Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition

TL;DR

This paper develops a periodic homogenization approach for pseudo-parabolic equations modeling fluid flow in porous media, using spatio-temporal decomposition and two-scale convergence, including advection effects and non-local temporal terms.

Contribution

It introduces a novel upscaling method for pseudo-parabolic equations with advection, employing spatio-temporal decomposition and periodic homogenization techniques.

Findings

Successfully upscaled pseudo-parabolic equations with advection.

Proved well-posedness of the extended pseudo-parabolic system.

Identified conditions for non-local-in-time term emergence.

Abstract

Pseudo-parabolic equations have been used to model unsaturated fluid flow in porous media. In this paper it is shown how a pseudo-parabolic equation can be upscaled when using a spatio-temporal decomposition employed in the Peszyn'ska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of periodic homogenization via two-scale convergence. The well-posedness of the extended pseudo-parabolic equation is shown as well. Moreover, we argue that under certain conditions, a non-local-in-time term arises from the elimination of an unknown.