Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

Takayoshi Ogawa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider singular limit problems of the Cauchy problem for the Patlak-Keller-Segel equation and related problems appeared in the theory of medical and biochemical dynamics. It is shown that the solution to the Patlak-Keller-Segel equation in a scaling critical function class converges strongly to a solution of the drift-diffusion system of parabolic-elliptic equations as the relaxation time parameter τ→ ∞. Analogous problem related to the Chaplain-Anderson model for cancer growth model is also presented as well as Arzhimer’s model that involves the multi-component drift-diffusion system. For the proof, we use generalized maximal regularity for the heat equations and systematically apply embeddings between the interpolation spaces shown in [40, 41]. The argument requires generalized version of maximal regularity developed in [40, 61], for the Cauchy problem of the heat equation.

Original languageEnglish
Title of host publicationNonlinear Partial Differential Equations for Future Applications
EditorsShigeaki Koike, Hideo Kozono, Takayoshi Ogawa, Shigeru Sakaguchi
PublisherSpringer
Pages103-182
Number of pages80
ISBN (Print)9789813348219
DOIs
Publication statusPublished - 2021
EventWorkshops on Nonlinear Partial Differential Equations for Future Applications, 2017 - Sendai, Japan
Duration: 2017 Oct 22017 Oct 6

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume346
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceWorkshops on Nonlinear Partial Differential Equations for Future Applications, 2017
Country/TerritoryJapan
CitySendai
Period17/10/217/10/6

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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