On the well-posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces

2011 ◽  
Vol 35 (2) ◽  
pp. 158-173 ◽  
Author(s):  
Junyu Lin ◽  
Shijin Ding
Keyword(s):  
Liquid Crystals ◽  
Heat Flow ◽  
Harmonic Maps ◽  
Nematic Liquid Crystals ◽  
Hydrodynamic Flow ◽  
Well Posedness ◽  
Critical Spaces

scholarly journals Entropy inequality and energy dissipation of inertial Qian–Sheng model for nematic liquid crystals

2021 ◽  
Vol 18 (01) ◽  
pp. 221-256
Author(s):  
Ning Jiang ◽  
Yi-Long Luo ◽  
Yangjun Ma ◽  
Shaojun Tang
Keyword(s):  
Liquid Crystals ◽  
Energy Dissipation ◽  
Nematic Liquid Crystals ◽  
Entropy Inequality ◽  
Small Data ◽  
Well Posedness ◽  
Smooth Solutions ◽  
Large Initial Data ◽  
Decay Of Solutions ◽  
Local Solutions

For the inertial Qian–Sheng model of nematic liquid crystals in the [Formula: see text]-tensor framework, we illustrate the roles played by the entropy inequality and energy dissipation in the well-posedness of smooth solutions when we employ energy method. We first derive the coefficients requirements from the entropy inequality, and point out the entropy inequality is insufficient to guarantee energy dissipation. We then introduce a novel Condition (H) which ensures the energy dissipation. We prove that when both the entropy inequality and Condition (H) are obeyed, the local in time smooth solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Furthermore, to extend the solutions globally in time and obtain the decay of solutions, we require at least one of the two conditions: entropy inequality, or [Formula: see text], which significantly enlarge the range of the coefficients in previous works.

INFLUENCE OF HYDRODYNAMIC FLOWS ON THE OPTICAL PROPERTIES OF HYBRID ALIGNED NEMATIC LIQUID CRYSTALS

2018 ◽  
Vol 52 (3 (247)) ◽  
pp. 213-216
Author(s):  
K.A. Petrosyan
Keyword(s):  
Optical Properties ◽  
Angular Distribution ◽  
Liquid Crystals ◽  
Flow Velocity ◽  
Volume Expansion ◽  
Nematic Liquid Crystals ◽  
Polarized Light ◽  
Hydrodynamic Flow ◽  
Light Passing ◽  
Expansion Mechanism

In this work we study the influence of hydrodynamic flows on the optical properties of hybrid aligned nematic liquid crystals (NLC) structure caused by direct volume expansion mechanism for the case, when the direction of flow velocity is perpendicular to the angular distribution of the molecules in the cell. It has been shown that the hydrodynamic flow leads to reorientation of NLC molecules. The behavior of the polarized light passing through the NLC layer for two opposite directions of the flow was observed.

scholarly journals Sphere-valued harmonic maps with surface energy and the K13 problem

2019 ◽  
Vol 12 (4) ◽  
pp. 363-392
Author(s):  
Stuart Day ◽  
Arghir Dani Zarnescu
Keyword(s):  
Liquid Crystals ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Critical Point ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Nematic Liquid Crystals ◽  
Energy Functional ◽  
Dirichlet Energy ◽  
Surface Term

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

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