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

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 ->infinity 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 B & x2d9;q,sigma s(Rn) and F & x2d9;q,sigma s(Rn)