Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

TL;DR

This paper investigates the homogenization of parabolic systems with periodic coefficients, providing operator error estimates for the exponential of elliptic operators and applying these results to initial boundary-value problems.

Contribution

It introduces new operator error estimates for the homogenization of the exponential of elliptic operators with periodic coefficients, specifically in the context of parabolic systems.

Findings

Derived operator norm approximations for $e^{-B_{D, ext{}\varepsilon}t}$

Established error estimates in Sobolev spaces for homogenized solutions

Applied results to initial boundary-value problems for parabolic systems

Abstract

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix second order elliptic differential operator $B_{D,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator $B_{D,\varepsilon}$ is positive definite; its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of the operator exponential $e^{-B_{D,\varepsilon}t}$, $t>0$, as $\varepsilon\rightarrow 0$. We obtain approximations for the exponential $e^{-B_{D,\varepsilon}t}$ in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.