Analysis of improved Nernst-Planck-Poisson models for incompressible electrolytic mixtures subject to chemical reactions

• Applicant: Dr. Pierre-Etienne Druet
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 388362093

The project is concerned with new existence theory for thermodynamically motivated nonlinear partial differential equations. The PDE model was introduced by Dreyer, Guhlke and Müller in 2013 in order to describe the charge transport in incompressible electro-chemical systems subject to a macroscopic velocity. The model not only remedies the deficiencies of the classical Nernst-Planck-Poisson theory of electrolytes, but it also leads to PDE structures that are completely new in the context of mathematical analysis.In this model the convection-diffusion-reaction equations describing the mass transfer of the species are coupled to the Navier-Stokes equations for the barycentric velocity of the mixture, and to the Poisson equation for the electrical potential. The constitutive equations for the chemical potentials, the diffusion fluxes and the reaction rates are derived from one single free energy function as to guaranty the nonnegativity of the entropy production. Thermodynamics helps creating new mathematical contents, in particular the following features: At first, the pressure contribution to the chemical potentials induces a coupling between diffusion flux and pressure gradient; At second, the incompressibility of the mixture is expressed via a constraint on its volume and not on its mass, so that the velocity field is not necessarily solenoidal; At third, the special thermodynamical structure of the reaction rates yield estimates in Orlicz classes.The study of the model promises new insights into the role of pressure by incompressible mixtures and, in particular, new pressure estimates. The cross-diffusion, that must appear due to the conservation of total mass, will also be investigated. These are questions of general interest for the analysis of incompressible multicomponent flow models. Moreover a new analytical approach of highly non-linear reactions shall be developed.These challenges give an uncommonly original mathematical content to the project. Its funding will contribute to keeping and developing an important field of competence opened by Dreyer, Druet, Gajewski and Guhlke in 2016 with the breakthrough analysis of the compressible model. It will answer the expectations towards the analysis of the more delicate incompressible case. For the investigator, this challenging project will act as an incomparable occasion to strengthen his experience in the analysis of non-linear partial differential equations of hydrodynamics and electrodynamics, and to develop the cooperation with national/international colleagues on this complex research subject.

• DFG Programme: Research Grants