On local weak solutions to Nernst–Planck–Poisson system

Abstract

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well as electrons and holes transport in semiconductors. The considered boundary conditions allow the physical system to be not only closed but also open. Theorems on existence, uniqueness, and nonnegativity of local weak solutions are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

Keywords:

• Parabolic–elliptic system
• existence
• uniqueness
• nonnegativity
• local weak solution
• strong and weak topologies

AMS Subject Classifications:

• Primary: 35M33
• 35D30
• 35A16
• Secondary: 35A01
• 35A02
• 35B09

Acknowledgements

The work of Piotr Kalita was supported by the National Science Center of Poland under Maestro Project DEC-2012/06/A/ST1/00262.

Notes

No potential conflict of interest was reported by the authors.