scholarly journals Global well-posedness for the dynamical Q-tensor model of liquid crystals

2015 ◽  
Vol 58 (6) ◽  
pp. 1349-1366 ◽  
Author(s):  
JinRui Huang ◽  
ShiJin Ding
Keyword(s):  
Liquid Crystals ◽  
Tensor Model ◽  
Well Posedness

A stable scheme and its convergence analysis for a 2D dynamic Q-tensor model of nematic liquid crystals

2017 ◽  
Vol 27 (08) ◽  
pp. 1459-1488 ◽  
Author(s):  
Yongyong Cai ◽  
Jie Shen ◽  
Xiang Xu
Keyword(s):  
Free Energy ◽  
Liquid Crystals ◽  
Convergence Analysis ◽  
Nematic Liquid Crystals ◽  
Gradient Flow ◽  
Tensor Model ◽  
Well Posedness ◽  
Unconditionally Stable ◽  
Numerical Studies ◽  
Stable Scheme

We propose an unconditionally stable numerical scheme for a 2D dynamic [Formula: see text]-tensor model of nematic liquid crystals. This dynamic [Formula: see text]-tensor model is an [Formula: see text]-gradient flow generated by the liquid crystal free energy that contains a cubic term, which is physically relevant but makes the free energy unbounded from below, and for this reason, has been avoided in other numerical studies. The unboundedness of the energy brings significant difficulty in analyzing the model and designing numerical schemes. By using a stabilizing technique, we construct an unconditionally stable scheme, and establish its unique solvability and convergence. Our convergence analysis also leads to, as a byproduct, the well-posedness of the original PDE system for the 2D [Formula: see text]-tensor model. Several numerical examples are presented to validate and demonstrate the effectiveness of the scheme.

scholarly journals Dynamic cubic instability in a 2D Q-tensor model for liquid crystals

2015 ◽  
Vol 25 (08) ◽  
pp. 1477-1517 ◽  
Author(s):  
Gautam Iyer ◽  
Xiang Xu ◽  
Arghir D. Zarnescu
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Initial Data ◽  
Elastic Constant ◽  
Nematic Liquid Crystal ◽  
Gradient Flow ◽  
Two Dimensions ◽  
Tensor Model ◽  
Well Posedness

We consider a four-elastic-constant Landau–de Gennes energy characterizing nematic liquid crystal configurations described using the Q-tensor formalism. The energy contains a cubic term and is unbounded from below. We study dynamical effects produced by the presence of this cubic term by considering an L2 gradient flow generated by this energy. We work in two dimensions and concentrate on understanding the relations between the physicality of the initial data and the global well-posedness of the system.

scholarly journals A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals

2020 ◽  
Vol 145 (4) ◽  
pp. 837-881 ◽  
Author(s):  
Juan Pablo Borthagaray ◽  
Ricardo H. Nochetto ◽  
Shawn W. Walker
Keyword(s):  
Liquid Crystals ◽  
Nematic Liquid Crystals ◽  
Tensor Model ◽  
Structure Preserving
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