GLOBAL WEAK SOLUTIONS TO A CHEMOTAXIS-NAVIER-STOKES SYSTEM IN ℝ3

Kyungkeun Kang; Jihoon Lee; Michael Winkler

doi:10.3934/dcds.2022091

GLOBAL WEAK SOLUTIONS TO A CHEMOTAXIS-NAVIER-STOKES SYSTEM IN ℝ³

Kyungkeun Kang, Jihoon Lee, Michael Winkler

Department of Mathematics

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

3 Citations (Scopus)

Abstract

The Cauchy problem in ℝ³for the chemotaxis-Navier 'Equation Presented' is considered. Under suitable conditions on the initial data (n₀, c₀, u₀), with regard to the crucial first component requiring that n₀∈ L¹(ℝ³) be nonnegative and such that (n₀+ 1) ln(n₀+ 1) ∈ L¹(ℝ³), a globally defined weak solution with (n. c, u)|_t=0= (n₀, c₀, u₀) is constructed. Apart from that, assuming that moreover ∫_ℝ3n₀(x) ln(1+|x|²)dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.

Original language	English
Pages (from-to)	5201-5222
Number of pages	22
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	42
Issue number	11
DOIs	https://doi.org/10.3934/dcds.2022091
Publication status	Published - 2022 Nov

Bibliographical note

Funding Information:
2020 Mathematics Subject Classification. 92C17, 35Q30, 35D30, 35K55. Key words and phrases. Chemotaxis, Navier-Stokes, quasi-energy inequality, weak solutions. K. Kang is supported by NRF Grant No. 2019R1A2C1084685. J. Lee is supported by NRF Grant No. 2021R1A2C1092830. M. Winkler acknowledges support of the Deutsche Forschungs-gemeinschaft (Project No. 462888149). ∗Corresponding author: Kyungkeun Kang.

Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2022091

Cite this

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title = "GLOBAL WEAK SOLUTIONS TO A CHEMOTAXIS-NAVIER-STOKES SYSTEM IN ℝ3",

abstract = "The Cauchy problem in ℝ3for the chemotaxis-Navier 'Equation Presented' is considered. Under suitable conditions on the initial data (n0, c0, u0), with regard to the crucial first component requiring that n0∈ L1(ℝ3) be nonnegative and such that (n0+ 1) ln(n0+ 1) ∈ L1(ℝ3), a globally defined weak solution with (n. c, u)|t=0= (n0, c0, u0) is constructed. Apart from that, assuming that moreover ∫ℝ3n0(x) ln(1+|x|2)dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.",

author = "Kyungkeun Kang and Jihoon Lee and Michael Winkler",

note = "Funding Information: 2020 Mathematics Subject Classification. 92C17, 35Q30, 35D30, 35K55. Key words and phrases. Chemotaxis, Navier-Stokes, quasi-energy inequality, weak solutions. K. Kang is supported by NRF Grant No. 2019R1A2C1084685. J. Lee is supported by NRF Grant No. 2021R1A2C1092830. M. Winkler acknowledges support of the Deutsche Forschungs-gemeinschaft (Project No. 462888149). ∗Corresponding author: Kyungkeun Kang. Publisher Copyright: {\textcopyright} 2022 American Institute of Mathematical Sciences. All rights reserved.",

year = "2022",

month = nov,

doi = "10.3934/dcds.2022091",

language = "English",

volume = "42",

pages = "5201--5222",

journal = "Discrete and Continuous Dynamical Systems",

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publisher = "Southwest Missouri State University",

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AU - Lee, Jihoon

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N1 - Funding Information: 2020 Mathematics Subject Classification. 92C17, 35Q30, 35D30, 35K55. Key words and phrases. Chemotaxis, Navier-Stokes, quasi-energy inequality, weak solutions. K. Kang is supported by NRF Grant No. 2019R1A2C1084685. J. Lee is supported by NRF Grant No. 2021R1A2C1092830. M. Winkler acknowledges support of the Deutsche Forschungs-gemeinschaft (Project No. 462888149). ∗Corresponding author: Kyungkeun Kang. Publisher Copyright: © 2022 American Institute of Mathematical Sciences. All rights reserved.

PY - 2022/11

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N2 - The Cauchy problem in ℝ3for the chemotaxis-Navier 'Equation Presented' is considered. Under suitable conditions on the initial data (n0, c0, u0), with regard to the crucial first component requiring that n0∈ L1(ℝ3) be nonnegative and such that (n0+ 1) ln(n0+ 1) ∈ L1(ℝ3), a globally defined weak solution with (n. c, u)|t=0= (n0, c0, u0) is constructed. Apart from that, assuming that moreover ∫ℝ3n0(x) ln(1+|x|2)dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.

AB - The Cauchy problem in ℝ3for the chemotaxis-Navier 'Equation Presented' is considered. Under suitable conditions on the initial data (n0, c0, u0), with regard to the crucial first component requiring that n0∈ L1(ℝ3) be nonnegative and such that (n0+ 1) ln(n0+ 1) ∈ L1(ℝ3), a globally defined weak solution with (n. c, u)|t=0= (n0, c0, u0) is constructed. Apart from that, assuming that moreover ∫ℝ3n0(x) ln(1+|x|2)dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.

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