Asymptotic Analysis - Volume 67, issue 1-2

Authors: Kirkinis, Eleftherios

Article Type: Research Article

Abstract: In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. …We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. Show more

Keywords: perturbation methods, asymptotic analysis, boundary layers, nonlinear oscillators, partial differential equations

DOI: 10.3233/ASY-2009-0964

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 1-16, 2010

Authors: Cattiaux, Patrick | Chafaï, Djalil | Motsch, Sébastien

Article Type: Research Article

Abstract: The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al. in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This diffusion solves a kinetic Fokker–Planck equation based on an Ornstein–Uhlenbeck Gaussian process. The long time “diffusive” behavior of this model was recently studied by Degond and Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The …approach can be adapted to many other kinetic “probabilistic” models. Show more

Keywords: Mathematical Biology, animal behavior, hypo-elliptic diffusions, kinetic Fokker–Planck equations, Poisson equation, invariance principles, central limit theorems, Gaussian and Markov processes

DOI: 10.3233/ASY-2009-0969

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 17-31, 2010

Authors: Gruais, Isabelle | Poliševski, Dan

Article Type: Research Article

Abstract: The present paper deals with the homogenization of the heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a rectangular honeycomb structure formed by a set of thin layers crossing orthogonally and periodically. We consider the case when the conductivity of the thin layers is in inverse proportion to the vanishing volume of the rectangular honeycomb structure. We find the system that governs the asymptotic behaviour of the temperature distribution of this binary medium. The dependence with respect to the thicknesses of the layers is also emphasized. We …use an energetic method associated to a natural control-zone of the vanishing domain. Show more

Keywords: homogenization, conduction, fine-scale, honeycomb structure

DOI: 10.3233/ASY-2009-0970

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 33-43, 2010

Authors: Griso, G. | Sánchez, M.T.

Article Type: Research Article

Abstract: The goal of this work is to study the asymptotic behaviour of catalyst supports in a linear elasticity problem. The catalyst support is a structure made of beams, placed periodically and with inner holes. We introduce a decomposition of the displacements in such a structure and we prove some convergence results. Finally, we obtain the limit problem when the elasticity parameter e and the periodicity parameter ε go to zero.

Keywords: homogenization, perforated beams, unfolding method

DOI: 10.3233/ASY-2009-0971

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 45-84, 2010

Authors: Volkmer, Hans

Article Type: Research Article

Abstract: Let v(t, x) and u(t, x) be solutions of the heat equation vt −Δv=0 and dissipative wave equation utt +ut −Δu=0, respectively. The paper finds the asymptotic expansions of the squared L2 -norms of v, u and u−v as well as of their derivatives as t→∞. Suitable conditions on the initial values u(0, x), ut (0, x) and v(0, x) lead to cancellation of the leading terms of the asymptotic expansion of u−v explaining the diffusion phenomenon for linear hyperbolic waves.

Keywords: heat equation, dissipative wave equation, diffusion phenomenon, asymptotic expansion of L^2-norm

DOI: 10.3233/ASY-2010-0980

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 85-100, 2010

Authors: Tate, Tatsuya

Article Type: Research Article

Abstract: We discuss asymptotics of the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold. We obtain an asymptotic formula for the average of the number of states. Furthermore, when the operator has integer eigenvalues and the periodic orbits of period less than 2π of the classical mechanics form clean submanifolds of lower dimensions, we give an asymptotic formula for the number of states itself. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer. We …use the Meinardus saddle point method of obtaining the asymptotics of the number of partitions, combined with a theorem due to Duistermaat–Guillemin and other authors on the singularities of the trace of the wave operators. Show more

Keywords: number of states of Boson gas, number of partitions, Meinardus suddle point method, singularities of traces of wave operators

DOI: 10.3233/ASY-2009-0973

Citation: Asymptotic Analysis, vol. 67, no. 1-2, pp. 101-123, 2010