Global existence for diffusion-electromigration systems in space dimension three and higher

We prove existence of global weak solutions for the Nernst-Planck-Poisson problem which describes the evolution of concentrations of charged species $X_1, ..., X_P$ subject to Fickian diffusion and chemical reactions in the presence of an electrical field, including in particular the Boltzmann statistics case. In contrast to the existing literature, existence is proved in any dimension. Moreover, we do not need the assumption $P = 2$ nor the assumption of equal diffusivities for all $P$ components. Our approach relies on the intrinsic energy structure and on an adequate nonlinear and curiously more regular approximate problem. The delicate passing to the limit is done in adequate functional spaces which lead to only weak solutions.