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  and corresponds to the critical case for multiple system treated in Kurokiba . 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.
"Two dimensional drift-diffusion system in a critical weighted space." Differential Integral Equations 28 (7/8) 753 - 776, July/August 2015. https://doi.org/10.57262/die/1431347862