---
_id: '63293'
abstract:
- lang: eng
text: "<p style='text-indent:20px;'>The Cauchy problem
in <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\mathbb{R}^3
$\\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes
system</p><p style='text-indent:20px;'><disp-formula> <label/>
<tex-math id=\"FE1\"> \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l}
\ n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t
+ u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta
u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array}
\\right. \\end{eqnarray*} $\\end{document} </tex-math></disp-formula></p><p
style='text-indent:20px;'>is considered. Under suitable conditions on the initial
data <inline-formula><tex-math id=\"M3\">\\begin{document}$ (n_0,
c_0, u_0) $\\end{document}</tex-math></inline-formula>, with regard
to the crucial first component requiring that <inline-formula><tex-math
id=\"M4\">\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>
be nonnegative and such that <inline-formula><tex-math id=\"M5\">\\begin{document}$
(n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>,
a globally defined weak solution with <inline-formula><tex-math id=\"M6\">\\begin{document}$
(n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula>
is constructed. Apart from that, assuming that moreover <inline-formula><tex-math
id=\"M7\">\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx
$\\end{document}</tex-math></inline-formula> is finite, it is shown
that a weak solution exists which enjoys further regularity features and preserves
mass in an appropriate sense.</p>"
article_number: '5201'
author:
- first_name: Kyungkeun
full_name: Kang, Kyungkeun
last_name: Kang
- first_name: Jihoon
full_name: Lee, Jihoon
last_name: Lee
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $. Discrete and Continuous Dynamical Systems.
2022;42(11). doi:10.3934/dcds.2022091
apa: Kang, K., Lee, J., & Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $. Discrete and Continuous Dynamical Systems,
42(11), Article 5201. https://doi.org/10.3934/dcds.2022091
bibtex: '@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $}, volume={42}, DOI={10.3934/dcds.2022091},
number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American
Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon
and Winkler, Michael}, year={2022} }'
chicago: Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions
to a Chemotaxis-Navier-Stokes System in $ \mathbb{R}^3 $.” Discrete and Continuous
Dynamical Systems 42, no. 11 (2022). https://doi.org/10.3934/dcds.2022091.
ieee: 'K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes
system in $ \mathbb{R}^3 $,” Discrete and Continuous Dynamical Systems,
vol. 42, no. 11, Art. no. 5201, 2022, doi: 10.3934/dcds.2022091.'
mla: Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes
System in $ \mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems,
vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022,
doi:10.3934/dcds.2022091.
short: K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42
(2022).
date_created: 2025-12-18T19:22:04Z
date_updated: 2025-12-18T20:08:21Z
doi: 10.3934/dcds.2022091
intvolume: ' 42'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
issn:
- 1078-0947
- 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global weak solutions to a chemotaxis-Navier-Stokes system in $ \mathbb{R}^3
$
type: journal_article
user_id: '31496'
volume: 42
year: '2022'
...
---
_id: '63320'
author:
- first_name: Youshan
full_name: Tao, Youshan
last_name: Tao
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Tao Y, Winkler M. Critical mass for infinite-time blow-up in a haptotaxis system
with nonlinear zero-order interaction. Discrete & Continuous Dynamical
Systems - A. 2020;41(1):439-454. doi:10.3934/dcds.2020216
apa: Tao, Y., & Winkler, M. (2020). Critical mass for infinite-time blow-up
in a haptotaxis system with nonlinear zero-order interaction. Discrete &
Continuous Dynamical Systems - A, 41(1), 439–454. https://doi.org/10.3934/dcds.2020216
bibtex: '@article{Tao_Winkler_2020, title={Critical mass for infinite-time blow-up
in a haptotaxis system with nonlinear zero-order interaction}, volume={41}, DOI={10.3934/dcds.2020216}, number={1},
journal={Discrete & Continuous Dynamical Systems - A}, publisher={American
Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler,
Michael}, year={2020}, pages={439–454} }'
chicago: 'Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
in a Haptotaxis System with Nonlinear Zero-Order Interaction.” Discrete &
Continuous Dynamical Systems - A 41, no. 1 (2020): 439–54. https://doi.org/10.3934/dcds.2020216.'
ieee: 'Y. Tao and M. Winkler, “Critical mass for infinite-time blow-up in a haptotaxis
system with nonlinear zero-order interaction,” Discrete & Continuous
Dynamical Systems - A, vol. 41, no. 1, pp. 439–454, 2020, doi: 10.3934/dcds.2020216.'
mla: Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
in a Haptotaxis System with Nonlinear Zero-Order Interaction.” Discrete &
Continuous Dynamical Systems - A, vol. 41, no. 1, American Institute of Mathematical
Sciences (AIMS), 2020, pp. 439–54, doi:10.3934/dcds.2020216.
short: Y. Tao, M. Winkler, Discrete & Continuous Dynamical Systems - A 41
(2020) 439–454.
date_created: 2025-12-18T19:33:59Z
date_updated: 2025-12-18T20:04:09Z
doi: 10.3934/dcds.2020216
intvolume: ' 41'
issue: '1'
language:
- iso: eng
page: 439-454
publication: Discrete & Continuous Dynamical Systems - A
publication_identifier:
issn:
- 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear
zero-order interaction
type: journal_article
user_id: '31496'
volume: 41
year: '2020'
...