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].
Accepté le :
DOI : 10.1051/cocv/2015043
Mots clés : Optimal transport, systems of parabolic PDEs, nonlocal equations
@article{COCV_2017__23_1_137_0, author = {Kinderlehrer, David and Monsaingeon, L\'eonard and Xu, Xiang}, title = {A {Wasserstein} gradient flow approach to {Poisson\ensuremath{-}Nernst\ensuremath{-}Planck} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {137--164}, publisher = {EDP-Sciences}, volume = {23}, number = {1}, year = {2017}, doi = {10.1051/cocv/2015043}, mrnumber = {3601019}, zbl = {1372.35167}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2015043/} }
TY - JOUR AU - Kinderlehrer, David AU - Monsaingeon, Léonard AU - Xu, Xiang TI - A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 137 EP - 164 VL - 23 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015043/ DO - 10.1051/cocv/2015043 LA - en ID - COCV_2017__23_1_137_0 ER -
%0 Journal Article %A Kinderlehrer, David %A Monsaingeon, Léonard %A Xu, Xiang %T A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 137-164 %V 23 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015043/ %R 10.1051/cocv/2015043 %G en %F COCV_2017__23_1_137_0
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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