Liouville Theorem for 3D Steady Q-Tensor System of Liquid Crystal in Mixed Lorentz Spaces

Mengru WAN; Xuemei DENG; Qunyi BIE

doi:10.1051/wujns/2025303235

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Volume 30 / No 3 (June 2025)

Wuhan Univ. J. Nat. Sci., 30 3 (2025) 235-240

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Issue		Wuhan Univ. J. Nat. Sci. Volume 30, Number 3, June 2025


Page(s)		235 - 240
DOI		https://doi.org/10.1051/wujns/2025303235
Published online		16 July 2025

Wuhan University Journal of Natural Sciences, 2025, Vol.30 No.3, 235-240

Mathematics

CLC number: O175

Liouville Theorem for 3D Steady Q-Tensor System of Liquid Crystal in Mixed Lorentz Spaces

三维稳态Q-tensor液晶流系统在混合Lorentz空间中的Liouville型定理

Mengru WAN (万梦茹)¹, Xuemei DENG (邓雪梅)¹^,2^† and Qunyi BIE (别群益)¹^,2

¹ College of Science, China Three Gorges University, Yichang 443002, Huibei, China
² Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, Huibei, China

^† Corresponding author. E-mail: dxmeisx@126.com

Received: 25 June 2024

Abstract

In this paper, we study Liouville theorem for 3D steady $Q$ -tensor system of liquid crystal in mixed Lorentz spaces. We obtain $u = 0, Q = 0$ on the conditions that $u \in L_{x_{1}}^{p, \infty} L_{x_{2}}^{q, \infty} L_{x_{3}}^{r, \infty} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3}), Q \in H^{2} (R^{3}), p, q,$ $r \in (3, \infty],$ and $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} \geq \frac{2}{3},$ which extends some known results.

摘要

本文在混合Lorentz空间中研究了三维稳态 $Q$ -tensor液晶系统的Liouville定理。我们在条件 $u \in L_{x_{1}}^{p, \infty} L_{x_{2}}^{q, \infty} L_{x_{3}}^{r, \infty} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3}), Q \in H^{2} (R^{3}), p, q, r \in (3, \infty]$ 和 $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} \geq \frac{2}{3}$ 下得到 $u = 0, Q = 0$ 这一结论，它推广了一些已有的结果。

Key words: mixed Lorentz spaces / Q-tensor system of liquid crystal / Liouville theorem

关键字 : 混合Lorentz空间 / Q-tensor液晶流系统 / Liouville定理

Cite this article: WAN Mengru, DENG Xuemei, BIE Qunyi. Liouville Theorem for 3D Steady Q-Tensor System of Liquid Crystal in Mixed Lorentz Spaces[J]. Wuhan Univ J of Nat Sci, 2025, 30(3): 235-240.

Biography: WAN Mengru, female, Master candidate, research direction: partial differential equations. E-mail: W1254873919@126.com

Foundation item: Supported by the National Natural Science Foundation of China (11871305)

© Wuhan University 2025

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

In this paper, we study the following 3D stationary $Q$ -tensor system of liquid crystal:

${\begin{array}{l} u \cdot \nabla u - μ Δ u + \nabla P = - \nabla \cdot (\nabla Q ⊙ \nabla Q) - λ \nabla \cdot (| Q | H) + \nabla \cdot (Q Δ Q - Δ Q Q), \\ \nabla \cdot u = 0, \\ u \cdot \nabla Q + Q Ω - Ω Q - λ | Q | D = Γ H, \end{array}$ (1)

with

$H = Δ Q - a Q + b (Q^{2} - \frac{t r (Q^{2})}{3} I_{3 \times 3}) - c Q t r (Q^{2}) .$

Here $u \in R^{3}, P \in R$ and

$Q \in S_{0}^{3} ≜ {A = (a_{i j})_{3 \times 3} | a_{i j} = a_{j i}, t r (A) = 0}$

stand for the flow velocity, the scalar pressure and the nematic tensor order parameter, respectively. The parameters $μ > 0,$ $Γ^{- 1} > 0$ and $λ \in R$ represent the viscosity coefficient, the rotational viscosity and the nematic alignment, respectively. The coefficients $a, b, c \in R$ with $c > 0$ are constants. ${(\nabla Q ⊙ \nabla Q)}_{i, j} = \partial_{x_{i}} Q : \partial_{x_{j}} Q$ is the symmetric additional stress tensor. $D ≜ \frac{1}{2} (\nabla u + \nabla u^{T})$ , and $Ω ≜ \frac{1}{2} (\nabla u - \nabla u^{T})$ are the symmetric and skew symmetric, respectively, where the notation $T$ represents the transposition of a matrix.

When $Q = 0$ , system (1) reduces to the 3D stationary Navier-Stokes system

${\begin{array}{l} u \cdot \nabla u - Δ u + \nabla P = 0, \\ \nabla \cdot u = 0 . \end{array}$ (2)

For system (2), a well-known result on the Liouville theorem is given by Galdi^[1], which concludes if $u \in L^{\frac{9}{2}} (R^{3}),$ then $u = 0$ . Chae-Wölf^[2] gave the following logarithmic improvement

$\int_{R^{3}} | u (x) |^{\frac{9}{2}} {l o g (2 + \frac{1}{| u (x) |})}^{- 1} d x < \infty .$

Kozono-Terasawa-Wakasugi^[3] extended Galdi's result^[1] to the Lorentz spaces $L^{\frac{9}{2}, \infty} (R^{3})$ . Luo and Yin^[4] showed that if the solution to system (2) satisfies

$u_{i} \in L_{x_{1}}^{p_{i}} L_{x_{2}}^{q_{i}} L_{x_{3}}^{r_{i}} (R^{3}), p_{i}, q_{i}, r_{i} \in [1, \infty), \frac{1}{p_{i}} + \frac{1}{q_{i}} + \frac{1}{r_{i}} = \frac{2}{3} (i = 1,2, 3),$ (3)

then $u = 0$ , where

${‖ {‖ {‖ f ‖}_{L_{x_{1}}^{p}} ‖}_{L_{x_{2}}^{q}} ‖}_{L_{x_{3}}^{r}} ≜ {(\int_{R} {(\int_{R} {(\int_{R} | f |^{p} d x_{1})}^{\frac{q}{p}} d x_{2})}^{\frac{r}{q}} d x_{3})}^{\frac{1}{r}} .$

For more Liouville theorem results of system (2), one could refer to Refs. [5-9] and references therein.

In recent years, the $Q$ -tensor system of liquid crystal (1) has received much attention. However, there are few results on its Liouville theorem. Gong et al^[10] proved that if

$u \in L^{\frac{9}{2}} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3}), Q \in H^{2} (R^{3}), b^{2} - 24 a c \leq 0,$

then $u = 0, Q = 0$ . Later, Lai and Wu^[11] generalized the conditions to

$u \in L^{\frac{9}{2}, \infty} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3}), Q \in H^{2} (R^{3}), b^{2} - 24 a c \leq 0 .$ (4)

On Liouville theorem for many other models, there are also numerous results (see for example Refs. [12-15]).

Inspired by the works mentioned above in this paper, we will prove that there is only the trivial solution for the steady $Q$ -tensor system of liquid crystal model in mixed Lorentz spaces. First of all, we recall the definition of mixed Lorentz spaces.

Definition 1 Assume the indexes $p = (p_{1}, p_{2}, p_{3})$ and $q = (q_{1}, q_{2}, q_{3})$ satisfying $1 \leq p_{i} < \infty$ , $1 \leq q_{i} \leq \infty$ , or $p_{i} = q_{i} = \infty (i = 1,2, 3)$ . A mixed Lorentz space $L^{p_{1}, q_{1}} (R_{x_{1}}; L^{p_{2}, q_{2}} (R_{x_{2}}; L^{p_{3}, q_{3}} (R_{x_{3}})))$ is the set of functions for which the following norm is finite:

${‖ {‖ {‖ f ‖}_{L_{x_{1}}^{p_{1}, q_{1}}} ‖}_{L_{x_{2}}^{p_{2}, q_{2}}} ‖}_{L_{x_{3}}^{p_{3}, q_{3}}} ≜ {\begin{array}{l} {(\int_{0}^{\infty} {(\int_{0}^{\infty} {(\int_{0}^{\infty} {| t_{1}^{\frac{1}{p_{1}}} t_{2}^{\frac{1}{p_{2}}} t_{3}^{\frac{1}{p_{3}}} f (t_{1}, t_{2}, t_{3}) |}^{q_{1}} \frac{d t_{1}}{t_{1}})}^{\frac{q_{2}}{q_{1}}} \frac{d t_{2}}{t_{2}})}^{\frac{p_{3}}{p_{2}}} \frac{d t_{3}}{t_{3}})}^{\frac{1}{p_{3}}}, 1 \leq p_{i} < \infty, \\ \underset{t > 0}{s u p} {| {x_{3} : \underset{γ > 0}{s u p γ} {| {x_{2} : \underset{λ > 0}{s u p λ} {| {x_{1} : | f (x_{1}, x_{2}, x_{3}) | λ} |}^{\frac{1}{p_{1}}} γ} |}^{\frac{1}{p_{2}}} t} |}^{\frac{1}{p_{3}}}, p_{i} = \infty . \end{array}$

The main result of the paper is stated in the following theorem.

Theorem 1 Let $(u, Q)$ be a smooth solution to system (1) in $R^{3}$ . If

$u \in L_{x_{1}}^{p, \infty} L_{x_{2}}^{q, \infty} L_{x_{3}}^{r, \infty} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3}), Q \in H^{2} (R^{3}), p, q, r \in (3, \infty], \frac{1}{p} + \frac{1}{q} + \frac{1}{r} \geq \frac{2}{3}, a n d b^{2} - 24 a c \leq 0,$

then $u = 0, Q = 0$ .

Remark 1 In the case of $p = q = r = \frac{9}{2}$ in Theorem 1, the sufficient condition coincides with (4) obtained in previous work^[11]. In addition, due to the following embedding (see for example Ref. [16])

$L^{\frac{9}{2}} (R^{3}) \subset L^{\frac{9}{2}, q} (R^{3}) (\frac{9}{2} \leq q \leq \infty),$ (5)

we deduce that our result extends Gong et al 's result in Ref. [10].

When $Q = 0$ , we obtain the following Liouville theorem for the Navier-Stokes equations (2), which is a supplement to the result of Ref. [4] (see (2)).

Corollary 1 Let $u \in L_{x_{1}}^{p, \infty} L_{x_{2}}^{q, \infty} L_{x_{3}}^{r, \infty} (R^{3}) ⋂ {\dot{H}}^{1} (R^{3})$ be a smooth solution to the Navier-Stokes equations (2) in $R^{3}$ . If $p, q, r \in (3, \infty], \frac{1}{p} + \frac{1}{q} + \frac{1}{r} \geq \frac{2}{3},$ then $u = 0$ .

1 Proof of Theorem 1

In this section, we give the proof of Theorem 1. To streamline the presentation, set

$B_{1} ≜ {x \in R^{3} | x \in (R, 2 R)}, B_{2} ≜ {x_{i} \in R | R < | x_{i} | < 2 R, i = 1,2, 3},$

${‖ {‖ {‖ \cdot ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} ≜ {‖ {‖ {‖ \cdot ‖}_{L_{x_{1}}^{p, \infty} (R | x_{1} | 2 R)} ‖}_{L_{x_{2}}^{q, \infty} (R | x_{2} | 2 R)} ‖}_{L_{x_{3}}^{r, \infty} (R | x_{3} | 2 R)} .$

We firstly recall the following lemma.

Lemma 1^[17] Let $β (Q) = 1 - 6 \frac{[t r (Q^{3} {)]}^{2}}{| Q |^{6}}, Q \in S_{0}^{3},$ then $0 \leq β (Q) \leq 1$ .

Proof We consider a smooth cut-off function $ϕ \in C_{c}^{\infty} (R)$ such that

$ϕ (y) = {\begin{matrix} 1, | y | < 1, \\ 0, | y | \geq 2 . \end{matrix}$

For each $R > 0$ , defining

$ϕ_{R} (x) = ϕ (\frac{x_{1}}{R}) ϕ (\frac{x_{2}}{R}) ϕ (\frac{x_{3}}{R}), x = (x_{1}, x_{2}, x_{3}) \in R^{3},$

then, one has

$ϕ_{R} (x) = {\begin{array}{l} 1, | x | < R, \\ 0, | x | \geq 2 R . \end{array}$

Moreover, there is a constant $C$ independent of $R$ such that $| \nabla^{k} ϕ_{R} | \leq \frac{C}{R^{k}}$ for $k \in {0,1, 2,3} .$

Multiplying the first equation and the second equation of (1) by $u (x) ϕ_{R} (x)$ and $- H (x) ϕ_{R} (x),$ respectively, we know

$\begin{array}{l} μ \int_{ℝ^{3}} | \nabla u |^{2} ϕ_{R} (x) d x + Γ \int_{ℝ^{3}} | H |^{2} ϕ_{R} (x) d x \\ = \int_{ℝ^{3}} (u \cdot \nabla Q) : Δ Q ϕ_{R} (x) d x - \int_{ℝ^{3}} (Ω Q - Q Ω) : Δ Q ϕ_{R} (x) d x - \int_{ℝ^{3}} (u \cdot \nabla Q) : [a Q + b (Q^{2} - \frac{t r (Q^{2})}{3} I_{3 \times 3}) + c Q | Q |^{2}] ϕ_{R} (x) d x \\ + \int_{ℝ^{3}} (Ω Q - Q Ω) : [a Q + b (Q^{2} - \frac{t r (Q^{2})}{3} I_{3 \times 3}) + c Q | Q |^{2}] ϕ_{R} (x) d x - λ \int_{ℝ^{3}} | Q | D : H ϕ_{R} (x) d x - \int_{ℝ^{3}} \nabla \cdot (\nabla Q ⊙ \nabla Q) \cdot u ϕ_{R} (x) d x \\ + λ \int_{ℝ^{3}} | x | H : \nabla (u ϕ_{R} (x)) d x - \int_{ℝ^{3}} (Q Δ Q - Δ Q Q) : \nabla (u ϕ_{R} (x)) d x + \int_{R^{3}} \frac{1}{2} | u |^{2} u \cdot \nabla ϕ_{R} (x) d x + \int_{ℝ^{3}} P u \cdot \nabla ϕ_{R} (x) + \frac{μ}{2} | u |^{2} Δ ϕ_{R} (x) d x \\ ≜ \sum_{i = 1}^{10} I_{i} . \end{array}$ (6)

Firstly, since $Q$ is symmetric and $Ω$ is skew symmetric, we deduce $I_{4} = 0$ . For $I_{1} + I_{6}, I_{2} + I_{8}$ and $I_{5} + I_{7}$ , using Hölder inequality, we obtain

$\begin{array}{l} I_{1} + I_{6} = \int_{R^{3}} u \nabla Q : Δ Q ϕ_{R} (x) d x - \int_{ℝ^{3}} \nabla \cdot (\nabla Q ⊙ \nabla Q) \cdot u ϕ_{R} (x) d x \\ = \int_{ℝ^{3}} (u \cdot \nabla Q) : \nabla Q ϕ_{R} (x) d x - \int_{ℝ^{3}} (u \cdot \nabla Q) Δ Q ϕ_{R} (x) d x - \int_{R^{3}} u \cdot \nabla \frac{1}{2} {| \nabla Q |}^{2} ϕ_{R} (x) d x = \int_{R^{3}} \frac{1}{2} {| \nabla Q |}^{2} u \cdot \nabla ϕ_{R} (x) d x \end{array}$

$\begin{array}{l} \leq \frac{C}{R} {‖ {| \nabla Q |}^{2} ‖}_{L^{2, \infty} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} \\ \leq C R^{\frac{1}{2} - (\frac{1}{p} + \frac{1}{q} + \frac{1}{r})} {‖ {| \nabla Q |}^{2} ‖}_{L^{2} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})}, \end{array}$ (7)

$\begin{array}{l} I_{2} + I_{8} = - \int_{ℝ^{3}} (Ω Q - Q Ω) : Δ Q ϕ_{R} (x) d x - \int_{ℝ^{3}} \nabla \cdot (\nabla Q ⊙ \nabla Q) \cdot u ϕ_{R} (x) d x = - \int_{ℝ^{3}} (Q Δ Q - Δ Q Q) : u ⊙ \nabla ϕ_{R} (x) d x \\ \leq \frac{C}{R} {‖ \nabla^{2} Q ‖}_{L^{2, \infty} (B_{1})} {‖ Q ‖}_{L^{\infty} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} \\ \leq C R^{\frac{1}{2} - (\frac{1}{p} + \frac{1}{q} + \frac{1}{r})} {‖ \nabla^{2} Q ‖}_{L^{2} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})}, \end{array}$ (8)

$\begin{array}{l} I_{5} + I_{7} = - λ \int_{ℝ^{3}} | Q | D : H ϕ_{R} (x) d x + λ \int_{ℝ^{3}} | Q | H : \nabla (u ϕ_{R} (x)) d x = \int_{ℝ^{3}} | Q | H : u \otimes \nabla ϕ_{R} (x) d x \\ \leq \frac{C}{R} {‖ H ‖}_{L^{2} (B_{1})} {‖ Q ‖}_{L^{\infty} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} \\ \leq C R^{\frac{1}{2} - (\frac{1}{p} + \frac{1}{q} + \frac{1}{r})} {‖ H ‖}_{L^{2} (B_{1})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} . \end{array}$ (9)

For $I_{3}$ and $I_{9}$ , along the same line, we have

$\begin{array}{l} I_{3} = - \int_{ℝ^{3}} (u \cdot \nabla Q) : [a Q + b (Q^{2} - \frac{t r (Q^{2})}{3} I_{3 \times 3}) + c Q | Q |^{2}] ϕ_{R} (x) d x = - \int_{ℝ^{3}} (Q Δ Q - Δ Q Q) : u ⊙ \nabla ϕ_{R} (x) d x \\ \leq \frac{C}{R} (1 + {‖ Q ‖}_{L^{\infty} (B_{1})} + {‖ Q ‖}_{L^{\infty} (B_{1})}^{2}) {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ Q ‖}_{L^{4, \infty} (B_{1})}^{2} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} \\ \leq C R^{\frac{1}{2} - (\frac{1}{p} + \frac{1}{q} + \frac{1}{r})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ Q ‖}_{L^{4} (B_{1})}^{2} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})}, \end{array}$ (10)

$\begin{array}{l} I_{9} = \int_{R^{3}} \frac{1}{2} | u |^{2} u \cdot \nabla ϕ_{R} (x) d x \leq \frac{C}{R} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{3} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{p}{p - 3}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 3}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 3}, 1} (B_{2})} \\ \leq C R^{2 - (\frac{3}{p} + \frac{3}{q} + \frac{3}{r})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{3} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{p}{p - 3}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 3}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 3}, 1} (B_{2})} . \end{array}$ (11)

For $I_{10}$ , note that

$Δ P = - d i v d i v (u \otimes u + \nabla Q ⊙ \nabla Q + λ | Q | H + Q Δ Q - Δ Q Q) .$

Let $P = P_{1} + P_{2}$ such that $Δ P_{1} = - d i v d i v (f_{1})$ and $Δ P_{2} = - d i v d i v (f_{2})$ , where

$f_{1} ≜ u \otimes u, f_{2} ≜ \nabla Q ⊙ \nabla Q + λ | Q | H + Q Δ Q - Δ Q Q .$

In view of the conditions $u \in L_{x_{1}}^{p, s_{1}} L_{x_{2}}^{q, s_{2}} L_{x_{3}}^{r, s_{3}} (R^{3})$ and $Q \in H^{2} (R^{3})$ , we obtain that

$f_{1} \in L_{x_{1}}^{\frac{p}{2}, \infty} L_{x_{2}}^{\frac{q}{2}, \infty} L_{x_{3}}^{\frac{r}{2}, \infty} (R^{3}), f_{2} \in L^{2} (R^{3}) .$

By Calderron-Zygmund theorem, it follows that

$P_{1} \in L_{x_{1}}^{\frac{p}{2}, \infty} L_{x_{2}}^{\frac{q}{2}, \infty} L_{x_{3}}^{\frac{r}{2}, \infty} (R^{3}), P_{2} \in L^{2} (R^{3}) .$

Then,

$\begin{array}{l} I_{10} = \int_{R^{3}} P u \cdot \nabla ϕ_{R} (x) + \frac{μ}{2} | u |^{2} Δ ϕ_{R} (x) d x = \int_{R^{3}} P_{1} u \cdot \nabla ϕ_{R} (x) d x + \int_{R^{3}} P_{2} u \cdot \nabla ϕ_{R} (x) d x + \int_{R^{3}} \frac{μ}{2} | u |^{2} Δ ϕ_{R} (x) d x \\ \leq \frac{C}{R} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{3} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{p}{p - 3}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 3}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 3}, 1} (B_{2})} \\ + \frac{C}{R} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} {‖ P_{2} ‖}_{L^{2, \infty} (B_{1})} \\ + \frac{C}{R^{2}} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{2} {‖ {‖ {‖ Δ ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 2}, 1} (B_{2})} \\ \leq C R^{2 - (\frac{3}{p} + \frac{3}{q} + \frac{3}{r})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{3} {‖ {‖ {‖ \nabla ϕ ‖}_{L_{x_{1}}^{\frac{p}{p - 3}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 3}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 3}, 1} (B_{2})} \\ + C R^{\frac{1}{2} - (\frac{1}{p} + \frac{1}{q} + \frac{1}{r})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})} {‖ P_{2} ‖}_{L^{2} (B_{1})} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{2 p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{2 q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{2 r}{r - 2}, 1} (B_{2})} \\ + C R^{1 - (\frac{2}{p} + \frac{2}{q} + \frac{2}{r})} {‖ {‖ {‖ u ‖}_{L_{x_{1}}^{p, \infty}} ‖}_{L_{x_{2}}^{q, \infty}} ‖}_{L_{x_{3}}^{r, \infty} (B_{2})}^{2} {‖ {‖ {‖ \nabla ϕ (\frac{\cdot}{R}) ‖}_{L_{x_{1}}^{\frac{p}{p - 2}, 1}} ‖}_{L_{x_{2}}^{\frac{q}{q - 2}, 1}} ‖}_{L_{x_{3}}^{\frac{r}{r - 2}, 1} (B_{2})} . \end{array}$ (12)

Therefore, when $R \to \infty$ , we can obtain $I_{i} \to 0 (i = 1, \dots, 10)$ . Thanks to the Sobolev embedding, we have

${‖ u ‖}_{L^{6} (R^{3})} \leq C {‖ \nabla u ‖}_{L^{2} (R^{3})},$

then $u = 0$ and $H = 0$ . By the definition of $H$ , we observe

$- Δ Q = - a Q + b [Q^{2} - \frac{t r (Q^{2})}{3} I_{3 \times 3}] - c Q t r (Q^{2}) .$ (13)

Taking the inner product of the equation (13) with $Q ϕ_{R} (x)$ , and by Lemma 1, one yields

$\begin{array}{l} \int_{R^{3}} | \nabla Q |^{2} ϕ_{R} (x) d x = \int_{ℝ^{3}} | Q |^{2} Δ ϕ_{R} (x) d x - \int_{ℝ^{3}} [a | Q |^{2} - b t r (| Q |^{3}) + c | Q |^{4}] ϕ_{R} (x) d x \\ \leq \frac{C}{R^{2}} \int_{ℝ^{3}} | Q |^{2} d x + \int_{R^{3}} (\frac{b^{2} - 24 a c}{24 c}) | Q |^{2} ϕ_{R} (x) d x \leq \frac{C}{R^{2}} \int_{ℝ^{3}} | Q |^{2} d x \to 0 (R \to \infty), \end{array}$

which combined with ${‖ Q ‖}_{L^{2}} < \infty$ , concludes $Q = 0$ . The proof of Theorem 1 is completed.

References

Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems[M]. Second Edition. New York: Springer-Verlag, 2011. [Google Scholar]
Chae D, Wolf J. On Liouville type theorems for the steady Navier-Stokes equations in [J]. Journal of Differential Equations, 2016, 261(10): 5541-5560. [Google Scholar]
Kozono H, Terasawa Y, Wakasugi Y. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions[J]. Journal of Functional Analysis, 2017, 272(2): 804-818. [Google Scholar]
Luo W, Yin Z Y. The Liouville theorem and the L2 decay for the FENE dumbbell model of polymeric flows[J]. Archive for Rational Mechanics and Analysis, 2017, 224(1): 209-231. [Google Scholar]
Chae D. Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations[J]. Communications in Mathematical Physics, 2014, 326(1): 37-48. [Google Scholar]
Chae D, Yoneda T. On the Liouville theorem for the stationary Navier-Stokes equations in a critical space[J]. Journal of Mathematical Analysis and Applications, 2013, 405(2): 706-710. [Google Scholar]
Superiore S, Gilbarg D, Weinberger H, et al. Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[J]. Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1978, 5: 381-404. [Google Scholar]
Koch G, Nadirashvili N, Seregin G A, et al. Liouville theorems for the Navier-Stokes equations and applications[J]. Acta Mathematica, 2009, 203(1): 83-105. [Google Scholar]
Korobkov M, Pileckas K, Russo R. The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl[J]. Journal of Mathematical Fluid Mechanics, 2015, 17(2): 287-293. [Google Scholar]
Gong H J, Liu X G, Zhang X T. Liouville theorem for the steady-state solutions of Q-tensor system of liquid crystal[J]. Applied Mathematics Letters, 2018, 76: 175-180. [Google Scholar]
Lai N, Wu J. A note on Liouville theorem for steady Q-tensor system of liquid crystal[J]. Journal of Zhejiang University (Science Edition), 2022, 49(4): 418-421(Ch). [Google Scholar]
Schulz S. Liouville-type theorem for the stationary equations of magnetohydrodynamics[J]. Acta Mathematica Scientia, 2019, 39(2): 491-497. [Google Scholar]
Wu F. Liouville-type theorems for the 3D compressible magnetohydrodynamics equations[J]. Nonlinear Analysis: Real World Applications, 2022, 64: 103429. [Google Scholar]
Yuan B Q, Xiao Y M. Liouville-type theorems for the 3D stationary Navier-Stokes, MHD and Hall-MHD equations[J]. Journal of Mathematical Analysis and Applications, 2020, 491(2): 124343. [Google Scholar]
Yuan B, Wang F. The Liouville theorems for 3D stationary tropical climate model in local Morrey spaces[J]. Applied Mathematics Letters, 2023, 138: 108533. [Google Scholar]
Jarrín O. A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces[J]. Journal of Mathematical Analysis and Applications, 2020, 486(1): 123871. [Google Scholar]
Majumdar A. Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory[J]. European Journal of Applied Mathematics, 2010, 21(2): 181-203. [Google Scholar]

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[1] Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems[M]. Second Edition. New York: Springer-Verlag, 2011. [Google Scholar]

[2] Chae D, Wolf J. On Liouville type theorems for the steady Navier-Stokes equations in [J]. Journal of Differential Equations, 2016, 261(10): 5541-5560. [Google Scholar]

[3] Kozono H, Terasawa Y, Wakasugi Y. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions[J]. Journal of Functional Analysis, 2017, 272(2): 804-818. [Google Scholar]

[4] Luo W, Yin Z Y. The Liouville theorem and the L2 decay for the FENE dumbbell model of polymeric flows[J]. Archive for Rational Mechanics and Analysis, 2017, 224(1): 209-231. [Google Scholar]

[5] Chae D. Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations[J]. Communications in Mathematical Physics, 2014, 326(1): 37-48. [Google Scholar]

[6] Chae D, Yoneda T. On the Liouville theorem for the stationary Navier-Stokes equations in a critical space[J]. Journal of Mathematical Analysis and Applications, 2013, 405(2): 706-710. [Google Scholar]

[7] Superiore S, Gilbarg D, Weinberger H, et al. Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral[J]. Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1978, 5: 381-404. [Google Scholar]

[8] Koch G, Nadirashvili N, Seregin G A, et al. Liouville theorems for the Navier-Stokes equations and applications[J]. Acta Mathematica, 2009, 203(1): 83-105. [Google Scholar]

[9] Korobkov M, Pileckas K, Russo R. The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl[J]. Journal of Mathematical Fluid Mechanics, 2015, 17(2): 287-293. [Google Scholar]

[10] Gong H J, Liu X G, Zhang X T. Liouville theorem for the steady-state solutions of Q-tensor system of liquid crystal[J]. Applied Mathematics Letters, 2018, 76: 175-180. [Google Scholar]

[11] Lai N, Wu J. A note on Liouville theorem for steady Q-tensor system of liquid crystal[J]. Journal of Zhejiang University (Science Edition), 2022, 49(4): 418-421(Ch). [Google Scholar]

[12] Schulz S. Liouville-type theorem for the stationary equations of magnetohydrodynamics[J]. Acta Mathematica Scientia, 2019, 39(2): 491-497. [Google Scholar]

[13] Wu F. Liouville-type theorems for the 3D compressible magnetohydrodynamics equations[J]. Nonlinear Analysis: Real World Applications, 2022, 64: 103429. [Google Scholar]

[14] Yuan B Q, Xiao Y M. Liouville-type theorems for the 3D stationary Navier-Stokes, MHD and Hall-MHD equations[J]. Journal of Mathematical Analysis and Applications, 2020, 491(2): 124343. [Google Scholar]

[15] Yuan B, Wang F. The Liouville theorems for 3D stationary tropical climate model in local Morrey spaces[J]. Applied Mathematics Letters, 2023, 138: 108533. [Google Scholar]

[16] Jarrín O. A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces[J]. Journal of Mathematical Analysis and Applications, 2020, 486(1): 123871. [Google Scholar]

[17] Majumdar A. Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory[J]. European Journal of Applied Mathematics, 2010, 21(2): 181-203. [Google Scholar]