A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations. (English) Zbl 1372.35167
Summary: 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 D. Matthes et al. [Commun. Partial Differ. Equations 34, No. 11, 1352–1397 (2009; Zbl 1187.35131)].
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K40 | Second-order parabolic systems |
| 47J30 | Variational methods involving nonlinear operators |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B33 | Critical exponents in context of PDEs |
Citations:
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