TY - GEN

T1 - Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

AU - Ogawa, Takayoshi

N1 - Funding Information:
Acknowledgements The author would like to thank Professor Kentaro Fujie for his useful discussion. The author is partially supported by JSPS Grant in aid for Scientific Research S #19H05597 and Challenging Research (Pioneering) #20K20284.
Publisher Copyright:
© 2021, Springer Nature Singapore Pte Ltd.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Bounded mean oscillation

KW - Critical space

KW - Drift-diffusion system

KW - Global well-posedness

KW - Keller-Segel equation

KW - Maximal regularity

KW - Scaling invariance

KW - Singular limit problem

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U2 - 10.1007/978-981-33-4822-6_4

DO - 10.1007/978-981-33-4822-6_4

M3 - Conference contribution

AN - SCOPUS:85105942293

SN - 9789813348219

T3 - Springer Proceedings in Mathematics and Statistics

SP - 103

EP - 182

BT - Nonlinear Partial Differential Equations for Future Applications

A2 - Koike, Shigeaki

A2 - Kozono, Hideo

A2 - Ogawa, Takayoshi

A2 - Sakaguchi, Shigeru

PB - Springer

T2 - Workshops on Nonlinear Partial Differential Equations for Future Applications, 2017

Y2 - 2 October 2017 through 6 October 2017

ER -