A refined convergence result in homogenization of second order parabolic systems
type: Item
instance of: scholarly article
title: A refined convergence result in homogenization of second order parabolic systems
zbMATH Open document ID: 1415.35027
author: Weisheng Niu
author: Yao Xu
published in: Journal of Differential Equations
publication date: 2019-03-21T00:00:00Z
review text: In the paper under review, the authors investigate the convergence rate in periodic homogenization of second order parabolic systems. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ and $T>0$. The authors consider the homogenization of the parabolic system \[ \begin{cases} \partial_t u_\epsilon+\mathcal{L}_\epsilon u_\epsilon =f & \text{in } \Omega \times (0,T),\\ u_\epsilon=0 & \text{on } \partial \Omega \times (0,T),\\ u_\epsilon=h & \text{on } \Omega \times \{t=0\}, \end{cases} \] where \[ \mathcal{L}_\epsilon=-\partial_\alpha (A_{ij}^{\alpha\beta}(x/\epsilon,t/\epsilon^2)\partial_\beta), \qquad 1\leq i,j\leq n,1\leq \alpha,\beta\leq d,\] where the coefficient matrix $A(y,s)=(A_{ij}^{\alpha\beta}(y,s))$ is real, bounded, measurable, $1$-periodic in $(y,s)$ and \[\mu |\xi|^2\leq A^{\alpha \beta}(y,s)\xi_\alpha \xi_\beta,\qquad |A(y,s)|\leq 1/\mu \quad \text{for a.e. } (y,s)\in \mathbb{R}^{d+1}. \] The unique solution $u^\epsilon$ of the problem above is well known to converge to the unique solution $u_0$ of the following parabolic equation with constant coefficients \[ \begin{cases} \partial_t u_0+\mathcal{L}_0 u_0 =f & \text{in } \Omega \times (0,T),\\ u_0=0 & \text{on } \partial \Omega \times (0,T),\\ u_0=h & \text{on } \Omega \times \{t=0\}, \end{cases} \] where $\mathcal{L}_0$ is the homogenized operator. The authors show the optimal $O(\epsilon)$ convergence rate of $u_\epsilon$ to $u_0$ in $L^2(0,T;L^{q_0}(\Omega))$, with $q_0=2d/(d-1)$, $d\geq 2$.
reviewed by: Paolo Musolino
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 7039978
zbMATH Keywords: periodic homogenization
zbMATH Keywords: convergence rates
zbMATH Keywords: time-dependent parabolic systems
MaRDI profile type: Publication
full work available at URL: j.jde.2018.12.033
OpenAlex ID: W2910753439
cites work: Quantitative stochastic homogenization and regularity theory of parabolic equations
cites work: Q4197852
cites work: The Method of Layer Potentials for the Heat Equation in Lipschitz Cylinders
cites work: Uniform regularity estimates in parabolic homogenization
cites work: Convergence rates in parabolic homogenization with time-dependent periodic coefficients
cites work: Error estimate and unfolding for periodic homogenization
cites work: Interior error estimate for periodic homogenization
cites work: Homogenization of Stokes Systems and Uniform Regularity Estimates
cites work: Heat kernels and maximal lp—lqestimates for parabolic evolution equations
cites work: Convergence rates in \(L ^{2}\) for elliptic homogenization problems
cites work: Estimates of eigenvalues and eigenfunctions in periodic homogenization
cites work: Periodic Homogenization of Green and Neumann Functions
cites work: Homogenization of solutions of initial boundary value problems for parabolic systems
cites work: Homogenization of initial boundary value problems for parabolic systems with periodic coefficients
cites work: Convergence rates and interior estimates in homogenization of higher order elliptic systems
cites work: Convergence rates in homogenization of higher-order parabolic systems
cites work: Boundary Value Problems for Parabolic Lame Systems and a Nonstationary Linearized System of Navier-Stokes Equations in Lipschitz Cylinders
cites work: Boundary estimates in elliptic homogenization
cites work: Periodic homogenization of elliptic systems
cites work: Convergence rates in periodic homogenization of systems of elasticity
cites work: Q5607484
cites work: Homogenization of the parabolic Cauchy problem in the Sobolev class \(H^1(\mathbb R^d)\)
cites work: HOMOGENIZATION OF THE DIRICHLET PROBLEM FOR ELLIPTIC SYSTEMS: ‐OPERATOR ERROR ESTIMATES
cites work: Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients
cites work: Convergence rates for general elliptic homogenization problems in Lipschitz domains
cites work: Estimates of homogenization for a parabolic equation with periodic coefficients
DOI: 10.1016/J.JDE.2018.12.033
description: scientific article; zbMATH DE number 7039978