On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a‐priori estimates and stability

Kovtunenko, Victor A.; Zubkova, Anna V.

doi:10.1002/mma.4140

Cited by 12 publications

(15 citation statements)

References 22 publications

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“…Well-posedness analysis. Existence and uniqueness of the weak solution of the PNP problem were proved rigorously in [20,21]. Therefore, here we formulate the main results omitting proofs.…”

Section: 2mentioning

confidence: 99%

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium

Kovtunenko¹,

Zubkova²

2018

Kinetic &Amp; Related Models

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In this paper a mathematical model generalizing Poisson-Nernst-Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

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“…Well-posedness analysis. Existence and uniqueness of the weak solution of the PNP problem were proved rigorously in [20,21]. Therefore, here we formulate the main results omitting proofs.…”

Section: 2mentioning

confidence: 99%

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium

Kovtunenko¹,

Zubkova²

2018

Kinetic &Amp; Related Models

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“…We formulate a generalized Poisson-Nernst-Planck system depending on a fixed parameter ε > 0, see [9][10][11]. We consider the number n of charged species with specific charges z i , molar masses m i > 0, volume factors β i > 0, and unknown concentrations c ε i for i = 1, .…”

Section: Problem Formulationmentioning

confidence: 99%

“…The solvability of classic PNP systems was studied in [5,6]. Based on a general approach from [7,8], in the previous works [9][10][11], we proved existence theorem for the generalized PNP problem and derived a-priori estimates.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of the generalized Poisson–Nernst–Planck problem in a two-phase medium: correctors and estimates

Kovtunenko

Zubkova

2019

Applicable Analysis

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The paper provides a rigorous homogenization of the Poisson-Nernst-Planck problem stated in an inhomogeneous domain composed of two, solid and pore, phases. The generalized PNP model is constituted of the Fickian cross-diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials, and Darcy's flow model. At the interface between two phases inhomogeneous boundary conditions describing electrochemical reactions are considered. The resulting doubly non-linear problem admits discontinuous solutions caused by jumps of field variables. Using an averaged problem and first-order asymptotic correctors, the homogenization procedure gives us an asymptotic expansion of the solution which is justified by residual error estimates.

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“…9 For the treatment of oscillating third boundary conditions, we refer to Belyaev et al 10 and Oleinik and Shaposhnikova. 11 Within elecktrokinetic modeling (see Allaire et al 12 ), in previous studies, [13][14][15][16] there were considered generalized Poisson-Nernst-Planck (PNP) models over two-phase domains accounting for interface reactions. The corresponding PDE system obeys a structure of the gradient flow; see, eg, other works.…”

Section: Introductionmentioning

confidence: 99%

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

Kovtunenko

Reichelt

Zubkova

2019

Math Methods in App Sciences

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This paper is devoted to the homogenization of a nonlinear transmission problem stated in a two-phase domain. We consider a system of linear diffusion equations defined in a periodic domain consisting of two disjoint phases that are both connected sets separated by a thin interface. Depending on the field variables, at the interface, nonlinear conditions are imposed to describe interface reactions. In the variational setting of the problem, we prove the homogenization theorem and a bidomain averaged model. The periodic unfolding technique is used to obtain the residual error estimate with a first-order corrector. KEYWORDS bidomain model, corrector estimates, diffusion problem, nonlinear transmission conditions, periodic unfolding technique MSC CLASSIFICATION 35B27; 35M10; 82C24This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a‐priori estimates and stability

Cited by 12 publications

References 22 publications

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium

Homogenization of the generalized Poisson–Nernst–Planck problem in a two-phase medium: correctors and estimates

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

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