Asymptotic Analysis - Volume 51, issue 3-4

Authors: Le Rousseau, Jérôme

Article Type: Research Article

Abstract: An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ∂z +a(z,x,Dx ) with Re(a)≥0, is constructed as the composition of global Fourier integral operators with complex phases. The symbol a(z,·) is assumed to have a regularity as low as Hölder, 𝒞0,α , with respect to the evolution parameter z. We prove a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞, with a convergence of order α. We also study the consequences of some truncation approximations of the symbol a(z,·) in the …construction of the Ansatz. Show more

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 189-207, 2007

Authors: Benachour, Saïd | Dăbuleanu-Hapca, Simona | Laurençot, Philippe

Article Type: Research Article

Abstract: Global classical solutions to the viscous Hamilton–Jacobi equation ut −Δu=a|∇u|p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W1,∞ (Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 209-229, 2007

Authors: Tone, Florentina

Article Type: Research Article

Abstract: In this article we discretize the two-dimensional space-periodic Navier–Stokes equations in time using the implicit Euler scheme and, with the aid of the discrete Gronwall lemma and the uniform discrete lemma, we prove that the scheme is H2 -uniformly stable in time. Moreover, in a final remark, we describe how the above result can be extended to show that the implicit Euler scheme is uniformly stable with respect to the H3 -norm, for all time.

Keywords: Navier–Stokes equations, discrete Gronwall lemmas, implicit Euler scheme

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 231-245, 2007

Authors: Gustafsson, Björn | Mossino, Jacqueline

Article Type: Research Article

Abstract: We give a criterion for H-convergence of elasticity tensors in terms of ordinary weak convergence of the factors in certain quotient representations of the tensors.

Keywords: nonperiodic homogenization, compensated compactness, corrector

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 247-269, 2007

Authors: Thomann, Laurent

Article Type: Research Article

Abstract: Using variational methods, we construct approximate solutions for the Gross–Pitaevski equation which concentrate on circles in $\mathbb{R}$ 3 . These solutions will help to show that the L2 flow is unstable for the usual topology and for the projective distance.

Keywords: nonlinear Schrödinger equation, instability

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 271-287, 2007

Authors: Luca, Rodica

Article Type: Research Article

Abstract: The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system supplemented with an extreme condition and initial data are investigated in a real Hilbert space.

Keywords: discrete hyperbolic system, multivalued operators, extreme conditions, maximal monotone operator, Cauchy problem, strong solution, weak solution

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 289-302, 2007

Authors: Borghol, Rouba

Article Type: Research Article

Abstract: The aim of this paper is to present some new characterizations of the Sobolev spaces Wk,p (Ω) where 1<p<∞, k≥2 and Ω is a bounded smooth domain of $\mathbb{R}$ N . For p=1, we prove same results by replacing Wk,1 (Ω) by the BVk (Ω) space. As an application we show how to recognize polynomial functions among locally integrable functions.

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 303-318, 2007

Authors: Valente, V. | Vergara Caffarelli, G.

Article Type: Research Article

Abstract: The dynamics of magneto-elastic materials is described by a nonlinear system which couples the equations of the magnetization and the displacements. We study the three-dimensional case and establish the existence of weak solutions. Our starting point is the Gilbert–Landau–Lifschitz equation introduced for describing the dynamics of micro-magnetic processes. Three terms of the total free energy are taken into account: the exchange energy, the elastic energy and the magneto-elastic energy usually adopted for cubic crystals, neglecting, in this approach, the contributions due to the anisotropy and the demagnetization effects. The existence theorem for the proposed differential system is proved combining the …Faedo–Galerkin approximations and the penalty method (FGP method). The asymptotic behavior are deduced from compactness properties. Show more

Keywords: magnetoelastic materials, nonlinear differential systems

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 319-333, 2007

Authors: Aganović, Ibrahim | Tambača, Josip | Tutek, Zvonimir

Article Type: Research Article

Abstract: In this paper we derive a two-dimensional model of elastic shell-like body from the three-dimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the shell. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain two-dimensional problem well defined for W1,∞ shells. Appropriate convergence results are proved.

Keywords: elastic shells, micropolar elasticity, asymptotic method, two-dimensional model, justification

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 335-361, 2007

Article Type: Other

Citation: Asymptotic Analysis, vol. 51, no. 3-4, pp. 363-364, 2007