Estimates of homogenization for a parabolic equation with periodic coefficients
type: Item
instance of: scholarly article
title: Estimates of homogenization for a parabolic equation with periodic coefficients
zbMATH Open document ID: 1129.35012
DOI: 10.1134/S1061920806020087
author: Svetlana E. Pastukhova
author: V. V. Zhikov
published in: Russian Journal of Mathematical Physics
publication date: 2007-03-26T00:00:00Z
review text: The authors consider in the half-space \({\mathbb R}^{n+1}_{+}=\{(x,t): x\in {\mathbb R}^n\), \(t\geqslant 0\}\) the family of the Cauchy problems \[ \partial_tu^\varepsilon - \text{ div}\,a(\varepsilon^{-1}x)\nabla u^\varepsilon=0 \quad \forall t>0, \qquad u^\varepsilon(\cdot,0)=f(x)\in L^2({\mathbb R}^n), \] where \(\varepsilon \in (0,1]\) and \(a(y)\) is a measurable symmetric periodic matrix having a periodicity cell \(Y=[-1/2,1/2)^n\) and satisfying for every \(\xi\in {\mathbb R}^n\) the inequality \(\lambda\xi^2\leqslant a(y)\xi\cdot\xi\leqslant \lambda^{-1}\xi^2\) with a positive constant \(\lambda\). The main results of the article concern estimates of \(L^2({\mathbb R}^n)\)-norms of the differences \(u^\varepsilon(\cdot,t)-u^0(\cdot,t)\) and \(\nabla u^\varepsilon(\cdot,t)-\nabla v^\varepsilon(\cdot,t)\) for every \(t>0\), where \(u^0\) is the solution of the corresponding homogenized problem and \(v^\varepsilon\) is the so-called first approximation.
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5136624
zbMATH Keywords: homogenization
zbMATH Keywords: parabolic equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: periodic coefficients
zbMATH Keywords: first approximation
reviewed by: Alexander A. Kovalevsky
MaRDI profile type: Publication
cites work: Non-homogeneous media and vibration theory
cites work: On operator estimates in homogenization theory
cites work: Some estimates from homogenized elasticity problems
cites work: On operator estimates for some problems in homogenization theory
cites work: Q3470799
full work available at URL: s1061920806020087
OpenAlex ID: W2084619856
description: scientific article; zbMATH DE number 5136624