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 \(\dot{F}^s_{q,\sigma }(\mathbb {R}^n)\).

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Notes

  1. 1.

    Such a difficulty was avoided in [3] by splitting the time integration.

  2. 2.

    The choice of T is independent of \(\tau >1\).

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Acknowledgements

The authors would like to thank Professor Kenji Yajima for his pointing out a mistake in the original paper. The work of T. Ogawa is partially supported by JSPS Grant in aid for Scientific Research S #25270702, A #19H00638 and Challenging Research (Pioneering) #17H06199. The work of M. Kurokiba is partially supported by JSPS Grant in aid for Scientific Research C #16K05219.

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Correspondence to Masaki Kurokiba.

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Kurokiba, M., Ogawa, T. Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces. J. Evol. Equ. 20, 421–457 (2020). https://doi.org/10.1007/s00028-019-00527-3

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Keywords

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

Mathematics Subject Classification

  • Primary 35K15
  • Secondary 35K55
  • 35Q60
  • 78A38