Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

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