scholarly journals A review of mathematical analysis of nematic and smectic-A liquid crystal models

2013 ◽  
Vol 25 (1) ◽  
pp. 133-153 ◽  
Author(s):  
BLANCA CLIMENT-EZQUERRA ◽  
FRANCISCO GUILLÉN-GONZÁLEZ
Keyword(s):  
Liquid Crystal ◽  
Mathematical Analysis ◽  
Parabolic Problems ◽  
Strongly Nonlinear ◽  
Smectic A ◽  
Nonlinear Parabolic ◽  
Time Periodicity ◽  
Temporal Profiles ◽  
Uniaxial Liquid Crystal ◽  
Smectic A Liquid Crystals

We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data

2016 ◽  
Vol 23 (3) ◽  
pp. 303-321 ◽  
Author(s):  
Youssef Akdim ◽  
Abdelmoujib Benkirane ◽  
Mostafa El Moumni ◽  
Hicham Redwane
Keyword(s):  
Growth Condition ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Measure Data ◽  
Renormalized Solution ◽  
Unbounded Function ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
The Right

AbstractWe study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$where the right-hand side belongs to ${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and ${b(x,u)}$ is unbounded function of u, ${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth ${|\nabla u|^{p-1}}$ in ${\nabla u}$. The critical growth condition on g is with respect to ${\nabla u}$ and there is no growth condition with respect to u, while the function ${H(x,t,\nabla u)}$ grows as ${|\nabla u|^{p-1}}$.

scholarly journals Existence of solutions for some strongly nonlinear parabolic problems involving lower order terms in divergence form in Musielak-Orlicz spaces

2019 ◽  
Vol 38 (6) ◽  
pp. 99-126
Author(s):  
Abdeslam Talha ◽  
Abdelmoujib Benkirane
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Orlicz Spaces ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Strongly Nonlinear ◽  
L1 Data ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this work, we prove an existence result of entropy solutions in Musielak-Orlicz-Sobolev spaces for a class of nonlinear parabolic equations with two lower order terms and L1-data.

scholarly journals Global in time solution and time-periodicity for a smectic-A liquid crystal model

2010 ◽  
Vol 9 (6) ◽  
pp. 1473-1493 ◽  
Author(s):  
Blanca Climent-Ezquerra ◽  
◽  
Francisco Guillén-González
Keyword(s):  
Liquid Crystal ◽  
Smectic A ◽  
Time Solution ◽  
Crystal Model ◽  
Time Periodicity

scholarly journals Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces

2014 ◽  
Vol 33 (1) ◽  
pp. 191 ◽  
Author(s):  
Mohamed Leimne Ahmed Oubeid ◽  
A. Benkirane ◽  
M. Sidi El Vally
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Parabolic Problems ◽  
Nonlinear Parabolic Problems ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic

We prove in this paper the existence of solutions of strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces. An approximation and a compactness results in inhomogeneous Musielak-Orlicz-Sobolev spaces have also been provided.

On a class of nonlinear anisotropic parabolic problems

2015 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
M. H. Abdou ◽  
M. Chrif ◽  
S. El Manouni ◽  
H. Hjiaj
Keyword(s):  
Sobolev Space ◽  
Parabolic Problem ◽  
Nonlinear Term ◽  
Parabolic Problems ◽  
Anisotropic Sobolev Space ◽  
Existence Of Weak Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
Sign Conditions ◽  
Image Position

We prove the existence of weak solutions for the strongly nonlinear parabolic problem in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

scholarly journals Effect of ferroelectric BaTiO3 particles on the threshold voltage of a smectic A liquid crystal

2018 ◽  
Vol 9 ◽  
pp. 824-828 ◽  
Author(s):  
Abbas Rahim Imamaliyev ◽  
Mahammadali Ahmad Ramazanov ◽  
Shirkhan Arastun Humbatov
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Threshold Voltage ◽  
Strong Electric Field ◽  
Dielectric Anisotropy ◽  
Smectic A ◽  
Capacitance Voltage ◽  
Two Factors ◽  
Smectic A Liquid Crystals

The influence of small ferroelectric BaTiO3 particles on the planar–homeotropic transition threshold voltage in smectic A liquid crystals consisting of p-nitrophenyl p-decyloxybenzoate and 4-cyano-4′-pentylbiphenyl were studied by using capacitance–voltage (C–V) measurements. It was shown that the BaTiO3 particles significantly reduce the threshold voltage. The obtained result is explained by two factors: an increase of dielectric anisotropy of the liquid crystals and the formation of a strong electric field near polarized particles of BaTiO3. It was shown that the role of the second factor is dominant. The explanations of some features observed in the C–V characteristics are given.

Spatially Resolved Characterization of the Electroclinic Effect in Chiral Smectic a Liquid Crystals

1996 ◽  
Vol 425 ◽  
Author(s):  
J. R. Lindle ◽  
S. R. Flom ◽  
F. J. Bartoli ◽  
A. T. Harter ◽  
R. E. Geer ◽  
...  
Keyword(s):  
Optical Properties ◽  
Angular Distribution ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
Tilt Angle ◽  
X Ray ◽  
Smectic A ◽  
Spatially Resolved ◽  
Smectic A Liquid Crystals

AbstractThe electro-optical properties of an electroclinic liquid crystal have been investigated using a spatially resolved technique which measures the birefringence, optical tilt angle and extinction within adjacent stripe domains. A comparison of these results with the data obtained from standard optical and X-ray measurements suggests the presence of substripes or an angular distribution of molecular directors within the stripe domains.

Freeze-fracture TEM of the helical smectic A* phase

1992 ◽  
Vol 50 (1) ◽  
pp. 278-279
Author(s):  
K.J. Ihn ◽  
R. Pindak ◽  
J. A. N. Zasadzinski
Keyword(s):  
Liquid Crystal ◽  
Grain Boundary ◽  
Grain Boundaries ◽  
Freeze Fracture ◽  
Smectic Phase ◽  
Layer Spacing ◽  
Screw Dislocations ◽  
Smectic A ◽  
Smectic Phases ◽  
Discrete Amount

A new liquid crystal (called the smectic-A* phase) that combines cholesteric twist and smectic layering was a surprise as smectic phases preclude twist distortions. However, the twist grain boundary (TGB) model of Renn and Lubensky predicted a defect-mediated smectic phase that incorporates cholesteric twist by a lattice of screw dislocations. The TGB model for the liquid crystal analog of the Abrikosov phase of superconductors consists of regularly spaced grain boundaries of screw dislocations, parallel to each other within the grain boundary, but rotated by a fixed angle with respect to adjacent grain boundaries. The dislocations divide the layers into blocks which rotate by a discrete amount, Δθ, given by the ratio of the layer spacing, d, to the distance between grain boundaries, lb; Δθ ≈ d/lb (Fig. 1).

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