Asymptotic Analysis - Volume 128, issue 3

Authors: Allaire, Grégoire | Lamacz-Keymling, Agnes | Rauch, Jeffrey

Article Type: Research Article

Abstract: This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, ϵ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times t of order ϵ − 2 + δ for any δ > 0 . Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of ϵ , yields an approximation …of the exact solution with error O ( ϵ N ) for times ϵ − N with N as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms. Show more

Keywords: Homogenization, secular growth, dispersive effects, asymptotic crimes, wave equations

DOI: 10.3233/ASY-211707

Citation: Asymptotic Analysis, vol. 128, no. 3, pp. 295-336, 2022

Authors: Chipot, Michel

Article Type: Research Article

Abstract: The goal of this note is to present a simple proof of existence and uniqueness of the solution of the Leray problem for high viscosities and homogeneous or non homogeneous boundary conditions. Furthermore we address the issue of uniqueness of the Poiseuille flow in a pipe.

Keywords: Nonlinear elliptic equations, Navier Stokes, stationary problem, Poiseuille flow

DOI: 10.3233/ASY-211708

Citation: Asymptotic Analysis, vol. 128, no. 3, pp. 337-350, 2022

Authors: Niu, Weisheng | Shen, Zhongwei

Article Type: Research Article

Abstract: We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale κ that represents the strength of the singular perturbation and on the length scale ε of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale ε and independent of κ . This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both ε and κ .

Keywords: Homogenization, singular perturbation, convergence rate, uniform Lipschitz estimate

DOI: 10.3233/ASY-211709

Citation: Asymptotic Analysis, vol. 128, no. 3, pp. 351-384, 2022

Authors: Barroso, Ana Cristina | Zappale, Elvira

Article Type: Research Article

Abstract: We obtain an integral representation for certain functionals arising in the context of optimal design and damage evolution problems under non-standard growth conditions and perimeter penalisation. Under our hypotheses, the integral representation includes a term which is absolutely continuous with respect to the Lebesgue measure and a perimeter term, but no additional singular term. We also study some dimension reduction problems providing results for the optimal design of thin films.

Keywords: Non-standard growth conditions, optimal design, damage, dimension reduction, thin films, sets of finite perimeter, convexity

DOI: 10.3233/ASY-211711

Citation: Asymptotic Analysis, vol. 128, no. 3, pp. 385-412, 2022

Authors: Rodiac, Rémy | Ubillús, Paúl

Article Type: Research Article

Abstract: In this article we derive the expression of renormalized energies for unit-valued harmonic maps defined on a smooth bounded domain in R 2 whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel–Brezis–Hélein in order to describe the position of limiting Ginzburg–Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

Keywords: Renormalized energies, Harmonic maps, Ginzburg–Landau vortices

DOI: 10.3233/ASY-211712

Citation: Asymptotic Analysis, vol. 128, no. 3, pp. 413-444, 2022