The effective conductivity of a randomly inhomogeneous medium

Using a representation of the solution to the diffusion equation in a randomly inhomogeneous medium in the form of a Feynman path integral an explicit expression is obtained for the effective conductivity in a space of arbitrary dimension. A calculation of the path integral only turns out to be possible in the case of a large-scale limit. In particular, it is shown that in the three-dimensional case the expression for the effective conductivity does not admit of an expansion in terms of the conductivity variance. This indicates that the use of standard perturbation theory in the form of an expansion in terms of the conductivity fluctuations is incorrect.     

Homogenization of a non‐linear degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time‐dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors a_j, j=1,2,3 are non‐linear, monotone functions of the temperature gradient. The heat capacities c_j(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=lim_ε→0ε^p/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.     

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

In this work we study the existence of wave solutions for a scalar reaction-diffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen–Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting \(u = 0\) and \(u = 1\). Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction.     

一类非线性微分方程的概周期解

运用Leray-Schauder不动点定理和Liapunov函数方法,研究了一类非线性微分方程的概周期解,得到了该微分方程概周期解存在的充分条件.     

The conductivity of a medium interrupted by voids

Résumé On calcule la conductivité d'un milieu en forme de parallelépipède (ou rectangle) percé par une répartition de trous et cadré par des surfaces de conductivité infini ou nul. On emploie un modèle stochastique où on associe une variable aléatoire au processus physique. Les résultats obtenus sont comparés avec les théories de Maxwell, Rayleigh, Wiener et Hashin-Shtrikman.     

Global existence for a new periodic integrable equation

We establish the local well-posedness for a new periodic integrable equation. We show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time.     

Wave breaking for a periodic shallow water equation

The focus of this paper is on the blow-up of a recently derived one-dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler's equations in the shallow water regime. Some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case.     

The pressure equation for fluid flow in a stochastic medium

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R ^d of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

The porous medium equation in a two-component domain

We consider a system of two porous medium equations defined on two different components of the real line, which are connected by the nonlinear contact condition

     

Multiple periodic solutions for a nonlinear suspension bridge equation

We investigate nonlinear oscillations in a fourth-order partialdifferential equation which models a suspension bridge. Previouswork establishes multiple periodic solutions when a parameterexceeds a certain eigenvalue. In this paper, we use Leray-Schauderdegree theory to prove that if the parameter is increased further,beyond a second eigenvalue, then additional solutions are created.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.     

Two-particle scattering in a periodic medium

We consider a two-particle Schrödinger difference operator with a periodic potential perturbed by an exponentially decaying interaction potential for particles on a one-dimensional lattice. We obtain rigorous results for the two-particle scattering problem in the case of a small interaction and low velocities. Here, as in other quasi-one-dimensional models, small interactions can significantly affect the scattering pattern. In particular, we find the probability that the velocities of two particles in a periodic medium (e.g., they can be ultracold atoms in a one-dimensional optical lattice) change their signs during a collision. This probability increases as the relative velocity decreases and also as the absolute value of the matrix element between single-particle unperturbed Bloch states increases.     

On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces

Effective models are derived for layered periodic elastic media'with slide contacts on all interfaces. In the case where each period consists of n layers with different plate velocities, the effective model has n phases. These models are investigated for typical media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 192–212. Translated by L. A. Molotkov.