Constructing two powerful methods to solve the Thomas–Fermi equation

Nonlinear Dynamics - We implement the reproducing kernel method and SL(2, R)-shooting method to solve the Thomas–Fermi equation. Powerful techniques are demonstrated by reproducing...     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation

In a recent article (Cancès et al. in Commun Math Phys 281:129–177, 2008), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree–Fock model, the ground state electronic density matrix is decomposed as ${\gamma = \gamma^0_{\rm per} + Q_{\nu,\varepsilon_{\rm F}}}$ , where ${\gamma^0_{\rm per}}$ is the ground state density matrix of the host crystal and ${Q_{\nu,\varepsilon_{\rm F}}}$ the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level ε _F being kept fixed. The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix ${Q_{\nu,\varepsilon_{\rm F}}}$ (which is known to be a self-adjoint Hilbert–Schmidt operator on ${L^2(\mathbb{R}^3)}$ ). We show in particular that if ${\int_{\mathbb{R}^3}\,\nu \neq 0, Q_{\nu,\varepsilon_{\rm F}}}$ is not trace-class. Moreover, the associated density of charge is not in ${L^1(\mathbb{R}^3)}$ if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect ν, the linear and nonlinear terms of the resolvent expansion of ${Q_{\nu,\varepsilon_{\rm F}}}$ . Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler–Wiser formula.     

On the Theory of the α–β Phase Transition in Quartz

The – phase transition in quartz has features which workers have long found puzzling. My purpose is to explore using a theory for this that is rather different conceptually from those used by other workers. It does suggest different kinds of experiments, likely to shed light on aspects of the transition, and a possible modification of theory which might be helpful in understanding some subtleties.     

Local instability of plates with pressed–in annular inclusions at the elastoplastic behavior of materials

The local instability of plates with annular inclusions is studied within the framework of exact three–dimensional equations. A numerical experiment is performed for the case where two rings are pressed in a plate made of the same material as the rings. The effect of the physicomechanical parameters of a medium on the critical contact pressures is studied.     

A Revised Exposition of the Green–Naghdi Theory of Heat Propagation

We offer a revised exposition of the three types of heat-propagation theories proposed by Green and Naghdi. Those theories, which make use of the notion of thermal displacement and allow for heat waves, are at variance with the standard Fourier theory; they have attracted considerable interest, and have been applied in a number of disparate physical circumstances, where heat propagation is coupled with elasticity, viscous flows, etc. (Straughan in Heat waves. Applied mathematical sciences, vol. 177. Springer, Berlin, 2011). However, their derivation is not exempt from criticisms, that we here detail, in hopes of opening the way to reconsideration of old applications and proposition of new ones.     

On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

We study the nonhomogeneous boundary value problem for Navier–Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional, bounded, multiply connected domain ${\Omega = \Omega_1 \backslash \overline{\Omega}_2, \overline\Omega_2\subset \Omega_1}$ . We prove that this problem has a solution if the flux ${\mathcal{F}}$ of the boundary value through ?Ω ₂ is nonnegative (inflow condition). The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-sided maximum principle for the total head pressure corresponding to this solution.     

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.     

Robin Boundary Effects in the Darcy–Rayleigh Problem with Local Thermal Non-equilibrium Model

Transport in Porous Media - Permeability is one of the key parameters for quantitatively evaluating groundwater resources and accurately predicting the rates of water inflows into coal mines. This...     

Stability of the Camassa–Holm Multi-peakons in the Dynamics of a Shallow-Water-Type System

Consideration herein is the stability issue of a variety of superpositions of the Camassa–Holm peakons and antipeakons in the dynamics of the two-component Camassa–Holm system, which is derived in the shallow water theory. These wave configurations accommodate the ordered trains of the Camassa–Holm peakons, the ordered trains of Camassa–Holm antipeakons and peakons as well as the Camassa–Holm multi-peakons. Using the features of conservation laws and the monotonicity properties of the local energy, we prove the orbital stability of these wave profiles in the energy space by the modulation argument.     

Local Null Controllability of the N-Dimensional Navier–Stokes System with N − 1 Scalar Controls in an Arbitrary Control Domain

In this paper we deal with the local null controllability of the N-dimensional Navier–Stokes system with internal controls having one vanishing component. The novelty of this work is that no condition is imposed on the control domain.     

Creation–annihilation process of limit cycles in the Rayleigh–Duffing oscillator

The present paper examines the creation?Cannihilation process of limit cycles in the Rayleigh?CDuffing oscillator with negative linear damping and negative linear stiffness. It is obtained by the perturbation method, in which the number of limit cycles in the Rayleigh?CDuffing oscillator varies with the linear damping and stiffness. Numerical simulations are performed in order to confirm the analytically obtained creation?Cannihilation process of limit cycles. Moreover, we compare the process of limit cycles in the Rayleigh?CDuffing oscillator to that of limit cycles in the van der Pol?CDuffing oscillator. The difference in these oscillator is only in nonlinear forces, which causes a qualitative difference in the creation?Cannihilation processes.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

High-Frequency Asymptotics and One-Dimensional Stability of Zel’dovich–von Neumann–D?ring Detonations in the Small-Heat Release and High-Overdrive Limits

We establish one-dimensional spectral, or “normal modes”, stability of Zel’dovich–von Neumann–D?ring detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed, verifying numerical observations made by Erpenbeck in the 1960s. The key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite, regular perturbation problem, and a careful high-frequency analysis depending uniformly on model parameters. The latter recovers and extends to arbitrary amplitudes the important result of high-frequency stability established by Erpenbeck by somewhat different techniques. Notably, the techniques used here yield quantitative estimates well suited for numerical stability investigation.     

The homogenization method of Bakhvalov–Pobedrya in the composite mechanics

The role of Professor Pobedrya in the development of the homogenization method in the mechanics of composite materials with periodic structure is discussed. A generalization of the homogenization method is proposed to the case of heterogeneous bodies whose structure is not periodic.     

On the Wellposedness of Three-Dimensional Inhomogeneous Navier–Stokes Equations in the Critical Spaces

We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is small enough in the critical Besov space [(B)dot]^1/2_2,1(mathbbR³){dot B^{1/2}_{2,1}(mathbb{R}^3)} , this system has a unique global solution.