Mathematics and liquid crystals

Ball, J. M.

doi:10.1080/15421406.2017.1289425

Cited by 87 publications

(129 citation statements)

References 174 publications

(200 reference statements)

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“…As described, for example, in [4,6,18], it is not sufficient to specify the energy functional, as part of the model is also the function space in which minimizers (and critical points etc) are to be sought. The larger this function space, the wilder potential singularities of minimizers and critical points may be.…”

Section: Function Spacesmentioning

confidence: 99%

“…Changing the function space can change the predicted minimizers, as well as the minimum value of the energy (the Lavrentiev phenomenon) as described both for nonlinear elasticity and liquid crystals in [4].…”

Section: Function Spacesmentioning

confidence: 99%

“…Here (see also [4]) we confine ourselves to showing that in general, for W (n, ∇n) having the Oseen Frank form (59) with the Frank constants satisfying the Ericksen inequalities (64), for sufficiently small δ the infimum I inf of I(n) among n ∈ SBV (Ω δ , S…”

Section: Order Reconstructionmentioning

confidence: 99%

“…These issues are perhaps better understood for solid mechanics than for liquid crystals (for more discussion see [4]), a particularly strong analogy being with function spaces used to describe different kinds of fracture in solids.…”

Section: Introductionmentioning

confidence: 99%

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Liquid Crystals and Their Defects

Ball

2017

Mathematical Thermodynamics of Complex Fluids

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Section: Function Spacesmentioning

confidence: 99%

Section: Function Spacesmentioning

confidence: 99%

Section: Order Reconstructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Liquid Crystals and Their Defects

Ball

2017

Mathematical Thermodynamics of Complex Fluids

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“…The Landau-de Gennes theory is a fundamental continuum theory that describes the state of nematic liquid crystals [4,16]. In this framework, the local orientation and degree of ordering for the liquid crystal molecules are modelled by a symmetric, traceless d × d matrix Q in R d (d = 2, 3), known as the Q-tensor order parameter [5,6].…”

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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