Study of a perforated thin plate according to the relative sizes of its different parameters

We are interested in the study of a thin plate, periodicially perforated by cylindrical holes, the axes of which are perpendicular to the plane of the plate. A horizontal section of the plate specifies its geometry, and shows a periodicity in the order of ?. The thickness of the plate is equal to e. The ratio of material is small, and is characterized by the parameter δ, the thickness of the bars being equal to ?δ. In this paper, we study the dependence of displacements on e, ? and δ, and to give equivalent limits when e, then ?, and finally δ, tend towards zero. An interesting result obtained in this work is the negative Poisson coefficient of the final equivalent material. Although this coefficient is theoretically between ?½ and 1, most materials encountered in practice have a positive one.     

On the number of singular points of weak solutions to the Navier‐Stokes equations

We consider a suitable weak solution to the three‐dimensional Navier‐Stokes equations in the space‐time cylinder Ω × ]0, T[. Let Σ be the set of singular points for this solution and Σ (t) ≡ {(x, t) ∈ Σ}. For a given open subset ω ? Ω and for a given moment of time t ∈]0, T[, we obtain an upper bound for the number of points of the set Σ(t) ? ω. © 2001 John Wiley & Sons, Inc.     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Hamiltonian Dynamic Equations for Fluid Films

Two‐dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid film Here δ/δ t is the invariant time derivative, ρ is the two‐dimensional density of the film, C is the normal component of the velocity field, V^α are the tangential components, B_αβ is the curvature tensor, and ?_α is the covariant surface derivative. The surface energy density e(ρ) is a generalization of the common surface tension and e_ρ is its derivative. The Laplace model corresponds to e(ρ) =σ/ρ , where σ is the surface tension density. The proper choice of e(ρ) in paramount in capturing particular effects displayed by fluid films. The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text.     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Domain separation by means of sign changing eigenfunctions of p-laplacians

We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p-Laplacians,p→1 under humogeneous Neumann houndary conditions. These eigenfunctions are proven to be limtes of a steepest descent method applied to suitable norm quotients. Finally we use these ideas for the construction of separators on simplex grids.     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

On the random 2‐stage minimum spanning tree

It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47–56] that if the edge costs of the complete graph K_n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to . Here we consider the following stochastic two‐stage version of this optimization problem. There are two sets of edge costs c_M: E → ? and c_T: E → ?, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c_M(e) and c_T(e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c_M(e), or to wait until its Tuesday price c_T(e) appears. The set of edges X_M bought on Monday is then completed by the set of edges X_T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o(1). We show that, in the case of two‐stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant ε > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) ? 1/2 + o(1). The threshold heuristic is shown to be sub‐optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning out‐arborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 ? 1/e + o(1). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

Robust family of exponential attractors for isotropic crystal models

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter β > 0), of two phase‐field equations, namely, the Allen–Cahn and the Cahn–Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

A new shallow water model with polynomial dependence on depth

In this paper, we study two‐dimensional Euler equations in a domain with small depth. With this aim, we introduce a small non‐dimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. We obtain a model for ε small that, after coming back to the original domain, gives us a shallow water model that considers the possibility of a non‐constant bottom, and the horizontal velocity has a dependence on z introduced by the vorticity when it is not zero. This represents an interesting novelty with respect to shallow water models found in the literature. We stand out that we do not need to make a priori assumptions about velocity or pressure behaviour to obtain the model. The new model is able to approximate the solutions to Euler equations with dependence on z (reobtaining the same velocities profile), whereas the classic model just obtains the average velocity. Copyright © 2007 John Wiley & Sons, Ltd.     

On the instability of resonances in parallel-plane waveguides

It has been observed¹³ that the propagation of acoustic waves in the region Ω₀= ?² × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown⁹ in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω₀. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω₀ ?B, where B is a smooth bounded domain with B??Ω₀. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x₁, x₂, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω₀ will be discussed in a subsequent paper.     

Mathematical and numerical modellization of large‐scale oceanic waves

This paper is devoted to the theoretical and numerical study of a method which computes the variability of current and density in an oceanic domain. The equations are of Navier–Stokes type for the velocity and of transport‐diffusion type for the density. They are linearized around a given mean circulation and modified by physical assumptions including hydrostatic approximation. The existence and uniqueness of a solution are proved for two sets of equations: first the three‐dimensional problem and then the two‐dimensional cyclic problem derived by assuming a sinusoïdal x‐dependence for the perturbation of the mean flow. The latter corresponds to a modellization of tropical instability waves which are illustrated by the ‘El Nino’ phenomenon. These two problems differ from classical ones because of hydrostatic approximation, boundary conditions imposed by the oceanic domain and complex‐valued functions for the cyclic case. A numerical model is developed for the two‐dimensional cyclic equations. Time discretization is performed by the characteristics method; space discretization uses Q₁ finite elements. Numerical results are presented in a realistic case corresponding to the tropical Pacific Ocean. Copyright © 1999 John Wiley & Sons. Ltd.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Asymptotic Localization of Energy in Nondisordered Oscillator Chains

We study two popular one‐dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ? > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ?, and we provide strong evidence that it decays faster than any power law in ? as ? → 0. The weak coupling regime also translates into a high‐temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM‐like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.     

Relative influence of two small parameters in non uniformly elliptic homogenization problems

We study three elliptic problems depending on two small parameters (? = homogenization parameter and δ = perturbation parameter which causes non uniform ellipticity). In each case, the homogenized operator corresponding to the second order operator is independent of the way (?, δ) → (0, 0), but the convergence results and the limit solution do depend on the relative size of ? and δ; the relevant parameter is δ?^?2.     

A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(D ² u) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng (J Sci Comput 38(1):74–98, 2009). The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation -eD²u^e + det(D²u^e) = f{-\epsilon\Delta^2u^\epsilon + {\rm det}(D^2u^\epsilon) = f} with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103:155–169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.