Asymptotic Analysis - Volume 79, issue 3-4

Authors: Andreucci, Daniele | Bellaveglia, Dario

Article Type: Research Article

Abstract: We study a parabolic problem set in a domain divided by a perforated interface. The pores alternate between an open and a closed state, periodically in time. We consider the asymptotics of the solution for vanishingly small size of the pores and time period. The interface condition prevailing in the limit is a linear relation between the flux (on either side) and the jump of the limiting solution across the interface. More exactly this behaviour only takes place when the relative sizes of the relevant geometrical and temporal parameters are connected by suitable relations. With respect to the stationary …version of this problem, which is known in the literature, we demonstrate the appearance of a new admissible asymptotic standard. More in general, we describe the precise interplay between the geometrical and temporal parameters leading to the quoted interface condition. This work represents a preliminary mathematical investigation of a model of selective transport of chemical species through biological membranes. Show more

Keywords: homogenization, alternating pores, parabolic equations, effective permeability of interfaces, diffusion, gating

DOI: 10.3233/ASY-2012-1093

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 189-227, 2012

Authors: Zaki, Rachad

Article Type: Research Article

Abstract: We consider the Stokes problem in a domain with holes periodically distributed with a period ε. The size of the holes is of the order of ε, a small parameter going to zero. On the boundary of the holes we prescribe a Robin-type condition depending on a parameter γ. The aim is to give the asymptotic behavior of the velocity and of the pressure of the fluid as ε goes to zero. The study for a problem of this type was done in Math. Meth. Appl. Sci. 19 (1996), 857–881, via classical homogenization methods. In this work we use the …periodic unfolding method in perforated domains (see C. R. Acad. Sci. Paris, Série 1 342 (2006), 469–474; Portugaliae Mathematica 63(4) (2006), 467–496; Asymptotic Analysis 53(4) (2007), 209–235; in: Multiple Scales in Problems in Biomath., Mech., Physics and Numeric, Gakuto Int. Series, Math. Sci. App., Vol. 31, Gakkokotosho, 2009, pp. 37–68, and SIAM J. Math. Anal. 44(2) (2012), 718–760), which allows us to consider a general geometric framework. We give the limit problems corresponding to different values of γ (Darcy, Brinkmann or Stokes type problems). Show more

Keywords: homogenization, periodic unfolding, porous media, Stokes system

DOI: 10.3233/ASY-2012-1094

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 229-250, 2012

Authors: Razafimandimby, Paul André | Sango, Mamadou

Article Type: Research Article

Abstract: We study the limit of the stochastic model for two dimensional second grade fluids subjected to the periodic boundary conditions as the stress modulus tends to zero. We show that under suitable conditions on the data the whole sequence of strong probabilistic solutions (uα ) of the stochastic second grade fluid converges to the unique strong probabilistic solution of the stochastic Navier–Stokes equations.

Keywords: stochastic second-grade fluids, tightness, strong and weak probabilistic solutions, asymptotic behavior

DOI: 10.3233/ASY-2012-1095

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 251-272, 2012

Authors: Punzo, Fabio

Article Type: Research Article

Abstract: We study, on weighted Riemannian model manifolds, well posedness of the Cauchy problem for a class of quasilinear parabolic equations with a coefficient which can be singular at infinity. We establish uniqueness or non-uniqueness of bounded solutions, under suitable assumptions on the behavior at infinity of the singular coefficient and on the Green function for the weighted Laplace–Beltrami operator.

Keywords: singular non-linear parabolic equations, weighted Riemannian manifolds, weighted Laplace–Beltrami operator, well-posedness, sub-supersolutions

DOI: 10.3233/ASY-2012-1098

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 273-301, 2012

Authors: Bessoud, Anne-Laure | Juntharee, Pongpol | Licht, Christian | Michaille, Gérard

Article Type: Research Article

Abstract: We show that the variational limit of a ε-soft and thin junction problem (𝒫ε ) with sources concentrated in the junction gives rise to a surface energy mixing the internal energy and sources. The surface energy functional possesses an integral representation with respect to the gradient Young-concentration measures generated by sequences (ūε )ε>0 of minimizers of (𝒫ε ).

Keywords: Γ-convergence, gradient Young measures, concentration measures, minimization problems

DOI: 10.3233/ASY-2012-1102

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 303-323, 2012

Authors: Miranda, Pablo | Raikov, Georgi

Article Type: Research Article

Abstract: We consider the unperturbed operator H0 :=(−i∇−A)2 +W, self-adjoint in L2 (R2 ). Here A is a magnetic potential which generates a constant magnetic field b>0, and the edge potential W=W¯ is a 𝒯-periodic non-constant bounded function depending only on the first coordinate x∈R of (x,y)∈R2 . Then the spectrum σ(H0 ) of H0 has a band structure, the band functions are b𝒯-periodic, and generically there are infinitely many open gaps in σ(H0 ). We establish explicit sufficient conditions which guarantee that a given band of σ(H0 ) has a positive length, and all the extremal points of the …corresponding band function are non-degenerate. Under these assumptions we consider the perturbed operators H± =H0 ±V where the electric potential V∈L∞ (R2 ) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H± in the spectral gaps of H0 . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schrödinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum σ(H0 ), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian. Show more

Keywords: magnetic Schrödinger operators, spectral gaps, eigenvalue distribution

DOI: 10.3233/ASY-2012-1103

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 325-345, 2012

Authors: Bouche, Daniel | Lafitte, Olivier

Article Type: Research Article

Abstract: In this paper we consider the high frequency diffraction of electromagnetic waves by a strictly convex obstacle. We restrict ourselves to the case of an obstacle in 2D and to the diffraction of a TE or TM wave. The aim of this paper is to present simultaneously with the same notations, unknowns and special functions the boundary layer analysis of this problem with stretched variables using the intuition of the solutions and the microlocal analysis way of construction of an adapted parametrix. We concentrate on the case of diffraction, i.e. studying the representation of a creeping wave on the …boundary, which occurs in the case of the study of a glancing point. We use both methods to derive the exact solution near a diffractive point, obtaining results in terms of suitable Airy functions. Show more

Keywords: diffraction, high frequency, glancing, parametrix, boundary layers

DOI: 10.3233/ASY-2012-1137

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 347-378, 2012

Article Type: Other

Citation: Asymptotic Analysis, vol. 79, no. 3-4, pp. 379-380, 2012