Issue |
ESAIM: COCV
Volume 23, Number 1, January-March 2017
|
|
---|---|---|
Page(s) | 137 - 164 | |
DOI | https://doi.org/10.1051/cocv/2015043 | |
Published online | 10 October 2016 |
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.
x2xu@odu.edu
Received:
22
January
2015
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
© EDP Sciences, SMAI 2016
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