### 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 τ→ ∞. 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˙q,σs(Rn) and F˙q,σs(Rn).

Original language | English |
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Pages (from-to) | 421-457 |

Number of pages | 37 |

Journal | Journal of Evolution Equations |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2020 Jun 1 |

### Keywords

- Critical space
- Drift–diffusion system
- Global well-posedness
- Keller–Segel equation
- Maximal regularity
- Scaling invariance
- Singular limit problem

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

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## Cite this

Kurokiba, M., & Ogawa, T. (2020). Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces.

*Journal of Evolution Equations*,*20*(2), 421-457. https://doi.org/10.1007/s00028-019-00527-3