Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces

Abstract

We consider a singular limit problem for the Cauchy problem of the Keller–Segel equation in a critical function space. We show that a solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift–diffusion system of parabolic–elliptic type (the simplified Keller–Segel model) in the critical space strongly as the relaxation time $$\tau \rightarrow \infty $$ . For the proof of singular limit problem, we employ generalized maximal regularity for the heat equation and use it systematically with the sequence of embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }(\mathbb {R}^n)$$ and $$\dot{F}^s_{q,\sigma }(\mathbb {R}^n)$$ .