Two-scale homogenization of abstract linear time-dependent PDEs

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.

Stochastic two-scale convergence and Young measures

“…In the following we briefly introduce the stochastic unfolding operator and provide its main properties, for the proofs and detailed studies we refer to [30,19,34,31].…”

Stochastic homogenization of $Λ$-convex gradient flows

“…Also, similarly as in the periodic case, stochastic two-scale convergence in the mean from [9] might be equivalently characterized as weak convergence of the unfolded sequence. In this respect, we develop a general procedure for stochastic homogenization problems, see also [48] for a detailed analysis of this method, and [36] for an extension to abstract, linear evolution systems in an operator theoretic framework. Stochastic unfolding has first been introduced by the second and third author in a discrete version in [35] where the discrete-tocontinuum limit of a rate-independent evolution is analyzed.…”

Stochastic two-scale convergence and Young measures

“…where V hom is a deterministic, convex integrand and characterized by a homogenization formula, see (31) below. There are different natural choices for the topology when passing to this limit:…”

“…Stochastic unfolding and two-scale convergence in the mean. In the following we briefly introduce the stochastic unfolding operator and provide its main properties, for the proofs and detailed studies we refer to [30,19,34,31].…”