scholarly journals A novel Landau-de Gennes model with quartic elastic terms

2020 ◽  
Vol 32 (1) ◽  
pp. 177-198
Author(s):  
DMITRY GOLOVATY ◽  
MICHAEL NOVACK ◽  
PETER STERNBERG
Keyword(s):  
Phase Transitions ◽  
Strong Convergence ◽  
Correlation Length ◽  
Elastic Constants ◽  
Isotropic Phase ◽  
Well Posedness ◽  
Clear Advantage ◽  
Minimisation Problem ◽  
Well Posed ◽  
Gennes Theory

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

scholarly journals Phase transitions in nematics: textures with tactoids and disclinations

2020 ◽  
Vol 15 ◽  
pp. 8 ◽  
Author(s):  
Dmitry Golovaty ◽  
Young-Ki Kim ◽  
Oleg D. Lavrentovich ◽  
Michael Novack ◽  
Peter Sternberg
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Liquid Crystals ◽  
Numerical Simulations ◽  
Elastic Constants ◽  
Nematic Phase ◽  
Topological Defects ◽  
First Order ◽  
Gennes Theory

We demonstrate that a first order isotropic-to-nematic phase transition in liquid crystals can be succesfully modeled within the generalized Landau-de Gennes theory by selecting an appropriate combination of elastic constants. The numerical simulations of the model established in this paper qualitatively reproduce the experimentally observed configurations that include interfaces and topological defects in the nematic phase.

HIGH RESOLUTION CALORIMETRIC STUDIES AT PHASE TRANSITIONS OF ALKYLCYANOBIPHENYL LIQUID CRYSTALS CONFINED TO SUBMICRON SIZE CYLINDRICAL CAVITIES

1995 ◽  
Vol 09 (18n19) ◽  
pp. 2247-2283 ◽  
Author(s):  
DANIELE FINOTELLO ◽  
GERMANO S. IANNACCHIONE
Keyword(s):  
Phase Transitions ◽  
Liquid Crystals ◽  
High Resolution ◽  
Specific Heat ◽  
Pore Wall ◽  
Isotropic Phase ◽  
First Order ◽  
Smectic A ◽  
Nematic Director ◽  
Scalar Order Parameter

We review results of a high resolution systematic study of the specific heat for alkyl-cyanobiphenyl liquid crystals confined to the 0.2µm diameter cylindrical pores Anopore membranes. The nematic director alignment at the pore wall is varied from homeotropic to tangential by pore surface treatment. Several phenomena are uncovered by these studies which probed the weakly first order nematic to isotropic, the continuous smectic-A to nematic and the first order smectic-A to isotropic phase transitions. The specific heat is strongly dependent on the nematic director configuration, and confinement effects are remarkably distinct according to the order of the phase transition. The influence of elastic distortions and surface ordering and disordering effects are evident. Despite considerable departures from bulk behavior with regards to specific heat peaks size, rounding and width, and transition temperature shifts, a bulk-like critical behavior appears to be retained. The formation of smectic translational order within the pores is hindered for those liquid crystals that also possess a nematic phase. The average scalar order parameter temperature dependence is extracted from the specific heat results using a simplified Landau-de Gennes type of model, and is shown to be consistent with nuclear magnetic resonance results.

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

Tykhonov well-posedness of a frictionless unilateral contact problem

2019 ◽  
Vol 25 (6) ◽  
pp. 1294-1311 ◽  
Author(s):  
Zhenhai Liu ◽  
Mircea Sofonea ◽  
Yi-bin Xiao
Keyword(s):  
Contact Problem ◽  
Variational Formulation ◽  
Unilateral Contact ◽  
Well Posedness ◽  
Unilateral Constraints ◽  
The Family ◽  
Contact Models ◽  
Mechanical Interpretation ◽  
Well Posed ◽  
Natural Way

We consider a frictionless contact problem, Problem [Formula: see text], for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem [Formula: see text]. Then we consider a perturbation of Problem [Formula: see text], which could be frictional, governed by a small parameter [Formula: see text]. This perturbation leads in a natural way to a family of sets [Formula: see text]. We prove that Problem [Formula: see text] is well-posed in the sense of Tykhonov with respect to the family [Formula: see text]. The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem [Formula: see text]. Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.

scholarly journals On the local well-posedness of a Benjamin-Ono-Boussinesq system

2005 ◽  
Vol 2005 (22) ◽  
pp. 3609-3630
Author(s):  
Ruying Xue
Keyword(s):  
Sobolev Space ◽  
Boussinesq System ◽  
Well Posedness ◽  
Well Posed

Consider a Benjamin-Ono-Boussinesq systemηt+ux+auxxx+(uη)x=0,ut+ηx+uux+cηxxx−duxxt=0, wherea,c, anddare constants satisfyinga=c>0,d>0ora<0,c<0,d>0. We prove that this system is locally well posed in Sobolev spaceHs(ℝ)×Hs+1(ℝ), withs>1/4.

Sign in / Sign up

Export Citation Format

Share Document