A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations

Kinderlehrer, David; Monsaingeon, Léonard; Xu, Xiang

doi:10.1051/cocv/2015043

Kinderlehrer, David ¹ ; Monsaingeon, Léonard ² ; Xu, Xiang ³

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.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 137-164.

Résumé

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].

Reçu le : 2015-01-22
Accepté le : 2015-08-06

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv/2015043

Classification : 35K65, 35K40, 47J30, 35Q92, 35B33
Mots clés : Optimal transport, systems of parabolic PDEs, nonlocal equations

Affiliations des auteurs :

Kinderlehrer, David ¹ ; Monsaingeon, Léonard ² ; Xu, Xiang ³

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     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {137--164},
     publisher = {EDP-Sciences},
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Kinderlehrer, David; Monsaingeon, Léonard; Xu, Xiang. A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 137-164. doi : 10.1051/cocv/2015043. http://www.numdam.org/articles/10.1051/cocv/2015043/

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