分数阶扩散的三维液晶方程的整体正则性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 918-924.

分数阶扩散的三维液晶方程的整体正则性

李强()

^{河南理工大学数学与信息科学学院河南焦作 454000}

收稿日期:2019-06-20 出版日期:2020-08-26 发布日期:2020-08-20
作者简介:李强, E-mail:1220494943@qq.com

Global Regularity for the 3D Liquid Crystal Equations with Fractional Diffusion

Qiang Li()

^{School of Mathematics and Information Science, Henan Polytechnic University, Henan Jiaozuo 454000}

Received:2019-06-20 Online:2020-08-26 Published:2020-08-20

摘要/Abstract

摘要：

该文考虑的是带有分数阶耗散项$(-\Delta)^{\alpha}u$和$(-\Delta)^{\beta}d$的广义不可压缩液晶模型.目标是在需要最小的耗散情况下建立整体正则性.在初值充分光滑的情况下，若耗散指标$\alpha\geq\frac{5}{4}$, $\beta\geq\frac{5}{4}$,方程组有唯一的整体光滑解.

关键词: 液晶方程, 分数阶耗散, 整体正则性

Abstract:

In this paper, the focus is the global regularity of three-dimensional liquid crystal equations with fractional dissipations $(-\Delta)^{\alpha}u$ and $(-\Delta)^{\beta}d$. The objective is to establish the global regularity of the fractional liquid crystal equations with the minimal amount of dissipations. And it is obtained that the equations have a global classical solution with sufficiently smooth data if $\alpha\geq\frac{5}{4}$ and $\beta\geq\frac{5}{4}$.

Key words: Liquid crystal equations, Fractional dissipation, Global regularity

中图分类号:

O175.29

李强. 分数阶扩散的三维液晶方程的整体正则性[J]. 数学物理学报, 2020, 40(4): 918-924.

Qiang Li. Global Regularity for the 3D Liquid Crystal Equations with Fractional Diffusion[J]. Acta mathematica scientia,Series A, 2020, 40(4): 918-924.

参考文献 10

1	Ericksen J L . Continuum theory of nematic liquid crystals. Res Mech, 1987, 21, 381- 392
2	Leslie F M . Theory of flow phenomenon in liquid crystals. Advances in Liquid Crystals, 1979, 4, 1- 81 doi: 10.1016/B978-0-12-025004-2.50008-9
3	Lin F H . Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun Pure Appl Math, 1989, 42, 789- 814 doi: 10.1002/cpa.3160420605
4	Lin F H , Liu C . Existence of solutions for the Ericksen-Leslie system. Arch Ration Mech Anal, 2000, 154 (2): 135- 156 doi: 10.1007/s002050000102
5	Wu J H . Generalized MHD equations. J Differential Equations, 2003, 195, 284- 312 doi: 10.1016/j.jde.2003.07.007
6	Wang Y N , Zhou Y , Alsaedi A , et al. Global regularity for the incompressible 2D generalized liquid crystal model with fractional diffusions. Appl Math Lett, 2014, 35, 18- 23 doi: 10.1016/j.aml.2014.03.014
7	Yuan B Q , Wei C Z . Global regularity of the generalized liquid crystal model with fractional diffusion. J Math Anal Appl, 2018, 467, 948- 958 doi: 10.1016/j.jmaa.2018.07.047
8	Li H M . Global regularity for the 3D micropolar equations. Appl Math Lett, 2019, 92, 70- 75 doi: 10.1016/j.aml.2019.01.011
9	Wang D H, Wu J H, Ye Z. Global regularity of the three-dimensional fractional micropolar equations. 2019, arXiv: 1903.03220vl[math.Ap]
10	Majda A , Bertozzi A . Vorticity and Incompressible Flow. Cambridge: Cambridge University Press, 2002