Maximal regularity and a singular limit problem for the Patlak\u2013Keller\u2013Segel system in the scaling critical space involving BMO

Abstract

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter tau rightarrow infty . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces dot{B}^s_{q,sigma }({mathbb {R}}^n) and dot{F}^s_{q,sigma }({mathbb {R}}^n) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.