Asymptotic Analysis - Volume 101, issue 3

Authors: Bonfoh, Ahmed | Enyi, Cyril D.

Article Type: Research Article

Abstract: Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016 , 1077–1105], we considered the conserved phase-field system τ ϕ t − Δ ( δ ϕ t − Δ ϕ + g ( ϕ ) − u ) = 0 , ε u t + ϕ t − Δ u = 0 , in a bounded domain of R d , …d = 1 , 2 , 3 , where τ > 0 is a relaxation time, δ > 0 is the viscosity parameter, ε ∈ ( 0 , 1 ] is the heat capacity, ϕ is the order parameter, u is the absolute temperature and g : R → R is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at ε = 0 . Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at ( ε , δ ) = ( 0 , 0 ) . To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε , we require ε to be dominated from above by δ . This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn–Hilliard equation, improving and extending some previous results. Show more

Keywords: Conserved phase-field equations, global attractors, exponential attractors, inertial manifolds, continuity

DOI: 10.3233/ASY-161395

Citation: Asymptotic Analysis, vol. 101, no. 3, pp. 97-148, 2017

Authors: Hanke, Hauke | Knees, Dorothee

Article Type: Research Article

Abstract: In this paper we discuss a damage model that is based on microstructure evolution. In the context of evolutionary Γ-convergence we derive a corresponding effective macroscopic model. In this model, the damage state of a given material point is related to a unit cell problem incorporating a specific microscopic defect. The size and shape of this underlying microscopic defect is determined by the evolution. According to the small intrinsic length scale inherent to the original models a numerical simulation of damage progression in a device of realistic size is hopeless. Due to the scale separation in the effective model, its …numerical treatment seems promising. Show more

Keywords: Two-scale convergence, folding and unfolding operator, Γ-convergence, discrete gradient, state dependent coefficient, damage model

DOI: 10.3233/ASY-161396

Citation: Asymptotic Analysis, vol. 101, no. 3, pp. 149-180, 2017

Authors: Ben Ayed, Ines

Article Type: Research Article

Abstract: In this paper, we investigate the behavior of solutions to 2D Klein–Gordon equation in the framework of Orlicz norm. The analysis we conducted in this article, which is based on profiles decompositions, emphasizes the distinguished role played by the 1 -oscillating component of the sequence of the Cauchy data. This phenomenon is strikingly different from those obtained in previous works, such as in Bahouri and Gérard [American Journal of Mathematics 121 (1999 ), 131–175] and Merle and Vega [International Mathematics Research Notices 8 (1998 ), 399–425].

Keywords: Free Klein–Gordon equation, Olicz space, profile decomposition

DOI: 10.3233/ASY-161398

Citation: Asymptotic Analysis, vol. 101, no. 3, pp. 181-206, 2017