AMaSiS 2018 Workshop: Abstracts

Entropy method for generalized Poisson–Nernst–Planck equations

Victor A. Kovtunenko(1,2) and José Rodrigo González Granada(3)

(1) Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz

(2) Lavrent’ev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Novosibirsk

(3) Department of Mathematics, Universidad Tecnológica de Pereira

To describe electro-kinetic transport phenomena occurring for micro-structures in many physical, chemical, and biological applications, a proper mathematical model adhering to the law of conservation of mass is suggested following the approach [1, 4]. The reference two-phase medium composed or pore and particle parts is described by nonlinear Poisson–Nernst–Planck (PNP) equations for concentrations of charged species and overall electrostatic potential. For physical consistency, they are generalized with entropy variables associating the pressure and quasi-Fermi electrochemical potentials.

Based on a suitable free energy, in [9] a variational principle is established within the Gibbs simplex, thus preserving the mass balance and positive species concentrations. The generalized PNP problem takes into account for nonlinear interface reactions which are of primary importance in applications. We provided the problem by rigorous asymptotic analysis in [2, 3], and by a-priori energy and entropy estimates in [7, 8]. Based on the entropy variables and following the formalism given in [5, 6], further the PNP system is endowed with the structure of a gradient flow.

Acknowledgments: The work is supported by the Austrian Science Fund (FWF) project P26147-N26: “Object identification problems: numerical analysis” (PION), the Austrian Academy of Sciences (OeAW), Colombian Institute for Education and Technical Studies Abroad (ICETEX) and Universidad Tecnológica de Pereira.

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