Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in $\mathbb{R}^{3}$

Jing Loong School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China [email protected] and Guoxu Yang School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China [email protected]

Abstract.

In this paper, we investigate a Liouville-type theorem for the MHD equations using Saint-Venant type estimates. We show that $(u,B)$ is a trivial solution if the growth of the $L^{s}$ mean oscillation of the potential functions for both the velocity and magnetic fields are controlled. Our growth assumption is weaker than those previously known for similar results. The main idea is to refine the Saint-Venant type estimates using the Froullani integral.

Keywords: Liouville-type theorems; MHD system; Saint-Venant type estimates; Froullani integral

1. Introduction

Consider the following general stationary MHD system in $\mathbb{R}^{3}$ :

\left\{\begin{array}[]{l}-\kappa\Delta u+u\cdot\nabla u+\nabla p=B\cdot\nabla B,\\ -\nu\Delta B+u\cdot\nabla B-B\cdot\nabla u=0,\\ \operatorname{div}u=0,\quad\operatorname{div}B=0,\end{array}\right.

(1.1)

where $u=(u_{1},u_{2},u_{3})$ is the velocity field of the fluid flows, $B=(B_{1},B_{2},B_{3})$ is the magnetic field, and $p$ is the pressure of the flows. In addition, $\kappa>0$ and $\nu>0$ are given parameters denoting the fluid viscosity and the magnetic resistivity, respectively. Without loss of generality, let $\kappa=\nu=1$ now on.

In this paper, we focus on the Liouville-type properties of the system (1.1), which is motivated by the development of Navier-Stokes equations. When $B\equiv 0$ , (1.1) is reduced to the Navier-Stokes system in $\mathbb{R}^{3}$ :

\left\{\begin{array}[]{l}u\cdot\nabla u+\nabla p-\Delta u=0,\\ \nabla\cdot u=0,\end{array}\right.

(1.2)

and a very challenging open question is whether there exists a nontrivial solution when the Dirichlet integral $\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx$ is finite, which dates back to Leray’s celebrated paper [L1933] and is explicitly written in Galdi’s book ([Galdi], Remark X. 9.4; see also Tsai’s book [T2018]). The Liouville type problem without any other assumptions remains widely open. Galdi proved the above Liouville type theorem by assuming $u\in L^{\frac{9}{2}}(\mathbb{R}^{3})$ in [Galdi]. Chae [Chae2014] showed the condition $\triangle u\in L^{\frac{6}{5}}(\mathbb{R}^{3})$ is sufficient for the vanishing property of $u$ by exploring the maximum principle of the head pressure. Chae-Wolf [ChaeWolf2016] gave an improvement of logarithmic form for Galdi’s result by assuming that $\int_{\mathbb{R}^{3}}|u|^{\frac{9}{2}}\{\ln(2+\frac{1}{|u|})\}^{-1}dx<\infty$ . Seregin obtained the conditional criterion $u\in BMO^{-1}(\mathbb{R}^{3})$ in [Seregin2016], where $u\in BMO^{-1}\left(\mathbb{R}^{3}\right)$ means $u=\operatorname{div}\boldsymbol{V}$ for some anti-symmetric tensor $\boldsymbol{V}\in BMO\left(\mathbb{R}^{3}\right)$ . The $BMO^{-1}$ condition was later relaxed to growth conditions on a mean oscillation of $\boldsymbol{V}$ in [S2018, CW2019, BY2024]. For more references on conditional Liouville properties or Liouville properties on special domains, we refer to [bang2022liouville, CPZZ2020, Chae2021, KTW2021, LRZ2022, SW2019] and the references therein. Relatively speaking, the two-dimensional case is easier because the vorticity of the 2D Navier-Stokes equations satisfies a nice elliptic equation, to which the maximum principle applies. For example, Gilbarg-Weinberger [GW1978] obtained Liouville type theorem provided the Dirichlet energy is finite. As a different type of Liouville property for the 2D Navier-Stokes equations, Koch-Nadirashvili-Seregin-Sverak [KNSS2009] showed that any bounded solution is a trivial solution as a byproduct of their results on the non-stationary case (see also [BFZ2013] for the unbounded velocity).

However, for the MHD system, the situation is quite different. Due to the lack of maximum principle, there is not much progress in the study of MHD equation. For the 2D MHD equations, Liouville type theorems were proved by assuming the smallness of the norm of the magnetic field in [WW2019], and De Nitti et al. [DHS2022] removed the smallness assumption. For the 3D MHD equations, Chae-Weng [CW2016] proved that if the smooth solution satisfies D-condition ( $\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla B|^{2}\right)dx<\infty$ ) and $u\in L^{3}(\mathbb{R}^{3})$ , then the solutions $(u,B)$ are identically zero. In [S2019], Schulz proved that if the smooth solution $(u,B)$ of the stationary MHD equations is in $L^{6}(\mathbb{R}^{3})$ and $u,B\in BMO^{-1}(\mathbb{R}^{3})$ then it is identically zero. Chae-Wolf [CW2021] showed that $L^{6}$ mean oscillations of the potential function of the velocity and magnetic field have certain linear growth by using the technique of [CW2019]. For more references, we refer to [LP2021, LLN2020, FW2021, WY2024] and the references therein.

Here are our main results.

Theorem 1.1.

Let ${u}$ and $B$ be smooth solutions of (1.1). Assume that there exists an $s\in(3,6]$ and smooth anti-symmetric potentials $\boldsymbol{V},\boldsymbol{W}\in C^{\infty}\left(\mathbb{R}^{3};\mathbb{R}^{3\times 3}\right)$ such that $\nabla\cdot\boldsymbol{V}=u$ , $\nabla\cdot\boldsymbol{W}=B$ , and

\displaystyle\left\|\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}\right\|_{L^{s}(B_{R})}+\left\|\boldsymbol{W}-(\boldsymbol{W})_{B_{R}}\right\|_{L^{s}(B_{R})}\lesssim R^{\frac{s+6}{3s}}(\log R)^{\beta},

(1.3)

where $\beta=\frac{s+6}{6s}$ , for all $R>2$ . Then $u=B\equiv 0$ .

Corollary 1.1.

Let ${u}$ be a smooth solution of (1.2). Assume that there exists an $s\in(3,6]$ and smooth anti-symmetric potential $\boldsymbol{V}\in C^{\infty}\left(\mathbb{R}^{3};\mathbb{R}^{3\times 3}\right)$ such that $\nabla\cdot\boldsymbol{V}=u$ , and

\displaystyle\left\|\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}\right\|_{L^{s}(B_{R})}\lesssim R^{\frac{s+6}{3s}}(\log R)^{\frac{s+6}{6s}},

for all $R>2$ . Then $u\equiv 0$ .

Remark 1.1.

We emphasize that for any divergence-free vector field $u$ , there always exists an anti-symmetric potential $\boldsymbol{V}$ such that $u=\operatorname{div}\boldsymbol{V}$ . See, e.g., [Seregin2016, CW2019, CW2021].

Remark 1.2.

In Theorem 1.1, $s\in(3,6]$ , but it can be generalized to $s\in(3,\infty)$ by replacing (1.3) with

\left\|\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}\right\|_{L^{s}(B_{R})}+\left\|\boldsymbol{W}-(\boldsymbol{W})_{B_{R}}\right\|_{L^{s}(B_{R})}\lesssim R^{\min\left\{\frac{s+6}{3s},\frac{s+18}{6s}\right\}}(\log R)^{\min\left\{\frac{s+6}{6s},\frac{1}{3}\right\}},

for all $R>2$ . Indeed, if $s>6$ , by Hölder inequality,

		$\displaystyle\quad\left\\|\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}\right\\|_{L^{6}(B_{R})}+\left\\|\boldsymbol{W}-(\boldsymbol{W})_{B_{R}}\right\\|_{L^{6}(B_{R})}$
		$\displaystyle\lesssim R^{\frac{s-6}{2s}}\left(\left\\|\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}\right\\|_{L^{s}(B_{R})}+\left\\|\boldsymbol{W}-(\boldsymbol{W})_{B_{R}}\right\\|_{L^{s}(B_{R})}\right)\lesssim R^{\frac{2}{3}}(\log R)^{\frac{1}{3}},$

which implies (1.3) when $s=6$ .

Remark 1.3.

For $s=3$ , we can not control the local Dirichlet energy (see (2.20) for details). However, if we assume $\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x<\infty$ , we could set $s=3$ . In fact, we only consider Case I in the proof of Theorem 1.1, and the proof by contradiction still holds due to we do not need the growth estimates for the local Dirichlet energy.

Remark 1.4.

The $\log R$ factor in (1.3) and (1.1) can be relaxed to $\prod_{k=1}^{m}l^{k}(R)$ for any $m\in\mathbb{N}$ , where $l^{k}$ is the composition of $|\log|$ function $k$ times. Indeed, the analytical method of the Froullani integral remains valid for $\prod_{k=1}^{m}l^{k}(R)$ .

Remark 1.5.

Theorem 1.1 relaxes the assumption in Theorem 1.1 of [CW2021] from $R^{\frac{s+6}{3s}}$ to $R^{\frac{s+6}{3s}}(\log R)^{\frac{s+6}{6s}}$ . Also, Corollary 1.1 relaxes the assumption in Theorem 1.1 of [BY2024] from $R^{\frac{s+6}{3s}}(\log R)^{\frac{s-3}{3s}}$ to $R^{\frac{s+6}{3s}}(\log R)^{\frac{s+6}{6s}}$ . We improved the coefficient of $\log R$ by refining the Saint-Venant type estimates using the Froullani integral. See (3.3)–(3.6) for more details. We also note that improving the coefficient of $R$ is difficult due to limitations of the method.

This paper is organized as follows: some notations and some necessary lemmas are presented in Sect.2; The Liouville type theorem on the MHD system is obtained in Sect.3, where we give the detailed proof of Theorem 1.1.

2. Preliminaries

First we introduce some notations. Denote by $\nabla^{\gamma}:=\partial_{x_{1}}^{\gamma_{1}}\partial_{x_{2}}^{\gamma_{2}}\partial_{x_{3}}^{\gamma_{3}}$ , where $\gamma=\left(\gamma_{1},\gamma_{2},\gamma_{3}\right)$ , $\gamma_{1},\gamma_{2},\gamma_{3}\in\mathbb{N}\cup\{0\}$ , $\partial_{i}=\frac{\partial}{\partial x_{i}}$ and $|\gamma|=\gamma_{1}+\gamma_{2}+\gamma_{3}$ . We denote $L^{p}(\Omega)$ by the usual Lebesgue space with the norm

\|f\|_{L^{p}(\Omega)}:=\begin{cases}\left(\int_{\Omega}|f(x)|^{p}dx\right)^{1/p},&1\leq p<\infty,\\ \underset{x\in\Omega}{\operatorname{ess\,sup}}|f(x)|,&p=\infty,\end{cases}

where $\Omega\subset\mathbb{R}^{3},1\leq p\leq\infty$ . $W^{k,p}(\Omega)$ and $\dot{W}^{k,p}(\Omega)$ are defined as follows:

	$\displaystyle\\|f\\|_{W^{k,p}(\Omega)}$	$\displaystyle:=\sum_{0\leq\|\gamma\|\leq k}\left\\|\nabla^{\gamma}f\right\\|_{L^{p}(\Omega)},$
	$\displaystyle\\|f\\|_{\dot{W}^{k,p}(\Omega)}$	$\displaystyle:=\sum_{\|\gamma\|=k}\left\\|\nabla^{\gamma}f\right\\|_{L^{p}(\Omega)},$

respectively. Let $C_{c}^{\infty}(\Omega)$ denote the space of infinitely differentiable functions with compact support in $\Omega$ . We denote by $W_{0}^{k,p}(\Omega)$ the closure of $C_{c}^{\infty}(\Omega)$ in $W^{k,p}(\Omega)$ . Throughout this paper, we denote $\overline{\boldsymbol{V}}:=\boldsymbol{V}-(\boldsymbol{V})_{B_{R}}$ , $\overline{\boldsymbol{W}}:=\boldsymbol{W}-(\boldsymbol{W})_{B_{R}}$ and denote $A\leq CB$ by $A\lesssim B$ for convenience. We denote that $B_{R}$ is the ball in $\mathbb{R}^{3}$ of radius $R$ centered at the origin, $B_{R}^{+}=B_{R}\cap\mathbb{R}_{+}^{3}$ . We denote by

\left\|\left(f,g\right)\right\|^{2}_{L^{2}(\Omega)}:=\left\|f\right\|^{2}_{L^{2}(\Omega)}+\left\|g\right\|^{2}_{L^{2}(\Omega)}.

Let $0<\rho<R$ , we denote $\eta_{1}$ by a cut-off function such that

\eta_{1}(x)=\eta_{1}(|x|)=\left\{\begin{array}[]{lll}1,&\text{ if }&|x|<\frac{R+\rho}{2};\\ 0,&\text{ if }&|x|>R,\end{array}\right.

and $0\leq\eta_{1}(x)\leq 1$ for any $\frac{R+\rho}{2}\leq|x|\leq R$ . We denote $\eta_{2}$ by a cut-off function such that

\eta_{2}(x)=\eta_{2}(|x|)=\left\{\begin{array}[]{lll}1,&\text{ if }&|x|<\rho;\\ 0,&\text{ if }&|x|>\frac{R+\rho}{2},\end{array}\right.

and $0\leq\eta_{2}(x)\leq 1$ for any $\rho\leq|x|\leq\frac{R+\rho}{2}$ . We denote $\eta$ by a cut-off function given by

\eta(x)=\begin{cases}1&\text{ if }x<1,\\ -x+2&\text{ if }1\leq x\leq 2,\\ 0&\text{ if }x>2,\end{cases}

and denote $\varphi_{R}(x):=\eta(\frac{|x|}{R})$ . It obviously holds

\displaystyle|\nabla\eta_{1}|+|\nabla\eta_{2}|\lesssim\left(R-\rho\right)^{-1}.

Next, we apply the following two lemmas to demonstrate that $\left\|u\eta_{1}\right\|_{L^{2}(B_{R})}$ and $\left\|u\eta_{1}\right\|_{L^{\frac{4s}{s+2}}(B_{R})}$ are controlled by $\|\overline{\boldsymbol{V}}\|_{L^{s}(B_{R})}$ and $\|\nabla u\|_{L^{2}(B_{R})}$ , with the same result holding for $B$ .

lemma 2.1 ([BY2024], Lemma 2.1).

Assume $s\geq 2$ . It holds

	$\displaystyle\left\\|{u}\eta_{1}\right\\|_{L^{2}(B_{R})}$	$\displaystyle\lesssim R^{\frac{3(s-2)}{4s}}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{u}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})},$		(2.1)
	$\displaystyle\left\\|B\eta_{1}\right\\|_{L^{2}(B_{R})}$	$\displaystyle\lesssim R^{\frac{3(s-2)}{4s}}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla B\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}.$		(2.1)

lemma 2.2 ([BY2024], Lemma 2.2 ).

Assume $s\geq 2$ . It holds

	$\displaystyle\left\\|{u}\eta_{1}\right\\|_{L^{\frac{4s}{s+2}}(B_{R})}$	$\displaystyle\lesssim\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{u}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+(R-\rho)^{-1}R^{\frac{3s-6}{4s}}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})},$		(2.2)
	$\displaystyle\left\\|{B}\eta_{1}\right\\|_{L^{\frac{4s}{s+2}}(B_{R})}$	$\displaystyle\lesssim\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{B}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+(R-\rho)^{-1}R^{\frac{3s-6}{4s}}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}.$		(2.2)

We recall the Bogovskii map (see [Galdi], Lemma III.3.1 and [T2018], §2.8). For a domain $\Omega\subset\mathbb{R}^{3}$ , denote

L_{0}^{q}(\Omega)=\left\{f\in L^{q}(\Omega):\int_{\Omega}f~{}\mathrm{d}x=0\right\}.

Then we use the following lemma to deal with the pressure.

lemma 2.3 ([T2021], Lemma 3).

Let $R>0$ , $1<L<\infty$ and $A=B_{LR}\backslash\bar{B}_{R}$ or $A=B_{LR}^{+}\backslash\bar{B}_{R}^{+}$ be an annulus or a half-annulus in $\mathbb{R}^{3}$ . There is a linear Bogovskii map $\operatorname{Bog}$ that maps a scalar function $f\in L_{0}^{q}(A),1<q<\infty$ , to a vector field $v=\operatorname{Bog}f\in W_{0}^{1,q}(A)$ and

\operatorname{div}v=f,\quad\|\nabla v\|_{L^{q}(A)}\leq\frac{C_{q}}{(L-1)L^{1-1/q}}\|f\|_{L^{q}(A)}.

The constant $C_{q}$ is independent of $L$ and $R$ .

Using the above three lemmas, we obtain the following energy inequality, which is crucial for proving the main theorem.

lemma 2.4.

Assume $2<\rho<R$ , $3\leq s\leq 6$ and $\beta>0$ . Let $\boldsymbol{V}$ and $\boldsymbol{W}$ satisfy (1.3). Then

			$\displaystyle\left\\|\left(\nabla u,\nabla B\right)\sqrt{\eta_{2}}\right\\|_{L^{2}(B_{R})}^{2}$
		$\displaystyle\lesssim$	$\displaystyle R^{10}(R-\rho)^{-10}(\log R)^{\frac{3s}{s+6}\beta}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{\frac{\rho+R}{2}}\setminus B_{\rho})}\left(\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}}+1\right).$

Proof.

We divide the proof into two steps.
Step I. Dealing with the pressure. Using Lemma 2.3 with $L=\frac{R+\rho}{2\rho}$ , for any $q\in(1,\infty)$ , there exists a constant $C_{q}>0$ and a function ${w}\in W_{0}^{1,q}\left(B_{(R+\rho)/2}\backslash B_{\rho}\right)$ such that $\operatorname{div}{w}={u}\cdot\nabla\eta_{2}$ and

\displaystyle\int_{B_{(R+\rho)/2}\backslash B_{\rho}}|\nabla{w}|^{q}~{}\mathrm{d}x\leq C_{q}\left(\frac{\rho}{R-\rho}\right)^{q}\int_{B_{(R+\rho)/2}\backslash B_{\rho}}\left|{u}\cdot\nabla\eta_{2}\right|^{q}~{}\mathrm{d}x.

(2.3)

We then extend ${w}=0$ in $B_{\rho}$ . Integrating by part, we find

\displaystyle\int_{B_{R}}\nabla p\cdot\left(u\eta_{2}-w\right)~{}\mathrm{d}x=-\int_{B_{R}}pu\cdot\nabla\eta_{2}~{}\mathrm{d}x+\int_{B_{R}}p\nabla\cdot w~{}\mathrm{d}x=0.

(2.4)

Step II. $L^{2}$ -estimate. Multiplying the equation $\eqref{eq:MHD}_{1}$ by ${u}\eta_{2}-{w}$ , multiplying the equation $\eqref{eq:MHD}_{2}$ by $B\eta_{2}$ and integrating by parts in $B_{R}$ , we have

$\displaystyle\int_{B_{R}}\|\nabla{u}\|^{2}\eta_{2}+\|\nabla{B}\|^{2}\eta_{2}~{}\mathrm{d}x=$	$\displaystyle-\int_{B_{R}}u\cdot\nabla u\cdot\nabla\eta_{2}+B\cdot\nabla B\cdot\nabla\eta_{2}~{}\mathrm{d}x+\int_{B_{R}}\nabla{u}\cdot\nabla{w}~{}\mathrm{d}x$	(2.5)
	$\displaystyle+\int_{B_{R}}u\cdot\nabla u\cdot w-B\cdot\nabla B\cdot w~{}\mathrm{d}x-\int_{B_{R}}u\cdot\nabla u\cdot u\eta_{2}+u\cdot\nabla B\cdot B\eta_{2}~{}\mathrm{d}x$
	$\displaystyle+\int_{B_{R}}B\cdot\nabla B\cdot u\eta_{2}+B\cdot\nabla u\cdot B\eta_{2}~{}\mathrm{d}x$
$\displaystyle=$	$\displaystyle:I+II+III+IV+V,$

where the pressure vanishes due to (2.4). For I, by Hölder inequality and Lemma 2.1, it holds

$\displaystyle\|I\|$	$\displaystyle\lesssim(R-\rho)^{-1}\left(\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}+\\|\nabla B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\right)$	(2.6)
	$\displaystyle\lesssim(R-\rho)^{-1}\left(\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\eta_{1}\\|_{L^{2}(B_{R})}+\\|\nabla B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|B\eta_{1}\\|_{L^{2}(B_{R})}\right)$
	$\displaystyle\lesssim(R-\rho)^{-1}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left[R^{\frac{3(s-2)}{4s}}\\|\left(\nabla{u},\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\right)\right.$
	$\displaystyle\left.\quad+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\right].$

For II, by Hölder inequality and (2.3) with $q=2$ , we get

	$\displaystyle\|II\|$	$\displaystyle\lesssim(R-\rho)^{-1}\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|\nabla w\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}$		(2.7)
		$\displaystyle\lesssim R(R-\rho)^{-2}\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}.$		(2.7)

Combining (2.6) and (2.7), we have

		$\displaystyle\|I\|+\|II\|\lesssim R(R-\rho)^{-2}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}$		(2.8)
		$\displaystyle\cdot\left[R^{\frac{3(s-2)}{4s}}\\|\left(\nabla{u},\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\right)+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\right].$		(2.8)

For III, by integration by parts, it holds

	$\displaystyle III$	$\displaystyle=\int_{B_{R}}\partial_{k}\overline{\boldsymbol{V}}_{ki}\partial_{i}u_{j}w_{j}-\partial_{k}\overline{\boldsymbol{W}}_{ki}\partial_{i}B_{j}w_{j}~{}\mathrm{d}x$
		$\displaystyle=-\int_{B_{R}}\overline{\boldsymbol{V}}_{ki}\partial_{i}u_{j}\partial_{k}w_{j}-\overline{\boldsymbol{W}}_{ki}\partial_{i}B_{j}\partial_{k}w_{j}~{}\mathrm{d}x,$

where we use the fact that $\overline{\boldsymbol{V}}$ and $\overline{\boldsymbol{W}}$ are anti-symmetric, $\nabla^{2}(u\cdot w)$ and $\nabla^{2}(B\cdot w)$ are symmetric. Then by Hölder inequality and (2.3) with $q=\frac{2s}{s-2}$ , we have

	$\displaystyle\|III\|$	$\displaystyle\lesssim\\|\nabla w\\|_{L^{\frac{2s}{s-2}}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)$		(2.9)
		$\displaystyle\lesssim\rho\left(R-\rho\right)^{-2}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}.$		(2.9)

For IV, by integration by parts, it holds

	$\displaystyle IV$	$\displaystyle=-\frac{1}{2}\int_{B_{R}}\partial_{k}\overline{\boldsymbol{V}}_{ki}\partial_{i}\|u\|^{2}\eta_{2}-\partial_{k}\overline{\boldsymbol{V}}_{ki}\partial_{i}\|B\|^{2}\eta_{2}~{}\mathrm{d}x$		(2.10)
		$\displaystyle=\frac{1}{2}\int_{B_{R}}\overline{\boldsymbol{V}}_{ki}\partial_{i}\|u\|^{2}\partial_{k}\eta_{2}-\overline{\boldsymbol{V}}_{ki}\partial_{i}\|B\|^{2}\partial_{k}\eta_{2}~{}\mathrm{d}x,$		(2.10)

where we use the fact that $\nabla^{2}|u|^{2}$ and $\nabla^{2}|B|^{2}$ are symmetric. Then by Hölder inequality, we have

\displaystyle|IV|\lesssim\left(R-\rho\right)^{-1}\|\overline{\boldsymbol{V}}\|_{L^{s}(B_{R})}\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left(\|u\eta_{1}^{2}\|_{L^{\frac{2s}{s-2}}(B_{R})}+\|B\eta_{1}^{2}\|_{L^{\frac{2s}{s-2}}(B_{R})}\right).

(2.11)

For V, by integration by parts, it holds

	$\displaystyle V$	$\displaystyle=\int_{B_{R}}\partial_{k}\overline{\boldsymbol{W}}_{ki}\partial_{i}B_{j}u_{j}\eta_{2}+\partial_{k}\overline{\boldsymbol{W}}_{ki}\partial_{i}u_{j}B_{j}\eta_{2}~{}\mathrm{d}x$
		$\displaystyle=-\int_{B_{R}}\overline{\boldsymbol{W}}_{ki}\partial_{i}B_{j}u_{j}\partial_{k}\eta_{2}+\overline{\boldsymbol{W}}_{ki}\partial_{i}u_{j}B_{j}\partial_{k}\eta_{2}~{}\mathrm{d}x,$

where we use the fact that $\left[\partial_{i}B_{j}\partial_{k}u_{j}+\partial_{i}u_{j}\partial_{k}B_{j}\right]$ is symmetric. Then by Hölder inequality, we have

\displaystyle|V|\lesssim\left(R-\rho\right)^{-1}\|\overline{\boldsymbol{W}}\|_{L^{s}(B_{R})}\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left(\|u\eta_{1}^{2}\|_{L^{\frac{2s}{s-2}}(B_{R})}+\|B\eta_{1}^{2}\|_{L^{\frac{2s}{s-2}}(B_{R})}\right).

(2.12)

Combining (2.9), (2.11) and (2.12), we get

		$\displaystyle\|III\|+\|IV\|+\|V\|$		(2.13)
		$\displaystyle\lesssim R\left(R-\rho\right)^{-2}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left(\\|u\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}+\\|B\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}\right).$		(2.13)

By interpolation inequality and Sobolev inequality, we get

$\displaystyle\\|u\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}$	$\displaystyle\leq\\|u\eta_{1}^{2}\\|^{\frac{4s-12}{s+6}}_{L^{\frac{4s}{s+2}}(B_{R})}\\|u\eta_{1}^{2}\\|^{\frac{18-3s}{s+6}}_{L^{6}(B_{R})}$	(2.14)
	$\displaystyle\lesssim\\|u\eta_{1}^{2}\\|^{\frac{4s-12}{s+6}}_{L^{\frac{4s}{s+2}}(B_{R})}\\|\nabla(u\eta_{1}^{2})\\|^{\frac{18-3s}{s+6}}_{L^{2}(B_{R})}$
	$\displaystyle\lesssim\\|u\eta_{1}^{2}\\|^{\frac{4s-12}{s+6}}_{L^{\frac{4s}{s+2}}(B_{R})}\left(\\|\nabla u\\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}+(R-\rho)^{-\frac{18-3s}{s+6}}\\|u\eta_{1}\\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}\right).$

Similarly, we get

\displaystyle\|B\eta_{1}^{2}\|_{L^{\frac{2s}{s-2}}(B_{R})}\lesssim\|B\eta_{1}^{2}\|^{\frac{4s-12}{s+6}}_{L^{\frac{4s}{s+2}}(B_{R})}\left(\|\nabla B\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}+(R-\rho)^{-\frac{18-3s}{s+6}}\|B\eta_{1}\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}\right).

(2.15)

Combining (2.14) and (2.15), using Lemma 2.1 and Lemma 2.2, we have

			$\displaystyle\\|u\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}+\\|B\eta_{1}^{2}\\|_{L^{\frac{2s}{s-2}}(B_{R})}$
		$\displaystyle\lesssim$	$\displaystyle\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{2s-6}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{2s-6}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}}$
			$\displaystyle+(R-\rho)^{-\frac{18-3s}{s+6}}R^{\frac{9(s-2)(6-s)}{4s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}$
			$\displaystyle+(R-\rho)^{-\frac{6(6-s)}{s+6}}R^{\frac{9(s-2)(6-s)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{12-s}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{12-s}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{2s-6}{s+6}}$
			$\displaystyle+(R-\rho)^{-\frac{4s-12}{s+6}}R^{\frac{3(s-2)(s-3)}{s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{4s-12}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{4s-12}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}$
			$\displaystyle+(R-\rho)^{-1}R^{\frac{3(s-2)}{4s}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{2(s+6)}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{2(s+6)}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{2(s+6)}}$
			$\displaystyle+(R-\rho)^{-\frac{2(12-s)}{s+6}}R^{\frac{3(12-s)(s-2)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right).$

Hence, we get

	$\displaystyle\|III\|+\|IV\|+\|V\|$	(2.16)
$\displaystyle\lesssim$	$\displaystyle R\left(R-\rho\right)^{-2}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left[\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{3s}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{3s}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{18-3s}{s+6}}R^{\frac{9(s-2)(6-s)}{4s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{3}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{3}{2}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{6(6-s)}{s+6}}R^{\frac{9(s-2)(6-s)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{18}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{18}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{2s-6}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{4s-12}{s+6}}R^{\frac{3(s-2)(s-3)}{s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-1}R^{\frac{3(s-2)}{4s}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{7s+6}{2(s+6)}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{7s+6}{2(s+6)}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{2(s+6)}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{2(12-s)}{s+6}}R^{\frac{3(12-s)(s-2)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{2}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{2}\right)\right].$

Putting (2.5), (LABEL:eq:I+) and (2.16) together, we get

	$\displaystyle\int_{B_{R}}\|\nabla{u}\|^{2}\eta_{2}+\|\nabla{B}\|^{2}\eta_{2}~{}\mathrm{d}x$	(2.17)
$\displaystyle\lesssim$	$\displaystyle R\left(R-\rho\right)^{-2}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left[\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{3s}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{3s}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{18-3s}{s+6}}R^{\frac{9(s-2)(6-s)}{4s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{3}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{3}{2}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{6(6-s)}{s+6}}R^{\frac{9(s-2)(6-s)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{18}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{18}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{2s-6}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{4s-12}{s+6}}R^{\frac{3(s-2)(s-3)}{s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{s+6}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{5s-6}{s+6}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{s+6}}\right.$
	$\displaystyle\left.+(R-\rho)^{-1}R^{\frac{3(s-2)}{4s}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{7s+6}{2(s+6)}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{7s+6}{2(s+6)}}\right)\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18-3s}{2(s+6)}}\right.$
	$\displaystyle\left.+(R-\rho)^{-\frac{2(12-s)}{s+6}}R^{\frac{3(12-s)(s-2)}{2s(s+6)}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{2}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{2}\right)\right.$
	$\displaystyle\left.+R^{\frac{3(s-2)}{4s}}\\|\left(\nabla{u},\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\right)\right.$
	$\displaystyle\left.+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\right]$
$\displaystyle:=$	$\displaystyle R\left(R-\rho\right)^{-2}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\sum_{i=1}^{8}I_{i}.$

We point out that $I_{1}$ is the main term, since

\displaystyle I_{1}\lesssim R(\log R)^{\frac{3s}{s+6}\beta}\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}},

(2.18)

by (1.3). For $I_{2}$ , by (1.3), we have

	$\displaystyle I_{2}$	$\displaystyle\lesssim R^{\frac{5s-6}{4s}}\left(\frac{R}{R-\rho}\right)^{\frac{18-3s}{s+6}}(\log R)^{\frac{3}{2}\beta}\\|\left(\nabla u,\nabla B\right)\\|_{2,R}^{\frac{1}{2}}$		(2.19)
		$\displaystyle\lesssim R\left(\frac{R}{R-\rho}\right)^{8}(\log R)^{\frac{3s}{s+6}\beta}\max\left\{\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}},1\right\}.$		(2.19)

Similarly, we have

\displaystyle\sum_{i=3}^{8}I_{i}\lesssim R\left(\frac{R}{R-\rho}\right)^{8}(\log R)^{\frac{3s}{s+6}\beta}\max\left\{\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{R})}^{\frac{12-s}{s+6}},1\right\}.

which, combined with (2.17), (2.18) and (2.19), completes the proof. ∎

With the lemma above, we obtain a growth estimate for the local Dirichlet energy via an iterative argument.

lemma 2.5 (Growth estimates of the local Dirichlet energy).

Assuming that $\boldsymbol{V}$ and $\boldsymbol{W}$ satisfy (1.3) for $3<s\leq 6$ , it holds

\int_{B_{R}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x\lesssim(\log R)^{\frac{3s}{s-3}\beta},

for all $R>2$ .

Proof.

We may assume that $\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{R})}>1$ when $R>R^{\prime}$ for some positive $R^{\prime}$ . Fix $R^{\prime}<\rho<R$ . By Lemma 2.4 and Young inequality, we have

	$\displaystyle\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{\rho})}^{2}$	$\displaystyle\lesssim R^{10}(R-\rho)^{-10}(\log R)^{{\frac{3s}{s+6}\beta}}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{18}{s+6}}$		(2.20)
		$\displaystyle\leq\frac{1}{2}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{R})}^{2}+CR^{\frac{10(s+6)}{s-3}}(R-\rho)^{-\frac{10(s+6)}{s-3}}(\log R)^{\frac{3s}{s-3}\beta},$		(2.20)

where we note that $s>3$ ensures the validity of Young inequality. Denote $t_{0}=\rho$ , $t_{i+1}-t_{i}=2^{-i-1}(R-\rho)$ . Using above inequality with $\rho,R$ replaced by $t_{i},t_{i+1}$ , and iterating $k$ times, we have

\displaystyle\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{t_{0}})}^{2}

\displaystyle\leq\frac{1}{2^{k}}\|\left(\nabla u,\nabla B\right)\|_{L^{2}(B_{t_{k}})}^{2}+CR^{\frac{10(s+6)}{s-3}}(R-\rho)^{-\frac{10(s+6)}{s-3}}(\log R)^{\frac{3s}{s-3}\beta}\sum_{i=0}^{k-1}\frac{1}{2^{i}}.

Letting $k\rightarrow\infty$ and choosing $\rho=R/2$ , we complete the proof. ∎

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

We define

g(R):=\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+|\nabla B|^{2}\right)\varphi_{R}~{}\mathrm{d}x,

which implies

g^{\prime}(R)=\int_{B_{2R}\backslash B_{R}}\left(|\nabla u|^{2}+|\nabla B|^{2}\right)\eta^{\prime}\left(\frac{|x|}{R}\right)\cdot\left(-\frac{|x|}{R^{2}}\right)~{}\mathrm{d}x\gtrsim R^{-1}\int_{B_{2R}\backslash B_{R}}\left(|\nabla u|^{2}+|\nabla B|^{2}\right)~{}\mathrm{d}x.

Replacing $\rho,R$ by $R,3R$ and choosing $\eta_{2}=\varphi_{R}$ in Lemma 2.4, we have

\displaystyle g(R)\lesssim R^{\frac{1}{2}}(\log R)^{\frac{3s}{s+6}\beta}g^{\prime}(R)^{\frac{1}{2}}\left(g(3R)^{\frac{12-s}{2(s+6)}}+1\right).

(3.1)

Recall that

\displaystyle\int_{B_{R}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x\lesssim(\log R)^{\frac{3s}{s-3}\beta},

for all $R>2$ in Lemma 2.5. Then by the definition of $g(R)$ , we have

\displaystyle g(R)\lesssim(\log R)^{\frac{3s}{s-3}\beta},

(3.2)

for all $R>2$ . We then prove $\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x=0$ by contradiction. Assuming $\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x\neq 0$ , we divide the proof into two cases.

Case I. $0\neq\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x\leq 2$ . There exists $R_{0}>e$ , such that $g(R)>0$ for all $R\geq R_{0}$ . Then by (3.1), choosing $\beta=\frac{s+6}{6s}$ , we directly obtain

\displaystyle g(R)\lesssim R^{\frac{1}{2}}(\log R)^{\frac{1}{2}}g^{\prime}(R)^{\frac{1}{2}},

which implies

\displaystyle\int_{R_{0}}^{+\infty}\frac{~{}\mathrm{d}R}{R\log R}\lesssim\int_{R_{0}}^{+\infty}\frac{g^{\prime}(R)}{g(R)^{2}}~{}\mathrm{d}R.

We derive a contradiction from the divergence of the left-hand side and the convergence of the right-hand side of the above inequality.

Case II. $\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x>2$ . There exists $R_{1}>9$ , such that $g(3R)\geq g(R)>1$ for all $R>R_{1}$ . Then by (3.1), choosing $\beta=\frac{s+6}{6s}$ , we obtain

\displaystyle g(R)\lesssim R^{\frac{1}{2}}(\log R)^{\frac{1}{2}}g^{\prime}(R)^{\frac{1}{2}}g(3R)^{\frac{12-s}{2(s+6)}},

which implies

\displaystyle\frac{\left(\frac{g(R)}{g(3R)}\right)^{\frac{12-s}{s+6}}}{R\log R}\lesssim\frac{{g}^{\prime}({R})}{{g}({R})^{\frac{3{s}}{{s}+6}}}.

Integrating both sides on $(R_{1},r)$ , we get

\displaystyle\int_{R_{1}}^{r}\frac{\left(\frac{g(R)}{g(3R)}\right)^{\frac{12-s}{s+6}}}{R\log R}~{}\mathrm{d}R\lesssim\int_{R_{1}}^{r}\frac{g^{\prime}(R)}{{g}({R})^{\frac{3{s}}{{s}+6}}}~{}\mathrm{d}R.

(3.3)

For the left side of (3.3), we note that

		$\displaystyle\quad\int_{R_{1}}^{r}\frac{\left(\frac{g(R)}{g(3R)}\right)^{\frac{12-s}{s+6}}}{R\log R}~{}\mathrm{d}R\gtrsim\int_{R_{1}}^{r}\frac{\log\frac{g(R)}{g(3R)}+1}{R\log R}~{}\mathrm{d}R$		(3.4)
		$\displaystyle=\int_{R_{1}}^{r}\frac{\log g(R)}{R\log R}~{}\mathrm{d}R-\int_{3R_{1}}^{3r}\frac{\log g(R)}{R\left(\log R-\log 3\right)}~{}\mathrm{d}R+\log\log r-\log\log R_{1}$
		$\displaystyle=\int_{R_{1}}^{r}\frac{\log g(R)}{R}\left(\frac{1}{\log R}-\frac{1}{\log R-\log 3}\right)~{}\mathrm{d}R-\int_{r}^{3r}\frac{\log g(R)}{R\left(\log R-\log 3\right)}~{}\mathrm{d}R$
		$\displaystyle\quad+\int_{R_{1}}^{3R_{1}}\frac{\log g(R)}{R\left(\log R-\log 3\right)}~{}\mathrm{d}R-\log\log R_{1}+\log\log r$
		$\displaystyle=I_{a}+I_{b}+I_{c}+\log\log r,$

where

I_{a}:=\int_{R_{1}}^{r}\frac{\log g(R)}{R}\left(\frac{1}{\log R}-\frac{1}{\log R-\log 3}\right)~{}\mathrm{d}R,\quad I_{b}:=-\int_{r}^{3r}\frac{\log g(R)}{R\left(\log R-\log 3\right)}~{}\mathrm{d}R

and

I_{c}:=\int_{R_{1}}^{3R_{1}}\frac{\log g(R)}{R\left(\log R-\log 3\right)}~{}\mathrm{d}R-\log\log R_{1}.

Noting $R_{1}>9$ implies $\log R-\log 3>\frac{1}{2}\log R$ for all $R>R_{1}$ and using (3.2), we have for $I_{a}$ :

\displaystyle|I_{a}|\lesssim\int_{R_{1}}^{r}\frac{\log g(R)}{R(\log R)^{2}}~{}\mathrm{d}R\lesssim\int_{R_{1}}^{r}\frac{\log\log R}{R(\log R)^{2}}~{}\mathrm{d}R,

(3.5)

for $I_{b}$ :

$\displaystyle\|I_{b}\|$	$\displaystyle\lesssim\int_{r}^{3r}\frac{\log\log R}{R\log R}~{}\mathrm{d}R=\left.\frac{1}{2}\left(\log\log R\right)^{2}\right\|_{r}^{3r}=\frac{1}{2}\left(\log\log 3r\right)^{2}-\frac{1}{2}\left(\log\log r\right)^{2}$	(3.6)
	$\displaystyle=\frac{1}{2}\left(\log\log r+\log\left(1+\frac{\log 3}{\log r}\right)\right)^{2}-\frac{1}{2}\left(\log\log r\right)^{2}$
	$\displaystyle\lesssim 1-\frac{1}{\log r}+\left(\log\left(1+\frac{\log 3}{\log r}\right)\right)^{2}.$

By(3.5) and (3.6), we have

\displaystyle\lim\limits_{r\rightarrow+\infty}I_{a}+I_{b}+I_{c}\gtrsim-1.

(3.7)

Letting $r\rightarrow+\infty$ in (3.3) and combining (LABEL:eq:leftside), (3.7), we derive a contradiction from the divergence of the left-hand side and the convergence of the right-hand side of (3.3). Hence

\int_{\mathbb{R}^{3}}|\nabla u|^{2}+|\nabla B|^{2}~{}\mathrm{d}x=0,

which implies $u\equiv u_{0}$ and $B\equiv B_{0}$ for all $x\in\mathbb{R}^{3}$ , where ${u}_{0}$ and $B_{0}$ are two constants. By Lemma 2.1 with $\rho=\frac{R}{2}$ , we have

\|u\|_{L^{2}(B_{\frac{R}{2}})}+\|B\|_{L^{2}(B_{\frac{R}{2}})}\lesssim R^{\frac{5s-6}{6s}}(\log R)^{\frac{s+6}{6s}},

which implies

\displaystyle|u_{0}|+|B_{0}|\lesssim R^{-\frac{2s+3}{3s}}(\log R)^{\frac{s+6}{6s}},

(3.8)

for all $R>2$ . Therefore, $u=B\equiv 0$ . The proof is completed. ∎

Remark 3.1.

It is evident that $0<\frac{g(R)}{g(3R)}\leq 1$ , but this is insufficient for estimating its lower bound. However, the result of Lemma 2.5 indicates that the growth rate of $g(R)$ is well-controlled, suggesting that its value might be increased through integration using the Froullani integral.

Acknowledgments. The authors would like to thank Professors Wendong Wang for some helpful communications and the support from the Key R&D Program Project No. 2023YFA1009200.

Declaration of competing interest. The authors state that there is no conflict of interest.

Data availability. No data was used in this paper.

	$\displaystyle\\|f\\|_{W^{k,p}(\Omega)}$	$\displaystyle:=\sum_{0\leq\|\gamma\|\leq k}\left\\|\nabla^{\gamma}f\right\\|_{L^{p}(\Omega)},$
	$\displaystyle\\|f\\|_{\dot{W}^{k,p}(\Omega)}$	$\displaystyle:=\sum_{\|\gamma\|=k}\left\\|\nabla^{\gamma}f\right\\|_{L^{p}(\Omega)},$

	$\displaystyle\left\\|{u}\eta_{1}\right\\|_{L^{2}(B_{R})}$	$\displaystyle\lesssim R^{\frac{3(s-2)}{4s}}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{u}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})},$		(2.1)
	$\displaystyle\left\\|B\eta_{1}\right\\|_{L^{2}(B_{R})}$	$\displaystyle\lesssim R^{\frac{3(s-2)}{4s}}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla B\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}.$		(2.1)

	$\displaystyle\left\\|{u}\eta_{1}\right\\|_{L^{\frac{4s}{s+2}}(B_{R})}$	$\displaystyle\lesssim\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{u}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+(R-\rho)^{-1}R^{\frac{3s-6}{4s}}\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})},$		(2.2)
	$\displaystyle\left\\|{B}\eta_{1}\right\\|_{L^{\frac{4s}{s+2}}(B_{R})}$	$\displaystyle\lesssim\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\\|\nabla{B}\\|_{L^{2}(B_{R})}^{\frac{1}{2}}+(R-\rho)^{-1}R^{\frac{3s-6}{4s}}\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}.$		(2.2)

$\displaystyle\|I\|$	$\displaystyle\lesssim(R-\rho)^{-1}\left(\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}+\\|\nabla B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\right)$	(2.6)
	$\displaystyle\lesssim(R-\rho)^{-1}\left(\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\eta_{1}\\|_{L^{2}(B_{R})}+\\|\nabla B\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|B\eta_{1}\\|_{L^{2}(B_{R})}\right)$
	$\displaystyle\lesssim(R-\rho)^{-1}\\|\left(\nabla u,\nabla B\right)\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\left[R^{\frac{3(s-2)}{4s}}\\|\left(\nabla{u},\nabla B\right)\\|_{L^{2}(B_{R})}^{\frac{1}{2}}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}^{\frac{1}{2}}\right)\right.$
	$\displaystyle\left.\quad+R^{\frac{3(s-2)}{2s}}(R-\rho)^{-1}\left(\\|\overline{\boldsymbol{V}}\\|_{L^{s}(B_{R})}+\\|\overline{\boldsymbol{W}}\\|_{L^{s}(B_{R})}\right)\right].$

	$\displaystyle\|II\|$	$\displaystyle\lesssim(R-\rho)^{-1}\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|\nabla w\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}$		(2.7)
		$\displaystyle\lesssim R(R-\rho)^{-2}\\|\nabla u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}\\|u\\|_{L^{2}(B_{(R+\rho)/2}\backslash B_{\rho})}.$		(2.7)

Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in ℝ3\mathbb{R}^{3}

Abstract.

1. Introduction

Theorem 1.1.

Corollary 1.1.

Remark 1.1.

Remark 1.2.

Remark 1.3.

Remark 1.4.

Remark 1.5.

2. Preliminaries

lemma 2.1 ([BY2024], Lemma 2.1).

lemma 2.2 ([BY2024], Lemma 2.2 ).

lemma 2.3 ([T2021], Lemma 3).

lemma 2.4.

Proof.

lemma 2.5 (Growth estimates of the local Dirichlet energy).

Proof.

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Remark 3.1.

References

Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in $\mathbb{R}^{3}$