Asymptotic Analysis - Volume 115, issue 3-4

Authors: Chennouf, Selwa | Bentalha, Fadila

Article Type: Research Article

Abstract: Our work deals with the homogenization of a diffusion process which take place in a domain formed by an ambient connected phase surrounding an ε -periodical network of small spherical particles and holes, ε is a small parameter ε > 0 . The asymptotic behavior is determined as ε → 0 , assuming that the total volume of the holes and particles vanishes as ε → 0 , while the total mass of the particles remains of the unity order.

Keywords: Diffusion, homogenization, perforated domains, particles, fine-scale substructure

DOI: 10.3233/ASY-191529

Citation: Asymptotic Analysis, vol. 115, no. 3-4, pp. 127-145, 2019

Authors: Karek, Chafia | Ould-Hammouda, Amar

Article Type: Research Article

Abstract: We consider the Stokes problem in a domain Ω ε δ 1 δ 2 of R N , N ⩾ 3 ε -periodically perforated by holes of size r 1 ( ε δ 1 ) and r 2 ( ε δ 2 ) (ε , δ 1 = δ 1 ( ε ) , δ …2 = δ 2 ( ε ) being small parameters), with r 1 ( ε δ 1 ) / ε → 0 and r 2 ( ε δ 2 ) / ε → 0 as ε → 0 . Our aim is to describe the asymptotic behavior of the velocity and the pressure of the fluid as ε → 0 and give, if possible, a limit (“homogenized”) problem. To do so, we use the periodic unfolding method introduced by Cioranescu, Damlamian and Griso (C. R. Acad. Sci. Paris, Ser. I 335 (2002 ) 99–104; SIAM J. of Math. Anal. 40 (4) (2008 ) 1585–1620). It allow us to consider a general geometric framework and so, to extend the results from Cioranescu, Donato and Ene (Math. Models and Methods in Appl. Sciences 19 (1996 ) 857–881) and Capatina and Ene (European J. of Math. 22 (2011 ) 333–345). Show more

Keywords: Stokes problem, Robin condition, unfolding method

DOI: 10.3233/ASY-191530

Citation: Asymptotic Analysis, vol. 115, no. 3-4, pp. 147-167, 2019

Authors: Anza Hafsa, Omar | Mandallena, Jean Philippe | Michaille, Gérard

Article Type: Research Article

Abstract: We establish a convergence theorem for a class of nonlinear reaction–diffusion equations when the diffusion term is the subdifferential of a convex functional in a class of functionals of the calculus of variations equipped with the Mosco-convergence. The reaction term, which is not globally Lipschitz with respect to the state variable, gives rise to bounded solutions, and cover a wide variety of models. As a consequence we prove a homogenization theorem for this class under a stochastic homogenization framework.

Keywords: Convergence of reaction–diffusion equations, stochastic homogenization, comparison principle

DOI: 10.3233/ASY-191531

Citation: Asymptotic Analysis, vol. 115, no. 3-4, pp. 169-221, 2019

Authors: Samovol, V.S.

Article Type: Research Article

Abstract: We study the Riccati equation with coefficients having power asymptotic forms in a neighbourhood of infinity. Also, we examine the solutions to these equations and describe their asymptotic forms.

Keywords: Riccati equation, continuable solution, power geometry, Newton polygon, asymptotic form

DOI: 10.3233/ASY-191534

Citation: Asymptotic Analysis, vol. 115, no. 3-4, pp. 223-239, 2019

Authors: Piatnitski, A. | Zhizhina, E.

Article Type: Research Article

Abstract: This paper deals with homogenization of parabolic problems for integral convolution type operators with a non-symmetric jump kernel in a periodic elliptic medium. It is shown that the homogenization result holds in moving coordinates. We determine the corresponding effective velocity and prove that the limit operator is a second order parabolic operator with constant coefficients. We also consider the behaviour of the effective velocity in the case of small antisymmetric perturbations of a symmetric kernel, in particular we show that the Einstein relation holds for the studied periodic environment.

Keywords: Homogenization in moving coordinates, periodic medium, non-local operator, non-symmetric convolution kernel

DOI: 10.3233/ASY-191533

Citation: Asymptotic Analysis, vol. 115, no. 3-4, pp. 241-262, 2019