On the homogenization of partial integro-differential-algebraic equations

TL;DR

This paper develops a unified Hilbert space framework for homogenizing various linear evolutionary boundary value problems across physics, including memory effects and differential-algebraic equations, ensuring well-posedness and causality of the limit equations.

Contribution

It introduces a general operator-theoretic approach to homogenization that encompasses non-periodic, memory, and differential-algebraic systems in a unified manner.

Findings

The limit equations are proven to be well-posed.

The approach applies to thermodynamics, elasticity, and electromagnetism.

Memory and differential-algebraic problems are included.

Abstract

We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics, elasticity, electro-magnetism and coupled systems thereof. The approach permits the consideration of memory problems as well as differential-algebraic equations. We show that the limit equation is well-posed and causal. We rely on techniques from functional analysis and operator theory only.