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

Abstract

A singular limit problem for the Cauchy problem of the Keller–Segel equation is considered in a critical function space based on Morrey spaces. 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 in the critical space strongly as the relaxation time tends to infinity. The key ingredient is a Banach space-valued Lorentz space in time. By the careful use of the real interpolation, a sharp estimate for the heat semigroup is obtained.