---
_id: '63254'
abstract:
- lang: eng
text: "AbstractThe chemotaxis-Navier–Stokes system
$$\\begin{aligned} \\left\\{
\\begin{array}{rcl} n_t+u\\cdot \\nabla n & =& \\Delta \\big (n c^{-\\alpha
} \\big ), \\\\ c_t+ u\\cdot \\nabla c & =& \\Delta c -nc,\\\\ u_t +
(u\\cdot \\nabla ) u & =& \\Delta u+\\nabla P + n\\nabla \\Phi , \\qquad
\\nabla \\cdot u=0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n
\ \r\n \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n n\r\n
\ t\r\n \r\n
\ +\r\n u\r\n
\ ·\r\n ∇\r\n
\ n\r\n \r\n
\ \r\n \r\n
\ =\r\n \r\n
\ \r\n \r\n
\ Δ\r\n \r\n
\ (\r\n \r\n
\ n\r\n \r\n
\ c\r\n \r\n
\ -\r\n α\r\n
\ \r\n \r\n
\ \r\n )\r\n
\ \r\n ,\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ c\r\n t\r\n
\ \r\n +\r\n
\ u\r\n ·\r\n
\ ∇\r\n c\r\n
\ \r\n \r\n
\ \r\n =\r\n
\ \r\n \r\n
\ \r\n Δ\r\n
\ c\r\n -\r\n
\ n\r\n c\r\n
\ ,\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n u\r\n
\ t\r\n \r\n
\ +\r\n \r\n
\ (\r\n u\r\n
\ ·\r\n ∇\r\n
\ )\r\n \r\n
\ u\r\n \r\n
\ \r\n \r\n
\ =\r\n \r\n
\ \r\n \r\n
\ Δ\r\n u\r\n
\ +\r\n ∇\r\n
\ P\r\n +\r\n
\ n\r\n ∇\r\n
\ Φ\r\n ,\r\n
\ \r\n ∇\r\n
\ ·\r\n u\r\n
\ =\r\n 0\r\n
\ ,\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n \r\n
\ modelling the
behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded
domain $$\\Omega \\subset
\\mathbb R^2$$\r\n
\ \r\n Ω\r\n ⊂\r\n
\ \r\n R\r\n
\ 2\r\n \r\n
\ \r\n .
For all $$\\alpha >
0$$\r\n
\ \r\n α\r\n >\r\n
\ 0\r\n \r\n
and all sufficiently regular $$\\Phi
$$\r\n
\ Φ\r\n ,
we construct global classical solutions and thereby extend recent results for
the fluid-free analogue to the system coupled to a Navier–Stokes system. As a
crucial new challenge, our analysis requires a priori estimates for u
at a point in the proof when knowledge about n is essentially
limited to the observation that the mass is conserved. To overcome this problem,
we also prove new uniform-in-time $$L^p$$\r\n \r\n
\ L\r\n p\r\n
\ \r\n
estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending
on the space-time $$L^2$$\r\n \r\n
\ L\r\n 2\r\n
\ \r\n
norm of the force term raised to an arbitrary small power."
article_number: '60'
author:
- first_name: Mario
full_name: Fuest, Mario
last_name: Fuest
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Fuest M, Winkler M. Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous
2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local
Sensing. Journal of Mathematical Fluid Mechanics. 2024;26(4). doi:10.1007/s00021-024-00899-8
apa: Fuest, M., & Winkler, M. (2024). Uniform $$L^p$$ Estimates for Solutions
to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid
System with Local Sensing. Journal of Mathematical Fluid Mechanics, 26(4),
Article 60. https://doi.org/10.1007/s00021-024-00899-8
bibtex: '@article{Fuest_Winkler_2024, title={Uniform $$L^p$$ Estimates for Solutions
to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid
System with Local Sensing}, volume={26}, DOI={10.1007/s00021-024-00899-8},
number={460}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer
Science and Business Media LLC}, author={Fuest, Mario and Winkler, Michael}, year={2024}
}'
chicago: Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions
to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid
System with Local Sensing.” Journal of Mathematical Fluid Mechanics 26,
no. 4 (2024). https://doi.org/10.1007/s00021-024-00899-8.
ieee: 'M. Fuest and M. Winkler, “Uniform $$L^p$$ Estimates for Solutions to the
Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid
System with Local Sensing,” Journal of Mathematical Fluid Mechanics, vol.
26, no. 4, Art. no. 60, 2024, doi: 10.1007/s00021-024-00899-8.'
mla: Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions
to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid
System with Local Sensing.” Journal of Mathematical Fluid Mechanics, vol.
26, no. 4, 60, Springer Science and Business Media LLC, 2024, doi:10.1007/s00021-024-00899-8.
short: M. Fuest, M. Winkler, Journal of Mathematical Fluid Mechanics 26 (2024).
date_created: 2025-12-18T19:05:09Z
date_updated: 2025-12-18T20:13:58Z
doi: 10.1007/s00021-024-00899-8
intvolume: ' 26'
issue: '4'
language:
- iso: eng
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
issn:
- 1422-6928
- 1422-6952
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes
Equations and Application to a Chemotaxis–Fluid System with Local Sensing
type: journal_article
user_id: '31496'
volume: 26
year: '2024'
...
---
_id: '34669'
article_number: '1'
author:
- first_name: Tobias
full_name: Black, Tobias
id: '23686'
last_name: Black
orcid: 0000-0001-9963-0800
citation:
ama: Black T. The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System.
Journal of Mathematical Fluid Mechanics. 2019;22(1). doi:10.1007/s00021-019-0464-z
apa: Black, T. (2019). The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
System. Journal of Mathematical Fluid Mechanics, 22(1), Article
1. https://doi.org/10.1007/s00021-019-0464-z
bibtex: '@article{Black_2019, title={The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
System}, volume={22}, DOI={10.1007/s00021-019-0464-z},
number={11}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer
Science and Business Media LLC}, author={Black, Tobias}, year={2019} }'
chicago: Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
System.” Journal of Mathematical Fluid Mechanics 22, no. 1 (2019). https://doi.org/10.1007/s00021-019-0464-z.
ieee: 'T. Black, “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
System,” Journal of Mathematical Fluid Mechanics, vol. 22, no. 1, Art.
no. 1, 2019, doi: 10.1007/s00021-019-0464-z.'
mla: Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
System.” Journal of Mathematical Fluid Mechanics, vol. 22, no. 1, 1, Springer
Science and Business Media LLC, 2019, doi:10.1007/s00021-019-0464-z.
short: T. Black, Journal of Mathematical Fluid Mechanics 22 (2019).
date_created: 2022-12-21T09:47:56Z
date_updated: 2022-12-21T10:04:29Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00021-019-0464-z
intvolume: ' 22'
issue: '1'
keyword:
- Applied Mathematics
- Computational Mathematics
- Condensed Matter Physics
- Mathematical Physics
language:
- iso: eng
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
issn:
- 1422-6928
- 1422-6952
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System
type: journal_article
user_id: '23686'
volume: 22
year: '2019'
...
---
_id: '63375'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Does Fluid Interaction Affect Regularity in the Three-Dimensional
Keller–Segel System with Saturated Sensitivity? Journal of Mathematical Fluid
Mechanics. 2018;20(4):1889-1909. doi:10.1007/s00021-018-0395-0
apa: Winkler, M. (2018). Does Fluid Interaction Affect Regularity in the Three-Dimensional
Keller–Segel System with Saturated Sensitivity? Journal of Mathematical Fluid
Mechanics, 20(4), 1889–1909. https://doi.org/10.1007/s00021-018-0395-0
bibtex: '@article{Winkler_2018, title={Does Fluid Interaction Affect Regularity
in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?}, volume={20},
DOI={10.1007/s00021-018-0395-0},
number={4}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1889–1909}
}'
chicago: 'Winkler, Michael. “Does Fluid Interaction Affect Regularity in the Three-Dimensional
Keller–Segel System with Saturated Sensitivity?” Journal of Mathematical Fluid
Mechanics 20, no. 4 (2018): 1889–1909. https://doi.org/10.1007/s00021-018-0395-0.'
ieee: 'M. Winkler, “Does Fluid Interaction Affect Regularity in the Three-Dimensional
Keller–Segel System with Saturated Sensitivity?,” Journal of Mathematical Fluid
Mechanics, vol. 20, no. 4, pp. 1889–1909, 2018, doi: 10.1007/s00021-018-0395-0.'
mla: Winkler, Michael. “Does Fluid Interaction Affect Regularity in the Three-Dimensional
Keller–Segel System with Saturated Sensitivity?” Journal of Mathematical Fluid
Mechanics, vol. 20, no. 4, Springer Science and Business Media LLC, 2018,
pp. 1889–909, doi:10.1007/s00021-018-0395-0.
short: M. Winkler, Journal of Mathematical Fluid Mechanics 20 (2018) 1889–1909.
date_created: 2025-12-19T11:06:02Z
date_updated: 2025-12-19T11:06:09Z
doi: 10.1007/s00021-018-0395-0
intvolume: ' 20'
issue: '4'
language:
- iso: eng
page: 1889-1909
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
issn:
- 1422-6928
- 1422-6952
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel
System with Saturated Sensitivity?
type: journal_article
user_id: '31496'
volume: 20
year: '2018'
...