Homogenization of a convection–diffusion equation in perforated domains with a weak adsorption

Abstract

The aim of this paper is to study the asymptotic behavior of the solution of a convection-diffusion equation in perforated domains with oscillating velocity and a Robin boundary condition which describes the adsorption on the bord of the obstacles. Without any periodicity assumption, for a large range of perforated media and by mean of variational homogenization, we find the global behavior when the characteristic size ε of the perforations tends to zero. The homogenized model, is a convection-diffusion equation but with an extra term coming from the weak adsorption boundary condition. An example is presented to illustrate the methodology.

Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

“…This simplified approach is based on the generalized Besicovitch type spaces (see Sect. 3), and widely opens the scope of applications of our result, Theorem 3.10, as can be easily seen in Sect. 4.…”

“…(H) 1 Each H ε maps continuously R N into itself; (H) 2 lim ε→0 |H ε (x)| = +∞ for any x ∈ R N with x = 0, where | · | denotes the Euclidean norm in R N ; (H) 3 The Lebesgue measure λ on R N is quasi-invariant under H, i.e., to each ε > 0 there is attached some γ (ε) > 0 such that H ε (λ) = γ (ε)λ. Now, given ε > 0, let…”

“…But here again, the size of the holes and the period of oscillations are of the same order, and moreover, the problem he studied is a linear elliptic equation. Similarly, see e.g., [3,31]. In [33] the author considers the situation of the present paper but with a monotone elliptic operator.…”

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

“…But here again, the size of the holes and the period of oscillations are of the same order. Similarly, see, e.g., [2], [27].…”

“…(1. 2) There are four constants c 1 , c 2 > 0, 0 < α 1 min(1, p − 1) (1. 3) and α 2 max(p, 2) such that, a.e.…”