A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations
1 Department of Mathematical Sciences, Carnegie Mellon
University, Pittsburgh, PA 15213, USA.
2 CAMGSD Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal.
3 Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA.
Revised: 24 July 2015
Accepted: 6 August 2015
The Poisson−Nernst−Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and nonlinear diffusion. The proof of the main results relies on the derivation of additional estimates based on the flow interchange technique developed by Matthes et al. in [D. Matthes, R.J. McCann and G. Savaré, Commun. Partial Differ. Equ. 34 (2009) 1352–1397].
Mathematics Subject Classification: 35K65 / 35K40 / 47J30 / 35Q92 / 35B33
Key words: Optimal transport / systems of parabolic PDEs / nonlocal equations
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