A Fractional Parabolic-elliptic Chemotaxis-fluid System

Kerui Jiang; Zuhan Liu; Ling Zhou

doi:doi:10.11648/j.acm.20251403.13

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A Fractional Parabolic-elliptic Chemotaxis-fluid System

Kerui Jiang^*, Zuhan Liu, Ling Zhou

Published in Applied and Computational Mathematics (Volume 14, Issue 3)

Received: 17 April 2025 Accepted: 3 May 2025 Published: 13 June 2025

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Abstract

The fractional diffusion can describe possible singularities and other anomalies, and the non-local system constructed by the fractional chemotaxis-fluid equations can reveal more colorful, realistic and effective biological phenomena. The theoretical research on the fractional chemotaxis-fluid system is still at the initial stage, and new methods and technologies are needed to overcome the difficulties brought by the fractional operator, which has important scientific value. As an exploration, a fractional parabolic-elliptic chemotaxis system coupled with the Navier-Stokes equation is considered in the whole space ℝ² in this paper. Our main objective is to investigate the existence and asymptotic behavior of solutions to system (1). By the aid of L^p-L^q-estimates of the fractional heat semigroup and Kato-Ponce commutator estimate, we show the existence of local solution for large initial data and the existence of global mild solution to system (1) for small initial data in the scale invariant class demonstrating that and . Furthermore, under the rest state of the fluid motion, by studying moments of lower order , we establish a blow-up criterion of solution to system (1) with the help of the proof by contradiction.

Published in	Applied and Computational Mathematics (Volume 14, Issue 3)
DOI	10.11648/j.acm.20251403.13
Page(s)	120-163
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Fractional Chemotaxis, Fractional Navier-Stokes, Blow-up, Mild Solution

References

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APA Style

Jiang, K., Liu, Z., Zhou, L. (2025). A Fractional Parabolic-elliptic Chemotaxis-fluid System. Applied and Computational Mathematics, 14(3), 120-163. https://doi.org/10.11648/j.acm.20251403.13

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ACS Style

Jiang, K.; Liu, Z.; Zhou, L. A Fractional Parabolic-elliptic Chemotaxis-fluid System. Appl. Comput. Math. 2025, 14(3), 120-163. doi: 10.11648/j.acm.20251403.13

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AMA Style

Jiang K, Liu Z, Zhou L. A Fractional Parabolic-elliptic Chemotaxis-fluid System. Appl Comput Math. 2025;14(3):120-163. doi: 10.11648/j.acm.20251403.13

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@article{10.11648/j.acm.20251403.13,
  author = {Kerui Jiang and Zuhan Liu and Ling Zhou},
  title = {A Fractional Parabolic-elliptic Chemotaxis-fluid System
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {3},
  pages = {120-163},
  doi = {10.11648/j.acm.20251403.13},
  url = {https://doi.org/10.11648/j.acm.20251403.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251403.13},
  abstract = {The fractional diffusion can describe possible singularities and other anomalies, and the non-local system constructed by the fractional chemotaxis-fluid equations can reveal more colorful, realistic and effective biological phenomena. The theoretical research on the fractional chemotaxis-fluid system is still at the initial stage, and new methods and technologies are needed to overcome the difficulties brought by the fractional operator, which has important scientific value. As an exploration, a fractional parabolic-elliptic chemotaxis system coupled with the Navier-Stokes equation is considered in the whole space ℝ2 in this paper. Our main objective is to investigate the existence and asymptotic behavior of solutions to system (1). By the aid of Lp-Lq-estimates of the fractional heat semigroup and Kato-Ponce commutator estimate, we show the existence of local solution for large initial data and the existence of global mild solution to system (1) for small initial data in the scale invariant class demonstrating that  and . Furthermore, under the rest state of the fluid motion, by studying moments  of lower order , we establish a blow-up criterion of solution to system (1) with the help of the proof by contradiction.},
 year = {2025}
}

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TY  - JOUR
T1  - A Fractional Parabolic-elliptic Chemotaxis-fluid System

AU  - Kerui Jiang
AU  - Zuhan Liu
AU  - Ling Zhou
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251403.13
DO  - 10.11648/j.acm.20251403.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 120
EP  - 163
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251403.13
AB  - The fractional diffusion can describe possible singularities and other anomalies, and the non-local system constructed by the fractional chemotaxis-fluid equations can reveal more colorful, realistic and effective biological phenomena. The theoretical research on the fractional chemotaxis-fluid system is still at the initial stage, and new methods and technologies are needed to overcome the difficulties brought by the fractional operator, which has important scientific value. As an exploration, a fractional parabolic-elliptic chemotaxis system coupled with the Navier-Stokes equation is considered in the whole space ℝ2 in this paper. Our main objective is to investigate the existence and asymptotic behavior of solutions to system (1). By the aid of Lp-Lq-estimates of the fractional heat semigroup and Kato-Ponce commutator estimate, we show the existence of local solution for large initial data and the existence of global mild solution to system (1) for small initial data in the scale invariant class demonstrating that  and . Furthermore, under the rest state of the fluid motion, by studying moments  of lower order , we establish a blow-up criterion of solution to system (1) with the help of the proof by contradiction.
VL  - 14
IS  - 3
ER  -

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Author Information

Kerui Jiang

College of Mathematical Science, Yangzhou University, Yangzhou, China

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Zuhan Liu

College of Mathematical Science, Yangzhou University, Yangzhou, China

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Ling Zhou

College of Mathematical Science, Yangzhou University, Yangzhou, China

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Jiang, K., Liu, Z., Zhou, L. (2025). A Fractional Parabolic-elliptic Chemotaxis-fluid System. Applied and Computational Mathematics, 14(3), 120-163. https://doi.org/10.11648/j.acm.20251403.13

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ACS Style

Jiang, K.; Liu, Z.; Zhou, L. A Fractional Parabolic-elliptic Chemotaxis-fluid System. Appl. Comput. Math. 2025, 14(3), 120-163. doi: 10.11648/j.acm.20251403.13

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AMA Style

Jiang K, Liu Z, Zhou L. A Fractional Parabolic-elliptic Chemotaxis-fluid System. Appl Comput Math. 2025;14(3):120-163. doi: 10.11648/j.acm.20251403.13

Copy | Download

@article{10.11648/j.acm.20251403.13,
  author = {Kerui Jiang and Zuhan Liu and Ling Zhou},
  title = {A Fractional Parabolic-elliptic Chemotaxis-fluid System
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {3},
  pages = {120-163},
  doi = {10.11648/j.acm.20251403.13},
  url = {https://doi.org/10.11648/j.acm.20251403.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251403.13},
  abstract = {The fractional diffusion can describe possible singularities and other anomalies, and the non-local system constructed by the fractional chemotaxis-fluid equations can reveal more colorful, realistic and effective biological phenomena. The theoretical research on the fractional chemotaxis-fluid system is still at the initial stage, and new methods and technologies are needed to overcome the difficulties brought by the fractional operator, which has important scientific value. As an exploration, a fractional parabolic-elliptic chemotaxis system coupled with the Navier-Stokes equation is considered in the whole space ℝ2 in this paper. Our main objective is to investigate the existence and asymptotic behavior of solutions to system (1). By the aid of Lp-Lq-estimates of the fractional heat semigroup and Kato-Ponce commutator estimate, we show the existence of local solution for large initial data and the existence of global mild solution to system (1) for small initial data in the scale invariant class demonstrating that  and . Furthermore, under the rest state of the fluid motion, by studying moments  of lower order , we establish a blow-up criterion of solution to system (1) with the help of the proof by contradiction.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Fractional Parabolic-elliptic Chemotaxis-fluid System

AU  - Kerui Jiang
AU  - Zuhan Liu
AU  - Ling Zhou
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251403.13
DO  - 10.11648/j.acm.20251403.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 120
EP  - 163
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251403.13
AB  - The fractional diffusion can describe possible singularities and other anomalies, and the non-local system constructed by the fractional chemotaxis-fluid equations can reveal more colorful, realistic and effective biological phenomena. The theoretical research on the fractional chemotaxis-fluid system is still at the initial stage, and new methods and technologies are needed to overcome the difficulties brought by the fractional operator, which has important scientific value. As an exploration, a fractional parabolic-elliptic chemotaxis system coupled with the Navier-Stokes equation is considered in the whole space ℝ2 in this paper. Our main objective is to investigate the existence and asymptotic behavior of solutions to system (1). By the aid of Lp-Lq-estimates of the fractional heat semigroup and Kato-Ponce commutator estimate, we show the existence of local solution for large initial data and the existence of global mild solution to system (1) for small initial data in the scale invariant class demonstrating that  and . Furthermore, under the rest state of the fluid motion, by studying moments  of lower order , we establish a blow-up criterion of solution to system (1) with the help of the proof by contradiction.
VL  - 14
IS  - 3
ER  -

Copy | Download