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Homogenization of Three Species Reaction Diffusion Equation in Perforated Domains

Saumyajit Das, Kshitij Sinha

Abstract

In this article we study the asymptotic behaviour of the solution of the three species chemical reaction-diffusion model with non-homogeneous Neumann boundary condition in a perforated domain. We investigate how the mass inflow at the microscale affects the three-species reaction-diffusion system at the macroscale using two-scale convergence. As the size of the perforations vanishes, the microscale effects are captured by a global source term in the homogenized equation, which remains a three-species reaction-diffusion system but with modified diffusion coefficients.

Homogenization of Three Species Reaction Diffusion Equation in Perforated Domains

Abstract

In this article we study the asymptotic behaviour of the solution of the three species chemical reaction-diffusion model with non-homogeneous Neumann boundary condition in a perforated domain. We investigate how the mass inflow at the microscale affects the three-species reaction-diffusion system at the macroscale using two-scale convergence. As the size of the perforations vanishes, the microscale effects are captured by a global source term in the homogenized equation, which remains a three-species reaction-diffusion system but with modified diffusion coefficients.

Paper Structure

This paper contains 10 sections, 21 theorems, 161 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Description of Domain
  3. Plan of the Paper
  4. Uniform Estimates
  5. Homogenization
  6. Existence Theory
  7. Extension Operator and its time derivative
  8. Trace Inequality and Anisotropic Sobolev Inequality
  9. Two-Scale Convergence
  10. Well-prepared Initial Condition

Key Result

Theorem 1.1

Let $a^1_{\varepsilon}, a^2_{\varepsilon}$ and $a^3_{\varepsilon}$ satisfies the system eq: aeps-eq: ceps respectively and the initial condition satisfies compatibility condition compatibility condition, uniform estimate CI and Then we have the following: for $i=1,2,3$ where $\Tilde{.}$ denotes the extension by zero outside $\Omega_{\varepsilon}$ and $a^i$ is the solution of the following equatio

Figures (3)

  • Figure 1: Domain with periodically distributed holes.
  • Figure 2: Perforated unit cell
  • Figure 3: $\varepsilon$- cell

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • ...and 25 more