Large-time behavior of solutions to a thermo-diffusion system with Smoluchowski interactions

TL;DR

This paper analyzes the long-term behavior of solutions to a coupled thermo-diffusion system modeling hot colloidal particles in porous media, addressing mathematical challenges posed by thermodynamic cross-coupling effects.

Contribution

It establishes the large-time behavior and uniqueness of solutions for a complex thermo-diffusion system with thermodynamic interactions.

Findings

Proved the solutions' large-time stability.

Established the uniqueness of solutions.

Addressed mathematical difficulties from thermodynamic cross terms.

Abstract

We prove the large time behavior of solutions to a coupled thermo-diffusion arising in the modelling of the motion of hot colloidal particles in porous media. Additionally, we also ensure the uniqueness of solutions of the target problem. The main mathematical difficulty is due to the presence in the right-hand side of the equations of products between temperature and concentration gradients. Such terms mimic the so-called thermodynamic Soret and Dufour effects. These are cross-coupling terms emphasizing in this context a strong interplay between heat conduction and molecular diffusion.