不可压液晶方程组的Serrin解

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1671-1683.

不可压液晶方程组的Serrin解

闵建中^1,*,刘宪高²(),刘子轩²()

¹ 上海健康医学院文理教学部上海 201318
² 复旦大学数学科学学院上海 200433

收稿日期:2020-10-23 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 闵建中 E-mail:xgliu@fudan.edu.cn;16110180007@fudan.edu.cn
作者简介:刘宪高, E-mail: xgliu@fudan.edu.cn|刘子轩, E-mail: 16110180007@fudan.edu.cn
基金资助:
国家自然科学基金(11631011);国家自然科学基金(11971113)

Serrin's Type Solutions of the Incompressible Liquid Crystals System

Jianzhong Min^1,*,Xiangao Liu²(),Zixuan Liu²()

¹ Science and Arts Faculty, Shanghai University of Medicine and Health Sciences, Shanghai 201318
² School of Mathematical Sciences, Fudan University, Shanghai 200433

Received:2020-10-23 Online:2021-12-26 Published:2021-12-02
Contact: Jianzhong Min E-mail:xgliu@fudan.edu.cn;16110180007@fudan.edu.cn
Supported by:
Supported by the NSFC(11631011);Supported by the NSFC(11971113)

摘要/Abstract

摘要：

该文研究了用简化的Ginzburg-Landau模型刻画的不可压液晶方程组的解的适定性问题，该模型是目前为止保持不可压液晶方程的非线性性质的最简单的模型（参见文献[1]）.该文得到了在初始资料满足以下条件$u_{0}\in L^{p}\cap H，$$d_{0}\in W^{1，p}，p\geq n$时，不可压液晶方程组的解具有存在唯一性.根据文献[2]中不可压液晶方程组的解的正则性的Serrin判定准则，该文得到了小初值光滑解的整体存在性和大初值光滑解的局部存在性.

关键词: 适定性, 不可压液晶方程组, Serrin准则, 唯一性

Abstract:

In this paper, we study the nematic liquid crystals system under the simplified Ginzburg-Landau model, which is probably the simplest mathematical model that one can derive, without destroying the basic nonlinear structure [1]. We get the local existence and uniquness of the Serrin's type of solutions provided the initial data $u_{0}\in L^{p}\cap H, $ $d_{0}\in W^{1, p}, p\geq n$. According to the Serrin's regularity criteria for the incompressible liquid crystals system [2], we actually prove the local existence of smooth solutions to liquid crystals system for big data and global existence of smooth solutions for small data.

Key words: Existence, Liquid crystal, Serrin's criterion, Uniqueness

中图分类号:

O175

闵建中,刘宪高,刘子轩. 不可压液晶方程组的Serrin解[J]. 数学物理学报, 2021, 41(6): 1671-1683.

Jianzhong Min,Xiangao Liu,Zixuan Liu. Serrin's Type Solutions of the Incompressible Liquid Crystals System[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1671-1683.

参考文献

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3	Liu X G , Min J , Zhang X . $L^{3, \infty}$ solutions of the liquid crystals system. Journal of Differential Equations, 2019, 267, 2643- 2670
4	Foia C . Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension n. Bulletin De La Societe Mathematique De France, 1961, 89, 1- 8
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