Very weak solutions for Poisson–Nernst–Planck system. (March 2015)

• Record Type: Journal Article
• Authors: Hineman, Jay L.; Ryham, Rolf J.
• Is Part Of: Nonlinear analysis. Volume 115(2015)
• Page Start: 12
• Page End: 24
• Publication Date: 2015-03
• Subjects: Poisson–Nernst–Planck -- Drift–diffusion–Poisson -- Stationary diffuse charge; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.11.018
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We formulate a notion of very weak solution for the Poisson–Nernst–Planck system. The stationary system possesses a local monotonicity formula. Iterative application of the formula reveals improvement in estimates for ion density and potential, and leads to a local boundedness result. Local boundedness extends to steady-state systems for multiple ions and variable coefficients. The formulation applies to the related Keller–Segel system where stationary very weak solutions in two dimensions are regular. Examples illustrate how structure influences this regularity in higher dimensions.