, letting $\varepsilon\to 0$ and applying Lemma 2.1, we deduce that

$\|\nabla\times h^{\varepsilon}\|_{L^{r_{2}}(0,T;L^{q}(\mathbb{T}^{d}))}\leq
O(\varepsilon^{\beta-1}).$ (3.16)

Then substituting (3.15) and (3.16) into (3.14), we can obtain

$|L|=\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\Big{|}\leq
o(\varepsilon^{\alpha+2\beta-1}),\ \text{as}\ \varepsilon\to 0.$

Therefore, a combination of $\alpha+2\beta-1\geq 0$ and (3.2) yields the
desired results.

(3) A slight modified the proof of (2) allows us to complete the proof of this
part. We omit the details here.

(4) In the same manner of derivation of (3.3), due to $\vec{j}=\nabla\times
h$, we get

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times
h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt=\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h^{\varepsilon}\otimes
h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\Big{]}\nabla\vec{j}^{\varepsilon}dxdt.$ (3.17)

Then in the light of the Hölder inequality, we arrive at

$\displaystyle\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\Big{|}$ (3.18)
$\displaystyle\leq$ $\displaystyle\|(h^{\varepsilon}\otimes
h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3d}{2d-2}}(\mathbb{T}^{d}))}\|\nabla\vec{j}^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3d}{d+2}}(\mathbb{T}^{d}))}.$

With the help of (3) in Lemma 2.3 and $\vec{j}\in
L^{3}(0,T;B^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}(\mathbb{T}^{d}))$, we
infer that, as $\varepsilon\rightarrow 0$,

$\|(h^{\varepsilon}\otimes h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3d}{2d-2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\frac{2}{3}}).$

Moreover, by means of Lemma 2.1, we know that, as $\varepsilon\rightarrow 0$,

$\|\nabla\vec{j}^{\varepsilon}\|_{L^{3}(0,T;L^{\frac{3d}{d+2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{-\frac{2}{3}}),$

which turns out that

$\Big{|}\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\Big{|}\leq o(1).$

Consequently, we immediately finish the proof of this part by $\vec{j}\in
L^{3}(0,T;B^{\frac{1}{3}}_{\frac{3d}{d+2},c(\mathbb{N})}(\mathbb{T}^{d}))$.

∎

Next, we turn our attentions to the proof of magnetic helicity balance for the
inviscid E-MHD equations (LABEL:E-MHD).

###### Proof of Theorem 1.2.

Recalling the magnetic potential equation for E-MHD equations (2.17) that

$H_{t}+\text{div}(h\otimes h)+\nabla p=0,$ (3.19)

where $H=\text{curl}^{-1}h$. Mollifying the equations (3.19) and (LABEL:E-MHD)
in spatial direction, we observe that

$\displaystyle H_{t}^{\varepsilon}+\text{div}(h\otimes h)^{\varepsilon}+\nabla
p^{\varepsilon}=0,$ $\displaystyle
h_{t}^{\varepsilon}+\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}=0.$

Then it follows from a straightforward computation that

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H^{\varepsilon}dx=$
$\displaystyle\int_{\mathbb{T}^{3}}h_{t}^{\varepsilon}\cdot
H^{\varepsilon}dx+\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H_{t}^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{3}}\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}\cdot
H^{\varepsilon}dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot\Big{[}\text{div}(h\otimes
h)^{\varepsilon}+\nabla p^{\varepsilon}\Big{]}dx$ $\displaystyle=$
$\displaystyle-2\int_{\mathbb{T}^{3}}\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}h^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle-2\int_{\mathbb{T}^{3}}\Big{[}\text{div}(h\otimes
h)^{\varepsilon}-\mathrm{div}\,(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}h^{\varepsilon}dx$ $\displaystyle=$ $\displaystyle
2\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\Big{]}\nabla
h^{\varepsilon}dx,$ (3.20)

where in the fourth equality we have used the divergence-free condition that
$\mathrm{div}\,h=0$. Hence, integrating the above equation over $[0,T]$, we
know that

$\int_{\mathbb{T}^{3}}h^{\varepsilon}(x,T)\cdot
H^{\varepsilon}(x,T)dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}(x,0)\cdot
H^{\varepsilon}(x,0)dx=2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\Big{]}\nabla
h^{\varepsilon}dxdt.$ (3.21)

For the convenience of computation, we denote the term on the right-hand side
of above equation (3.21) by $R$. Consequently, to obtain the desired magnetic
helicity equality, we need to show that the limit of term $R$ is zero as
$\varepsilon\to 0$. Indeed, by the Hölder inequality, we have

$\displaystyle|R|=$
$\displaystyle\Big{|}2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-h^{\varepsilon}\otimes h^{\varepsilon}\Big{]}\nabla
h^{\varepsilon}dxdt\Big{|}$ (3.22) $\displaystyle\leq$ $\displaystyle
C\|(h\otimes h)^{\varepsilon}-h^{\varepsilon}\otimes
h^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{3}))}\|\nabla
h^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{3}))}.$

Since $h\in L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{3})),$
then aplying Lemma 2.2 we have, as $\varepsilon\to 0$,

$\|(h\otimes h)^{\varepsilon}-h^{\varepsilon}\otimes
h^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{3}))}\leq
o(\varepsilon^{\frac{2}{3}})$

On the other hand, by virtue of Lemma 2.1, as $\varepsilon\to 0$, we find

$\|\nabla h^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{3}))}\leq
o(\varepsilon^{-\frac{2}{3}})$

Hence, letting $\varepsilon\to 0$, we conclude that

$|R|=\Big{|}2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-h^{\varepsilon}\otimes h^{\varepsilon}\Big{]}\nabla
h^{\varepsilon}dxdt\Big{|}\leq o(1).$ (3.23)

Thus, taking the limits in (3.21) as $\varepsilon\rightarrow 0$ and using
(3.23), we finish the proof of this theorem.

∎

##  4 Energy and magnetic helicity conservation for the inviscid H-MHD system

In this section, we are concerned with the energy and magnetic helicity
conservation for the inviscid H-MHD equations.

###### Proof of Theorem 1.3.

Mollifying the equations (LABEL:hallMHD) in spatial direction (see the
notations in Section 2), we have

$\displaystyle\partial_{t}{u^{\varepsilon}}+\text{div}(u\otimes
u)^{\varepsilon}-\text{div}(h\otimes
h)^{\varepsilon}+\nabla\Pi^{\varepsilon}=0,$ (4.1)
$\displaystyle\partial_{t}{h^{\varepsilon}}+\text{div}(u\otimes
h)^{\varepsilon}-\text{div}(h\otimes
u)^{\varepsilon}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}=0.$

Multiplying $\eqref{rmhd}_{1}$ by $u^{\varepsilon}$ and $\eqref{rmhd}_{2}$ by
$h^{\varepsilon}$ respectively, then integrating them over
$[0,T]\times\mathbb{T}^{d}$, we have

$\displaystyle\frac{1}{2}\|u^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}+\frac{1}{2}\|h^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|u^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|h^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}$
(4.2) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(u\otimes
u)^{\varepsilon}u^{\varepsilon}-\text{div}(h\otimes
h)^{\varepsilon}u^{\varepsilon}+\text{div}(\Pi
u^{\varepsilon})+\text{div}(u\otimes h)^{\varepsilon}h^{\varepsilon}$
$\displaystyle-\text{div}(h\otimes
u)^{\varepsilon}h^{\varepsilon}+\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(u\otimes
u)^{\varepsilon}u^{\varepsilon}-\text{div}(h\otimes
h)^{\varepsilon}u^{\varepsilon}+\text{div}(u\otimes
h)^{\varepsilon}h^{\varepsilon}$ $\displaystyle-\text{div}(h\otimes
u)^{\varepsilon}h^{\varepsilon}+\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle\,I+II+III+IV+V,$

where the divergence-free condition and (2.15) have been used.

First, by means of divergence-free condition and the integration by parts, the
term $I$ can be rewritten as

$\displaystyle I=$
$\displaystyle-\int_{\mathbb{T}^{d}}\text{div}\big{(}(u\otimes
u)^{\varepsilon}-(u^{\varepsilon}\otimes
u^{\varepsilon})\big{)}u^{\varepsilon}dxdt-\int_{\mathbb{T}^{d}}\text{div}(u^{\varepsilon}\otimes
u^{\varepsilon})u^{\varepsilon}dxdt$ (4.3) $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{d}}\text{div}\big{(}(u\otimes
u)^{\varepsilon}-(u^{\varepsilon}\otimes
u^{\varepsilon})\big{)}u^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{d}}\big{(}(u\otimes
u)^{\varepsilon}-(u^{\varepsilon}\otimes u^{\varepsilon})\big{)}\nabla
u^{\varepsilon}dxdt.$

Likewise, for the second term $II$, the integration by parts helps us to get

$\displaystyle II=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}\big{(}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\big{)}u^{\varepsilon}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(h^{\varepsilon}\otimes
h^{\varepsilon})u^{\varepsilon}dxdt$ (4.4)
$\displaystyle=-\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\big{)}\nabla
u^{\varepsilon}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(h^{\varepsilon}\otimes
h^{\varepsilon})u^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle\,II_{1}+II_{2}.$

We will show that the terms $II_{2}$ can be canceled in term $IV$. Indeed, in
light of the integration by parts and $\text{div}\,h=0$, we arrive at

$\displaystyle IV=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}\big{(}(h\otimes
u)^{\varepsilon}-(h^{\varepsilon}\otimes
u^{\varepsilon})\big{)}h^{\varepsilon}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(h^{\varepsilon}\otimes
u^{\varepsilon})h^{\varepsilon}dxdt$ (4.5) $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{d}}\big{(}(h\otimes
u)^{\varepsilon}-(h^{\varepsilon}\otimes u^{\varepsilon})\big{)}\nabla
h^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}h^{\varepsilon}\cdot\nabla
h^{\varepsilon}\cdot u^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(h\otimes
u)^{\varepsilon}-(h^{\varepsilon}\otimes u^{\varepsilon})\big{)}\nabla
h^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(h^{\varepsilon}\otimes
h^{\varepsilon})u^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle\,IV_{1}-II_{2}.$

Next, by the same token, we can control the term $III$ as

$\displaystyle III=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}\big{(}(u\otimes
h)^{\varepsilon}-(u^{\varepsilon}\otimes
h^{\varepsilon})\big{)}h^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\text{div}(u^{\varepsilon}\otimes
h^{\varepsilon})h^{\varepsilon}dxdt$ (4.6) $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(u\otimes
h)^{\varepsilon}-(u^{\varepsilon}\otimes h^{\varepsilon})\big{)}\nabla
h^{\varepsilon}dxdt.$

Finally, for the term $V$, recalling (3.3), we can obtain

$\displaystyle V=$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$ (4.7) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}\nabla(\nabla\times h^{\varepsilon})dxdt.$

Then substrituting (4.3)-(4.7) into (LABEL:4.2), we have

$\displaystyle\frac{1}{2}\|u^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}+\frac{1}{2}\|h^{\varepsilon}(T)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|u^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}-\frac{1}{2}\|h^{\varepsilon}(0)\|^{2}_{L^{2}(\mathbb{T}^{d})}$
(4.8) $\displaystyle=$ $\displaystyle\int_{\mathbb{T}^{d}}\big{(}(u\otimes
u)^{\varepsilon}-(u^{\varepsilon}\otimes u^{\varepsilon})\big{)}\nabla
u^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\big{)}\nabla
u^{\varepsilon}dxdt$
$\displaystyle+\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(u\otimes
h)^{\varepsilon}-(u^{\varepsilon}\otimes h^{\varepsilon})\big{)}\nabla
h^{\varepsilon}dxdt-\int_{0}^{T}\int_{\mathbb{T}^{d}}\big{(}(h\otimes
u)^{\varepsilon}-(h^{\varepsilon}\otimes u^{\varepsilon})\big{)}\nabla
h^{\varepsilon}dxdt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}\nabla(\nabla\times h^{\varepsilon})dxdt$
$\displaystyle=$ $\displaystyle I+II_{1}+III+IV_{1}+V.$

Now, to obtain the desired energy conservation, we are in a position to prove
the terms on the right-hand side of (4.8) tend to zero as $\varepsilon\to 0$.
However, since the last term $V$ is same as that in the proof of Theorem 1.1,
here we oly focus on the rest terms. Actually, for the term $I$, by virtue of
the Hölder inequality, we can get

$\displaystyle|I|\leq$ $\displaystyle C\|(u^{\varepsilon}\otimes
u^{\varepsilon})-(u\otimes
u)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\|\nabla
u^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}.$ (4.9)

On the other hand, applying Lemma 2.2 and Lemma 2.1 with $u\in
L^{3}(0,T;B^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{d}))$, we discover

$\|(u^{\varepsilon}\otimes u^{\varepsilon})-(u\otimes
u)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\frac{2}{3}}),$

and

$\|\nabla u^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{-\frac{2}{3}}),$

which together with (4.9) imply

$|I|\leq o(1),\ \text{as}\ \varepsilon\to 0.$ (4.10)

To control the terms $II_{1}$, $III$ and $IV_{1}$, in light of the Hölder
inequality, we have

$\displaystyle|II_{1}+III+IV_{1}|$ (4.11) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}|\big{(}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\big{)}||\nabla
u^{\varepsilon}|dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}|\big{(}(h\otimes
u)^{\varepsilon}-(h^{\varepsilon}\otimes u^{\varepsilon})\big{)}||\nabla
h^{\varepsilon}|dxdt$ $\displaystyle\leq$ $\displaystyle
C\|(u^{\varepsilon}\otimes h^{\varepsilon})-(u\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\|\nabla
h^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}$ $\displaystyle\leq$
$\displaystyle C\|(h\otimes h)^{\varepsilon}-(h^{\varepsilon}\otimes
h^{\varepsilon})\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\|\nabla
u^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}$
$\displaystyle+C\|(h\otimes u)^{\varepsilon}-(h^{\varepsilon}\otimes
u^{\varepsilon})\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\|\nabla
h^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))},$

Then due to $h\in
L^{3}(0,T;\underline{B}^{\frac{2}{3}}_{3,VMO}(\mathbb{T}^{d}))$ and $u\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{d}))$, by virtue of
(1) in Lemma 2.2, as $\varepsilon\to 0$, we remark that

$\|(h^{\varepsilon}\otimes h^{\varepsilon})-(h\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{\frac{4}{3}}),$ $\|(h^{\varepsilon}\otimes
u^{\varepsilon})-(h\otimes
u)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{1}),$ $\|(u^{\varepsilon}\otimes h^{\varepsilon})-(u\otimes
h)^{\varepsilon}\|_{L^{\frac{3}{2}}(0,T;L^{\frac{3}{2}}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{1}).$

And it follows from Lemma 2.1 that

$\|\nabla h^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{-\frac{1}{3}}),\ \text{and}\ \|\nabla
u^{\varepsilon}\|_{L^{3}(0,T;L^{3}(\mathbb{T}^{d}))}\leq
o(\varepsilon^{-\frac{2}{3}}).$

Thus, we end up with

$|II_{1}+III+IV_{1}|\leq o(\varepsilon^{\frac{2}{3}}).$

Recalling (3.10), we have

$|V|\leq o(1).$

With these estimates in hand, by taking $\varepsilon\rightarrow 0$ in
(LABEL:4.2), we can achieve the proof of Theorem 1.3.

∎

Next, we consider the magnetic helicity conservation for the inviscid H-MHD
equations (LABEL:hallMHD).

###### Proof of Theorem 1.4.

Mollifying the equations (2.18) and (2.19) in spatial direction, we have

$\displaystyle h_{t}^{\varepsilon}+\nabla\times(h\times
u)^{\varepsilon}+\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}=0,$ $\displaystyle H_{t}^{\varepsilon}+(h\times
u)^{\varepsilon}+\text{div}(h\otimes h)^{\varepsilon}+\nabla
p^{\varepsilon}=0.$

By a straightforward computation, it gives

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H^{\varepsilon}dx=\int_{\mathbb{T}^{3}}h_{t}^{\varepsilon}\cdot
H^{\varepsilon}dx+\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H_{t}^{\varepsilon}dx$ (4.12) $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{3}}\Big{[}\nabla\times(h\times
u)^{\varepsilon}+\nabla\times\Big{[}\text{div}(h\otimes
h)\Big{]}^{\varepsilon}\Big{]}\cdot
H^{\varepsilon}dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot\Big{[}(h\times
u)^{\varepsilon}+\text{div}(h\otimes h)^{\varepsilon}+\nabla
p^{\varepsilon}\Big{]}dx$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{3}}\Big{[}(h\times
u)^{\varepsilon}+\mathrm{div}\,(h\otimes h)^{\varepsilon}\Big{]}\cdot
h^{\varepsilon}dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot\Big{[}(h\times
u)^{\varepsilon}+\mathrm{div}\,(h\otimes h)^{\varepsilon}\Big{]}dx$
$\displaystyle=$ $\displaystyle-2\int_{\mathbb{T}^{3}}(h\times
u)^{\varepsilon}\cdot
h^{\varepsilon}dx-2\int_{\mathbb{T}^{3}}\mathrm{div}\,(h\otimes
h)^{\varepsilon}\cdot h^{\varepsilon}dx,$

where the integration by parts and the divergence-free condition have been
used. Then we conclude by performing a time integral that

$\displaystyle\int_{\mathbb{T}^{3}}h^{\varepsilon}(x,T)\cdot
B^{\varepsilon}(x,T)dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}(x,0)\cdot
B^{\varepsilon}(x,0)dx$ (4.13) $\displaystyle=$
$\displaystyle-2\int_{0}^{T}\int_{\mathbb{T}^{3}}(h\times
u)^{\varepsilon}\cdot
h^{\varepsilon}dxdt-2\int_{0}^{T}\int_{\mathbb{T}^{3}}\text{div}(h\otimes
h)^{\varepsilon}\cdot h^{\varepsilon}dxdt$ $\displaystyle=$
$\displaystyle-2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\times
u)^{\varepsilon}-(h^{\varepsilon}\times
u^{\varepsilon})\Big{]}h^{\varepsilon}dxdt+2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-h^{\varepsilon}\otimes h^{\varepsilon}\Big{]}\nabla
h^{\varepsilon}dxdt$ $\displaystyle=$ $\displaystyle I+II,$

where we have used the divergence-free condition and the fact that
$h^{\varepsilon}\cdot(u^{\varepsilon}\times
h^{\varepsilon})=u^{\varepsilon}\cdot(h^{\varepsilon}\times
h^{\varepsilon})=0$.

Now, to obtain the desired the magnetic helicity conservation, we need to take
the limits in $\varepsilon\to 0$ in the above equation. Precisely, we have to
show that the terms $I$, $II$ tend to zero as $\varepsilon\to 0$. Since the
second term $II$ is exactly the same as (3.21) in proof of Theorem 1.2, hence,
we here only need to focus on the term $I$. Actually, applying Lemma 2.4 with
$u\in L^{3}(0,T;L^{3}(\mathbb{T}^{3}))$ and $h\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{3})),$ we see that
$I$ tends to zero as $\varepsilon\rightarrow 0.$ By arguing as what was done
to prove (3.23), we have

$|II|\leq o(1).$

Thus, the proof of this theorem is completed. ∎

##  5 Refined sufficient conditions of conserved quantities in the viscous
E-MHD and H-MHD equations

In this section, making use of the energy
$L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{T}^{3})))$,
we will reduce the proof of Corollary 1.6-1.5 to the results in Theorem
1.3-1.4.

###### Proof of Corollary 1.5.

First, Notice that for any weak solutions of the system (LABEL:visE-MHD),
there holds that $h\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$.

1.(Energy conservation) Multiplying the viscid E-MHD equations (LABEL:visE-
MHD) by $(h^{\varepsilon})^{\varepsilon}$ and integrating the resualant over
time-space, we have

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\frac{1}{2}\frac{d}{dt}|h^{\varepsilon}|^{2}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}|\nabla
h^{\varepsilon}|^{2}dxdt+\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt=0.$ (5.1)

Recalling (3.13) and $\vec{j}=\nabla\times h$, we have

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(\vec{j}\times
h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\Big{]}\nabla\times h^{\varepsilon}dxdt$ (5.2)
$\displaystyle=\int_{0}^{T}\int_{\mathbb{T}^{d}}\Big{[}(\vec{j}\times
h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\Big{]}\vec{j}^{\varepsilon}dxdt$

To control this term, first in light of Hölder inequality, we see that

$\displaystyle|\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times
h)\times h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt|\leq$
$\displaystyle\|(\vec{j}\times h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\|_{L^{\frac{r_{1}r_{2}}{r_{1}+r_{2}}}(0,T;L^{\frac{pq}{p+q}}(\mathbb{T}^{d}))}\|\vec{j}^{\varepsilon}\|_{L^{r_{1}}(0,T;L^{p}(\mathbb{T}^{d}))},$
(5.3)

where $\frac{2}{r_{1}}+\frac{1}{r_{2}}=1$ and $\frac{2}{p}+\frac{1}{q}=1$.
Moreover, since $\vec{j}\in L^{r_{1}}(0,T;L^{p}(\mathbb{T}^{d}))$ and $h\in
L^{r_{2}}(0,T;L^{q}(\mathbb{T}^{d}))$, then it follows from Lemma 2.4 that

$\|(\vec{j}\times h)^{\varepsilon}-(\vec{j}^{\varepsilon}\times
h^{\varepsilon})\|_{L^{\frac{r_{1}r_{2}}{r_{1}+r_{2}}}(0,T;L^{\frac{pq}{p+q}}(\mathbb{T}^{d}))}\to
0,\text{as}\ \varepsilon\to 0,$ (5.4)

which immediately implies

$\int_{0}^{T}\int_{\mathbb{T}^{d}}\nabla\times\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}h^{\varepsilon}dxdt\to 0,\text{as}\ \varepsilon\to 0.$
(5.5)

Hence, taking the limits in (LABEL:5.1) as $\varepsilon\to 0$, we can conclude
the desired energy conservation.

2.(Magnetic helicity conservation) First, recalling the indentity $\Delta
h=\nabla(\mathrm{div}\,h)-\nabla\times(\nabla\times h)$, we can get the
magnetic vector potential equation for the viscous E-MHD equations
(LABEL:visE-MHD) as follows

$H_{t}+(\nabla\times h)\times h+(\nabla\times h)+\nabla p=0,\ \
\mathrm{div}\,H=0,$ (5.6)

where $H=\text{curl}^{-1}$. Then multiplying (LABEL:visE-MHD) by
$(H^{\varepsilon})^{\varepsilon}$ and (5.6) by
$(h^{\varepsilon})^{\varepsilon}$, respectively, we have

$\displaystyle\frac{d}{dt}\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H^{\varepsilon}dx=\int_{\mathbb{T}^{3}}h_{t}^{\varepsilon}\cdot
H^{\varepsilon}dx+\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot
H_{t}^{\varepsilon}dx$ (5.7) $\displaystyle=$
$\displaystyle-\int_{\mathbb{T}^{3}}\Big{[}\nabla\times\Big{(}(\nabla\times
h)\times h\Big{)}^{\varepsilon}+\nabla\times(\nabla\times
h)^{\varepsilon}\Big{]}\cdot
H^{\varepsilon}dx-\int_{\mathbb{T}^{3}}h^{\varepsilon}\cdot\Big{[}\Big{(}(\nabla\times
h)\times h\Big{)}^{\varepsilon}+\nabla\times h^{\varepsilon}+\nabla
p^{\varepsilon}\Big{]}dx$ $\displaystyle=$
$\displaystyle-2\int_{\mathbb{T}^{3}}\Big{[}(\nabla\times h)\times
h\Big{]}^{\varepsilon}h^{\varepsilon}dx-2\int_{\mathbb{T}^{3}}(\nabla\times
h)^{\varepsilon}\cdot h^{\varepsilon}dx$ $\displaystyle=$
$\displaystyle-2\int_{\mathbb{T}^{3}}\Big{[}\text{div}(h\otimes
h)^{\varepsilon}-\mathrm{div}\,(h^{\varepsilon}\otimes
h^{\varepsilon})\Big{]}h^{\varepsilon}dx-2\int_{\mathbb{T}^{3}}(\nabla\times
h)^{\varepsilon}\cdot h^{\varepsilon}dx$ $\displaystyle=$ $\displaystyle
2\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\Big{]}\nabla
h^{\varepsilon}dx-2\int_{\mathbb{T}^{3}}(\nabla\times h)^{\varepsilon}\cdot
h^{\varepsilon}dx,$

where we have used (2.15) and the divergence-free condition. Hence,
integrating (5.7) over time-space, we can obtain

$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{3}}\frac{d}{dt}h^{\varepsilon}\cdot
H^{\varepsilon}dxdt+2\int_{0}^{T}\int_{\mathbb{T}^{3}}(\nabla\times
h)^{\varepsilon}\cdot
h^{\varepsilon}dxdt=2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-(h^{\varepsilon}\otimes h^{\varepsilon})\Big{]}\nabla
h^{\varepsilon}dxdt.$ (5.8)

Now, to obtain the desired magnetic helicity conservation, it suffices to show
that the term on the right-hand side of (5.8) tends to zero as $\varepsilon\to
0$. For the convenience of computation, we let this term by $R$.

(1) To control the term $R$, it follows from the Hölder inequality that

$\displaystyle|R|=$
$\displaystyle\Big{|}2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-h^{\varepsilon}\otimes h^{\varepsilon}\Big{]}\nabla
h^{\varepsilon}dxdt\Big{|}$ (5.9) $\displaystyle\leq$ $\displaystyle
C\|(h\otimes h)^{\varepsilon}-h^{\varepsilon}\otimes
h^{\varepsilon}\|_{L^{\frac{2p}{p+1}}(0,T;L^{\frac{2q}{q+1}}(\mathbb{T}^{3}))}\|\nabla
h^{\varepsilon}\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}$

Since $\vec{j}\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$ and $h\in
L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3})),$ making use of
Lemma (2.5), we have

$\displaystyle\|(h\otimes h)^{\varepsilon}-h^{\varepsilon}\otimes
h^{\varepsilon}\|_{L^{\frac{2p}{p+1}}(0,T;L^{\frac{2q}{q+1}}(\mathbb{T}^{3}))}\leq
C\varepsilon\|h\|_{L^{p}(W^{1,q}(0,T;\mathbb{T}^{3}))}\|h\|_{L^{\frac{2p}{p-1}}(L^{\frac{2q}{q-1}}(0,T;\mathbb{T}^{3}))},$
(5.10) $\displaystyle\Big{\|}\nabla
h^{\varepsilon}\Big{\|}_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}\leq
C\varepsilon^{-1}\|h\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))},$
$\displaystyle\limsup_{\varepsilon\rightarrow 0}\varepsilon\Big{\|}\nabla
h\Big{\|}_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}=0.$

This implies that

$\limsup_{\varepsilon\rightarrow
0}\Big{|}2\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{[}(h\otimes
h)^{\varepsilon}-h^{\varepsilon}\otimes h^{\varepsilon}\Big{]}\nabla
h^{\varepsilon}dxdt\Big{|}=0.$ (5.11)

Thus, taking the limits in (3.21) as $\varepsilon\rightarrow 0$ and using the
Lebesgue dominated convergence theorem and (5.11), we complete the proof of
this part.

(2) We derive from $\vec{j}\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$ that $\nabla
h\in L^{p}(0,T;L^{q}(\mathbb{T}^{3}))$ via the classical Calderón-Zygmund
Theorem. A natural choice in the first part of Corollary 1.5 is $p=q=2$,
hence, we know that $h\in L^{4}(0,T;L^{4}(\mathbb{T}^{3}))$ guarantee magnetic
helicity relation (1.10). Then, we have completed the second part of this
Corollary.

(3) In light of the interpolation inequality, we have

$\displaystyle\|h\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{3}))}\leq$ $\displaystyle
C\|h\|_{L^{\infty}\left(0,T;L^{2}(\mathbb{T}^{3})\right)}^{\frac{(q-4)}{2q-4}}\|h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2q-4}}\leq
C.$ (5.12)

where $\frac{2}{p}+\frac{2}{q}=1$ and $q\geq 4$ was used.

Likewise, we also have

$\displaystyle\|h\|_{L^{4}(0,T;L^{4}(\mathbb{T}^{3}))}$ $\displaystyle\leq
C\|h\|_{L^{2}(0,T;L^{6}(\mathbb{T}^{3}))}^{\frac{3(4-q)}{2(6-q)}}\|h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2(6-q)}}$
(5.13) $\displaystyle\leq C\left(\|\nabla
h\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{3}))}+\|h\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\right)^{\frac{3(4-q)}{2(6-q)}}\|h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{q}{2(6-q)}}\leq
C.$

where $3<q<4$ and $\frac{1}{p}+\frac{3}{q}=1$ was used.

Hence, based on the result in part (2), we conclude the third part of this
Corollary.

(4) For $\frac{1}{p}+\frac{6}{5q}=1$ and $q\geq\frac{9}{5}$, we deduce from
the Gagliardo-Nirenberg inequality that

$\displaystyle\|h\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}\leq$
$\displaystyle
C\|h\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}^{\frac{5q-9}{5q-6}}\|\nabla
h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{3}{5q-6}}$ (5.14)
$\displaystyle\leq$ $\displaystyle
C\|h\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}^{\frac{5q-9}{5q-6}}\|\vec{j}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{3}{5q-6}}\leq
C.$

For $\frac{3}{2}<q<\frac{9}{5}$ and $\frac{1}{p}+\frac{3}{q}=2$, using the
Gagliardo-Nirenberg inequality once again, we find

$\displaystyle\|h\|_{L^{\frac{2p}{p-1}}(0,T;L^{\frac{2q}{q-1}}(\mathbb{T}^{3}))}$
$\displaystyle\leq
C\|h\|_{L^{2}(0,T;L^{6}(\mathbb{T}^{3}))}^{\frac{9-5q}{6-3q}}\|\nabla
h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{2q-3}{6-3q}}$ (5.15)
$\displaystyle\leq\left(\|\nabla
h\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{3}))}+\|h\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\right)^{\frac{9-5q}{6-3q}}\|\nabla
h\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{2q-3}{6-3q}}$
$\displaystyle\leq\left(\|\vec{j}\|_{L^{2}(0,T;L^{2}(\mathbb{T}^{3}))}+\|h\|_{L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))}\right)^{\frac{9-5q}{6-3q}}\|\vec{j}\|_{L^{p}(0,T;L^{q}(\mathbb{T}^{3}))}^{\frac{2q-3}{6-3q}}\leq
C.$

Then we conclude the desired result from the first part of this Corollary 1.5.
Then the proof of this corollary is completed. ∎

###### Proof of Corollary 1.6.

(1) We derive from the $u\in L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$ and Sobolev
embedding theorem that $u\in L^{2}(0,T;H^{\frac{5}{6}}(\mathbb{T}^{3}))$.
Since $H^{\frac{5}{6}}(\mathbb{T}^{3})\approx B^{\frac{5}{6}}_{2,2}$, we
conclude by $B^{\frac{5}{6}}_{2,2}\subseteq B^{\frac{1}{3}}_{3,2}$ that $u\in
L^{2}(0,T;B^{\frac{1}{3}}_{3,2})$. This together with the Dominated-
Convergence-Theorem and $u\in L^{3}(0,T;B_{3,\infty}^{\frac{1}{3}})$ means
that $u\in L^{3}(0,T;B_{3,c(\mathbb{N})}^{\frac{1}{3}})$. This further enables
us to get $u\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{d}))$. Theorem 1.3
helps us to complete the proof of this part.

(2) Exactly as in the above derivation, we deduce from $h\in
L^{3}(0,T;B^{\frac{1}{3}}_{3,\infty}(\mathbb{T}^{3}))$ that $h\in
L^{3}(0,T;\underline{B}^{\frac{1}{3}}_{3,VMO}(\mathbb{T}^{d}))$. The
interpolation inequality and $u\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{3}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{3}))$ leads to $u\in
L^{3}(0,T;L^{3}(\mathbb{T}^{3}))$. With this in hand, we immediately finish
the proof of this part by Theorem 1.4. ∎

## Acknowledgements

Wang was partially supported by the National Natural Science Foundation of
China under grant (No. 11971446, No. 12071113 and No. 11601492) and sponsored
by Natural Science Foundation of Henan Province (No. 232300421077). Ye was
partially supported by the National Natural Science Foundation of China under
grant (No.11701145) and sponsored by Natural Science Foundation of Henan
Province (No. 232300420111).

Author Declarations statement

The authors have no conflicts to disclose.

Data Availability

This publication is supported by multiple datasets, which are openly available
at locations cited in the reference section.

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