三维稳态向列型液晶方程各向异性的Liouville定理

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 661-669.

三维稳态向列型液晶方程各向异性的Liouville定理

陈浩¹(),邓雪梅^1,²,别群益^1,^2,^*()

1.三峡大学理学院湖北宜昌 443002
2.三峡大学三峡数学研究中心湖北宜昌 443002

收稿日期:2023-05-17 修回日期:2023-10-07 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *别群益，E-mail：qybie@126.com
作者简介:陈浩，E-mail：chenhao215@outlook.com
基金资助:
国家自然科学基金(11871305)

Anisotropic Liouville Type Theorem for the Stationary Nematic Liquid Crystal Equations in $\mathbb{R}^{3}$

Chen Hao¹(),Deng Xuemei^1,²,Bie Qunyi^1,^2,^*()

1. College of Science, China Three Gorges University, Hubei Yichang 443002
2. Three Gorges Mathematical research Center, China Three Gorges University, Hubei Yichang 443002

Received:2023-05-17 Revised:2023-10-07 Online:2024-06-26 Published:2024-05-17
Supported by:
NSFC(11871305)

摘要/Abstract

摘要：

该文研究了三维稳态向列型液晶方程的Liouville定理, 证明了如果 $\nabla d\in{L^2}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3})$, $u\in{L^6}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3})$, 以及 $u_i$ 满足各向异性的可积条件 ${u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}),\forall i = 1,2,3$, 其中 $2 < q < \infty, 1 \le s \le \infty $ 且 $\frac{2}{q} + \frac{1}{s} \ge \frac{1}{2}$, 则 $u=0, \nabla d=0$.

关键词: 向列型液晶方程, Liouville定理, 各向异性

Abstract:

This paper investigates a Liouville type theorem for three-dimensional stationary liquid crystal equations. We show that if $u\in{L^6}({\mathbb{R}^3}) \cap {L^ q}({\mathbb{R}^3})$, $\nabla d\in{L^2}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3})$ and the anisotropic integrability conditions of ${u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}), \forall i = 1,2,3$, $\frac{2}{q} + \frac{1}{s} \ge \frac{1}{2}$, $2 < q < \infty,1 \le s \le \infty $ are satisfied, then $u=0, \nabla d = 0$.

Key words: Nematic liquid crystal equations, The Liouville problem, Anisotropy

中图分类号:

O175.2

陈浩, 邓雪梅, 别群益. 三维稳态向列型液晶方程各向异性的Liouville定理[J]. 数学物理学报, 2024, 44(3): 661-669.

Chen Hao, Deng Xuemei, Bie Qunyi. Anisotropic Liouville Type Theorem for the Stationary Nematic Liquid Crystal Equations in $\mathbb{R}^{3}$[J]. Acta mathematica scientia,Series A, 2024, 44(3): 661-669.

参考文献 13

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