# Two-scale homogenization of abstract linear time-dependent PDEs

## TL;DR

This paper develops a unified Hilbert space framework for homogenizing a broad class of linear time-dependent PDEs, including stochastic cases, by combining operator theory with periodic unfolding techniques.

## Contribution

It introduces a novel operator-theoretic approach to homogenization of abstract evolutionary systems, extending periodic unfolding to stochastic settings.

## Key findings

- Established homogenization results for elliptic PDEs, Maxwell's equations, and the wave equation.
- Unified framework simplifies analysis of oscillatory coefficients in PDEs.
- Applicable to both periodic and stochastic homogenization scenarios.

## 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.