# Liouville type Theorems for the stationary compressible 3D MHD equations

Jae-Myoung Kim

###### Abstract.

In this paper, we investigate the three dimensional stationary compressible
$3D$ MHD equations, and obtain Liouville type theorems if a smooth solution
$(\rho,\mathbf{u})$ satisfies some suitable conditions. In particular, our
results improve and generalize the corresponding result.

Department of Mathematics Education, Andong National University, Andong 36729,
Republic of Korea

E-mail address<EMAIL_ADDRESS>
Keywords: Liouville type theorem; compressible 3D MHD equations; Lorentz
space; (local) Morrey space

AMS Subject Classification (2000): 35Q30, 76N10.

##  1\. Introduction

The system of barotropic compressible magnetohydrodynamic(MHD) equations in
$\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$ can be read as follows:

(1.1) $\begin{cases}\operatorname{div}(\rho\mathbf{u})=0,\\\
-\nu\Delta\mathbf{u}-(\lambda+\nu)\nabla\operatorname{div}\mathbf{u}+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})-\operatorname{div}(\mathbf{b}\otimes\mathbf{b})+\nabla(P+\frac{|\mathbf{b}|^{2}}{2})=0,\\\
-\Delta\mathbf{b}+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{b})-\operatorname{div}(\mathbf{b}\otimes\mathbf{u})=0,\\\
\operatorname{div}\mathbf{b}=0,\end{cases}$

where the vector $\mathbf{u}$ denotes the flow velocity field and the scalar
function $\rho$ represents the density of the fluid. Without loss of
generality, we consider the flows with $\gamma-$law pressure:

$P(\rho):=a\rho^{\gamma},\quad a>0,\quad\gamma>1.$

The shear viscosity $\nu$ and bulk viscosity $\lambda$ are both constants and
satisfy

$\nu>0\ ,\ \lambda+\frac{2}{3}\nu>0.$

We consider the initial value problem of (1.1), which requires initial
conditions

(1.2)
$\textbf{u}(x,0)=\textbf{u}_{0}(x)\quad\text{and}\quad\textbf{b}(x,0)=\textbf{b}_{0}(x),\qquad
x\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}.$

Among them, we briefly recall the previous results concerned with the
compressible 3D MHD equations. The local strong solutions to the compressible
MHD with large initial data were obtained, by Vol’pert and Khudiaev [27] as
the initial density is strictly positive and also by Fan and Yu [5] as the
initial density may contain vacuum, respectively. Hu and Wang [6, 7, 8, 9]
proved the global existence and uniqueness of classical solutions to the
Cauchy problem with smooth initial data which are of small energy but possibly
large oscillations. They obtain the global existence of weak solutions for
initial data which may be discontinuous and contain vacuum states, and where
the initial temperature is allowed to be zero. Later, Li, Xu and Zhang [15]
shown if the initial energy is sufficiently small, the global existence of a
regular solution is shown. This solution is classical with arbitrarily large
oscillations, and contain vacuum states.

On the contrary, to the best of one’s knowledge, There is not much result
related to the equations (1.1). Yang et. al. [29] prove existence and
uniqueness of strong solutions to barotropic compressible magnetohydrodynamic
(see also [28] and [22]). Comparing to the steady incompressible MHD
equations, we refer to [32] and [34]. The long-time behavior of a small global
solution can be found in [9], [31].

Regarding Liouville theorems for compressible $3D$ Navier-Stokes equations,
Chae [2] showed if the smooth solution $(\rho,\mathbf{u})$ satisfies

$\begin{split}\|\rho\|_{L^{\infty}(\mathbb{R}^{3})}+\|\nabla\mathbf{u}\|_{L^{2}(\mathbb{R}^{3})}+\|\mathbf{u}\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})}<\infty.\end{split}$

then $u\equiv 0$ and $\rho=constant$. After that, through the suitable
decomposition for pressure (Lemma 2.3 below), Li and Yu in [20] shown

$\begin{split}\|\rho\|_{L^{\infty}(\mathbb{R}^{3})}+\|\nabla\mathbf{u}\|_{L^{2}(\mathbb{R}^{3})}+\|\mathbf{u}\|_{L^{\frac{9}{2}}(\mathbb{R}^{3})}<\infty.\end{split}$

Later, in the Lorentz framework, Li and Niu [17] proved if $(\rho,\mathbf{u})$
is a smooth solution with $\rho\in L^{\infty}(\mathbb{R}^{3})$,
$\nabla\mathbf{u}\in L^{2}(\mathbb{R}^{3})$ and $\mathbf{u}\in
L^{p,q}(\mathbb{R}^{3})$ for $3<p<\frac{9}{2}$, $3\leq q\leq\infty$ or
$p=q=3$. Then $\mathbf{u}\equiv 0$ and $\rho=constant$ on $\mathbb{R}^{3}$. It
is a relaxation result to the corresponding result [20]. In mathematics
analysis, Liouville theorems is classical problem. However, it has not been
resolved so far in a fluid equations.

In present paper, we study Liouviile Theorem for the compressible 3D MHD
equations in a whose space in viewpoint of Lorentz space or Morrey space based
on the Cacciopoly-type estimates. In our analysis, in particular, we use
pressure decomposition (2.1) based on Lemma 2.3 below, which is a essential
tool to deal with the pressure.

Before we look at the main results, we give some definitions for functional
spaces. Given $1\leq p<\infty,1\leq q\leq\infty$, we say that a measurable
function $f\in L^{p,q}(\mathbb{R}^{3})$ if
$\|f\|_{L^{p,q}(\mathbb{R}^{3})}<\infty,$ where

$\|f\|_{L^{p,q}(\mathbb{R}^{3})}:=\begin{cases}\left(\int_{0}^{\infty}t^{q-1}|\\{x\in\mathbb{R}^{3}:|f(x)|>t\\}|^{\frac{q}{p}}\,dt\right)^{\frac{1}{q}},\quad\mathrm{if}\,\quad
q<+\infty,\\\
\sup\limits_{t>0}t|\\{x\in\mathbb{R}^{3}:|f(x)|>t\\}|^{\frac{1}{p}},\quad\mathrm{if}\,\quad
q=+\infty.\end{cases}$

The space satisfies the continuous embedding (see e.g. [21])

(1.3) $L^{p}(\mathbb{R}^{3})=L^{p,p}(\mathbb{R}^{3})\hookrightarrow
L^{p,q}(\mathbb{R}^{3})\hookrightarrow L^{p,\infty}(\mathbb{R}^{3})\ ,\ p\leq
q<\infty.$

In this direction, parallel to the result of Li and Niu, first result is
stated as

######  Theorem 1.1.

Suppose that $(\rho,\mathbf{u},\mathbf{b})$ is a smooth solution to (1.1) with
$\rho\in L^{\infty}(\mathbb{R}^{3})$, $\nabla\mathbf{u},\nabla\mathbf{b}\in
L^{2}(\mathbb{R}^{3})$ and $\mathbf{u},\ \mathbf{b}\in
L^{p,q}(\mathbb{R}^{3})$ for $3<p<\frac{9}{2}$, $3\leq q\leq\infty$ or
$p=q=3$. Then $\mathbf{u}\equiv 0$ and $\rho=constant$ on $\mathbb{R}^{3}$.

In light of [16, Theorem 1.2], for $p\geq\frac{9}{2}$, we are easily checked
to obtain a following results and omit a proof.

######  Theorem 1.2.

Let Let $\mathcal{C}(R/2,R)=\\{x\in\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}:R/2<|x|<R\\}$ and

$M_{p,q}(R):=R^{\frac{2}{3}-\frac{3}{p}}(\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|_{L^{p,q}(\mathcal{C}(R/2,R))}).$

Suppose that $(\rho,\mathbf{u},\mathbf{b})$ is a smooth solution to (1.1) with
$\rho\in L^{\infty}(\mathbb{R}^{3})$ and $\nabla\mathbf{u},\
\nabla\mathbf{b}\in L^{2}(\mathbb{R}^{3}).$ For $p\geq\frac{9}{2}$, $3\leq
q\leq\infty$, assume that

$\begin{split}\liminf_{R\rightarrow\infty}M_{\,p,\,q}(R)<\infty,\end{split}$

then

(1.4)
$\begin{split}D(\mathbf{u},\mathbf{b}):=\int_{\mathbb{R}^{3}}|\nabla\mathbf{u}|^{2}+|\nabla\mathbf{b}|^{2}\,dx\leq\mathcal{C}\liminf_{R\rightarrow\infty}M^{3}_{\,p,\,q}(R).\end{split}$

If moreover assume

(1.5) $\begin{split}\liminf_{R\rightarrow\infty}M^{3}_{\,p,\,q}(R)\leq\delta
D(\mathbf{u})\end{split}$

for some $0<\delta<\frac{1}{\mathcal{C}},$ then $\mathbf{u}\equiv 0$ and
$\rho=constant$ on $\mathbb{R}^{3}$.

Next, we also introduce the definition of the (homogeneous) Morrey and local
Morrey spaces (see e.g. [14] for more details). Let $1<p<r<+\infty$, the
homogeneous Morrey space $\dot{M}^{p,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ is the set of functions $f\in
L^{p}_{loc}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ such that

$\|f\|_{\dot{M}^{p,r}}=\sup_{R>0,\,\,x_{0}\in\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}R^{\frac{3}{r}}\left(\frac{1}{R^{3}}\int_{B(x_{0},R)}|f(x)|^{p}dx\right)^{\frac{1}{p}}<+\infty,$

where $B(x_{0},R)$ denotes the ball centered at $x_{0}$ and with radio $R$. It
is well known $L^{r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\subset
L^{r,q}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset\dot{M}^{p,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$, where, for $r\leq q\leq+\infty$.

For $\gamma\geq 0$ and $1<p<+\infty$, we define the _local Morrey space_
$M^{p}_{\gamma}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ as the Banach
space of functions $f\in L^{p}_{loc}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ such that

$\|f\|_{M^{p}_{\gamma}}=\sup_{R\geq
1}\left(\frac{1}{R^{\gamma}}\int_{B(0,R)}|f(x)|^{p}dx\right)^{1/p}<+\infty.$

Moreover,

$\gamma_{1}\leq\gamma_{2}\Rightarrow
M^{p}_{\gamma_{1}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\subset
M^{p}_{\gamma_{2}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),$

and also

$1<p<r<+\infty\Rightarrow\displaystyle{\dot{M}^{p,r}}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})=M^{p}_{3(1-p/r)}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset M^{p}_{\gamma}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}),$

where $\gamma>0$ is constant with $3(1-p/r)<\gamma$.

Finally, we define the space $M^{p}_{\gamma,0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ as the set of functions $f\in
M^{p}_{\gamma}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ such that

$\lim_{R\to+\infty}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|f(x)|^{p}dx\right)^{1/p}=0.$

Relationship between these spaces [13] is simply known that for $3<r<9/2$ and
for $3(1-3/r)<\delta<1$,

$L^{r}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\subset
L^{r,\infty}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset\dot{M}^{3,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset M^{3}_{\delta}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset M^{3}_{1,0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}),$

and for $r=9/2$ and $9/2<q<+\infty$, $L^{9/2}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset L^{9/2,q}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})\subset M^{3}_{1,0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$.

In the framework of Morrey space, third result is followed as

######  Theorem 1.3.

Let $\textbf{u},\textbf{b}\in L^{2}_{loc}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ be a weak solution of (1.1). If
$\textbf{u},\textbf{b}\in\dot{M}^{p,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ with $3\leq p<r<\frac{9}{2}$, then $\textbf{u}\equiv
0\equiv\textbf{b}$ in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$.

In the framework of local Morrey space, third result is followed as

######  Theorem 1.4.

Let $(\rho,\textbf{u},\mathbf{b})$ be a smooth solution of (1.1).

  1. $1)$

Let $\gamma=1$. If $\textbf{u}\in M^{3}_{1,0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ and $\nabla\mathbf{b}\in
M^{3}_{1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ then we have
$\textbf{u}=0$ and $\nabla\mathbf{b}=0$.

  2. $2)$

Let $1<\gamma<3/2$. If $\textbf{u}\in M^{3}_{\gamma,0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ and $\nabla\mathbf{b}\in
M^{3}_{\gamma}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, and moreover if
the velocity u verifies:

$\lim_{R\to+\infty}R^{\gamma-1}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}(x)|^{3}dx\right)^{1/3}=0,$

then we have $\textbf{u}=0$ and $\mathbf{b}=0$.

When $\rho=constant$ , the system of (1.1) reduce as follows:

(1.6)
$\begin{cases}-\nu\Delta\mathbf{u}+\operatorname{div}(\mathbf{u}\otimes\mathbf{u})+\nabla\pi=0,\\\
-\Delta\mathbf{b}+\operatorname{div}(\mathbf{u}\otimes\mathbf{b})-\operatorname{div}(\mathbf{b}\otimes\mathbf{u})=0,\\\
\operatorname{div}\mathbf{u}=0=\operatorname{div}\mathbf{b},\end{cases}$

where $\pi$ is a total pressure.

For the incompressible 3D MHD equations (1.6), Schulz [26] proved a Liouville
theorem without the finite Dirichlet integral, which is if the smooth solution

$(u,b)\in(L^{p}\cap BMO^{-1})(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}),\quad p\in(2,6],$

then $u=b=0$. Recently, Yuan and Xiao [30] shown if

$(u,b)\in L^{p}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),\quad 2\leq p\leq
9/2,$

then $u=b=0$. For a interesting results, we also refer to [19] in spite of
results in [11] and [16] inspired by the work in [4] in the Lorentz framework.

In this directions, two results are stated in the framework of Lebesgue and
Lorentz spaces.

######  Theorem 1.5.

Let $\textbf{u},\textbf{b}\in L^{2}_{loc}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ be a weak solution of (1.6).

  1. 1)

If $\textbf{u},\textbf{b}\in L^{9/2,q}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$, with $9/2\leq q<+\infty$, then we have
$\textbf{u}=0=\textbf{b}$.

  2. 2)

If $\textbf{u}\in L^{r,q}$ with $9/2<r\leq q<+\infty$ and $\textbf{b}\in
L^{9/2,q}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, with $9/2\leq
q<+\infty$, and in addition, if

$\sup_{R>1}R^{2-\frac{9}{r}}\|\textbf{u}\|_{L^{r,q}(\mathcal{C}(R/4,2R))}<+\infty,$

then we have $\textbf{u}=0=\textbf{b}$.

Investigating solutions of the equations (1.1) that may decay to zero in
different rates as $|x|\rightarrow\infty$ with respect to different
directions, we introduce a new functional space [24]. For two given numbers
$q,r\in(1,\infty)$ and $f:\mathbb{R}^{3}\rightarrow\mathbb{R}$ is a measurable
function, _the mixed-norm Lebesgue space_ $L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ is the space that is equipped with the following norm

$\|f\|_{L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})}=\left[\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits}\left(\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{2}}|f(x_{1},x_{2},x_{3})|^{q}dx_{1}dx_{2}\right)^{r/q}dx_{3}\right]^{1/r}.$

######  Theorem 1.6.

Let $q,r\in[3,\infty)$ be two numbers satisfying

$\frac{2}{q}+\frac{1}{r}\geq\frac{2}{3}.$

and $(\textbf{u},\textbf{b})\in H^{1}_{\textup{loc}}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ be a weak solution of (1.6) and assume that
$(\textbf{u},\textbf{b})\in L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$. Then $\textbf{u}\equiv 0\equiv\textbf{b}$ in
$\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$.

######  Remark 1.1.

The result given in 1.5 is of particular interest since this result can be
regarded as a improvement of the results given in [10] and [25].

######  Remark 1.2.

In case of the compressible fluids, we don’t know if Theorem 1.5 or 1.6 are
true yet. For a proof of Theorem 1.5, because it is necessary to get the
estimate of $\|\nabla u\|_{L^{p}}$. More speaking, to control a estimate of
$\|\nabla u\|_{L^{p}}$, it needs a higher regularity of $\rho$, in succession,
which yields that it needs a higher regularity of $u$. On the other hands, for
a proof of Theorem, because it is necessary to use Lemma 1.6 based on the
weight function.

Lastly, we make a few comments.

######  Remark 1.3.

The present paper focuses on Liouville theorem for the compressible MHD
equation, not Hall-MHD equation. We think it is secondary to consider the Hall
term in the system (1.1). Considering the Hall term,
$\mbox{curl}\times(\text{curl}\mathbf{b}\times\mathbf{b})$ in (1.1), namely,
(compressible) Hall-MHD equations, we can see that additional conditions are
required for the magnetic filed $b$ in light of the references [33], [30] and
[18]. Proofs of this part is not difficult, and thus leave it to the readers.
For this equation, we refer to [3] and [34] for interested readers.

######  Remark 1.4.

For a various coupled equation with (compressible) Naiver-Stokes equations,
the above Theorems are applicable through Cacciopoly type estimate.

##  2\. Proof of Theorem 1.1

The following inequalities in Lorentz spaces are useful.

######  Lemma 2.1 ([23]).

Let $f\in L^{p_{1},q_{1}}(\mathbb{R}^{3})$ and $g\in
L^{p_{2},q_{2}}(\mathbb{R}^{3})$ with $1\leq p_{1},p_{2}\leq\infty$, $1\leq
q_{1},q_{2}\leq\infty.$ Then $fg\in L^{p,q}(\mathbb{R}^{3})$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$,
$\frac{1}{q}\leq\frac{1}{q_{1}}+\frac{1}{q_{2}}$ and

$\begin{split}\|fg\|_{L^{p,q}(\mathbb{R}^{3})}\leq
C\|f\|_{L^{p_{1},q_{1}}(\mathbb{R}^{3})}\|g\|_{L^{p_{2},q_{2}}(\mathbb{R}^{3})}\end{split}$

for a constant $C>0.$

######  Lemma 2.2 ([B1976]).

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $1<p<\infty$,
$1<q\leq\infty$ and $f\in L^{p,q}(\Omega)$. Then

$\begin{split}\|\nabla^{2}(-\Delta)^{-1}f\|_{L^{p,q}(\Omega)}\leq
C\|f\|_{L^{p,q}(\Omega)},\end{split}$

where the constant $C>0$ is independent of $\Omega$.

To obtain the estimate for the pressure term, we recall the following lemma
from [20].

######  Lemma 2.3.

Let $P\in L^{\infty}(\mathbb{R}^{3}),$ $p_{1}\in L^{r_{1}}(\mathbb{R}^{3}),$
$p_{2}\in L^{r_{2}}(\mathbb{R}^{3})$ with $1\leq r_{1},r_{2}<\infty.$ Suppose
that $P-p_{1}-p_{2}$ is weakly harmonic, that is

$\Delta(P-p_{1}-p_{2})=0$

in the sense of distribution, then there exists a constant $c$ such that

$\begin{split}P-p_{1}-p_{2}=c\quad\,\mbox{a.e.}\quad\,x\in\mathbb{R}^{3}\end{split}$

If furthermore $P(x)\geq 0$ a.e., then we also have $c\geq 0$.

Taking the div operation on both sides of $\eqref{MHD}_{2}$, we have
$\Delta(P-p_{1}-p_{2})=0$, where

(2.1) $p_{1}:=(-\Delta)^{-1}\partial_{i}\partial_{j}(\rho
u_{i}u_{j})\quad\mbox{and}\quad
p_{2}:=(\lambda+2\nu)\operatorname{div}\mathbf{u}.$

Using the assumption $\nabla\mathbf{u}\in L^{2}{(\mathbb{R}^{3})}$ and the
Sobolev embedding $\dot{H}^{1}(\mathbb{R}^{3})\hookrightarrow
L^{6}(\mathbb{R}^{3})$, it follows

$p_{1}\in L^{3}(\mathbb{R}^{3})\ ,\ p_{2}\in L^{2}(\mathbb{R}^{3}).$

Due to Lemma 2.3, there exists a constant $c\geq 0$ such that

$a\rho^{\gamma}=P=c+p_{1}+p_{2}.$

Considering the function
$P_{1}:=\rho^{\gamma-1}-(\frac{c}{a})^{\frac{\gamma-1}{\gamma}}=(\frac{c+p_{1}+p_{2}}{a})^{\frac{\gamma-1}{\gamma}}-(\frac{c}{a})^{\frac{\gamma-1}{\gamma}}$,
we have

(2.2) $\nabla
P=\nabla(a\rho^{\gamma})=\frac{a\gamma}{\gamma-1}\rho\nabla(\rho^{\gamma-1})=\frac{a\gamma}{\gamma-1}\rho\nabla
P_{1}$

and

(2.3) $\begin{split}|P_{1}\rho|\leq(|p_{1}|+|p_{2}|).\end{split}$

###### Proof of Theorem 1.1.

Following the same arguments in [18] or [20], we are aimed to show a suitable
Caccioppoli-type inequality. For the convenience of our readers, we give a
sketch of the proof. Let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ be a
radial cut-off function satisfying

$\varphi(|x|)=\begin{cases}1,\quad\mathrm{if}\,|x|<\frac{1}{2},\\\
0,\quad\mathrm{if}\,|x|>1,\end{cases}$

and $0\leq\varphi(|x|)\leq 1$ for $\frac{1}{2}\leq|x|\leq 1.$ For each given
$R>0$, we define $\varphi_{R}(x):=\varphi(\frac{|x|}{R})$ satisfying

$\|\nabla^{k}\varphi_{R}\|_{L^{\infty}}\leqslant CR^{-k},\quad k=0,1,2$

for some constant $C>0$ independent of $x\in\mathbb{R}^{3}$.

Taking the inner product of $\eqref{MHD}_{2}$ with $\mathbf{u}\varphi_{R}$ and
$\eqref{MHD}_{3}$ with $\mathbf{b}\varphi_{R}$ and after integrating by parts,
it follows that

(2.4)
$\begin{split}&\int_{B_{\frac{R}{2}}}\Big{(}\nu|\nabla\mathbf{u}|^{2}+|\nabla\mathbf{b}|^{2}\Big{)}\varphi_{R}\,dx+(\lambda+\nu)\int_{\frac{R}{2}}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&=-\nu\int_{\mathcal{C}(R/2,R)}(|\mathbf{u}|^{2}+|\mathbf{b}|^{2})\Delta\varphi_{R}\,dx-2(\lambda+\nu)\int_{\mathcal{C}(R/2,R)}\operatorname{div}\mathbf{u}\mathbf{u}\cdot\nabla\varphi_{R}\,dx\\\
&\quad-\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{2}\nabla\varphi_{R}\cdot\rho\mathbf{u}\,dx+\int_{\mathcal{C}(R/2,R)}|\mathbf{b}|^{2}\nabla\varphi_{R}\cdot\mathbf{u}\,dx-\int_{\mathcal{C}(R/2,R)}\nabla
P\cdot\mathbf{u}\varphi_{R}\,dx:=\sum_{i=1}^{6}\mathcal{I}_{i},\end{split}$

where we use $\operatorname{div}(\rho\mathbf{u})=0$ and by the integration by
parts and chain rule,

$\begin{split}&-2\nu\int_{\mathbb{R}^{3}}\nabla\mathbf{u}:\mathbf{u}\otimes\nabla\varphi_{R}\,dx=\nu\int_{\mathbb{R}^{n}}|\mathbf{u}|^{2}\Delta(\varphi_{R})\,dx.\end{split}$

Now, we estimate $\mathcal{I}_{i}$ term by term. we assume $p>3$, $3\leq
q\leq\infty$ or $p=q=3$. For $\mathcal{I}_{1}$, since

$\begin{split}\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{2}|\Delta\varphi_{R}|\,dx\lesssim
R^{1-\frac{6}{p}}\|\mathbf{u}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))},\end{split}$

we have

$\begin{split}|\mathcal{I}_{1}|\lesssim
R^{1-\frac{6}{p}}\Big{(}\|\mathbf{u}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))}\Big{)}.\end{split}$

For $\mathcal{I}_{2}$, by Hölder and Young inequalities, it yields

$\begin{split}|\mathcal{I}_{2}|&\leq
C(\lambda+\nu)\int_{\mathcal{C}(R/2,R)}|\operatorname{div}\mathbf{u}||\mathbf{u}||\nabla\varphi_{R}|\,dx\\\
&\leq\frac{(\lambda+\nu)}{2}\|\operatorname{div}\mathbf{u}\|_{L^{2}(\mathbb{R}^{3})}^{2}+CR^{1-\frac{6}{p}}\|\mathbf{u}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))}.\\\
\end{split}$

For $\mathcal{I}_{3}$, we have

$\begin{split}|\mathcal{I}_{3}|&\lesssim
R^{-1}\|\rho\|_{L^{\infty}}\||\mathbf{u}|^{3}\|_{L^{\frac{p}{3},\frac{q}{3}}(\mathcal{C}(R/2,R))}\|1\|_{L^{\frac{p}{p-3},\frac{q}{q-3}}(\mathcal{C}(R/2,R))}\leq
CR^{2-\frac{9}{p}}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}^{3}.\end{split}$

In a same manner as $\mathcal{I}_{3}$, by the integration by parts,
$\mathcal{I}_{4}$ and $\mathcal{I}_{5}$ are rewritten by

$\begin{split}|\mathcal{I}_{4}|+|\mathcal{I}_{5}|\lesssim
R^{2-\frac{9}{p}}\Big{(}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}^{3}+\|\mathbf{b}\|_{L^{p,q}(\mathcal{C}(R/2,R))}^{3}\Big{)}.\end{split}$

Integrating by parts with (2.1) and (2.2), $\mathcal{I}_{6}$ becomes

$\begin{split}\mathcal{I}_{6}&=-\int_{\mathbb{R}^{3}}\frac{a\gamma}{\gamma-1}\rho\nabla
P_{1}\cdot\mathbf{u}\varphi_{R}\,dx\\\
&=\frac{a\gamma}{\gamma-1}\int_{\mathbb{R}^{3}}P_{1}\operatorname{div}(\rho\mathbf{u})\varphi_{R}\,dx+\frac{a\gamma}{\gamma-1}\int_{\mathbb{R}^{3}}P_{1}\rho\mathbf{u}\cdot\nabla(\varphi_{R})\,dx\\\
&=\frac{2a\gamma}{\gamma-1}\int_{\mathbb{R}^{3}}P_{1}\rho\mathbf{u}\cdot(\varphi_{R}\nabla\varphi_{R})\,dx,\end{split}$

and thus we have

$\begin{split}|\mathcal{I}_{6}|\leq
R^{-1}\int_{\mathcal{C}(R/2,R)}(|p_{1}|+|p_{2}|)|\mathbf{u}|\,dx.\end{split}$

By Lemmas 2.1 and 2.2, we have

$\begin{split}R^{-1}\int_{\mathcal{C}(R/2,R)}|p_{1}||\mathbf{u}|\,dx&\leq
CR^{-1}\|p_{1}\|_{L^{\frac{p}{2},\frac{q}{2}}(\mathcal{C}(R/2,R))}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}\|1\|_{L^{\frac{p}{p-3},\frac{q}{q-3}}(\mathcal{C}(R/2,R))}\\\
&\leq
CR^{2-\frac{9}{p}}\|\mathbf{u}\|^{3}_{L^{p,q}(\mathcal{C}(R/2,R))}\end{split}$

and

$\begin{split}R^{-1}\int_{\mathcal{C}(R/2,R)}|p_{2}||\mathbf{u}|\,dx&\leq
R^{-1}\|p_{2}\|_{L^{\frac{p}{p-1},\frac{q}{q-1}}(\mathcal{C}(R/2,R))}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}\\\
&\leq
CR^{\frac{1}{2}-\frac{3}{p}}\|\operatorname{div}\mathbf{u}\|_{{L^{2}}(\mathbb{R}^{3})}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}\\\
&\leq\frac{\nu}{2}\|\nabla\mathbf{u}\|^{2}_{{L^{2}}(|x|\leq
2R)}+CR^{1-\frac{6}{p}}\|\mathbf{u}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))}.\end{split}$

Therefore,

$\begin{split}|\mathcal{I}_{6}|&\lesssim
R^{2-\frac{9}{p}}\|\mathbf{u}\|^{3}_{L^{p,q}(\mathcal{C}(R/2,R))}+CR^{1-\frac{6}{p}}\|\mathbf{u}\|_{L^{p,q}(\mathcal{C}(R/2,R))}.\end{split}$

Substituting all estimates of $\mathcal{I}_{1}$–$\mathcal{I}_{6}$ into (2.4)
leads to

(2.5) $\begin{split}&\nu\int_{|x|\leq
R}|\nabla\mathbf{u}|^{2}\,dx+\frac{(\lambda+\nu)}{2}\int_{|x|\leq
R}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&\lesssim\left(R^{1-\frac{6}{p}}(\|\mathbf{u}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{p,q}(\mathcal{C}(R/2,R))})+R^{2-\frac{9}{p}}(\|\mathbf{u}\|^{3}_{L^{p,q}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{3}_{L^{p,q}(\mathcal{C}(R/2,R))})\right)\end{split}$

for all $R>0$.

Passing $R\rightarrow+\infty$ in (2.5), due to the assumptions, we have

$\begin{split}\lim_{R\rightarrow+\infty}(\int_{B_{\frac{R}{2}}}|\nabla\mathbf{u}|^{2}\,dx+\int_{\frac{R}{2}}|\operatorname{div}\mathbf{u}|^{2}\,dx)=0,\end{split}$

which implies that $\mathbf{u}\equiv 0\equiv\mathbf{b}$. On the other hand, we
know that $\nabla(a\rho^{\gamma})=0$, which implies that $\rho$=constant on
$\mathbb{R}^{3}$ by means of $\eqref{MHD}_{2}$. The proof of Theorem 1.1 is
ended. ∎

##  3\. Morrey space-Proof of Theorem 1.3

From (2.5) with the relationship between spaces (1.3), for all $R>0$, we know

(3.1) $\begin{split}&\nu\int_{|x|\leq
R}|\nabla\mathbf{u}|^{2}\,dx+\frac{(\lambda+\nu)}{2}\int_{|x|\leq
R}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&\lesssim\left(R^{1-\frac{6}{p}}(\|\mathbf{u}\|^{2}_{L^{p}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{p}(\mathcal{C}(R/2,R))})+R^{2-\frac{9}{p}}(\|\mathbf{u}\|^{3}_{L^{p}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{3}_{L^{p}(\mathcal{C}(R/2,R))})\right).\end{split}$

Using the definition of Morrey space, the inequality (3.1) becomes

$\begin{split}&\int_{B_{\frac{R}{2}}}\Big{(}|\nabla\mathbf{u}|^{2}+|\nabla\mathbf{b}|^{2}\Big{)}\varphi_{R}\,dx+\int_{\frac{R}{2}}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&\lesssim
R^{2-\frac{9}{r}}(\|\mathbf{u}\|^{2}_{\dot{M}^{p,r}}+\|\mathbf{b}\|^{2}_{\dot{M}^{p,r}})\|\mathbf{u}\|_{\dot{M}^{p,r}}+R^{1-\frac{6}{r}}\Big{(}\|\mathbf{u}\|^{2}_{\dot{M}^{p,r}}+\|\mathbf{b}\|^{2}_{\dot{M}^{p,r}}\Big{)},\end{split}$

which implies $\mathbf{u}\equiv 0\equiv\mathbf{b}$ in $\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}$ as $R\rightarrow\infty$. The proof of Theorem 1.3 is
complete.

##  4\. Proof of Theorem 1.4

From (2.5), for all $R>0$, we know

(4.2) $\begin{split}&\nu\int_{|x|\leq
R}|\nabla\mathbf{u}|^{2}\,dx+\frac{(\lambda+\nu)}{2}\int_{|x|\leq
R}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&\lesssim\left(R^{1-\frac{6}{p}}(\|\mathbf{u}\|^{2}_{L^{p}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{p}(\mathcal{C}(R/2,R))})+R^{2-\frac{9}{p}}(\|\mathbf{u}\|^{3}_{L^{p}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{3}_{L^{p}(\mathcal{C}(R/2,R))})\right).\end{split}$

Note that by definition of the local Morrey spaces
$M^{3}_{\gamma,0}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, we have
$\textbf{u},\mathbf{b}\in L^{3}_{loc}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ and thus, letting $p=3$, the estimate (4.2) becomes

$\begin{split}&\int_{B_{\frac{R}{2}}}\Big{(}|\nabla\mathbf{u}|^{2}+|\nabla\mathbf{b}|^{2}\Big{)}\varphi_{R}\,dx+\int_{\frac{R}{2}}|\operatorname{div}\mathbf{u}|^{2}\,dx\\\
&\lesssim
R^{-1}(\|\mathbf{u}\|^{3}_{L^{3}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{3}_{L^{3}(\mathcal{C}(R/2,R))})+CR^{-1}(\|\mathbf{u}\|^{2}_{L^{3}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{3}(\mathcal{C}(R/2,R))}).\end{split}$

And thus, we again write for $1\leq\gamma<3/2$

(4.3)
$\displaystyle\int_{B_{\frac{R}{2}}}\Big{(}|\nabla\mathbf{u}|^{2}+|\nabla\mathbf{b}|^{2}\Big{)}\varphi_{R}\,dx+\int_{\frac{R}{2}}|\operatorname{div}\mathbf{u}|^{2}\,dx$
$\displaystyle\lesssim$ $\displaystyle
R^{-1}(\|\mathbf{u}\|^{2}_{L^{2}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{2}(\mathcal{C}(R/2,R))})+\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{3}dx\right)^{2/3}R^{\gamma-1}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{3}dx\right)^{1/3}$
$\displaystyle+$
$\displaystyle\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{b}|^{3}dx\right)^{2/3}R^{\gamma-1}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{b}|^{3}dx\right)^{1/3}$
$\displaystyle\lesssim$ $\displaystyle
R^{-1}\Big{(}\|\mathbf{u}\|^{2}_{L^{2}(\mathcal{C}(R/2,R))}+\|\mathbf{b}\|^{2}_{L^{2}(\mathcal{C}(R/2,R))}\Big{)}$
$\displaystyle+$ $\displaystyle
R^{\gamma-1}\left[\|\textbf{u}\|^{2}_{M^{3}_{\gamma}}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{3}dx\right)^{1/3}+\|\mathbf{b}\|^{2}_{M^{3}_{\gamma}}\left(\frac{1}{R^{\gamma}}\int_{\mathcal{C}(R/2,R)}|\textbf{b}|^{3}dx\right)^{1/3}\right].$

We shall study each term in the right-hand side. For the first and second term
in (4.3), we have

(4.4)
$\lim_{R\to+\infty}\frac{c}{R^{2}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{2}dx=0,\quad\mbox{and}\quad\lim_{R\to+\infty}\frac{c}{R}\Big{(}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{3}dx\Big{)}^{2/3}=0.$

Indeed,

$\frac{c}{R^{2}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{2}dx\leq
c\,R^{-1}\left(\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{3}\right)^{2/3}\leq
c\,R^{\frac{2}{3}\gamma-1}\|\textbf{u}\|^{2}_{M^{3}_{\gamma}}.$

and also through a same computation, we have

$R^{-1}\|\mathbf{u}\|^{2}_{L^{3}(\mathcal{C}(R/2,R))}\leq
c\,R^{\frac{2}{3}\gamma-1}\|\textbf{u}\|^{2}_{M^{3}_{\gamma}}.$

Due to $\frac{2}{3}\gamma-1<0$, hence (4.4) follows. Also as
$R\rightarrow\infty$, the remains term in the estimate above vanish depending
on the parameter $\gamma$ (see [13] for a detailed proof). Hence, we have the
identities $\textbf{u}=0$ and $\mathbf{b}=0$ and therefore, Theorem 1.4 is
proven.

##  5\. Proof of Theorem 1.5

Before proving Theorem 1.5, we see the following Caccioppoli type estimate:

######  Lemma 5.1.

Let $\mathcal{C}(R/2,R)=\\{x\in\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}:R/2<|x|<R\\}$. If the solution $(u,b)$ verifies $u\in
L^{p}_{loc}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ and $\nabla u\in
L^{\frac{p}{2}}_{loc}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ with
$3\leq p<+\infty$, then for all $R>1$ we have

$\begin{split}&\int_{B_{R/2}}|\nabla\mathbf{u}|^{2}dx+\int_{B_{R/2}}|\nabla\mathbf{b}|^{2}dx\lesssim\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}\\\
&+\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{b}|^{p}dx\right)^{\frac{1}{p}}\\\
&+\Big{[}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}\otimes\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}+\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{b}\otimes\mathbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}\Big{]}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}.\end{split}$

###### Proof.

Note that

$-\sum_{i,j=1}^{3}\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}\partial_{jj}v_{i}\cdot
v_{i}\varphi_{R}\,dx=\sum_{i,j=1}^{3}\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}\partial_{j}v_{i}\cdot\partial_{j}v_{i}\varphi_{R}\,dx+\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}\partial_{j}v_{i}\cdot v_{i}\partial_{j}\varphi_{R}\,dx$
(5.5) $=\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}|\nabla\mathbf{v}|^{2}\varphi_{R}\,dx+\sum_{i,j=1}^{3}\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}\partial_{j}v_{i}\cdot v_{i}\partial_{j}\varphi_{R}\,dx.$

Using (5.5),

(5.6) $\displaystyle\int_{B_{R/2}}(|\nabla\mathbf{u}|^{2}$ $\displaystyle+$
$\displaystyle|\nabla\mathbf{b}|^{2})\,dx=-\sum_{i,j=1}^{3}\int_{B_{R}}\Big{[}(\partial_{j}u_{i})(\partial_{j}\varphi_{R})u_{i}+(\partial_{j}b_{i})(\partial_{j}\varphi_{R})b_{i}\Big{]}dx$
$\displaystyle-2(\lambda+\nu)\int_{\mathcal{C}(R/2,R)}\operatorname{div}\mathbf{u}\mathbf{u}\cdot\nabla\varphi_{R}\,dx+\int_{B_{R}}(u\cdot\nabla)u\cdot\varphi_{R}udx$
$\displaystyle-\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{2}\nabla\varphi_{R}\cdot\rho\mathbf{u}\,dx+\int_{\mathcal{C}(R/2,R)}|\mathbf{b}|^{2}\nabla\varphi_{R}\cdot\mathbf{u}\,dx,$
$\displaystyle-\int_{\mathcal{C}(R/2,R)}\nabla
P\cdot\mathbf{u}\varphi_{R}\,dx:=\sum_{i=1}^{6}\mathcal{M}_{i}.$

We study now these four terms above. In term $I_{1}$ remark that we have the
function $\partial_{i}\varphi_{R}$, but since the test function $\varphi_{R}$
verifies $\varphi_{R}(1)$ if $|x|<\frac{R}{2}$ and $\varphi_{R}(x)=0$ if
$|x|>R$ then we have $supp\,(\nabla\varphi_{R})\subset\mathcal{C}(R/2,R)$, and
thus we can write

$\mathcal{M}_{1}=-\sum_{i,j=1}^{3}\int_{\mathcal{C}(R/2,R)}\partial_{j}u_{i}(\partial_{j}\varphi_{R})u_{i}dx-\sum_{i,j=1}^{3}\int_{\mathcal{C}(R/2,R)}\partial_{j}b_{i}(\partial_{j}\varphi_{R})b_{i}dx.$

Then, applying the Holdër inequality, we have

$\displaystyle\int_{\mathcal{C}(R/2,R)}\partial_{j}u_{i}(\partial_{j}\varphi_{R})u_{i}dx$
$\displaystyle\lesssim$ $\displaystyle\left(\int_{\mathcal{C}(R/2,R)}|\nabla
u|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}\frac{1}{R}\left(\int_{\mathcal{C}(R/2,R)}|u|^{q}dx\right)^{\frac{1}{q}}$
$\displaystyle\lesssim$ $\displaystyle\left(\int_{\mathcal{C}(R/2,R)}|\nabla
u|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|u|^{p}dx\right)^{\frac{1}{p}}.$

With this estimate at hand $\mathcal{M}_{1}$ is bounded by

$\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}+\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{b}|^{p}dx\right)^{\frac{1}{p}}$

In a same manner as $\mathcal{M}_{1}$, $\mathcal{M}_{2}$ is estimated by

$\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}$

We study the term $\mathcal{M}_{4}$ and $\mathcal{M}_{5}$ in (5.6). we have

$\mathcal{M}_{3}+\mathcal{M}_{4}+\mathcal{M}_{5}\lesssim\Big{[}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}\otimes\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}+\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{b}\otimes\mathbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}\Big{]}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}.$

Here, for the estimate $\mathcal{M}_{3}$, in addition, we use
$\|\rho\|_{L^{\infty}}<\infty$. Finally, From (2.1) and (2.2), we recall that

$\mathcal{M}_{6}=\frac{2a\gamma}{\gamma-1}\int_{\mathbb{R}^{3}}P_{1}\rho\mathbf{u}\cdot(\varphi_{R}\nabla\varphi_{R})\,dx,$

and from (2.3).

$|P_{1}\rho|\leq C(a,\|\rho\|_{L^{\infty}})(|p_{1}|+|p_{2}|).$

in a similar way,

$\mathcal{M}_{6}\lesssim\Big{[}\left(\int_{\mathcal{C}(R/2,R)}|\nabla\mathbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}+\left(\int_{\mathcal{C}(R/2,R)}||\mathbf{u}|^{2}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}\Big{]}R^{2-\frac{9}{p}}\left(\int_{\mathcal{C}(R/2,R)}|\mathbf{u}|^{p}dx\right)^{\frac{1}{p}}.$

Collecting all $\mathcal{M}_{i}$, we proved Lemma (5.1). ∎

###### Proof of Theorem 1.5.

Following the arguments in [12], for $1<p<r\leq q<+\infty$ and for $R>1$ we
have

(5.7) $\int_{B_{R}}|\textbf{u}|^{p}dx\leq
c\,\,R^{3(1-\frac{p}{r})}\|\textbf{u}\|^{p}_{L^{r,\infty}}\leq
c\,R^{3(1-\frac{p}{r})}\|\textbf{u}\|^{p}_{L^{r,q}}.$

On the other hand, since

$\textbf{u}=-\frac{1}{\Delta}\left(\mathop{\mathbb{P}\kern
0.0pt}\nolimits\left((\textbf{u}\cdot\vec{\nabla})\textbf{u}\right)-\mathop{\mathbb{P}\kern
0.0pt}\nolimits\left((\textbf{b}\cdot\vec{\nabla})\textbf{b}\right)\right),$

we have for $i=1,2,3$

$\partial_{i}\textbf{u}=-\sum_{j=1}^{3}\frac{1}{\Delta}\left(\mathop{\mathbb{P}\kern
0.0pt}\nolimits\left(\partial_{i}\partial_{j}(u_{j}\textbf{u})\right)-\mathop{\mathbb{P}\kern
0.0pt}\nolimits\left(\partial_{i}\partial_{j}(B_{j}\textbf{b})\right)\right)=\sum_{j=1}^{3}\mathop{\mathbb{P}\kern
0.0pt}\nolimits\left(\mathcal{R}_{i}\mathcal{R}_{j}(u_{j}\textbf{u})-\mathcal{R}_{i}\mathcal{R}_{j}(B_{j}\textbf{b})\right),$

where $\mathcal{R}_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ denotes the i-th
Riesz transform. Thus,

(5.8) $\int_{B_{R}}|\vec{\nabla}\textbf{u}|^{\frac{p}{2}}dx\leq
c\,R^{3(1-\frac{p/2}{r/2})}\Big{(}\|\textbf{u}\otimes\textbf{u}\|^{\frac{p}{2}}_{L^{\frac{r}{2},\frac{q}{2}}}+\|\textbf{b}\otimes\textbf{b}\|^{\frac{p}{2}}_{L^{\frac{r}{2},\frac{q}{2}}}\Big{)},\quad
3\leq p<+\infty.$

Similarly, we can know

(5.9) $\int_{B_{R}}|\vec{\nabla}\textbf{b}|^{\frac{p}{2}}dx\leq
c\,R^{3(1-\frac{p/2}{r/2})}\Big{(}\|\textbf{u}\otimes\textbf{b}\|^{\frac{p}{2}}_{L^{\frac{r}{2},\frac{q}{2}}}+\|\textbf{b}\otimes\textbf{u}\|^{\frac{p}{2}}_{L^{\frac{r}{2},\frac{q}{2}}}\Big{)},\quad
3\leq p<+\infty.$

Through Lemma 5.1, we write for all $R>1$

(5.10)
$\int_{B_{R/2}}|\nabla\textbf{u}|^{2}dx+\int_{B_{R/2}}|\nabla\textbf{b}|^{2}dx\lesssim
G_{u}P_{u}+G_{b}P_{b}+(S_{u}+S_{b})P_{u},$

where

$G_{u}:=c\left(R^{\frac{6}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\nabla\otimes\textbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}},\quad
S_{u}:=R^{\frac{6}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}\otimes\textbf{u}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}}\right),$
$G_{b}:=cR^{\frac{6}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\nabla\otimes\textbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}},\quad
S_{b}:=R^{\frac{6}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\textbf{b}\otimes\textbf{b}|^{\frac{p}{2}}dx\right)^{\frac{2}{p}},$
$P_{u}:=R^{2-\frac{9}{r}}\left(R^{\frac{3}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\textbf{u}|^{p}dx\right)^{\frac{1}{p}}\right),\quad
P_{b}:=R^{2-\frac{9}{r}}\left(R^{\frac{3}{r}}\left(\frac{1}{R^{3}}\int_{\mathcal{C}(R/2,R)}|\textbf{b}|^{p}dx\right)^{\frac{1}{p}}\right).$

For this we introduce the cut-off function
$\theta_{R}\in\mathcal{C}^{\infty}_{0}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$ such that $\theta_{R}=1$ on $\mathcal{C}(R/2,R)$,
$supp\,(\theta_{R})\subset\mathcal{C}(R/4,2R)$ and
$\|\nabla\theta_{R}\|_{L^{\infty}}\leq\frac{c}{R}$. In the same approach in
[12] with (5.7)–(5.9), (5.10) yields

$\int_{B_{\frac{B}{2}}}\Big{(}|\nabla\textbf{u}|^{2}dx+|\nabla\textbf{b}|^{2}dx\Big{)}\leq
c\left(\|\theta_{R}(\nabla\textbf{u})\|_{L^{\frac{r}{2},\frac{q}{2}}}+\|\theta_{R}\textbf{u}\|^{2}_{L^{r,q}}\right)\,R^{2-\frac{9}{r}}\|\textbf{u}\|_{L^{r,q}(\mathcal{C}(R/4,R))},$
$+c\|\theta_{R}(\nabla\textbf{b})\|_{L^{\frac{r}{2},\frac{q}{2}}}R^{2-\frac{9}{r}}\|\textbf{b}\|_{L^{r,q}(\mathcal{C}(R/4,R))}+c\|\theta_{R}\textbf{b}\|^{2}_{L^{r,q}}R^{2-\frac{9}{r}}\|\textbf{u}\|_{L^{r,q}(\mathcal{C}(R/4,R))}.$

This implies $\textbf{u}\equiv 0\equiv\textbf{b}$ in $\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}$ under the assumptions in Theorem 1.5. ∎

##  6\. Proof of Theorem 1.6

For each $q\in[1,\infty)$, a non-negative measurable function
$\omega:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is said to be in the Muckenhoupt
$A_{q}(\mathbb{R}^{n})$-class if

$[\omega]_{A_{q}}:=\displaystyle{\sup_{R>0,x_{0}\in\mathbb{R}^{n}}\left(\frac{1}{|B_{R}(x_{0})|}\int_{B_{R}(x_{0})}\omega(x)dx\right)\left(\frac{1}{B_{R}(x_{0})}\int_{B_{R}(x_{0})}\omega(x)^{-\frac{1}{q-1}}dx\right)^{q-1}}<\infty$

for $q\in(1,\infty)$, and

$[\omega]_{A_{1}}:=\displaystyle{\sup_{R>0,x_{0}\in\mathbb{R}^{n}}\left(\frac{1}{|B_{R}(x_{0})|}\int_{B_{R}(x_{0})}\omega(x)dx\right)\|\frac{1}{\omega}\|_{L_{\infty}(B_{R}(x_{0})}}<\infty,$

where $B_{R}(x_{0})$ denotes the ball in $\mathbb{R}^{3}$ with the radius $R$
and centered at $x_{0}\in\mathbb{R}^{3}$.

We introduce the following lemma on weighted mixed norm estimates for the
pressure of the equations (1.1). This lemma is an key ingredient in the paper.

######  Lemma 6.1.

Let $q,r\in(2,\infty)$ and $M_{0}\geq 1$. Assume that
$\textbf{u},\textbf{b}\in L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3},\omega)$ with
$\omega(x)=\omega_{1}(x^{\prime})\omega_{2}(x_{3})$ for all a.e.
$x=(x^{\prime},x_{3})\in\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{2}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits$, and for
$\omega_{1}\in A_{\frac{q}{2}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2})$,
$\omega_{2}\in A_{\frac{r}{2}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits)$ with

$[\omega_{1}]_{A_{\frac{q}{2}}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{2})}\leq
M_{0},\quad[\omega_{2}]_{A_{\frac{r}{2}}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits)}\leq M_{0}.$

Then, there exists $N=N(q,r,M_{0})>0$ such that

$\|P\|_{L_{\frac{q}{2},\frac{r}{2}}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3},\omega)}\leq
N\Big{(}\|\textbf{u}\|_{L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3},\omega)}^{2}+\|\textbf{b}\|_{L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3},\omega)}^{2}\Big{)},$

where $P$ is defined as

(6.11)
$P=\sum_{i,j=1}^{3}\Big{(}\mathcal{R}_{i}\mathcal{R}_{j}(u_{i}u_{j})+\mathcal{R}_{i}\mathcal{R}_{j}(b_{i}b_{j})\Big{)},$

in which $\mathcal{R}_{i}$ denotes the $i$-th Riesz transform.

###### Proof.

This proof is almost similar to that in [24, Theorem 1.4] and so we omit the
proof. ∎

###### Proof of Theorem 1.6.

Let $\phi\in C^{\infty}_{0}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits)$ be a
standard cut-off function with $0\leq\phi\leq 1$ and

$\phi=1\quad\text{on}\quad[-1/2,1/2]\quad\text{and}\quad\phi=0\quad\text{on}\quad\mathop{\mathbb{R}\kern
0.0pt}\nolimits\setminus[-1,1].$

For each $R>0$, let $Q_{R}=[-R,R]^{3}$ and

$\phi_{R}(x)=\phi(\frac{x_{1}}{R})\phi(\frac{x_{2}}{R})\phi(\frac{x_{3}}{R}),\quad
x=(x_{1},x_{2},x_{3})\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}.$

Then, we see that $\phi_{R}(x)=1\quad\text{on}\quad
Q_{R/2},\quad\phi=0\quad\text{on}\quad\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}\setminus Q_{R}$. Then, there is a constant $C>0$
independent on $R$ such that $|\nabla^{k}\phi_{R}|\leq\frac{C}{R^{k}},\quad
k=0,1,2$. Testing $\textbf{u}\phi_{R}$ and $b\phi_{R}$ as a test function for
the system (1.1), respectively, we have

$\displaystyle\int_{Q_{R/2}}$
$\displaystyle(|\nabla\textbf{u}|^{2}+|\nabla\textbf{b}|^{2})dx$
$\displaystyle\leq\frac{1}{2}\int_{Q_{R}\setminus
Q_{R/2}}(|\textbf{u}|^{2}+|\textbf{b}|^{2})|\Delta\phi_{R}|dx+\frac{1}{2}\int_{Q_{R}\setminus
Q_{R/2}}(|\textbf{u}|^{3}+|\textbf{b}|^{3})|\nabla\phi_{R}|dx$ (6.12)
$\displaystyle+\int_{Q_{R}\setminus
Q_{R/2}}|p||\textbf{u}||\nabla\phi_{R}|dx,$

where we use

$\int_{Q_{R}\setminus
Q_{R/2}}|\textbf{u}||\textbf{b}|^{2}|\nabla\phi_{R}|dx\leq\frac{1}{2}\int_{Q_{R}\setminus
Q_{R/2}}(|\textbf{u}|^{3}+|\textbf{b}|^{3})|\nabla\phi_{R}|dx$

From the arguments in [24], we can show that the three terms in (6.12) becomes
to $0$ as $R\rightarrow\infty$ under the assumptions and Lemma 6.11. Indeed,
the proof of this part is almost same that in [24] and thus we skip a proof.
Hence, we obtain

$\int_{\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3}}\Big{(}|\nabla\textbf{u}|^{2}+|\nabla\textbf{b}|^{2}\Big{)}dx=\lim_{R\rightarrow\infty}\int_{Q_{R/2}}\Big{(}|\nabla\textbf{u}|^{2}+|\nabla\textbf{b}|^{2}\Big{)}dx=0.$

Therefore, $(\textbf{u},\textbf{b})$ is a constant function in
$\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$. From this and the fact that
$(\textbf{u},\textbf{b})\in L_{q,r}(\mathop{\mathbb{R}\kern
0.0pt}\nolimits^{3})$, we conclude that $\textbf{u}\equiv 0\equiv\textbf{b}$
in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$. The proof is then completed.
∎

## Acknowledgements

Jae-Myoung Kim was supported by a Research Grant of Andong National
University.

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