---
_id: '63342'
abstract:
- lang: eng
text: "AbstractIn a bounded planar domain $\\varOmega
$\r\n
\ Ω\r\n
with smooth boundary, the initial-boundary value problem of homogeneous Neumann
type for the Keller-Segel-fluid system \r\n\t\t\t
$$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t} +
\\nabla \\cdot (nu) = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in \\varOmega
, \\ t>0, \\\\ 0 = \\Delta c -c+n, & x\\in \\varOmega , \\ t>0, \\end{array}\\displaystyle
\\right . \\end{aligned}$$ \r\n
\ \r\n {\r\n \r\n
\ \r\n \r\n \r\n
\ n\r\n t\r\n
\ \r\n +\r\n
\ ∇\r\n ⋅\r\n
\ (\r\n n\r\n
\ u\r\n )\r\n
\ =\r\n Δ\r\n
\ n\r\n −\r\n
\ ∇\r\n ⋅\r\n
\ (\r\n n\r\n
\ ∇\r\n c\r\n
\ )\r\n ,\r\n
\ \r\n \r\n x\r\n
\ ∈\r\n Ω\r\n
\ ,\r\n \r\n
\ t\r\n >\r\n
\ 0\r\n ,\r\n
\ \r\n \r\n \r\n
\ \r\n 0\r\n
\ =\r\n Δ\r\n
\ c\r\n −\r\n
\ c\r\n +\r\n
\ n\r\n ,\r\n
\ \r\n \r\n x\r\n
\ ∈\r\n Ω\r\n
\ ,\r\n \r\n
\ t\r\n >\r\n
\ 0\r\n ,\r\n
\ \r\n \r\n \r\n
\ \r\n
is considered, where $u$\r\n u\r\n
\ is a given
sufficiently smooth velocity field on $\\overline
{\\varOmega }\\times [0,\\infty )$\r\n
\ \r\n Ω\r\n ‾\r\n
\ \r\n ×\r\n [\r\n
\ 0\r\n ,\r\n
\ ∞\r\n )\r\n
\ that is
tangential on $\\partial
\\varOmega $\r\n
\ ∂\r\n Ω\r\n
\ but not
necessarily solenoidal.It is firstly shown that for any choice
of $n_{0}\\in C^{0}(\\overline
{\\varOmega })$\r\n
\ \r\n n\r\n 0\r\n
\ \r\n ∈\r\n \r\n
\ C\r\n 0\r\n
\ \r\n (\r\n \r\n
\ Ω\r\n ‾\r\n
\ \r\n )\r\n
with $\\int _{\\varOmega}n_{0}<4\\pi
$\r\n
\ \r\n ∫\r\n Ω\r\n
\ \r\n \r\n n\r\n
\ 0\r\n \r\n <\r\n
\ 4\r\n π\r\n
\ , this problem
admits a global classical solution with $n(\\cdot
,0)=n_{0}$\r\n
\ n\r\n (\r\n
\ ⋅\r\n ,\r\n
\ 0\r\n )\r\n
\ =\r\n \r\n n\r\n
\ 0\r\n \r\n ,
and that this solution is even bounded whenever $u$\r\n u\r\n
\ is bounded
and $\\int _{\\varOmega}n_{0}<2\\pi
$\r\n
\ \r\n ∫\r\n Ω\r\n
\ \r\n \r\n n\r\n
\ 0\r\n \r\n <\r\n
\ 2\r\n π\r\n
\ . Secondly,
it is seen that for each $m>4\\pi
$\r\n
\ m\r\n >\r\n
\ 4\r\n π\r\n
\ one can
find a classical solution with $\\int
_{\\varOmega}n(\\cdot ,0)=m$\r\n
\ \r\n ∫\r\n Ω\r\n
\ \r\n n\r\n (\r\n
\ ⋅\r\n ,\r\n
\ 0\r\n )\r\n
\ =\r\n m\r\n
\ which blows
up in finite time, provided that $\\varOmega
$\r\n
\ Ω\r\n
satisfies a technical assumption requiring $\\partial
\\varOmega $\r\n
\ ∂\r\n Ω\r\n
\ to contain
a line segment.In particular, this indicates that the value $4\\pi
$\r\n
\ 4\r\n π\r\n
\ of the critical
mass for the corresponding fluid-free Keller-Segel system is left unchanged by
any fluid interaction of the considered type, thus marking a considerable contrast
to a recent result revealing some fluid-induced increase of critical blow-up masses
in a related Cauchy problem in the entire plane."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
Blow-up in Bounded Planar Domains? Acta Applicandae Mathematicae. 2020;169(1):577-591.
doi:10.1007/s10440-020-00312-2
apa: Winkler, M. (2020). Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
Blow-up in Bounded Planar Domains? Acta Applicandae Mathematicae, 169(1),
577–591. https://doi.org/10.1007/s10440-020-00312-2
bibtex: '@article{Winkler_2020, title={Can Fluid Interaction Influence the Critical
Mass for Taxis-Driven Blow-up in Bounded Planar Domains?}, volume={169}, DOI={10.1007/s10440-020-00312-2},
number={1}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science
and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={577–591}
}'
chicago: 'Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for
Taxis-Driven Blow-up in Bounded Planar Domains?” Acta Applicandae Mathematicae
169, no. 1 (2020): 577–91. https://doi.org/10.1007/s10440-020-00312-2.'
ieee: 'M. Winkler, “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
Blow-up in Bounded Planar Domains?,” Acta Applicandae Mathematicae, vol.
169, no. 1, pp. 577–591, 2020, doi: 10.1007/s10440-020-00312-2.'
mla: Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
Blow-up in Bounded Planar Domains?” Acta Applicandae Mathematicae, vol.
169, no. 1, Springer Science and Business Media LLC, 2020, pp. 577–91, doi:10.1007/s10440-020-00312-2.
short: M. Winkler, Acta Applicandae Mathematicae 169 (2020) 577–591.
date_created: 2025-12-18T19:47:51Z
date_updated: 2025-12-18T19:57:40Z
doi: 10.1007/s10440-020-00312-2
intvolume: ' 169'
issue: '1'
language:
- iso: eng
page: 577-591
publication: Acta Applicandae Mathematicae
publication_identifier:
issn:
- 0167-8019
- 1572-9036
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up
in Bounded Planar Domains?
type: journal_article
user_id: '31496'
volume: 169
year: '2020'
...
---
_id: '8753'
abstract:
- lang: eng
text: In a wide range of applications it is desirable to optimally control a dynamical
system with respect to concurrent, potentially competing goals. This gives rise
to a multiobjective optimal control problem where, instead of computing a single
optimal solution, the set of optimal compromises, the so-called Pareto set, has
to be approximated. When the problem under consideration is described by a partial
differential equation (PDE), as is the case for fluid flow, the computational
cost rapidly increases and makes its direct treatment infeasible. Reduced order
modeling is a very popular method to reduce the computational cost, in particular
in a multi query context such as uncertainty quantification, parameter estimation
or optimization. In this article, we show how to combine reduced order modeling
and multiobjective optimal control techniques in order to efficiently solve multiobjective
optimal control problems constrained by PDEs. We consider a global, derivative
free optimization method as well as a local, gradient-based approach for which
the optimality system is derived in two different ways. The methods are compared
with regard to the solution quality as well as the computational effort and they
are illustrated using the example of the flow around a cylinder and a backward-facing-step
channel flow.
author:
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
citation:
ama: Peitz S, Ober-Blöbaum S, Dellnitz M. Multiobjective Optimal Control Methods
for the Navier-Stokes Equations Using Reduced Order Modeling. Acta Applicandae
Mathematicae. 2018;161(1):171–199. doi:10.1007/s10440-018-0209-7
apa: Peitz, S., Ober-Blöbaum, S., & Dellnitz, M. (2018). Multiobjective Optimal
Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling.
Acta Applicandae Mathematicae, 161(1), 171–199. https://doi.org/10.1007/s10440-018-0209-7
bibtex: '@article{Peitz_Ober-Blöbaum_Dellnitz_2018, title={Multiobjective Optimal
Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling},
volume={161}, DOI={10.1007/s10440-018-0209-7},
number={1}, journal={Acta Applicandae Mathematicae}, author={Peitz, Sebastian
and Ober-Blöbaum, Sina and Dellnitz, Michael}, year={2018}, pages={171–199} }'
chicago: 'Peitz, Sebastian, Sina Ober-Blöbaum, and Michael Dellnitz. “Multiobjective
Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling.”
Acta Applicandae Mathematicae 161, no. 1 (2018): 171–199. https://doi.org/10.1007/s10440-018-0209-7.'
ieee: 'S. Peitz, S. Ober-Blöbaum, and M. Dellnitz, “Multiobjective Optimal Control
Methods for the Navier-Stokes Equations Using Reduced Order Modeling,” Acta
Applicandae Mathematicae, vol. 161, no. 1, pp. 171–199, 2018, doi: 10.1007/s10440-018-0209-7.'
mla: Peitz, Sebastian, et al. “Multiobjective Optimal Control Methods for the Navier-Stokes
Equations Using Reduced Order Modeling.” Acta Applicandae Mathematicae,
vol. 161, no. 1, 2018, pp. 171–199, doi:10.1007/s10440-018-0209-7.
short: S. Peitz, S. Ober-Blöbaum, M. Dellnitz, Acta Applicandae Mathematicae 161
(2018) 171–199.
date_created: 2019-03-29T13:30:41Z
date_updated: 2022-01-21T10:01:41Z
department:
- _id: '101'
doi: 10.1007/s10440-018-0209-7
intvolume: ' 161'
issue: '1'
language:
- iso: eng
page: 171–199
project:
- _id: '52'
name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication: Acta Applicandae Mathematicae
publication_identifier:
issn:
- 0167-8019
- 1572-9036
publication_status: published
status: public
title: Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using
Reduced Order Modeling
type: journal_article
user_id: '15694'
volume: 161
year: '2018'
...
---
_id: '63368'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics with
Mildly Saturated Chemotactic Sensitivity. Acta Applicandae Mathematicae.
2018;163(1):1-17. doi:10.1007/s10440-018-0211-0
apa: Winkler, M. (2018). Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
with Mildly Saturated Chemotactic Sensitivity. Acta Applicandae Mathematicae,
163(1), 1–17. https://doi.org/10.1007/s10440-018-0211-0
bibtex: '@article{Winkler_2018, title={Boundedness in a Chemotaxis-May-Nowak Model
for Virus Dynamics with Mildly Saturated Chemotactic Sensitivity}, volume={163},
DOI={10.1007/s10440-018-0211-0},
number={1}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science
and Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1–17}
}'
chicago: 'Winkler, Michael. “Boundedness in a Chemotaxis-May-Nowak Model for Virus
Dynamics with Mildly Saturated Chemotactic Sensitivity.” Acta Applicandae Mathematicae
163, no. 1 (2018): 1–17. https://doi.org/10.1007/s10440-018-0211-0.'
ieee: 'M. Winkler, “Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
with Mildly Saturated Chemotactic Sensitivity,” Acta Applicandae Mathematicae,
vol. 163, no. 1, pp. 1–17, 2018, doi: 10.1007/s10440-018-0211-0.'
mla: Winkler, Michael. “Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
with Mildly Saturated Chemotactic Sensitivity.” Acta Applicandae Mathematicae,
vol. 163, no. 1, Springer Science and Business Media LLC, 2018, pp. 1–17, doi:10.1007/s10440-018-0211-0.
short: M. Winkler, Acta Applicandae Mathematicae 163 (2018) 1–17.
date_created: 2025-12-19T11:02:13Z
date_updated: 2025-12-19T11:02:21Z
doi: 10.1007/s10440-018-0211-0
intvolume: ' 163'
issue: '1'
language:
- iso: eng
page: 1-17
publication: Acta Applicandae Mathematicae
publication_identifier:
issn:
- 0167-8019
- 1572-9036
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics with Mildly
Saturated Chemotactic Sensitivity
type: journal_article
user_id: '31496'
volume: 163
year: '2018'
...
---
_id: '39948'
author:
- first_name: Margit
full_name: Rösler, Margit
id: '37390'
last_name: Rösler
- first_name: Michael
full_name: Voit, Michael
last_name: Voit
citation:
ama: Rösler M, Voit M. SU(d)-Biinvariant Random Walks on SL(d,C) and their Euclidean
Counterparts. Acta Applicandae Mathematicae. 2006;90(1-2):179-195. doi:10.1007/s10440-006-9035-4
apa: Rösler, M., & Voit, M. (2006). SU(d)-Biinvariant Random Walks on SL(d,C)
and their Euclidean Counterparts. Acta Applicandae Mathematicae, 90(1–2),
179–195. https://doi.org/10.1007/s10440-006-9035-4
bibtex: '@article{Rösler_Voit_2006, title={SU(d)-Biinvariant Random Walks on SL(d,C)
and their Euclidean Counterparts}, volume={90}, DOI={10.1007/s10440-006-9035-4},
number={1–2}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science
and Business Media LLC}, author={Rösler, Margit and Voit, Michael}, year={2006},
pages={179–195} }'
chicago: 'Rösler, Margit, and Michael Voit. “SU(d)-Biinvariant Random Walks on SL(d,C)
and Their Euclidean Counterparts.” Acta Applicandae Mathematicae 90, no.
1–2 (2006): 179–95. https://doi.org/10.1007/s10440-006-9035-4.'
ieee: 'M. Rösler and M. Voit, “SU(d)-Biinvariant Random Walks on SL(d,C) and their
Euclidean Counterparts,” Acta Applicandae Mathematicae, vol. 90, no. 1–2,
pp. 179–195, 2006, doi: 10.1007/s10440-006-9035-4.'
mla: Rösler, Margit, and Michael Voit. “SU(d)-Biinvariant Random Walks on SL(d,C)
and Their Euclidean Counterparts.” Acta Applicandae Mathematicae, vol.
90, no. 1–2, Springer Science and Business Media LLC, 2006, pp. 179–95, doi:10.1007/s10440-006-9035-4.
short: M. Rösler, M. Voit, Acta Applicandae Mathematicae 90 (2006) 179–195.
date_created: 2023-01-25T09:57:30Z
date_updated: 2023-01-26T17:47:14Z
department:
- _id: '555'
doi: 10.1007/s10440-006-9035-4
extern: '1'
intvolume: ' 90'
issue: 1-2
keyword:
- Applied Mathematics
language:
- iso: eng
page: 179-195
publication: Acta Applicandae Mathematicae
publication_identifier:
issn:
- 0167-8019
- 1572-9036
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: SU(d)-Biinvariant Random Walks on SL(d,C) and their Euclidean Counterparts
type: journal_article
user_id: '37390'
volume: 90
year: '2006'
...