Corrector Estimates for a Thermodiffusion Model with Weak Thermal Coupling

Abstract

The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology "weak thermal coupling" refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conductiondiffusion interaction terms, while the "high-contrast" is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes.

Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains

“…There exist many articles on the derivation of error estimates for different classes of reaction-diffusion systems, eg, other works, [22][23][24][25] exploiting a higher regularity of the limit solution and the continuous extension operator from a perforated domain. Moreover, unfolding-based error estimates have been proven for linear, elliptic transmission problems in Reichelt,26 for reaction-diffusion systems with linear boundary conditions in perforated domains in Muntean and Reichelt,27 and for systems with nonlinear interface conditions in a two-phase domain in Fatima et al 28 The latter results are based on the quantification of the periodicity defect for the periodic unfolding operator in Griso, 29,30 and they hold without assuming higher regularity for the corrector problem.…”

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

“…The process is usually called thermo-diffusion and considerable phenomenological understanding is available (compare [11] or the more recent accounts by Wojnar [14,15]). Regarding the presence in the right-hand side of the model equaations of the products between temperature and concentration gradients -mimicking thermodynamic Soret and Dufour effects pointing out a strong interplay between heat conduction and molecular diffusion -we refer the reader to [8,9,13]. In these settings, such strongly nonlinear structures arising in the model equations play a decisive role in capturing the expected evolution of the physical system.…”