---
_id: '63362'
abstract:
- lang: eng
  text: "<jats:p>The system</jats:p>\r\n          <jats:p>\r\n            <jats:disp-formula>\r\n
    \             <jats:tex-math>\\left\\{\\begin{matrix} u_{t} = \\mathrm{\\Delta
    }u−\\chi \\mathrm{∇} \\cdot \\left(\\frac{u}{v}\\mathrm{∇}v\\right)−uv + B_{1}(x,t),
    \\\\ v_{t} = \\mathrm{\\Delta }v + uv−v + B_{2}(x,t), \\\\  \\end{matrix}\\right.\\:\\:(
    \\star )</jats:tex-math>\r\n            </jats:disp-formula>\r\n          </jats:p>\r\n
    \         <jats:p>\r\n            is considered in a disk \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\mathrm{\\Omega } \\subset \\mathbb{R}^{2}</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            , with a positive parameter
    \r\n            <jats:inline-formula>\r\n              <jats:tex-math>χ</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             and given nonnegative and suitably
    regular functions \r\n            <jats:inline-formula>\r\n              <jats:tex-math>B_{1}</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             and \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>B_{2}</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            defined on \r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\mathrm{\\Omega
    } \\times (0,\\infty )</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           . In the particular version obtained when \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\chi  = 2</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           ,  (\r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\star</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            ) was proposed in [31] as a
    model for crime propagation in urban regions.\r\n          </jats:p>\r\n          <jats:p>\r\n
    \           Within a suitable generalized framework, it is shown that under mild
    assumptions on the parameter functions and the initial data the no-flux initial-boundary
    value problem for (\r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\star</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            ) possesses at least one global
    solution in the case when all model ingredients are radially symmetric with respect
    to the center of \r\n            <jats:inline-formula>\r\n              <jats:tex-math>Ω</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n            . Moreover, under an additional
    hypothesis on stabilization of the given external source terms in both equations,
    these solutions are shown to approach the solution of an elliptic boundary value
    problem in an appropriate sense.\r\n          </jats:p>\r\n          <jats:p>The
    analysis is based on deriving a priori estimates for a family of approximate problems,
    in a first step achieving some spatially global but weak initial regularity information
    which in a series of spatially localized arguments is thereafter successively
    improved.</jats:p>\r\n          <jats:p>\r\n            To the best of our knowledge,
    this is the first result on global existence of solutions to the two-dimensional
    version of the full original system  (\r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\star</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           ) for arbitrarily large values of \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>χ</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           .\r\n          </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global solvability and stabilization in a two-dimensional cross-diffusion
    system modeling urban crime propagation. <i>Annales de l’Institut Henri Poincaré
    C, Analyse non linéaire</i>. 2019;36(6):1747-1790. doi:<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>
  apa: Winkler, M. (2019). Global solvability and stabilization in a two-dimensional
    cross-diffusion system modeling urban crime propagation. <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i>, <i>36</i>(6), 1747–1790. <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">https://doi.org/10.1016/j.anihpc.2019.02.004</a>
  bibtex: '@article{Winkler_2019, title={Global solvability and stabilization in a
    two-dimensional cross-diffusion system modeling urban crime propagation}, volume={36},
    DOI={<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>},
    number={6}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire},
    publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler,
    Michael}, year={2019}, pages={1747–1790} }'
  chicago: 'Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional
    Cross-Diffusion System Modeling Urban Crime Propagation.” <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i> 36, no. 6 (2019): 1747–90. <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">https://doi.org/10.1016/j.anihpc.2019.02.004</a>.'
  ieee: 'M. Winkler, “Global solvability and stabilization in a two-dimensional cross-diffusion
    system modeling urban crime propagation,” <i>Annales de l’Institut Henri Poincaré
    C, Analyse non linéaire</i>, vol. 36, no. 6, pp. 1747–1790, 2019, doi: <a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>.'
  mla: Winkler, Michael. “Global Solvability and Stabilization in a Two-Dimensional
    Cross-Diffusion System Modeling Urban Crime Propagation.” <i>Annales de l’Institut
    Henri Poincaré C, Analyse Non Linéaire</i>, vol. 36, no. 6, European Mathematical
    Society - EMS - Publishing House GmbH, 2019, pp. 1747–90, doi:<a href="https://doi.org/10.1016/j.anihpc.2019.02.004">10.1016/j.anihpc.2019.02.004</a>.
  short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire
    36 (2019) 1747–1790.
date_created: 2025-12-19T10:58:29Z
date_updated: 2025-12-19T10:58:37Z
doi: 10.1016/j.anihpc.2019.02.004
intvolume: '        36'
issue: '6'
language:
- iso: eng
page: 1747-1790
publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
publication_identifier:
  issn:
  - 0294-1449
  - 1873-1430
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Global solvability and stabilization in a two-dimensional cross-diffusion system
  modeling urban crime propagation
type: journal_article
user_id: '31496'
volume: 36
year: '2019'
...
---
_id: '63363'
abstract:
- lang: eng
  text: '<jats:p> This work is concerned with a prototypical model for the spatio-temporal
    evolution of a forager–exploiter system, consisting of two species which simultaneously
    consume a common nutrient, and which interact through a taxis-type mechanism according
    to which individuals from the exploiter subpopulation move upward density gradients
    of the forager subgroup. Specifically, the model [Formula: see text] for the population
    densities [Formula: see text] and [Formula: see text] of foragers and exploiters,
    as well as the nutrient concentration [Formula: see text], is considered in smoothly
    bounded domains [Formula: see text], [Formula: see text]. It is first shown that
    under an explicit condition linking the sizes of the resource production rate
    [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type
    initial-boundary value problem admits a global solution within an appropriate
    generalized concept. The second of the main results asserts stabilization of these
    solutions toward spatially homogeneous equilibria in the large time limit, provided
    that [Formula: see text] satisfies a mild assumption on temporal decay. To the
    best of our knowledge, these are the first rigorous analytical results addressing
    taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).
    </jats:p>'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global generalized solutions to a multi-dimensional doubly tactic
    resource consumption model accounting for social interactions. <i>Mathematical
    Models and Methods in Applied Sciences</i>. 2019;29(03):373-418. doi:<a href="https://doi.org/10.1142/s021820251950012x">10.1142/s021820251950012x</a>
  apa: Winkler, M. (2019). Global generalized solutions to a multi-dimensional doubly
    tactic resource consumption model accounting for social interactions. <i>Mathematical
    Models and Methods in Applied Sciences</i>, <i>29</i>(03), 373–418. <a href="https://doi.org/10.1142/s021820251950012x">https://doi.org/10.1142/s021820251950012x</a>
  bibtex: '@article{Winkler_2019, title={Global generalized solutions to a multi-dimensional
    doubly tactic resource consumption model accounting for social interactions},
    volume={29}, DOI={<a href="https://doi.org/10.1142/s021820251950012x">10.1142/s021820251950012x</a>},
    number={03}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2019}, pages={373–418}
    }'
  chicago: 'Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional
    Doubly Tactic Resource Consumption Model Accounting for Social Interactions.”
    <i>Mathematical Models and Methods in Applied Sciences</i> 29, no. 03 (2019):
    373–418. <a href="https://doi.org/10.1142/s021820251950012x">https://doi.org/10.1142/s021820251950012x</a>.'
  ieee: 'M. Winkler, “Global generalized solutions to a multi-dimensional doubly tactic
    resource consumption model accounting for social interactions,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 29, no. 03, pp. 373–418, 2019,
    doi: <a href="https://doi.org/10.1142/s021820251950012x">10.1142/s021820251950012x</a>.'
  mla: Winkler, Michael. “Global Generalized Solutions to a Multi-Dimensional Doubly
    Tactic Resource Consumption Model Accounting for Social Interactions.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 29, no. 03, World Scientific
    Pub Co Pte Ltd, 2019, pp. 373–418, doi:<a href="https://doi.org/10.1142/s021820251950012x">10.1142/s021820251950012x</a>.
  short: M. Winkler, Mathematical Models and Methods in Applied Sciences 29 (2019)
    373–418.
date_created: 2025-12-19T10:59:03Z
date_updated: 2025-12-19T10:59:10Z
doi: 10.1142/s021820251950012x
intvolume: '        29'
issue: '03'
language:
- iso: eng
page: 373-418
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  issn:
  - 0218-2025
  - 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Global generalized solutions to a multi-dimensional doubly tactic resource
  consumption model accounting for social interactions
type: journal_article
user_id: '31496'
volume: 29
year: '2019'
...
---
_id: '63366'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Instantaneous regularization of distributions from&#60;mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"
    id="d1e19" altimg="si17.gif"&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;(&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;C&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;0&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;)&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;⋆&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;×&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;L&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;2&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;/mml:math&#62;in
    the one-dimensional parabolic Keller–Segel system. <i>Nonlinear Analysis</i>.
    2019;183:102-116. doi:<a href="https://doi.org/10.1016/j.na.2019.01.017">10.1016/j.na.2019.01.017</a>
  apa: Winkler, M. (2019). Instantaneous regularization of distributions from&#60;mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"
    id="d1e19" altimg="si17.gif"&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;(&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;C&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;0&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;)&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;⋆&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;×&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;L&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;2&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;/mml:math&#62;in
    the one-dimensional parabolic Keller–Segel system. <i>Nonlinear Analysis</i>,
    <i>183</i>, 102–116. <a href="https://doi.org/10.1016/j.na.2019.01.017">https://doi.org/10.1016/j.na.2019.01.017</a>
  bibtex: '@article{Winkler_2019, title={Instantaneous regularization of distributions
    from&#60;mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"
    overflow="scroll" id="d1e19" altimg="si17.gif"&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;(&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;C&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;0&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;)&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;⋆&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;×&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;L&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;2&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;/mml:math&#62;in
    the one-dimensional parabolic Keller–Segel system}, volume={183}, DOI={<a href="https://doi.org/10.1016/j.na.2019.01.017">10.1016/j.na.2019.01.017</a>},
    journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Winkler, Michael},
    year={2019}, pages={102–116} }'
  chicago: 'Winkler, Michael. “Instantaneous Regularization of Distributions From&#60;mml:Math
    Xmlns:Mml="http://Www.W3.Org/1998/Math/MathML" Display="inline" Overflow="scroll"
    Id="d1e19" Altimg="si17.Gif"&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mo&#62;(&#60;/Mml:Mo&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mi&#62;C&#60;/Mml:Mi&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mn&#62;0&#60;/Mml:Mn&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;mml:Mo&#62;)&#60;/Mml:Mo&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mo&#62;⋆&#60;/Mml:Mo&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;mml:Mo&#62;×&#60;/Mml:Mo&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mi&#62;L&#60;/Mml:Mi&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mn&#62;2&#60;/Mml:Mn&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;/Mml:Math&#62;in
    the One-Dimensional Parabolic Keller–Segel System.” <i>Nonlinear Analysis</i>
    183 (2019): 102–16. <a href="https://doi.org/10.1016/j.na.2019.01.017">https://doi.org/10.1016/j.na.2019.01.017</a>.'
  ieee: 'M. Winkler, “Instantaneous regularization of distributions from&#60;mml:math
    xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"
    id="d1e19" altimg="si17.gif"&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;(&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;C&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;0&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;)&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mo&#62;⋆&#60;/mml:mo&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;mml:mo&#62;×&#60;/mml:mo&#62;&#60;mml:msup&#62;&#60;mml:mrow&#62;&#60;mml:mi&#62;L&#60;/mml:mi&#62;&#60;/mml:mrow&#62;&#60;mml:mrow&#62;&#60;mml:mn&#62;2&#60;/mml:mn&#62;&#60;/mml:mrow&#62;&#60;/mml:msup&#62;&#60;/mml:math&#62;in
    the one-dimensional parabolic Keller–Segel system,” <i>Nonlinear Analysis</i>,
    vol. 183, pp. 102–116, 2019, doi: <a href="https://doi.org/10.1016/j.na.2019.01.017">10.1016/j.na.2019.01.017</a>.'
  mla: Winkler, Michael. “Instantaneous Regularization of Distributions From&#60;mml:Math
    Xmlns:Mml="http://Www.W3.Org/1998/Math/MathML" Display="inline" Overflow="scroll"
    Id="d1e19" Altimg="si17.Gif"&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mo&#62;(&#60;/Mml:Mo&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mi&#62;C&#60;/Mml:Mi&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mn&#62;0&#60;/Mml:Mn&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;mml:Mo&#62;)&#60;/Mml:Mo&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mo&#62;⋆&#60;/Mml:Mo&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;mml:Mo&#62;×&#60;/Mml:Mo&#62;&#60;mml:Msup&#62;&#60;mml:Mrow&#62;&#60;mml:Mi&#62;L&#60;/Mml:Mi&#62;&#60;/Mml:Mrow&#62;&#60;mml:Mrow&#62;&#60;mml:Mn&#62;2&#60;/Mml:Mn&#62;&#60;/Mml:Mrow&#62;&#60;/Mml:Msup&#62;&#60;/Mml:Math&#62;in
    the One-Dimensional Parabolic Keller–Segel System.” <i>Nonlinear Analysis</i>,
    vol. 183, Elsevier BV, 2019, pp. 102–16, doi:<a href="https://doi.org/10.1016/j.na.2019.01.017">10.1016/j.na.2019.01.017</a>.
  short: M. Winkler, Nonlinear Analysis 183 (2019) 102–116.
date_created: 2025-12-19T11:01:12Z
date_updated: 2025-12-19T11:01:21Z
doi: 10.1016/j.na.2019.01.017
intvolume: '       183'
language:
- iso: eng
page: 102-116
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: Instantaneous regularization of distributions from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
  display="inline" overflow="scroll" id="d1e19" altimg="si17.gif"><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>in
  the one-dimensional parabolic Keller–Segel system
type: journal_article
user_id: '31496'
volume: 183
year: '2019'
...
---
_id: '63359'
article_number: '196'
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 fast signal diffusion limit in Keller–Segel(-fluid)
    systems. <i>Calculus of Variations and Partial Differential Equations</i>. 2019;58(6).
    doi:<a href="https://doi.org/10.1007/s00526-019-1656-3">10.1007/s00526-019-1656-3</a>
  apa: Wang, Y., Winkler, M., &#38; Xiang, Z. (2019). The fast signal diffusion limit
    in Keller–Segel(-fluid) systems. <i>Calculus of Variations and Partial Differential
    Equations</i>, <i>58</i>(6), Article 196. <a href="https://doi.org/10.1007/s00526-019-1656-3">https://doi.org/10.1007/s00526-019-1656-3</a>
  bibtex: '@article{Wang_Winkler_Xiang_2019, title={The fast signal diffusion limit
    in Keller–Segel(-fluid) systems}, volume={58}, DOI={<a href="https://doi.org/10.1007/s00526-019-1656-3">10.1007/s00526-019-1656-3</a>},
    number={6196}, journal={Calculus of Variations and Partial Differential Equations},
    publisher={Springer Science and Business Media LLC}, author={Wang, Yulan and Winkler,
    Michael and Xiang, Zhaoyin}, year={2019} }'
  chicago: Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “The Fast Signal Diffusion
    Limit in Keller–Segel(-Fluid) Systems.” <i>Calculus of Variations and Partial
    Differential Equations</i> 58, no. 6 (2019). <a href="https://doi.org/10.1007/s00526-019-1656-3">https://doi.org/10.1007/s00526-019-1656-3</a>.
  ieee: 'Y. Wang, M. Winkler, and Z. Xiang, “The fast signal diffusion limit in Keller–Segel(-fluid)
    systems,” <i>Calculus of Variations and Partial Differential Equations</i>, vol.
    58, no. 6, Art. no. 196, 2019, doi: <a href="https://doi.org/10.1007/s00526-019-1656-3">10.1007/s00526-019-1656-3</a>.'
  mla: Wang, Yulan, et al. “The Fast Signal Diffusion Limit in Keller–Segel(-Fluid)
    Systems.” <i>Calculus of Variations and Partial Differential Equations</i>, vol.
    58, no. 6, 196, Springer Science and Business Media LLC, 2019, doi:<a href="https://doi.org/10.1007/s00526-019-1656-3">10.1007/s00526-019-1656-3</a>.
  short: Y. Wang, M. Winkler, Z. Xiang, Calculus of Variations and Partial Differential
    Equations 58 (2019).
date_created: 2025-12-19T10:56:58Z
date_updated: 2025-12-19T10:57:05Z
doi: 10.1007/s00526-019-1656-3
intvolume: '        58'
issue: '6'
language:
- iso: eng
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
  issn:
  - 0944-2669
  - 1432-0835
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: The fast signal diffusion limit in Keller–Segel(-fluid) systems
type: journal_article
user_id: '31496'
volume: 58
year: '2019'
...
---
_id: '63364'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. How strong singularities can be regularized by logistic degradation
    in the Keller–Segel system? <i>Annali di Matematica Pura ed Applicata (1923 -)</i>.
    2019;198(5):1615-1637. doi:<a href="https://doi.org/10.1007/s10231-019-00834-z">10.1007/s10231-019-00834-z</a>
  apa: Winkler, M. (2019). How strong singularities can be regularized by logistic
    degradation in the Keller–Segel system? <i>Annali Di Matematica Pura Ed Applicata
    (1923 -)</i>, <i>198</i>(5), 1615–1637. <a href="https://doi.org/10.1007/s10231-019-00834-z">https://doi.org/10.1007/s10231-019-00834-z</a>
  bibtex: '@article{Winkler_2019, title={How strong singularities can be regularized
    by logistic degradation in the Keller–Segel system?}, volume={198}, DOI={<a href="https://doi.org/10.1007/s10231-019-00834-z">10.1007/s10231-019-00834-z</a>},
    number={5}, journal={Annali di Matematica Pura ed Applicata (1923 -)}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2019}, pages={1615–1637}
    }'
  chicago: 'Winkler, Michael. “How Strong Singularities Can Be Regularized by Logistic
    Degradation in the Keller–Segel System?” <i>Annali Di Matematica Pura Ed Applicata
    (1923 -)</i> 198, no. 5 (2019): 1615–37. <a href="https://doi.org/10.1007/s10231-019-00834-z">https://doi.org/10.1007/s10231-019-00834-z</a>.'
  ieee: 'M. Winkler, “How strong singularities can be regularized by logistic degradation
    in the Keller–Segel system?,” <i>Annali di Matematica Pura ed Applicata (1923
    -)</i>, vol. 198, no. 5, pp. 1615–1637, 2019, doi: <a href="https://doi.org/10.1007/s10231-019-00834-z">10.1007/s10231-019-00834-z</a>.'
  mla: Winkler, Michael. “How Strong Singularities Can Be Regularized by Logistic
    Degradation in the Keller–Segel System?” <i>Annali Di Matematica Pura Ed Applicata
    (1923 -)</i>, vol. 198, no. 5, Springer Science and Business Media LLC, 2019,
    pp. 1615–37, doi:<a href="https://doi.org/10.1007/s10231-019-00834-z">10.1007/s10231-019-00834-z</a>.
  short: M. Winkler, Annali Di Matematica Pura Ed Applicata (1923 -) 198 (2019) 1615–1637.
date_created: 2025-12-19T10:59:58Z
date_updated: 2025-12-19T11:00:04Z
doi: 10.1007/s10231-019-00834-z
intvolume: '       198'
issue: '5'
language:
- iso: eng
page: 1615-1637
publication: Annali di Matematica Pura ed Applicata (1923 -)
publication_identifier:
  issn:
  - 0373-3114
  - 1618-1891
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: How strong singularities can be regularized by logistic degradation in the
  Keller–Segel system?
type: journal_article
user_id: '31496'
volume: 198
year: '2019'
...
---
_id: '63367'
article_number: '48'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Does repulsion-type directional preference in chemotactic migration
    continue to regularize Keller–Segel systems when coupled to the Navier–Stokes
    equations? <i>Nonlinear Differential Equations and Applications NoDEA</i>. 2019;26(6).
    doi:<a href="https://doi.org/10.1007/s00030-019-0600-8">10.1007/s00030-019-0600-8</a>
  apa: Winkler, M. (2019). Does repulsion-type directional preference in chemotactic
    migration continue to regularize Keller–Segel systems when coupled to the Navier–Stokes
    equations? <i>Nonlinear Differential Equations and Applications NoDEA</i>, <i>26</i>(6),
    Article 48. <a href="https://doi.org/10.1007/s00030-019-0600-8">https://doi.org/10.1007/s00030-019-0600-8</a>
  bibtex: '@article{Winkler_2019, title={Does repulsion-type directional preference
    in chemotactic migration continue to regularize Keller–Segel systems when coupled
    to the Navier–Stokes equations?}, volume={26}, DOI={<a href="https://doi.org/10.1007/s00030-019-0600-8">10.1007/s00030-019-0600-8</a>},
    number={648}, journal={Nonlinear Differential Equations and Applications NoDEA},
    publisher={Springer Science and Business Media LLC}, author={Winkler, Michael},
    year={2019} }'
  chicago: Winkler, Michael. “Does Repulsion-Type Directional Preference in Chemotactic
    Migration Continue to Regularize Keller–Segel Systems When Coupled to the Navier–Stokes
    Equations?” <i>Nonlinear Differential Equations and Applications NoDEA</i> 26,
    no. 6 (2019). <a href="https://doi.org/10.1007/s00030-019-0600-8">https://doi.org/10.1007/s00030-019-0600-8</a>.
  ieee: 'M. Winkler, “Does repulsion-type directional preference in chemotactic migration
    continue to regularize Keller–Segel systems when coupled to the Navier–Stokes
    equations?,” <i>Nonlinear Differential Equations and Applications NoDEA</i>, vol.
    26, no. 6, Art. no. 48, 2019, doi: <a href="https://doi.org/10.1007/s00030-019-0600-8">10.1007/s00030-019-0600-8</a>.'
  mla: Winkler, Michael. “Does Repulsion-Type Directional Preference in Chemotactic
    Migration Continue to Regularize Keller–Segel Systems When Coupled to the Navier–Stokes
    Equations?” <i>Nonlinear Differential Equations and Applications NoDEA</i>, vol.
    26, no. 6, 48, Springer Science and Business Media LLC, 2019, doi:<a href="https://doi.org/10.1007/s00030-019-0600-8">10.1007/s00030-019-0600-8</a>.
  short: M. Winkler, Nonlinear Differential Equations and Applications NoDEA 26 (2019).
date_created: 2025-12-19T11:01:41Z
date_updated: 2025-12-19T11:01:47Z
doi: 10.1007/s00030-019-0600-8
intvolume: '        26'
issue: '6'
language:
- iso: eng
publication: Nonlinear Differential Equations and Applications NoDEA
publication_identifier:
  issn:
  - 1021-9722
  - 1420-9004
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Does repulsion-type directional preference in chemotactic migration continue
  to regularize Keller–Segel systems when coupled to the Navier–Stokes equations?
type: journal_article
user_id: '31496'
volume: 26
year: '2019'
...
---
_id: '63354'
author:
- first_name: Philippe
  full_name: Souplet, Philippe
  last_name: Souplet
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Souplet P, Winkler M. Blow-up Profiles for the Parabolic–Elliptic Keller–Segel
    System in Dimensions                                                         
             $${n\geq 3}$$                                                       
                        n                      ≥                      3. <i>Communications
    in Mathematical Physics</i>. 2018;367(2):665-681. doi:<a href="https://doi.org/10.1007/s00220-018-3238-1">10.1007/s00220-018-3238-1</a>
  apa: Souplet, P., &#38; Winkler, M. (2018). Blow-up Profiles for the Parabolic–Elliptic
    Keller–Segel System in Dimensions                                             
                         $${n\geq 3}$$                                           
                                    n                      ≥                     
    3. <i>Communications in Mathematical Physics</i>, <i>367</i>(2), 665–681. <a href="https://doi.org/10.1007/s00220-018-3238-1">https://doi.org/10.1007/s00220-018-3238-1</a>
  bibtex: '@article{Souplet_Winkler_2018, title={Blow-up Profiles for the Parabolic–Elliptic
    Keller–Segel System in Dimensions                                             
                         $${n\geq 3}$$                                           
                                    n                      ≥                     
    3}, volume={367}, DOI={<a href="https://doi.org/10.1007/s00220-018-3238-1">10.1007/s00220-018-3238-1</a>},
    number={2}, journal={Communications in Mathematical Physics}, publisher={Springer
    Science and Business Media LLC}, author={Souplet, Philippe and Winkler, Michael},
    year={2018}, pages={665–681} }'
  chicago: 'Souplet, Philippe, and Michael Winkler. “Blow-up Profiles for the Parabolic–Elliptic
    Keller–Segel System in Dimensions                                             
                         $${n\geq 3}$$                                           
                                    n                      ≥                     
    3.” <i>Communications in Mathematical Physics</i> 367, no. 2 (2018): 665–81. <a
    href="https://doi.org/10.1007/s00220-018-3238-1">https://doi.org/10.1007/s00220-018-3238-1</a>.'
  ieee: 'P. Souplet and M. Winkler, “Blow-up Profiles for the Parabolic–Elliptic Keller–Segel
    System in Dimensions                                                         
             $${n\geq 3}$$                                                       
                        n                      ≥                      3,” <i>Communications
    in Mathematical Physics</i>, vol. 367, no. 2, pp. 665–681, 2018, doi: <a href="https://doi.org/10.1007/s00220-018-3238-1">10.1007/s00220-018-3238-1</a>.'
  mla: Souplet, Philippe, and Michael Winkler. “Blow-up Profiles for the Parabolic–Elliptic
    Keller–Segel System in Dimensions                                             
                         $${n\geq 3}$$                                           
                                    n                      ≥                     
    3.” <i>Communications in Mathematical Physics</i>, vol. 367, no. 2, Springer Science
    and Business Media LLC, 2018, pp. 665–81, doi:<a href="https://doi.org/10.1007/s00220-018-3238-1">10.1007/s00220-018-3238-1</a>.
  short: P. Souplet, M. Winkler, Communications in Mathematical Physics 367 (2018)
    665–681.
date_created: 2025-12-19T10:52:55Z
date_updated: 2025-12-19T10:53:03Z
doi: 10.1007/s00220-018-3238-1
intvolume: '       367'
issue: '2'
language:
- iso: eng
page: 665-681
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Blow-up Profiles for the Parabolic–Elliptic Keller–Segel System in Dimensions                                                                   $${n\geq
  3}$$                                                                            n                      ≥                      3
type: journal_article
user_id: '31496'
volume: 367
year: '2018'
...
---
_id: '63361'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. How unstable is spatial homogeneity in Keller-Segel systems? A new
    critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases.
    <i>Mathematische Annalen</i>. 2018;373(3-4):1237-1282. doi:<a href="https://doi.org/10.1007/s00208-018-1722-8">10.1007/s00208-018-1722-8</a>
  apa: Winkler, M. (2018). How unstable is spatial homogeneity in Keller-Segel systems?
    A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic
    cases. <i>Mathematische Annalen</i>, <i>373</i>(3–4), 1237–1282. <a href="https://doi.org/10.1007/s00208-018-1722-8">https://doi.org/10.1007/s00208-018-1722-8</a>
  bibtex: '@article{Winkler_2018, title={How unstable is spatial homogeneity in Keller-Segel
    systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic
    cases}, volume={373}, DOI={<a href="https://doi.org/10.1007/s00208-018-1722-8">10.1007/s00208-018-1722-8</a>},
    number={3–4}, journal={Mathematische Annalen}, publisher={Springer Science and
    Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1237–1282}
    }'
  chicago: 'Winkler, Michael. “How Unstable Is Spatial Homogeneity in Keller-Segel
    Systems? A New Critical Mass Phenomenon in Two- and Higher-Dimensional Parabolic-Elliptic
    Cases.” <i>Mathematische Annalen</i> 373, no. 3–4 (2018): 1237–82. <a href="https://doi.org/10.1007/s00208-018-1722-8">https://doi.org/10.1007/s00208-018-1722-8</a>.'
  ieee: 'M. Winkler, “How unstable is spatial homogeneity in Keller-Segel systems?
    A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic
    cases,” <i>Mathematische Annalen</i>, vol. 373, no. 3–4, pp. 1237–1282, 2018,
    doi: <a href="https://doi.org/10.1007/s00208-018-1722-8">10.1007/s00208-018-1722-8</a>.'
  mla: Winkler, Michael. “How Unstable Is Spatial Homogeneity in Keller-Segel Systems?
    A New Critical Mass Phenomenon in Two- and Higher-Dimensional Parabolic-Elliptic
    Cases.” <i>Mathematische Annalen</i>, vol. 373, no. 3–4, Springer Science and
    Business Media LLC, 2018, pp. 1237–82, doi:<a href="https://doi.org/10.1007/s00208-018-1722-8">10.1007/s00208-018-1722-8</a>.
  short: M. Winkler, Mathematische Annalen 373 (2018) 1237–1282.
date_created: 2025-12-19T10:57:59Z
date_updated: 2025-12-19T10:58:06Z
doi: 10.1007/s00208-018-1722-8
intvolume: '       373'
issue: 3-4
language:
- iso: eng
page: 1237-1282
publication: Mathematische Annalen
publication_identifier:
  issn:
  - 0025-5831
  - 1432-1807
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: How unstable is spatial homogeneity in Keller-Segel systems? A new critical
  mass phenomenon in two- and higher-dimensional parabolic-elliptic cases
type: journal_article
user_id: '31496'
volume: 373
year: '2018'
...
---
_id: '63365'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global classical solvability and generic infinite-time blow-up in
    quasilinear Keller–Segel systems with bounded sensitivities. <i>Journal of Differential
    Equations</i>. 2018;266(12):8034-8066. doi:<a href="https://doi.org/10.1016/j.jde.2018.12.019">10.1016/j.jde.2018.12.019</a>
  apa: Winkler, M. (2018). Global classical solvability and generic infinite-time
    blow-up in quasilinear Keller–Segel systems with bounded sensitivities. <i>Journal
    of Differential Equations</i>, <i>266</i>(12), 8034–8066. <a href="https://doi.org/10.1016/j.jde.2018.12.019">https://doi.org/10.1016/j.jde.2018.12.019</a>
  bibtex: '@article{Winkler_2018, title={Global classical solvability and generic
    infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities},
    volume={266}, DOI={<a href="https://doi.org/10.1016/j.jde.2018.12.019">10.1016/j.jde.2018.12.019</a>},
    number={12}, journal={Journal of Differential Equations}, publisher={Elsevier
    BV}, author={Winkler, Michael}, year={2018}, pages={8034–8066} }'
  chicago: 'Winkler, Michael. “Global Classical Solvability and Generic Infinite-Time
    Blow-up in Quasilinear Keller–Segel Systems with Bounded Sensitivities.” <i>Journal
    of Differential Equations</i> 266, no. 12 (2018): 8034–66. <a href="https://doi.org/10.1016/j.jde.2018.12.019">https://doi.org/10.1016/j.jde.2018.12.019</a>.'
  ieee: 'M. Winkler, “Global classical solvability and generic infinite-time blow-up
    in quasilinear Keller–Segel systems with bounded sensitivities,” <i>Journal of
    Differential Equations</i>, vol. 266, no. 12, pp. 8034–8066, 2018, doi: <a href="https://doi.org/10.1016/j.jde.2018.12.019">10.1016/j.jde.2018.12.019</a>.'
  mla: Winkler, Michael. “Global Classical Solvability and Generic Infinite-Time Blow-up
    in Quasilinear Keller–Segel Systems with Bounded Sensitivities.” <i>Journal of
    Differential Equations</i>, vol. 266, no. 12, Elsevier BV, 2018, pp. 8034–66,
    doi:<a href="https://doi.org/10.1016/j.jde.2018.12.019">10.1016/j.jde.2018.12.019</a>.
  short: M. Winkler, Journal of Differential Equations 266 (2018) 8034–8066.
date_created: 2025-12-19T11:00:24Z
date_updated: 2025-12-19T11:00:31Z
doi: 10.1016/j.jde.2018.12.019
intvolume: '       266'
issue: '12'
language:
- iso: eng
page: 8034-8066
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global classical solvability and generic infinite-time blow-up in quasilinear
  Keller–Segel systems with bounded sensitivities
type: journal_article
user_id: '31496'
volume: 266
year: '2018'
...
---
_id: '63360'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. A three-dimensional Keller–Segel–Navier–Stokes system with logistic
    source: Global weak solutions and asymptotic stabilization. <i>Journal of Functional
    Analysis</i>. 2018;276(5):1339-1401. doi:<a href="https://doi.org/10.1016/j.jfa.2018.12.009">10.1016/j.jfa.2018.12.009</a>'
  apa: 'Winkler, M. (2018). A three-dimensional Keller–Segel–Navier–Stokes system
    with logistic source: Global weak solutions and asymptotic stabilization. <i>Journal
    of Functional Analysis</i>, <i>276</i>(5), 1339–1401. <a href="https://doi.org/10.1016/j.jfa.2018.12.009">https://doi.org/10.1016/j.jfa.2018.12.009</a>'
  bibtex: '@article{Winkler_2018, title={A three-dimensional Keller–Segel–Navier–Stokes
    system with logistic source: Global weak solutions and asymptotic stabilization},
    volume={276}, DOI={<a href="https://doi.org/10.1016/j.jfa.2018.12.009">10.1016/j.jfa.2018.12.009</a>},
    number={5}, journal={Journal of Functional Analysis}, publisher={Elsevier BV},
    author={Winkler, Michael}, year={2018}, pages={1339–1401} }'
  chicago: 'Winkler, Michael. “A Three-Dimensional Keller–Segel–Navier–Stokes System
    with Logistic Source: Global Weak Solutions and Asymptotic Stabilization.” <i>Journal
    of Functional Analysis</i> 276, no. 5 (2018): 1339–1401. <a href="https://doi.org/10.1016/j.jfa.2018.12.009">https://doi.org/10.1016/j.jfa.2018.12.009</a>.'
  ieee: 'M. Winkler, “A three-dimensional Keller–Segel–Navier–Stokes system with logistic
    source: Global weak solutions and asymptotic stabilization,” <i>Journal of Functional
    Analysis</i>, vol. 276, no. 5, pp. 1339–1401, 2018, doi: <a href="https://doi.org/10.1016/j.jfa.2018.12.009">10.1016/j.jfa.2018.12.009</a>.'
  mla: 'Winkler, Michael. “A Three-Dimensional Keller–Segel–Navier–Stokes System with
    Logistic Source: Global Weak Solutions and Asymptotic Stabilization.” <i>Journal
    of Functional Analysis</i>, vol. 276, no. 5, Elsevier BV, 2018, pp. 1339–401,
    doi:<a href="https://doi.org/10.1016/j.jfa.2018.12.009">10.1016/j.jfa.2018.12.009</a>.'
  short: M. Winkler, Journal of Functional Analysis 276 (2018) 1339–1401.
date_created: 2025-12-19T10:57:28Z
date_updated: 2025-12-19T10:57:36Z
doi: 10.1016/j.jfa.2018.12.009
intvolume: '       276'
issue: '5'
language:
- iso: eng
page: 1339-1401
publication: Journal of Functional Analysis
publication_identifier:
  issn:
  - 0022-1236
publication_status: published
publisher: Elsevier BV
status: public
title: 'A three-dimensional Keller–Segel–Navier–Stokes system with logistic source:
  Global weak solutions and asymptotic stabilization'
type: journal_article
user_id: '31496'
volume: 276
year: '2018'
...
---
_id: '63369'
abstract:
- lang: eng
  text: '<jats:p>The paper studies large time behaviour of solutions to the Keller–Segel
    system with quadratic degradation in a liquid environment, as given by</jats:p><jats:p><jats:disp-formula><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif"
    mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210518000057_equ01"
    /></jats:disp-formula></jats:p><jats:p>under Neumann boundary conditions in a
    bounded domain <jats:italic>Ω ⊂</jats:italic> ℝ<jats:sup><jats:italic>n</jats:italic></jats:sup>,
    where <jats:italic>n</jats:italic> ≥ 1 is arbitrary. It is shown that whenever
    <jats:italic>U</jats:italic> : <jats:italic>Ω ×</jats:italic> (0,<jats:italic>∞</jats:italic>)
    <jats:italic>→</jats:italic> ℝ<jats:sup><jats:italic>n</jats:italic></jats:sup>
    is a bounded and sufficiently regular solenoidal vector field any non-trivial
    global bounded solution of (<jats:italic>⋆</jats:italic>) approaches the trivial
    equilibrium at a rate that, with respect to the norm in either of the spaces <jats:italic>L</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Ω</jats:italic>)
    and <jats:italic>L<jats:sup>∞</jats:sup></jats:italic>(<jats:italic>Ω</jats:italic>),
    can be controlled from above and below by appropriate multiples of 1<jats:italic>/</jats:italic>(<jats:italic>t</jats:italic>
    + 1). This underlines that, even up to this quantitative level of accuracy, the
    large time behaviour in (<jats:italic>⋆</jats:italic>) is essentially independent
    not only of the particular fluid flow, but also of any effect originating from
    chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy
    problem, for which known results show that in the <jats:italic>n</jats:italic>
    = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing
    the respective spatial <jats:italic>L</jats:italic><jats:sup>1</jats:sup> norms
    of solutions.</jats:p>'
author:
- first_name: Xinru
  full_name: Cao, Xinru
  last_name: Cao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Cao X, Winkler M. Sharp decay estimates in a bioconvection model with quadratic
    degradation in bounded domains. <i>Proceedings of the Royal Society of Edinburgh:
    Section A Mathematics</i>. 2018;148(5):939-955. doi:<a href="https://doi.org/10.1017/s0308210518000057">10.1017/s0308210518000057</a>'
  apa: 'Cao, X., &#38; Winkler, M. (2018). Sharp decay estimates in a bioconvection
    model with quadratic degradation in bounded domains. <i>Proceedings of the Royal
    Society of Edinburgh: Section A Mathematics</i>, <i>148</i>(5), 939–955. <a href="https://doi.org/10.1017/s0308210518000057">https://doi.org/10.1017/s0308210518000057</a>'
  bibtex: '@article{Cao_Winkler_2018, title={Sharp decay estimates in a bioconvection
    model with quadratic degradation in bounded domains}, volume={148}, DOI={<a href="https://doi.org/10.1017/s0308210518000057">10.1017/s0308210518000057</a>},
    number={5}, journal={Proceedings of the Royal Society of Edinburgh: Section A
    Mathematics}, publisher={Cambridge University Press (CUP)}, author={Cao, Xinru
    and Winkler, Michael}, year={2018}, pages={939–955} }'
  chicago: 'Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection
    Model with Quadratic Degradation in Bounded Domains.” <i>Proceedings of the Royal
    Society of Edinburgh: Section A Mathematics</i> 148, no. 5 (2018): 939–55. <a
    href="https://doi.org/10.1017/s0308210518000057">https://doi.org/10.1017/s0308210518000057</a>.'
  ieee: 'X. Cao and M. Winkler, “Sharp decay estimates in a bioconvection model with
    quadratic degradation in bounded domains,” <i>Proceedings of the Royal Society
    of Edinburgh: Section A Mathematics</i>, vol. 148, no. 5, pp. 939–955, 2018, doi:
    <a href="https://doi.org/10.1017/s0308210518000057">10.1017/s0308210518000057</a>.'
  mla: 'Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection
    Model with Quadratic Degradation in Bounded Domains.” <i>Proceedings of the Royal
    Society of Edinburgh: Section A Mathematics</i>, vol. 148, no. 5, Cambridge University
    Press (CUP), 2018, pp. 939–55, doi:<a href="https://doi.org/10.1017/s0308210518000057">10.1017/s0308210518000057</a>.'
  short: 'X. Cao, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section
    A Mathematics 148 (2018) 939–955.'
date_created: 2025-12-19T11:02:55Z
date_updated: 2025-12-19T11:03:03Z
doi: 10.1017/s0308210518000057
intvolume: '       148'
issue: '5'
language:
- iso: eng
page: 939-955
publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics'
publication_identifier:
  issn:
  - 0308-2105
  - 1473-7124
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Sharp decay estimates in a bioconvection model with quadratic degradation in
  bounded domains
type: journal_article
user_id: '31496'
volume: 148
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. <i>Acta Applicandae Mathematicae</i>.
    2018;163(1):1-17. doi:<a href="https://doi.org/10.1007/s10440-018-0211-0">10.1007/s10440-018-0211-0</a>
  apa: Winkler, M. (2018). Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
    with Mildly Saturated Chemotactic Sensitivity. <i>Acta Applicandae Mathematicae</i>,
    <i>163</i>(1), 1–17. <a href="https://doi.org/10.1007/s10440-018-0211-0">https://doi.org/10.1007/s10440-018-0211-0</a>
  bibtex: '@article{Winkler_2018, title={Boundedness in a Chemotaxis-May-Nowak Model
    for Virus Dynamics with Mildly Saturated Chemotactic Sensitivity}, volume={163},
    DOI={<a href="https://doi.org/10.1007/s10440-018-0211-0">10.1007/s10440-018-0211-0</a>},
    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.” <i>Acta Applicandae Mathematicae</i>
    163, no. 1 (2018): 1–17. <a href="https://doi.org/10.1007/s10440-018-0211-0">https://doi.org/10.1007/s10440-018-0211-0</a>.'
  ieee: 'M. Winkler, “Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
    with Mildly Saturated Chemotactic Sensitivity,” <i>Acta Applicandae Mathematicae</i>,
    vol. 163, no. 1, pp. 1–17, 2018, doi: <a href="https://doi.org/10.1007/s10440-018-0211-0">10.1007/s10440-018-0211-0</a>.'
  mla: Winkler, Michael. “Boundedness in a Chemotaxis-May-Nowak Model for Virus Dynamics
    with Mildly Saturated Chemotactic Sensitivity.” <i>Acta Applicandae Mathematicae</i>,
    vol. 163, no. 1, Springer Science and Business Media LLC, 2018, pp. 1–17, doi:<a
    href="https://doi.org/10.1007/s10440-018-0211-0">10.1007/s10440-018-0211-0</a>.
  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: '63370'
author:
- first_name: Elio
  full_name: Espejo, Elio
  last_name: Espejo
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Espejo E, Winkler M. Global classical solvability and stabilization in a two-dimensional
    chemotaxis-Navier–Stokes system modeling coral fertilization. <i>Nonlinearity</i>.
    2018;31(4):1227-1259. doi:<a href="https://doi.org/10.1088/1361-6544/aa9d5f">10.1088/1361-6544/aa9d5f</a>
  apa: Espejo, E., &#38; Winkler, M. (2018). Global classical solvability and stabilization
    in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization.
    <i>Nonlinearity</i>, <i>31</i>(4), 1227–1259. <a href="https://doi.org/10.1088/1361-6544/aa9d5f">https://doi.org/10.1088/1361-6544/aa9d5f</a>
  bibtex: '@article{Espejo_Winkler_2018, title={Global classical solvability and stabilization
    in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization},
    volume={31}, DOI={<a href="https://doi.org/10.1088/1361-6544/aa9d5f">10.1088/1361-6544/aa9d5f</a>},
    number={4}, journal={Nonlinearity}, publisher={IOP Publishing}, author={Espejo,
    Elio and Winkler, Michael}, year={2018}, pages={1227–1259} }'
  chicago: 'Espejo, Elio, and Michael Winkler. “Global Classical Solvability and Stabilization
    in a Two-Dimensional Chemotaxis-Navier–Stokes System Modeling Coral Fertilization.”
    <i>Nonlinearity</i> 31, no. 4 (2018): 1227–59. <a href="https://doi.org/10.1088/1361-6544/aa9d5f">https://doi.org/10.1088/1361-6544/aa9d5f</a>.'
  ieee: 'E. Espejo and M. Winkler, “Global classical solvability and stabilization
    in a two-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization,”
    <i>Nonlinearity</i>, vol. 31, no. 4, pp. 1227–1259, 2018, doi: <a href="https://doi.org/10.1088/1361-6544/aa9d5f">10.1088/1361-6544/aa9d5f</a>.'
  mla: Espejo, Elio, and Michael Winkler. “Global Classical Solvability and Stabilization
    in a Two-Dimensional Chemotaxis-Navier–Stokes System Modeling Coral Fertilization.”
    <i>Nonlinearity</i>, vol. 31, no. 4, IOP Publishing, 2018, pp. 1227–59, doi:<a
    href="https://doi.org/10.1088/1361-6544/aa9d5f">10.1088/1361-6544/aa9d5f</a>.
  short: E. Espejo, M. Winkler, Nonlinearity 31 (2018) 1227–1259.
date_created: 2025-12-19T11:03:26Z
date_updated: 2025-12-19T11:03:32Z
doi: 10.1088/1361-6544/aa9d5f
intvolume: '        31'
issue: '4'
language:
- iso: eng
page: 1227-1259
publication: Nonlinearity
publication_identifier:
  issn:
  - 0951-7715
  - 1361-6544
publication_status: published
publisher: IOP Publishing
status: public
title: Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier–Stokes
  system modeling coral fertilization
type: journal_article
user_id: '31496'
volume: 31
year: '2018'
...
---
_id: '63377'
article_number: '40'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Finite-time blow-up in low-dimensional Keller–Segel systems with
    logistic-type superlinear degradation. <i>Zeitschrift für angewandte Mathematik
    und Physik</i>. 2018;69(2). doi:<a href="https://doi.org/10.1007/s00033-018-0935-8">10.1007/s00033-018-0935-8</a>
  apa: Winkler, M. (2018). Finite-time blow-up in low-dimensional Keller–Segel systems
    with logistic-type superlinear degradation. <i>Zeitschrift Für Angewandte Mathematik
    Und Physik</i>, <i>69</i>(2), Article 40. <a href="https://doi.org/10.1007/s00033-018-0935-8">https://doi.org/10.1007/s00033-018-0935-8</a>
  bibtex: '@article{Winkler_2018, title={Finite-time blow-up in low-dimensional Keller–Segel
    systems with logistic-type superlinear degradation}, volume={69}, DOI={<a href="https://doi.org/10.1007/s00033-018-0935-8">10.1007/s00033-018-0935-8</a>},
    number={240}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2018} }'
  chicago: Winkler, Michael. “Finite-Time Blow-up in Low-Dimensional Keller–Segel
    Systems with Logistic-Type Superlinear Degradation.” <i>Zeitschrift Für Angewandte
    Mathematik Und Physik</i> 69, no. 2 (2018). <a href="https://doi.org/10.1007/s00033-018-0935-8">https://doi.org/10.1007/s00033-018-0935-8</a>.
  ieee: 'M. Winkler, “Finite-time blow-up in low-dimensional Keller–Segel systems
    with logistic-type superlinear degradation,” <i>Zeitschrift für angewandte Mathematik
    und Physik</i>, vol. 69, no. 2, Art. no. 40, 2018, doi: <a href="https://doi.org/10.1007/s00033-018-0935-8">10.1007/s00033-018-0935-8</a>.'
  mla: Winkler, Michael. “Finite-Time Blow-up in Low-Dimensional Keller–Segel Systems
    with Logistic-Type Superlinear Degradation.” <i>Zeitschrift Für Angewandte Mathematik
    Und Physik</i>, vol. 69, no. 2, 40, Springer Science and Business Media LLC, 2018,
    doi:<a href="https://doi.org/10.1007/s00033-018-0935-8">10.1007/s00033-018-0935-8</a>.
  short: M. Winkler, Zeitschrift Für Angewandte Mathematik Und Physik 69 (2018).
date_created: 2025-12-19T11:06:58Z
date_updated: 2025-12-19T11:07:05Z
doi: 10.1007/s00033-018-0935-8
intvolume: '        69'
issue: '2'
language:
- iso: eng
publication: Zeitschrift für angewandte Mathematik und Physik
publication_identifier:
  issn:
  - 0044-2275
  - 1420-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type
  superlinear degradation
type: journal_article
user_id: '31496'
volume: 69
year: '2018'
...
---
_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? <i>Journal of Mathematical Fluid
    Mechanics</i>. 2018;20(4):1889-1909. doi:<a href="https://doi.org/10.1007/s00021-018-0395-0">10.1007/s00021-018-0395-0</a>
  apa: Winkler, M. (2018). Does Fluid Interaction Affect Regularity in the Three-Dimensional
    Keller–Segel System with Saturated Sensitivity? <i>Journal of Mathematical Fluid
    Mechanics</i>, <i>20</i>(4), 1889–1909. <a href="https://doi.org/10.1007/s00021-018-0395-0">https://doi.org/10.1007/s00021-018-0395-0</a>
  bibtex: '@article{Winkler_2018, title={Does Fluid Interaction Affect Regularity
    in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?}, volume={20},
    DOI={<a href="https://doi.org/10.1007/s00021-018-0395-0">10.1007/s00021-018-0395-0</a>},
    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?” <i>Journal of Mathematical Fluid
    Mechanics</i> 20, no. 4 (2018): 1889–1909. <a href="https://doi.org/10.1007/s00021-018-0395-0">https://doi.org/10.1007/s00021-018-0395-0</a>.'
  ieee: 'M. Winkler, “Does Fluid Interaction Affect Regularity in the Three-Dimensional
    Keller–Segel System with Saturated Sensitivity?,” <i>Journal of Mathematical Fluid
    Mechanics</i>, vol. 20, no. 4, pp. 1889–1909, 2018, doi: <a href="https://doi.org/10.1007/s00021-018-0395-0">10.1007/s00021-018-0395-0</a>.'
  mla: Winkler, Michael. “Does Fluid Interaction Affect Regularity in the Three-Dimensional
    Keller–Segel System with Saturated Sensitivity?” <i>Journal of Mathematical Fluid
    Mechanics</i>, vol. 20, no. 4, Springer Science and Business Media LLC, 2018,
    pp. 1889–909, doi:<a href="https://doi.org/10.1007/s00021-018-0395-0">10.1007/s00021-018-0395-0</a>.
  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'
...
---
_id: '63381'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes
    system with rotational flux components. <i>Journal of Evolution Equations</i>.
    2018;18(3):1267-1289. doi:<a href="https://doi.org/10.1007/s00028-018-0440-8">10.1007/s00028-018-0440-8</a>
  apa: Winkler, M. (2018). Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes
    system with rotational flux components. <i>Journal of Evolution Equations</i>,
    <i>18</i>(3), 1267–1289. <a href="https://doi.org/10.1007/s00028-018-0440-8">https://doi.org/10.1007/s00028-018-0440-8</a>
  bibtex: '@article{Winkler_2018, title={Global mass-preserving solutions in a two-dimensional
    chemotaxis-Stokes system with rotational flux components}, volume={18}, DOI={<a
    href="https://doi.org/10.1007/s00028-018-0440-8">10.1007/s00028-018-0440-8</a>},
    number={3}, journal={Journal of Evolution Equations}, publisher={Springer Science
    and Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1267–1289}
    }'
  chicago: 'Winkler, Michael. “Global Mass-Preserving Solutions in a Two-Dimensional
    Chemotaxis-Stokes System with Rotational Flux Components.” <i>Journal of Evolution
    Equations</i> 18, no. 3 (2018): 1267–89. <a href="https://doi.org/10.1007/s00028-018-0440-8">https://doi.org/10.1007/s00028-018-0440-8</a>.'
  ieee: 'M. Winkler, “Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes
    system with rotational flux components,” <i>Journal of Evolution Equations</i>,
    vol. 18, no. 3, pp. 1267–1289, 2018, doi: <a href="https://doi.org/10.1007/s00028-018-0440-8">10.1007/s00028-018-0440-8</a>.'
  mla: Winkler, Michael. “Global Mass-Preserving Solutions in a Two-Dimensional Chemotaxis-Stokes
    System with Rotational Flux Components.” <i>Journal of Evolution Equations</i>,
    vol. 18, no. 3, Springer Science and Business Media LLC, 2018, pp. 1267–89, doi:<a
    href="https://doi.org/10.1007/s00028-018-0440-8">10.1007/s00028-018-0440-8</a>.
  short: M. Winkler, Journal of Evolution Equations 18 (2018) 1267–1289.
date_created: 2025-12-19T11:08:43Z
date_updated: 2025-12-19T11:08:50Z
doi: 10.1007/s00028-018-0440-8
intvolume: '        18'
issue: '3'
language:
- iso: eng
page: 1267-1289
publication: Journal of Evolution Equations
publication_identifier:
  issn:
  - 1424-3199
  - 1424-3202
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system
  with rotational flux components
type: journal_article
user_id: '31496'
volume: 18
year: '2018'
...
---
_id: '63380'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Global existence and stabilization in a degenerate chemotaxis-Stokes
    system with mildly strong diffusion enhancement. <i>Journal of Differential Equations</i>.
    2018;264(10):6109-6151. doi:<a href="https://doi.org/10.1016/j.jde.2018.01.027">10.1016/j.jde.2018.01.027</a>
  apa: Winkler, M. (2018). Global existence and stabilization in a degenerate chemotaxis-Stokes
    system with mildly strong diffusion enhancement. <i>Journal of Differential Equations</i>,
    <i>264</i>(10), 6109–6151. <a href="https://doi.org/10.1016/j.jde.2018.01.027">https://doi.org/10.1016/j.jde.2018.01.027</a>
  bibtex: '@article{Winkler_2018, title={Global existence and stabilization in a degenerate
    chemotaxis-Stokes system with mildly strong diffusion enhancement}, volume={264},
    DOI={<a href="https://doi.org/10.1016/j.jde.2018.01.027">10.1016/j.jde.2018.01.027</a>},
    number={10}, journal={Journal of Differential Equations}, publisher={Elsevier
    BV}, author={Winkler, Michael}, year={2018}, pages={6109–6151} }'
  chicago: 'Winkler, Michael. “Global Existence and Stabilization in a Degenerate
    Chemotaxis-Stokes System with Mildly Strong Diffusion Enhancement.” <i>Journal
    of Differential Equations</i> 264, no. 10 (2018): 6109–51. <a href="https://doi.org/10.1016/j.jde.2018.01.027">https://doi.org/10.1016/j.jde.2018.01.027</a>.'
  ieee: 'M. Winkler, “Global existence and stabilization in a degenerate chemotaxis-Stokes
    system with mildly strong diffusion enhancement,” <i>Journal of Differential Equations</i>,
    vol. 264, no. 10, pp. 6109–6151, 2018, doi: <a href="https://doi.org/10.1016/j.jde.2018.01.027">10.1016/j.jde.2018.01.027</a>.'
  mla: Winkler, Michael. “Global Existence and Stabilization in a Degenerate Chemotaxis-Stokes
    System with Mildly Strong Diffusion Enhancement.” <i>Journal of Differential Equations</i>,
    vol. 264, no. 10, Elsevier BV, 2018, pp. 6109–51, doi:<a href="https://doi.org/10.1016/j.jde.2018.01.027">10.1016/j.jde.2018.01.027</a>.
  short: M. Winkler, Journal of Differential Equations 264 (2018) 6109–6151.
date_created: 2025-12-19T11:08:16Z
date_updated: 2025-12-19T11:08:22Z
doi: 10.1016/j.jde.2018.01.027
intvolume: '       264'
issue: '10'
language:
- iso: eng
page: 6109-6151
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Global existence and stabilization in a degenerate chemotaxis-Stokes system
  with mildly strong diffusion enhancement
type: journal_article
user_id: '31496'
volume: 264
year: '2018'
...
---
_id: '63376'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A critical blow-up exponent in a chemotaxis system with nonlinear
    signal production. <i>Nonlinearity</i>. 2018;31(5):2031-2056. doi:<a href="https://doi.org/10.1088/1361-6544/aaaa0e">10.1088/1361-6544/aaaa0e</a>
  apa: Winkler, M. (2018). A critical blow-up exponent in a chemotaxis system with
    nonlinear signal production. <i>Nonlinearity</i>, <i>31</i>(5), 2031–2056. <a
    href="https://doi.org/10.1088/1361-6544/aaaa0e">https://doi.org/10.1088/1361-6544/aaaa0e</a>
  bibtex: '@article{Winkler_2018, title={A critical blow-up exponent in a chemotaxis
    system with nonlinear signal production}, volume={31}, DOI={<a href="https://doi.org/10.1088/1361-6544/aaaa0e">10.1088/1361-6544/aaaa0e</a>},
    number={5}, journal={Nonlinearity}, publisher={IOP Publishing}, author={Winkler,
    Michael}, year={2018}, pages={2031–2056} }'
  chicago: 'Winkler, Michael. “A Critical Blow-up Exponent in a Chemotaxis System
    with Nonlinear Signal Production.” <i>Nonlinearity</i> 31, no. 5 (2018): 2031–56.
    <a href="https://doi.org/10.1088/1361-6544/aaaa0e">https://doi.org/10.1088/1361-6544/aaaa0e</a>.'
  ieee: 'M. Winkler, “A critical blow-up exponent in a chemotaxis system with nonlinear
    signal production,” <i>Nonlinearity</i>, vol. 31, no. 5, pp. 2031–2056, 2018,
    doi: <a href="https://doi.org/10.1088/1361-6544/aaaa0e">10.1088/1361-6544/aaaa0e</a>.'
  mla: Winkler, Michael. “A Critical Blow-up Exponent in a Chemotaxis System with
    Nonlinear Signal Production.” <i>Nonlinearity</i>, vol. 31, no. 5, IOP Publishing,
    2018, pp. 2031–56, doi:<a href="https://doi.org/10.1088/1361-6544/aaaa0e">10.1088/1361-6544/aaaa0e</a>.
  short: M. Winkler, Nonlinearity 31 (2018) 2031–2056.
date_created: 2025-12-19T11:06:33Z
date_updated: 2025-12-19T11:06:40Z
doi: 10.1088/1361-6544/aaaa0e
intvolume: '        31'
issue: '5'
language:
- iso: eng
page: 2031-2056
publication: Nonlinearity
publication_identifier:
  issn:
  - 0951-7715
  - 1361-6544
publication_status: published
publisher: IOP Publishing
status: public
title: A critical blow-up exponent in a chemotaxis system with nonlinear signal production
type: journal_article
user_id: '31496'
volume: 31
year: '2018'
...
---
_id: '63382'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
- first_name: Tomomi
  full_name: Yokota, Tomomi
  last_name: Yokota
citation:
  ama: Winkler M, Yokota T. Stabilization in the logarithmic Keller–Segel system.
    <i>Nonlinear Analysis</i>. 2018;170:123-141. doi:<a href="https://doi.org/10.1016/j.na.2018.01.002">10.1016/j.na.2018.01.002</a>
  apa: Winkler, M., &#38; Yokota, T. (2018). Stabilization in the logarithmic Keller–Segel
    system. <i>Nonlinear Analysis</i>, <i>170</i>, 123–141. <a href="https://doi.org/10.1016/j.na.2018.01.002">https://doi.org/10.1016/j.na.2018.01.002</a>
  bibtex: '@article{Winkler_Yokota_2018, title={Stabilization in the logarithmic Keller–Segel
    system}, volume={170}, DOI={<a href="https://doi.org/10.1016/j.na.2018.01.002">10.1016/j.na.2018.01.002</a>},
    journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Winkler, Michael
    and Yokota, Tomomi}, year={2018}, pages={123–141} }'
  chicago: 'Winkler, Michael, and Tomomi Yokota. “Stabilization in the Logarithmic
    Keller–Segel System.” <i>Nonlinear Analysis</i> 170 (2018): 123–41. <a href="https://doi.org/10.1016/j.na.2018.01.002">https://doi.org/10.1016/j.na.2018.01.002</a>.'
  ieee: 'M. Winkler and T. Yokota, “Stabilization in the logarithmic Keller–Segel
    system,” <i>Nonlinear Analysis</i>, vol. 170, pp. 123–141, 2018, doi: <a href="https://doi.org/10.1016/j.na.2018.01.002">10.1016/j.na.2018.01.002</a>.'
  mla: Winkler, Michael, and Tomomi Yokota. “Stabilization in the Logarithmic Keller–Segel
    System.” <i>Nonlinear Analysis</i>, vol. 170, Elsevier BV, 2018, pp. 123–41, doi:<a
    href="https://doi.org/10.1016/j.na.2018.01.002">10.1016/j.na.2018.01.002</a>.
  short: M. Winkler, T. Yokota, Nonlinear Analysis 170 (2018) 123–141.
date_created: 2025-12-19T11:09:11Z
date_updated: 2025-12-19T11:09:19Z
doi: 10.1016/j.na.2018.01.002
intvolume: '       170'
language:
- iso: eng
page: 123-141
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: Stabilization in the logarithmic Keller–Segel system
type: journal_article
user_id: '31496'
volume: 170
year: '2018'
...
---
_id: '63371'
abstract:
- lang: eng
  text: <jats:p>Adhesion between cells and other cells (cell–cell adhesion) or other
    tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon.
    Consequently, a number of recently developed mathematical models for cell adhesion
    have taken the form of non-local partial differential equations, where the non-local
    term arises inside a spatial derivative. The mathematical properties of such a
    non-local gradient term are not yet well understood. Here we use sophisticated
    estimation techniques to show local and global existence of classical solutions
    for such examples of adhesion-type models, and we provide a uniform upper bound
    for the solutions. Further, we discuss the significance of these results to applications
    in cell sorting and in cancer invasion and support the theoretical results through
    numerical simulations.</jats:p>
author:
- first_name: T.
  full_name: HILLEN, T.
  last_name: HILLEN
- first_name: K. J.
  full_name: PAINTER, K. J.
  last_name: PAINTER
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: HILLEN T, PAINTER KJ, Winkler M. Global solvability and explicit bounds for
    non-local adhesion models. <i>European Journal of Applied Mathematics</i>. 2017;29(4):645-684.
    doi:<a href="https://doi.org/10.1017/s0956792517000328">10.1017/s0956792517000328</a>
  apa: HILLEN, T., PAINTER, K. J., &#38; Winkler, M. (2017). Global solvability and
    explicit bounds for non-local adhesion models. <i>European Journal of Applied
    Mathematics</i>, <i>29</i>(4), 645–684. <a href="https://doi.org/10.1017/s0956792517000328">https://doi.org/10.1017/s0956792517000328</a>
  bibtex: '@article{HILLEN_PAINTER_Winkler_2017, title={Global solvability and explicit
    bounds for non-local adhesion models}, volume={29}, DOI={<a href="https://doi.org/10.1017/s0956792517000328">10.1017/s0956792517000328</a>},
    number={4}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
    University Press (CUP)}, author={HILLEN, T. and PAINTER, K. J. and Winkler, Michael},
    year={2017}, pages={645–684} }'
  chicago: 'HILLEN, T., K. J. PAINTER, and Michael Winkler. “Global Solvability and
    Explicit Bounds for Non-Local Adhesion Models.” <i>European Journal of Applied
    Mathematics</i> 29, no. 4 (2017): 645–84. <a href="https://doi.org/10.1017/s0956792517000328">https://doi.org/10.1017/s0956792517000328</a>.'
  ieee: 'T. HILLEN, K. J. PAINTER, and M. Winkler, “Global solvability and explicit
    bounds for non-local adhesion models,” <i>European Journal of Applied Mathematics</i>,
    vol. 29, no. 4, pp. 645–684, 2017, doi: <a href="https://doi.org/10.1017/s0956792517000328">10.1017/s0956792517000328</a>.'
  mla: HILLEN, T., et al. “Global Solvability and Explicit Bounds for Non-Local Adhesion
    Models.” <i>European Journal of Applied Mathematics</i>, vol. 29, no. 4, Cambridge
    University Press (CUP), 2017, pp. 645–84, doi:<a href="https://doi.org/10.1017/s0956792517000328">10.1017/s0956792517000328</a>.
  short: T. HILLEN, K.J. PAINTER, M. Winkler, European Journal of Applied Mathematics
    29 (2017) 645–684.
date_created: 2025-12-19T11:03:50Z
date_updated: 2025-12-19T11:03:57Z
doi: 10.1017/s0956792517000328
intvolume: '        29'
issue: '4'
language:
- iso: eng
page: 645-684
publication: European Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0956-7925
  - 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: Global solvability and explicit bounds for non-local adhesion models
type: journal_article
user_id: '31496'
volume: 29
year: '2017'
...