# Some Liouville-type theorems for the stationary 3D magneto-micropolar fluids

Jae-Myoung Kim and Seungchan Ko Department of Mathematics Education, Andong
National University, Andong, 36729, Korea (Republic of). Email:
<EMAIL_ADDRESS>of Mathematics, The University of Hong Kong,
Pokfulam Road, Hong Kong. Email<EMAIL_ADDRESS>
###### Abstract

In this paper we prove some Liouville-type theorems for the stationary
magneto-micropolar fluids under suitable conditions in three space dimensions.
We first prove that the solutions are trivial under the assumption of certain
growth conditions for the mean oscillations of the potentials. And then we
show similar results assuming that the the solutions are contained in
$L^{p}(\mathbb{R}^{3})$ with $p\in[2,9/2)$. Finally we show the same result
for lower values of $p\in[1,9/4)$ with the further assumption that the
solutions vanish at infinity.

AMS Subject Classification Number: 35Q30, 76D05, 76D03
keywords: stationary magneto-micropolar equations, Liouville-type theorem

##  1 Introduction

In the present paper, we consider the stationary magneto-micropolar fluid
equations in $\mathbb{R}^{3}$, which consists of the following partial
differential equations:

$\left\\{\begin{aligned} -\Delta u+(u\cdot\nabla)u+\nabla\Pi&=\chi\nabla\times
w+(b\cdot\nabla)b,\\\ -\gamma\Delta w+(u\cdot\nabla)w&=\nabla(\nabla\cdot
w)+\chi\nabla\times u-2\chi w,\\\ -\nu\Delta
b+(u\cdot\nabla)b&=(b\cdot\nabla)u,\\\ \nabla\cdot u&=\nabla\cdot
b\,=0,\end{aligned}\right.$ (1.1)

where $u=(u_{1},u_{2},u_{3})$, $w=(w_{1},w_{2},w_{3})$,
$b=(b_{1},b_{2},b_{3})$ and $\Pi$ denote the fluid velocity, the angular
velocity of the rotation of the fluid particles, the magnetic fields and
pressure respectively. The positive constant $\gamma$ in (1.1) correspond to
the angular viscosity, $\nu$ is the inverse of the magnetic Reynolds number
and $\chi$ is the micro-rotational viscosity. In this paper, we assume that
$\gamma$=$\nu$=$\chi$=1 for simplicity. Equation $\eqref{mag-mipolar}_{1}$ is
similar to the classical Navier–Stokes equations, but here it is coupled with
equations $\eqref{mag-mipolar}_{2}$ for $w$ and $\eqref{mag-mipolar}_{3}$ for
$b$. Equation $\eqref{mag-mipolar}_{2}$ describes the motion in the macro-
volumes as they go through micro-rotational effects, represented by the micro-
rotational velocity vector $w$. If the fluids have no micro structure, $w$
vanishes and the system (1.1) becomes a magneto-hydrodynamics system. Equation
$\eqref{mag-mipolar}_{3}$ is the Maxwell system for the electric field. This
model was first introduced by Ahmadi and Shahinpoor [1]. After that, Rojas-
Medar [15] proved the local-in-time existence and uniqueness of strong
solutions in a bounded domain based on the spectral Galerkin method.
Furthermore, Rojas-Medar and Boldrini [16] established the existence of weak
solutions to the model (1.1) in a bounded domain and in particular, the
uniqueness was also proved for a two-dimensional domain. The existence of
global-in-time strong solutions was addressed by Ortega-Torres and Rojas-Medar
[14].

After Galdi’s work in [9], Liouville-type problems for the stationary fluid
equations has been extensively studied and there are a large number of works
on the Liouville type-problems even to these dates (see e.g. [7, 2, 5, 11, 20]
and a review paper [22]). Here, we shall study some Liouville-type results
under the assumptions with regard to the potential functions. We say that
$\Phi\in L^{1}_{loc}({\mathbb{R}}^{3};{\mathbb{R}}^{3\times 3})$ is the
potential functions for the vector fields $u\in
L^{1}_{loc}({\mathbb{R}}^{3})$, if $\mathrm{div}\,\Phi=u$. In [18], Seregin
obtained Liouville-type theorems for the steady-state Navier-Stokes equations
under the assumptions that the potential $\Phi\in BMO({\mathbb{R}}^{3})$ and
$u\in L^{6}(\mathbb{R}^{3})$, and in [19] the integrability condition for the
velocity was dropped. After that, very recently, Chae and Wolf [6] showed
Liouville-type theorem for the stationary Navier-Stokes equations under the
assumption

$\left(\frac{1}{|B_{r}|}\int_{B_{r}}|\Phi-\Phi_{B_{r}}|^{s}{\,{\mathrm{d}}x}\right)^{\frac{1}{s}}\lesssim
r^{\frac{1}{3}-\frac{1}{s}}\qquad\forall 1<r<+\infty$

for some $3<s<+\infty$, and similar results were proved for MHD equations in
[3]. The first theorem of the present paper is the extension of the result of
[3]. Here however, we shall adopt a different approach to control the pressure
term by introducing an auxiliary function and utilizing it as a test function.
In specific, we first aim in this paper to prove the following Liouville-type
result.

###### Theorem 1

Let $(u,b,w,\Pi)$ be a smooth solution to the equations (1.1). Assume that
there exist potentials $\Phi,\Psi,\Upsilon\in
C^{\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3\times 3})$ such that
$\nabla\cdot\Phi=u$, $\nabla\cdot\Psi=b$, $\nabla\cdot\Upsilon=w$ and

$\left(\frac{1}{|B_{r}|}\int_{B_{r}}\big{|}\Phi-\Phi_{B_{r}}\big{|}^{s}{\,{\mathrm{d}}x}\right)^{\frac{1}{s}}+\left(\frac{1}{|B_{r}|}\int_{B_{r}}\big{|}\Psi-\Psi_{B(r)}\big{|}^{s}{\,{\mathrm{d}}x}\right)^{\frac{1}{s}}$
$+\left(\frac{1}{|B_{r}|}\int_{B_{r}}\big{|}\Upsilon-\Upsilon_{B_{r}}\big{|}^{s}{\,{\mathrm{d}}x}\right)^{\frac{1}{s}}\leq
Cr^{\frac{1}{3}-\frac{1}{s}},\qquad r>1$ (1.2)

for some $3<s\leq 6$. Then $u\equiv b\equiv w\equiv 0$.

###### Remark 1

In the case of $w\equiv 0$ Theorem 1 reduces to [3, Theorem 1.1].

Later, Zhang et. al. [23] proved that if smooth solutions of the stationary
MHD equations are bounded in $L^{\frac{9}{2}}({\mathbb{R}}^{3})$ and have
finite Dirichlet integral, then they are also identically zero. After that,
Schulz [17] obtained the Liouville theorem for this equations provided that
the smooth solution $(u,b)$ are contained in $L^{p}(\mathbb{R}^{3})\cap
BMO^{-1}({\mathbb{R}}^{3})$ with $p\in(2,6]$. Recently, Yuan and Xiao [21]
proved that if smooth solution $(u,b)\in L^{p}({\mathbb{R}}^{3})$ with $2\leq
p\leq\frac{9}{2}$, then $u=b=0$. In this direction, the second objective of
this paper is as follows.

###### Theorem 2

Let $p\in[2,\frac{9}{2})$. Assume that $(u,b,w,\Pi)$ is a smooth solution to
the equations (1.1) with $u,b,w\in L^{p}({\mathbb{R}}^{3})$. Then $u\equiv
b\equiv w\equiv 0$.

Furthermore, parallel to the result of Liu and Liu [13], we shall also prove
the following theorem.

###### Theorem 3

Let $p\in[1,\frac{9}{4})$. Assume that $(u,b,w,\Pi)$ is a smooth solution to
the equations (1.1) with $u,b,w\in L^{p}({\mathbb{R}}^{3})$ satisfying
$\lim_{|x|\rightarrow\infty}\ u(x)=\lim_{|x|\rightarrow\infty}\
b(x)=\lim_{|x|\rightarrow\infty}\ w(x)=0$. Then $u\equiv b\equiv w\equiv 0$.

###### Remark 2

Even if we consider the model with the variable density, that is, density
dependent models, Theorem 1, 2 and 3 still hold under suitable additional
assumptions (see, for example, [13]).

###### Remark 3

In the light of the work of Liu and Liu [13], through the similar approach, we
can also obtain the Liouville-type results in Lorentz spaces (see e.g. [12,
8]).

##  2 Preliminaries

In this section, we introduce some notations and auxiliary results which will
be used throughout the paper. We denote the ball with center $x_{0}$ and
radius $R$ by $B_{R}(x_{0})$. If $x_{0}=0$, we simply write $B_{R}=B_{R}(0)$.
Throughout the paper, the notation $P\lesssim Q$ implies that there exists
some constant $C>0$ such that $P\leq CQ$. Also, $C$ denotes a generic positive
constant, which may change at each appearance.

Let us also define a family of cut-off functions. For $0<r<r^{\prime}$, we let
$\xi=\xi_{r,r^{\prime}}\in C^{\infty}_{c}(B_{r^{\prime}})$ be a radially non-
increasing scalar function such that

$\xi_{r,r^{\prime}}(x)=\begin{cases}1,\quad x\in B_{r},\\\ 0,\quad x\in
B^{c}_{r^{\prime}},\end{cases}$ (2.1)

with the properties $|\nabla\xi_{r,r^{\prime}}|<C_{1}/(r^{\prime}-r)$, and
$|\nabla^{2}\xi_{r,r^{\prime}}|<C_{2}/(r^{\prime}-r)^{2}$ for some constant
$C_{1},$ $C_{2}>0$.

Next, for a bounded domain $\Omega\subset\mathbb{R}^{3}$, we consider the
following problem: for given $f\in L^{p}(\Omega)$ with

$\int_{\Omega}f(x){\,{\mathrm{d}}x}=0,$ (2.2)

find a vector-valued function $v\in W^{1,p}_{0}(\Omega)^{d}$ satisfying

$\displaystyle\nabla\cdot v$ $\displaystyle=f,$ (2.3) $\displaystyle\|\nabla
v\|_{p}$ $\displaystyle\leq C\|f\|_{p}$

for some constant $C=C(d,p,\Omega)$. For this matter, we have the following
theorem which is quoted from [9].

###### Theorem 4

[9, Theorem III.3.1] Assume that $\Omega$ satisfies the cone condition. Then
for given $f\in L^{p}(\Omega)$ with $1<p<\infty$ satisfying (2.2), there
exists at least one solution for the problem (2.3).

We will use this result to construct an auxiliary function to control the
pressure term in the next section. We will also use the following iteration
lemma frequently in our analysis.

###### Lemma 1

[10, Lemma 3.1] Let $f(r)$ be a non-negative bounded function on
$[r_{0},r_{1}]\subset{\mathbb{R}}_{\geq 0}$. Suppose that there exist non-
negative constants $A$, $B$, $D$, $E$ and positive numbers $d<b<a$ and a
parameter $\theta\in(0,1)$ such that for any $r_{0}\leq s<t\leq r_{1}$,

$f(s)\leq\theta
f(t)+\frac{A}{(t-s)^{a}}+\frac{B}{(t-s)^{b}}+\frac{D}{(t-s)^{d}}+E.$

Then we have

$f(s)\leq
C(a,b,d,\theta)\left[\frac{A}{(t-s)^{a}}+\frac{B}{(t-s)^{b}}+\frac{D}{(t-s)^{d}}+E\right].$

We shall also use the following lemma for the proof of Theorem 1.

###### Lemma 2

[4, Lemma 2.1 and Lemma 2.2] Suppose that $R>1$ and $f\in
W^{1,2}(B_{R};{\mathbb{R}}^{3})$. For $0<\rho<R$, we let $\psi\in
C^{\infty}_{c}(B_{R})$ such that $0\leq\psi\leq 1$ and $|\nabla\psi|\leq
C/(R-\rho)$ for some constant $C>0$. Assume further that there exists the
potential $F\in W^{2,2}(B_{R};{\mathbb{R}}^{3\times 3})$ with $\nabla\cdot
F=f$ and the growth condition

$\left(\frac{1}{|B_{r}|}\int_{B_{r}}\big{|}F-F_{B_{r}}\big{|}^{s}{\,{\mathrm{d}}x}\right)^{\frac{1}{s}}\lesssim
r^{\frac{1}{3}-\frac{1}{s}},\qquad r>1$

for some $3<s\leq 6$. Then there holds

$\|\psi^{2}f\|_{L^{2}(B_{R})}^{2}\lesssim
R^{\frac{11}{6}-\frac{1}{s}}\|\psi\nabla
f\|_{L^{2}(B_{R})}+R^{\frac{11}{3}-\frac{2}{s}}(R-\rho)^{-2}$ (2.4)

and

$\|\psi^{3}f\|_{L^{3}(B_{R})}^{3}\lesssim R\|\psi\nabla
f\|_{L^{2}(B_{R})}^{\frac{18}{s+6}}+R^{4-\frac{3}{s}}(R-\rho)^{-3}+R\left((R-\rho)^{-1}\|\psi^{2}f\|_{L^{2}(B_{R})}\right)^{\frac{18}{s+6}}.$
(2.5)

##  3 Proof of Theorem 1

Let $\varphi_{R}$ be a cut-off function in $C^{\infty}_{c}({\mathbb{R}}^{3})$
given by $\varphi_{R}=\xi_{\rho,\tau}$ for
$1<\frac{R}{2}<\rho<\frac{2}{3}R<R<\tau<2R$. We begin with some estimates for
the terms related to $w$. By using the Hölder and Young’s inequality, we note
that

$\displaystyle-\int_{\mathbb{R}^{3}}\nabla(\nabla\cdot
w)(w\varphi_{R}^{2}){\,{\mathrm{d}}x}$
$\displaystyle=\int_{\mathbb{R}^{3}}(\nabla\cdot
w)\nabla\cdot(w\varphi_{R}^{2}){\,{\mathrm{d}}x}$
$\displaystyle=\int_{\mathbb{R}^{3}}|\nabla\cdot
w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}+2\int_{\mathbb{R}^{3}}(\nabla\cdot
w)w\varphi_{R}\cdot\varphi_{R}{\,{\mathrm{d}}x}$
$\displaystyle\geq\int_{\mathbb{R}^{3}}|\nabla\cdot
w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}-2\bigg{|}\int_{\mathbb{R}^{3}}(\nabla\cdot
w)w\varphi_{R}\cdot\varphi_{R}{\,{\mathrm{d}}x}\bigg{|}$
$\displaystyle\geq\int_{\mathbb{R}^{3}}|\nabla\cdot
w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}-\varepsilon\int_{\mathbb{R}^{3}}|\nabla\cdot
w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}-C(\varepsilon)\int_{\mathbb{R}^{3}}|w|^{2}|\nabla\varphi_{R}|^{2}{\,{\mathrm{d}}x}.$

Next, using the vector identity
$\nabla\times(u\varphi^{2}_{R})=\varphi^{2}_{R}\nabla\times
u+\nabla\varphi^{2}_{R}\times u$ and Young’s inequality, we also note that

$\displaystyle\int_{\mathbb{R}^{3}}\nabla\times w\cdot
u\varphi^{2}_{R}{\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}\nabla\times u\cdot
w\varphi^{2}_{R}{\,{\mathrm{d}}x}$
$\displaystyle=\int_{\mathbb{R}^{3}}w\cdot\nabla\times(u\varphi^{2}_{R}){\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}\nabla\times
u\cdot w\varphi^{2}_{R}{\,{\mathrm{d}}x}$
$\displaystyle=2\int_{\mathbb{R}^{3}}w\varphi^{2}_{R}\cdot\nabla\times
u{\,{\mathrm{d}}x}+2\int_{\mathbb{R}^{3}}w\cdot\varphi_{R}\nabla\varphi\times
u{\,{\mathrm{d}}x}$ $\displaystyle\leq\frac{2}{3}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}+\frac{3}{2}\int_{\mathbb{R}^{3}}|w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}$
$\displaystyle\hskip
11.38109pt+\varepsilon\int_{\mathbb{R}^{3}}|w|^{2}\varphi_{R}^{2}{\,{\mathrm{d}}x}+C(\varepsilon)\int_{\mathbb{R}^{3}}|u|^{2}|\nabla\varphi_{R}|^{2}{\,{\mathrm{d}}x}.$

Before proceeding more, we introduce an auxiliary function which is needed to
handle the pressure term. We set $\Omega=B_{\tau}\setminus B_{\rho}$ and
$f=\varphi_{R}u$. As we know by Green’s Theorem that

$\int_{B_{\tau}\setminus
B_{\rho}}\nabla\cdot(\varphi_{R}u){\,{\mathrm{d}}x}=\int_{\partial
B_{\tau}}\varphi u\cdot\nu{\,{\mathrm{d}}x}=0,$

we can apply Theorem 4 to show the existence of vector-valued function
$W_{R}\in W^{1,p}_{0}(B_{\tau}\setminus B_{\rho})$ satisfying

$\nabla\cdot
W_{R}=\nabla\cdot(\varphi_{R}u)\quad{\rm{in}}\,\,B_{\tau}\setminus B_{\rho},$

with

$\|\nabla W_{R}\|_{L^{p}(B_{\tau}\setminus
B_{\rho})}\lesssim\|\nabla\varphi_{R}\cdot u\|_{L^{p}(B_{\tau}\setminus
B_{\rho})}.$ (3.1)

Now, we multiply the equations $\eqref{mag-mipolar}_{1}$, $\eqref{mag-
mipolar}_{2}$ and $\eqref{mag-mipolar}_{3}$ by $u\varphi^{2}_{R}-W_{R}$,
$b\varphi^{2}_{R}$ and $w\varphi^{2}_{R}$, respectively and integrate over
${\mathbb{R}}^{3}$. Then integration by parts with the use of divergence-free
conditions yields that

$\displaystyle\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right)\varphi_{R}^{2}{\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}|\nabla\cdot
w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}|w|^{2}\varphi^{2}_{R}{\,{\mathrm{d}}x}$
$\displaystyle\lesssim\int_{\mathbb{R}^{3}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right)|\nabla\varphi_{R}|^{2}{\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right)u\cdot\varphi_{R}\nabla\varphi_{R}{\,{\mathrm{d}}x}$
$\displaystyle\hskip 11.38109pt+\int_{\mathbb{R}^{3}}\nabla u\cdot\nabla
W_{R}{\,{\mathrm{d}}x}-\int_{\mathbb{R}^{3}}(u\cdot\nabla)u\cdot
W_{R}{\,{\mathrm{d}}x}-\int_{\mathbb{R}^{3}}(b\cdot\nabla)b\cdot
W_{R}{\,{\mathrm{d}}x}$ $+\int_{\mathbb{R}^{3}}\nabla\times w\cdot
W_{R}{\,{\mathrm{d}}x}+\int_{\mathbb{R}^{3}}(u\cdot
B)(B\cdot\nabla)\varphi^{2}_{R}{\,{\mathrm{d}}x}$ (3.2)

where we have used the above estimates. We shall estimate the terms on the
right-hand side of (3.2). First, by (3.1), Hölder’s inequality and Young’s
inequalities we have

$\int_{{\mathbb{R}}^{3}}\nabla u\cdot\nabla W_{R}\,\mathrm{d}x\leq\|\nabla
u\|_{L^{2}(B_{\tau})}\|\nabla W_{R}\|_{L^{2}(B_{\tau})}\leq\frac{1}{4}\|\nabla
u\|^{2}_{L^{2}(B_{\tau})}+C(\tau-\rho)^{-2}\|u\|^{2}_{L^{2}(B_{\tau}\setminus
B_{\rho})}.$

Next, by (3.1) and Hölder’s inequality, we obtain

$\displaystyle\int_{\mathbb{R}^{3}}(u\cdot\nabla)u\cdot
W_{R}{\,{\mathrm{d}}x}$ $\displaystyle=\sum_{i,j}\int_{B_{\tau}\setminus
B_{\rho}}u_{i}\partial_{i}u_{j}(W_{R})_{j}{\,{\mathrm{d}}x}=-\sum_{i,j}\int_{B_{\tau}\setminus
B_{\rho}}u_{i}u_{j}\partial_{i}(W_{R})_{j}{\,{\mathrm{d}}x}$
$\displaystyle\lesssim\left(\int_{B_{\tau}\setminus
B_{\rho}}|u|^{3}{\,{\mathrm{d}}x}\right)^{2/3}\left(\int_{B_{\tau}\setminus
B_{R}}|\nabla W_{R}|^{3}{\,{\mathrm{d}}x}\right)^{1/3}$
$\displaystyle\lesssim(\tau-\rho)^{-1}\left(\int_{B_{\tau}\setminus
B_{\rho}}|u|^{3}{\,{\mathrm{d}}x}\right)^{2/3}\left(\int_{B_{\tau}\setminus
B_{R}}|u|^{3}{\,{\mathrm{d}}x}\right)^{1/3}$
$\displaystyle\lesssim(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{3}{\,{\mathrm{d}}x},$

and similarly, we get

$\int_{{\mathbb{R}}^{3}}(b\cdot\nabla)b\cdot
W_{R}{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}|b|^{3}{\,{\mathrm{d}}x}.$

Furthermore, note that

$\int_{{\mathbb{R}}^{3}}\nabla\times w\cdot
W_{R}\leq\|w\|_{L^{2}(B_{\tau})}\|\nabla
W_{R}\|_{L^{2}(B_{\tau})}\leq\frac{1}{4}\|w\|^{2}_{L^{2}(B_{\tau})}+C(\tau-\rho)^{-2}\|u\|^{2}_{L^{2}(B_{\tau}\setminus
B_{\rho})}.$

Finally, by Hölder’s inequality and Young’s inequality, we note that

$\int_{\mathbb{R}^{3}}(u\cdot
b)(b\cdot\nabla)\varphi^{2}_{R}{\,{\mathrm{d}}x}\lesssim\int_{B_{\tau}\setminus
B_{\rho}}|u||b|^{2}|\nabla\varphi|{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}\right){\,{\mathrm{d}}x}.$

Altogether, we obtain from (3.2) that

$\displaystyle\hskip 11.38109pt\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+\int_{B_{\rho}}|w|^{2}{\,{\mathrm{d}}x}$
$\displaystyle\lesssim(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}+(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$
$\displaystyle\hskip 11.38109pt+\frac{1}{4}\int_{B_{\tau}}\left(|\nabla
u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+\frac{1}{4}\int_{B_{\tau}}|w|^{2}{\,{\mathrm{d}}x}.$

Then by Lemma 1, we conclude that

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+\int_{B_{\rho}}|w|^{2}{\,{\mathrm{d}}x}$
$\lesssim(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}+(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$ (3.3)

Before proceeding further, let us briefly describe the strategy of the proof.
We set $\tau=2\rho$ for convenience and we shall firstly show

$\rho^{-1}\int_{B_{2\rho}\setminus
B_{\rho}}(|u|^{3}+|b|^{3}+|w|^{3})\,\mathrm{d}x\to
0\qquad\mbox{as}\,\,\,\rho\rightarrow\infty.$ (3.4)

For the remaining part in (3.3), by the Hölder’s inequality, we note that

$\rho^{-2}\int_{B_{2\rho}\setminus
B_{\rho}}(|u|^{2}+|b|^{2}+|w|^{2}){\,{\mathrm{d}}x}\lesssim\rho^{-\frac{1}{3}}\left(\rho^{-1}\int_{B_{2\rho}\setminus
B_{\rho}}(|u|^{3}+|b|^{3}+|w|^{3}){\,{\mathrm{d}}x}\right)^{\frac{2}{3}}.$

Hence, if we show

$\left(\rho^{-1}\int_{B_{2\rho}\setminus
B_{\rho}}(|u|^{3}+|b|^{3}+|w|^{3}){\,{\mathrm{d}}x}\right)<C,$ (3.5)

for some constant $C>0$, we can get

$\rho^{-2}\int_{B_{2\rho}\setminus
B_{\rho}}(|u|^{2}+|b|^{2}+|w|^{2}){\,{\mathrm{d}}x}\to
0\qquad\mbox{as}\,\,\,\rho\to\infty.$ (3.6)

Due to (3.4) and (3.6), we have from (3.3) that

$\int_{B_{\rho}}(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}){\,{\mathrm{d}}x}\to 0\qquad\mbox{as}\qquad\rho\rightarrow\infty,$

which implies that $u$, $b$ and $w$ must be constants. Thanks to (3.4), we
finally conclude that $u\equiv b\equiv w\equiv 0$.

As described above, we first aim to prove (3.4). Recall that $R>\rho>R/4>1$
and set $\psi=\xi_{\rho,R}-\xi_{\rho/4,R/4}$. By Lemma 2, we have from Young’s
inequality that

$\displaystyle\int_{B_{R}}|\psi^{3}u|^{3}{\,{\mathrm{d}}x}$
$\displaystyle\lesssim R\|\psi\nabla
u\|_{L^{2}(B_{R})}^{\frac{18}{s+6}}+R^{4-\frac{3}{s}}(R-\rho)^{-3}$
$\displaystyle\hskip
11.38109pt+R\left((R-\rho)^{-2}R^{\frac{11}{6}-\frac{1}{s}}\|\psi\nabla
u\|_{L^{2}(B_{R})}+R^{\frac{11}{3}-\frac{2}{s}}(R-\rho)^{-4}\right)^{\frac{9}{s+6}}$
$\displaystyle\lesssim R\|\psi\nabla
u\|_{L^{2}(B_{R})}^{\frac{18}{s+6}}+R^{4-\frac{3}{s}}(R-\rho)^{-3}+R\big{(}\|\psi\nabla
u\|_{L^{2}(B_{R})}^{2}+\rho^{\frac{11}{3}-\frac{2}{s}}(R-\rho)^{-4}\big{)}^{\frac{9}{s+6}}.$

By taking $\rho=2r$ and $R=4r$ for $r>1$, we deduce that

$r^{-1}\int_{B_{2r}\setminus B_{r}}|u|^{3}{\,{\mathrm{d}}x}\lesssim\|\nabla
u\|_{L^{2}(B_{4r}\setminus B_{r/2})}^{\frac{18}{s+6}}+r^{-\frac{3}{s}}.$ (3.7)

Similarly, we can also obtain

$r^{-1}\int_{B_{2r}\setminus
B_{r}}\left(|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}\lesssim\left(\|\nabla
b\|_{L^{2}(B_{4r}\setminus B_{r/2})}^{\frac{18}{s+6}}+\|\nabla
w\|_{L^{2}(B_{4r}\setminus
B_{r/2})}^{\frac{18}{s+6}}\right)+r^{-\frac{3}{s}}.$ (3.8)

Next, we shall show that

$\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}\leq C,$ (3.9)

for some constant $C>0$. We set $R>\rho>1$ and $\overline{R}=(R+\rho)/2$. If
we take $\varphi=\xi_{\rho,\overline{R}}$ as a cut-off function and proceed
with the same argument used to derive (3.3), we obtain that

$\displaystyle\hskip 11.38109pt\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla w|^{2}\right){\,{\mathrm{d}}x}$
$\displaystyle\lesssim(R-\rho)^{-2}\int_{B_{\overline{R}}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}+(R-\rho)^{-1}\int_{B_{\overline{R}}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}.$

And then we set $\psi=\xi_{\overline{R},R}$. Then as $\psi=1$ on
$B(\overline{R})$, we note that

$\int_{B_{\rho}}(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}){\,{\mathrm{d}}x}\leq C({\rm{I}}_{1}+{\rm{I}}_{2}),$

where

${\rm{I}}_{1}:=(R-\rho)^{-2}\int_{B_{R}}\left(|\psi^{2}u|^{2}+|\psi^{2}b|^{2}+|\psi^{2}w|^{2}\right){\,{\mathrm{d}}x},$

and

${\rm{I}}_{2}:=(R-\rho)^{-1}\int_{B_{R}}\left(|\psi^{3}u|^{3}+|\psi^{3}b|^{3}+|\psi^{3}w|^{3}\right){\,{\mathrm{d}}x}.$

Then by Lemma 2 together with the assumption (1.2), and Young’s inequality, we
have that

$\displaystyle(R-\rho)^{-1}\int_{B_{R}}|\psi^{3}u|^{3}{\,{\mathrm{d}}x}$
$\displaystyle\leq CR(R-\rho)^{-1}\|\psi\nabla
u\|_{L^{2}(B_{R})}^{\frac{18}{s+6}}+CR^{4-\frac{3}{s}}(R-\rho)^{-4}$
$\displaystyle\hskip
11.38109pt+CR(R-\rho)^{-1}\big{(}(R-\rho)^{-1}\|\psi^{2}u\|_{L^{2}(B_{R})}\big{)}^{\frac{18}{s+6}}$
$\displaystyle\leq\frac{1}{4}\|\psi\nabla
u\|_{L^{2}(B(R))}^{2}+CR^{\frac{s+6}{s-3}}(R-\rho)^{-\frac{s+6}{s-3}}+{\rm{I}}_{1}.$

In the same way, if we proceed with the above argument for $b$ and $w$, we
obtain

${\rm{I}}_{2}\leq\frac{1}{4}\int_{B_{R}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+CR^{\frac{s+6}{s-3}}(R-\rho)^{-\frac{s+6}{s-3}}+{\rm{I}}_{1}.$

Next, for ${\rm{I}}_{1}$, using (2.4), we have by Young’s inequality that

$\displaystyle(R-\rho)^{-2}\int_{B_{R}}|\psi^{2}u|^{2}{\,{\mathrm{d}}x}$
$\displaystyle\leq CR^{\frac{11}{6}-\frac{1}{s}}(R-\rho)^{-2}\|\psi\nabla
u\|_{L^{2}(B_{R})}+CR^{\frac{11}{3}-\frac{2}{s}}(R-\rho)^{-4}$
$\displaystyle\leq CR^{2}(R-\rho)^{-2}\|\psi\nabla
u\|_{L^{2}(B_{R})}+R^{4}(R-\rho)^{-4}$
$\displaystyle\leq\frac{1}{4}\|\psi\nabla
u\|_{L^{2}(B_{R})}^{2}+CR^{4}(R-\rho)^{-4}.$

If we use the same method with $b$ and $w$, it follows that

${\rm{I}}_{1}\leq\frac{1}{4}\int_{B_{R}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla w|^{2}\right){\,{\mathrm{d}}x}+CR^{4}(R-\rho)^{-4}.$

Collecting the estimates for ${\rm{I}}_{1}$ and ${\rm{I}}_{2}$ yields

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}\leq\frac{1}{2}\int_{B_{R}}\left(|\nabla
u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+CR^{\frac{s+6}{s-3}}(R-\rho)^{-\frac{s+6}{s-3}},$

where we have used the facts $R(R-\rho)^{-1}>1$ and $3<s\leq 6$. Then applying
Lemma 1 gives us the estimate

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}\leq
CR^{\frac{s+6}{s-3}}(R-\rho)^{-\frac{s+6}{s-3}}.$

If we take $R=2\rho$ and let $\rho\rightarrow\infty$, we obtain (3.9), and
hence from (3.7) and (3.8), we conclude that (3.4) holds.

It remains to show (3.5).

By direct computation we observe that

$\displaystyle
r^{-1}\int_{B_{r}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$
$\displaystyle=\sum_{j=1}^{\infty}2^{-j}(2^{-j}r)^{-1}\int_{B_{2^{-(j-1)}r}\setminus
B_{2^{-j}r}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$
$\displaystyle\leq\sup_{1/2\leq\rho\leq r/2}\rho^{-1}\int_{B_{2\rho}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$
$\displaystyle\hskip
11.38109pt+\int_{B_{1}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}.$

Therefore from (3.4) we have (3.5), and consequently, we deduce that the
convergence (3.6) holds.

Now we are ready to conclude $u\equiv b\equiv w\equiv 0$. From (3.3) together
with (3.4) and (3.6), we have

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}\to
0\qquad\mbox{as}\,\,\,\rho\rightarrow\infty,$

which means that $u$, $b$ and $w$ are constants. By (3.4), we finally obtain
that $u\equiv b\equiv w\equiv 0$.

##  4 Proof of Theorem 2

From (3.2) in the proof in Theorem 1, we know that

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}$
$\lesssim(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}+(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|u||b|^{2}+|u||w|^{2}\right){\,{\mathrm{d}}x}$ (4.1)

In order to deal with the right-hand side of (4.1), let us consider two cases
as follows:

(Case 1) $p\in[3,\frac{9}{2})$: Then we have by Hölder’s inequality,

$\displaystyle\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}$
$\displaystyle\lesssim(\tau-\rho)^{-2}\|(u,b,w)\|^{2}_{L^{2}(B_{2R})}+(\tau-\rho)^{-1}\|(u,b,w)\|^{3}_{L^{3}(B_{2R})}$
$\displaystyle\lesssim\rho^{1-\frac{6}{p}}\|(u,b,w)\|^{2}_{L^{p}(B_{2\rho})}+\rho^{2-\frac{9}{p}}\|(u,b,w)\|^{3}_{L^{p}(B_{2\rho})},$

where we have chosen $\tau=2\rho$. Hence if we let $\rho\rightarrow\infty$, we
have

$\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}=0.$ (4.2)

(Case 2) $p\in[2,\frac{18}{5})$: For this case, let us consider a non-negative
cut-off function $\theta(x)=\xi_{\tau,2R}(x)-\xi_{\frac{R}{2},\rho}(x)$ with
$\|\nabla\theta\|_{L^{\infty}}\lesssim\max\\{\frac{1}{\rho-\frac{R}{2}},\frac{1}{2R-\tau}\\}$.
Note that by the interpolation inequality,

$\displaystyle\|w\|^{2}_{L^{4}(B_{\tau}\setminus
B_{\rho})}\lesssim\|w\theta\|^{2}_{L^{4}(\mathbb{R}^{3})}$
$\displaystyle\lesssim\|w\theta\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\|\nabla(w\theta)\|_{L^{2}(\mathbb{R}^{3})}^{\frac{3}{2}}$
$\displaystyle\lesssim\|w\theta\|^{\frac{1}{2}}_{L^{2}(\mathbb{R}^{3})}\left(\|(\nabla
w)\theta\|^{\frac{3}{2}}_{L^{2}(\mathbb{R}^{3})}+\|w(\nabla\theta)\|^{\frac{3}{2}}_{L^{2}(\mathbb{R}^{3})}\right)$
$\displaystyle\lesssim\|w\|^{\frac{1}{2}}_{L^{2}(B_{2R}\setminus
B_{R/2})}\left(\|\nabla w\|^{\frac{3}{2}}_{L^{2}(B_{2R}\setminus
B_{R/2})}+\frac{1}{R^{\frac{3}{2}}}\|w\|^{\frac{3}{2}}_{L^{2}(B_{2R}\setminus
B_{R/2})}\right)$
$\displaystyle\lesssim\|w\|^{\frac{1}{2}}_{L^{2}(B_{2R}\setminus
B_{R/2})}\|\nabla w\|^{\frac{3}{2}}_{L^{2}(B_{2R}\setminus
B_{R/2})}+\frac{1}{R^{\frac{3}{2}}}\|w\|^{2}_{L^{2}(B_{2R}\setminus
B_{R/2})}.$

And thus, by Hölder’s inequality and Young’s inequality,

$\displaystyle(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}|u||w|^{2}{\,{\mathrm{d}}x}$
$\displaystyle\lesssim(\tau-\rho)^{-1}\|u\|_{L^{2}(B_{\tau}\setminus
B_{\rho})}\|w\|^{2}_{L^{4}(B_{\tau}\setminus B_{\rho})}$
$\displaystyle\lesssim(\tau-\rho)^{-1}\|u\|_{L^{2}(B_{2R})}\left(\|w\|^{\frac{1}{2}}_{L^{2}(B_{2R})}\|\nabla
w\|^{\frac{3}{2}}_{L^{2}(B_{2R})}+\frac{1}{R^{3/2}}\|w\|^{2}_{L^{2}(B_{2R})}\right)$
$\displaystyle\lesssim(\tau-\rho)^{-4}\|(u,w)\|^{6}_{L^{2}(B_{2R})}+\frac{1}{4}\|\nabla
w\|^{2}_{L^{2}(B_{2R})}+\frac{(\tau-\rho)^{-1}}{R^{3/2}}\|(u,w)\|^{3}_{L^{2}(B_{2R})}.$

Therefore, we have

$\displaystyle\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}$
$\displaystyle\leq\frac{1}{4}\int_{B_{2R}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+\frac{C}{(\tau-\rho)^{2}}\|(u,b,w)\|^{2}_{L^{2}(B_{2R})}$
$\displaystyle\hskip
11.38109pt+\frac{C}{(\tau-\rho)^{4}}\|(u,b,w)\|^{6}_{L^{2}(B_{2R})}+\frac{C}{(\tau-\rho)R^{\frac{3}{2}}}\|(u,b,w)\|^{3}_{L^{2}(B_{2R})}$
$\displaystyle\leq\frac{1}{4}\int_{B_{2R}}\left(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}+CR^{1-\frac{6}{p}}\|(u,b,w)\|^{2}_{L^{p}(B_{2R})}$
$\displaystyle\hskip
11.38109pt+CR^{5-\frac{18}{p}}\|(u,b,w)\|^{6}_{L^{p}(B_{2R})}+CR^{2-\frac{9}{p}}\|(u,b,w)\|^{3}_{L^{p}(B_{2R})}$

If we apply Lemma 1 with $f(r):=\int_{B_{r}}(|\nabla u|^{2}+|\nabla
b|^{2}+|\nabla w|^{2})\,\mathrm{d}x$, we get

$\displaystyle\int_{B_{R/2}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}$ $\displaystyle\lesssim
R^{1-\frac{6}{p}}\|(u,b,w)\|^{2}_{L^{p}(B_{2R})}+R^{5-\frac{18}{p}}\|(u,b,w)\|^{6}_{L^{p}(B_{2R})}$
$\displaystyle\hskip
11.38109pt+R^{2-\frac{9}{p}}\|(u,b,w)\|^{3}_{L^{p}(B_{2R})}$

If we let $R\rightarrow\infty$ we can immediately find that

$\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}=0,$

which implies that $u$, $b$ and $w$ are a constant vectors in $R^{3}$. Since
$u,b,w\in L^{p}(\mathbb{R}^{3})$ for $p\in[2,\frac{9}{2})$, we conclude that
$u\equiv b\equiv w\equiv 0$ in $\mathbb{R}^{3}$.

##  5 Proof of Theorem 3

We first recall the fact that any continuous functions vanishing at infinity
must be bounded; thus, from the assumption of the theorem, we have $u,b,w\in
L^{\infty}(\mathbb{R}^{3})$. From (3.3) in the proof in Theorem 1, we know
that

$\int_{B_{\rho}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}$
$\lesssim\frac{1}{(\tau-\rho)^{2}}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}+\frac{1}{(\tau-\rho)}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}$ (5.1)

In order to control the right-hand side in (5.1), let us consider it in two
cases as follows:

(Case 1) $p\in[1,3/2)$: We note that with the choice $\tau=2R$ and $\rho=R$,

$(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{2}{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-2}\|u\|_{L^{\infty}(\mathbb{R}^{3})}\int_{B_{\tau}\setminus
B_{\rho}}|u|{\,{\mathrm{d}}x}\lesssim R^{1-\frac{3}{p}}\|u\|_{L^{p}(B_{2R})}.$

And thus, with the same arguments for $b$ and $w$, we get

$(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}\lesssim
R^{1-\frac{3}{p}}\|(u,b,w)\|_{L^{p}(B_{2R})}.$

For the second term on the right-hand side of (5.1),

$(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{3}{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-1}\|u\|^{2}_{L^{\infty}(\mathbb{R}^{3})}\int_{B_{\tau}\setminus
B_{\rho}}|u|{\,{\mathrm{d}}x}\lesssim R^{2-\frac{3}{p}}\|u\|_{L^{p}(B_{2R})},$

and similarly for $b$ and $w$, we have

$(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}\lesssim
R^{2-\frac{3}{p}}\|(u,b,w)\|_{L^{p}(B_{2R})}.$ (5.2)

(Case 2) $p\in[\frac{3}{2},\frac{9}{4})$: In this case, we note that

$(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{2}{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-2}\|w\|^{1/2}_{L^{\infty}(\mathbb{R}^{3})}\int_{B_{\tau}\setminus
B_{\rho}}|w|^{\frac{3}{2}}{\,{\mathrm{d}}x}\lesssim
R^{1-\frac{9}{2p}}\|w\|^{\frac{3}{2}}_{L^{p}(B_{2R})},$

and in the same way for $b$ and $w$, we have

$(\tau-\rho)^{-2}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{2}+|b|^{2}+|w|^{2}\right){\,{\mathrm{d}}x}\lesssim
R^{1-\frac{9}{2p}}\|(u,b,w)\|^{\frac{3}{2}}_{L^{p}(B_{2R})}.$

For the second term, in a similar way, we also get,

$(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{3}{\,{\mathrm{d}}x}\lesssim(\tau-\rho)^{-1}\|u\|^{\frac{3}{2}}_{L^{\infty}(\mathbb{R}^{3})}\int_{B_{\tau}\setminus
B_{\rho}}|u|^{\frac{3}{2}}{\,{\mathrm{d}}x}\lesssim
R^{1-\frac{3}{p}}\|u\|_{L^{p}(B_{2R})},$

which directly implies that

$(\tau-\rho)^{-1}\int_{B_{\tau}\setminus
B_{\rho}}\left(|u|^{3}+|b|^{3}+|w|^{3}\right){\,{\mathrm{d}}x}\lesssim
R^{1-\frac{3}{p}}\|(u,b,w)\|_{L^{p}(B_{2R})}.$ (5.3)

Collecting (5.2) and (5.3), we have

$\int_{B_{R}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right){\,{\mathrm{d}}x}\lesssim\begin{cases}(R^{1-\frac{3}{p}}+R^{2-\frac{3}{p}})\|(u,b,w)\|_{L^{p}(B_{2R})},\quad\,p\in[1,\frac{3}{2}),\\\
(R^{1-\frac{9}{2p}}+R^{2-\frac{9}{2p}})\|(u,b,w)\|^{3/2}_{L^{p}(B_{2R})},\quad
p\in[\frac{3}{2},\frac{9}{4}).\end{cases}$

If we let $R\rightarrow+\infty$ in the above estimate, we can immediately find
that

$\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla b|^{2}+|\nabla
w|^{2}\right)\,dx=0,$

which implies that $u$, $b$ and $w$ are a constant vectors in
$\mathbb{R}^{3}$. As we know $u,b,w\in L^{p}(\mathbb{R}^{3})$ for
$p\in[1,\frac{9}{4})$, we finally conclude that $u\equiv b\equiv w\equiv 0$ in
${\mathbb{R}}^{3}$.

## Acknowledgments

Jae-Myoung Kim was supported by National Research Foundation of Korea Grant
funded by the Korean Government (NRF-2020R1C1C1A01006521).

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