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

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 $\mathit{u}=\mathrm{0},\mathit{Q}=\mathrm{0}$ on the conditions that $\mathit{u}\in L_{x_{\mathrm{1}}}^{p,\mathrm{\infty }}{L}_{x_{\mathrm{2}}}^{q,\mathrm{\infty }}{L}_{x_{\mathrm{3}}}^{r,\mathrm{\infty }}({\mathbb{R}}^{\mathrm{3}})\bigcap {\dot{H}}^{\mathrm{1}}({\mathbb{R}}^{\mathrm{3}}),\mathit{Q}\in H^{\mathrm{2}}({\mathbb{R}}^{\mathrm{3}}),p,q,$$r\in (\mathrm{3},\mathrm{\infty }],$ and $\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}+\frac{\mathrm{1}}{r}\ge \frac{\mathrm{2}}{\mathrm{3}},$ 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$\mathit{u}\in L_{x_{\mathrm{1}}}^{p,\mathrm{\infty }}{L}_{x_{\mathrm{2}}}^{q,\mathrm{\infty }}{L}_{x_{\mathrm{3}}}^{r,\mathrm{\infty }}({\mathbb{R}}^{\mathrm{3}})\bigcap {\dot{H}}^{\mathrm{1}}({\mathbb{R}}^{\mathrm{3}}),\mathit{Q}\in H^{\mathrm{2}}({\mathbb{R}}^{\mathrm{3}}),p,q,r\in (\mathrm{3},\mathrm{\infty }]$和 $\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}+\frac{\mathrm{1}}{r}\ge \frac{\mathrm{2}}{\mathrm{3}}$下得到 $\mathit{u}=\mathrm{0},\mathit{Q}=\mathrm{0}$这一结论，它推广了一些已有的结果。