CPAA
The existence and blow-up criterion of liquid crystals system in critical Besov space
Yi-hang Hao Xian-Gao Liu
Communications on Pure & Applied Analysis 2014, 13(1): 225-236 doi: 10.3934/cpaa.2014.13.225
We consider the existence of strong solution to liquid crystals system in critical Besov space, and give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.
keywords: liquid crystals strong solution. Littlewood-Paley theory
DCDS
Globally weak solutions to the flow of compressible liquid crystals system
Xian-Gao Liu Jie Qing
Discrete & Continuous Dynamical Systems - A 2013, 33(2): 757-788 doi: 10.3934/dcds.2013.33.757
We study a simplified system for the compressible fluid of Nematic Liquid Crystals in a bounded domain in three Euclidean space and prove the global existence of the finite energy weak solutions.
keywords: global existence. compressible Nematic liquid crystals
CPAA
Liouville theorem for MHD system and its applications
Xian-gao Liu Xiaotao Zhang
Communications on Pure & Applied Analysis 2018, 17(6): 2329-2350 doi: 10.3934/cpaa.2018111

In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. And we mainly study weak axi-symmetric solutions of MHD system in $\mathbb{R}^3× (0, T)$.

keywords: Incompressible magnetohydrodynamics(MHD) Liouville theorem singularties
DCDS
Serrin's regularity results for the incompressible liquid crystals system
Xian-Gao Liu Jianzhong Min Kui Wang Xiaotao Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5579-5594 doi: 10.3934/dcds.2016045
In this paper, we study the simplified system of the original Ericksen--Leslie equations for the flow of liquid crystals [10]. Under Serrin criteria [13], we prove a partial interior regularity result of weak solutions for the three-dimensional incompressible liquid crystal system.
keywords: partial regularity Serrin criteria heat kernel theory. Incompressible liquid crystals Calderón-Zygmund inequality

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