Andrey Piatnitski

Quenched invariance principles for random walks on percolation clusters

We consider a supercritical Bernoulli percolation model in , d≥2, and study the simple symmetric random walk on the infinite percolation cluster. The aim of this paper is to prove the almost sure (quenched) invariance principle for this random walk.

Homogenization of a nonlinear random parabolic partial differential equation

The aim of this work is to show how to homogenize a semilinear parabolic second-order partial differential equation, whose coefficients are periodic functions of the space variable, and are perturbed by an ergodic diffusion process, the nonlinear term being highly oscillatory. Our homogenized equation is a parabolic stochastic partial differential equation.

Homogenization of a singular random one-dimensional PDE

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zeroorder term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.

Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

This paper is devoted to the homogenization of a coupled system of diffusion-convection equations in a domain with periodic microstructure, modeling the flow and transport of immiscible compressible, such as water-gas, fluids through porous media. The problem is formulated in terms of a nonlinear parabolic equation for the nonwetting phase pressure and a nonlinear degenerate parabolic diffusion-convection equation for the wetting saturation phase with rapidly oscillating porosity function and absolute permeability tensor. We obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using two-scale convergence. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

Homogenization of the linearized ionic transport equations in rigid periodic porous media

In this paper we undertake the rigorous homogenization of a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, which allows us to use O’Briens linearized equations as the starting model. We establish convergence of the homogenization procedure and discuss the homogenized equations. Even if the symmetry of the effective tensor is known from the literature [J. R. Looker and S. L. Carnie, Transp. Porous Media, 65, 107 (2006)], its positive definiteness does not seem to be known. Based on the rigorous study of the underlying equations, we prove that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. This result justifies the approach of many authors who use Onsager theory as starting point.

HOMOGENIZATION OF A NONLINEAR CONVECTION-DIFFUSION EQUATION WITH RAPIDLY OSCILLATING COEFFICIENTS AND STRONG CONVECTION

A Cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied. Under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects. The convergence of the homogenization procedure is justified by using a new version of a two-scale convergence technique adapted to rapidly moving coordinates.

UNIFORM SPECTRAL ASYMPTOTICS FOR SINGULARLY PERTURBED LOCALLY PERIODIC OPERATORS

ABSTRACT We consider the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficients are scaled as ε2 and vary both on the macroscopic scale and on the periodic microscopic scale. We make a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior. We then prove an exponential localization phenomena at this minimum point. Namely, the k-th original eigenfunction is shown to be asymptotically given by the product of the first cell eigenfunction (at the ϵ scale) times the k-th eigenfunction of an homogenized problem (at the scale). The homogenized problem is a diffusion equation with quadratic potential in the whole space. We first perform asymptotic expansions, and then prove convergence by using a factorization strategy.

Homogenization of Elliptic Difference Operators

We develop some aspects of general homogenization theory for second order elliptic difference operators and consider several models of homogenization problems for random discrete elliptic operators with rapidly oscillating coefficients. More precisely, we study the asymptotic behavior of effective coefficients for a family of random difference schemes whose coefficients can be obtained by the discretization of random high-contrast checker-board structures. Then we compare, for various discretization methods, the effective coefficients obtained with the homogenized coefficients for corresponding differential operators.

HOMOGENIZATION OF BOUNDARY-VALUE PROBLEM IN A LOCALLY PERIODIC PERFORATED DOMAIN

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.

Singular Double Porosity Model

We consider the linear parabolic equation describing the transport of a contaminant in a porous media crossed by a net of infinitely thin fractures. The permeability is very high in the fractures but very low in the porous blocks. We derive the homogenized model corresponding to a net of infinitely thin fractures, by means of the singular measures technique. We assume that these singular measures are supported by hyperplanes of codimension one. We prove in a second step that this homogenized model could be obtained indistinctly either by letting the fracture thickness, in the standard double porosity model, tend to zero, or by homogenizing a model with infinitely thin fractures.