Andrey Piatnitski

UiT - The Arctic University of Norway, campus Narvik · Department of Technology

The paper studies a bounded symmetric operator Aε in L2(Rd) with(Aεu)(x)=ε−d−2∫Rda((x−y)/ε)μ(x/ε,y/ε)(u(x)−u(y))dy; here ε is a small positive parameter. It is assumed that a(x) is a non-negative L1(Rd) function such that a(−x)=a(x) and the moments Mk=∫Rd|x|ka(x)dx, k=1,2,3, are finite. It is also assumed that μ(x,y) is Zd-periodic both in x and y...

The paper studies a bounded symmetric operator ${\mathbf{A}}_\varepsilon$ in $L_2(\mathbf{R}^d)$ with $$ ({\mathbf{A}}_\varepsilon u) (x) = \varepsilon^{-d-2} \int_{\mathbf{R}^d} a((x-y)/\varepsilon) \mu(x/\varepsilon, y/\varepsilon) \left( u(x) - u(y) \right)\,dy; $$ here $\varepsilon$ is a small positive parameter. It is assumed that $a(x)$ is a...

In this work we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic length scale we describe our model in terms of a lattice random walk in a high-contrast periodic medium with absorption. Applying then the upscaling procedure we obtain the macroscopic model for...

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system co...

In this paper, we establish a homogenization result for a nonlinear degenerate system arising from two-phase flow through fractured porous media with periodic microstructure taking into account the temperature effects. The mathematical model is given by a coupled system of two-phase flow equations, and an energy balance equation. The microscopic mo...

This paper deals with the homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies certain symmetry and uniform ellipticity conditions, we prove the almost sure homogenization result and show that the limit operato...

Prolongating our previous paper on the Einstein relation, we study the motion of a particle diffusing in a random reversible environment when subject to a small external forcing. In order to describe the long time behavior of the particle, we introduce the notions of steady state and weak steady state. We establish the continuity of weak steady sta...

The goal of the paper is to describe the large time behaviour of a symmetric diffusion in a high-contrast periodic environment and to characterize the limit process under the diffusive scaling. We consider separately the and settings.

In this paper, we consider nonisothermal two-phase flows through heterogeneous porous media with periodic microstructure. Examples of such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste, thermally enhanced oil recovery and geothermal systems. The mathematical model is given by a...

We study the large time behaviour of the fundamental solution of parabolic equations with an elliptic part being non-local convolution type operator. We assume that this operator is a generator of a Markov jump process, and that its convolution kernel decays at least exponentially at infinity. The fundamental solution shows rather different asympto...

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially, we prove a number of lower and upper bounds for the resolvent of such operators. In particular we focus on sharp estimates of the resolvent kernel for small values of the spectral parameter. We consider two applicati...

The paper gives a rigorous description, based on mathematical homogenization theory, for flows of solvents with not charged solute particles under osmotic pressure for periodic porous media permeable for solute particles. The effective Darcy type equations for the flow under osmotic pressure distributed within the porous media are derived. The effe...

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially we prove a number of lower and upper bounds for the resolvent of such operators. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators pertur...

The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of re...

A two-dimensional Steklov-type spectral problem for the Laplacian in a domain divided into two parts by a perforated interface with a periodic microstructure is considered. The Steklov boundary condition is set on the lateral sides of the channels, a Neumann condition is specified on the rest of the interface, and a Dirichlet and Neumann condition...

Prolongating our previous paper on the Einstein relation, we study the motion of a particle diffusing in a random reversible environment when subject to a small external forcing. In order to describe the long time behavior of the particle, we introduce the notions of steady state and weak steady state. We establish the continuity of weak steady sta...

In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equ...

The paper deals with homogenization of Navier-Stokes-type system describing electrorheologial fluid with random characteristics. Under non-standard growth conditions we construct the homogenized model and prove the convergence result. The structure of the limit equations is also studied

In the paper, we consider the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure. Our goal is to construct an asymptotic expansion of its solution. We provide error estimates and also introduce and justify the asymptotic partial domain decomposition for this problem.

The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue...

In this paper, we present the 3D-1D asymptotic analysis of the Dirichlet spectral problem associated with an elliptic operator with axial periodic heterogeneities. We extend to the 3D-1D case previous 3D-2D results (see [10]) and we analyze the special case where the scale of thickness is much smaller than the scale of the heterogeneities and the p...

This paper focuses on deriving double-porosity models from simple high-contrast atomistic interactions. Using the variational approach and F-convergence techniques we derive the effective double-porosity type problem and prove the convergence. We also consider the dynamical case and study the asymptotic behavior of solutions for the gradient flow o...

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed convection-diffusion operators with oscillating locally periodic coefficients. It follows from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint www.arxiv.org, arXiv:1206....

We study a discrete-to-continuous Gamma-limit of a family of high-contrast double porosity type functionals defined on a scaled integer lattice. Under periodicity and p-growth conditions we prove the homogenization result and describe the structure of the limit functional. Also, we study the convergence of the corresponding gradient flow.

This paper presents a study of immiscible compressible two-phase, such as water and gas, flow through highly heterogeneous porous media with periodic microstructure. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. We will consider a domain made up of several zones with diff...

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model...

The paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic in spatial variables and random stationary in time. Under proper mixing assumptions, we study the limit behaviour of the normalized difference between solutions of the original and the homogenized problems. The asymptotic behaviou...

We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media taking into account capillary and gravity effects. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This...

The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we...

We study the homogenization of lattice energies related to Ising systems of the formEε(u)=−∑ijcijεuiuj, with uiui a spin variable indexed on the portion of a cubic lattice Ω∩εZdΩ∩εZd, by computing their Γ-limit in the framework of surface energies in a BV setting. We introduce a notion of homogenizability of the system {cijε} that allows to treat p...

In this article, we consider the problem of homogenising the linear heat equation perturbed by a rapidly oscillating random potential. We consider the situation where the space-time scaling of the potential's oscillations is \textit{not} given by the diffusion scaling that leaves the heat equation invariant. Instead, we treat the case where spatial...

In this work we undertake a numerical study of the effective co-efficients arising in the upscaling of a system of partial differential equations describing transport of a dilute N -component electrolyte in a Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, arou...

We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of $N$ chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behavior of its solution depending on a parameter $\beta$ which is the...

We consider a homogenization of elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of a localization of the eigenfunctions. Moreover, the limit behaviour of the lower part o...

We study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of Γ-convergence and techniques mixing Geometric Measure Theory and Percolation while scaling to zero the lattice spacing. The limit is not trivial above a percolation threshold. Since the lattice energies a...

We study the asymptotic behavior of solutions to a boundary value problem for the Poisson equation with a singular right-hand side, singular potential and with alternating type of the boundary condition. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem by means of the unfoldi...

The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behaviour of the first eigenpair (ground state) of this problem. The main tool in deriving the limit (effective) problem is the viscosity solutions technique for Hamil...

In this paper we study the homogenization of a nonautonomous parabolic equation with a large random rapidly oscillating potential in the case of one-dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables, then, under proper mixing assumptions, th...

We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in periodically perforated domain under the assumption that the spectral weight function changes sign. We show that the limit behaviour of the spectrum depends essentially on wether the average of the weight function over the boundary of holes is positive...

We consider reversible diffusions in random environment and prove the Einstein relation for this model. It says that the derivative of the effective velocity under an additional local drift equals the diffusivity of the model without drift. The Einstein relation is conjectured to hold for a variety of models but is proved insofar only in particular...

We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us t...

We deal with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary...

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neuman...

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The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodi...

The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which con...

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and es...

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’Brien's linearized eq...

The paper deals with homogenization of an elliptic boundary value problem stated in a domain which consists of two connected components sep-arated by a rapidly oscillating interface with a periodic microstructure, the interface being situated in a small neighbourhood of a hyperplane. At the interface we suppose the following transmission conditions...

We study the homogenization problem for a convection-diffusion equation in a periodic porous medium in the presence of chemical reaction on the pores surface. Mathematically this model is described in terms of a solution to a system of convection-diffusion equation in the medium and ordinary differential equation defined on the pores surface. These...

The aim of the paper is to study the asymptotic behavior of solutions to a Neumann boundary value problem for a nonlinear elliptic equation with nonstandard growth condition of the form -div(|∇u ε | p ε (x)-2 ∇u ε )+|u ε | p ε (x)-2 u ε =f(x) in a perforated domain Ω ε ,ε being a small parameter that characterizes the microscopic length scale of th...

The paper deals with the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients in the case when the density function (the factor on the spectral parameter) changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum of this problem is di...

This paper deals with the homogenization of a second order parabolic operator with a large nonlinear potential and periodically oscillating coefficients of both spatial and temporal variables. Under a centering condition for the nonlinear zero-order term, we obtain the effective problem and prove a convergence result. The main feature of the homoge...

In this work we study reactive flows through porous media. We suppose dominant Peclet's number, dominant Damköhler's number and general linear reactions at the pore boundaries. Our goal is to obtain the dispersion tensor and the upscaled model. We introduce the multiple scale expansions with drift for the problem and use this technique to upscale t...

We consider a simplified model for the radionuclides migration in an underground nuclear waste repository, based on a linear partial differential equation of diffusion convection type. This partial differential equation has a source term constituted by a large number of "local" sources spatially periodically distributed and lying on the porous doma...

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 formu-lated in terms of a nonlinear parabolic equation for the nonwetting phase pressur...

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ...

We derive a macroscopic model for an underground nuclear waste repository consisting of long storage cells linked by a possibly damaged drifts. As the first result we find a simple first-order approximation. Secondly, we compute a corrector using a matched expansion around the drift. We prove an appropriate convergence result.

The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium and combining the bulk (volume distributed) energy and the surface energy distributed on the perforation boundary. It is assumed that the mean value of surface energy at each level set of test function is equal to...

This chapter is devoted to the homogenization of a stationary convection diffusion model problem in a thin rod structure. More precisely, we study the asymptotic behavior of solutions to a boundary value problem for a convection diffusion equation defined in a thin cylinder that is the union of two nonintersecting cylinders with a junction at the o...

We study the homogenization problem for a convection-diffusion equation in a periodic porous medium in the presence of chemical reaction on the pores surface. Mathematically this model is described in terms of a solution to a system of convection-diffusion equation in the medium and ordinary differ-ential equation defined on the pores surface. Thes...

We study the homogenization of the following nonlinear Dirichlet variational problem:inf{∫Ωε{1pε(x)|∇u|pε(x)+1pε(x)|u|pε(x)−f(x)u}dx:u∈W01,pε(⋅)(Ωε)} in a perforated domain Ωε=Ω∖Fε⊂Rn, n⩾2, where ε is a small positive parameter that characterizes the scale of the microstructure. The non-standard exponent pε(x) is assumed to be an oscillating contin...

We study the asymptotic behaviour of the displacement of a thin periodically perforated rod under the action of forces applied to one of the rod ends, another end of the rod is clamped. We show that, up to boundary layer functions arising in the vicinity of the end points of the rod, the set of solutions forms a finite dimensional space, and that i...

We studied the asymptotic behavior of the solution of a nonlinear parabolic equation with nonstandard growth in a ε-periodic fractured medium, where ε is the parameter that characterizes the scale of the microstructure tending to zero. We consider a double porosity type model describing the flow of a compressible fluid in a heterogeneous anisotropi...

The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neu- mann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove th...

In this paper we study periodic elastic rod-structures which are lo- cally anisotropic and symmetric with respect to some plane. In order to find the effective behavior and approximate local behavior (so-called corrector-results) of such structures, one has to solve a finite number of boundary-value problems on one period of the rod-structure, the...

A prototype for variational percolation problems with surface energies is considered: the description of the macroscopic properties of a (two-dimensional) discrete membrane with randomly distributed defects in the spirit of the weak membrane model of Blake and Zisserman (quadratic springs that may break at a critical length of the elongation). Afte...

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order 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 pr...

This paper is devoted to homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with continuous variable exponents. We assume that the sequence of exponents converges in the uniform metric and that the Lagrangian has a periodic microstructure. Then under natural coerciveness assumptions we prove a Ga...

We consider the damped-driven KdV equation: where 0<ν⩽1 and the random process η is smooth in x and white in t. For any periodic function u(x) let I=(I1,I2,…) be the vector, formed by the KdV integrals of motion, calculated for the potential u(x). We prove that if u(t,x) is a solution of the equation above, then for 0⩽t≲ν−1 and ν→0 the vector I(t)=...

We are interested to study $u(x,t)$ , the evolution in time of the concentration, which is transported by diffusion and convection from a "sources site" made of a large number of similar "local sources". For this we consider a "local model" based on a general diffusion convection equation: \begin{eqnarray} \label{intro_eq} \partial_t u^\eps-\mathrm...

The paper deals with homogenization of stationary and non-stationary high contrast periodic double porosity type problem stated in a porous medium containing a 2D or 3D thin layer. We consider two different types of high contrast medium. The medium of the first type is characterized by the asymptotically vanishing volume fraction of fractures (high...

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the form − div(a ε (x)|∇u ε | p−2 ∇u ε) + g(x)|u ε | p−2 u ε = S ε (x) in Ω, with a high-contrast discontinuous coefficient a ε (x), where ε is the parameter characterizing the scale of the microstucture. The coefficient a ε (x) is assume...

The goal of the paper is to study the asymptotic behaviour of solutions to a high contrast quasilinear equation of the form where with n ≥ 2, 1 < p ≤ n, and the coefficient Gε(x) is assumed to blow up as ε → 0 on a set of Nε isolated inclusions of asymptotically small measure. Here as ε → 0. It is shown that the asymptotic behaviour, as ε → 0, of...

This paper deals with homogenization of random nonlin- ear monotone operators in divergence form. We assume that the structure conditions (strict monotonicity and continuity conditions) degenerate and are given in terms of a weight function. Under proper integrability assump- tions on the weight function we construct the eectiv e operator and prove...

We study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential, in a general stationary and mixing random media, which is periodic in space. It is shown that a solution of the corresponding Cauchy problem converges...

We study a parabolic operator in a perforated medium with random rapidly pulsating perforation. Assuming that the geometry of the perforations is spatially periodic and stationary random in time with good mixing properties, we show that this problem admits homogenization in moving coordinates, and derive the homogenized problem.

We study the homogenization of a Schrödinger equation with a large periodic potential: denoting by ∈ the period, the potential is scaled as ∈ −2. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solutio...

In this note we study the homogenization problem for a singularly perturbed non-stationary parabolic operator with lower order terms. We assume a self-similar scaling of spatial and temporal variables and prove the existence of rapidly moving coordinates in which a solution of the corresponding Cauchy problem is asymptotically given as the product...

We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by ε the period, the potential is scaled as ε −2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscilla...

In this paper we outline an approach by Γ-convergence to some problems related to ‘double-porosity’ homogenization. Various such models have been discussed in the mathematical literature, the first rigorous result for a linear double-porosity model having been obtained by Arbogast, Douglas and Hornung in [7]. The two-scale convergence approach to d...

This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other “cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample [0,ρ]d of the random medium, exten...

We study nonstationary linearized reaction-diusion problem in a medium with locally periodic microstructure. Under the assumption that the characteristics of the medium are random stationary rapidly oscillating functions of time, we construct and justify a homogenized problem.

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the p...

We consider the linearized equations of slightly compressible single fluid flow through a highly heterogeneous random porous medium, consisting of two types of material. Due to the high heterogeneity of the two materials the ratio of their permeability coefficients is of order ε 2 , where ε is the characteristic scale of heterogeneities. Supposing...

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...