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 -
