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

[1]	Lin F. Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena. Comm Pure Appl Math, 1989, 42(6): 789-814
[2]	Galdi G P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations:Steady-State Problems. New York: Springer, 2011
[3]	Chae D, Wolf J. On Liouville type theorems for the steady Navier-Stokes equations in $\mathbb{R}^3$ . J Dyn Differ Equ, 2016, 261(10): 5541-560
[4]	Seregin G. A Liouville type theorem for steady-state Navier-Stokes equations. arXiv:1611.01563, 2016
[5]	Chae D. Anisotropic Liouville type theorem for the stationary Navier-Stokes equations in $\mathbb{R}^3$ . Appl Math Lett, 2023, 142: 108655
[6]	Lin F, Lin J, Wang C. Liquid crystal flows in two dimensions. Arch Ration Mech Anal, 2010, 197(1): 297-336
[7]	Hong M C. Global existence of solutions of the simplified Ericksen-Leslie system in dimension two. Calc Var Partial Differ Equ, 2011, 40(1/2): 15-36
[8]	Xu X, Zhang Z. Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J Dyn Differ Equ, 2012, 252(2): 1169-1181
[9]	Lin F, Wang C. On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin Ann Math Ser B, 2010, 31: 921-938
[10]	Lin F, Wang C. Global existence of weak solutions of the nematic liquid crystal flow in dimension three. Comm Pure Appl Math, 2016, 69(8): 1532-1571
[11]	Jarrín O. Liouville theorems for a stationary and non-stationary coupled system of liquid crystal flows in local Morrey spaces. J Math Fluid Mech, 2022, 24: Article number 50
[12]	Hao Y, Liu X, Zhang X. Liouville theorem for steady-state solutions of simplified Ericksen-Leslie system. arXiv:1906.06318v1, 2019
[13]	Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970

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