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.     

Existence of surface smectic states of liquid crystals

The Landau–de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state.     

Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.     

Freedericksz transitions in circular toroidal layers of smectic C liquid crystals

The aim of this paper is to consider theoretically a Freedericksztransition for concentric toroidal layers of smectic C liquidcrystal arising from a simple geometric setup, thereby extendingthe results of Atkin & Stewart [Q. Jl Mech. Appl. Math.,47, 1994] who considered spherical layers of smectic C in theusual cone and plate geometry. Application of smectic continuumtheory leads, after suitable approximations are made, to a lineargoverning equilibrium equation which is satisfied by both thetrivial solution and a variable solution involving Bessel functions.We are able to determine the critical magnitude _cH of the magneticfield H at which this variable solution exists, and a standardenergy comparison reveals that the variable solution is expectedto be more energetically favourable than the zero solution providedH > _cH. Numerical examples of critical thresholds are given,which are comparable to those in the literature for nematics.The paper ends with a discussion section and some indicationof possible future work.     

Configurations of few lines in 3-space. Isotopy,chirality and planar layouts

We give a characterization of the planar layouts of configurations with at most five lines. From this we obtain a new proof of Viro's theorem that the isotopy type of such configurations is completely determined by chirality. We extend this result to labelled configurations. We also give an infinite family of non-realizable line diagrams, called alternatingC-angles, not containing non-realizable subdiagrams.     

可压液晶流之解的渐近行为分析

本文考察可压液晶流的长时间行为. 如果稳态函数d_s 极小化Ginzburg-Landau 逼近能量, 则可以证明有限能量弱解(ρ, u, d)(t, x) 收敛于(ρ_s, 0, d_s)(x), 这里ρ_s 是由总质量守恒唯一确定的密度函数.     

Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals

In this paper, we consider the weak solution of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in ${\mathbb{R}}^3$. When the initial data are of small energy and initial density is positive and essentially bounded, we prove the existence of a global weak solution in ${\mathbb{R}}^3$. The large-time behavior of a global weak solution is also established.     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.     

Regularity of solutions to the liquid crystal flows with rough initial data

In this paper, we are concerned with the regularity of solutions to the liquid crystal flows with rough initial data in Rⁿ$R^n$. We prove that the solution constructed by Wang (2011) in [23] has higher regularity. Moreover we obtain a decay estimate in time for any space derivative.     

Regularity of the solutions to the liquid crystal equations with small rough data

In [30], the author has proved the existence of Koch–Tataru [12] type solution to the incompressible nematic liquid crystal flow with initial data (_u0,_d0)$(u_0,d_0)$ in BMO⁻¹×BMO${\mathit{BMO}}^{-1}\times \mathit{BMO}$. In this paper, we shall present that the solution in [30] has arbitrary spatial regularity. Furthermore, we shall give the decay estimates for any space derivative.     

Growth morphology and structural characteristic of C70 single crystals

Large size C₇₀ single crystals with the dimension of more than 5 mm are grown from the vapor phase by controlling nucleation. X-ray diffraction and electron diffraction confirm that in the C₇₀ single crystal a phase of the hexagonal close-packed (hcp) structure coexists with a minor face-center-cubic (fcc) phase at room temperature. The morphologies and their formation mechanism of the C₇₀ single crystals are investigated by means of scanning electron micrascopy and optical microscopy. The influence of growth conditions on the morphologies of C₇₀ single crystals is discussed. Project supported by the National Natural Science Foundation of China (Grant No. 59772026).     

Are Branch and Bound and A* Algorithms Identical?

Heuristic Search and Branch and Bound algorithms have many similarities. In this paper, we address the question of the extent to which they are similar. We firstly show that these algorithms apply the same principles, although generating graphs with different properties: Heuristic Search can explore any kind of graphs, whereas the Branch and Bound algorithm generates particular graphs with a restrictive inheritance property. Nevertheless, we show that the Branch and Bound principles can be used to perform Heuristic Search. We conclude that the two types of algorithms are therefore essentially identical; they only differ at the interpretation level.     

Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞

We study hypercyclicity and supercyclicity of weighted shifts on ℓ^∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c₀ or ℓ^p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ^∞ is norm nowhere dense. Finally, we show that ℓ^∞ supports an isometry that is weak * sequentially supercyclic.     

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.     

Existence and uniqueness of analytic solution for unsteady crystals with zero surface tension

We study the initial value problem for two-dimensional dendritic crystal growth with zero surface tension. If the initial data is analytic and close to Ivantsov steady solution, it is proved that unique analytic solution exists locally in time. The analysis is based on a Nirenberg Theorem on abstract Cauchy-Kovalevsky problem in properly chosen Banach spaces.     

Bounds for the fundamental solutions of elliptic and parabolic equations

It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

A class of functional equation and fractal interpolation functions

A new class of functional equation in C0(I) is investigated. It is proved that some class of FIF satisfies the functional equation. Another functional equation is constructed. Theirsolutions can approximate FIF arbitrarily. And a new approximate estimate between FIF andinterpolated function is given.