---
_id: '63621'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Refining Hölder regularity theory in degenerate drift-diffusion equations.
    <i>Annali di Matematica Pura ed Applicata (1923 -)</i>. Published online 2026.
    doi:<a href="https://doi.org/10.1007/s10231-025-01642-4">10.1007/s10231-025-01642-4</a>
  apa: Black, T. (2026). Refining Hölder regularity theory in degenerate drift-diffusion
    equations. <i>Annali Di Matematica Pura Ed Applicata (1923 -)</i>. <a href="https://doi.org/10.1007/s10231-025-01642-4">https://doi.org/10.1007/s10231-025-01642-4</a>
  bibtex: '@article{Black_2026, title={Refining Hölder regularity theory in degenerate
    drift-diffusion equations}, DOI={<a href="https://doi.org/10.1007/s10231-025-01642-4">10.1007/s10231-025-01642-4</a>},
    journal={Annali di Matematica Pura ed Applicata (1923 -)}, publisher={Springer
    Science and Business Media LLC}, author={Black, Tobias}, year={2026} }'
  chicago: Black, Tobias. “Refining Hölder Regularity Theory in Degenerate Drift-Diffusion
    Equations.” <i>Annali Di Matematica Pura Ed Applicata (1923 -)</i>, 2026. <a href="https://doi.org/10.1007/s10231-025-01642-4">https://doi.org/10.1007/s10231-025-01642-4</a>.
  ieee: 'T. Black, “Refining Hölder regularity theory in degenerate drift-diffusion
    equations,” <i>Annali di Matematica Pura ed Applicata (1923 -)</i>, 2026, doi:
    <a href="https://doi.org/10.1007/s10231-025-01642-4">10.1007/s10231-025-01642-4</a>.'
  mla: Black, Tobias. “Refining Hölder Regularity Theory in Degenerate Drift-Diffusion
    Equations.” <i>Annali Di Matematica Pura Ed Applicata (1923 -)</i>, Springer Science
    and Business Media LLC, 2026, doi:<a href="https://doi.org/10.1007/s10231-025-01642-4">10.1007/s10231-025-01642-4</a>.
  short: T. Black, Annali Di Matematica Pura Ed Applicata (1923 -) (2026).
date_created: 2026-01-15T10:09:43Z
date_updated: 2026-01-15T10:10:58Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s10231-025-01642-4
language:
- iso: eng
publication: Annali di Matematica Pura ed Applicata (1923 -)
publication_identifier:
  issn:
  - 0373-3114
  - 1618-1891
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Refining Hölder regularity theory in degenerate drift-diffusion equations
type: journal_article
user_id: '23686'
year: '2026'
...
---
_id: '63672'
article_number: '24'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Shohei
  full_name: Kohatsu, Shohei
  last_name: Kohatsu
- first_name: Duan
  full_name: Wu, Duan
  last_name: Wu
citation:
  ama: Black T, Kohatsu S, Wu D. Global solvability and large-time behavior in a doubly
    degenerate migration model involving saturated signal consumption. <i>Journal
    of Evolution Equations</i>. 2026;26(1). doi:<a href="https://doi.org/10.1007/s00028-025-01163-w">10.1007/s00028-025-01163-w</a>
  apa: Black, T., Kohatsu, S., &#38; Wu, D. (2026). Global solvability and large-time
    behavior in a doubly degenerate migration model involving saturated signal consumption.
    <i>Journal of Evolution Equations</i>, <i>26</i>(1), Article 24. <a href="https://doi.org/10.1007/s00028-025-01163-w">https://doi.org/10.1007/s00028-025-01163-w</a>
  bibtex: '@article{Black_Kohatsu_Wu_2026, title={Global solvability and large-time
    behavior in a doubly degenerate migration model involving saturated signal consumption},
    volume={26}, DOI={<a href="https://doi.org/10.1007/s00028-025-01163-w">10.1007/s00028-025-01163-w</a>},
    number={124}, journal={Journal of Evolution Equations}, publisher={Springer Science
    and Business Media LLC}, author={Black, Tobias and Kohatsu, Shohei and Wu, Duan},
    year={2026} }'
  chicago: Black, Tobias, Shohei Kohatsu, and Duan Wu. “Global Solvability and Large-Time
    Behavior in a Doubly Degenerate Migration Model Involving Saturated Signal Consumption.”
    <i>Journal of Evolution Equations</i> 26, no. 1 (2026). <a href="https://doi.org/10.1007/s00028-025-01163-w">https://doi.org/10.1007/s00028-025-01163-w</a>.
  ieee: 'T. Black, S. Kohatsu, and D. Wu, “Global solvability and large-time behavior
    in a doubly degenerate migration model involving saturated signal consumption,”
    <i>Journal of Evolution Equations</i>, vol. 26, no. 1, Art. no. 24, 2026, doi:
    <a href="https://doi.org/10.1007/s00028-025-01163-w">10.1007/s00028-025-01163-w</a>.'
  mla: Black, Tobias, et al. “Global Solvability and Large-Time Behavior in a Doubly
    Degenerate Migration Model Involving Saturated Signal Consumption.” <i>Journal
    of Evolution Equations</i>, vol. 26, no. 1, 24, Springer Science and Business
    Media LLC, 2026, doi:<a href="https://doi.org/10.1007/s00028-025-01163-w">10.1007/s00028-025-01163-w</a>.
  short: T. Black, S. Kohatsu, D. Wu, Journal of Evolution Equations 26 (2026).
date_created: 2026-01-20T14:13:53Z
date_updated: 2026-01-20T14:14:50Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00028-025-01163-w
intvolume: '        26'
issue: '1'
language:
- iso: eng
publication: Journal of Evolution Equations
publication_identifier:
  issn:
  - 1424-3199
  - 1424-3202
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Global solvability and large-time behavior in a doubly degenerate migration
  model involving saturated signal consumption
type: journal_article
user_id: '23686'
volume: 26
year: '2026'
...
---
_id: '56960'
article_number: '109361'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Absence of dead-core formations in chemotaxis systems with degenerate
    diffusion. <i>Applied Mathematics Letters</i>. 2025;161. doi:<a href="https://doi.org/10.1016/j.aml.2024.109361">10.1016/j.aml.2024.109361</a>
  apa: Black, T. (2025). Absence of dead-core formations in chemotaxis systems with
    degenerate diffusion. <i>Applied Mathematics Letters</i>, <i>161</i>, Article
    109361. <a href="https://doi.org/10.1016/j.aml.2024.109361">https://doi.org/10.1016/j.aml.2024.109361</a>
  bibtex: '@article{Black_2025, title={Absence of dead-core formations in chemotaxis
    systems with degenerate diffusion}, volume={161}, DOI={<a href="https://doi.org/10.1016/j.aml.2024.109361">10.1016/j.aml.2024.109361</a>},
    number={109361}, journal={Applied Mathematics Letters}, publisher={Elsevier BV},
    author={Black, Tobias}, year={2025} }'
  chicago: Black, Tobias. “Absence of Dead-Core Formations in Chemotaxis Systems with
    Degenerate Diffusion.” <i>Applied Mathematics Letters</i> 161 (2025). <a href="https://doi.org/10.1016/j.aml.2024.109361">https://doi.org/10.1016/j.aml.2024.109361</a>.
  ieee: 'T. Black, “Absence of dead-core formations in chemotaxis systems with degenerate
    diffusion,” <i>Applied Mathematics Letters</i>, vol. 161, Art. no. 109361, 2025,
    doi: <a href="https://doi.org/10.1016/j.aml.2024.109361">10.1016/j.aml.2024.109361</a>.'
  mla: Black, Tobias. “Absence of Dead-Core Formations in Chemotaxis Systems with
    Degenerate Diffusion.” <i>Applied Mathematics Letters</i>, vol. 161, 109361, Elsevier
    BV, 2025, doi:<a href="https://doi.org/10.1016/j.aml.2024.109361">10.1016/j.aml.2024.109361</a>.
  short: T. Black, Applied Mathematics Letters 161 (2025).
date_created: 2024-11-08T13:28:04Z
date_updated: 2024-11-08T13:30:02Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1016/j.aml.2024.109361
intvolume: '       161'
language:
- iso: eng
publication: Applied Mathematics Letters
publication_identifier:
  issn:
  - 0893-9659
publication_status: published
publisher: Elsevier BV
status: public
title: Absence of dead-core formations in chemotaxis systems with degenerate diffusion
type: journal_article
user_id: '23686'
volume: 161
year: '2025'
...
---
_id: '60205'
article_number: '113555'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: 'Black T. Very mild diffusion enhancement and singular sensitivity: Existence
    of bounded weak solutions in a two-dimensional chemotaxis-Navier–Stokes system.
    <i>Journal of Differential Equations</i>. 2025;443. doi:<a href="https://doi.org/10.1016/j.jde.2025.113555">10.1016/j.jde.2025.113555</a>'
  apa: 'Black, T. (2025). Very mild diffusion enhancement and singular sensitivity:
    Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier–Stokes
    system. <i>Journal of Differential Equations</i>, <i>443</i>, Article 113555.
    <a href="https://doi.org/10.1016/j.jde.2025.113555">https://doi.org/10.1016/j.jde.2025.113555</a>'
  bibtex: '@article{Black_2025, title={Very mild diffusion enhancement and singular
    sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier–Stokes
    system}, volume={443}, DOI={<a href="https://doi.org/10.1016/j.jde.2025.113555">10.1016/j.jde.2025.113555</a>},
    number={113555}, journal={Journal of Differential Equations}, publisher={Elsevier
    BV}, author={Black, Tobias}, year={2025} }'
  chicago: 'Black, Tobias. “Very Mild Diffusion Enhancement and Singular Sensitivity:
    Existence of Bounded Weak Solutions in a Two-Dimensional Chemotaxis-Navier–Stokes
    System.” <i>Journal of Differential Equations</i> 443 (2025). <a href="https://doi.org/10.1016/j.jde.2025.113555">https://doi.org/10.1016/j.jde.2025.113555</a>.'
  ieee: 'T. Black, “Very mild diffusion enhancement and singular sensitivity: Existence
    of bounded weak solutions in a two-dimensional chemotaxis-Navier–Stokes system,”
    <i>Journal of Differential Equations</i>, vol. 443, Art. no. 113555, 2025, doi:
    <a href="https://doi.org/10.1016/j.jde.2025.113555">10.1016/j.jde.2025.113555</a>.'
  mla: 'Black, Tobias. “Very Mild Diffusion Enhancement and Singular Sensitivity:
    Existence of Bounded Weak Solutions in a Two-Dimensional Chemotaxis-Navier–Stokes
    System.” <i>Journal of Differential Equations</i>, vol. 443, 113555, Elsevier
    BV, 2025, doi:<a href="https://doi.org/10.1016/j.jde.2025.113555">10.1016/j.jde.2025.113555</a>.'
  short: T. Black, Journal of Differential Equations 443 (2025).
date_created: 2025-06-13T11:12:23Z
date_updated: 2025-06-13T11:13:22Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1016/j.jde.2025.113555
intvolume: '       443'
language:
- iso: eng
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: 'Very mild diffusion enhancement and singular sensitivity: Existence of bounded
  weak solutions in a two-dimensional chemotaxis-Navier–Stokes system'
type: journal_article
user_id: '23686'
volume: 443
year: '2025'
...
---
_id: '43105'
article_number: '103868'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Mario
  full_name: Fuest, Mario
  last_name: Fuest
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Masaaki
  full_name: Mizukami, Masaaki
  last_name: Mizukami
citation:
  ama: 'Black T, Fuest M, Lankeit J, Mizukami M. Possible points of blow-up in chemotaxis
    systems with spatially heterogeneous logistic source. <i>Nonlinear Analysis: Real
    World Applications</i>. 2023;73. doi:<a href="https://doi.org/10.1016/j.nonrwa.2023.103868">10.1016/j.nonrwa.2023.103868</a>'
  apa: 'Black, T., Fuest, M., Lankeit, J., &#38; Mizukami, M. (2023). Possible points
    of blow-up in chemotaxis systems with spatially heterogeneous logistic source.
    <i>Nonlinear Analysis: Real World Applications</i>, <i>73</i>, Article 103868.
    <a href="https://doi.org/10.1016/j.nonrwa.2023.103868">https://doi.org/10.1016/j.nonrwa.2023.103868</a>'
  bibtex: '@article{Black_Fuest_Lankeit_Mizukami_2023, title={Possible points of blow-up
    in chemotaxis systems with spatially heterogeneous logistic source}, volume={73},
    DOI={<a href="https://doi.org/10.1016/j.nonrwa.2023.103868">10.1016/j.nonrwa.2023.103868</a>},
    number={103868}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier
    BV}, author={Black, Tobias and Fuest, Mario and Lankeit, Johannes and Mizukami,
    Masaaki}, year={2023} }'
  chicago: 'Black, Tobias, Mario Fuest, Johannes Lankeit, and Masaaki Mizukami. “Possible
    Points of Blow-up in Chemotaxis Systems with Spatially Heterogeneous Logistic
    Source.” <i>Nonlinear Analysis: Real World Applications</i> 73 (2023). <a href="https://doi.org/10.1016/j.nonrwa.2023.103868">https://doi.org/10.1016/j.nonrwa.2023.103868</a>.'
  ieee: 'T. Black, M. Fuest, J. Lankeit, and M. Mizukami, “Possible points of blow-up
    in chemotaxis systems with spatially heterogeneous logistic source,” <i>Nonlinear
    Analysis: Real World Applications</i>, vol. 73, Art. no. 103868, 2023, doi: <a
    href="https://doi.org/10.1016/j.nonrwa.2023.103868">10.1016/j.nonrwa.2023.103868</a>.'
  mla: 'Black, Tobias, et al. “Possible Points of Blow-up in Chemotaxis Systems with
    Spatially Heterogeneous Logistic Source.” <i>Nonlinear Analysis: Real World Applications</i>,
    vol. 73, 103868, Elsevier BV, 2023, doi:<a href="https://doi.org/10.1016/j.nonrwa.2023.103868">10.1016/j.nonrwa.2023.103868</a>.'
  short: 'T. Black, M. Fuest, J. Lankeit, M. Mizukami, Nonlinear Analysis: Real World
    Applications 73 (2023).'
date_created: 2023-03-27T07:25:58Z
date_updated: 2023-03-27T07:27:03Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1016/j.nonrwa.2023.103868
intvolume: '        73'
keyword:
- Applied Mathematics
- Computational Mathematics
- General Economics
- Econometrics and Finance
- General Engineering
- General Medicine
- Analysis
language:
- iso: eng
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Possible points of blow-up in chemotaxis systems with spatially heterogeneous
  logistic source
type: journal_article
user_id: '23686'
volume: 73
year: '2023'
...
---
_id: '34677'
article_number: '96'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Chunyan
  full_name: Wu, Chunyan
  last_name: Wu
citation:
  ama: 'Black T, Wu C. Prescribed signal concentration on the boundary: eventual smoothness
    in a chemotaxis-Navier–Stokes system with logistic proliferation. <i>Calculus
    of Variations and Partial Differential Equations</i>. 2022;61(3). doi:<a href="https://doi.org/10.1007/s00526-022-02201-y">10.1007/s00526-022-02201-y</a>'
  apa: 'Black, T., &#38; Wu, C. (2022). Prescribed signal concentration on the boundary:
    eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation.
    <i>Calculus of Variations and Partial Differential Equations</i>, <i>61</i>(3),
    Article 96. <a href="https://doi.org/10.1007/s00526-022-02201-y">https://doi.org/10.1007/s00526-022-02201-y</a>'
  bibtex: '@article{Black_Wu_2022, title={Prescribed signal concentration on the boundary:
    eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation},
    volume={61}, DOI={<a href="https://doi.org/10.1007/s00526-022-02201-y">10.1007/s00526-022-02201-y</a>},
    number={396}, journal={Calculus of Variations and Partial Differential Equations},
    publisher={Springer Science and Business Media LLC}, author={Black, Tobias and
    Wu, Chunyan}, year={2022} }'
  chicago: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the
    Boundary: Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic
    Proliferation.” <i>Calculus of Variations and Partial Differential Equations</i>
    61, no. 3 (2022). <a href="https://doi.org/10.1007/s00526-022-02201-y">https://doi.org/10.1007/s00526-022-02201-y</a>.'
  ieee: 'T. Black and C. Wu, “Prescribed signal concentration on the boundary: eventual
    smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation,”
    <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no.
    3, Art. no. 96, 2022, doi: <a href="https://doi.org/10.1007/s00526-022-02201-y">10.1007/s00526-022-02201-y</a>.'
  mla: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary:
    Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic Proliferation.”
    <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no.
    3, 96, Springer Science and Business Media LLC, 2022, doi:<a href="https://doi.org/10.1007/s00526-022-02201-y">10.1007/s00526-022-02201-y</a>.'
  short: T. Black, C. Wu, Calculus of Variations and Partial Differential Equations
    61 (2022).
date_created: 2022-12-21T09:50:59Z
date_updated: 2023-07-10T11:37:27Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00526-022-02201-y
intvolume: '        61'
issue: '3'
keyword:
- Applied Mathematics
- Analysis
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: 'Prescribed signal concentration on the boundary: eventual smoothness in a
  chemotaxis-Navier–Stokes system with logistic proliferation'
type: journal_article
user_id: '23686'
volume: 61
year: '2022'
...
---
_id: '34673'
article_number: '96'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Mario
  full_name: Fuest, Mario
  last_name: Fuest
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
citation:
  ama: Black T, Fuest M, Lankeit J. Relaxed parameter conditions for chemotactic collapse
    in logistic-type parabolic–elliptic Keller–Segel systems. <i>Zeitschrift für angewandte
    Mathematik und Physik</i>. 2021;72(3). doi:<a href="https://doi.org/10.1007/s00033-021-01524-8">10.1007/s00033-021-01524-8</a>
  apa: Black, T., Fuest, M., &#38; Lankeit, J. (2021). Relaxed parameter conditions
    for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems.
    <i>Zeitschrift Für Angewandte Mathematik Und Physik</i>, <i>72</i>(3), Article
    96. <a href="https://doi.org/10.1007/s00033-021-01524-8">https://doi.org/10.1007/s00033-021-01524-8</a>
  bibtex: '@article{Black_Fuest_Lankeit_2021, title={Relaxed parameter conditions
    for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems},
    volume={72}, DOI={<a href="https://doi.org/10.1007/s00033-021-01524-8">10.1007/s00033-021-01524-8</a>},
    number={396}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer
    Science and Business Media LLC}, author={Black, Tobias and Fuest, Mario and Lankeit,
    Johannes}, year={2021} }'
  chicago: Black, Tobias, Mario Fuest, and Johannes Lankeit. “Relaxed Parameter Conditions
    for Chemotactic Collapse in Logistic-Type Parabolic–Elliptic Keller–Segel Systems.”
    <i>Zeitschrift Für Angewandte Mathematik Und Physik</i> 72, no. 3 (2021). <a href="https://doi.org/10.1007/s00033-021-01524-8">https://doi.org/10.1007/s00033-021-01524-8</a>.
  ieee: 'T. Black, M. Fuest, and J. Lankeit, “Relaxed parameter conditions for chemotactic
    collapse in logistic-type parabolic–elliptic Keller–Segel systems,” <i>Zeitschrift
    für angewandte Mathematik und Physik</i>, vol. 72, no. 3, Art. no. 96, 2021, doi:
    <a href="https://doi.org/10.1007/s00033-021-01524-8">10.1007/s00033-021-01524-8</a>.'
  mla: Black, Tobias, et al. “Relaxed Parameter Conditions for Chemotactic Collapse
    in Logistic-Type Parabolic–Elliptic Keller–Segel Systems.” <i>Zeitschrift Für
    Angewandte Mathematik Und Physik</i>, vol. 72, no. 3, 96, Springer Science and
    Business Media LLC, 2021, doi:<a href="https://doi.org/10.1007/s00033-021-01524-8">10.1007/s00033-021-01524-8</a>.
  short: T. Black, M. Fuest, J. Lankeit, Zeitschrift Für Angewandte Mathematik Und
    Physik 72 (2021).
date_created: 2022-12-21T09:48:36Z
date_updated: 2023-07-10T11:39:23Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00033-021-01524-8
intvolume: '        72'
issue: '3'
keyword:
- Applied Mathematics
- General Physics and Astronomy
- General Mathematics
language:
- iso: eng
publication: Zeitschrift für angewandte Mathematik und Physik
publication_identifier:
  issn:
  - 0044-2275
  - 1420-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic
  Keller–Segel systems
type: journal_article
user_id: '23686'
volume: 72
year: '2021'
...
---
_id: '34675'
article_number: '135'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Chunyan
  full_name: Wu, Chunyan
  last_name: Wu
citation:
  ama: 'Black T, Wu C. Prescribed signal concentration on the boundary: Weak solvability
    in a chemotaxis-Stokes system with proliferation. <i>Zeitschrift für angewandte
    Mathematik und Physik</i>. 2021;72(4). doi:<a href="https://doi.org/10.1007/s00033-021-01565-z">10.1007/s00033-021-01565-z</a>'
  apa: 'Black, T., &#38; Wu, C. (2021). Prescribed signal concentration on the boundary:
    Weak solvability in a chemotaxis-Stokes system with proliferation. <i>Zeitschrift
    Für Angewandte Mathematik Und Physik</i>, <i>72</i>(4), Article 135. <a href="https://doi.org/10.1007/s00033-021-01565-z">https://doi.org/10.1007/s00033-021-01565-z</a>'
  bibtex: '@article{Black_Wu_2021, title={Prescribed signal concentration on the boundary:
    Weak solvability in a chemotaxis-Stokes system with proliferation}, volume={72},
    DOI={<a href="https://doi.org/10.1007/s00033-021-01565-z">10.1007/s00033-021-01565-z</a>},
    number={4135}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer
    Science and Business Media LLC}, author={Black, Tobias and Wu, Chunyan}, year={2021}
    }'
  chicago: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the
    Boundary: Weak Solvability in a Chemotaxis-Stokes System with Proliferation.”
    <i>Zeitschrift Für Angewandte Mathematik Und Physik</i> 72, no. 4 (2021). <a href="https://doi.org/10.1007/s00033-021-01565-z">https://doi.org/10.1007/s00033-021-01565-z</a>.'
  ieee: 'T. Black and C. Wu, “Prescribed signal concentration on the boundary: Weak
    solvability in a chemotaxis-Stokes system with proliferation,” <i>Zeitschrift
    für angewandte Mathematik und Physik</i>, vol. 72, no. 4, Art. no. 135, 2021,
    doi: <a href="https://doi.org/10.1007/s00033-021-01565-z">10.1007/s00033-021-01565-z</a>.'
  mla: 'Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary:
    Weak Solvability in a Chemotaxis-Stokes System with Proliferation.” <i>Zeitschrift
    Für Angewandte Mathematik Und Physik</i>, vol. 72, no. 4, 135, Springer Science
    and Business Media LLC, 2021, doi:<a href="https://doi.org/10.1007/s00033-021-01565-z">10.1007/s00033-021-01565-z</a>.'
  short: T. Black, C. Wu, Zeitschrift Für Angewandte Mathematik Und Physik 72 (2021).
date_created: 2022-12-21T09:48:45Z
date_updated: 2023-07-10T11:39:30Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00033-021-01565-z
intvolume: '        72'
issue: '4'
keyword:
- Applied Mathematics
- General Physics and Astronomy
- General Mathematics
language:
- iso: eng
publication: Zeitschrift für angewandte Mathematik und Physik
publication_identifier:
  issn:
  - 0044-2275
  - 1420-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: 'Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes
  system with proliferation'
type: journal_article
user_id: '23686'
volume: 72
year: '2021'
...
---
_id: '63291'
abstract:
- lang: eng
  text: <jats:p> An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes
    model with porous medium type diffusion is considered. Previous related literature
    has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip
    boundary conditions for the density of cells, the chemical concentration and the
    fluid velocity field, respectively; in particular, available qualitative results
    strongly indicate that only trivial solution behavior can be expected on large
    time scales. In line with refined modeling approaches to oxygen evolution near
    fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant
    concentration on the boundary. Despite an apparent loss of mathematically favorable
    energy structures thereby induced, by means of an alternative variational approach
    a basic theory of global existence is developed in a natural framework of weak
    solvability. Beyond this, some additional qualitative information on the large
    time behavior of these solutions is derived by identifying a certain global relaxation
    property. Specifically, a second result asserts, within a suitable topological
    setting, the existence of a bounded set which eventually absorbs each individual
    of the obtained trajectories, and the diameter of which is bounded only by the
    physically relevant quantities of total population size and prescribed boundary
    concentration of the chemical signal. </jats:p>
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Black T, Winkler M. Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
    system with prescribed signal concentration on the boundary. <i>Mathematical Models
    and Methods in Applied Sciences</i>. 2021;32(01):137-173. doi:<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>
  apa: Black, T., &#38; Winkler, M. (2021). Global weak solutions and absorbing sets
    in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the
    boundary. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>32</i>(01),
    137–173. <a href="https://doi.org/10.1142/s021820252250004x">https://doi.org/10.1142/s021820252250004x</a>
  bibtex: '@article{Black_Winkler_2021, title={Global weak solutions and absorbing
    sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration
    on the boundary}, volume={32}, DOI={<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>},
    number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Black, Tobias and Winkler, Michael}, year={2021},
    pages={137–173} }'
  chicago: 'Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing
    Sets in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration
    on the Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i> 32,
    no. 01 (2021): 137–73. <a href="https://doi.org/10.1142/s021820252250004x">https://doi.org/10.1142/s021820252250004x</a>.'
  ieee: 'T. Black and M. Winkler, “Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes
    system with prescribed signal concentration on the boundary,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 32, no. 01, pp. 137–173, 2021,
    doi: <a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>.'
  mla: Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing Sets
    in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration on the
    Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32,
    no. 01, World Scientific Pub Co Pte Ltd, 2021, pp. 137–73, doi:<a href="https://doi.org/10.1142/s021820252250004x">10.1142/s021820252250004x</a>.
  short: T. Black, M. Winkler, Mathematical Models and Methods in Applied Sciences
    32 (2021) 137–173.
date_created: 2025-12-18T19:20:48Z
date_updated: 2025-12-18T20:08:01Z
doi: 10.1142/s021820252250004x
intvolume: '        32'
issue: '01'
language:
- iso: eng
page: 137-173
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  issn:
  - 0218-2025
  - 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system
  with prescribed signal concentration on the boundary
type: journal_article
user_id: '31496'
volume: 32
year: '2021'
...
---
_id: '34670'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Global generalized solutions to a forager–exploiter model with superlinear
    degradation and their eventual regularity properties. <i>Mathematical Models and
    Methods in Applied Sciences</i>. 2020;30(06):1075-1117. doi:<a href="https://doi.org/10.1142/s0218202520400072">10.1142/s0218202520400072</a>
  apa: Black, T. (2020). Global generalized solutions to a forager–exploiter model
    with superlinear degradation and their eventual regularity properties. <i>Mathematical
    Models and Methods in Applied Sciences</i>, <i>30</i>(06), 1075–1117. <a href="https://doi.org/10.1142/s0218202520400072">https://doi.org/10.1142/s0218202520400072</a>
  bibtex: '@article{Black_2020, title={Global generalized solutions to a forager–exploiter
    model with superlinear degradation and their eventual regularity properties},
    volume={30}, DOI={<a href="https://doi.org/10.1142/s0218202520400072">10.1142/s0218202520400072</a>},
    number={06}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Lt}, author={Black, Tobias}, year={2020}, pages={1075–1117}
    }'
  chicago: 'Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model
    with Superlinear Degradation and Their Eventual Regularity Properties.” <i>Mathematical
    Models and Methods in Applied Sciences</i> 30, no. 06 (2020): 1075–1117. <a href="https://doi.org/10.1142/s0218202520400072">https://doi.org/10.1142/s0218202520400072</a>.'
  ieee: 'T. Black, “Global generalized solutions to a forager–exploiter model with
    superlinear degradation and their eventual regularity properties,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 06, pp. 1075–1117, 2020,
    doi: <a href="https://doi.org/10.1142/s0218202520400072">10.1142/s0218202520400072</a>.'
  mla: Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model with
    Superlinear Degradation and Their Eventual Regularity Properties.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 06, World Scientific
    Pub Co Pte Lt, 2020, pp. 1075–117, doi:<a href="https://doi.org/10.1142/s0218202520400072">10.1142/s0218202520400072</a>.
  short: T. Black, Mathematical Models and Methods in Applied Sciences 30 (2020) 1075–1117.
date_created: 2022-12-21T09:48:11Z
date_updated: 2023-07-10T11:39:59Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1142/s0218202520400072
intvolume: '        30'
issue: '06'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 1075-1117
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  issn:
  - 0218-2025
  - 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Lt
status: public
title: Global generalized solutions to a forager–exploiter model with superlinear
  degradation and their eventual regularity properties
type: journal_article
user_id: '23686'
volume: 30
year: '2020'
...
---
_id: '34672'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Global generalized solutions to a parabolic-elliptic Keller-Segel
    system with singular sensitivity. <i>Discrete &#38;amp; Continuous Dynamical Systems
    - S</i>. 2019;13(2):119-137. doi:<a href="https://doi.org/10.3934/dcdss.2020007">10.3934/dcdss.2020007</a>
  apa: Black, T. (2019). Global generalized solutions to a parabolic-elliptic Keller-Segel
    system with singular sensitivity. <i>Discrete &#38;amp; Continuous Dynamical Systems
    - S</i>, <i>13</i>(2), 119–137. <a href="https://doi.org/10.3934/dcdss.2020007">https://doi.org/10.3934/dcdss.2020007</a>
  bibtex: '@article{Black_2019, title={Global generalized solutions to a parabolic-elliptic
    Keller-Segel system with singular sensitivity}, volume={13}, DOI={<a href="https://doi.org/10.3934/dcdss.2020007">10.3934/dcdss.2020007</a>},
    number={2}, journal={Discrete &#38;amp; Continuous Dynamical Systems - S}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Black, Tobias}, year={2019},
    pages={119–137} }'
  chicago: 'Black, Tobias. “Global Generalized Solutions to a Parabolic-Elliptic Keller-Segel
    System with Singular Sensitivity.” <i>Discrete &#38;amp; Continuous Dynamical
    Systems - S</i> 13, no. 2 (2019): 119–37. <a href="https://doi.org/10.3934/dcdss.2020007">https://doi.org/10.3934/dcdss.2020007</a>.'
  ieee: 'T. Black, “Global generalized solutions to a parabolic-elliptic Keller-Segel
    system with singular sensitivity,” <i>Discrete &#38;amp; Continuous Dynamical
    Systems - S</i>, vol. 13, no. 2, pp. 119–137, 2019, doi: <a href="https://doi.org/10.3934/dcdss.2020007">10.3934/dcdss.2020007</a>.'
  mla: Black, Tobias. “Global Generalized Solutions to a Parabolic-Elliptic Keller-Segel
    System with Singular Sensitivity.” <i>Discrete &#38;amp; Continuous Dynamical
    Systems - S</i>, vol. 13, no. 2, American Institute of Mathematical Sciences (AIMS),
    2019, pp. 119–37, doi:<a href="https://doi.org/10.3934/dcdss.2020007">10.3934/dcdss.2020007</a>.
  short: T. Black, Discrete &#38;amp; Continuous Dynamical Systems - S 13 (2019) 119–137.
date_created: 2022-12-21T09:48:28Z
date_updated: 2022-12-21T10:04:44Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.3934/dcdss.2020007
intvolume: '        13'
issue: '2'
keyword:
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Analysis
language:
- iso: eng
page: 119-137
publication: Discrete &amp; Continuous Dynamical Systems - S
publication_identifier:
  issn:
  - 1937-1179
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global generalized solutions to a parabolic-elliptic Keller-Segel system with
  singular sensitivity
type: journal_article
user_id: '23686'
volume: 13
year: '2019'
...
---
_id: '34669'
article_number: '1'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System.
    <i>Journal of Mathematical Fluid Mechanics</i>. 2019;22(1). doi:<a href="https://doi.org/10.1007/s00021-019-0464-z">10.1007/s00021-019-0464-z</a>
  apa: Black, T. (2019). The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
    System. <i>Journal of Mathematical Fluid Mechanics</i>, <i>22</i>(1), Article
    1. <a href="https://doi.org/10.1007/s00021-019-0464-z">https://doi.org/10.1007/s00021-019-0464-z</a>
  bibtex: '@article{Black_2019, title={The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
    System}, volume={22}, DOI={<a href="https://doi.org/10.1007/s00021-019-0464-z">10.1007/s00021-019-0464-z</a>},
    number={11}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer
    Science and Business Media LLC}, author={Black, Tobias}, year={2019} }'
  chicago: Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
    System.” <i>Journal of Mathematical Fluid Mechanics</i> 22, no. 1 (2019). <a href="https://doi.org/10.1007/s00021-019-0464-z">https://doi.org/10.1007/s00021-019-0464-z</a>.
  ieee: 'T. Black, “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
    System,” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 22, no. 1, Art.
    no. 1, 2019, doi: <a href="https://doi.org/10.1007/s00021-019-0464-z">10.1007/s00021-019-0464-z</a>.'
  mla: Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes
    System.” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 22, no. 1, 1, Springer
    Science and Business Media LLC, 2019, doi:<a href="https://doi.org/10.1007/s00021-019-0464-z">10.1007/s00021-019-0464-z</a>.
  short: T. Black, Journal of Mathematical Fluid Mechanics 22 (2019).
date_created: 2022-12-21T09:47:56Z
date_updated: 2022-12-21T10:04:29Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00021-019-0464-z
intvolume: '        22'
issue: '1'
keyword:
- Applied Mathematics
- Computational Mathematics
- Condensed Matter Physics
- Mathematical Physics
language:
- iso: eng
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
  issn:
  - 1422-6928
  - 1422-6952
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System
type: journal_article
user_id: '23686'
volume: 22
year: '2019'
...
---
_id: '34668'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Masaaki
  full_name: Mizukami, Masaaki
  last_name: Mizukami
citation:
  ama: 'Black T, Lankeit J, Mizukami M. A Keller‐Segel‐fluid system with singular
    sensitivity: Generalized solutions. <i>Mathematical Methods in the Applied Sciences</i>.
    2019;42(9):3002-3020. doi:<a href="https://doi.org/10.1002/mma.5561">10.1002/mma.5561</a>'
  apa: 'Black, T., Lankeit, J., &#38; Mizukami, M. (2019). A Keller‐Segel‐fluid system
    with singular sensitivity: Generalized solutions. <i>Mathematical Methods in the
    Applied Sciences</i>, <i>42</i>(9), 3002–3020. <a href="https://doi.org/10.1002/mma.5561">https://doi.org/10.1002/mma.5561</a>'
  bibtex: '@article{Black_Lankeit_Mizukami_2019, title={A Keller‐Segel‐fluid system
    with singular sensitivity: Generalized solutions}, volume={42}, DOI={<a href="https://doi.org/10.1002/mma.5561">10.1002/mma.5561</a>},
    number={9}, journal={Mathematical Methods in the Applied Sciences}, publisher={Wiley},
    author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2019},
    pages={3002–3020} }'
  chicago: 'Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “A Keller‐Segel‐fluid
    System with Singular Sensitivity: Generalized Solutions.” <i>Mathematical Methods
    in the Applied Sciences</i> 42, no. 9 (2019): 3002–20. <a href="https://doi.org/10.1002/mma.5561">https://doi.org/10.1002/mma.5561</a>.'
  ieee: 'T. Black, J. Lankeit, and M. Mizukami, “A Keller‐Segel‐fluid system with
    singular sensitivity: Generalized solutions,” <i>Mathematical Methods in the Applied
    Sciences</i>, vol. 42, no. 9, pp. 3002–3020, 2019, doi: <a href="https://doi.org/10.1002/mma.5561">10.1002/mma.5561</a>.'
  mla: 'Black, Tobias, et al. “A Keller‐Segel‐fluid System with Singular Sensitivity:
    Generalized Solutions.” <i>Mathematical Methods in the Applied Sciences</i>, vol.
    42, no. 9, Wiley, 2019, pp. 3002–20, doi:<a href="https://doi.org/10.1002/mma.5561">10.1002/mma.5561</a>.'
  short: T. Black, J. Lankeit, M. Mizukami, Mathematical Methods in the Applied Sciences
    42 (2019) 3002–3020.
date_created: 2022-12-21T09:47:47Z
date_updated: 2022-12-21T10:04:36Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1002/mma.5561
intvolume: '        42'
issue: '9'
keyword:
- General Engineering
- General Mathematics
language:
- iso: eng
page: 3002-3020
publication: Mathematical Methods in the Applied Sciences
publication_identifier:
  issn:
  - 0170-4214
  - 1099-1476
publication_status: published
publisher: Wiley
status: public
title: 'A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions'
type: journal_article
user_id: '23686'
volume: 42
year: '2019'
...
---
_id: '34671'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Masaaki
  full_name: Mizukami, Masaaki
  last_name: Mizukami
citation:
  ama: Black T, Lankeit J, Mizukami M. Stabilization in the Keller–Segel system with
    signal-dependent sensitivity. <i>Applicable Analysis</i>. 2019;99(16):2877-2891.
    doi:<a href="https://doi.org/10.1080/00036811.2019.1585534">10.1080/00036811.2019.1585534</a>
  apa: Black, T., Lankeit, J., &#38; Mizukami, M. (2019). Stabilization in the Keller–Segel
    system with signal-dependent sensitivity. <i>Applicable Analysis</i>, <i>99</i>(16),
    2877–2891. <a href="https://doi.org/10.1080/00036811.2019.1585534">https://doi.org/10.1080/00036811.2019.1585534</a>
  bibtex: '@article{Black_Lankeit_Mizukami_2019, title={Stabilization in the Keller–Segel
    system with signal-dependent sensitivity}, volume={99}, DOI={<a href="https://doi.org/10.1080/00036811.2019.1585534">10.1080/00036811.2019.1585534</a>},
    number={16}, journal={Applicable Analysis}, publisher={Informa UK Limited}, author={Black,
    Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2019}, pages={2877–2891}
    }'
  chicago: 'Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “Stabilization
    in the Keller–Segel System with Signal-Dependent Sensitivity.” <i>Applicable Analysis</i>
    99, no. 16 (2019): 2877–91. <a href="https://doi.org/10.1080/00036811.2019.1585534">https://doi.org/10.1080/00036811.2019.1585534</a>.'
  ieee: 'T. Black, J. Lankeit, and M. Mizukami, “Stabilization in the Keller–Segel
    system with signal-dependent sensitivity,” <i>Applicable Analysis</i>, vol. 99,
    no. 16, pp. 2877–2891, 2019, doi: <a href="https://doi.org/10.1080/00036811.2019.1585534">10.1080/00036811.2019.1585534</a>.'
  mla: Black, Tobias, et al. “Stabilization in the Keller–Segel System with Signal-Dependent
    Sensitivity.” <i>Applicable Analysis</i>, vol. 99, no. 16, Informa UK Limited,
    2019, pp. 2877–91, doi:<a href="https://doi.org/10.1080/00036811.2019.1585534">10.1080/00036811.2019.1585534</a>.
  short: T. Black, J. Lankeit, M. Mizukami, Applicable Analysis 99 (2019) 2877–2891.
date_created: 2022-12-21T09:48:19Z
date_updated: 2022-12-21T10:04:50Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1080/00036811.2019.1585534
intvolume: '        99'
issue: '16'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
page: 2877-2891
publication: Applicable Analysis
publication_identifier:
  issn:
  - 0003-6811
  - 1563-504X
publication_status: published
publisher: Informa UK Limited
status: public
title: Stabilization in the Keller–Segel system with signal-dependent sensitivity
type: journal_article
user_id: '23686'
volume: 99
year: '2019'
...
---
_id: '34664'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear
    Diffusion. <i>SIAM Journal on Mathematical Analysis</i>. 2018;50(4):4087-4116.
    doi:<a href="https://doi.org/10.1137/17m1159488">10.1137/17m1159488</a>
  apa: Black, T. (2018). Global Very Weak Solutions to a Chemotaxis-Fluid System with
    Nonlinear Diffusion. <i>SIAM Journal on Mathematical Analysis</i>, <i>50</i>(4),
    4087–4116. <a href="https://doi.org/10.1137/17m1159488">https://doi.org/10.1137/17m1159488</a>
  bibtex: '@article{Black_2018, title={Global Very Weak Solutions to a Chemotaxis-Fluid
    System with Nonlinear Diffusion}, volume={50}, DOI={<a href="https://doi.org/10.1137/17m1159488">10.1137/17m1159488</a>},
    number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society
    for Industrial &#38; Applied Mathematics (SIAM)}, author={Black, Tobias}, year={2018},
    pages={4087–4116} }'
  chicago: 'Black, Tobias. “Global Very Weak Solutions to a Chemotaxis-Fluid System
    with Nonlinear Diffusion.” <i>SIAM Journal on Mathematical Analysis</i> 50, no.
    4 (2018): 4087–4116. <a href="https://doi.org/10.1137/17m1159488">https://doi.org/10.1137/17m1159488</a>.'
  ieee: 'T. Black, “Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear
    Diffusion,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 4, pp.
    4087–4116, 2018, doi: <a href="https://doi.org/10.1137/17m1159488">10.1137/17m1159488</a>.'
  mla: Black, Tobias. “Global Very Weak Solutions to a Chemotaxis-Fluid System with
    Nonlinear Diffusion.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no.
    4, Society for Industrial &#38; Applied Mathematics (SIAM), 2018, pp. 4087–116,
    doi:<a href="https://doi.org/10.1137/17m1159488">10.1137/17m1159488</a>.
  short: T. Black, SIAM Journal on Mathematical Analysis 50 (2018) 4087–4116.
date_created: 2022-12-21T09:47:01Z
date_updated: 2022-12-21T10:04:57Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1137/17m1159488
intvolume: '        50'
issue: '4'
keyword:
- Applied Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
page: 4087-4116
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  issn:
  - 0036-1410
  - 1095-7154
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion
type: journal_article
user_id: '23686'
volume: 50
year: '2018'
...
---
_id: '34666'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes
    system in 2D. <i>Journal of Differential Equations</i>. 2018;265(5):2296-2339.
    doi:<a href="https://doi.org/10.1016/j.jde.2018.04.035">10.1016/j.jde.2018.04.035</a>
  apa: Black, T. (2018). Eventual smoothness of generalized solutions to a singular
    chemotaxis-Stokes system in 2D. <i>Journal of Differential Equations</i>, <i>265</i>(5),
    2296–2339. <a href="https://doi.org/10.1016/j.jde.2018.04.035">https://doi.org/10.1016/j.jde.2018.04.035</a>
  bibtex: '@article{Black_2018, title={Eventual smoothness of generalized solutions
    to a singular chemotaxis-Stokes system in 2D}, volume={265}, DOI={<a href="https://doi.org/10.1016/j.jde.2018.04.035">10.1016/j.jde.2018.04.035</a>},
    number={5}, journal={Journal of Differential Equations}, publisher={Elsevier BV},
    author={Black, Tobias}, year={2018}, pages={2296–2339} }'
  chicago: 'Black, Tobias. “Eventual Smoothness of Generalized Solutions to a Singular
    Chemotaxis-Stokes System in 2D.” <i>Journal of Differential Equations</i> 265,
    no. 5 (2018): 2296–2339. <a href="https://doi.org/10.1016/j.jde.2018.04.035">https://doi.org/10.1016/j.jde.2018.04.035</a>.'
  ieee: 'T. Black, “Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes
    system in 2D,” <i>Journal of Differential Equations</i>, vol. 265, no. 5, pp.
    2296–2339, 2018, doi: <a href="https://doi.org/10.1016/j.jde.2018.04.035">10.1016/j.jde.2018.04.035</a>.'
  mla: Black, Tobias. “Eventual Smoothness of Generalized Solutions to a Singular
    Chemotaxis-Stokes System in 2D.” <i>Journal of Differential Equations</i>, vol.
    265, no. 5, Elsevier BV, 2018, pp. 2296–339, doi:<a href="https://doi.org/10.1016/j.jde.2018.04.035">10.1016/j.jde.2018.04.035</a>.
  short: T. Black, Journal of Differential Equations 265 (2018) 2296–2339.
date_created: 2022-12-21T09:47:19Z
date_updated: 2022-12-21T10:05:11Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1016/j.jde.2018.04.035
intvolume: '       265'
issue: '5'
keyword:
- Analysis
- Applied Mathematics
language:
- iso: eng
page: 2296-2339
publication: Journal of Differential Equations
publication_identifier:
  issn:
  - 0022-0396
publication_status: published
publisher: Elsevier BV
status: public
title: Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes
  system in 2D
type: journal_article
user_id: '23686'
volume: 265
year: '2018'
...
---
_id: '34667'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Global solvability of chemotaxis–fluid systems with nonlinear diffusion
    and matrix-valued sensitivities in three dimensions. <i>Nonlinear Analysis</i>.
    2018;180:129-153. doi:<a href="https://doi.org/10.1016/j.na.2018.10.003">10.1016/j.na.2018.10.003</a>
  apa: Black, T. (2018). Global solvability of chemotaxis–fluid systems with nonlinear
    diffusion and matrix-valued sensitivities in three dimensions. <i>Nonlinear Analysis</i>,
    <i>180</i>, 129–153. <a href="https://doi.org/10.1016/j.na.2018.10.003">https://doi.org/10.1016/j.na.2018.10.003</a>
  bibtex: '@article{Black_2018, title={Global solvability of chemotaxis–fluid systems
    with nonlinear diffusion and matrix-valued sensitivities in three dimensions},
    volume={180}, DOI={<a href="https://doi.org/10.1016/j.na.2018.10.003">10.1016/j.na.2018.10.003</a>},
    journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Black, Tobias},
    year={2018}, pages={129–153} }'
  chicago: 'Black, Tobias. “Global Solvability of Chemotaxis–Fluid Systems with Nonlinear
    Diffusion and Matrix-Valued Sensitivities in Three Dimensions.” <i>Nonlinear Analysis</i>
    180 (2018): 129–53. <a href="https://doi.org/10.1016/j.na.2018.10.003">https://doi.org/10.1016/j.na.2018.10.003</a>.'
  ieee: 'T. Black, “Global solvability of chemotaxis–fluid systems with nonlinear
    diffusion and matrix-valued sensitivities in three dimensions,” <i>Nonlinear Analysis</i>,
    vol. 180, pp. 129–153, 2018, doi: <a href="https://doi.org/10.1016/j.na.2018.10.003">10.1016/j.na.2018.10.003</a>.'
  mla: Black, Tobias. “Global Solvability of Chemotaxis–Fluid Systems with Nonlinear
    Diffusion and Matrix-Valued Sensitivities in Three Dimensions.” <i>Nonlinear Analysis</i>,
    vol. 180, Elsevier BV, 2018, pp. 129–53, doi:<a href="https://doi.org/10.1016/j.na.2018.10.003">10.1016/j.na.2018.10.003</a>.
  short: T. Black, Nonlinear Analysis 180 (2018) 129–153.
date_created: 2022-12-21T09:47:30Z
date_updated: 2022-12-21T10:05:04Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1016/j.na.2018.10.003
intvolume: '       180'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
page: 129-153
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: Global solvability of chemotaxis–fluid systems with nonlinear diffusion and
  matrix-valued sensitivities in three dimensions
type: journal_article
user_id: '23686'
volume: 180
year: '2018'
...
---
_id: '34663'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
citation:
  ama: Black T. Global existence and asymptotic stability in a competitive two-species
    chemotaxis system with two signals. <i>Discrete &#38;amp; Continuous Dynamical
    Systems - B</i>. 2017;22(4):1253-1272. doi:<a href="https://doi.org/10.3934/dcdsb.2017061">10.3934/dcdsb.2017061</a>
  apa: Black, T. (2017). Global existence and asymptotic stability in a competitive
    two-species chemotaxis system with two signals. <i>Discrete &#38;amp; Continuous
    Dynamical Systems - B</i>, <i>22</i>(4), 1253–1272. <a href="https://doi.org/10.3934/dcdsb.2017061">https://doi.org/10.3934/dcdsb.2017061</a>
  bibtex: '@article{Black_2017, title={Global existence and asymptotic stability in
    a competitive two-species chemotaxis system with two signals}, volume={22}, DOI={<a
    href="https://doi.org/10.3934/dcdsb.2017061">10.3934/dcdsb.2017061</a>}, number={4},
    journal={Discrete &#38;amp; Continuous Dynamical Systems - B}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Black, Tobias}, year={2017},
    pages={1253–1272} }'
  chicago: 'Black, Tobias. “Global Existence and Asymptotic Stability in a Competitive
    Two-Species Chemotaxis System with Two Signals.” <i>Discrete &#38;amp; Continuous
    Dynamical Systems - B</i> 22, no. 4 (2017): 1253–72. <a href="https://doi.org/10.3934/dcdsb.2017061">https://doi.org/10.3934/dcdsb.2017061</a>.'
  ieee: 'T. Black, “Global existence and asymptotic stability in a competitive two-species
    chemotaxis system with two signals,” <i>Discrete &#38;amp; Continuous Dynamical
    Systems - B</i>, vol. 22, no. 4, pp. 1253–1272, 2017, doi: <a href="https://doi.org/10.3934/dcdsb.2017061">10.3934/dcdsb.2017061</a>.'
  mla: Black, Tobias. “Global Existence and Asymptotic Stability in a Competitive
    Two-Species Chemotaxis System with Two Signals.” <i>Discrete &#38;amp; Continuous
    Dynamical Systems - B</i>, vol. 22, no. 4, American Institute of Mathematical
    Sciences (AIMS), 2017, pp. 1253–72, doi:<a href="https://doi.org/10.3934/dcdsb.2017061">10.3934/dcdsb.2017061</a>.
  short: T. Black, Discrete &#38;amp; Continuous Dynamical Systems - B 22 (2017) 1253–1272.
date_created: 2022-12-21T09:46:50Z
date_updated: 2022-12-21T10:05:19Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.3934/dcdsb.2017061
intvolume: '        22'
issue: '4'
keyword:
- Applied Mathematics
- Discrete Mathematics and Combinatorics
language:
- iso: eng
page: 1253-1272
publication: Discrete &amp; Continuous Dynamical Systems - B
publication_identifier:
  issn:
  - 1553-524X
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Global existence and asymptotic stability in a competitive two-species chemotaxis
  system with two signals
type: journal_article
user_id: '23686'
volume: 22
year: '2017'
...
---
_id: '34665'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Masaaki
  full_name: Mizukami, Masaaki
  last_name: Mizukami
citation:
  ama: Black T, Lankeit J, Mizukami M. Singular sensitivity in a Keller–Segel-fluid
    system. <i>Journal of Evolution Equations</i>. 2017;18(2):561-581. doi:<a href="https://doi.org/10.1007/s00028-017-0411-5">10.1007/s00028-017-0411-5</a>
  apa: Black, T., Lankeit, J., &#38; Mizukami, M. (2017). Singular sensitivity in
    a Keller–Segel-fluid system. <i>Journal of Evolution Equations</i>, <i>18</i>(2),
    561–581. <a href="https://doi.org/10.1007/s00028-017-0411-5">https://doi.org/10.1007/s00028-017-0411-5</a>
  bibtex: '@article{Black_Lankeit_Mizukami_2017, title={Singular sensitivity in a
    Keller–Segel-fluid system}, volume={18}, DOI={<a href="https://doi.org/10.1007/s00028-017-0411-5">10.1007/s00028-017-0411-5</a>},
    number={2}, journal={Journal of Evolution Equations}, publisher={Springer Science
    and Business Media LLC}, author={Black, Tobias and Lankeit, Johannes and Mizukami,
    Masaaki}, year={2017}, pages={561–581} }'
  chicago: 'Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “Singular Sensitivity
    in a Keller–Segel-Fluid System.” <i>Journal of Evolution Equations</i> 18, no.
    2 (2017): 561–81. <a href="https://doi.org/10.1007/s00028-017-0411-5">https://doi.org/10.1007/s00028-017-0411-5</a>.'
  ieee: 'T. Black, J. Lankeit, and M. Mizukami, “Singular sensitivity in a Keller–Segel-fluid
    system,” <i>Journal of Evolution Equations</i>, vol. 18, no. 2, pp. 561–581, 2017,
    doi: <a href="https://doi.org/10.1007/s00028-017-0411-5">10.1007/s00028-017-0411-5</a>.'
  mla: Black, Tobias, et al. “Singular Sensitivity in a Keller–Segel-Fluid System.”
    <i>Journal of Evolution Equations</i>, vol. 18, no. 2, Springer Science and Business
    Media LLC, 2017, pp. 561–81, doi:<a href="https://doi.org/10.1007/s00028-017-0411-5">10.1007/s00028-017-0411-5</a>.
  short: T. Black, J. Lankeit, M. Mizukami, Journal of Evolution Equations 18 (2017)
    561–581.
date_created: 2022-12-21T09:47:13Z
date_updated: 2022-12-21T10:05:25Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1007/s00028-017-0411-5
intvolume: '        18'
issue: '2'
keyword:
- Mathematics (miscellaneous)
language:
- iso: eng
page: 561-581
publication: Journal of Evolution Equations
publication_identifier:
  issn:
  - 1424-3199
  - 1424-3202
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Singular sensitivity in a Keller–Segel-fluid system
type: journal_article
user_id: '23686'
volume: 18
year: '2017'
...
---
_id: '34660'
author:
- first_name: Tobias
  full_name: Black, Tobias
  id: '23686'
  last_name: Black
  orcid: 0000-0001-9963-0800
- first_name: Johannes
  full_name: Lankeit, Johannes
  last_name: Lankeit
- first_name: Masaaki
  full_name: Mizukami, Masaaki
  last_name: Mizukami
citation:
  ama: Black T, Lankeit J, Mizukami M. On the weakly competitive case in a two-species
    chemotaxis model. <i>IMA Journal of Applied Mathematics</i>. 2016;81(5):860-876.
    doi:<a href="https://doi.org/10.1093/imamat/hxw036">10.1093/imamat/hxw036</a>
  apa: Black, T., Lankeit, J., &#38; Mizukami, M. (2016). On the weakly competitive
    case in a two-species chemotaxis model. <i>IMA Journal of Applied Mathematics</i>,
    <i>81</i>(5), 860–876. <a href="https://doi.org/10.1093/imamat/hxw036">https://doi.org/10.1093/imamat/hxw036</a>
  bibtex: '@article{Black_Lankeit_Mizukami_2016, title={On the weakly competitive
    case in a two-species chemotaxis model}, volume={81}, DOI={<a href="https://doi.org/10.1093/imamat/hxw036">10.1093/imamat/hxw036</a>},
    number={5}, journal={IMA Journal of Applied Mathematics}, publisher={Oxford University
    Press (OUP)}, author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki},
    year={2016}, pages={860–876} }'
  chicago: 'Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “On the Weakly
    Competitive Case in a Two-Species Chemotaxis Model.” <i>IMA Journal of Applied
    Mathematics</i> 81, no. 5 (2016): 860–76. <a href="https://doi.org/10.1093/imamat/hxw036">https://doi.org/10.1093/imamat/hxw036</a>.'
  ieee: 'T. Black, J. Lankeit, and M. Mizukami, “On the weakly competitive case in
    a two-species chemotaxis model,” <i>IMA Journal of Applied Mathematics</i>, vol.
    81, no. 5, pp. 860–876, 2016, doi: <a href="https://doi.org/10.1093/imamat/hxw036">10.1093/imamat/hxw036</a>.'
  mla: Black, Tobias, et al. “On the Weakly Competitive Case in a Two-Species Chemotaxis
    Model.” <i>IMA Journal of Applied Mathematics</i>, vol. 81, no. 5, Oxford University
    Press (OUP), 2016, pp. 860–76, doi:<a href="https://doi.org/10.1093/imamat/hxw036">10.1093/imamat/hxw036</a>.
  short: T. Black, J. Lankeit, M. Mizukami, IMA Journal of Applied Mathematics 81
    (2016) 860–876.
date_created: 2022-12-21T09:46:18Z
date_updated: 2022-12-21T10:05:33Z
department:
- _id: '34'
- _id: '10'
- _id: '90'
doi: 10.1093/imamat/hxw036
intvolume: '        81'
issue: '5'
keyword:
- Applied Mathematics
language:
- iso: eng
page: 860-876
publication: IMA Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0272-4960
  - 1464-3634
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: On the weakly competitive case in a two-species chemotaxis model
type: journal_article
user_id: '23686'
volume: 81
year: '2016'
...