Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

In Jee Jeong; Kyungkeun Kang

doi:10.1007/s00220-021-04292-8

Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

In Jee Jeong, Kyungkeun Kang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its C²-norm with certain initial data.

Original language	English
Pages (from-to)	1175-1217
Number of pages	43
Journal	Communications in Mathematical Physics
Volume	390
Issue number	3
DOIs	https://doi.org/10.1007/s00220-021-04292-8
Publication status	Published - 2022 Mar

Bibliographical note

Funding Information:
IJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant No. 2019R1F1A1058486. KK has been supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. We are grateful to the anonymous referees for various comments and suggestions, which have significantly improved the manuscript. Especially, we are grateful for pointing to us the possibility of the interesting generalization given in Remark .

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s00220-021-04292-8

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title = "Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type",

abstract = "We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its C2-norm with certain initial data.",

author = "Jeong, {In Jee} and Kyungkeun Kang",

note = "Funding Information: IJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant No. 2019R1F1A1058486. KK has been supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. We are grateful to the anonymous referees for various comments and suggestions, which have significantly improved the manuscript. Especially, we are grateful for pointing to us the possibility of the interesting generalization given in Remark . Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

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AU - Kang, Kyungkeun

N1 - Funding Information: IJ has been supported by the New Faculty Startup Fund from Seoul National University, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant No. 2019R1F1A1058486. KK has been supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. We are grateful to the anonymous referees for various comments and suggestions, which have significantly improved the manuscript. Especially, we are grateful for pointing to us the possibility of the interesting generalization given in Remark . Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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N2 - We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its C2-norm with certain initial data.

AB - We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its C2-norm with certain initial data.

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Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

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