Two dimensional drift-diffusion system in a critical weighted space

Abstract

We establish the time local well-posedness of a multi-component parabolic-elliptic drift-diffusion model in two space dimensions in a scaling critical weighted space. This is a generalization of former result Kurokiba-Ogawa [20] and corresponds to the critical case for multiple system treated in Kurokiba [18]. In order to show the critical well-posedness, we introduce a refined version of Brezis-Gallouët inequality in term of the Besov space and apply it to the some weighted estimate in two dimensions.