A note on a nonlocal nonlinear reaction–diffusion model

We give an application of the Crandall–Rabinowitz theorem on local bifurcation to a system of nonlinear parabolic equations with nonlocal reaction and cross-diffusion terms as well as nonlocal initial conditions. The system arises as steady-state equations of two interacting age-structured populations.     

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

A new method for solution of the evolution of plane curves satisfying the geometric equation v=β(x,k,ν), where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ? ?² at the point x∈Γ, is proposed. We derive a governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non‐trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to find a numerical solution for 2D anisotropic interface motions and image segmentation problems. Copyright © 2004 John Wiley & Sons, Ltd.     

On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D’Alembert–Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results.     

Chaos in fluid-conveying NSGT nanotubes with geometric imperfections

A scale-dependent model of nanobeams with large deformations is developed to investigate the influences of a geometric imperfection on the chaotic response of nanotubes. In order to comprehensively simulate the effects of being at nanoscales, a nonlocal strain gradient theory (NSGT) is utilised. To model a geometric imperfection, an initial deflection is taken into account for the nanosystem. Since the relative motion between the nanofluid and nanotube at the interface is not negligible, Karniadakis–Beskok assumptions are employed to incorporate the effects of this relative motion. Utilising an energy-work balance technique, the nonlinear governing equations are derived for the coupled motion of the nanofluid-conveying NSGT nanotube. Finally, the influences of the geometric imperfection on the motion response are analysed using a direct-time-integration approach and a Galerkin scheme.     

Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using Potentials

New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.     

A Third-Order Dispersive Flow for Closed Curves into Kähler Manifolds

This article is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into Kähler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method.     

Boundary stabilization of vibration of nonlocal micropolar elastic media

In this paper, we study the stabilization problem of vibration of linearized three-dimensional nonlocal micropolar elasticity. For this purpose, we need to demonstrate the well-posedness of the system of equations governing the vibration of three-dimensional nonlocal micropolar media for both forced (i.e. with boundary feedback) and unforced cases. We assume the non-homogeneous system of equations for the unforced (uncontrolled) case to establish the well-posedness. It should be pointed out that the well-posedness of the evolution equations in micropolar case has been studied by many authors; but, the well-posedness in the nonlocal micropolar is an open problem. Our tools in well-posedness analysis are the semigroup techniques. Afterwards, we pursue the stabilization problem and show that the vibration of the nonlocal micropolar elastic media will be eventually dissipated under boundary feedback actions consisting of stress and couple stress feedback laws. These control laws are simple, linear and can be easily implemented in practical applications. The stabilization proof is accomplished using Lyapunov stability and LaSalle’s invariant set theorems.     

Asymptotic Analysis of a Hyperbolic Mixed Problem in One-Dimensional Viscoelasticity

We present an asymptotic analysis of the boundary-generated, small-amplitude, high-frequency waves in a one-dimensional, semi-infinite, viscoelastic solid characterized by a single-integral constitutive functional. The equations governing the wave motion constitute a 2×2 system of hyperbolic Volterra integrodifferential equations. The method of analysis is based on a single-wave expansion of nonlinear geometric optics.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

Nonlocal Geometric Expansion of Convex Planar Curves

We consider a class of nonlocal geometric equations for expanding curves in the plane, arising in the study of evolutions governed by Monge-Kantorovich mass transfer. We construct convex solutions, given convex initial data. In order to obtain such solutions, we develop a new version of Perron's method. We give applications to the problem of characterizing fast/slow diffusion limits.     

Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

Dynamics of Poisson-Nernst-Planck Systems and Ionic Flows Through Ion Channels: A Review

Poisson-Nernst-Planck systems are basic models for electrodiffusion process, particularly, for ionic flows through ion channels embedded in cell membranes. In this article, we present a brief review on a geometric singular perturbation framework for analyzing the steady-state of a quasi-one-dimensional Poisson-Nernst-Planck model. The framework is based on the general geometric singular perturbed theory from nonlinear dynamical system theory and, most crucially, on the reveal of two specific structures of Poisson-Nernst-Planck systems. As a result of the geometric framework, one obtains a governing system–an algebraic system of equations that involves all physical quantities such as protein structures of membrane channels as well as boundary conditions, and hence, provides a complete platform for studying the interplay between protein structure and boundary conditions and effects on ionic flow properties. As an illustration, we will present concrete applications of the theory to several topics of biologically significant based on collaboration works with many excellent researchers.     

Uniqueness results for nonlocal Hamilton-Jacobi equations

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.     

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.     

A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.     

Nonlocal symmetries in two-field divergent evolutionary systems

We evaluate nonlocal symmetries for third-order exactly integrable two-field divergent evolutionary equations. These symmetries, regarded as evolutionary equations, commute with higher analogues of the underlying original equations and seem to be exactly integrable. By differentiating nonlocal systems and changing the variables, we obtain local hyperbolic systems and third-order nonevolutionary systems. We find a zero-curvature representation for some of the new systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 351–363, September, 2008.     

Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh–Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.     

Analysis of a system related to a model for phase transitions in dissipative isochoric thermoviscoelastic materials

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Kinematics of coupler curves for spherical four-bar linkages based on new spherical adjoint approach

The local and global geometric properties of spherical coupler curves constitute spherical kinematics of spherical four-bar linkages, which can be adopted to reveal distribution characteristics of spherical coupler curves. New unified spherical adjoint approach is established in the paper to study both the local and global geometric properties in order to enrich the atlas of spherical coupler curves with geometric characteristics. Since the constraint curve of spherical four-bar linkage is a simple spherical circle and the spherical centrodes imply intrinsic properties of spherical motion of the coupler link, they are in their turn taken as the original curves in spherical adjoint approach to derive the geodesic curvature and analyze the local geometric characteristics of the spherical coupler curves. The conditions for different spherical double points, such as spherical crunodes, tacnodes and cusps of the spherical coupler curve are derived through the spherical adjoint approach. The spherical surface of the coupler link can be divided into several areas by the spherical moving centrode and the spherical tacnode's tracer curve. The points in each area trace spherical coupler curves with a specific shape. The characteristic points, which trace spherical coupler curves with cusp, geodesic inflection point, spherical Ball point, spherical Burmester point, crunode and tacnode can be readily located in the coupler link by the modelling procedure and the derived condition equations. In the end the distribution of spherical coupler curves with both local and global characteristics is elaborated. The research proposes systematic geometric properties of spherical coupler curves based on the new established approach, and provides a solid theoretical basis for the kinematic analysis and synthesis of the spherical four-bar linkages.