Homogenization of Poisson-Nernst-Planck equations for multiple species in a porous medium

Abstract

We rigorously derive a homogenized model for the Poisson--Nernst--Planck (PNP) equations for the case of multiple species defined on a periodic porous medium in spatial dimensions two and three. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by constructing suitable cut-off functions. The cut-off function, together with the application of a previously known energy functional, yields strong convergence of the microscopic concentrations in $L^1_t L^r_x$, for some $r>2$, enabling us to pass to the limit in the nonlinear drift term. Finally, we derive the homogenized equations by means of two-scale convergence in $L^p_t L^q_x$ setting.