Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence

We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the Aubin--Lions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear