Correctors justification for a Smoluchowski--Soret--Dufour model posed in perforated domains

TL;DR

This paper derives corrector estimates for a coupled thermo-diffusion model in perforated domains, analyzing the convergence rate of the homogenization process involving complex nonlinear aggregation, deposition, and micro-surface interactions.

Contribution

It provides the first rigorous corrector estimates for a nonlinear coupled thermo-diffusion system with micro-surface effects in perforated media.

Findings

Quantified convergence rate of the homogenization limit.

Established estimates for nonlinear aggregation and deposition processes.

Applied energy methods to handle high-dimensional convergence challenges.

Abstract

We study a coupled thermo-diffusion system that accounts for the dynamics of hot colloids in periodically heterogeneous media. Our model describes the joint evolution of temperature and colloidal concentrations in a saturated porous structure, where the Smoluchowski interactions are responsible for aggregation and fragmentation processes in the presence of Soret-Dufour type effects. Additionally, we allow for deposition and depletion on internal micro-surfaces. In this work, we derive corrector estimates quantifying the rate of convergence of the periodic homogenization limit process performed in \cite{KAM14} via two-scale convergence arguments. The major technical difficulties in the proof are linked to the estimates between nonlinear processes of aggregation and deposition and to the convergence arguments of the \emph{a priori} information of the oscillating weak solutions and cell functions in high dimensions. Essentially, we circumvent the arisen difficulties by a suitable use of the energy method and of fine integral estimates controlling interactions at the level of micro-surfaces.