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February  2018, 11(1): 119-135. doi: 10.3934/krm.2018007

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

 1 Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36,8010 Graz, Austria 2 Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia

* Corresponding author: anna.zubkova@uni-graz.at

Received  November 2016 Revised  March 2017 Published  August 2017

Fund Project: The authors are supported by the Austrian Science Fund (FWF) Project P26147-N26: "Object identification problems: numerical analysis" (PION).

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.

Citation: Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007
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An example domain $\Omega = Q \cup \omega \cup \partial\omega$ with two phases $Q$ and $\omega$, the boundary $\partial\Omega$, and two faces $\partial\omega^+$ and $\partial\omega^-$ of the interface $\partial\omega$ shown in zoom.