Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that inclusions are separated by distances of the same order ɛ as their size, we find a limiting functional as ɛ approaches zero. We generalize the variational method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg–Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.     

A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Homogenization of a Ginzburg–Landau functional

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ?, we obtain a limiting functional as $?\to 0$. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

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.     

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two‐dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design. We split the infinite sum over the lattice of images into a directly summed “near” part plus a small number of auxiliary sources that represent the (smooth) remaining “far” contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank‐1 or rank‐3 correction and a Schur complement, leaves a well‐conditioned square system that can be solved iteratively using fast multipole acceleration plus a low‐rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension‐ and PDE‐independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close‐to‐touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against high‐accuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 10⁴ randomly located inclusions per unit cell. © 2018 Wiley Periodicals, Inc.     

Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

This is the first of a series of papers devoted to the initial value problem for the one‐dimensional Euler system of compressible fluids and augmented versions containing higher‐order terms. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity‐capillarity limit associated with the Navier‐Stokes‐Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. Our method of proof combines energy and effective energy estimates, a nonlinear Sobolev inequality, high‐integrability properties for the mass density and for the velocity, and compactness properties based on entropies.© 2015 Wiley Periodicals, Inc.     

Smectic A liquid crystal configurations with interface defects

We study planar energy minimizing configurations of smectic A liquid crystal materials and classify the corresponding defect structures. We investigate focal conic configurations in wedge, non‐parallel plates, funnel‐shaped domains, and non‐concentric annuli. The application of the stability condition for focal conics is relevant to the specification of the location of the interfacial defects. Self‐similar structures are discussed for a class of solutions with the same bulk energy. We propose surface energies terms to serve as selection mechanisms of particular self‐similar configurations. We also show how the modelling of chevron texture naturally arises in the present framework. Copyright © 2001 John Wiley & Sons, Ltd.     

Effective Elastic Characteristics of a Piecewise-Homogeneous Isotropic Body with a System of Rectilinear Thin Inclusions under Conditions of Longitudinal Shear

We study the problem of determination of effective elastic characteristics of a piecewise-homogeneous isotropic body containing a twice-periodic system of tunnel thin rectilinear inclusions. The body is subjected to antiplane strain, and the conditions of ideal mechanical contact between the matrix and inclusions are satisfied. We reduce the solution of the problem to a system of integro-differential equations which enable us to find the expressions for the effective elastic characteristics of the composite. We also present the results of numerical analysis of these characteristics for a rectangular grid of periods and for various geometric and mechanical parameters of the problem.     

Instanton in the Field of a Pointlike Source of a Euclidean Non-Abelian Field

We consider the behavior of an (anti)instanton in the field of a pointlike source of a Euclidean non-Abelian field and investigate (anti)instanton deformations described by variations of its characteristic parameters. We formulate the variational problem of seeking the corresponding “crumpled” topological configurations and solve it algebraically using the Ritz method (of multipole decomposition of deformation fields). We investigate the domain of parameters specific for the instanton liquid model. We propose a simple method of taking contributions of such configurations in the functional integral into account approximately. In the framework of superposition analysis, we obtain an estimate for the mean energy of the source in the instanton liquid, and the estimate increases linearly with distance. In the case of a dipole in the color-singlet state, the energy is a linear function of the distance between the sources with the “strength” coefficient agreeing well with the known model and lattice estimates. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 267–298, February, 2006.     

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq–Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.     

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

We consider a nematic liquid crystal occupying the exterior region in \({\mathbb {R}}^3\) outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to an external field, modeled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough, we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.     

On integro‐differential inclusions with operator‐valued kernels

We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.     

On an Isoperimetric Problem with a Competing Nonlocal Term I: The Planar Case

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a nonlocal repulsive term. We characterize existence, nonexistence, and radial symmetry of the minimizers as a function of mass in the situation where the nonlocal term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low‐energy configuration to split into smaller pieces when the nonlocal term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass, suggesting that for large masses low‐energy configurations consist of many roughly equal‐size pieces far apart. In the case of slowly decaying kernels, we give a complete characterization of the minimizers. © 2012 Wiley Periodicals, Inc.     

Effective moduli of particulate solids: Lubrication approximation method

To efficiently calculate the effective properties of a composite, which consists of rigid spherical inclusions not necessarily of the same sizes in a homogeneous isotropic elastic matrix, a method based on the lubrication forces between neighbouring particles has been developed. The method is used to evaluate the effective Lamé moduli and the Poisson's ratio of the composite, for the particles in random configurations and in cubic lattices. A good agreement with experimental results given by Smith (1975) for particles in random configurations is observed, and also the numerical results on the effective moduli agree well with the results given by Nunan & Keller (1984) for particles in cubic lattices.     

Conservative solutions to a system of variational wave equations

We investigate a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The main difficulty arises from the possible concentration of energy. We construct the solution by introducing a new set of variables depending on the energy, whereby all singularities are resolved.     

Linear and non‐linear stability thresholds for thermal convection in a box

We analyse the problem of finding instability thresholds and global non‐linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non‐uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non‐Linear Mech. 1999; 34 :603–613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible sub‐critical instabilities may be found. Copyright © 2006 John Wiley & Sons, Ltd.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

Nontopological Condensates for the Self‐Dual Chern‐Simons‐Higgs Model

For the abelian self‐dual Chern‐Simons‐Higgs model we address existence issues of periodic vortex configurations—the so‐called condensates—of nontopological type as k → 0, where k > 0 is the Chern‐Simons parameter. We provide a positive answer to the longstanding problem on the existence of nontopological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as k → 0, a question that is of definite physical interest. © 2015 Wiley Periodicals, Inc.