Two-scale homogenization of abstract linear time-dependent PDEs

Article type: Research Article

Authors: Neukamm, Stefan^a | Varga, Mario^a | Waurick, Marcus^{b; c; *}

Affiliations: [a] Faculty of Mathematics, TU Dresden, Germany. E-mails: stefan.neukamm@tu-dresden.de, mario.varga@tu-dresden.de | [b] Chair Applied Analysis, Institute of Mathematics (E-10), Hamburg University of Technology, Germany. E-mail: marcus.waurick@tuhh.de | [c] Department of Mathematics and Statistics, University of Strathclyde, Glasgow, Scotland

Correspondence: [*] Corresponding author. E-mail: marcus.waurick@tuhh.de.

Abstract: Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

Keywords: Periodic and stochastic homogenization, unfolding, abstract evolutionary equations, Maxwell’s equations

DOI: 10.3233/ASY-201654

Journal: Asymptotic Analysis, vol. 125, no. 3-4, pp. 247-287, 2021