# Homogenization of a pseudo-parabolic system via a spatial-temporal decoupling: upscaling and corrector estimates for perforated domains

## TL;DR

This paper develops a method to upscale a pseudo-parabolic system with drift in perforated domains by using a spatial-temporal decoupling, providing convergence speed estimates and extending applicability to various space-time domains.

## Contribution

It introduces a novel spatial-temporal decomposition approach for homogenizing pseudo-parabolic systems with drift in perforated domains, including convergence speed analysis.

## Key findings

- Established convergence speeds for the upscaled system.
- Extended applicability to space-time domains like perforated and time-independent domains.
- Provided convergence results for global times as epsilon approaches zero.

## Abstract

In this paper, we determine the convergence speed of an upscaling of a pseudo-parabolic system containing drift terms with scale separation of size $\epsilon \ll 1$. Both the upscaling and convergence speed determination exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into a spatial elliptic partial differential equation and a temporal ordinary differential equation. We extend the applicability to space-time domains that are a product of spatial and temporal domains, such as a time-independent perforated spatial domain. Finally, for special cases we show convergence speeds for global times, i.e. $t \in \mathbf{R}_+$, by using time intervals that converge to $\mathbf{R}_+$ as $\epsilon\downarrow 0$.