A Wasserstein Gradient Flow Approach to Poisson-Nernst-Planck Equations

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 non-linear 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. Savare, Commun. Partial Differ. Equ. 34 (2009) 1352-1397]

Recent Developments in Liquid Crystal Theory

Kinderlehrer, David. (1989). Recent Developments in Liquid Crystal Theory. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/4959

A theory and challenges for coarsening in microstructure

Non UBCUnreviewedAuthor affiliation: Carnegie Mellon UniversityFacult

Remarks about the Poisson-Nernst-Planck equations

The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasser- stein distance and a free energy in an appropriate space of probability measures. The interaction term between the species arising from the Gauss law is singular which gives rise to some challenging issues. We give a description of this situation attempting to maintain a minimal technical level including the basic format of the Wasserstein-type im- plicit scheme. This is joint work with L ́eonard Monsaingeon and Xiang Xu.Non UBCUnreviewedAuthor affiliation: Carnegie Mellon UniversityFacult

Towards a gradient flow for microstructure

A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe very briefly the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. Its identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a mass transport JKO implicit scheme, which we found astonishing. Consequently the GBCD is the solution, in some sense, of a Fokker-Planck Equation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process. (joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, X.-Y. Lu and S. Ta'asan).Non UBCUnreviewedAuthor affiliation: Carnegie Mellon UniversityFacult