On Liouville-type theorems for the stationary nematic liquid crystal equations

Abstract

This paper is concerned with 3D steady compressible nematic liquid crystal flows. We establish a Liouville-type theorem when the smooth solution $(\rho , u, d)$ satisfies some suitable conditions.

1 Introduction

      In this paper, we consider the following simplified version of Ericksen–Leslie system modeling the hydrodynamic flow of stationary compressible nematic liquid crystals in ${\mathbb {R}^{3}}$( [8, 12])

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {div}(\rho u)=0,\\ \text {div}(\rho u\otimes u)-\mu \Delta u-(\lambda +\mu )\nabla \text {div}u+\nabla P(\rho )=-\Delta d\cdot \nabla d,\\ u\cdot \nabla d =\Delta d+|\nabla d|^{2}d, \end{array}\right. } \end{aligned}$$

(1.1)

where $\rho$, u and d denote the density, the velocity field and the macroscopic average of the nematic liquid crystal orientation field, respectively. The shear viscosity coefficient $\mu$ and the bulk viscosity coefficient $\lambda$ satisfy the physical conditions: $\mu >0$, $2\mu +3 \lambda >0$. The $P(\rho )$ is the pressure, which satisfies the so-called $\gamma$-law:

$$\begin{aligned} P(\rho )=a\rho ^{\gamma } \quad with \quad a>0, \quad \gamma >1, \end{aligned}$$

(1.2)

where a is a physical constant and $\gamma$ is the adiabatic exponent.

When $\nabla d=0$, the system (1.1) simplifies to the stationary Navier–Stokes equations, and significant progress has been made in studying the Liouville-type problems associated with these equations.

For the stationary incompressible Navier–Stokes equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} u \cdot \nabla u-\Delta u+\nabla P =0,\\ \text {div}u=0. \end{array}\right. } \end{aligned}$$

(1.3)

A well-known result on the Liouville theorem is presented in G.Galdi’s book [9]. Galdi established that if $u\in L^{\tfrac{9}{2}}(\mathbb {R}^{3})$, then it holds that $u=0$. Later, Chae showed an interesting result in [3], stating that the condition $\Delta u\in L^{\tfrac{6}{5}}(\mathbb {R}^{3})$ is sufficient for $u=0$ in $\mathbb {R}^{3}$. Furthermore, G.Seregin [18] provided an improved condition $u\in L^{6}(\mathbb {R}^{3})\cap BMO^{-1}$. Recently, Chae-Wolf [4] building upon a refined Caccioppoli-type inequality, achieved a logarithmic enhancement of the Liouville result, under the assumption that:

$$\begin{aligned} N(u):={\mathop {\int }\limits _{\mathbb {R}^{3}}}|u|^{\tfrac{9}{2}}\{ln(2+1/|u|)\}^{-1}dx <\infty . \end{aligned}$$

There exist numerous significant results concerning Liouville-type results in the study of incompressible fluids, we can refer to [5, 11, 18, 21] and the references therein.

For stationary compressible Navier–Stokes equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {div}(\rho u)=0,\\ \text {div}(\rho u\otimes u)-\mu \Delta u-(\lambda +\mu )\nabla \text {div}u+\nabla P(\rho )=0,\\ P(\rho )=a\rho ^{\gamma },\quad a>0, \gamma >1. \end{array}\right. } \end{aligned}$$

(1.4)

Chae [2] was the first to establish Liouville-type results for stationary solutions in the context of the compressible Navier–Stokes equations in $\mathbb {R}^{d}, d\ge 2$. He demonstrated that if the smooth solution $(\rho , u)$ satisfies:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \rho \Vert _{{L^{\infty }(\mathbb {R}^{d})}}+\Vert \nabla u\Vert _{{L^{2}(\mathbb {R}^{d})}}+\Vert u\Vert _{L^{_{\tfrac{d}{d-1}}}(\mathbb {R}^{d})}<\infty ,\quad when\quad d\le 6,\\ \Vert \rho \Vert _{{L^{\infty }(\mathbb {R}^{d})}}+\Vert \nabla u\Vert _{{L^{2}(\mathbb {R}^{d})}}+\Vert u\Vert _{L^{_{\tfrac{d}{d-1}}}(\mathbb {R}^{d})}+\Vert u\Vert _{L^{_{\tfrac{3d}{d-1}}}(\mathbb {R}^{d})}<\infty , \quad when\quad d\le 7, \end{array}\right. } \end{aligned}$$

then the system (1.4) only have a trivial solution that $u=0$, $\rho =constant$. Later, Li and Yu in their work [14] got the same conclusion under the condition

$$\begin{aligned} \Vert \rho \Vert _{{L^{\infty }(\mathbb {R}^{d})}}+\Vert u\Vert _{L^{_{\tfrac{d}{d-1}}}(\mathbb {R}^{d})}+\Vert u\Vert _{L^{_{\tfrac{3d}{d-1}}}(\mathbb {R}^{d})}<\infty ,\quad d \ge 2. \end{aligned}$$

Owing to $L^{\tfrac{9}{2}}(\mathbb {R}^{3})\hookrightarrow L^{\tfrac{9}{2},\infty }(\mathbb {R}^{3})$, Zhong in [22] used the Lorentz space properties to improve the result of Li and Yu [14], here he assumed that

$$\begin{aligned} \Vert \rho \Vert _{{L^{\infty }(\mathbb {R}^{3})}}+\Vert \nabla u\Vert _{{L^{2}(\mathbb {R}^{3})}}+\Vert u\Vert _{L^{\tfrac{9}{2},\infty }(\mathbb {R}^{3})}<\infty . \end{aligned}$$

However, to further weaken the condition $u\in L^{\tfrac{9}{2}}(\mathbb {R}^{3})$, Li and Niu [15] proved a Liouville-type theorem by deriving an a priori estimate that

$$\begin{aligned}{} & {} \mu {\mathop {\int }\limits _{|x|\le R}}|\nabla u|^{2}dx+\frac{(\lambda +\mu )}{2}{\mathop {\int }\limits _{|x|\le R}}|\text {div}u|^{2}dx\\ {}{} & {} \le C(R^{\tfrac{1}{2}-\tfrac{3}{p}}\Vert u\Vert _{L^{p,q}({R\le |x|\le 2R})}+R^{1-\tfrac{6}{p}}\Vert u\Vert ^{2}_{L^{p,q}({R\le |x|\le 2R})}+R^{2-\tfrac{9}{p}}\Vert u\Vert ^{3}_{L^{p,q}({R\le |x|\le 2R})}). \end{aligned}$$

under the hypotheses

$$\begin{aligned} \Vert \rho \Vert _{{L^{\infty }(\mathbb {R}^{3})}}+\Vert \nabla u\Vert _{{L^{2}(\mathbb {R}^{3})}}+\Vert u\Vert _{L^{p,q}(\mathbb {R}^{3})}<\infty , \end{aligned}$$

where $3<p<\frac{9}{2}$, $3\le q\le \infty$ or $p=q=3$. In the case of $p\ge \frac{9}{2}$, additional conditions are required:

$$\begin{aligned} \liminf _{R\rightarrow \infty }R^{\tfrac{2}{3}-\tfrac{3}{p}}\Vert u\Vert _{L^{p,q}({R\le |x|\le 2R})}<\infty , \;\liminf _{R\rightarrow \infty } R^{2-\tfrac{9}{p}}\Vert u\Vert ^{3}_{L^{p,q}({R\le |x|\le 2R})}<\delta {\mathop {\int }\limits _{\mathbb {R}^{3}}}|\nabla u|^{2}dx, \end{aligned}$$

where $\delta$ is a sufficiently small positive constant. These conditions are derived from the ideas presented in the works [11] and [19].

Now, we study the Liouville-type theorems for compressible nematic liquid crystal equations in $\mathbb {R}^{3}$, which have limited existing results and can be regarded as the Navier–Stokes system with the external force $-\Delta d\cdot \nabla d$. When $u=0$, (1.1)$_{3}$ represents the heat flow of the harmonic map, we present the following results:

Theorem 1.1

Let $(\rho , u, d)$ be a smooth solution to (1.1)–(1.2) with $\rho \in L^{\infty }({\mathbb {R}^{3}})$ and $(u, \nabla d)\in L^{\tfrac{9}{2},q}({\mathbb {R}^{3}})(0<q<\infty )$. Then, $u=0$, $\rho =constant$, and d satisfies the harmonic map equation of

$$\begin{aligned} \Delta d+|\nabla d|^{2}d=0. \end{aligned}$$

(1.5)

Theorem 1.2

Let d: $\mathbb {R}^{3}\rightarrow \mathbb {S}^{2}$ be a solution of (1.5), and $\nabla d\in L^{p}(2\le p\le 3)$, then d is a constant map.

Remark 1.3

[11] Based on Lebesgue’s dominated convergence theorem, we know when $h\in L^{\tfrac{9}{2},^{q}}$, then $\lim _{R\rightarrow \infty }\Vert h\Vert _{L^{{\tfrac{9}{2}},^{q}}({R\le |x|\le 2R})}=0(0<q<\infty )$, but we do not know whether

$$\begin{aligned} \lim _{R\rightarrow \infty }\Vert h\Vert _{L^{{\tfrac{9}{2}},^{\infty }}({R\le |x|\le 2R})}=0. \end{aligned}$$

Such as $h\in C^{\infty }(\mathbb {R}^{3})$ and behaves like

$$\begin{aligned} |h(x)|=O(|x|^{-2/3}),\; as |x|\rightarrow \infty . \end{aligned}$$

Combining the definition of weak norm, we get that $h(x)\in L^{{\tfrac{9}{2}},^{\infty }}$, but

$$\begin{aligned} \begin{aligned} \Vert h(x)\Vert _{L^{{\tfrac{9}{2}},^{\infty }}({R\le |x|\le 2R})}&=\sup _{\alpha>0} \alpha |\{|h(x)|>\alpha ,x\in [R,2R]\}|^{2/9}\\&\ge O(1)|h(2R)|\cdot ((2R)^{3})^{2/9}\\&=O(1). \end{aligned} \end{aligned}$$

Remark 1.4

When studying the harmonic mapping from an open subset of $\mathbb {R}^{3}$ into a sphere $\mathbb {S}^{2}$, various results have been obtained under certain geometric conditions. For instance, Choi [6] demonstrated that if the image of the harmonic map d lies in a hemisphere, then d is a constant map. Additionally, Jin [17] proved that when $\nabla d$ decays rapidly with respect to spatial variations, the Liouville theorem for d can be derived. Taking inspiration from their work, we speculate that if d decays rapidly at infinity, a Liouville theorem for d may also be established. Therefore, by combining the techniques proposed in [7, 16, 20] for minimizing the energy functional of harmonic mappings, we can obtain the Liouville theorem for d under the condition $\nabla d\in L^{p}(2\le p\le 3)$.

Remark 1.5

When d is a constant vector, Theorem 1.1 can be viewed as an extension of the works presented in [15] and [22].

The rest of this paper is organized as follows: In Sect. 2, we prepare some elementary facts, which are important for the proof. Finally, we will give the proofs of Theorem 1.1 and Theorem 1.2.

2 Preliminaries

We will recall some definitions and lemmas that will be used later.

Definition 2.1

Given $1\le p<\infty$, $1\le q<\infty$, the Lorentz space $L^{p,q}({\mathbb {R}^{3}})$ consists of all measurable functions f for which the quantity

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{L^{p,q}(\mathbb {R}^{3})}:=\left\{ \begin{array}{lr} ({\mathop {\int }\limits _{0}^{\infty }}t^{q-1}|\{x\in {\mathbb {R}^{3}}:|f(x)|>t\}|^{\tfrac{q}{p}}dt )^{\tfrac{1}{q}}, &{} q<+\infty ,\\ sup_{t>0}t|\{x\in {\mathbb {R}^{3}}:|f(x)|>t\}|^{\tfrac{1}{p}} , &{} q=+\infty , \end{array} \right. \end{aligned} \end{aligned}$$

is finite.

Lemma 2.1

[10, 13] Let $1<p<\infty$ and $1\le s\le \infty$ with $\frac{1}{p'}+\frac{1}{p}=1$ and $\frac{1}{s'}+\frac{1}{s}=1$. Then, pointwise multiplication is a bounded bilinear operator:

• (i) from $L^{p,s}(\mathbb {R}^{3})\times L^{\infty }(\mathbb {R}^{3})$ to $L^{p,s}(\mathbb {R}^{3})$;

• (ii) from $L^{p,s}(\mathbb {R}^{3})\times L^{p',s'}(\mathbb {R}^{3})$ to $L^{1}(\mathbb {R}^{3})$;

• (iii) from $L^{p_{1},s_{1}}(\mathbb {R}^{3})\times L^{p_{2},s_{2}}(\mathbb {R}^{3})$ to $L^{p,s}(\mathbb {R}^{3})$ for $1<p_{1},p_{2}<\infty$ with $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $1\le s_{1},s_{2}\le \infty$ with $s=min\{s_{1},s_{2}\}$.

Lemma 2.2

[11] Let $1\le r<\infty$, $1\le s\le \infty$ and $f\in L^{r,s}(\mathbb {R}^{3})$. For a parameter $R>0$, we put $f_{R}(x)=f({\frac{x}{R}})$. Then, it holds that $\Vert f_R\Vert _{L^{r,s}}=R^{\tfrac{3}{r}}\Vert f\Vert _{L^{r,s}}$.

Lemma 2.3

[14] 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 $1\le r_{1}, r_{2}<\infty$. Assuming that $P-p_{1}-p_{2}$ is weakly harmonic, that is $\Delta (P-p_{1}-p_{2})=0$ in the sense of distribution, there is a constant c such that $P-p_{1}-p_{2}=c$, for a.e. $x\in \mathbb {R}^{3}$.

If more $P(x)\ge 0$ a.e., then we get $c\ge 0$ too.

Lemma 2.4

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

$$\begin{aligned} \Vert \nabla ^{2}(-\Delta )^{-1}f\Vert _{L^{p,q}(\Omega )}\le C\Vert f\Vert _{L^{p,q}(\Omega )}, \end{aligned}$$

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

3 The proof of Theorem 1.1

Let $\phi \in C_{c}^{\infty }(\mathbb {R}^{3})$ be a radial cutoff function such that

$$\begin{aligned} \begin{aligned} \phi (|x|) \left\{ \begin{array}{lr} =1, &{} |x|<1,\\ =0, &{}|x|\ge 2.\\ \end{array} \right. \end{aligned} \end{aligned}$$

For each $R>0$, let

$$\begin{aligned} \phi _{R}(|x|)\triangleq \phi (\frac{|x|}{R}). \end{aligned}$$

(3.1)

Moreover, there exists a constant C independent on R and $k=0,1,2,3$ on $\mathbb {R}^{3}$ such that

$$\begin{aligned} |\nabla ^{k}\phi _{R}|\le \frac{C}{R^{k}}. \end{aligned}$$

(3.2)

Firstly, taking the inner product of (1.1)$_{2}$ with $u\phi ^{2}_{R}$ and integrating by parts over $\mathbb {R}^{3}$, we get

$$\begin{aligned} \begin{aligned}&\mu {\mathop {\int }\limits _{\mathbb {R}^{3}}}|\nabla u|^{2}\phi ^{2}_{R}dx+(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}|\text {div}u|^{2}\phi ^{2}_{R}dx\\ =&\mu {\mathop {\int }\limits _{\mathbb {R}^{3}}}|u|^{2}(\phi _{R}\Delta \phi _{R}+|\nabla \phi ^{2}_{R}|)dx-2(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}\text {div}u(u\cdot \nabla \phi _{R} \cdot \phi _{R})dx\\&-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\nabla P\cdot u\phi ^{2}_{R}dx-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\rho (u\cdot \nabla u)\phi ^{2}_{R}dx-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\Delta d\cdot \nabla d\cdot u\phi ^{2}_{R}dx. \end{aligned} \end{aligned}$$

(3.3)

Similarly, taking the inner product of (1.1)$_{3}$ with $(\Delta d+|\nabla d|^{2}d)\phi ^{2}_{R}$, we have

$$\begin{aligned} \begin{aligned} {\mathop {\int }\limits _{\mathbb {R}^{3}}}|\Delta d+|\nabla d|^{2}d|^{2}\phi ^{2}_{R}dx={\mathop {\int }\limits _{\mathbb {R}^{3}}}(u\cdot \nabla d)(\Delta d+|\nabla d|^{2}d)\phi ^{2}_{R}dx. \end{aligned} \end{aligned}$$

(3.4)

Combining (3.3) and (3.4) yields

$$\begin{aligned} \begin{aligned}&\mu {\mathop {\int }\limits _{\mathbb {R}^{3}}}|\nabla u|^{2}\phi ^{2}_{R}dx+(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}|\text {div}u|^{2}\phi ^{2}_{R}dx+{\mathop {\int }\limits _{\mathbb {R}^{3}}}|\Delta d+|\nabla d|^{2}d|^{2}\phi ^{2}_{R}dx\\&\quad =\,\mu {\mathop {\int }\limits _{\mathbb {R}^{3}}}|u|^{2}(\phi _{R}\Delta \phi _{R}+|\nabla \phi ^{2}_{R}|)dx-2(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}\text {div}u(u\cdot \nabla \phi _{R} \cdot \phi _{R})dx\\&\qquad -{\mathop {\int }\limits _{\mathbb {R}^{3}}}\nabla P\cdot u\phi ^{2}_{R}dx-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\rho (u\cdot \nabla u)u\phi ^{2}_{R}dx-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\Delta d\cdot \nabla d\cdot u\phi ^{2}_{R}dx\\&\qquad +{\mathop {\int }\limits _{\mathbb {R}^{3}}}(u\cdot \nabla d)(\Delta d+|\nabla d|^{2}d)\phi ^{2}_{R}dx\\&\quad \triangleq \sum _{i=1}^{6}J_{i}(t). \end{aligned} \end{aligned}$$

(3.5)

Secondly, we will estimate $J_{i}$ one by one. It follows from Lemma 2.1, Lemma 2.2 and (3.2) that

$$\begin{aligned} \begin{aligned} J_{1}\le&\,C(\mu ){\mathop {\int }\limits _{R\le |x|\le 2R}}|u|^{2}(|\phi _{R}\Delta \phi _{R}|+|\nabla \phi ^{2}_{R}|)dx\\ \le ~&CR^{-2}\Vert |u|^{2}\Vert _{L^{\tfrac{9}{4},{\tfrac{q}{2}}}({R\le |x|\le 2R})}\Vert 1\Vert _{L^{\tfrac{9}{5},{\tfrac{q}{q-2}}}({R\le |x|\le 2R})}\\ \le ~&CR^{-2+\tfrac{5}{3}}\Vert u\Vert ^{2}_{L^{\tfrac{9}{2},^{q}}({R\le |x|\le 2R})}\\ \le ~&CR^{-\tfrac{1}{3}}\Vert u\Vert ^{2}_{L^{{\tfrac{9}{2}},^{q}}({R\le |x|\le 2R})}\rightarrow 0 \quad as\quad R\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.6)

The estimate of $J_{2}$ is similar to that of $J_{1}$ and we get

$$\begin{aligned} \begin{aligned} J_{2}=&-2(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}\text {div} u(u\cdot \nabla \phi _{R}\cdot \phi _{R})dx\\&\le C(\lambda +\mu )R^{-1}{\mathop {\int }\limits _{R\le |x|\le 2R}}|\phi _{R}\text {div}u||u|dx\\&\le C(\lambda +\mu )R^{-1}\Vert u\Vert _{L^{\tfrac{9}{2},^{q}}({R\le |x|\le 2R})}\Vert \phi _{R}\text {div}u\Vert _{L^{2,q}({R\le |x|\le 2R})}\Vert 1\Vert _{L^{\tfrac{18}{5},\tfrac{q}{q-2}}({R\le |x|\le 2R})}\\&\le \frac{(\lambda +\mu )}{4}\Vert \phi _{R}\text {div}u\Vert ^{2}_{L^{2,q}({R\le |x|\le 2R})}+CR^{-\tfrac{1}{3}}\Vert u\Vert ^{2}_{L^{\tfrac{9}{2},^{q}}({R\le |x|\le 2R})}\\&\rightarrow \frac{(\lambda +\mu )}{4}\Vert \phi _{R}\text {div}u\Vert ^{2}_{L^{2,q}({R\le |x|\le 2R})} \quad as\quad R\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.7)

As for $J_{3}$, we incorporate some ideas from [14]. Taking divergence on both sides of (1.1)$_{2}$, we get

$$\begin{aligned} \Delta (P-p_{1}-p_{2})=0, \end{aligned}$$

where

$$\begin{aligned} p_{1}\triangleq (-\Delta )^{-1}\partial _{i}\partial _{j}(\rho u_{i}u_{j}),\quad p_{2}\triangleq (\lambda +2\mu )\text {div} u-\partial _{i}d\cdot \partial _{j}d. \end{aligned}$$

(3.8)

According to Lemma 2.3, there exists a nonnegative constant c such that

$$\begin{aligned} a\rho ^{\gamma }=P=c+p_{1}+p_{2},\quad a.e \quad x\in \mathbb {R}^{3}. \end{aligned}$$

Setting

$$\begin{aligned} P_{1}\triangleq \rho ^{\gamma -1}-(\frac{c}{a})^{\tfrac{\gamma -1}{\gamma }}=(\frac{c+p_{1}+p_{2}}{a})^{\tfrac{\gamma -1}{\gamma }}-(\frac{c}{a})^{\tfrac{\gamma -1}{\gamma }}. \end{aligned}$$

(3.9)

By combining (1.2) and (3.9), we can deduce that

$$\begin{aligned} \nabla P=\nabla (a\rho ^{\gamma })={\frac{a\gamma }{\gamma -1}}\rho \nabla (\rho ^{\gamma -1})={\frac{a\gamma }{\gamma -1}}\rho \nabla P_{1}, \end{aligned}$$

and

$$\begin{aligned} |P_{1}\rho |\le C(a,\Vert \rho \Vert _{L^{\infty }})(|p_{1}|+|p_{2}|). \end{aligned}$$

(3.10)

Therefore, by applying integration by parts and combining (1.1)$_{1}$ and (3.10), we can conclude that

$$\begin{aligned} \begin{aligned} J_{3}=&\,-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\nabla P\cdot u\phi ^{2}_{R}dx\\ =~&\,-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\frac{a\gamma }{\gamma -1}\rho \nabla P_{1}\cdot u\phi ^{2}_{R}dx\\ =~&\,2\frac{a\gamma }{\gamma -1}{\mathop {\int }\limits _{\mathbb {R}^{3}}}P_{1}\rho u\cdot \phi _{R} \nabla \phi _{R}dx\\ \le ~&\,C(a,\gamma )R^{-1}{\mathop {\int }\limits _{\mathbb {R}^{3}}}|P_{1}\rho ||u||\phi _{R}|dx\\ \le ~&\,C(a,\gamma ,\Vert \rho \Vert _{L^{\infty }})R^{-1}{\mathop {\int }\limits _{\mathbb {R}^{3}}}(|p_{1}||u|+|p_{2}||u|)|\phi _{R}|dx\\ \triangleq&I_{1}+I_{2}. \end{aligned} \end{aligned}$$

(3.11)

By $\rho \in L^{\infty }$, Lemma 2.1, Lemma 2.2 and Lemma 2.4, we get that

$$\begin{aligned} \begin{aligned} I_{1}\le&\,C(a,\gamma ,\Vert \rho \Vert _{L^{\infty }})R^{-1}\Vert p_{1}\Vert _{L^{\tfrac{9}{4}, \tfrac{q}{2}}(R\le |x|\le 2R)}\Vert u\Vert _{L^{\tfrac{9}{2},^q}(R\le |x|\le 2R)}\Vert 1\Vert _{L^{3,\tfrac{q}{q-3}}(R\le |x|\le 2R)}\\ \le&\,C\Vert (-\Delta )^{-1}\partial _{i}\partial _{j}\rho u_{i}u_{j}\Vert _{L^{\tfrac{9}{4},\tfrac{q}{2}}(R\le |x|\le 2R)}\Vert u\Vert _{L^{\tfrac{9}{2}, ^q}(R\le |x|\le 2R)}\\ \le&\,C\Vert u\Vert _{L^{\tfrac{9}{2},^q}(R\le |x|\le 2R)}^{3}\rightarrow 0 \quad as\quad R\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.12)

and

$$\begin{aligned} I_{2}\le&{} C(a,\gamma ,\Vert \rho \Vert _{L^{\infty }})R^{-1}{\mathop {\int }\limits _{\mathbb {R}^{3}}}(\lambda +2\mu )|\text{ div } u||u||\phi _{R}|+|\partial _{i}d\cdot \partial _{j}d||u||\phi _{R}|dx\nonumber \\ \le&{} C(a,\gamma ,\mu ,\Vert \rho \Vert _{L^{\infty }})R^{-1}{\mathop {\int }\limits _{({R\le |x|\le 2R})}}(|\phi _{R}\text{ div } u||u|+|\nabla d|^{2}|u|)dx\nonumber \\ \le&{} CR^{-1}\Vert \phi _{R}\text{ div } u\Vert _{L^{2},^q(R\le |x|\le 2R)}\Vert u\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\Vert 1\Vert _{L^{\tfrac{18}{5},\tfrac{q}{q-2}}({R\le |x|\le 2R})}\nonumber \\{}&{} +CR^{-1}\Vert \nabla d\Vert _{L^{\tfrac{9}{2}, ^q}(R\le |x|\le 2R)}^{2}\Vert u\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\Vert 1\Vert _{L^{3}, \tfrac{q}{q-3}({R\le |x|\le 2R})}\nonumber \\ \le&{} CR^{-\tfrac{1}{6}}\Vert \phi _{R}\text{ div } u\Vert _{L^{2},^q({R\le |x|\le 2R})}\Vert u\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\nonumber \\{}&{} +C\Vert \nabla d\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}^{2}\Vert u\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\nonumber \\ \le&{} \frac{(\lambda +\mu )}{4}\Vert \phi _{R}\text{ div }u\Vert ^{2}_{L^{2,q}({R\le |x|\le 2R})}+CR^{-\tfrac{1}{3}}\Vert u\Vert ^{2}_{L^{\tfrac{9}{2},^q}(R\le |x|\le 2R)}\nonumber \\{}&{} +C\Vert \nabla d\Vert _{L^{\tfrac{9}{2}, ^q}({R\le |x|\le 2R})}^{2}\Vert u\Vert _{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\nonumber \\ \rightarrow {}&{} \frac{(\lambda +\mu )}{4}\Vert \phi _{R}\text{ div }u\Vert ^{2}_{L^{2,q}({R\le |x|\le 2R})} \quad as\quad R\rightarrow \infty . \end{aligned}$$

(3.13)

As for $J_{4}$, integrating by parts and utilizing (1.1)$_{1}$, Lemma 2.1 and Lemma 2.3, we get the following results that

$$\begin{aligned} \begin{aligned} J_{4}=&-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\rho u\cdot \nabla u\cdot u \phi ^{2}_{R}dx\\ =&-\frac{1}{2}{\mathop {\int }\limits _{\mathbb {R}^{3}}}\rho u\cdot \nabla |u|^{2}\cdot \phi ^{2}_{R}dx\\ =&\frac{1}{2}{\mathop {\int }\limits _{\mathbb {R}^{3}}}\text {div}(\rho u)|u|^{2}\cdot \phi ^{2}_{R}+\rho |u|^{3}\cdot \nabla (\phi ^{2}_{R})dx\\ =&{\mathop {\int }\limits _{\mathbb {R}^{3}}}\rho |u|^{3}\cdot \phi _{R}\nabla \phi _{R}dx\\ \le&CR^{-1}\Vert |u|^{3}\Vert _{L^{\tfrac{3}{2},\tfrac{q}{3}}({R\le |x|\le 2R})}\Vert 1\Vert _{L^{3,\tfrac{q}{q-3}}({R\le |x|\le 2R})}\\ \le&C\Vert u\Vert ^{3}_{L^{\tfrac{9}{2},^q}({R\le |x|\le 2R})}\rightarrow 0 \quad as\quad R\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.14)

For the estimation of $J_{5}+J_{6}$, direct computation yields

$$\begin{aligned} \begin{aligned} J_{5}+J_{6}&=-{\mathop {\int }\limits _{\mathbb {R}^{3}}}\Delta d\cdot \nabla d\cdot u\phi ^{2}_{R}dx+{\mathop {\int }\limits _{\mathbb {R}^{3}}}(u\cdot \nabla d)(\Delta d+|\nabla d|^{2}d)\phi ^{2}_{R}dx\\&={\mathop {\int }\limits _{\mathbb {R}^{3}}}u\cdot \nabla d\cdot d|\nabla d|^{2}d\phi ^{2}_{R}dx\\&=\frac{1}{2}{\mathop {\int }\limits _{\mathbb {R}^{3}}}u\cdot \nabla |d|^{2}|\nabla d|^{2}d\phi ^{2}_{R}dx\\&=0, \end{aligned} \end{aligned}$$

(3.15)

where we have used $|d|=1$. Inserting (3.6), (3.7), (3.11), (3.12), (3.13), (3.14) and (3.15) into (3.5), we deduce that

$$\begin{aligned} \mu {\mathop {\int }\limits _{\mathbb {R}^{3}}}|\nabla u|^{2}dx+(\lambda +\mu ){\mathop {\int }\limits _{\mathbb {R}^{3}}}|\text {div}u|^{2}dx+{\mathop {\int }\limits _{\mathbb {R}^{3}}}|\Delta d+|\nabla d|^{2}d|^{2}dx=0. \end{aligned}$$

Since $\mu >0$, $2\mu +3\lambda >0$ and taking $\Vert u\Vert _{L^{6}}\le \Vert \nabla u\Vert _{L^{2}}$ into consideration, yields

$$\begin{aligned} u=0,\quad \Delta d+|\nabla d|^{2}d=0. \end{aligned}$$

Now, based on (1.1)$_{2}$ and $|d|=1$, the pressure P satisfies

$$\begin{aligned} \begin{aligned} \nabla P&=-\Delta d\nabla d=|\nabla d|^{2}d\nabla d\\&=\frac{1}{2}|\nabla d|^{2}\nabla |d|^{2}\\&=0. \end{aligned} \end{aligned}$$

Thus, we have showed that $\rho =constant$ and finished the proof of Theorem 1.1.

4 The proof of Theorem 1.2

By [7], we know that a function $d\in H^{1}(\mathbb {R}^{3};\mathbb {S}^{2})$ is a weakly harmonic mapping of $\mathbb {R}^{3}$ into $\mathbb {S}^{2}$ provided:

$$\begin{aligned} -\Delta d=|\nabla d|^{2}d \quad in \quad \mathbb {R}^{3}, \end{aligned}$$

(4.1)

now the solution d of this Euler–Lagrange equation satisfies the minima of the energy function

$$\begin{aligned} E(d)=\int |\nabla d|^{2}dy. \end{aligned}$$

(4.2)

This system (4.1) is to hold in the weak sense, that is,

$$\begin{aligned} {\mathop {\int }\limits _{\mathbb {R}^{3}}}\nabla d: \nabla wdx={\mathop {\int }\limits _{\mathbb {R}^{3}}}|\nabla d|^{2}d \cdot wdx, \end{aligned}$$

(4.3)

for each test function $w\in H^{1}(\mathbb {R}^{3})\cap L^{\infty }(\mathbb {R}^{3})$ having compact support, which shows that d is stationary with respect to variations of the target $\mathbb {S}^{2}$. So the stationary mapping d satisfies the monotonicity inequalities

$$\begin{aligned} \frac{1}{r}{\mathop {\int }\limits _{B(x,r)}}|\nabla d|^{2}dy\le \frac{1}{R}{\mathop {\int }\limits _{B(x,R)}}|\nabla d|^{2}dy, \end{aligned}$$

(4.4)

for all concentric balls $B(x,r)\subset B(x,R)\subset \mathbb {R}^{3}$, which was apparently first proved by Price [16]. By (4.4), thus we have

$$\begin{aligned} (\frac{1}{r}-\frac{1}{R}){\mathop {\int }\limits _{B(x,r)}}|\nabla d|^{2}dy\le \frac{1}{R}{\mathop {\int }\limits _{B_{R}\backslash B_{r}}}|\nabla d|^{2}dy. \end{aligned}$$

(4.5)

Now we observe that by $\nabla d\in L^{p}$ and $H\ddot{o}lder$ inequality

$$\begin{aligned} \frac{1}{R}{\mathop {\int }\limits _{B_{R}\backslash B_{r}}}|\nabla d|^{2}dy\le \frac{1}{R}\left( ({\mathop {\int }\limits _{B_{R}\backslash B_{r}}}|\nabla d|^{p}dy)^{\tfrac{2}{p}}\cdot (R^{3}-r^{3})^{\tfrac{2}{q}}\right) \le CR^{\tfrac{6}{q}-1}({\mathop {\int }\limits _{B_{R}\backslash B_{r}}}|\nabla d|^{p}dy)^{\tfrac{2}{p}}, \end{aligned}$$

(4.6)

where p, q are positive values satisfying $\frac{2}{p}+\frac{2}{q}=1$ and $\frac{6}{q}-1\le 0$. Now, we can establish the following result:

$$\begin{aligned} 6\le q,\;2 \le p\le 3. \end{aligned}$$

When $R\rightarrow \infty$, together with (4.5) and (4.6), which leads to

$$\begin{aligned} \frac{1}{r}{\mathop {\int }\limits _{B(x,r)}}|\nabla d|^{2}dy\le C({\mathop {\int }\limits _{\mathbb {R}^{3}\backslash B_{r}}}|\nabla d|^{p}dy)^{\tfrac{2}{p}}. \end{aligned}$$

Because of the monotonicity of u, implies that

$$\begin{aligned} \frac{1}{r}{\mathop {\int }\limits _{B(x,r)}}|\nabla d|^{2}dy\le \lim _{r\rightarrow \infty }\frac{1}{r}{\mathop {\int }\limits _{B(x,r)}}|\nabla d|^{2}dy\le \lim _{r\rightarrow \infty }C({\mathop {\int }\limits _{\mathbb {R}^{3}\backslash B_{r}}}|\nabla d|^{p}dy)^{\tfrac{2}{p}}=0. \end{aligned}$$

So $\nabla d\equiv 0$ and d is a constant map. This completes the proof of Theorem 1.2.