---
_id: '63372'
author:
- first_name: Yulan
full_name: Wang, Yulan
last_name: Wang
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
- first_name: Zhaoyin
full_name: Xiang, Zhaoyin
last_name: Xiang
citation:
ama: Wang Y, Winkler M, Xiang Z. The small-convection limit in a two-dimensional
chemotaxis-Navier–Stokes system. Mathematische Zeitschrift. 2017;289(1-2):71-108.
doi:10.1007/s00209-017-1944-6
apa: Wang, Y., Winkler, M., & Xiang, Z. (2017). The small-convection limit in
a two-dimensional chemotaxis-Navier–Stokes system. Mathematische Zeitschrift,
289(1–2), 71–108. https://doi.org/10.1007/s00209-017-1944-6
bibtex: '@article{Wang_Winkler_Xiang_2017, title={The small-convection limit in
a two-dimensional chemotaxis-Navier–Stokes system}, volume={289}, DOI={10.1007/s00209-017-1944-6},
number={1–2}, journal={Mathematische Zeitschrift}, publisher={Springer Science
and Business Media LLC}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin},
year={2017}, pages={71–108} }'
chicago: 'Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “The Small-Convection
Limit in a Two-Dimensional Chemotaxis-Navier–Stokes System.” Mathematische
Zeitschrift 289, no. 1–2 (2017): 71–108. https://doi.org/10.1007/s00209-017-1944-6.'
ieee: 'Y. Wang, M. Winkler, and Z. Xiang, “The small-convection limit in a two-dimensional
chemotaxis-Navier–Stokes system,” Mathematische Zeitschrift, vol. 289,
no. 1–2, pp. 71–108, 2017, doi: 10.1007/s00209-017-1944-6.'
mla: Wang, Yulan, et al. “The Small-Convection Limit in a Two-Dimensional Chemotaxis-Navier–Stokes
System.” Mathematische Zeitschrift, vol. 289, no. 1–2, Springer Science
and Business Media LLC, 2017, pp. 71–108, doi:10.1007/s00209-017-1944-6.
short: Y. Wang, M. Winkler, Z. Xiang, Mathematische Zeitschrift 289 (2017) 71–108.
date_created: 2025-12-19T11:04:38Z
date_updated: 2025-12-19T11:04:46Z
doi: 10.1007/s00209-017-1944-6
intvolume: ' 289'
issue: 1-2
language:
- iso: eng
page: 71-108
publication: Mathematische Zeitschrift
publication_identifier:
issn:
- 0025-5874
- 1432-1823
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system
type: journal_article
user_id: '31496'
volume: 289
year: '2017'
...
---
_id: '63374'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Singular structure formation in a degenerate haptotaxis model involving
myopic diffusion. Journal de Mathématiques Pures et Appliquées. 2017;112:118-169.
doi:10.1016/j.matpur.2017.11.002
apa: Winkler, M. (2017). Singular structure formation in a degenerate haptotaxis
model involving myopic diffusion. Journal de Mathématiques Pures et Appliquées,
112, 118–169. https://doi.org/10.1016/j.matpur.2017.11.002
bibtex: '@article{Winkler_2017, title={Singular structure formation in a degenerate
haptotaxis model involving myopic diffusion}, volume={112}, DOI={10.1016/j.matpur.2017.11.002},
journal={Journal de Mathématiques Pures et Appliquées}, publisher={Elsevier BV},
author={Winkler, Michael}, year={2017}, pages={118–169} }'
chicago: 'Winkler, Michael. “Singular Structure Formation in a Degenerate Haptotaxis
Model Involving Myopic Diffusion.” Journal de Mathématiques Pures et Appliquées
112 (2017): 118–69. https://doi.org/10.1016/j.matpur.2017.11.002.'
ieee: 'M. Winkler, “Singular structure formation in a degenerate haptotaxis model
involving myopic diffusion,” Journal de Mathématiques Pures et Appliquées,
vol. 112, pp. 118–169, 2017, doi: 10.1016/j.matpur.2017.11.002.'
mla: Winkler, Michael. “Singular Structure Formation in a Degenerate Haptotaxis
Model Involving Myopic Diffusion.” Journal de Mathématiques Pures et Appliquées,
vol. 112, Elsevier BV, 2017, pp. 118–69, doi:10.1016/j.matpur.2017.11.002.
short: M. Winkler, Journal de Mathématiques Pures et Appliquées 112 (2017) 118–169.
date_created: 2025-12-19T11:05:33Z
date_updated: 2025-12-19T11:05:40Z
doi: 10.1016/j.matpur.2017.11.002
intvolume: ' 112'
language:
- iso: eng
page: 118-169
publication: Journal de Mathématiques Pures et Appliquées
publication_identifier:
issn:
- 0021-7824
publication_status: published
publisher: Elsevier BV
status: public
title: Singular structure formation in a degenerate haptotaxis model involving myopic
diffusion
type: journal_article
user_id: '31496'
volume: 112
year: '2017'
...
---
_id: '63378'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: 'Winkler M. One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction
Revisited—Extinction at Spatial Infinity. Journal of Dynamics and Differential
Equations. 2017;30(1):331-358. doi:10.1007/s10884-017-9577-3'
apa: 'Winkler, M. (2017). One-Dimensional Super-Fast Diffusion: Persistence Versus
Extinction Revisited—Extinction at Spatial Infinity. Journal of Dynamics and
Differential Equations, 30(1), 331–358. https://doi.org/10.1007/s10884-017-9577-3'
bibtex: '@article{Winkler_2017, title={One-Dimensional Super-Fast Diffusion: Persistence
Versus Extinction Revisited—Extinction at Spatial Infinity}, volume={30}, DOI={10.1007/s10884-017-9577-3},
number={1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2017}, pages={331–358}
}'
chicago: 'Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus
Extinction Revisited—Extinction at Spatial Infinity.” Journal of Dynamics and
Differential Equations 30, no. 1 (2017): 331–58. https://doi.org/10.1007/s10884-017-9577-3.'
ieee: 'M. Winkler, “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction
Revisited—Extinction at Spatial Infinity,” Journal of Dynamics and Differential
Equations, vol. 30, no. 1, pp. 331–358, 2017, doi: 10.1007/s10884-017-9577-3.'
mla: 'Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus
Extinction Revisited—Extinction at Spatial Infinity.” Journal of Dynamics and
Differential Equations, vol. 30, no. 1, Springer Science and Business Media
LLC, 2017, pp. 331–58, doi:10.1007/s10884-017-9577-3.'
short: M. Winkler, Journal of Dynamics and Differential Equations 30 (2017) 331–358.
date_created: 2025-12-19T11:07:24Z
date_updated: 2025-12-19T11:07:30Z
doi: 10.1007/s10884-017-9577-3
intvolume: ' 30'
issue: '1'
language:
- iso: eng
page: 331-358
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: 'One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction
at Spatial Infinity'
type: journal_article
user_id: '31496'
volume: 30
year: '2017'
...
---
_id: '63379'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Renormalized radial large-data solutions to the higher-dimensional
Keller–Segel system with singular sensitivity and signal absorption. Journal
of Differential Equations. 2017;264(3):2310-2350. doi:10.1016/j.jde.2017.10.029
apa: Winkler, M. (2017). Renormalized radial large-data solutions to the higher-dimensional
Keller–Segel system with singular sensitivity and signal absorption. Journal
of Differential Equations, 264(3), 2310–2350. https://doi.org/10.1016/j.jde.2017.10.029
bibtex: '@article{Winkler_2017, title={Renormalized radial large-data solutions
to the higher-dimensional Keller–Segel system with singular sensitivity and signal
absorption}, volume={264}, DOI={10.1016/j.jde.2017.10.029},
number={3}, journal={Journal of Differential Equations}, publisher={Elsevier BV},
author={Winkler, Michael}, year={2017}, pages={2310–2350} }'
chicago: 'Winkler, Michael. “Renormalized Radial Large-Data Solutions to the Higher-Dimensional
Keller–Segel System with Singular Sensitivity and Signal Absorption.” Journal
of Differential Equations 264, no. 3 (2017): 2310–50. https://doi.org/10.1016/j.jde.2017.10.029.'
ieee: 'M. Winkler, “Renormalized radial large-data solutions to the higher-dimensional
Keller–Segel system with singular sensitivity and signal absorption,” Journal
of Differential Equations, vol. 264, no. 3, pp. 2310–2350, 2017, doi: 10.1016/j.jde.2017.10.029.'
mla: Winkler, Michael. “Renormalized Radial Large-Data Solutions to the Higher-Dimensional
Keller–Segel System with Singular Sensitivity and Signal Absorption.” Journal
of Differential Equations, vol. 264, no. 3, Elsevier BV, 2017, pp. 2310–50,
doi:10.1016/j.jde.2017.10.029.
short: M. Winkler, Journal of Differential Equations 264 (2017) 2310–2350.
date_created: 2025-12-19T11:07:52Z
date_updated: 2025-12-19T11:07:59Z
doi: 10.1016/j.jde.2017.10.029
intvolume: ' 264'
issue: '3'
language:
- iso: eng
page: 2310-2350
publication: Journal of Differential Equations
publication_identifier:
issn:
- 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Renormalized radial large-data solutions to the higher-dimensional Keller–Segel
system with singular sensitivity and signal absorption
type: journal_article
user_id: '31496'
volume: 264
year: '2017'
...
---
_id: '63383'
abstract:
- lang: eng
text: "This paper is concerned with radially symmetric solutions of the parabolic-elliptic
version of the Keller-Segel system with flux limitation, as given by \r\n\r\n \r\n \r\n \r\n \r\n
\ (\\star)\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 \r\n \r\n (\r\n \r\n \r\n
\ \r\n \r\n u\r\n
\ ∇\r\n
\ u\r\n \r\n
\ \r\n \r\n u\r\n
\ 2\r\n \r\n
\ +\r\n \r\n
\ |\r\n \r\n
\ ∇\r\n
\ u\r\n \r\n
\ \r\n |\r\n \r\n 2\r\n
\ \r\n \r\n \r\n
\ \r\n \r\n )\r\n \r\n \r\n
\ −\r\n χ\r\n \r\n
\ ∇\r\n ⋅\r\n \r\n \r\n (\r\n \r\n \r\n
\ \r\n \r\n u\r\n
\ ∇\r\n
\ v\r\n \r\n
\ \r\n 1\r\n
\ +\r\n \r\n
\ |\r\n \r\n
\ ∇\r\n
\ v\r\n \r\n
\ \r\n |\r\n \r\n 2\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 0\r\n
\ =\r\n Δ\r\n v\r\n −\r\n μ\r\n +\r\n
\ u\r\n ,\r\n
\ \r\n \r\n \r\n
\ \r\n
\ \r\n \r\n \r\n \r\n
\ \\begin{equation}\\tag {\\star
} \\begin {cases} u_t=\\nabla \\cdot \\Big (\\frac {u\\nabla u}{\\sqrt {u^2+|\\nabla
u|^2}}\\Big ) - \\chi \\, \\nabla \\cdot \\Big (\\frac {u\\nabla v}{\\sqrt {1+|\\nabla
v|^2}}\\Big ), \\\\[3pt] 0=\\Delta v - \\mu + u, \\end{cases} \\end{equation}\r\n
\ \r\n\r\n\r\n under the initial condition
\r\n\r\n \r\n \r\n u\r\n \r\n
\ \r\n |\r\n
\ \r\n \r\n t\r\n
\ =\r\n 0\r\n \r\n
\ \r\n =\r\n \r\n u\r\n
\ 0\r\n \r\n >\r\n
\ 0\r\n \r\n u|_{t=0}=u_0>0\r\n
\ \r\n\r\n and no-flux boundary conditions
in a ball \r\n\r\n
\ \r\n \r\n Ω\r\n ⊂\r\n \r\n \r\n R\r\n
\ \r\n n\r\n \r\n \r\n
\ \\Omega \\subset \\mathbb {R}^n\r\n
\ \r\n\r\n, where \r\n\r\n \r\n \r\n χ\r\n >\r\n 0\r\n
\ \r\n \\chi >0\r\n
\ \r\n\r\n and \r\n\r\n \r\n \r\n
\ μ\r\n :=\r\n \r\n
\ 1\r\n \r\n \r\n
\ |\r\n \r\n
\ Ω\r\n \r\n |\r\n
\ \r\n \r\n \r\n \r\n
\ ∫\r\n Ω\r\n \r\n \r\n u\r\n
\ 0\r\n \r\n \r\n \\mu :=\\frac {1}{|\\Omega |} \\int _\\Omega u_0\r\n
\ \r\n\r\n. A previous result of the
authors [Comm. Partial Differential Equations 42 (2017), 436–473] has asserted
global existence of bounded classical solutions for arbitrary positive radial
initial data \r\n\r\n \r\n \r\n \r\n
\ u\r\n 0\r\n \r\n
\ ∈\r\n \r\n C\r\n
\ 3\r\n \r\n (\r\n
\ \r\n \r\n Ω\r\n ¯\r\n \r\n \r\n )\r\n
\ \r\n u_0\\in
C^3(\\bar \\Omega )\r\n \r\n\r\n
when either \r\n\r\n \r\n \r\n
\ n\r\n ≥\r\n 2\r\n
\ \r\n n\\ge 2\r\n
\ \r\n\r\n and \r\n\r\n
\ \r\n \r\n χ\r\n
\ >\r\n 1\r\n \r\n
\ \\chi >1\r\n
\ \r\n\r\n, or \r\n\r\n \r\n
\ \r\n n\r\n =\r\n 1\r\n
\ \r\n n=1\r\n
\ \r\n\r\n and \r\n\r\n \r\n \r\n \r\n
\ ∫\r\n Ω\r\n \r\n \r\n u\r\n
\ 0\r\n \r\n >\r\n
\ \r\n 1\r\n \r\n (\r\n \r\n χ\r\n 2\r\n \r\n
\ −\r\n 1\r\n \r\n
\ )\r\n +\r\n
\ \r\n \r\n \r\n \r\n
\ \\int _\\Omega u_0>\\frac
{1}{\\sqrt {(\\chi ^2-1)_+}}\r\n \r\n\r\n.
\r\n\r\nThis
present paper shows that these conditions are essentially optimal: Indeed, it
is shown that if the taxis coefficient satisfies \r\n\r\n
\ \r\n \r\n χ\r\n
\ >\r\n 1\r\n \r\n
\ \\chi >1\r\n
\ \r\n\r\n, then for any choice of
\r\n\\[\r\n\r\n \r\n
\ \r\n {\r\n \r\n \r\n
\ \r\n m\r\n >\r\n
\ \r\n 1\r\n \r\n
\ \r\n χ\r\n
\ 2\r\n \r\n −\r\n 1\r\n \r\n
\ \r\n \r\n \r\n a\r\n
\ m\r\n p\r\n ;\r\n
\ \r\n if \r\n \r\n n\r\n =\r\n
\ 1\r\n \r\n \r\n
\ ,\r\n \r\n \r\n
\ \r\n \r\n \r\n \r\n m\r\n >\r\n
\ 0\r\n \r\n is
arbitrary\r\n \r\n \r\n \r\n
\ a\r\n m\r\n p\r\n
\ ;\r\n \r\n if \r\n
\ \r\n n\r\n
\ ≥\r\n 2\r\n
\ \r\n \r\n ,\r\n
\ \r\n \r\n \r\n \r\n \r\n
\ \\begin {cases} m>\\frac
{1}{\\sqrt {\\chi ^2-1}} & \\text {if $n=1$}, \\\\ \\text {$m>0$ is arbitrary}
& \\text {if $n\\ge 2$}, \\end {cases}\r\n \r\n\r\n\\]\r\n
there exist positive initial data \r\n\r\n \r\n
\ \r\n \r\n u\r\n 0\r\n
\ \r\n ∈\r\n \r\n
\ C\r\n 3\r\n \r\n
\ (\r\n \r\n
\ \r\n Ω\r\n
\ ¯\r\n \r\n
\ \r\n )\r\n \r\n
\ u_0\\in C^3(\\bar \\Omega )\r\n
\ \r\n\r\n satisfying \r\n\r\n
\ \r\n \r\n \r\n ∫\r\n Ω\r\n
\ \r\n \r\n u\r\n 0\r\n
\ \r\n =\r\n m\r\n
\ \r\n \\int _\\Omega
u_0=m\r\n \r\n\r\n
which are such that for some \r\n\r\n \r\n \r\n T\r\n >\r\n
\ 0\r\n \r\n T>0\r\n
\ \r\n\r\n, (\r\n\r\n \r\n
\ ⋆\r\n \\star\r\n
\ \r\n\r\n) possesses a uniquely determined
classical solution \r\n\r\n \r\n \r\n (\r\n u\r\n ,\r\n
\ v\r\n )\r\n \r\n
\ (u,v)\r\n
\ \r\n\r\n in \r\n\r\n \r\n
\ \r\n Ω\r\n
\ ×\r\n (\r\n
\ 0\r\n ,\r\n T\r\n
\ )\r\n \r\n \\Omega \\times (0,T)\r\n \r\n\r\n
blowing up at time \r\n\r\n \r\n
\ T\r\n T\r\n
\ \r\n\r\n in the sense that \r\n\r\n
\ \r\n \r\n \r\n lim sup\r\n \r\n
\ t\r\n ↗\r\n
\ T\r\n \r\n \r\n
\ ‖\r\n u\r\n
\ (\r\n ⋅\r\n
\ ,\r\n t\r\n )\r\n
\ \r\n ‖\r\n \r\n \r\n
\ L\r\n ∞\r\n \r\n (\r\n
\ Ω\r\n )\r\n \r\n \r\n
\ =\r\n ∞\r\n
\ \r\n \\limsup
_{t\\nearrow T} \\|u(\\cdot ,t)\\|_{L^\\infty (\\Omega )}=\\infty\r\n
\ \r\n\r\n.
\r\n\r\nThis result
is derived by means of a comparison argument applied to the doubly degenerate
scalar parabolic equation satisfied by the mass accumulation function associated
with (\r\n\r\n \r\n ⋆\r\n
\ \\star\r\n
\ \r\n\r\n).
"
author:
- first_name: Nicola
full_name: Bellomo, Nicola
last_name: Bellomo
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Bellomo N, Winkler M. Finite-time blow-up in a degenerate chemotaxis system
with flux limitation. Transactions of the American Mathematical Society, Series
B. 2017;4(2):31-67. doi:10.1090/btran/17
apa: Bellomo, N., & Winkler, M. (2017). Finite-time blow-up in a degenerate
chemotaxis system with flux limitation. Transactions of the American Mathematical
Society, Series B, 4(2), 31–67. https://doi.org/10.1090/btran/17
bibtex: '@article{Bellomo_Winkler_2017, title={Finite-time blow-up in a degenerate
chemotaxis system with flux limitation}, volume={4}, DOI={10.1090/btran/17},
number={2}, journal={Transactions of the American Mathematical Society, Series
B}, publisher={American Mathematical Society (AMS)}, author={Bellomo, Nicola and
Winkler, Michael}, year={2017}, pages={31–67} }'
chicago: 'Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate
Chemotaxis System with Flux Limitation.” Transactions of the American Mathematical
Society, Series B 4, no. 2 (2017): 31–67. https://doi.org/10.1090/btran/17.'
ieee: 'N. Bellomo and M. Winkler, “Finite-time blow-up in a degenerate chemotaxis
system with flux limitation,” Transactions of the American Mathematical Society,
Series B, vol. 4, no. 2, pp. 31–67, 2017, doi: 10.1090/btran/17.'
mla: Bellomo, Nicola, and Michael Winkler. “Finite-Time Blow-up in a Degenerate
Chemotaxis System with Flux Limitation.” Transactions of the American Mathematical
Society, Series B, vol. 4, no. 2, American Mathematical Society (AMS), 2017,
pp. 31–67, doi:10.1090/btran/17.
short: N. Bellomo, M. Winkler, Transactions of the American Mathematical Society,
Series B 4 (2017) 31–67.
date_created: 2025-12-19T11:09:53Z
date_updated: 2025-12-19T11:10:17Z
doi: 10.1090/btran/17
intvolume: ' 4'
issue: '2'
language:
- iso: eng
page: 31-67
publication: Transactions of the American Mathematical Society, Series B
publication_identifier:
issn:
- 2330-0000
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: Finite-time blow-up in a degenerate chemotaxis system with flux limitation
type: journal_article
user_id: '31496'
volume: 4
year: '2017'
...