Homogenization of a thermo-diffusion system with Smoluchowski interactions

Abstract

We analyze a coupled system of evolution equations that describes the effect of thermal gradients on the motion and deposition of N populations of colloidal species diffusing and interacting together through Smoluchowski production terms. This class of systems is particularly useful in studying drug delivery, contaminant transport in complex media, as well as heat shocks thorough permeable media. The particularity lies in the modeling of the nonlinear and nonlocal coupling between diffusion and thermal conduction. We investigate the semidiscrete as well as the fully discrete a priori error analysis of the finite elements approximation of the weak solution to a thermo-diffusion reaction system posed in a macroscopic domain. The mathematical techniques include energy-like estimates and compactness arguments.

A high-order corrector estimate for a semi-linear elliptic system in perforated domains

“…In [7], the investigated microscopic semi-linear system resembles a steady state-type of thermo-diffusion systems ( [10,9]). Essentially, we have analyzed the solvability of the microscopic system in [7], derived the upscaled equations as well as the corresponding effective coefficients, and proved the high-order corrector estimates for the differences of concentrations and their gradients in which the standard energy method has been used.…”

Corrector Estimates for a Thermodiffusion Model with Weak Thermal Coupling

“…Proof. The existence of solutions, non-negativity, and uniform boundedness follow from the Lemmata 3.2 -3.6 and Theorem 3.8 in [KAM14] by replacing κ ε and τ ε with ε 2 κ ε and ετ , respectively. Note that the proof can be generalized from diffusion coefficients d ε , κ ε ∈ R to symmetric matrices as in Assumption 2.3(i).…”

“…We investigate a system of reaction-diffusion equations which includes mollified crossdiffusion terms and different diffusion length scales. The cross-diffusion terms are motivated by the incorporation of Soret and Dufour effects as outlined in [KAM14]. For more information on phenomenological descriptions of thermo-diffusion, we refer the reader to [deM84].…”

“…Furthermore, the reaction term R(·) models the Smoluchovski interaction production. In the original model from [KAM14], the function v ε is an additional unknown modeling the mass of deposited species on the pore surface Γ ε , and it is shown to possess the regularity v ε ∈ H 1 (0, T ; L 2 (Γ ε ))∩L ∞ ((0, T )×Γ ε ).…”

“…In equation (35) Remark 2.5. Since our solutions are uniformly bounded in L ∞ ((0, T ) × Ω ε ), we may consider reaction terms with arbitrary growth as in [KAM14]. Also note that estimate (2.6) remains valid for all α, β ≥ 1 and β = 0.…”