Asymptotic Analysis - Volume 29, issue 3-4

Authors: Berlyand, Leonid | Khruslov, Evgen

Article Type: Research Article

Abstract: We consider a nonlinear homogenization problem for a Ginzburg–Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα , α>1, and the coefficient …in front of the surface energy term, is of the order of εβ . We found that the parametric half‐plane {(α,β): α>1, −∞<β+∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential. We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems. Show more

Keywords: Ginzburg–Landau functional, inclusions, liquid crystals homogenization, effective potential

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 185-219, 2002

Authors: Carbone, L. | Cioranescu, D. | De Arcangelis, R. | Gaudiello, A.

Article Type: Research Article

Abstract: The homogenization process for some energies of integral type arising in the modelling of rubber‐like elastomers is carried out. The main feature of the variational problems taken into account is that pointwise oscillating constraints on the admissible deformations, determined by general, not necessarily bounded, constraint sets are involved. The classical homogenization result is established also in this framework, both for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained, and with an explicitly computed homogeneous density. Some explicit computations for the homogenized integrands relative to energy densities …coming from models in literature are also discussed. Résumé. Le but de ce travail est d'étudier l'homogénéisation d'une classe d'énergies définies par une intégrale et intervenant dans la modélisation du caoutchouc. La principale caractéristique des problémes variationnels attachés à ces énergies est le fait que le gradient des déformations admissibles est soumis à des contraintes ponctuelles oscillantes (non nécessairement bornées). Nous prouvons des résultats d'homogénéisation pour différentes conditions aux limites (de Dirichlet affines, de Neumann et mixtes) et montrons que l'énergie limite est encore du type intégral avec une densité calculée explicitement, en ayant toujours des contraintes sur le gradient. Enfin, un calcul explicite donne l'intégrand homogénéisé pour quelques énergies relatives à la modélisation du caoutchouc. Show more

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 221-272, 2002

Authors: Leone, Chiara

Article Type: Research Article

Abstract: We deal with minima for convex functionals of the calculus of variations when the forcing term is a function of L1 (Ω), exhibiting a notion of minimum equivalent to that one given in [2]. This new formulation allows us to make easier the proofs of existence and uniqueness results already obtained in [2]. Moreover with this alternative definition we prove that minima are stable if we perturb the functionals under the notion of Γ‐convergence.

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 273-282, 2002

Authors: Tachim Medjo, T.

Article Type: Research Article

Abstract: In this article we study the existence and uniqueness of solutions of the stationary Navier–Stokes equations in two‐dimensional exterior domains. Using well‐known Lq ‐estimates for the Oseen problems and an iterative process based on repeated resolution of Oseen problems, we prove the existence and uniqueness of solutions of the Navier–Stokes equations in two‐dimensional exterior domains. Our results give another answer to a question recently raised in [2,3] concerning the summability with exponent p∈(1,2) of the gradient of the velocity of stationary plane flow of a viscous fluid in exterior domains.

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 283-291, 2002

Authors: Bouhennache, Tark

Article Type: Research Article

Abstract: Nous étudions le comportement asymptotique à haute fréquence des ondes guidées dans une bande élastique isotrope et hétérogène Ω={(x,y); y∈(0,L)} ⊂ R2 , avec les conditions de surface libre en y=0 et de Dirichlet en y=L. Nous montrons en particulier l'existence d'une “onde de Rayleigh asymptotique”. Abstract. We study the asymptotic behavior at high frequency of the guided waves in an isotropic and heterogeneous elastic strip Ω={(x,y); y∈(0,L)} ⊂ R2 , with the free surface condition on y=0 and Dirichlet condition on y=L. We prove, in particular, the existence of an “asymptotic Rayleigh wave”.

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 293-308, 2002

Authors: Gordon, P.V. | Kamin, S. | Sivashinsky, G.I.

Article Type: Research Article

Abstract: The long time behavior of the system γΘt −(γ−1)Πt =Ω(Ψ,Θ), Ψt =−Ω(Ψ,Θ), Πt −Θt =Πxx is explored. The model describes gaseous detonation subjected to strong hydraulic resistance, e.g., gaseous explosion in inert porous media. It is shown that successful initiation of a self‐sustaining detonation wave is not always attainable and depends on the initial conditions. The current work is an extension of the previous study of the problem based on a simplified formulation assuming a similarity relation between the temperature and concentration fields.

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 309-321, 2002

Authors: Laitinen, Mika T.

Article Type: Research Article

Abstract: In highly absorptive semitransparent material, conductive‐radiative heat transfer is often approximated by either diffusion approximation or Stefan–Boltzmann boundary condition. Using singular perturbations and boundary layer analysis, we derive rigorously these two approximations from the radiative transport equation. We prove error estimates for temperature and propose a variant of diffusion approximation which effectively describes the boundary layer behavior. To obtain these estimates, we prove stability and boundedness of solutions for conductive‐radiative heat equation independently of the radiation parameters.

Keywords: conductive‐radiative heat transfer, diffusion approximation, semitransparent media, Stefan–Boltzmann law

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 323-342, 2002

Authors: Elias, Uri | Gingold, Harry

Article Type: Research Article

Abstract: A theorem of asymptotic integration is proven for linear systems of differential equations. The theorem is designed to fit a specialized family of differential systems which occur frequently in quantum mechanics. It is shown to be best possible in a certain sense. The method provided differs from an established trend that transforms the differential system, via a preparation theorem, to a differential system, where the coefficient matrix is the sum of a diagonal matrix and a remainder matrix that must be absolutely integrable at infinity. In this work the fundamental matrix solution is given as a product of a diagonal …matrix and a perturbation of the identity matrix. The perturbation of the identity matrix being on the right in the product rather than on the left as is common in the literature. Show more

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 343-357, 2002

Article Type: Other

Citation: Asymptotic Analysis, vol. 29, no. 3-4, pp. 359-360, 2002