Asymptotic Analysis - Volume 20, issue 1

Authors: Souganidis, Panagiotis E.

Article Type: Research Article

Abstract: Homogenization‐type results for the Cauchy problem for first‐order PDE (Hamilton–Jacobi equations) are presented. The main assumption is that the Hamiltonian is superlinear and convex with respect to the gradient and stationary and ergodic with respect to the spatial variable. Some of applications to related problems as well as to the asymptotics of reaction–diffusion equations and turbulent combustion are also presented.

Citation: Asymptotic Analysis, vol. 20, no. 1, pp. 1-11, 1999

Authors: Lahmar‐Benbernou, Amina | Martinez, André

Article Type: Research Article

Abstract: The aim of this study is to give complete semiclassical asymptotics of the residues {\rm Res}[S(\lambda,\omega,\omega^\prime),\rho] at some pole \rho of the distributional kernel of the scattering matrix S(\lambda) corresponding to a semiclassical two‐body Schrödinger operator P=-h^2\Delta + V , and considered as a meromorphic operator‐valued function with respect to the energy \lambda . We do it in the case where the pole \rho considered is a shape resonance of P . This is a continuation of A. Benbernou, Estimation des residus de la matrice de diffusion associés a …dès résonances de forme I (to appear in Ann. Inst. H. Poincaré), where an extra geometrical condition was assumed (namely the absence of caustics near the energy level {\rm Re}\ \rho ). Here we drop this assumption by using an FBI transform which permits to work in the complexified phase space. Then we show that some semiclassical WKB expansions are global, and this allows us to find out estimates for the residue of the type {\rm O}(h^N{\rm e}^{-2S_0/h}) , where S_0 is the Agmon width of the potential barrier, and N may be arbitrarily large depending on an explicit geometrical location of the incoming and outgoing waves \omega and \omega' one consider. Full asymptotic expansions are obtained under some additional generic geometric assumption on the potential V . Show more

Citation: Asymptotic Analysis, vol. 20, no. 1, pp. 13-38, 1999

Authors: Percivale, Danilo

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 20, no. 1, pp. 39-59, 1999

Authors: Tassa, Tamir

Article Type: Research Article

Abstract: We study the homogenization of oscillatory solutions of partial differential equations with a multiple number of small scales. We consider a variety of problems – nonlinear convection–diffusion equations with oscillatory initial and forcing data, the Carleman model for the discrete Boltzman equations, and two‐dimensional linear transport equations with oscillatory coefficients. In these problems, the initial values, force terms or coefficients are oscillatory functions with a multiple number of small scales – f(x,{x}/{{\varepsilon}_1},\ldots,{x}/{{\varepsilon}_n}) . The essential question in this context is what is the weak limit of such functions when {\varepsilon}_i \downarrow 0 and what is the corresponding …convergence rate. It is shown that the weak limit equals the average of f(x,\cdot) over an affine submanifold of the torus T^n ; the submanifold and its dimension are determined by the limit ratios between the scales, \alpha_i= \lim {{\varepsilon}_1}/{{\varepsilon}_i} , their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios {{\varepsilon}_1}/{{\varepsilon}_i} tend to their limit \alpha_i . These results and the accompanying convergence rate estimates are then used in deriving the homogenized equations in each of the abovementioned problems. Show more

Citation: Asymptotic Analysis, vol. 20, no. 1, pp. 61-96, 1999