19 research outputs found
The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary
AbstractWe study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain Ωε⊂Rnwith a very rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter ε is posed on the oscillating part of the boundary and that the (n−1)-dimensional volume of this part goes to infinity as ε→0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem
Homogenization of Biomechanical Models for Plant Tissues
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 equations for fluid flow inside the
cells. The chemical process in cells and the elastic properties of cell walls
and middle lamella are coupled because elastic moduli depend on densities
involved in chemical reactions, whereas chemical reactions depend on mechanical
stresses. Using homogenization techniques we derive rigorously a macroscopic
model for plant biomechanics. To pass to the limit in the nonlinear reaction
terms, which depend on elastic strain, we prove the strong two-scale
convergence of the displacement gradient and velocity field
Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media
Effective diffusion in vanishing viscosity
International audienceWe study the asymptotic behavior of effective diffusion for singular perturbed elliptic operators with potential first order terms. Assuming that the potential is a random perturbation of a fixed periodic function and that this perturbation does not affect essentially the structure of the potential, we prove the exponential decay of the effective diffusion. Moreover, we establish its logarithmic asymptotics in terms of proper percolation level for the random potential
Homogenization in perforated domains with rapidly pulsing perforations
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 period ε of the holes goes to zero.
Since standard conservation laws do not
hold in this model, a first difficulty is to get
a priori estimates of the solutions. We obtain them in a weighted space
where the
weight is the principal eigenfunction of an “adjoint” periodic
time-dependent eigenvalue problem. This problem is not a
classical one, and its investigation is an important
part of this work. Then, by using the multiple scale method,
we construct the
leading terms of a formal expansion (with respect to ε) of the solution and give the limit
“homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes,
a large convection term may appear in the limit equation
Contents
L. Piatnitski We consider the damped-driven KdV equation ˙u − νuxx + uxxx − 6uux = √ ν η(t, x), x ∈ S 1 ∫ ∫, u dx ≡ η dx ≡ 0, 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) = (I1(u(t, ·)), I2(u(t, ·)),...
Asymptotics of a spectral-sieve problem
International audienceIn a bounded domain with a thin periodically punctured interface we study the limit behavior of the bottom of spectrum for a Steklov type spectral problem, the Steklov boundary condition being imposed on the perforation surface. For a certain range of parameters we construct the effective spectral problem and justify the convergence of eigenpair