Asymptotic Analysis - Volume 83, issue 1-2

Authors: Capitanelli, R. | Vivaldi, M.A.

Article Type: Research Article

Abstract: Laplacean transport across and towards irregular interfaces have been used to model many phenomena in nature and technology. The peculiar aspect is that these phenomena take place in domains with small bulk and large interfaces in order to produce rapid and efficient transport. In this paper we perform the asymptotic homogenization analysis of Robin problems in domains with a fractal boundary.

Keywords: fractals, variational methods for second-order elliptic equations, boundary value problems for second-order elliptic equations, homogenization

DOI: 10.3233/ASY-2012-1149

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 1-33, 2013

Authors: Claeys, Xavier | Delourme, Bérangère

Article Type: Research Article

Abstract: This work deals with the scattering of acoustic waves by a thin ring that contains many regularly-spaced heterogeneities. We provide and justify a complete description of the solution with respect to the period and the thickness of the heterogeneities. Our approach mixes matched asymptotic expansions and homogenization theory.

Keywords: periodic thin interfaces, matched asymptotic expansions, homogenization, Helmholtz equation

DOI: 10.3233/ASY-2012-1150

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 35-82, 2013

Authors: Kurta, Vasilii V.

Article Type: Research Article

Abstract: We obtain a new Liouville comparison principle for entire weak solutions (u,v) of semilinear parabolic second-order partial differential inequalities of the form ut −ℒu−|u|q−1 u≥vt −ℒv−|v|q−1 v  (*) in the half-space S=R+ ×Rn , where n≥1 is a natural number, q>0 is a real number and ℒ=Σi,j=1 n ∂/∂xi [aij (t,x) ∂/∂xj ]. We assume that the coefficients aij (t,x), i,j=1,…,n, of the operator ℒ are functions that are defined, measurable and locally bounded in S. We also assume that aij (t,x)=aji (t,x), i,j=1,…,n, for almost all (t,x)∈S and that Σi,j=1 n aij (t,x)ξi ξj …≥0 for almost all (t,x)∈S and all ξ∈Rn . The critical exponents in the Liouville comparison principle obtained, which are responsible for the non-existence of non-trivial (i.e., such that u\not\equiv v) entire weak solutions to (*) in S, depend on the behavior of the coefficients of the operator ℒ at infinity. As direct corollaries we obtain a new Fujita comparison principle for entire weak solutions (u,v) of the Cauchy problem for the inequality (*), as well as new Liouville-type and Fujita-type theorems for non-negative entire weak solutions u of the inequality (*) in the case when v≡0. All the results obtained are new and sharp. Show more

Keywords: comparison principle, entire solution, Fujita blow-up critical exponent, Liouville theorem, semilinear parabolic inequality, weak solution

DOI: 10.3233/ASY-2012-1151

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 83-99, 2013

Authors: Zhang, Yanyan | Zheng, Songmu

Article Type: Research Article

Abstract: In this paper, a 1-d quasilinear nonuniform parabolic chemotaxis model with volume-filling effect is studied. The global existence and uniqueness of classical solution is proved. Furthermore, we prove that the global solution is uniformly bounded in time. With the help of a suitable non-smooth Simon–Łojasiewicz approach, we obtain the results on convergence of the solution to equilibrium and the convergence rate.

Keywords: chemotaxis, volume-filling effect, quasilinear nonuniform parabolic system, uniform boundedness, convergence to equilibrium, non-smooth Simon–Łojasiewicz approach

DOI: 10.3233/ASY-2012-1154

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 101-125, 2013

Authors: Alama, Stan | Bronsard, Lia | Galvão-Sousa, Bernardo

Article Type: Research Article

Abstract: We consider singular limits of the three-dimensional Ginzburg–Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain has characteristic thickness on the scale ε>0, and we consider the simultaneous limit as the thickness ε→0 and the Ginzburg–Landau parameter κ→∞. We assume that the applied field is strong (on the order of ε−1 in magnitude) in its components tangential to the film domain, and of order log κ in its dependence on κ. We prove that the Ginzburg–Landau energy Γ-converges to an energy associated with a two-obstacle problem, posed on the planar domain which …supports the thin film. The same limit is obtained regardless of the relationship between ε and κ in the limit. Two illustrative examples are presented, each of which demonstrating how the curvature of the film can induce the presence of both (positively oriented) vortices and (negatively oriented) antivortices coexisting in a global minimizer of the energy. Show more

Keywords: partial differential equations, calculus of variations, Ginzburg–Landau, superconductivity

DOI: 10.3233/ASY-2012-1155

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 127-156, 2013

Authors: Benamou, J.-D. | Collino, F. | Marmorat, S.

Article Type: Research Article

Abstract: We present a mathematical and numerical analysis of the stability and accuracy of the NMLA (Numerical MicroLocal Analysis) method [J. Comput. Phys. 199(2) (2004), 717–741] and its discretization. We restrict to homogeneous space and focus on the two simplest cases: (1) Noisy plane wave packets, (2) Noisy point source solutions. A stability result is obtained through the introduction of a new “impedance” observable. The analysis of the point source case leads to a modified second order (curvature dependent) correction of the algorithm. Since NMLA is local, this second order improved version can be applied to general data (heterogeneous media). See …[J. Comput. Phys. 231(14) (2012), 4643–4661] for a an application to a source discovery inverse problem. Show more

Keywords: plane waves, point source, inverse scattering

DOI: 10.3233/ASY-121157

Citation: Asymptotic Analysis, vol. 83, no. 1-2, pp. 157-187, 2013