The spectral method in asymptotic diffusion problems with drift
type: Item
instance of: scholarly article
title: The spectral method in asymptotic diffusion problems with drift
zbMATH Open document ID: 0884.35048
DOI: 10.1007/BF02308824
author: I. A. Aleksandrova
published in: Mathematical Notes
publication date: 1997-08-18T00:00:00Z
review text: Consider the diffusion equation \[ {\partial u\over \partial t}- \text{div} \bigl( {\mathbf a} (x)\nabla u\bigr) +b(x) \nabla u=0, \tag{1} \] where \({\mathbf a} (x)\) is a smooth periodic symmetric matrix satisfying the ellipticity condition \(l_1|\xi|^2 \leq {\mathbf a}_{ij} (x)\xi_i \xi_j\), \(l_1>0\), and \(b(x)\) is a smooth periodic vector field. Let \(K(x,y,t)\) be a fundamental solution of equation (1) bounded on \(\mathbb{R}^m \times \mathbb{R}^m\) for each \(t>0\). In the present paper we consider several asymptotic problems, namely homogenization, the central limit theorem, and stabilization, in terms of the behavior of the fundamental solution \(K(x,y,t)\) as \(t\to\infty\). We use the method suggested by \textit{V. V. Zhikov} in [Differ. Equations 25, No. 1, 33-39 (1989); transl. from Differ. Uravn. 25, No. 1, 44-50 (1989; Zbl 0695.35014)], where the diffusion equation without drift was considered. We derive all our results from an appropriate estimate for \(K(x,y,t)\), which is based on comparing the original equation (1) with a simple equation of the form \(\partial u/\partial t- \text{div} ({\mathbf a}^0 \nabla u)=0\), where \({\mathbf a}^0\) is a constant positive definite symmetric matrix.
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1037762
zbMATH Keywords: homogenization
zbMATH Keywords: central limit theorem
zbMATH Keywords: stabilization
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1037762