scholarly journals Regularity Criterion for the 3D Nematic Liquid Crystal Flows

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Nematic Liquid Crystal ◽  
Regularity Criterion ◽  
Hydrodynamic Theory ◽  
Nematic Liquid Crystal Flows

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

scholarly journals A regularity criterion for liquid crystal flows in terms of the component of velocity and the horizontal derivative components of orientation field

2021 ◽  
Vol 7 (3) ◽  
pp. 4168-4175
Author(s):  
Qiang Li ◽  
◽  
Baoquan Yuan ◽  
Keyword(s):  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Velocity Component ◽  
Smooth Solution ◽  
Regularity Criterion ◽  
Orientation Field ◽  
Horizontal Derivative ◽  
Nematic Liquid Crystal Flows ◽  
Vorticity Component

<abstract><p>In this paper, we establish a regularity criterion for the 3D nematic liquid crystal flows. More precisely, we prove that the local smooth solution $ (u, d) $ is regular provided that velocity component $ u_{3} $, vorticity component $ \omega_{3} $ and the horizontal derivative components of the orientation field $ \nabla_{h}d $ satisfy</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \int_{0}^{T}|| u_{3}||_{L^{p}}^{\frac{2p}{p-3}}+||\omega_{3}||_{L^{q}}^{\frac{2q}{2q-3}}+||\nabla_{h} d||_{L^{a}}^{\frac{2a}{a-3}} \mbox{d} t&lt;\infty,\nonumber \\ with\ \ 3&lt; p\leq\infty,\ \frac{3}{2}&lt; q\leq\infty,\ 3&lt; a\leq\infty. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> </abstract>

scholarly journals The Uniaxial Limit of the Non-Inertial Qian–Sheng Model for Liquid Crystals

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 912
Author(s):  
Sirui Li ◽  
Fangxin Zhao
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Elastic Constant ◽  
Nematic Liquid Crystal ◽  
Expansion Method ◽  
Inertial Effect ◽  
Rigorous Derivation ◽  
Leslie Model ◽  
Hilbert Expansion ◽  
Nematic Liquid Crystal Flows

In this article, we consider the Qian–Sheng model in the Landau–de Gennes framework describing nematic liquid crystal flows when the inertial effect is neglected. By taking the limit of elastic constant to zero (also called the uniaxial limit) and utilizing the so-called Hilbert expansion method, we provide a rigorous derivation from the non-inertial Qian–Sheng model to the Ericksen–Leslie model.

scholarly journals Regularity Criterion for the Nematic Liquid Crystal Flows in Terms of Velocity

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Ruiying Wei ◽  
Zheng-an Yao ◽  
Yin Li
Keyword(s):  
Liquid Crystal ◽  
Fractional Derivative ◽  
Nematic Liquid Crystal ◽  
Lebesgue Space ◽  
Sufficient Conditions ◽  
Regularity Criterion ◽  
Nematic Liquid Crystal Flows

We study the regularity criterion for the 3D nematic liquid crystal flows in the framework of anisotropic Lebesgue space. More precisely, we proved some sufficient conditions in terms of velocity or the fractional derivative of velocity in one direction.

scholarly journals Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals

2016 ◽  
Vol 14 (04) ◽  
pp. 523-536 ◽  
Author(s):  
Tao Huang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
Nematic Liquid Crystal ◽  
Weak Solutions ◽  
Nematic Liquid Crystals ◽  
Regularity Criterion ◽  
Liquid Crystal Flow ◽  
The Cauchy Problem

In this paper, we establish an [Formula: see text]-regularity criterion for any weak solution [Formula: see text] to the nematic liquid crystal flow (1.1) such that [Formula: see text] for some [Formula: see text] and [Formula: see text] satisfying the condition (1.2). As consequences, we prove the interior smoothness of any such a solution when [Formula: see text] and [Formula: see text]. We also show that uniqueness holds for the class of weak solutions [Formula: see text] the Cauchy problem of the nematic liquid crystal flow (1.1) that satisfy [Formula: see text] for some [Formula: see text] and [Formula: see text] satisfying (1.2).

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