Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

In this paper, we consider the uniform estimates of strong solutions in the Mach number ? and t ∈ [0,∞) for the compressible nematic liquid crystal flows in a 3‐D bounded domain , provided the initial data are small enough and the density is close to the constant state. Here, we consider the case that the velocity field satisfies the Dirichlet boundary condition. Based on the uniform estimates, we obtain the global convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.     

Studies of layering and chirality of smectic A* liquid crystals

We discuss mathematical and physical aspects of the phase transition from nematic to smectic A liquid crystals. The first approach deals with analyzing a model obtained from the Maier-Saupe theory of nematic by taking into account that elongated liquid crystal molecules present distinguishable ends. Moreover, we represent long range microscopic interactions by means of nonlocal free energy functionals. The smectic configurations emerge as solutions of the extended nematic theory, through a modulation process. The second part of the article deals with energy minimization of the de Gennes free energy for smectic A* liquid crystals, and with the study of uniform twist grain boundary (TGB) structures. The goal is to mathematically justify parameter regions of the phase diagram of the transition between nematic and smectic A liquid crystals. Both approaches complement each other from the point of view that, while the first one deals with mechanisms causing layer arrangements, the second approach focuses on how chirality and layer effects interact, in a system with preassumed periodicity. The A* notation refers to chiral liquid crystals.     

On the Weak Solutions to a Stochastic 2D Simplified Ericksen-Leslie Model

Potential Analysis - We study in this article a stochastic version of a 2D Ericksen-Leslie systems. The system model the dynamic of nematic liquid crystals under the influence of stochastic...     

Global existence of the finite energy weak solutions to a nematic liquid crystals model

In this paper, we are concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. For a bounded domain in Ropf³, under the assumption that initial density belongs to , we show the global existence of weak solutions to the nematic liquid crystals model with a penalized system. Furthermore, we also obtain the energy inequality for weak solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow-up of the Smooth Solution to the Compressible Nematic Liquid Crystal System

In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality.     

A Serrin criterion for compressible nematic liquid crystal flows

This paper is concerned with a simplified system, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. We establish a blowup criterion for three‐dimensional compressible nematic liquid crystal flows, which is analogous to the well‐known Serrin's blowup criterion for three‐dimensional incompressible viscous flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of solutions to the biaxial nematic liquid crystals with two order parameter tensors

The recent paper considers some mathematical problems on a tensor model with two order parameters to the biaxial nematic liquid crystals. The energy in this model contains a cubic term and is unbounded from below. We study the gradient flow generated by this energy in two dimensions. We focus on establishing global existence of classical and weak solutions.     

Reduced Landau-de Gennes functional and surface smectic state of liquid crystals

In [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382], based on the de Gennes analogy between liquid crystals and superconductivity [P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753-756], the second author introduced the critical wave number Qc₃ (which is an analog of the upper critical field Hc₃ for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analogy of the surface superconducting state. In a surface smectic state, the bulk liquid crystal is in the nematic state, and a thin layer of smectic appears in a helical strip on the surface of the sample. In this paper we study an approximate form of the Landau-de Gennes model of liquid crystals, and examine the behavior of minimizers, in particular the boundary layer behavior. Our work shows the importance of the joint chirality constant qτ, which is the product of wave number q and chirality τ and also appears in the work of [P. Bauman, M. Calderer, C. Liu, D. Phillips, The phase transition between chiral nematic and smectic A^∗ liquid crystals, Arch. Rational Mech. Anal. 165 (2002) 161-186] and [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382]. The joint chirality constant of a liquid crystal is useful to predict whether the liquid crystal is of type I or type II, and it is also useful to examine whether the liquid crystal is in a surface smectic state. The results in this paper suggest that a liquid crystal with large Ginzburg-Landau parameter κ and large joint chirality constant qτ exhibits type II behavior, and it will be in the surface smectic state if qτ∼bκ² for some β₀<b<1, where β₀ is the lowest eigenvalue of the Schrödinger operator with a unit magnetic field in the half space, and 0<β₀<1. We also show that a liquid crystal with small qτ exhibits type I behavior.     

Large deviations for nematic liquid crystals driven by pure jump noise

In this paper, we establish a large deviation principle for a stochastic evolution equation, which describes the system governing the nematic liquid crystals driven by pure jump noise. The proof is based on the weak convergence approach.     

On nematic liquid crystals with variable degree of orientation

Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.     

Local Well-Posedness for the Compressible Nematic Liquid Crystals Flow with Vacuum

In this paper we prove the local well-posedness of strong solutions to the compressible nematic liquid crystals flow with vacuum in a bounded domain $\Omega\subset \mathbb{R}^3$.     

Existence and blow-up of the solutions to the viscous quantum magnetohydrodynamic nematic liquid crystal model

In this paper, we investigate the coupled viscous quantum magnetohydrodynamic equations and nematic liquid crystal equations which describe the motion of the nematic liquid crystals under the magnetic field and the quantum effects in the two-dimensional case. We prove the existence of the global finite energy weak solutions by use of a singular pressure close to vacuum. Then we obtain the local-in-time existence of the smooth solution. In the final, the blow-up of the smooth solutions is studied. The main techniques are Faedo-Galerkin method, compactness theory, Arzela-Ascoli theorem and construction of the functional differential inequality.     

An Elementary Proof of Eigenvalue Preservation for the Co-rotational Beris-Edwards System

Journal of Nonlinear Science - We study the corotational Beris-Edwards system modeling nematic liquid crystals and revisit the eigenvalue preservation property discussed in Wu et al. (Arch Rational...     

Global well‐posedness of the nonhomogeneous incompressible liquid crystals systems

This paper examines the initial‐value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow‐up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena

This is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals. It also includes some new results concerning disclinations (line defects) which have not been published elsewhere.     

Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

We analyze the radially symmetric solution corresponding to the vortex defect (the so-called melting hedgehog) in the Landau–de Gennes theory for nematic liquid crystals. We prove the existence, uniqueness and stability results of the melting hedgehog.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

The stability of nematic liquid crystals under crossed electric and magnetic fields

The alignment of the director in nematic liquid crystals is known to be influenced by electric and/or magnetic fields. The stability of known traveling wave solutions is examined for a best fit cubic equation which approximates the nonlinear sinusoidal terms in the dynamic equation for the director when it is subjected to crossed electric and magnetic fields.