Asymptotic Analysis - Volume 128, issue 1

Authors: Ntsokongo, Armel Judice | Tathy, Christian

Article Type: Research Article

Abstract: The aim of this paper is to study higher-order Caginalp phase-field systems based on the Maxwell–Cattaneo law, instead of the classical Fourier law. More precisely, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.

Keywords: Phase-field systems, higher-order systems, Anisotropy, Maxwell–Cattaneo law, well-posedness, long-time behavior, dissipativity, global attractor, exponential attractor

DOI: 10.3233/ASY-211695

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 1-30, 2022

Authors: Gérard-Varet, David

Article Type: Research Article

Abstract: We present a gentle approach to the justification of effective media approximations, for PDE’s set outside the union of n ≫ 1 spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term , made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as n → + ∞ …by simple arguments. Show more

Keywords: Homogenisation, fluid mechanics

DOI: 10.3233/ASY-211696

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 31-53, 2022

Authors: Lu, Jianfeng | Zhang, Zihang | Zhou, Zhennan

Article Type: Research Article

Abstract: We derive the semiclassical Bloch dynamics with second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external electric field, the bi-characteristics system after a positional shift introduced by Berry connections agrees with the recent result in previous works. Moreover, for the case with a linear external electric field, we show that the extra terms arising in the bi-characteristics system after the positional shift are also gauge independent.

Keywords: Bloch dynamics, Berry phase, second-order correction, WKB analysis

DOI: 10.3233/ASY-211697

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 55-84, 2022

Authors: Li, Tatsien | Rao, Bopeng

Article Type: Research Article

Abstract: We show the sufficiency of Kalman’s rank condition for the uniqueness of solution to a coupled system of wave equations in a rectangular domain. The approach does not need any gap condition on the spectrum of the differential operator and the usual multiplier geometrical condition. Then, the study on the asymptotic synchronization by groups can be improved for the corresponding system.

Keywords: Fourier analysis, uniqueness theorem, Kalman’s rank condition, asymptotic synchronization, coupled system of wave equations

DOI: 10.3233/ASY-211699

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 85-102, 2022

Authors: Huh, Hyungjin

Article Type: Research Article

Abstract: We obtain the growth of Sobolev norms of the solution to the Maxwell–Dirac equations in R 1 + 1 by applying elementary techniques. In particular, we estimate L ∞ bound of the solution by making use of a local energy conservation. The similar idea can be applied to the Dirac–Klein–Gordon equations.

Keywords: Maxwell–Dirac, Dirac–Klein–Gordon, L∞ bound, growth of Sobolev norms

DOI: 10.3233/ASY-211700

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 103-111, 2022

Authors: Chiadò Piat, V. | D’Elia, L. | Nazarov, S.A.

Article Type: Research Article

Abstract: We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω ⊂ R d which is divided into two subdomains: an annulus Ω 1 and a core Ω 0 . The density and the stiffness constants are of order ε − 2 m and ε − 1 in Ω 0 , while they are of order 1 in Ω …1 . Here m ∈ R is fixed and ε > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε → 0 for any m . In dimension 2 the case when Ω 0 touches the exterior boundary ∂ Ω and Ω 1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x → O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given. Show more

Keywords: Neumann Laplacian, asymptotics of eigenvalues and eigenfunctions, stiff Neumann problem, domain with cuspidal point, kissing domains

DOI: 10.3233/ASY-211701

Citation: Asymptotic Analysis, vol. 128, no. 1, pp. 113-148, 2022