---
_id: '63139'
author:
- first_name: Srikanth B.
  full_name: Iyengar, Srikanth B.
  last_name: Iyengar
- first_name: Janina Carmen
  full_name: Letz, Janina Carmen
  id: '121953'
  last_name: Letz
  orcid: 0000-0002-5497-8296
- first_name: Jian
  full_name: Liu, Jian
  last_name: Liu
- first_name: Josh
  full_name: Pollitz, Josh
  last_name: Pollitz
citation:
  ama: Iyengar SB, Letz JC, Liu J, Pollitz J. Exceptional complete intersection maps
    of local rings. <i>Pacific J Math</i>. 2022;318(2):275-293. doi:<a href="https://doi.org/10.2140/pjm.2022.318.275">10.2140/pjm.2022.318.275</a>
  apa: Iyengar, S. B., Letz, J. C., Liu, J., &#38; Pollitz, J. (2022). Exceptional
    complete intersection maps of local rings. <i>Pacific J. Math.</i>, <i>318</i>(2),
    275–293. <a href="https://doi.org/10.2140/pjm.2022.318.275">https://doi.org/10.2140/pjm.2022.318.275</a>
  bibtex: '@article{Iyengar_Letz_Liu_Pollitz_2022, title={Exceptional complete intersection
    maps of local rings}, volume={318}, DOI={<a href="https://doi.org/10.2140/pjm.2022.318.275">10.2140/pjm.2022.318.275</a>},
    number={2}, journal={Pacific J. Math.}, author={Iyengar, Srikanth B. and Letz,
    Janina Carmen and Liu, Jian and Pollitz, Josh}, year={2022}, pages={275–293} }'
  chicago: 'Iyengar, Srikanth B., Janina Carmen Letz, Jian Liu, and Josh Pollitz.
    “Exceptional Complete Intersection Maps of Local Rings.” <i>Pacific J. Math.</i>
    318, no. 2 (2022): 275–93. <a href="https://doi.org/10.2140/pjm.2022.318.275">https://doi.org/10.2140/pjm.2022.318.275</a>.'
  ieee: 'S. B. Iyengar, J. C. Letz, J. Liu, and J. Pollitz, “Exceptional complete
    intersection maps of local rings,” <i>Pacific J. Math.</i>, vol. 318, no. 2, pp.
    275–293, 2022, doi: <a href="https://doi.org/10.2140/pjm.2022.318.275">10.2140/pjm.2022.318.275</a>.'
  mla: Iyengar, Srikanth B., et al. “Exceptional Complete Intersection Maps of Local
    Rings.” <i>Pacific J. Math.</i>, vol. 318, no. 2, 2022, pp. 275–93, doi:<a href="https://doi.org/10.2140/pjm.2022.318.275">10.2140/pjm.2022.318.275</a>.
  short: S.B. Iyengar, J.C. Letz, J. Liu, J. Pollitz, Pacific J. Math. 318 (2022)
    275–293.
date_created: 2025-12-16T14:28:18Z
date_updated: 2025-12-16T14:44:02Z
doi: 10.2140/pjm.2022.318.275
extern: '1'
intvolume: '       318'
issue: '2'
language:
- iso: eng
page: 275-293
publication: Pacific J. Math.
publication_identifier:
  issn:
  - 0030-8730
status: public
title: Exceptional complete intersection maps of local rings
type: journal_article
user_id: '121953'
volume: 318
year: '2022'
...
---
_id: '55135'
alternative_title:
- 'Themenheft zur 20. Jahrestagung der Gesellschaft für Aphasieforschung und –behandlung.
  Neurologie & Rehabilitation. '
citation:
  ama: Jonas K, Quinting J, Gerhards L, et al., eds. <i>Kommunikation – Sprache, Emotion,
    Kognition</i>.; 2022.
  apa: Jonas, K., Quinting, J., Gerhards, L., Hüsgen, A., Rubi-Fessen, I., Stenneken,
    P., &#38; Rosenkranz, A. (Eds.). (2022). <i>Kommunikation – Sprache, Emotion,
    Kognition</i>.
  bibtex: '@book{Jonas_Quinting_Gerhards_Hüsgen_Rubi-Fessen_Stenneken_Rosenkranz_2022,
    title={Kommunikation – Sprache, Emotion, Kognition}, year={2022} }'
  chicago: Jonas, Kristina, Jana Quinting, Lisa Gerhards, Anne Hüsgen, Ilona Rubi-Fessen,
    Prisca Stenneken, and Anna Rosenkranz, eds. <i>Kommunikation – Sprache, Emotion,
    Kognition</i>, 2022.
  ieee: K. Jonas <i>et al.</i>, Eds., <i>Kommunikation – Sprache, Emotion, Kognition</i>.
    2022.
  mla: Jonas, Kristina, et al., editors. <i>Kommunikation – Sprache, Emotion, Kognition</i>.
    2022.
  short: K. Jonas, J. Quinting, L. Gerhards, A. Hüsgen, I. Rubi-Fessen, P. Stenneken,
    A. Rosenkranz, eds., Kommunikation – Sprache, Emotion, Kognition, 2022.
date_created: 2024-07-08T11:24:56Z
date_updated: 2025-12-17T07:34:41Z
department:
- _id: '5'
- _id: '890'
editor:
- first_name: Kristina
  full_name: Jonas, Kristina
  id: '94540'
  last_name: Jonas
  orcid: 0000-0002-1067-9139
- first_name: Jana
  full_name: Quinting, Jana
  last_name: Quinting
- first_name: Lisa
  full_name: Gerhards, Lisa
  last_name: Gerhards
- first_name: Anne
  full_name: Hüsgen, Anne
  last_name: Hüsgen
- first_name: Ilona
  full_name: Rubi-Fessen, Ilona
  last_name: Rubi-Fessen
- first_name: Prisca
  full_name: Stenneken, Prisca
  last_name: Stenneken
- first_name: Anna
  full_name: Rosenkranz, Anna
  last_name: Rosenkranz
extern: '1'
language:
- iso: eng
status: public
title: Kommunikation – Sprache, Emotion, Kognition
type: conference_editor
user_id: '94540'
year: '2022'
...
---
_id: '63175'
author:
- first_name: S
  full_name: Barlovits, S
  last_name: Barlovits
- first_name: A
  full_name: Caldeira, A
  last_name: Caldeira
- first_name: G
  full_name: Fesakis, G
  last_name: Fesakis
- first_name: S
  full_name: Jablonski, S
  last_name: Jablonski
- first_name: DK
  full_name: Filippaki, DK
  last_name: Filippaki
- first_name: C
  full_name: Lázaro, C
  last_name: Lázaro
- first_name: M
  full_name: Ludwig, M
  last_name: Ludwig
- first_name: MF
  full_name: Mammana, MF
  last_name: Mammana
- first_name: A
  full_name: Moura, A
  last_name: Moura
- first_name: DXK
  full_name: Oehler, DXK
  last_name: Oehler
- first_name: T
  full_name: Recio, T
  last_name: Recio
- first_name: E
  full_name: Taranto, E
  last_name: Taranto
- first_name: S
  full_name: Volika, S
  last_name: Volika
citation:
  ama: 'Barlovits S, Caldeira A, Fesakis G, et al. Adaptive, Synchronous, and Mobile
    Online Education: Developing the ASYMPTOTE Learning Environment. 2022;10(10).'
  apa: 'Barlovits, S., Caldeira, A., Fesakis, G., Jablonski, S., Filippaki, D., Lázaro,
    C., Ludwig, M., Mammana, M., Moura, A., Oehler, D., Recio, T., Taranto, E., &#38;
    Volika, S. (2022). <i>Adaptive, Synchronous, and Mobile Online Education: Developing
    the ASYMPTOTE Learning Environment</i>. <i>10</i>(10).'
  bibtex: '@article{Barlovits_Caldeira_Fesakis_Jablonski_Filippaki_Lázaro_Ludwig_Mammana_Moura_Oehler_et
    al._2022, title={Adaptive, Synchronous, and Mobile Online Education: Developing
    the ASYMPTOTE Learning Environment}, volume={10}, number={10}, author={Barlovits,
    S and Caldeira, A and Fesakis, G and Jablonski, S and Filippaki, DK and Lázaro,
    C and Ludwig, M and Mammana, MF and Moura, A and Oehler, DXK and et al.}, year={2022}
    }'
  chicago: 'Barlovits, S, A Caldeira, G Fesakis, S Jablonski, DK Filippaki, C Lázaro,
    M Ludwig, et al. “Adaptive, Synchronous, and Mobile Online Education: Developing
    the ASYMPTOTE Learning Environment” 10, no. 10 (2022).'
  ieee: 'S. Barlovits <i>et al.</i>, “Adaptive, Synchronous, and Mobile Online Education:
    Developing the ASYMPTOTE Learning Environment,” vol. 10, no. 10, 2022.'
  mla: 'Barlovits, S., et al. <i>Adaptive, Synchronous, and Mobile Online Education:
    Developing the ASYMPTOTE Learning Environment</i>. no. 10, 2022.'
  short: S. Barlovits, A. Caldeira, G. Fesakis, S. Jablonski, D. Filippaki, C. Lázaro,
    M. Ludwig, M. Mammana, A. Moura, D. Oehler, T. Recio, E. Taranto, S. Volika, 10
    (2022).
date_created: 2025-12-17T08:53:34Z
date_updated: 2025-12-17T08:56:18Z
intvolume: '        10'
issue: '10'
publication_identifier:
  issn:
  - 2227-7390
publication_status: published
quality_controlled: '1'
status: public
title: 'Adaptive, Synchronous, and Mobile Online Education: Developing the ASYMPTOTE
  Learning Environment'
type: journal_article
user_id: '111489'
volume: 10
year: '2022'
...
---
_id: '63174'
author:
- first_name: A
  full_name: Barbosa, A
  last_name: Barbosa
- first_name: I
  full_name: Vale, I
  last_name: Vale
- first_name: S
  full_name: Jablonski, S
  last_name: Jablonski
- first_name: M
  full_name: Ludwig, M
  last_name: Ludwig
citation:
  ama: Barbosa A, Vale I, Jablonski S, Ludwig M. Walking through Algebraic Thinking
    with Theme-Based (Mobile) Math Trails. 2022;12(5).
  apa: Barbosa, A., Vale, I., Jablonski, S., &#38; Ludwig, M. (2022). <i>Walking through
    Algebraic Thinking with Theme-Based (Mobile) Math Trails</i>. <i>12</i>(5).
  bibtex: '@article{Barbosa_Vale_Jablonski_Ludwig_2022, title={Walking through Algebraic
    Thinking with Theme-Based (Mobile) Math Trails}, volume={12}, number={5}, author={Barbosa,
    A and Vale, I and Jablonski, S and Ludwig, M}, year={2022} }'
  chicago: Barbosa, A, I Vale, S Jablonski, and M Ludwig. “Walking through Algebraic
    Thinking with Theme-Based (Mobile) Math Trails” 12, no. 5 (2022).
  ieee: A. Barbosa, I. Vale, S. Jablonski, and M. Ludwig, “Walking through Algebraic
    Thinking with Theme-Based (Mobile) Math Trails,” vol. 12, no. 5, 2022.
  mla: Barbosa, A., et al. <i>Walking through Algebraic Thinking with Theme-Based
    (Mobile) Math Trails</i>. no. 5, 2022.
  short: A. Barbosa, I. Vale, S. Jablonski, M. Ludwig, 12 (2022).
date_created: 2025-12-17T08:53:34Z
date_updated: 2025-12-17T08:56:22Z
intvolume: '        12'
issue: '5'
publication_identifier:
  issn:
  - 2227-7102
publication_status: published
quality_controlled: '1'
status: public
title: Walking through Algebraic Thinking with Theme-Based (Mobile) Math Trails
type: journal_article
user_id: '111489'
volume: 12
year: '2022'
...
---
_id: '36424'
abstract:
- lang: eng
  text: "Das Oratorium kann als wichtige Gattung für die Analyse des Zusammenspiels
    von Religion und Politik im Medium der Musik gelten. Die Studie wendet sich dem
    Verhältnis dieser religiösen Musikform zum deutschen Nationalismus vom Beginn
    des Ersten bis zum Ende des Zweiten Weltkriegs zu. Im Fokus stehen verschiedene
    Weisen der Instrumentalisierung von Musik, etwa zur ‚Heldenehrung‘, zur Bildung
    einer ‚vorgestellten Gemeinschaft‘ oder zur Gewinnung der Arbeiterschaft. Bei
    der Analyse der Politisierung des Oratoriums sind vier Beobachtungsperspektiven
    leitend: Konfession, Säkularisierung, Erinnerungskultur und Vergemeinschaftung.\r\n\r\nEntsprechend
    liegt der Schwerpunkt der Studie auf der Analyse der sozialgeschichtlichen Funktion
    der Gattung. Daneben jedoch werden – in Form von Fallstudien zu bisher unerforschten
    Werken – auch musikalische Detailanalysen durchgeführt, die das in der jeweiligen
    Komposition realisierte Verhältnis von religiösen und nationalistischen Elementen
    offenlegen. "
author:
- first_name: Dominik
  full_name: Höink, Dominik
  id: '90389'
  last_name: Höink
citation:
  ama: Höink D. <i>Oratorium und Nation (1914–1945) </i>. Vol 8. Waxmann Verlag; 2022.
  apa: Höink, D. (2022). <i>Oratorium und Nation (1914–1945) </i> (Vol. 8). Waxmann
    Verlag.
  bibtex: '@book{Höink_2022, place={Münster}, series={ Münsteraner Schriften zur zeitgenössischen
    Musik}, title={Oratorium und Nation (1914–1945) }, volume={8}, publisher={Waxmann
    Verlag}, author={Höink, Dominik}, year={2022}, collection={ Münsteraner Schriften
    zur zeitgenössischen Musik} }'
  chicago: 'Höink, Dominik. <i>Oratorium und Nation (1914–1945) </i>. Vol. 8.  Münsteraner
    Schriften zur zeitgenössischen Musik. Münster: Waxmann Verlag, 2022.'
  ieee: 'D. Höink, <i>Oratorium und Nation (1914–1945) </i>, vol. 8. Münster: Waxmann
    Verlag, 2022.'
  mla: Höink, Dominik. <i>Oratorium und Nation (1914–1945) </i>. Waxmann Verlag, 2022.
  short: D. Höink, Oratorium und Nation (1914–1945) , Waxmann Verlag, Münster, 2022.
date_created: 2023-01-12T12:43:47Z
date_updated: 2025-12-17T09:00:36Z
department:
- _id: '233'
- _id: '716'
intvolume: '         8'
language:
- iso: ger
page: '590'
place: Münster
publication_identifier:
  isbn:
  - 978-3-8309-3984-9
publication_status: published
publisher: Waxmann Verlag
series_title: ' Münsteraner Schriften zur zeitgenössischen Musik'
status: public
title: 'Oratorium und Nation (1914–1945) '
type: book
user_id: '90389'
volume: 8
year: '2022'
...
---
_id: '34705'
abstract:
- lang: eng
  text: n 1789, Eberhard repudiated Kant’s claim expressed in the first edition of
    his Critique of Pure Reason to have delivered a new, namely transcendental turn
    in philosophy, as he was able to retrace our cognition to the origin of phenomena
    instead of delivering a “merely logical deduction”. Eberhard holds that there
    was nothing new, but all delivered in Leibniz and Wolff; to prove his claim he
    refers to a quote from Du Châtelet, taken from a paragraph where she determines
    the right understanding as to be able “to penetrate to the origin of phenomena”.
    This paper brings Du Châtelet into discourse with Kant via this Eberhard quote.
    In its first part, it investigates the use of her quote in the Kant-Eberhard controversy.
    The second part serves to ground the quote in Du Châtelet’s epistemology. It lays
    out how to understand Du Châtelet’s claim to penetrate to the origin of phenomena.
    Du Châtelet’s claim to have renewed philosophy must be taken seriously, and it
    is helpful for rethinking the German philosophical development from the rationalists
    to Kant through including Du Châtelet’s theory of cognition.
author:
- first_name: Ruth Edith
  full_name: Hagengruber, Ruth Edith
  id: '198'
  last_name: Hagengruber
  orcid: https://orcid.org/0000-0003-3360-6335
citation:
  ama: 'Hagengruber RE. Du Châtelet and Kant: Claiming the Renewal of Philosophy.
    In: Hagengruber RE, ed. <i>Époque Émilienne Philosophy and Science in the Age
    of Émilie Du Châtelet (1706-1749)</i>. Vol 10. Women in the History of Philosophy
    and Science . Springer International Publishing; 2022:57-84. doi:<a href="https://doi.org/10.1007/978-3-030-89921-9_3">10.1007/978-3-030-89921-9_3</a>'
  apa: 'Hagengruber, R. E. (2022). Du Châtelet and Kant: Claiming the Renewal of Philosophy.
    In R. E. Hagengruber (Ed.), <i>Époque Émilienne Philosophy and Science in the
    Age of Émilie Du Châtelet (1706-1749)</i> (Vol. 10, pp. 57–84). Springer International
    Publishing. <a href="https://doi.org/10.1007/978-3-030-89921-9_3">https://doi.org/10.1007/978-3-030-89921-9_3</a>'
  bibtex: '@inbook{Hagengruber_2022, place={Cham}, series={Women in the History of
    Philosophy and Science }, title={Du Châtelet and Kant: Claiming the Renewal of
    Philosophy}, volume={10}, DOI={<a href="https://doi.org/10.1007/978-3-030-89921-9_3">10.1007/978-3-030-89921-9_3</a>},
    booktitle={Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
    (1706-1749)}, publisher={Springer International Publishing}, author={Hagengruber,
    Ruth Edith}, editor={Hagengruber, Ruth Edith}, year={2022}, pages={57–84}, collection={Women
    in the History of Philosophy and Science } }'
  chicago: 'Hagengruber, Ruth Edith. “Du Châtelet and Kant: Claiming the Renewal of
    Philosophy.” In <i>Époque Émilienne Philosophy and Science in the Age of Émilie
    Du Châtelet (1706-1749)</i>, edited by Ruth Edith Hagengruber, 10:57–84. Women
    in the History of Philosophy and Science . Cham: Springer International Publishing,
    2022. <a href="https://doi.org/10.1007/978-3-030-89921-9_3">https://doi.org/10.1007/978-3-030-89921-9_3</a>.'
  ieee: 'R. E. Hagengruber, “Du Châtelet and Kant: Claiming the Renewal of Philosophy,”
    in <i>Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
    (1706-1749)</i>, vol. 10, R. E. Hagengruber, Ed. Cham: Springer International
    Publishing, 2022, pp. 57–84.'
  mla: 'Hagengruber, Ruth Edith. “Du Châtelet and Kant: Claiming the Renewal of Philosophy.”
    <i>Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet (1706-1749)</i>,
    edited by Ruth Edith Hagengruber, vol. 10, Springer International Publishing,
    2022, pp. 57–84, doi:<a href="https://doi.org/10.1007/978-3-030-89921-9_3">10.1007/978-3-030-89921-9_3</a>.'
  short: 'R.E. Hagengruber, in: R.E. Hagengruber (Ed.), Époque Émilienne Philosophy
    and Science in the Age of Émilie Du Châtelet (1706-1749), Springer International
    Publishing, Cham, 2022, pp. 57–84.'
date_created: 2022-12-21T11:35:14Z
date_updated: 2025-12-17T13:40:24Z
department:
- _id: '615'
- _id: '14'
- _id: '519'
doi: 10.1007/978-3-030-89921-9_3
editor:
- first_name: Ruth Edith
  full_name: Hagengruber, Ruth Edith
  last_name: Hagengruber
intvolume: '        10'
keyword:
- Émilie Du Châtelet
- History of Science
- Newton
- Kant
- Eberhard
- Wolff
- Leibniz
language:
- iso: eng
page: 57-84
place: Cham
publication: Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
  (1706-1749)
publication_identifier:
  isbn:
  - '9783030899202'
  - '9783030899219'
  issn:
  - 2523-8760
  - 2523-8779
publication_status: published
publisher: Springer International Publishing
quality_controlled: '1'
series_title: 'Women in the History of Philosophy and Science '
status: public
title: 'Du Châtelet and Kant: Claiming the Renewal of Philosophy'
type: book_chapter
user_id: '91827'
volume: 10
year: '2022'
...
---
_id: '63184'
author:
- first_name: Ruth
  full_name: Hagengruber, Ruth
  id: '198'
  last_name: Hagengruber
  orcid: https://orcid.org/0000-0003-3360-6335
citation:
  ama: 'Hagengruber R. An Introduction to the Volume. In: Hagengruber RE, ed. <i>Époque
    Émilienne Philosophy and Science in the Age of Émilie Du Châtelet (1706-1749)</i>.
    Vol 10. Women in the History of Philosophy and Science . Springer International
    Publishing; 2022:1-20. doi:<a href="https://doi.org/10.1007/978-3-030-89921-9_1">10.1007/978-3-030-89921-9_1</a>'
  apa: Hagengruber, R. (2022). An Introduction to the Volume. In R. E. Hagengruber
    (Ed.), <i>Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
    (1706-1749)</i> (Vol. 10, pp. 1–20). Springer International Publishing. <a href="https://doi.org/10.1007/978-3-030-89921-9_1">https://doi.org/10.1007/978-3-030-89921-9_1</a>
  bibtex: '@inbook{Hagengruber_2022, place={Cham}, series={Women in the History of
    Philosophy and Science }, title={An Introduction to the Volume}, volume={10},
    DOI={<a href="https://doi.org/10.1007/978-3-030-89921-9_1">10.1007/978-3-030-89921-9_1</a>},
    booktitle={Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
    (1706-1749)}, publisher={Springer International Publishing}, author={Hagengruber,
    Ruth}, editor={Hagengruber, Ruth Edith}, year={2022}, pages={1–20}, collection={Women
    in the History of Philosophy and Science } }'
  chicago: 'Hagengruber, Ruth. “An Introduction to the Volume.” In <i>Époque Émilienne
    Philosophy and Science in the Age of Émilie Du Châtelet (1706-1749)</i>, edited
    by Ruth Edith Hagengruber, 10:1–20. Women in the History of Philosophy and Science
    . Cham: Springer International Publishing, 2022. <a href="https://doi.org/10.1007/978-3-030-89921-9_1">https://doi.org/10.1007/978-3-030-89921-9_1</a>.'
  ieee: 'R. Hagengruber, “An Introduction to the Volume,” in <i>Époque Émilienne Philosophy
    and Science in the Age of Émilie Du Châtelet (1706-1749)</i>, vol. 10, R. E. Hagengruber,
    Ed. Cham: Springer International Publishing, 2022, pp. 1–20.'
  mla: Hagengruber, Ruth. “An Introduction to the Volume.” <i>Époque Émilienne Philosophy
    and Science in the Age of Émilie Du Châtelet (1706-1749)</i>, edited by Ruth Edith
    Hagengruber, vol. 10, Springer International Publishing, 2022, pp. 1–20, doi:<a
    href="https://doi.org/10.1007/978-3-030-89921-9_1">10.1007/978-3-030-89921-9_1</a>.
  short: 'R. Hagengruber, in: R.E. Hagengruber (Ed.), Époque Émilienne Philosophy
    and Science in the Age of Émilie Du Châtelet (1706-1749), Springer International
    Publishing, Cham, 2022, pp. 1–20.'
date_created: 2025-12-17T13:29:05Z
date_updated: 2025-12-17T13:38:49Z
department:
- _id: '14'
doi: 10.1007/978-3-030-89921-9_1
editor:
- first_name: Ruth Edith
  full_name: Hagengruber, Ruth Edith
  last_name: Hagengruber
intvolume: '        10'
language:
- iso: eng
page: 1-20
place: Cham
publication: Époque Émilienne Philosophy and Science in the Age of Émilie Du Châtelet
  (1706-1749)
publication_identifier:
  isbn:
  - '9783030899202'
  - '9783030899219'
  issn:
  - 2523-8760
  - 2523-8779
publication_status: published
publisher: Springer International Publishing
series_title: 'Women in the History of Philosophy and Science '
status: public
title: An Introduction to the Volume
type: book_chapter
user_id: '91827'
volume: 10
year: '2022'
...
---
_id: '63206'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>Pure iron is very attractive as a
    biodegradable implant material due to its high biocompatibility. In combination
    with additive manufacturing, which facilitates great flexibility of the implant
    design, it is possible to selectively adjust the microstructure of the material
    in the process, thereby control the corrosion and fatigue behavior. In the present
    study, conventional hot-rolled (HR) pure iron is compared to pure iron manufactured
    by electron beam melting (EBM). The microstructure, the corrosion behavior and
    the fatigue properties were studied comprehensively. The investigated sample conditions
    showed significant differences in the microstructures that led to changes in corrosion
    and fatigue properties. The EBM iron showed significantly lower fatigue strength
    compared to the HR iron. These different fatigue responses were observed under
    purely mechanical loading as well as with superimposed corrosion influence and
    are summarized in a model that describes the underlying failure mechanisms.</jats:p>
article_number: '18'
author:
- first_name: Steffen
  full_name: Wackenrohr, Steffen
  last_name: Wackenrohr
- first_name: Christof Johannes Jaime
  full_name: Torrent, Christof Johannes Jaime
  last_name: Torrent
- first_name: Sebastian
  full_name: Herbst, Sebastian
  last_name: Herbst
- first_name: Florian
  full_name: Nürnberger, Florian
  last_name: Nürnberger
- first_name: Philipp
  full_name: Krooss, Philipp
  last_name: Krooss
- first_name: Christoph
  full_name: Ebbert, Christoph
  id: '7266'
  last_name: Ebbert
- first_name: Markus
  full_name: Voigt, Markus
  id: '15182'
  last_name: Voigt
- first_name: Guido
  full_name: Grundmeier, Guido
  id: '194'
  last_name: Grundmeier
- first_name: Thomas
  full_name: Niendorf, Thomas
  last_name: Niendorf
- first_name: Hans Jürgen
  full_name: Maier, Hans Jürgen
  last_name: Maier
citation:
  ama: Wackenrohr S, Torrent CJJ, Herbst S, et al. Corrosion fatigue behavior of electron
    beam melted iron in simulated body fluid. <i>npj Materials Degradation</i>. 2022;6(1).
    doi:<a href="https://doi.org/10.1038/s41529-022-00226-4">10.1038/s41529-022-00226-4</a>
  apa: Wackenrohr, S., Torrent, C. J. J., Herbst, S., Nürnberger, F., Krooss, P.,
    Ebbert, C., Voigt, M., Grundmeier, G., Niendorf, T., &#38; Maier, H. J. (2022).
    Corrosion fatigue behavior of electron beam melted iron in simulated body fluid.
    <i>Npj Materials Degradation</i>, <i>6</i>(1), Article 18. <a href="https://doi.org/10.1038/s41529-022-00226-4">https://doi.org/10.1038/s41529-022-00226-4</a>
  bibtex: '@article{Wackenrohr_Torrent_Herbst_Nürnberger_Krooss_Ebbert_Voigt_Grundmeier_Niendorf_Maier_2022,
    title={Corrosion fatigue behavior of electron beam melted iron in simulated body
    fluid}, volume={6}, DOI={<a href="https://doi.org/10.1038/s41529-022-00226-4">10.1038/s41529-022-00226-4</a>},
    number={118}, journal={npj Materials Degradation}, publisher={Springer Science
    and Business Media LLC}, author={Wackenrohr, Steffen and Torrent, Christof Johannes
    Jaime and Herbst, Sebastian and Nürnberger, Florian and Krooss, Philipp and Ebbert,
    Christoph and Voigt, Markus and Grundmeier, Guido and Niendorf, Thomas and Maier,
    Hans Jürgen}, year={2022} }'
  chicago: Wackenrohr, Steffen, Christof Johannes Jaime Torrent, Sebastian Herbst,
    Florian Nürnberger, Philipp Krooss, Christoph Ebbert, Markus Voigt, Guido Grundmeier,
    Thomas Niendorf, and Hans Jürgen Maier. “Corrosion Fatigue Behavior of Electron
    Beam Melted Iron in Simulated Body Fluid.” <i>Npj Materials Degradation</i> 6,
    no. 1 (2022). <a href="https://doi.org/10.1038/s41529-022-00226-4">https://doi.org/10.1038/s41529-022-00226-4</a>.
  ieee: 'S. Wackenrohr <i>et al.</i>, “Corrosion fatigue behavior of electron beam
    melted iron in simulated body fluid,” <i>npj Materials Degradation</i>, vol. 6,
    no. 1, Art. no. 18, 2022, doi: <a href="https://doi.org/10.1038/s41529-022-00226-4">10.1038/s41529-022-00226-4</a>.'
  mla: Wackenrohr, Steffen, et al. “Corrosion Fatigue Behavior of Electron Beam Melted
    Iron in Simulated Body Fluid.” <i>Npj Materials Degradation</i>, vol. 6, no. 1,
    18, Springer Science and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1038/s41529-022-00226-4">10.1038/s41529-022-00226-4</a>.
  short: S. Wackenrohr, C.J.J. Torrent, S. Herbst, F. Nürnberger, P. Krooss, C. Ebbert,
    M. Voigt, G. Grundmeier, T. Niendorf, H.J. Maier, Npj Materials Degradation 6
    (2022).
date_created: 2025-12-18T11:55:16Z
date_updated: 2025-12-18T11:56:57Z
department:
- _id: '35'
- _id: '302'
- _id: '321'
doi: 10.1038/s41529-022-00226-4
intvolume: '         6'
issue: '1'
language:
- iso: eng
publication: npj Materials Degradation
publication_identifier:
  issn:
  - 2397-2106
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Corrosion fatigue behavior of electron beam melted iron in simulated body fluid
type: journal_article
user_id: '7266'
volume: 6
year: '2022'
...
---
_id: '33670'
article_number: '013701'
author:
- first_name: Timon
  full_name: Schapeler, Timon
  id: '55629'
  last_name: Schapeler
  orcid: 0000-0001-7652-1716
- first_name: Tim
  full_name: Bartley, Tim
  id: '49683'
  last_name: Bartley
citation:
  ama: Schapeler T, Bartley T. Information extraction in photon-counting experiments.
    <i>Physical Review A</i>. 2022;106(1). doi:<a href="https://doi.org/10.1103/physreva.106.013701">10.1103/physreva.106.013701</a>
  apa: Schapeler, T., &#38; Bartley, T. (2022). Information extraction in photon-counting
    experiments. <i>Physical Review A</i>, <i>106</i>(1), Article 013701. <a href="https://doi.org/10.1103/physreva.106.013701">https://doi.org/10.1103/physreva.106.013701</a>
  bibtex: '@article{Schapeler_Bartley_2022, title={Information extraction in photon-counting
    experiments}, volume={106}, DOI={<a href="https://doi.org/10.1103/physreva.106.013701">10.1103/physreva.106.013701</a>},
    number={1013701}, journal={Physical Review A}, publisher={American Physical Society
    (APS)}, author={Schapeler, Timon and Bartley, Tim}, year={2022} }'
  chicago: Schapeler, Timon, and Tim Bartley. “Information Extraction in Photon-Counting
    Experiments.” <i>Physical Review A</i> 106, no. 1 (2022). <a href="https://doi.org/10.1103/physreva.106.013701">https://doi.org/10.1103/physreva.106.013701</a>.
  ieee: 'T. Schapeler and T. Bartley, “Information extraction in photon-counting experiments,”
    <i>Physical Review A</i>, vol. 106, no. 1, Art. no. 013701, 2022, doi: <a href="https://doi.org/10.1103/physreva.106.013701">10.1103/physreva.106.013701</a>.'
  mla: Schapeler, Timon, and Tim Bartley. “Information Extraction in Photon-Counting
    Experiments.” <i>Physical Review A</i>, vol. 106, no. 1, 013701, American Physical
    Society (APS), 2022, doi:<a href="https://doi.org/10.1103/physreva.106.013701">10.1103/physreva.106.013701</a>.
  short: T. Schapeler, T. Bartley, Physical Review A 106 (2022).
date_created: 2022-10-11T07:13:12Z
date_updated: 2025-12-18T17:07:12Z
department:
- _id: '15'
- _id: '230'
- _id: '623'
doi: 10.1103/physreva.106.013701
intvolume: '       106'
issue: '1'
language:
- iso: eng
project:
- _id: '209'
  name: 'ISOQC: Quantenkommunikation mit integrierter Optik im Zusammenhang mit supraleitender
    Elektronik'
publication: Physical Review A
publication_identifier:
  issn:
  - 2469-9926
  - 2469-9934
publication_status: published
publisher: American Physical Society (APS)
status: public
title: Information extraction in photon-counting experiments
type: journal_article
user_id: '55629'
volume: 106
year: '2022'
...
---
_id: '63234'
article_number: '103452'
author:
- first_name: Jens
  full_name: Wiegmann, Jens
  last_name: Wiegmann
- first_name: Christian
  full_name: Leppin, Christian
  id: '117722'
  last_name: Leppin
- first_name: Arne
  full_name: Langhoff, Arne
  last_name: Langhoff
- first_name: Jan
  full_name: Schwaderer, Jan
  last_name: Schwaderer
- first_name: Sabine
  full_name: Beuermann, Sabine
  last_name: Beuermann
- first_name: Diethelm
  full_name: Johannsmann, Diethelm
  last_name: Johannsmann
- first_name: Alfred P.
  full_name: Weber, Alfred P.
  last_name: Weber
citation:
  ama: Wiegmann J, Leppin C, Langhoff A, et al. Influence of the solvent evaporation
    rate on the β-Phase content of electrosprayed PVDF particles and films studied
    by a fast Multi-Overtone QCM. <i>Advanced Powder Technology</i>. 2022;33(3). doi:<a
    href="https://doi.org/10.1016/j.apt.2022.103452">10.1016/j.apt.2022.103452</a>
  apa: Wiegmann, J., Leppin, C., Langhoff, A., Schwaderer, J., Beuermann, S., Johannsmann,
    D., &#38; Weber, A. P. (2022). Influence of the solvent evaporation rate on the
    β-Phase content of electrosprayed PVDF particles and films studied by a fast Multi-Overtone
    QCM. <i>Advanced Powder Technology</i>, <i>33</i>(3), Article 103452. <a href="https://doi.org/10.1016/j.apt.2022.103452">https://doi.org/10.1016/j.apt.2022.103452</a>
  bibtex: '@article{Wiegmann_Leppin_Langhoff_Schwaderer_Beuermann_Johannsmann_Weber_2022,
    title={Influence of the solvent evaporation rate on the β-Phase content of electrosprayed
    PVDF particles and films studied by a fast Multi-Overtone QCM}, volume={33}, DOI={<a
    href="https://doi.org/10.1016/j.apt.2022.103452">10.1016/j.apt.2022.103452</a>},
    number={3103452}, journal={Advanced Powder Technology}, publisher={Elsevier BV},
    author={Wiegmann, Jens and Leppin, Christian and Langhoff, Arne and Schwaderer,
    Jan and Beuermann, Sabine and Johannsmann, Diethelm and Weber, Alfred P.}, year={2022}
    }'
  chicago: Wiegmann, Jens, Christian Leppin, Arne Langhoff, Jan Schwaderer, Sabine
    Beuermann, Diethelm Johannsmann, and Alfred P. Weber. “Influence of the Solvent
    Evaporation Rate on the β-Phase Content of Electrosprayed PVDF Particles and Films
    Studied by a Fast Multi-Overtone QCM.” <i>Advanced Powder Technology</i> 33, no.
    3 (2022). <a href="https://doi.org/10.1016/j.apt.2022.103452">https://doi.org/10.1016/j.apt.2022.103452</a>.
  ieee: 'J. Wiegmann <i>et al.</i>, “Influence of the solvent evaporation rate on
    the β-Phase content of electrosprayed PVDF particles and films studied by a fast
    Multi-Overtone QCM,” <i>Advanced Powder Technology</i>, vol. 33, no. 3, Art. no.
    103452, 2022, doi: <a href="https://doi.org/10.1016/j.apt.2022.103452">10.1016/j.apt.2022.103452</a>.'
  mla: Wiegmann, Jens, et al. “Influence of the Solvent Evaporation Rate on the β-Phase
    Content of Electrosprayed PVDF Particles and Films Studied by a Fast Multi-Overtone
    QCM.” <i>Advanced Powder Technology</i>, vol. 33, no. 3, 103452, Elsevier BV,
    2022, doi:<a href="https://doi.org/10.1016/j.apt.2022.103452">10.1016/j.apt.2022.103452</a>.
  short: J. Wiegmann, C. Leppin, A. Langhoff, J. Schwaderer, S. Beuermann, D. Johannsmann,
    A.P. Weber, Advanced Powder Technology 33 (2022).
date_created: 2025-12-18T17:22:31Z
date_updated: 2025-12-18T17:37:31Z
doi: 10.1016/j.apt.2022.103452
extern: '1'
intvolume: '        33'
issue: '3'
language:
- iso: eng
publication: Advanced Powder Technology
publication_identifier:
  issn:
  - 0921-8831
publication_status: published
publisher: Elsevier BV
quality_controlled: '1'
status: public
title: Influence of the solvent evaporation rate on the β-Phase content of electrosprayed
  PVDF particles and films studied by a fast Multi-Overtone QCM
type: journal_article
user_id: '117722'
volume: 33
year: '2022'
...
---
_id: '63233'
author:
- first_name: Christian
  full_name: Leppin, Christian
  id: '117722'
  last_name: Leppin
- first_name: Arne
  full_name: Langhoff, Arne
  last_name: Langhoff
- first_name: Diethelm
  full_name: Johannsmann, Diethelm
  last_name: Johannsmann
citation:
  ama: Leppin C, Langhoff A, Johannsmann D. Square-Wave Electrogravimetry Combined
    with Voltammetry Reveals Reversible Submonolayer Adsorption of Redox-Active Ions.
    <i>Analytical Chemistry</i>. 2022;94(28):10227-10233. doi:<a href="https://doi.org/10.1021/acs.analchem.2c01763">10.1021/acs.analchem.2c01763</a>
  apa: Leppin, C., Langhoff, A., &#38; Johannsmann, D. (2022). Square-Wave Electrogravimetry
    Combined with Voltammetry Reveals Reversible Submonolayer Adsorption of Redox-Active
    Ions. <i>Analytical Chemistry</i>, <i>94</i>(28), 10227–10233. <a href="https://doi.org/10.1021/acs.analchem.2c01763">https://doi.org/10.1021/acs.analchem.2c01763</a>
  bibtex: '@article{Leppin_Langhoff_Johannsmann_2022, title={Square-Wave Electrogravimetry
    Combined with Voltammetry Reveals Reversible Submonolayer Adsorption of Redox-Active
    Ions}, volume={94}, DOI={<a href="https://doi.org/10.1021/acs.analchem.2c01763">10.1021/acs.analchem.2c01763</a>},
    number={28}, journal={Analytical Chemistry}, publisher={American Chemical Society
    (ACS)}, author={Leppin, Christian and Langhoff, Arne and Johannsmann, Diethelm},
    year={2022}, pages={10227–10233} }'
  chicago: 'Leppin, Christian, Arne Langhoff, and Diethelm Johannsmann. “Square-Wave
    Electrogravimetry Combined with Voltammetry Reveals Reversible Submonolayer Adsorption
    of Redox-Active Ions.” <i>Analytical Chemistry</i> 94, no. 28 (2022): 10227–33.
    <a href="https://doi.org/10.1021/acs.analchem.2c01763">https://doi.org/10.1021/acs.analchem.2c01763</a>.'
  ieee: 'C. Leppin, A. Langhoff, and D. Johannsmann, “Square-Wave Electrogravimetry
    Combined with Voltammetry Reveals Reversible Submonolayer Adsorption of Redox-Active
    Ions,” <i>Analytical Chemistry</i>, vol. 94, no. 28, pp. 10227–10233, 2022, doi:
    <a href="https://doi.org/10.1021/acs.analchem.2c01763">10.1021/acs.analchem.2c01763</a>.'
  mla: Leppin, Christian, et al. “Square-Wave Electrogravimetry Combined with Voltammetry
    Reveals Reversible Submonolayer Adsorption of Redox-Active Ions.” <i>Analytical
    Chemistry</i>, vol. 94, no. 28, American Chemical Society (ACS), 2022, pp. 10227–33,
    doi:<a href="https://doi.org/10.1021/acs.analchem.2c01763">10.1021/acs.analchem.2c01763</a>.
  short: C. Leppin, A. Langhoff, D. Johannsmann, Analytical Chemistry 94 (2022) 10227–10233.
date_created: 2025-12-18T17:21:21Z
date_updated: 2025-12-18T17:38:07Z
doi: 10.1021/acs.analchem.2c01763
extern: '1'
intvolume: '        94'
issue: '28'
language:
- iso: eng
page: 10227-10233
publication: Analytical Chemistry
publication_identifier:
  issn:
  - 0003-2700
  - 1520-6882
publication_status: published
publisher: American Chemical Society (ACS)
quality_controlled: '1'
status: public
title: Square-Wave Electrogravimetry Combined with Voltammetry Reveals Reversible
  Submonolayer Adsorption of Redox-Active Ions
type: journal_article
user_id: '117722'
volume: 94
year: '2022'
...
---
_id: '63310'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>The chemotaxis–Stokes system<jats:disp-formula
    id="j_ans-2022-0004_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_001.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"
    display="block"><m:mfenced open="{" close=""><m:mrow><m:mtable displaystyle="true"><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>n</m:mi><m:mo>=</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo
    stretchy="false">(</m:mo><m:mrow/></m:mrow><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>n</m:mi><m:mrow><m:mo
    stretchy="false">)</m:mo><m:mrow/></m:mrow><m:mo>−</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo
    stretchy="false">(</m:mo><m:mrow/></m:mrow><m:mi>n</m:mi><m:mi>S</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>c</m:mi><m:mrow><m:mo
    stretchy="false">)</m:mo><m:mrow/></m:mrow><m:mo>,</m:mo></m:mtd></m:mtr><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>+</m:mo><m:mi>u</m:mi><m:mo>⋅</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>c</m:mi><m:mo>=</m:mo><m:mi
    mathvariant="normal">Δ</m:mi><m:mi>c</m:mi><m:mo>−</m:mo><m:mi>n</m:mi><m:mi>c</m:mi><m:mo>,</m:mo></m:mtd></m:mtr><m:mtr><m:mtd
    columnalign="left"><m:msub><m:mrow><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>t</m:mi></m:mrow></m:msub><m:mo>=</m:mo><m:mi
    mathvariant="normal">Δ</m:mi><m:mi>u</m:mi><m:mo>+</m:mo><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi>P</m:mi><m:mo>+</m:mo><m:mi>n</m:mi><m:mrow><m:mo>∇</m:mo></m:mrow><m:mi
    mathvariant="normal">Φ</m:mi><m:mo>,</m:mo><m:mspace width="1.0em"/><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>⋅</m:mo><m:mi>u</m:mi><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mtd></m:mtr></m:mtable></m:mrow></m:mfenced></m:math><jats:tex-math>\left\{\begin{array}{l}{n}_{t}+u\cdot
    \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla
    c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla
    \Phi ,\hspace{1.0em}\nabla \cdot u=0,\end{array}\right.</jats:tex-math></jats:alternatives></jats:disp-formula>is
    considered in a smoothly bounded convex domain<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_002.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="normal">Ω</m:mi><m:mo>⊂</m:mo><m:msup><m:mrow><m:mi
    mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mn>3</m:mn></m:mrow></m:msup></m:math><jats:tex-math>\Omega
    \subset {{\mathbb{R}}}^{3}</jats:tex-math></jats:alternatives></jats:inline-formula>,
    with given suitably regular functions<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_003.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mo>:</m:mo><m:mrow><m:mo
    stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>→</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow></m:math><jats:tex-math>D:{[}0,\infty )\to {[}0,\infty
    )</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_004.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>S</m:mi><m:mo>:</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>×</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>→</m:mo><m:msup><m:mrow><m:mi
    mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mn>3</m:mn><m:mo>×</m:mo><m:mn>3</m:mn></m:mrow></m:msup></m:math><jats:tex-math>S:\overline{\Omega
    }\times {[}0,\infty )\times \left(0,\infty )\to {{\mathbb{R}}}^{3\times 3}</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_005.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="normal">Φ</m:mi><m:mo>:</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>→</m:mo><m:mi mathvariant="double-struck">R</m:mi></m:math><jats:tex-math>\Phi
    :\overline{\Omega }\to {\mathbb{R}}</jats:tex-math></jats:alternatives></jats:inline-formula>such
    that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_006.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>D\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>on<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_007.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>\left(0,\infty
    )</jats:tex-math></jats:alternatives></jats:inline-formula>. It is shown that
    if with some nondecreasing<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_008.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>:</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>→</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>{S}_{0}:\left(0,\infty
    )\to \left(0,\infty )</jats:tex-math></jats:alternatives></jats:inline-formula>we
    have<jats:disp-formula id="j_ans-2022-0004_eq_002"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_009.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mo>∣</m:mo><m:mi>S</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>∣</m:mo><m:mo>≤</m:mo><m:mfrac><m:mrow><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:msup><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mstyle
    displaystyle="false"><m:mfrac><m:mrow><m:mn>1</m:mn></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:mfrac></m:mstyle></m:mrow></m:msup></m:mrow></m:mfrac><m:mspace
    width="1.0em"/><m:mspace width="0.1em"/><m:mtext>for all</m:mtext><m:mspace width="0.1em"/><m:mspace
    width="0.33em"/><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>,</m:mo><m:mi>c</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>∈</m:mo><m:mover
    accent="true"><m:mrow><m:mi mathvariant="normal">Ω</m:mi></m:mrow><m:mrow><m:mo
    stretchy="true">¯</m:mo></m:mrow></m:mover><m:mo>×</m:mo><m:mrow><m:mo stretchy="false">[</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo
    stretchy="false">)</m:mo></m:mrow><m:mo>×</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mi>∞</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo></m:math><jats:tex-math>|
    S\left(x,n,c)| \le \frac{{S}_{0}\left(c)}{{c}^{\tfrac{1}{2}}}\hspace{1.0em}\hspace{0.1em}\text{for
    all}\hspace{0.1em}\hspace{0.33em}\left(x,n,c)\in \overline{\Omega }\times {[}0,\infty
    )\times \left(0,\infty ),</jats:tex-math></jats:alternatives></jats:disp-formula>then
    for all<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_010.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>M</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>M\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>there exists<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_011.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>L</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>M</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>L\left(M)\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>such that whenever<jats:disp-formula
    id="j_ans-2022-0004_eq_003"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_012.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"
    display="block"><m:munder><m:mrow><m:mi>liminf</m:mi></m:mrow><m:mrow><m:mi>n</m:mi><m:mo>→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mi>L</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>M</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mspace
    width="1.0em"/><m:mspace width="0.1em"/><m:mtext>and</m:mtext><m:mspace width="0.1em"/><m:mspace
    width="1.0em"/><m:munder><m:mrow><m:mi>liminf</m:mi></m:mrow><m:mrow><m:mi>n</m:mi><m:mo>↘</m:mo><m:mn>0</m:mn></m:mrow></m:munder><m:mfrac><m:mrow><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:mfrac><m:mo>&gt;</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:math><jats:tex-math>\mathop{\mathrm{liminf}}\limits_{n\to
    \infty }D\left(n)\gt L\left(M)\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{liminf}}\limits_{n\searrow
    0}\frac{D\left(n)}{n}\gt 0,</jats:tex-math></jats:alternatives></jats:disp-formula>for
    all sufficiently regular initial data<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_013.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mrow><m:msub><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>u</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub></m:mrow><m:mo>)</m:mo></m:mrow></m:math><jats:tex-math>\left({n}_{0},{c}_{0},{u}_{0})</jats:tex-math></jats:alternatives></jats:inline-formula>fulfilling<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_014.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo>‖</m:mo><m:msub><m:mrow><m:mi>c</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:msub><m:mrow><m:mo>‖</m:mo></m:mrow><m:mrow><m:msup><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>∞</m:mi></m:mrow></m:msup><m:mrow><m:mo>(</m:mo><m:mrow><m:mi
    mathvariant="normal">Ω</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:msub><m:mo>≤</m:mo><m:mi>M</m:mi></m:math><jats:tex-math>\Vert
    {c}_{0}{\Vert }_{{L}^{\infty }\left(\Omega )}\le M</jats:tex-math></jats:alternatives></jats:inline-formula>an
    associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global
    bounded weak solution, classical if additionally<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_015.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mn>0</m:mn></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>D\left(0)\gt
    0</jats:tex-math></jats:alternatives></jats:inline-formula>. When combined with
    previously known results, this particularly implies global existence of bounded
    solutions when<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_ans-2022-0004_eq_016.png"/><m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>D</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>n</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msup><m:mrow><m:mi>n</m:mi></m:mrow><m:mrow><m:mi>m</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow></m:msup></m:math><jats:tex-math>D\left(n)={n}^{m-1}</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_017.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>n</m:mi><m:mo>≥</m:mo><m:mn>0</m:mn></m:math><jats:tex-math>n\ge
    0</jats:tex-math></jats:alternatives></jats:inline-formula>, with arbitrary<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_018.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>m</m:mi><m:mo>&gt;</m:mo><m:mn>1</m:mn></m:math><jats:tex-math>m\gt
    1</jats:tex-math></jats:alternatives></jats:inline-formula>, but beyond this asserts
    global boundedness also in the presence of diffusivities which exhibit arbitrarily
    slow divergence to<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2022-0004_eq_019.png"/><m:math
    xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo>+</m:mo><m:mi>∞</m:mi></m:math><jats:tex-math>+\infty</jats:tex-math></jats:alternatives></jats:inline-formula>at
    large densities and of possibly singular chemotactic sensitivities.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. Chemotaxis-Stokes interaction with very weak diffusion enhancement:
    Blow-up exclusion via detection of absorption-induced entropy structures involving
    multiplicative couplings. <i>Advanced Nonlinear Studies</i>. 2022;22(1):88-117.
    doi:<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>'
  apa: 'Winkler, M. (2022). Chemotaxis-Stokes interaction with very weak diffusion
    enhancement: Blow-up exclusion via detection of absorption-induced entropy structures
    involving multiplicative couplings. <i>Advanced Nonlinear Studies</i>, <i>22</i>(1),
    88–117. <a href="https://doi.org/10.1515/ans-2022-0004">https://doi.org/10.1515/ans-2022-0004</a>'
  bibtex: '@article{Winkler_2022, title={Chemotaxis-Stokes interaction with very weak
    diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy
    structures involving multiplicative couplings}, volume={22}, DOI={<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>},
    number={1}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2022}, pages={88–117} }'
  chicago: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
    Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
    Involving Multiplicative Couplings.” <i>Advanced Nonlinear Studies</i> 22, no.
    1 (2022): 88–117. <a href="https://doi.org/10.1515/ans-2022-0004">https://doi.org/10.1515/ans-2022-0004</a>.'
  ieee: 'M. Winkler, “Chemotaxis-Stokes interaction with very weak diffusion enhancement:
    Blow-up exclusion via detection of absorption-induced entropy structures involving
    multiplicative couplings,” <i>Advanced Nonlinear Studies</i>, vol. 22, no. 1,
    pp. 88–117, 2022, doi: <a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>.'
  mla: 'Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion
    Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures
    Involving Multiplicative Couplings.” <i>Advanced Nonlinear Studies</i>, vol. 22,
    no. 1, Walter de Gruyter GmbH, 2022, pp. 88–117, doi:<a href="https://doi.org/10.1515/ans-2022-0004">10.1515/ans-2022-0004</a>.'
  short: M. Winkler, Advanced Nonlinear Studies 22 (2022) 88–117.
date_created: 2025-12-18T19:29:40Z
date_updated: 2025-12-18T20:05:30Z
doi: 10.1515/ans-2022-0004
intvolume: '        22'
issue: '1'
language:
- iso: eng
page: 88-117
publication: Advanced Nonlinear Studies
publication_identifier:
  issn:
  - 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: 'Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up
  exclusion via detection of absorption-induced entropy structures involving multiplicative
  couplings'
type: journal_article
user_id: '31496'
volume: 22
year: '2022'
...
---
_id: '63305'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>A no-flux initial-boundary value
    problem for the doubly degenrate parabolic system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big ( uv\\nabla u\\big ) +
    \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star
    ) \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:mfenced>\r\n                              <mml:mrow>\r\n
    \                               <mml:mtable>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:msub>\r\n                                          <mml:mi>u</mml:mi>\r\n
    \                                         <mml:mi>t</mml:mi>\r\n                                        </mml:msub>\r\n
    \                                       <mml:mo>=</mml:mo>\r\n                                        <mml:mi>∇</mml:mi>\r\n
    \                                       <mml:mo>·</mml:mo>\r\n                                        <mml:mrow>\r\n
    \                                         <mml:mo>(</mml:mo>\r\n                                        </mml:mrow>\r\n
    \                                       <mml:mi>u</mml:mi>\r\n                                        <mml:mi>v</mml:mi>\r\n
    \                                       <mml:mi>∇</mml:mi>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mo>)</mml:mo>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mo>+</mml:mo>\r\n
    \                                       <mml:mi>ℓ</mml:mi>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mi>v</mml:mi>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                    </mml:mtd>\r\n
    \                                 </mml:mtr>\r\n                                  <mml:mtr>\r\n
    \                                   <mml:mtd>\r\n                                      <mml:mrow>\r\n
    \                                       <mml:mrow/>\r\n                                        <mml:msub>\r\n
    \                                         <mml:mi>v</mml:mi>\r\n                                          <mml:mi>t</mml:mi>\r\n
    \                                       </mml:msub>\r\n                                        <mml:mo>=</mml:mo>\r\n
    \                                       <mml:mi>Δ</mml:mi>\r\n                                        <mml:mi>v</mml:mi>\r\n
    \                                       <mml:mo>-</mml:mo>\r\n                                        <mml:mi>u</mml:mi>\r\n
    \                                       <mml:mi>v</mml:mi>\r\n                                        <mml:mo>,</mml:mo>\r\n
    \                                     </mml:mrow>\r\n                                    </mml:mtd>\r\n
    \                                 </mml:mtr>\r\n                                </mml:mtable>\r\n
    \                             </mml:mrow>\r\n                            </mml:mfenced>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mo>⋆</mml:mo>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>is
    considered in a smoothly bounded convex domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega
    \\subset \\mathbb {R}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>⊂</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mrow>\r\n                        <mml:mi>R</mml:mi>\r\n
    \                     </mml:mrow>\r\n                      <mml:mi>n</mml:mi>\r\n
    \                   </mml:msup>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\ge 0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>ℓ</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    The first of the main results asserts that for nonnegative initial data <jats:inline-formula><jats:alternatives><jats:tex-math>$$(u_0,v_0)\\in
    (L^\\infty (\\Omega ))^2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:msub>\r\n                        <mml:mi>u</mml:mi>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                      </mml:msub>\r\n
    \                     <mml:mo>,</mml:mo>\r\n                      <mml:msub>\r\n
    \                       <mml:mi>v</mml:mi>\r\n                        <mml:mn>0</mml:mn>\r\n
    \                     </mml:msub>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:msup>\r\n                          <mml:mi>L</mml:mi>\r\n
    \                         <mml:mi>∞</mml:mi>\r\n                        </mml:msup>\r\n
    \                       <mml:mrow>\r\n                          <mml:mo>(</mml:mo>\r\n
    \                         <mml:mi>Ω</mml:mi>\r\n                          <mml:mo>)</mml:mo>\r\n
    \                       </mml:mrow>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mn>2</mml:mn>\r\n
    \                   </mml:msup>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0\\not \\equiv
    0$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>≢</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$v_0\\not \\equiv 0$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>v</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>≢</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sqrt{v_0}\\in W^{1,2}(\\Omega
    )$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msqrt>\r\n                      <mml:msub>\r\n
    \                       <mml:mi>v</mml:mi>\r\n                        <mml:mn>0</mml:mn>\r\n
    \                     </mml:msub>\r\n                    </mml:msqrt>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>W</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mn>1</mml:mn>\r\n
    \                       <mml:mo>,</mml:mo>\r\n                        <mml:mn>2</mml:mn>\r\n
    \                     </mml:mrow>\r\n                    </mml:msup>\r\n                    <mml:mrow>\r\n
    \                     <mml:mo>(</mml:mo>\r\n                      <mml:mi>Ω</mml:mi>\r\n
    \                     <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    there exists a global weak solution (<jats:italic>u</jats:italic>, <jats:italic>v</jats:italic>)
    which, inter alia, belongs to <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^0(\\overline{\\Omega
    }\\times (0,\\infty )) \\times C^{2,1}(\\overline{\\Omega }\\times (0,\\infty
    ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mover>\r\n                        <mml:mi>Ω</mml:mi>\r\n
    \                       <mml:mo>¯</mml:mo>\r\n                      </mml:mover>\r\n
    \                     <mml:mo>×</mml:mo>\r\n                      <mml:mrow>\r\n
    \                       <mml:mo>(</mml:mo>\r\n                        <mml:mn>0</mml:mn>\r\n
    \                       <mml:mo>,</mml:mo>\r\n                        <mml:mi>∞</mml:mi>\r\n
    \                       <mml:mo>)</mml:mo>\r\n                      </mml:mrow>\r\n
    \                     <mml:mo>)</mml:mo>\r\n                    </mml:mrow>\r\n
    \                   <mml:mo>×</mml:mo>\r\n                    <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mrow>\r\n                        <mml:mn>2</mml:mn>\r\n
    \                       <mml:mo>,</mml:mo>\r\n                        <mml:mn>1</mml:mn>\r\n
    \                     </mml:mrow>\r\n                    </mml:msup>\r\n                    <mml:mrow>\r\n
    \                     <mml:mo>(</mml:mo>\r\n                      <mml:mover>\r\n
    \                       <mml:mi>Ω</mml:mi>\r\n                        <mml:mo>¯</mml:mo>\r\n
    \                     </mml:mover>\r\n                      <mml:mo>×</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>(</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and satisfies <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sup _{t&gt;0}
    \\Vert u(\\cdot ,t)\\Vert _{L^p(\\Omega )}&lt;\\infty $$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mo>sup</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mi>t</mml:mi>\r\n
    \                       <mml:mo>&gt;</mml:mo>\r\n                        <mml:mn>0</mml:mn>\r\n
    \                     </mml:mrow>\r\n                    </mml:msub>\r\n                    <mml:msub>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>‖</mml:mo>\r\n
    \                       <mml:mi>u</mml:mi>\r\n                        <mml:mrow>\r\n
    \                         <mml:mo>(</mml:mo>\r\n                          <mml:mo>·</mml:mo>\r\n
    \                         <mml:mo>,</mml:mo>\r\n                          <mml:mi>t</mml:mi>\r\n
    \                         <mml:mo>)</mml:mo>\r\n                        </mml:mrow>\r\n
    \                       <mml:mo>‖</mml:mo>\r\n                      </mml:mrow>\r\n
    \                     <mml:mrow>\r\n                        <mml:msup>\r\n                          <mml:mi>L</mml:mi>\r\n
    \                         <mml:mi>p</mml:mi>\r\n                        </mml:msup>\r\n
    \                       <mml:mrow>\r\n                          <mml:mo>(</mml:mo>\r\n
    \                         <mml:mi>Ω</mml:mi>\r\n                          <mml:mo>)</mml:mo>\r\n
    \                       </mml:mrow>\r\n                      </mml:mrow>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>&lt;</mml:mo>\r\n                    <mml:mi>∞</mml:mi>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\in [1,p_0)$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>p</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:mo>[</mml:mo>\r\n                    <mml:mn>1</mml:mn>\r\n
    \                   <mml:mo>,</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>p</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>)</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$$p_0:=\\frac{n}{(n-2)_+}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:msub>\r\n                      <mml:mi>p</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>:</mml:mo>\r\n                    <mml:mo>=</mml:mo>\r\n
    \                   <mml:mfrac>\r\n                      <mml:mi>n</mml:mi>\r\n
    \                     <mml:msub>\r\n                        <mml:mrow>\r\n                          <mml:mo>(</mml:mo>\r\n
    \                         <mml:mi>n</mml:mi>\r\n                          <mml:mo>-</mml:mo>\r\n
    \                         <mml:mn>2</mml:mn>\r\n                          <mml:mo>)</mml:mo>\r\n
    \                       </mml:mrow>\r\n                        <mml:mo>+</mml:mo>\r\n
    \                     </mml:msub>\r\n                    </mml:mfrac>\r\n                  </mml:mrow>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>. It is next
    seen that for each of these solutions one can find <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_\\infty
    \\in \\bigcap _{p\\in [1,p_0)} L^p(\\Omega )$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mi>∞</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>∈</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mo>⋂</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mi>p</mml:mi>\r\n
    \                       <mml:mo>∈</mml:mo>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>1</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:msub>\r\n                          <mml:mi>p</mml:mi>\r\n
    \                         <mml:mn>0</mml:mn>\r\n                        </mml:msub>\r\n
    \                       <mml:mo>)</mml:mo>\r\n                      </mml:mrow>\r\n
    \                   </mml:msub>\r\n                    <mml:msup>\r\n                      <mml:mi>L</mml:mi>\r\n
    \                     <mml:mi>p</mml:mi>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mi>Ω</mml:mi>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    such that, within an appropriate topological setting, <jats:inline-formula><jats:alternatives><jats:tex-math>$$(u(\\cdot
    ,t),v(\\cdot ,t))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mo>(</mml:mo>\r\n                    <mml:mi>u</mml:mi>\r\n
    \                   <mml:mo>(</mml:mo>\r\n                    <mml:mo>·</mml:mo>\r\n
    \                   <mml:mo>,</mml:mo>\r\n                    <mml:mi>t</mml:mi>\r\n
    \                   <mml:mo>)</mml:mo>\r\n                    <mml:mo>,</mml:mo>\r\n
    \                   <mml:mi>v</mml:mi>\r\n                    <mml:mo>(</mml:mo>\r\n
    \                   <mml:mo>·</mml:mo>\r\n                    <mml:mo>,</mml:mo>\r\n
    \                   <mml:mi>t</mml:mi>\r\n                    <mml:mo>)</mml:mo>\r\n
    \                   <mml:mo>)</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    approaches the equilibrium <jats:inline-formula><jats:alternatives><jats:tex-math>$$(u_\\infty
    ,0)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mo>(</mml:mo>\r\n                    <mml:msub>\r\n
    \                     <mml:mi>u</mml:mi>\r\n                      <mml:mi>∞</mml:mi>\r\n
    \                   </mml:msub>\r\n                    <mml:mo>,</mml:mo>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                    <mml:mo>)</mml:mo>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    in the large time limit. Finally, in the case <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\le
    5$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>n</mml:mi>\r\n                    <mml:mo>≤</mml:mo>\r\n
    \                   <mml:mn>5</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    a result ensuring a certain stability property of any member in the uncountably
    large family of steady states <jats:inline-formula><jats:alternatives><jats:tex-math>$$(u_0,0)$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mo>(</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>,</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                   <mml:mo>)</mml:mo>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    with arbitrary and suitably regular <jats:inline-formula><jats:alternatives><jats:tex-math>$$u_0:\\Omega
    \\rightarrow [0,\\infty )$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>u</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>:</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                   <mml:mo>→</mml:mo>\r\n                    <mml:mrow>\r\n                      <mml:mo>[</mml:mo>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                      <mml:mo>,</mml:mo>\r\n
    \                     <mml:mi>∞</mml:mi>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    is derived. This provides some rigorous evidence for the appropriateness of (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    to model the emergence of a strikingly large variety of stable structures observed
    in experiments on bacterial motion in nutrient-poor environments. Essential parts
    of the analysis rely on the use of an apparently novel class of functional inequalities
    to suitably cope with the doubly degenerate diffusion mechanism in (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>).</jats:p>"
article_number: '108'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion
    system modeling bacterial motion on a nutrient-poor agar. <i>Calculus of Variations
    and Partial Differential Equations</i>. 2022;61(3). doi:<a href="https://doi.org/10.1007/s00526-021-02168-2">10.1007/s00526-021-02168-2</a>
  apa: Winkler, M. (2022). Stabilization of arbitrary structures in a doubly degenerate
    reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. <i>Calculus
    of Variations and Partial Differential Equations</i>, <i>61</i>(3), Article 108.
    <a href="https://doi.org/10.1007/s00526-021-02168-2">https://doi.org/10.1007/s00526-021-02168-2</a>
  bibtex: '@article{Winkler_2022, title={Stabilization of arbitrary structures in
    a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor
    agar}, volume={61}, DOI={<a href="https://doi.org/10.1007/s00526-021-02168-2">10.1007/s00526-021-02168-2</a>},
    number={3108}, journal={Calculus of Variations and Partial Differential Equations},
    publisher={Springer Science and Business Media LLC}, author={Winkler, Michael},
    year={2022} }'
  chicago: Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate
    Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.”
    <i>Calculus of Variations and Partial Differential Equations</i> 61, no. 3 (2022).
    <a href="https://doi.org/10.1007/s00526-021-02168-2">https://doi.org/10.1007/s00526-021-02168-2</a>.
  ieee: 'M. Winkler, “Stabilization of arbitrary structures in a doubly degenerate
    reaction-diffusion system modeling bacterial motion on a nutrient-poor agar,”
    <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no.
    3, Art. no. 108, 2022, doi: <a href="https://doi.org/10.1007/s00526-021-02168-2">10.1007/s00526-021-02168-2</a>.'
  mla: Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate
    Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.”
    <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no.
    3, 108, Springer Science and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1007/s00526-021-02168-2">10.1007/s00526-021-02168-2</a>.
  short: M. Winkler, Calculus of Variations and Partial Differential Equations 61
    (2022).
date_created: 2025-12-18T19:26:32Z
date_updated: 2025-12-18T20:04:43Z
doi: 10.1007/s00526-021-02168-2
intvolume: '        61'
issue: '3'
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: Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion
  system modeling bacterial motion on a nutrient-poor agar
type: journal_article
user_id: '31496'
volume: 61
year: '2022'
...
---
_id: '63311'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>The Cauchy problem in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb
    {R}^n$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msup>\r\n                    <mml:mrow>\r\n                      <mml:mi>R</mml:mi>\r\n
    \                   </mml:mrow>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                 </mml:msup>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\ge 1$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mi>n</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    for the degenerate parabolic equation <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:msub>\r\n
    \                             <mml:mi>u</mml:mi>\r\n                              <mml:mi>t</mml:mi>\r\n
    \                           </mml:msub>\r\n                            <mml:mo>=</mml:mo>\r\n
    \                           <mml:msup>\r\n                              <mml:mi>u</mml:mi>\r\n
    \                             <mml:mi>p</mml:mi>\r\n                            </mml:msup>\r\n
    \                           <mml:mi>Δ</mml:mi>\r\n                            <mml:mi>u</mml:mi>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mrow>\r\n                              <mml:mo>(</mml:mo>\r\n
    \                             <mml:mo>⋆</mml:mo>\r\n                              <mml:mo>)</mml:mo>\r\n
    \                           </mml:mrow>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>is
    considered for <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\ge
    1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>p</mml:mi>\r\n                    <mml:mo>≥</mml:mo>\r\n
    \                   <mml:mn>1</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.
    It is shown that given any positive <jats:inline-formula><jats:alternatives><jats:tex-math>$$f\\in
    C^0([0,\\infty ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>f</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$$g\\in C^0([0,\\infty
    ))$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>g</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:msup>\r\n                      <mml:mi>C</mml:mi>\r\n
    \                     <mml:mn>0</mml:mn>\r\n                    </mml:msup>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mrow>\r\n                        <mml:mo>[</mml:mo>\r\n
    \                       <mml:mn>0</mml:mn>\r\n                        <mml:mo>,</mml:mo>\r\n
    \                       <mml:mi>∞</mml:mi>\r\n                        <mml:mo>)</mml:mo>\r\n
    \                     </mml:mrow>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    satisfying <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    f(t)\\rightarrow + \\infty \\quad \\text{ and } \\quad g(t)\\rightarrow 0 \\qquad
    \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:mi>f</mml:mi>\r\n
    \                           <mml:mo>(</mml:mo>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>)</mml:mo>\r\n                            <mml:mo>→</mml:mo>\r\n
    \                           <mml:mo>+</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>and</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>g</mml:mi>\r\n
    \                           <mml:mo>(</mml:mo>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>)</mml:mo>\r\n                            <mml:mo>→</mml:mo>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>as</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mo>,</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>one
    can find positive and radially symmetric continuous initial data with the property
    that the initial value problem for (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\star
    $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mo>⋆</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>)
    admits a positive classical solution such that <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned}
    t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow
    \\infty \\qquad \\text{ and } \\qquad \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb
    {R}^n)} \\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n
    \                   <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n
    \                         <mml:mrow>\r\n                            <mml:msup>\r\n
    \                             <mml:mi>t</mml:mi>\r\n                              <mml:mfrac>\r\n
    \                               <mml:mn>1</mml:mn>\r\n                                <mml:mi>p</mml:mi>\r\n
    \                             </mml:mfrac>\r\n                            </mml:msup>\r\n
    \                           <mml:msub>\r\n                              <mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                                <mml:mi>u</mml:mi>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:mo>·</mml:mo>\r\n                                  <mml:mo>,</mml:mo>\r\n
    \                                 <mml:mi>t</mml:mi>\r\n                                  <mml:mo>)</mml:mo>\r\n
    \                               </mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                             </mml:mrow>\r\n                              <mml:mrow>\r\n
    \                               <mml:msup>\r\n                                  <mml:mi>L</mml:mi>\r\n
    \                                 <mml:mi>∞</mml:mi>\r\n                                </mml:msup>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>(</mml:mo>\r\n
    \                                 <mml:msup>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mi>R</mml:mi>\r\n                                    </mml:mrow>\r\n
    \                                   <mml:mi>n</mml:mi>\r\n                                  </mml:msup>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                             </mml:mrow>\r\n                            </mml:msub>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mtext>and</mml:mtext>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:msub>\r\n
    \                             <mml:mrow>\r\n                                <mml:mo>‖</mml:mo>\r\n
    \                               <mml:mi>u</mml:mi>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:mo>·</mml:mo>\r\n
    \                                 <mml:mo>,</mml:mo>\r\n                                  <mml:mi>t</mml:mi>\r\n
    \                                 <mml:mo>)</mml:mo>\r\n                                </mml:mrow>\r\n
    \                               <mml:mo>‖</mml:mo>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:msup>\r\n
    \                                 <mml:mi>L</mml:mi>\r\n                                  <mml:mi>∞</mml:mi>\r\n
    \                               </mml:msup>\r\n                                <mml:mrow>\r\n
    \                                 <mml:mo>(</mml:mo>\r\n                                  <mml:msup>\r\n
    \                                   <mml:mrow>\r\n                                      <mml:mi>R</mml:mi>\r\n
    \                                   </mml:mrow>\r\n                                    <mml:mi>n</mml:mi>\r\n
    \                                 </mml:msup>\r\n                                  <mml:mo>)</mml:mo>\r\n
    \                               </mml:mrow>\r\n                              </mml:mrow>\r\n
    \                           </mml:msub>\r\n                            <mml:mo>→</mml:mo>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                            <mml:mspace/>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mtext>as</mml:mtext>\r\n
    \                           <mml:mspace/>\r\n                            <mml:mi>t</mml:mi>\r\n
    \                           <mml:mo>→</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mo>,</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>but
    that <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\liminf
    _{t\\rightarrow \\infty } \\frac{t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty
    (\\mathbb {R}^n)}}{f(t)} =0 \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:munder>\r\n                              <mml:mo>lim
    inf</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:munder>\r\n
    \                           <mml:mfrac>\r\n                              <mml:mrow>\r\n
    \                               <mml:msup>\r\n                                  <mml:mi>t</mml:mi>\r\n
    \                                 <mml:mfrac>\r\n                                    <mml:mn>1</mml:mn>\r\n
    \                                   <mml:mi>p</mml:mi>\r\n                                  </mml:mfrac>\r\n
    \                               </mml:msup>\r\n                                <mml:msub>\r\n
    \                                 <mml:mrow>\r\n                                    <mml:mo>‖</mml:mo>\r\n
    \                                   <mml:mi>u</mml:mi>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mo>(</mml:mo>\r\n                                      <mml:mo>·</mml:mo>\r\n
    \                                     <mml:mo>,</mml:mo>\r\n                                      <mml:mi>t</mml:mi>\r\n
    \                                     <mml:mo>)</mml:mo>\r\n                                    </mml:mrow>\r\n
    \                                   <mml:mo>‖</mml:mo>\r\n                                  </mml:mrow>\r\n
    \                                 <mml:mrow>\r\n                                    <mml:msup>\r\n
    \                                     <mml:mi>L</mml:mi>\r\n                                      <mml:mi>∞</mml:mi>\r\n
    \                                   </mml:msup>\r\n                                    <mml:mrow>\r\n
    \                                     <mml:mo>(</mml:mo>\r\n                                      <mml:msup>\r\n
    \                                       <mml:mrow>\r\n                                          <mml:mi>R</mml:mi>\r\n
    \                                       </mml:mrow>\r\n                                        <mml:mi>n</mml:mi>\r\n
    \                                     </mml:msup>\r\n                                      <mml:mo>)</mml:mo>\r\n
    \                                   </mml:mrow>\r\n                                  </mml:mrow>\r\n
    \                               </mml:msub>\r\n                              </mml:mrow>\r\n
    \                             <mml:mrow>\r\n                                <mml:mi>f</mml:mi>\r\n
    \                               <mml:mo>(</mml:mo>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>)</mml:mo>\r\n                              </mml:mrow>\r\n
    \                           </mml:mfrac>\r\n                            <mml:mo>=</mml:mo>\r\n
    \                           <mml:mn>0</mml:mn>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>and
    <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\limsup
    _{t\\rightarrow \\infty } \\frac{\\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb
    {R}^n)}}{g(t)} =\\infty . \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mrow>\r\n
    \                           <mml:munder>\r\n                              <mml:mo>lim
    sup</mml:mo>\r\n                              <mml:mrow>\r\n                                <mml:mi>t</mml:mi>\r\n
    \                               <mml:mo>→</mml:mo>\r\n                                <mml:mi>∞</mml:mi>\r\n
    \                             </mml:mrow>\r\n                            </mml:munder>\r\n
    \                           <mml:mfrac>\r\n                              <mml:msub>\r\n
    \                               <mml:mrow>\r\n                                  <mml:mo>‖</mml:mo>\r\n
    \                                 <mml:mi>u</mml:mi>\r\n                                  <mml:mrow>\r\n
    \                                   <mml:mo>(</mml:mo>\r\n                                    <mml:mo>·</mml:mo>\r\n
    \                                   <mml:mo>,</mml:mo>\r\n                                    <mml:mi>t</mml:mi>\r\n
    \                                   <mml:mo>)</mml:mo>\r\n                                  </mml:mrow>\r\n
    \                                 <mml:mo>‖</mml:mo>\r\n                                </mml:mrow>\r\n
    \                               <mml:mrow>\r\n                                  <mml:msup>\r\n
    \                                   <mml:mi>L</mml:mi>\r\n                                    <mml:mi>∞</mml:mi>\r\n
    \                                 </mml:msup>\r\n                                  <mml:mrow>\r\n
    \                                   <mml:mo>(</mml:mo>\r\n                                    <mml:msup>\r\n
    \                                     <mml:mrow>\r\n                                        <mml:mi>R</mml:mi>\r\n
    \                                     </mml:mrow>\r\n                                      <mml:mi>n</mml:mi>\r\n
    \                                   </mml:msup>\r\n                                    <mml:mo>)</mml:mo>\r\n
    \                                 </mml:mrow>\r\n                                </mml:mrow>\r\n
    \                             </mml:msub>\r\n                              <mml:mrow>\r\n
    \                               <mml:mi>g</mml:mi>\r\n                                <mml:mo>(</mml:mo>\r\n
    \                               <mml:mi>t</mml:mi>\r\n                                <mml:mo>)</mml:mo>\r\n
    \                             </mml:mrow>\r\n                            </mml:mfrac>\r\n
    \                           <mml:mo>=</mml:mo>\r\n                            <mml:mi>∞</mml:mi>\r\n
    \                           <mml:mo>.</mml:mo>\r\n                          </mml:mrow>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n
    \                 </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula></jats:p>"
article_number: '47'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Oscillatory decay in a degenerate parabolic equation. <i>Partial
    Differential Equations and Applications</i>. 2022;3(4). doi:<a href="https://doi.org/10.1007/s42985-022-00186-z">10.1007/s42985-022-00186-z</a>
  apa: Winkler, M. (2022). Oscillatory decay in a degenerate parabolic equation. <i>Partial
    Differential Equations and Applications</i>, <i>3</i>(4), Article 47. <a href="https://doi.org/10.1007/s42985-022-00186-z">https://doi.org/10.1007/s42985-022-00186-z</a>
  bibtex: '@article{Winkler_2022, title={Oscillatory decay in a degenerate parabolic
    equation}, volume={3}, DOI={<a href="https://doi.org/10.1007/s42985-022-00186-z">10.1007/s42985-022-00186-z</a>},
    number={447}, journal={Partial Differential Equations and Applications}, publisher={Springer
    Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }'
  chicago: Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.”
    <i>Partial Differential Equations and Applications</i> 3, no. 4 (2022). <a href="https://doi.org/10.1007/s42985-022-00186-z">https://doi.org/10.1007/s42985-022-00186-z</a>.
  ieee: 'M. Winkler, “Oscillatory decay in a degenerate parabolic equation,” <i>Partial
    Differential Equations and Applications</i>, vol. 3, no. 4, Art. no. 47, 2022,
    doi: <a href="https://doi.org/10.1007/s42985-022-00186-z">10.1007/s42985-022-00186-z</a>.'
  mla: Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” <i>Partial
    Differential Equations and Applications</i>, vol. 3, no. 4, 47, Springer Science
    and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1007/s42985-022-00186-z">10.1007/s42985-022-00186-z</a>.
  short: M. Winkler, Partial Differential Equations and Applications 3 (2022).
date_created: 2025-12-18T19:30:04Z
date_updated: 2025-12-18T20:05:38Z
doi: 10.1007/s42985-022-00186-z
intvolume: '         3'
issue: '4'
language:
- iso: eng
publication: Partial Differential Equations and Applications
publication_identifier:
  issn:
  - 2662-2963
  - 2662-2971
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Oscillatory decay in a degenerate parabolic equation
type: journal_article
user_id: '31496'
volume: 3
year: '2022'
...
---
_id: '63312'
abstract:
- lang: eng
  text: "<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;The chemotaxis
    system&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt;
    &lt;tex-math id=\"FE1\"&gt; \\begin{document}$ \\begin{array}{l}\\left\\{ \\begin{array}{l}
    \tu_t = \\nabla \\cdot \\big( D(u) \\nabla u \\big) - \\nabla \\cdot \\big( uS(x,
    u, v)\\cdot \\nabla v\\big), \\\\ \tv_t = \\Delta v -uv, \\end{array} \\right.
    \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;is considered in a bounded domain &lt;inline-formula&gt;&lt;tex-math
    id=\"M1\"&gt;\\begin{document}$ \\Omega\\subset \\mathbb{R}^n $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    &lt;inline-formula&gt;&lt;tex-math id=\"M2\"&gt;\\begin{document}$ n\\ge 2 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    with smooth boundary.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;It is shown
    that if &lt;inline-formula&gt;&lt;tex-math id=\"M3\"&gt;\\begin{document}$ D:
    [0, \\infty) \\to [0, \\infty) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    and &lt;inline-formula&gt;&lt;tex-math id=\"M4\"&gt;\\begin{document}$ S: \\overline{\\Omega}\\times
    [0, \\infty)\\times (0, \\infty)\\to \\mathbb{R}^{n\\times n} $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    are suitably smooth functions satisfying&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt;
    &lt;label/&gt; &lt;tex-math id=\"FE2\"&gt; \\begin{document}$ \\begin{array}{l}D(u)
    \\ge k_D u^{m-1} \t\\qquad {\\rm{for\\; all}}\\; u\\ge 0 \\end{array} $\\end{document}
    &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;and&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math
    id=\"FE3\"&gt; \\begin{document}$ \\begin{array}{l}|S(x, u, v)| \\le \\frac{S_0(v)}{v^\\alpha}
    \\qquad {\\rm{for\\; all}}\\; (x, u, v)\\; \\in \\Omega\\times (0, \\infty)^2
    \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;with some&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt;
    &lt;label/&gt; &lt;tex-math id=\"FE4\"&gt; \\begin{document}$ \\begin{array}{l}m&amp;gt;\\frac{3n-2}{2n}
    \t\\qquad {\\rm{and}}\\;\\alpha\\in [0, 1), \\end{array} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;and with some &lt;inline-formula&gt;&lt;tex-math
    id=\"M5\"&gt;\\begin{document}$ k_D&amp;gt;0 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    and nondecreasing &lt;inline-formula&gt;&lt;tex-math id=\"M6\"&gt;\\begin{document}$
    S_0: (0, \\infty)\\to (0, \\infty) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    then for all suitably regular initial data a corresponding no-flux type initial-boundary
    value problem admits a global bounded weak solution which actually is smooth and
    classical if &lt;inline-formula&gt;&lt;tex-math id=\"M7\"&gt;\\begin{document}$
    D(0)&amp;gt;0 $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;</jats:p>"
article_number: '6565'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities. <i>Discrete and Continuous Dynamical
    Systems - B</i>. 2022;27(11). doi:<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>
  apa: Winkler, M. (2022). Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities. <i>Discrete and Continuous Dynamical
    Systems - B</i>, <i>27</i>(11), Article 6565. <a href="https://doi.org/10.3934/dcdsb.2022009">https://doi.org/10.3934/dcdsb.2022009</a>
  bibtex: '@article{Winkler_2022, title={Approaching logarithmic singularities in
    quasilinear chemotaxis-consumption systems with signal-dependent sensitivities},
    volume={27}, DOI={<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>},
    number={116565}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Winkler, Michael}, year={2022}
    }'
  chicago: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear
    Chemotaxis-Consumption Systems with Signal-Dependent Sensitivities.” <i>Discrete
    and Continuous Dynamical Systems - B</i> 27, no. 11 (2022). <a href="https://doi.org/10.3934/dcdsb.2022009">https://doi.org/10.3934/dcdsb.2022009</a>.
  ieee: 'M. Winkler, “Approaching logarithmic singularities in quasilinear chemotaxis-consumption
    systems with signal-dependent sensitivities,” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 27, no. 11, Art. no. 6565, 2022, doi: <a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>.'
  mla: Winkler, Michael. “Approaching Logarithmic Singularities in Quasilinear Chemotaxis-Consumption
    Systems with Signal-Dependent Sensitivities.” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 27, no. 11, 6565, American Institute of Mathematical Sciences
    (AIMS), 2022, doi:<a href="https://doi.org/10.3934/dcdsb.2022009">10.3934/dcdsb.2022009</a>.
  short: M. Winkler, Discrete and Continuous Dynamical Systems - B 27 (2022).
date_created: 2025-12-18T19:30:32Z
date_updated: 2025-12-18T20:05:47Z
doi: 10.3934/dcdsb.2022009
intvolume: '        27'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems - B
publication_identifier:
  issn:
  - 1531-3492
  - 1553-524X
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Approaching logarithmic singularities in quasilinear chemotaxis-consumption
  systems with signal-dependent sensitivities
type: journal_article
user_id: '31496'
volume: 27
year: '2022'
...
---
_id: '63309'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>This manuscript is concerned with
    the problem of efficiently estimating chemotactic gradients, as forming a ubiquitous
    issue of key importance in virtually any proof of boundedness features in Keller–Segel
    type systems. A strategy is proposed which at its core relies on bounds for such
    quantities, conditional in the sense of involving certain Lebesgue norms of solution
    components that explicitly influence the signal evolution.</jats:p><jats:p>Applications
    of this procedure firstly provide apparently novel boundedness results for two
    particular classes chemotaxis systems, and apart from that are shown to significantly
    condense proofs for basically well‐known statements on boundedness in two further
    Keller–Segel type problems.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
    systems via conditional L∞$L^\infty$ estimates for taxis gradients. <i>Mathematische
    Nachrichten</i>. 2022;295(9):1840-1862. doi:<a href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>
  apa: Winkler, M. (2022). A unifying approach toward boundedness in Keller–Segel
    type cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis
    gradients. <i>Mathematische Nachrichten</i>, <i>295</i>(9), 1840–1862. <a href="https://doi.org/10.1002/mana.202000403">https://doi.org/10.1002/mana.202000403</a>
  bibtex: '@article{Winkler_2022, title={A unifying approach toward boundedness in
    Keller–Segel type cross‐diffusion systems via conditional L∞$L^\infty$ estimates
    for taxis gradients}, volume={295}, DOI={<a href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>},
    number={9}, journal={Mathematische Nachrichten}, publisher={Wiley}, author={Winkler,
    Michael}, year={2022}, pages={1840–1862} }'
  chicago: 'Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel
    Type Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis
    Gradients.” <i>Mathematische Nachrichten</i> 295, no. 9 (2022): 1840–62. <a href="https://doi.org/10.1002/mana.202000403">https://doi.org/10.1002/mana.202000403</a>.'
  ieee: 'M. Winkler, “A unifying approach toward boundedness in Keller–Segel type
    cross‐diffusion systems via conditional L∞$L^\infty$ estimates for taxis gradients,”
    <i>Mathematische Nachrichten</i>, vol. 295, no. 9, pp. 1840–1862, 2022, doi: <a
    href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>.'
  mla: Winkler, Michael. “A Unifying Approach toward Boundedness in Keller–Segel Type
    Cross‐diffusion Systems via Conditional L∞$L^\infty$ Estimates for Taxis Gradients.”
    <i>Mathematische Nachrichten</i>, vol. 295, no. 9, Wiley, 2022, pp. 1840–62, doi:<a
    href="https://doi.org/10.1002/mana.202000403">10.1002/mana.202000403</a>.
  short: M. Winkler, Mathematische Nachrichten 295 (2022) 1840–1862.
date_created: 2025-12-18T19:28:46Z
date_updated: 2025-12-18T20:05:19Z
doi: 10.1002/mana.202000403
intvolume: '       295'
issue: '9'
language:
- iso: eng
page: 1840-1862
publication: Mathematische Nachrichten
publication_identifier:
  issn:
  - 0025-584X
  - 1522-2616
publication_status: published
publisher: Wiley
status: public
title: A unifying approach toward boundedness in Keller–Segel type cross‐diffusion
  systems via conditional L∞$L^\infty$ estimates for taxis gradients
type: journal_article
user_id: '31496'
volume: 295
year: '2022'
...
---
_id: '63306'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. A critical blow-up exponent for flux limiation in a Keller-Segel
    system. <i>Indiana University Mathematics Journal</i>. 2022;71(4):1437-1465. doi:<a
    href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>
  apa: Winkler, M. (2022). A critical blow-up exponent for flux limiation in a Keller-Segel
    system. <i>Indiana University Mathematics Journal</i>, <i>71</i>(4), 1437–1465.
    <a href="https://doi.org/10.1512/iumj.2022.71.9042">https://doi.org/10.1512/iumj.2022.71.9042</a>
  bibtex: '@article{Winkler_2022, title={A critical blow-up exponent for flux limiation
    in a Keller-Segel system}, volume={71}, DOI={<a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>},
    number={4}, journal={Indiana University Mathematics Journal}, publisher={Indiana
    University Mathematics Journal}, author={Winkler, Michael}, year={2022}, pages={1437–1465}
    }'
  chicago: 'Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a
    Keller-Segel System.” <i>Indiana University Mathematics Journal</i> 71, no. 4
    (2022): 1437–65. <a href="https://doi.org/10.1512/iumj.2022.71.9042">https://doi.org/10.1512/iumj.2022.71.9042</a>.'
  ieee: 'M. Winkler, “A critical blow-up exponent for flux limiation in a Keller-Segel
    system,” <i>Indiana University Mathematics Journal</i>, vol. 71, no. 4, pp. 1437–1465,
    2022, doi: <a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>.'
  mla: Winkler, Michael. “A Critical Blow-up Exponent for Flux Limiation in a Keller-Segel
    System.” <i>Indiana University Mathematics Journal</i>, vol. 71, no. 4, Indiana
    University Mathematics Journal, 2022, pp. 1437–65, doi:<a href="https://doi.org/10.1512/iumj.2022.71.9042">10.1512/iumj.2022.71.9042</a>.
  short: M. Winkler, Indiana University Mathematics Journal 71 (2022) 1437–1465.
date_created: 2025-12-18T19:26:56Z
date_updated: 2025-12-18T20:04:53Z
doi: 10.1512/iumj.2022.71.9042
intvolume: '        71'
issue: '4'
language:
- iso: eng
page: 1437-1465
publication: Indiana University Mathematics Journal
publication_identifier:
  issn:
  - 0022-2518
publication_status: published
publisher: Indiana University Mathematics Journal
status: public
title: A critical blow-up exponent for flux limiation in a Keller-Segel system
type: journal_article
user_id: '31496'
volume: 71
year: '2022'
...
---
_id: '63284'
abstract:
- lang: eng
  text: '<jats:p> A no-flux initial-boundary value problem for the cross-diffusion
    system [Formula: see text] is considered in smoothly bounded domains [Formula:
    see text] with [Formula: see text]. It is shown that whenever [Formula: see text]
    is positive on [Formula: see text] and such that [Formula: see text] for some
    [Formula: see text], for all suitably regular positive initial data a global very
    weak solution, particularly preserving mass in its first component, can be constructed.
    This extends previous results which either concentrate on non-degenerate analogs,
    or are restricted to the special case [Formula: see text]. </jats:p><jats:p> To
    appropriately cope with the considerably stronger cross-degeneracies thus allowed
    through [Formula: see text] when [Formula: see text] is large, in its core part
    the analysis relies on the use of the Moser–Trudinger inequality in controlling
    the respective diffusion rates [Formula: see text] from below. </jats:p>'
article_number: '2250012'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model. <i>Bulletin of Mathematical
    Sciences</i>. 2022;13(02). doi:<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>
  apa: Winkler, M. (2022). Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model. <i>Bulletin of Mathematical
    Sciences</i>, <i>13</i>(02), Article 2250012. <a href="https://doi.org/10.1142/s1664360722500126">https://doi.org/10.1142/s1664360722500126</a>
  bibtex: '@article{Winkler_2022, title={Application of the Moser–Trudinger inequality
    in the construction of global solutions to a strongly degenerate migration model},
    volume={13}, DOI={<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>},
    number={022250012}, journal={Bulletin of Mathematical Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }'
  chicago: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the
    Construction of Global Solutions to a Strongly Degenerate Migration Model.” <i>Bulletin
    of Mathematical Sciences</i> 13, no. 02 (2022). <a href="https://doi.org/10.1142/s1664360722500126">https://doi.org/10.1142/s1664360722500126</a>.
  ieee: 'M. Winkler, “Application of the Moser–Trudinger inequality in the construction
    of global solutions to a strongly degenerate migration model,” <i>Bulletin of
    Mathematical Sciences</i>, vol. 13, no. 02, Art. no. 2250012, 2022, doi: <a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>.'
  mla: Winkler, Michael. “Application of the Moser–Trudinger Inequality in the Construction
    of Global Solutions to a Strongly Degenerate Migration Model.” <i>Bulletin of
    Mathematical Sciences</i>, vol. 13, no. 02, 2250012, World Scientific Pub Co Pte
    Ltd, 2022, doi:<a href="https://doi.org/10.1142/s1664360722500126">10.1142/s1664360722500126</a>.
  short: M. Winkler, Bulletin of Mathematical Sciences 13 (2022).
date_created: 2025-12-18T19:18:11Z
date_updated: 2025-12-18T20:07:05Z
doi: 10.1142/s1664360722500126
intvolume: '        13'
issue: '02'
language:
- iso: eng
publication: Bulletin of Mathematical Sciences
publication_identifier:
  issn:
  - 1664-3607
  - 1664-3615
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Application of the Moser–Trudinger inequality in the construction of global
  solutions to a strongly degenerate migration model
type: journal_article
user_id: '31496'
volume: 13
year: '2022'
...
---
_id: '63286'
abstract:
- lang: eng
  text: '<jats:p> The chemotaxis system [Formula: see text] is considered in a ball
    [Formula: see text]. </jats:p><jats:p> It is shown that if [Formula: see text]
    suitably generalizes the prototype given by [Formula: see text] with some [Formula:
    see text], and if diffusion is suitably weak in the sense that [Formula: see text]
    is such that there exist [Formula: see text] and [Formula: see text] fulfilling
    [Formula: see text] then for appropriate choices of sufficiently concentrated
    initial data, an associated no-flux initial-boundary value problem admits a global
    classical solution [Formula: see text] which blows up in infinite time and satisfies
    [Formula: see text] A major part of the proof is based on a comparison argument
    involving explicitly constructed subsolutions to a scalar parabolic problem satisfied
    by mass accumulation functions corresponding to solutions of ( ⋆ ). </jats:p>'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Exponential grow-up rates in a quasilinear Keller–Segel system.
    <i>Asymptotic Analysis</i>. 2022;131(1):33-57. doi:<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>
  apa: Winkler, M. (2022). Exponential grow-up rates in a quasilinear Keller–Segel
    system. <i>Asymptotic Analysis</i>, <i>131</i>(1), 33–57. <a href="https://doi.org/10.3233/asy-221765">https://doi.org/10.3233/asy-221765</a>
  bibtex: '@article{Winkler_2022, title={Exponential grow-up rates in a quasilinear
    Keller–Segel system}, volume={131}, DOI={<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>},
    number={1}, journal={Asymptotic Analysis}, publisher={SAGE Publications}, author={Winkler,
    Michael}, year={2022}, pages={33–57} }'
  chicago: 'Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
    System.” <i>Asymptotic Analysis</i> 131, no. 1 (2022): 33–57. <a href="https://doi.org/10.3233/asy-221765">https://doi.org/10.3233/asy-221765</a>.'
  ieee: 'M. Winkler, “Exponential grow-up rates in a quasilinear Keller–Segel system,”
    <i>Asymptotic Analysis</i>, vol. 131, no. 1, pp. 33–57, 2022, doi: <a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>.'
  mla: Winkler, Michael. “Exponential Grow-up Rates in a Quasilinear Keller–Segel
    System.” <i>Asymptotic Analysis</i>, vol. 131, no. 1, SAGE Publications, 2022,
    pp. 33–57, doi:<a href="https://doi.org/10.3233/asy-221765">10.3233/asy-221765</a>.
  short: M. Winkler, Asymptotic Analysis 131 (2022) 33–57.
date_created: 2025-12-18T19:18:51Z
date_updated: 2025-12-18T20:07:19Z
doi: 10.3233/asy-221765
intvolume: '       131'
issue: '1'
language:
- iso: eng
page: 33-57
publication: Asymptotic Analysis
publication_identifier:
  issn:
  - 0921-7134
  - 1875-8576
publication_status: published
publisher: SAGE Publications
status: public
title: Exponential grow-up rates in a quasilinear Keller–Segel system
type: journal_article
user_id: '31496'
volume: 131
year: '2022'
...
---
_id: '63293'
abstract:
- lang: eng
  text: "<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;The Cauchy problem
    in &lt;inline-formula&gt;&lt;tex-math id=\"M2\"&gt;\\begin{document}$  \\mathbb{R}^3
    $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for the chemotaxis-Navier–Stokes
    system&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt;
    &lt;tex-math id=\"FE1\"&gt; \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l}
    \     n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t
    + u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta
    u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array}
    \\right. \\end{eqnarray*} $\\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p
    style='text-indent:20px;'&gt;is considered. Under suitable conditions on the initial
    data &lt;inline-formula&gt;&lt;tex-math id=\"M3\"&gt;\\begin{document}$ (n_0,
    c_0, u_0) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, with regard
    to the crucial first component requiring that &lt;inline-formula&gt;&lt;tex-math
    id=\"M4\"&gt;\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    be nonnegative and such that &lt;inline-formula&gt;&lt;tex-math id=\"M5\"&gt;\\begin{document}$
    (n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,
    a globally defined weak solution with &lt;inline-formula&gt;&lt;tex-math id=\"M6\"&gt;\\begin{document}$
    (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;
    is constructed. Apart from that, assuming that moreover &lt;inline-formula&gt;&lt;tex-math
    id=\"M7\"&gt;\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx
    $\\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is finite, it is shown
    that a weak solution exists which enjoys further regularity features and preserves
    mass in an appropriate sense.&lt;/p&gt;</jats:p>"
article_number: '5201'
author:
- first_name: Kyungkeun
  full_name: Kang, Kyungkeun
  last_name: Kang
- first_name: Jihoon
  full_name: Lee, Jihoon
  last_name: Lee
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $. <i>Discrete and Continuous Dynamical Systems</i>.
    2022;42(11). doi:<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>
  apa: Kang, K., Lee, J., &#38; Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $. <i>Discrete and Continuous Dynamical Systems</i>,
    <i>42</i>(11), Article 5201. <a href="https://doi.org/10.3934/dcds.2022091">https://doi.org/10.3934/dcds.2022091</a>
  bibtex: '@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $}, volume={42}, DOI={<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>},
    number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon
    and Winkler, Michael}, year={2022} }'
  chicago: Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions
    to a Chemotaxis-Navier-Stokes System in $  \mathbb{R}^3 $.” <i>Discrete and Continuous
    Dynamical Systems</i> 42, no. 11 (2022). <a href="https://doi.org/10.3934/dcds.2022091">https://doi.org/10.3934/dcds.2022091</a>.
  ieee: 'K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes
    system in $  \mathbb{R}^3 $,” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 42, no. 11, Art. no. 5201, 2022, doi: <a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>.'
  mla: Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes
    System in $  \mathbb{R}^3 $.” <i>Discrete and Continuous Dynamical Systems</i>,
    vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022,
    doi:<a href="https://doi.org/10.3934/dcds.2022091">10.3934/dcds.2022091</a>.
  short: K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42
    (2022).
date_created: 2025-12-18T19:22:04Z
date_updated: 2025-12-18T20:08:21Z
doi: 10.3934/dcds.2022091
intvolume: '        42'
issue: '11'
language:
- iso: eng
publication: Discrete and Continuous Dynamical Systems
publication_identifier:
  issn:
  - 1078-0947
  - 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global weak solutions to a chemotaxis-Navier-Stokes system in $  \mathbb{R}^3
  $
type: journal_article
user_id: '31496'
volume: 42
year: '2022'
...