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

Huang, Jinrui; Ding, Shijin

doi:10.1007/s11425-015-4990-8

Cited by 34 publications

(20 citation statements)

References 27 publications

(43 reference statements)

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“…As a starting point, we consider the famous Landau-de Gennes free energy (see Ref. [47,48]), which in the absence of external aligning fields and surfaces is given as…”

Section: The Q-tensor and Corresponding Free Energy Functionalsmentioning

confidence: 99%

Nonstationary models for liquid crystals: A fresh mathematical perspective

Emmrich

Klapp

Lasarzik

2018

Journal of Non-Newtonian Fluid Mechanics

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In this article we discuss nonstationary models for inhomogeneous liquid crystals driven out of equilibrium by flow. Emphasis is put on those models which are used in the mathematics as well as in the physics literature, the overall goal being to illustrate the mathematical progress on popular models which physicists often just solve numerically. Our discussion includes the Doi-Hess model for the orientational distribution function, the Q-tensor model and the Ericksen-Leslie model which focuses on the director dynamics. We survey particularly the mathematical issues (such as existence of solutions) and linkages between these models. Moreover, we introduce the new concept of relative energies measuring the distance between solutions of equation systems with nonconvex energy functionals and discuss possible applications of this concept for future studies.Nonstationary models for liquid crystals: A fresh mathematical perspective physicists, chemists, material scientists, and even (applied) mathematicians. This interest is recently also triggered by the important role of liquid-crystal physics in the fields of biophysics (e.g., for the structure of the cytoskeleton or the movement between actin and myosin, see Ahmadi et al. [6]), in active matter Ref.[7] and in astrophysics (emergence of topological defects). Many of these contexts involve physical situations outside thermal equilibrium, where the material properties generally depend on time. The purpose of the present article is to summarize modeling approaches for such nonstationary (out-of-equilibrium) liquid crystals from both, a mathematical and a physical perspective.Clearly, the presence of orientational degrees of freedom makes the theoretical description of liquid crystals more complex than that of ordinary (atomic) fluids. This holds for microscopic ("bottom-up") approaches such as classical density functional theory (see Ref. [8,9]), but also for coarse-grained approaches such as phase-field crystal modeling (see Ref.[10]) and for mesoscopic (continuum) approaches involving appropriate order parameter fields (see Ref. [11]) or even macroscopic variables, such as a stress tensor (see Sec. 4). Such mesoscopic approaches have become particularly popular for the description of liquid crystals under flow, a situation of major relevance for many applications (see Ref. [12]). Mathematically, continuum approaches for nonstationary liquid crystals involve typically nonlinear coupled (partial) differential equations. While physicists just tend to solve these equations numerically and explore the emerging physical behavior, there are many open problems from the mathematical (and numerical) side concerning, e.g., the existence and uniqueness of solutions. From the physical side, this poses the danger of overseeing important dynamical features, while from the mathematical side, there is a certain risk to concentrate on too simplistic (or even unphysical) models.It is in this spirit that we here aim at giving an overview over some of the most relevant nonstationary mode...

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“…As a starting point, we consider the famous Landau-de Gennes free energy (see Ref. [47,48]), which in the absence of external aligning fields and surfaces is given as…”

Section: The Q-tensor and Corresponding Free Energy Functionalsmentioning

confidence: 99%

Nonstationary models for liquid crystals: A fresh mathematical perspective

Emmrich

Klapp

Lasarzik

2018

Journal of Non-Newtonian Fluid Mechanics

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“…In [13,14], nonisothermal variants of the Beris-Edwards system were derived and the authors proved the existence of global weak solutions in the case of a singular potential under periodic boundary conditions for general ξ ∈ R and d = 3. In [19], the authors considered a general Beris-Edwards system where the Dirichlet type elastic functional as in (1.6) is replaced by three quadratic functionals. For the Cauchy problem in R 3 , they proved the existence of global weak solutions as well as the existence of a unique global strong solution provided that the fluid viscosity is sufficiently large.…”

Section: T) = U(x T) Q(x + E I T) = Q(x T) For (X T) ∈mentioning

confidence: 99%

Global Strong Solutions of the Full Navier--Stokes and $Q$-Tensor System for Nematic Liquid Crystal Flows in Two Dimensions

Cavaterra¹,

Rocca²,

Wu³

et al. 2016

SIAM J. Math. Anal.

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Abstract. We consider a full Navier-Stokes and Q-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter ξ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.

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“…Moreover, from (1.15) and (1.19) we easily infer the following relation ζ 2κ > 0. (1.21) Under the current assumption (1.18), the general system (1.5)-(1.7) differs from those that have been extensively studied in the literature, see for instance, [2,3,10,[13][14][15][17][18][19][20][21]30,34,35,38,39]. An important feature is that its free energy E(Q) now contains an unusual cubic term associated with the coefficient L 4 , which is physically meaningful but may cause the free energy functional E(Q) to be unbounded from below.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, initial boundary value problems subject to various boundary conditions have been investigated by several authors, see for instance, [3,19,20], where the existence of global weak solutions, existence and uniqueness of local strong solutions as well as some regularity criteria were established. For the Beris-Edwards system with three elastic constants L 1 , L 2 , L 3 but L 4 = ξ = 0, in [21] the authors studied the Cauchy problem in R 3 and proved the existence of global weak solutions as well as the existence and uniqueness of global strong solutions provided that the fluid viscosity is sufficiently large. Next, concerning the full Beris-Edward system with a general parameter ξ ∈ R, existence of global weak solutions for the Cauchy problem in R d with d = 2, 3 was established in [34] for sufficiently small |ξ|, while the uniqueness of weak solutions for d = 2 was given in [15].…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness of the two dimensional Beris–Edwards system with general Laudau–de Gennes free energy

Liu

2019

Journal of Differential Equations

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In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter Q-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial L ∞ -norm of the Q-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the L ∞ -norm of the Q-tensor along the time evolution.Here, a, b, c ∈ R are material-dependent and temperature-dependent constants. On the other hand, the elastic free energy characterizes the distortion effect of the liquid crystal and its density function F elastic gives the strain energy density due to spatial variations in the Q-tensor:( 1.4) for 1 i, j, k, l d. Throughout this paper, we use the Einstein summation convention over repeated indices. The coefficients L 1 , L 2 , L 3 , L 4 are material-dependent elastic constants. We note that F elastic consists of three independent terms associated with L 1 , L 2 , L 3 that are quadratic in the first order partial derivatives of the Q-tensor, plus a cubic term associated with the coefficient L 4 . In the literature, the case L 2 = L 3 = L 4 = 0 is usually called isotropic, otherwise anisotropic if at least one of L 2 , L 3 , L 4 does not vanish. In particular, the retention of the cubic term (i.e., L 4 = 0) is due to the physically relevant consideration that it allows a complete reduction of the Landau-de Gennes energy E(Q) to the classical Oseen-Frank energy for nematic liquid crystals [6,31] (see also [24, Appendix B]). There exists a vast recent literature on the mathematical study of the Landau-de Gennes theory, and we refer interested readers to [2-7, 9-15, 17-24, 30-32, 34, 35, 38-40] as well as the references cited therein.In this paper, we study a basic model for the evolution of an incompressible nematic liquid crystal flow, which was first proposed by Beris-Edwards [7]. The resulting PDE system consists of the Navier-Stokes equations for the fluid velocity u and nonlinear convection-diffusion equations of parabolic type for the Q-tensor (see, e.g., [2,10,34]). To simplify the mathematical setting, throughout this paper we confine ourselves to the two dimensional periodic case such that d = 2 and Ω = T 2 , where T 2 stands for the periodic box with period ℓ i in the i-th direction with O = (0, ℓ 1 ) × (0, ℓ 2 ) being the periodic cell. Without loss of generality, we simply set O = (0, 1) 2 . Then the coupled system we are going to study takes the following form:

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

Cited by 34 publications

References 27 publications

Nonstationary models for liquid crystals: A fresh mathematical perspective

Nonstationary models for liquid crystals: A fresh mathematical perspective

Global Strong Solutions of the Full Navier--Stokes and $Q$-Tensor System for Nematic Liquid Crystal Flows in Two Dimensions

Global well-posedness of the two dimensional Beris–Edwards system with general Laudau–de Gennes free energy

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