Asymptotic Analysis - Volume 72, issue 1-2

Authors: Heida, Martin

Article Type: Research Article

Abstract: The stochastic two-scale convergence method, recently developed in an article by Zhikov and Piatnitskii [Izv. Math. 70(1) (2006), 19–67] is extended to arbitrary probability spaces and is now based on the theory of stochastic geometry instead of random measures. It will be shown that former results on stochastic and periodic two-scale convergence fit into the new approach in a natural way. These results will be applied to functions with jumps on (n−1)-dimensional manifolds, in particular to a homogenization problem of heat transfer through a composite material or polycrystal.

Keywords: stochastic homogenization, two-scale convergence, singular structures, heat transfer

DOI: 10.3233/ASY-2010-1022

Citation: Asymptotic Analysis, vol. 72, no. 1-2, pp. 1-30, 2011

Authors: Afendikov, Andrei | Fiedler, Bernold | Liebscher, Stefan

Article Type: Research Article

Abstract: We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we obtain a dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. We investigate such a “Takens–Bogdanov bifurcation without parameters” by blow-up …and averaging techniques. In particular we describe the complete set ℬ of all small bounded solutions. In the case of a double symmetry of the external force, which leads to a bi-reversible problem, the authors have proved in Asymptot. Anal. 60(3,4) (2008), 185–211, that ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. In the present article we study the more complicated case where only one symmetry is present. Then ℬ consist entirely of trivial equilibria and multipulse heteroclinic pairs. The latter form a very complicated, albeit non-recurrent, set. Graphics of simplest case scenarios for ℬ are included. Show more

Keywords: planar fluid flow, bifurcation without parameters, Takens–Bogdanov bifurcation, spatial dynamics

DOI: 10.3233/ASY-2010-1026

Citation: Asymptotic Analysis, vol. 72, no. 1-2, pp. 31-76, 2011

Authors: Fournais, Søren | Persson, Mikael

Article Type: Research Article

Abstract: In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schrödinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to infinity. This asymptotic formula is used to prove that the eigenvalue is monotonically increasing for large values of the magnetic field.

Keywords: eigenvalue asymptotics, large magnetic field, unit ball, Ginzburg–Landau functional, surface superconductivity

DOI: 10.3233/ASY-2010-1023

Citation: Asymptotic Analysis, vol. 72, no. 1-2, pp. 77-123, 2011