Two dimensional drift-diffusion system in a critical weighted space. (English) Zbl 1363.35148

Summary: 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 [M. Kurokiba and T. Ogawa, Differ. Integral Equ. 16, No. 4, 427–452 (2003; Zbl 1161.35432)] and corresponds to the critical case for multiple system treated in M. Kurokiba [Differ. Integral Equ. 27, No. 5–6, 425–446 (2014; Zbl 1340.35121)]. In order to show the critical well-posedness, we introduce a refined version of Brezis-GallouĂ«t inequality in term of the Besov space and apply it to the some weighted estimate in two dimensions.


35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
78A35 Motion of charged particles