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Homogenization of Poisson-Nernst-Planck equations for multiple species in a porous medium

Apratim Bhattacharya

TL;DR

This work provides a rigorous homogenization framework for the Poisson--Nernst--Planck system with multiple ionic species in a periodic porous medium, valid in dimensions two and three. By introducing nonlinear diffusion via an app-PNP approximation and employing carefully designed cut-off functions, the authors obtain uniform estimates and strong convergence in $L^1_tL^r_x$, enabling passage to the limit in nonlinear drift terms. The homogenized model is characterized by a two-scale Nernst--Planck system coupled to a homogenized Poisson equation with a constant effective matrix $A_{hom}$ defined through cell problems; strong convergence and two-scale limits are established for concentrations and the electrostatic potential. The resulting macroscopic equations (valid for general $P$ species) provide a computationally tractable description of electro-diffusion in porous media and extend prior two-species results to the multi-species setting, with a discussion on uniqueness under additional regularity. This contributes a robust theoretical foundation for multiscale electro-diffusion modeling and supports numerical homogenization of complex ionic transport problems.

Abstract

We rigorously derive a homogenized model for the Poisson--Nernst--Planck (PNP) equations for the case of multiple species defined on a periodic porous medium in spatial dimensions two and three. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by constructing suitable cut-off functions. The cut-off function, together with the application of a previously known energy functional, yields strong convergence of the microscopic concentrations in $L^1_t L^r_x$, for some $r>2$, enabling us to pass to the limit in the nonlinear drift term. Finally, we derive the homogenized equations by means of two-scale convergence in $L^p_t L^q_x$ setting.

Homogenization of Poisson-Nernst-Planck equations for multiple species in a porous medium

TL;DR

This work provides a rigorous homogenization framework for the Poisson--Nernst--Planck system with multiple ionic species in a periodic porous medium, valid in dimensions two and three. By introducing nonlinear diffusion via an app-PNP approximation and employing carefully designed cut-off functions, the authors obtain uniform estimates and strong convergence in , enabling passage to the limit in nonlinear drift terms. The homogenized model is characterized by a two-scale Nernst--Planck system coupled to a homogenized Poisson equation with a constant effective matrix defined through cell problems; strong convergence and two-scale limits are established for concentrations and the electrostatic potential. The resulting macroscopic equations (valid for general species) provide a computationally tractable description of electro-diffusion in porous media and extend prior two-species results to the multi-species setting, with a discussion on uniqueness under additional regularity. This contributes a robust theoretical foundation for multiscale electro-diffusion modeling and supports numerical homogenization of complex ionic transport problems.

Abstract

We rigorously derive a homogenized model for the Poisson--Nernst--Planck (PNP) equations for the case of multiple species defined on a periodic porous medium in spatial dimensions two and three. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by constructing suitable cut-off functions. The cut-off function, together with the application of a previously known energy functional, yields strong convergence of the microscopic concentrations in , for some , enabling us to pass to the limit in the nonlinear drift term. Finally, we derive the homogenized equations by means of two-scale convergence in setting.
Paper Structure (26 sections, 20 theorems, 124 equations, 1 figure)

This paper contains 26 sections, 20 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. The mathematical model
  4. Previous contributions
  5. Mathematical challenge
  6. Main idea
  7. Outline of the paper
  8. Function spaces and notations
  9. The microscopic problem
  10. Geometry of the microscopic domain
  11. Assumptions on the data
  12. Existence of weak solutions for microscopic PNP
  13. The microscopic app-PNP
  14. Existence of weak solutions for microscopic app-PNP
  15. Convergence of microscopic app-PNP to microscopic PNP
  16. ...and 11 more sections

Key Result

Proposition 2.1

Suppose the assumptions $(A1)-(A5)$ are satisfied. Then there exist $c_{i,\epsilon} \in L^{\frac{5}{4}}(0,T; W^{1,\frac{5}{4}}(\Omega_\epsilon)) \cap L^{\frac{5}{4}}(0,T; L^\frac{15}{7}(\Omega_\epsilon))$, $c_{i, \epsilon} \geq 0$ a.e. in $(0,T) \times \Omega_\epsilon$ and $\phi_{\epsilon} \in L^\i for all $\psi_1 \in C^\infty ([0,T] \times \overline{\Omega}_\epsilon)$ with $\psi_1(T,.) =0$ and

Figures (1)

  • Figure 1: The standard cell $Y= Y^f \cup \overline{Y^s}$ (left) and the porous medium $\Omega$ with the fluid (pore) part $\Omega_\epsilon$ (right).

Theorems & Definitions (49)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • ...and 39 more