---
_id: '40364'
article_number: '013371'
author:
- first_name: Polina R.
  full_name: Sharapova, Polina R.
  id: '60286'
  last_name: Sharapova
- first_name: G.
  full_name: Frascella, G.
  last_name: Frascella
- first_name: M.
  full_name: Riabinin, M.
  last_name: Riabinin
- first_name: A. M.
  full_name: Pérez, A. M.
  last_name: Pérez
- first_name: O. V.
  full_name: Tikhonova, O. V.
  last_name: Tikhonova
- first_name: S.
  full_name: Lemieux, S.
  last_name: Lemieux
- first_name: R. W.
  full_name: Boyd, R. W.
  last_name: Boyd
- first_name: G.
  full_name: Leuchs, G.
  last_name: Leuchs
- first_name: M. V.
  full_name: Chekhova, M. V.
  last_name: Chekhova
citation:
  ama: Sharapova PR, Frascella G, Riabinin M, et al. Properties of bright squeezed
    vacuum at increasing brightness. <i>Physical Review Research</i>. 2020;2(1). doi:<a
    href="https://doi.org/10.1103/physrevresearch.2.013371">10.1103/physrevresearch.2.013371</a>
  apa: Sharapova, P. R., Frascella, G., Riabinin, M., Pérez, A. M., Tikhonova, O.
    V., Lemieux, S., Boyd, R. W., Leuchs, G., &#38; Chekhova, M. V. (2020). Properties
    of bright squeezed vacuum at increasing brightness. <i>Physical Review Research</i>,
    <i>2</i>(1), Article 013371. <a href="https://doi.org/10.1103/physrevresearch.2.013371">https://doi.org/10.1103/physrevresearch.2.013371</a>
  bibtex: '@article{Sharapova_Frascella_Riabinin_Pérez_Tikhonova_Lemieux_Boyd_Leuchs_Chekhova_2020,
    title={Properties of bright squeezed vacuum at increasing brightness}, volume={2},
    DOI={<a href="https://doi.org/10.1103/physrevresearch.2.013371">10.1103/physrevresearch.2.013371</a>},
    number={1013371}, journal={Physical Review Research}, publisher={American Physical
    Society (APS)}, author={Sharapova, Polina R. and Frascella, G. and Riabinin, M.
    and Pérez, A. M. and Tikhonova, O. V. and Lemieux, S. and Boyd, R. W. and Leuchs,
    G. and Chekhova, M. V.}, year={2020} }'
  chicago: Sharapova, Polina R., G. Frascella, M. Riabinin, A. M. Pérez, O. V. Tikhonova,
    S. Lemieux, R. W. Boyd, G. Leuchs, and M. V. Chekhova. “Properties of Bright Squeezed
    Vacuum at Increasing Brightness.” <i>Physical Review Research</i> 2, no. 1 (2020).
    <a href="https://doi.org/10.1103/physrevresearch.2.013371">https://doi.org/10.1103/physrevresearch.2.013371</a>.
  ieee: 'P. R. Sharapova <i>et al.</i>, “Properties of bright squeezed vacuum at increasing
    brightness,” <i>Physical Review Research</i>, vol. 2, no. 1, Art. no. 013371,
    2020, doi: <a href="https://doi.org/10.1103/physrevresearch.2.013371">10.1103/physrevresearch.2.013371</a>.'
  mla: Sharapova, Polina R., et al. “Properties of Bright Squeezed Vacuum at Increasing
    Brightness.” <i>Physical Review Research</i>, vol. 2, no. 1, 013371, American
    Physical Society (APS), 2020, doi:<a href="https://doi.org/10.1103/physrevresearch.2.013371">10.1103/physrevresearch.2.013371</a>.
  short: P.R. Sharapova, G. Frascella, M. Riabinin, A.M. Pérez, O.V. Tikhonova, S.
    Lemieux, R.W. Boyd, G. Leuchs, M.V. Chekhova, Physical Review Research 2 (2020).
date_created: 2023-01-26T13:45:35Z
date_updated: 2025-12-16T11:26:50Z
department:
- _id: '15'
- _id: '569'
- _id: '170'
- _id: '429'
- _id: '230'
- _id: '35'
doi: 10.1103/physrevresearch.2.013371
intvolume: '         2'
issue: '1'
keyword:
- General Engineering
language:
- iso: eng
project:
- _id: '53'
  name: 'TRR 142: TRR 142'
- _id: '56'
  name: 'TRR 142 - C: TRR 142 - Project Area C'
- _id: '72'
  name: 'TRR 142 - C2: TRR 142 - Subproject C2'
publication: Physical Review Research
publication_identifier:
  issn:
  - 2643-1564
publication_status: published
publisher: American Physical Society (APS)
status: public
title: Properties of bright squeezed vacuum at increasing brightness
type: journal_article
user_id: '16199'
volume: 2
year: '2020'
...
---
_id: '40381'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The phenomenon
    of entanglement is the basis of quantum information and quantum communication
    processes. Entangled systems with a large number of photons are of great interest
    at present because they provide a platform for streaming technologies based on
    photonics. In this paper we present a device which operates with four-photons
    and based on the Hong–Ou–Mandel interference. The presented device allows to maximize
    the degree of spatial entanglement and generate the highly entangled four-dimensional
    Bell states. Furthermore, the use of the interferometer in different regimes leads
    to fast interference fringes in the coincidence probability with period of oscillations
    twice smaller than the pump wavelength. We have a good agreement between theoretical
    simulations and experimental results.</jats:p>"
article_number: '045020'
author:
- first_name: A
  full_name: Ferreri, A
  last_name: Ferreri
- first_name: V
  full_name: Ansari, V
  last_name: Ansari
- first_name: Benjamin
  full_name: Brecht, Benjamin
  id: '27150'
  last_name: Brecht
  orcid: '0000-0003-4140-0556 '
- first_name: Christine
  full_name: Silberhorn, Christine
  id: '26263'
  last_name: Silberhorn
- first_name: Polina R.
  full_name: Sharapova, Polina R.
  id: '60286'
  last_name: Sharapova
citation:
  ama: Ferreri A, Ansari V, Brecht B, Silberhorn C, Sharapova PR. Spatial entanglement
    and state engineering via four-photon Hong–Ou–Mandel interference. <i>Quantum
    Science and Technology</i>. 2020;5(4). doi:<a href="https://doi.org/10.1088/2058-9565/abb411">10.1088/2058-9565/abb411</a>
  apa: Ferreri, A., Ansari, V., Brecht, B., Silberhorn, C., &#38; Sharapova, P. R.
    (2020). Spatial entanglement and state engineering via four-photon Hong–Ou–Mandel
    interference. <i>Quantum Science and Technology</i>, <i>5</i>(4), Article 045020.
    <a href="https://doi.org/10.1088/2058-9565/abb411">https://doi.org/10.1088/2058-9565/abb411</a>
  bibtex: '@article{Ferreri_Ansari_Brecht_Silberhorn_Sharapova_2020, title={Spatial
    entanglement and state engineering via four-photon Hong–Ou–Mandel interference},
    volume={5}, DOI={<a href="https://doi.org/10.1088/2058-9565/abb411">10.1088/2058-9565/abb411</a>},
    number={4045020}, journal={Quantum Science and Technology}, publisher={IOP Publishing},
    author={Ferreri, A and Ansari, V and Brecht, Benjamin and Silberhorn, Christine
    and Sharapova, Polina R.}, year={2020} }'
  chicago: Ferreri, A, V Ansari, Benjamin Brecht, Christine Silberhorn, and Polina
    R. Sharapova. “Spatial Entanglement and State Engineering via Four-Photon Hong–Ou–Mandel
    Interference.” <i>Quantum Science and Technology</i> 5, no. 4 (2020). <a href="https://doi.org/10.1088/2058-9565/abb411">https://doi.org/10.1088/2058-9565/abb411</a>.
  ieee: 'A. Ferreri, V. Ansari, B. Brecht, C. Silberhorn, and P. R. Sharapova, “Spatial
    entanglement and state engineering via four-photon Hong–Ou–Mandel interference,”
    <i>Quantum Science and Technology</i>, vol. 5, no. 4, Art. no. 045020, 2020, doi:
    <a href="https://doi.org/10.1088/2058-9565/abb411">10.1088/2058-9565/abb411</a>.'
  mla: Ferreri, A., et al. “Spatial Entanglement and State Engineering via Four-Photon
    Hong–Ou–Mandel Interference.” <i>Quantum Science and Technology</i>, vol. 5, no.
    4, 045020, IOP Publishing, 2020, doi:<a href="https://doi.org/10.1088/2058-9565/abb411">10.1088/2058-9565/abb411</a>.
  short: A. Ferreri, V. Ansari, B. Brecht, C. Silberhorn, P.R. Sharapova, Quantum
    Science and Technology 5 (2020).
date_created: 2023-01-26T14:06:23Z
date_updated: 2025-12-16T11:27:56Z
department:
- _id: '15'
- _id: '569'
- _id: '170'
- _id: '288'
- _id: '230'
- _id: '429'
- _id: '35'
doi: 10.1088/2058-9565/abb411
intvolume: '         5'
issue: '4'
keyword:
- Electrical and Electronic Engineering
- Physics and Astronomy (miscellaneous)
- Materials Science (miscellaneous)
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
project:
- _id: '53'
  name: 'TRR 142: TRR 142'
- _id: '56'
  name: 'TRR 142 - C: TRR 142 - Project Area C'
- _id: '72'
  name: 'TRR 142 - C2: TRR 142 - Subproject C2'
publication: Quantum Science and Technology
publication_identifier:
  issn:
  - 2058-9565
publication_status: published
publisher: IOP Publishing
status: public
title: Spatial entanglement and state engineering via four-photon Hong–Ou–Mandel interference
type: journal_article
user_id: '16199'
volume: 5
year: '2020'
...
---
_id: '63179'
citation:
  ama: Ludwig M, ed. <i>Research on Outdoor STEM Education in the Digital Age. Proceedings
    of the ROSETA Online Conference in June 2020</i>. WTM-Verlag; 2020. doi:<a href="https://doi.org/10.37626/ga9783959871440.0">10.37626/ga9783959871440.0</a>
  apa: Ludwig, M. (Ed.). (2020). <i>Research on Outdoor STEM Education in the digital
    Age. Proceedings of the ROSETA Online Conference in June 2020</i>. WTM-Verlag.
    <a href="https://doi.org/10.37626/ga9783959871440.0">https://doi.org/10.37626/ga9783959871440.0</a>
  bibtex: '@book{Ludwig_2020, title={Research on Outdoor STEM Education in the digital
    Age. Proceedings of the ROSETA Online Conference in June 2020}, DOI={<a href="https://doi.org/10.37626/ga9783959871440.0">10.37626/ga9783959871440.0</a>},
    publisher={WTM-Verlag}, year={2020} }'
  chicago: Ludwig, Matthias, ed. <i>Research on Outdoor STEM Education in the Digital
    Age. Proceedings of the ROSETA Online Conference in June 2020</i>. WTM-Verlag,
    2020. <a href="https://doi.org/10.37626/ga9783959871440.0">https://doi.org/10.37626/ga9783959871440.0</a>.
  ieee: M. Ludwig, Ed., <i>Research on Outdoor STEM Education in the digital Age.
    Proceedings of the ROSETA Online Conference in June 2020</i>. WTM-Verlag, 2020.
  mla: Ludwig, Matthias, editor. <i>Research on Outdoor STEM Education in the Digital
    Age. Proceedings of the ROSETA Online Conference in June 2020</i>. WTM-Verlag,
    2020, doi:<a href="https://doi.org/10.37626/ga9783959871440.0">10.37626/ga9783959871440.0</a>.
  short: M. Ludwig, ed., Research on Outdoor STEM Education in the Digital Age. Proceedings
    of the ROSETA Online Conference in June 2020, WTM-Verlag, 2020.
date_created: 2025-12-17T08:55:20Z
date_updated: 2025-12-17T08:56:05Z
doi: 10.37626/ga9783959871440.0
editor:
- first_name: Matthias
  full_name: Ludwig, Matthias
  last_name: Ludwig
publication_identifier:
  isbn:
  - '9783959871440'
publication_status: published
publisher: WTM-Verlag
status: public
title: Research on Outdoor STEM Education in the digital Age. Proceedings of the ROSETA
  Online Conference in June 2020
type: book_editor
user_id: '111489'
year: '2020'
...
---
_id: '63195'
author:
- first_name: Claudia
  full_name: Decker, Claudia
  id: '31046'
  last_name: Decker
- first_name: Maria
  full_name: Mochalova, Maria
  last_name: Mochalova
citation:
  ama: 'Decker C, Mochalova M. Vielfalt gegen die Einfalt: Praktische Umsetzung eines
    Projektkonzepts. In: ; 2020.'
  apa: 'Decker, C., &#38; Mochalova, M. (2020). <i>Vielfalt gegen die Einfalt: Praktische
    Umsetzung eines Projektkonzepts</i>. Bundekongress der Zentren für Lehrerbildung
    und Professional Schools of Education, Köln.'
  bibtex: '@inproceedings{Decker_Mochalova_2020, title={Vielfalt gegen die Einfalt:
    Praktische Umsetzung eines Projektkonzepts}, author={Decker, Claudia and Mochalova,
    Maria}, year={2020} }'
  chicago: 'Decker, Claudia, and Maria Mochalova. “Vielfalt Gegen Die Einfalt: Praktische
    Umsetzung Eines Projektkonzepts,” 2020.'
  ieee: 'C. Decker and M. Mochalova, “Vielfalt gegen die Einfalt: Praktische Umsetzung
    eines Projektkonzepts,” presented at the Bundekongress der Zentren für Lehrerbildung
    und Professional Schools of Education, Köln, 2020.'
  mla: 'Decker, Claudia, and Maria Mochalova. <i>Vielfalt Gegen Die Einfalt: Praktische
    Umsetzung Eines Projektkonzepts</i>. 2020.'
  short: 'C. Decker, M. Mochalova, in: 2020.'
conference:
  end_date: 28.2.2020
  location: Köln
  name: Bundekongress der Zentren für Lehrerbildung und Professional Schools of Education
  start_date: 26.2.2020
date_created: 2025-12-18T10:45:12Z
date_updated: 2025-12-18T10:45:20Z
department:
- _id: '33'
language:
- iso: eng
status: public
title: 'Vielfalt gegen die Einfalt: Praktische Umsetzung eines Projektkonzepts'
type: conference
user_id: '31046'
year: '2020'
...
---
_id: '37933'
abstract:
- lang: eng
  text: <jats:p>We present a time-over-threshold readout technique to count the number
    of activated pixels from an array of superconducting nanowire single photon detectors
    (SNSPDs). This technique places no additional heatload on the cryostat, and retains
    the intrinsic count rate of the time-tagger. We demonstrate proof-of-principle
    operation with respect to a four-pixel device. Furthermore, we show that, given
    some permissible error threshold, the number of pixels that can be reliably read
    out scales linearly with the intrinsic signal-to-noise ratio of the individual
    pixel response.</jats:p>
article_number: '5528'
author:
- first_name: Johannes
  full_name: Tiedau, Johannes
  last_name: Tiedau
- first_name: Timon
  full_name: Schapeler, Timon
  id: '55629'
  last_name: Schapeler
  orcid: 0000-0001-7652-1716
- first_name: Vikas
  full_name: Anant, Vikas
  last_name: Anant
- first_name: Helmut
  full_name: Fedder, Helmut
  last_name: Fedder
- first_name: Christine
  full_name: Silberhorn, Christine
  id: '26263'
  last_name: Silberhorn
- first_name: Tim
  full_name: Bartley, Tim
  id: '49683'
  last_name: Bartley
citation:
  ama: Tiedau J, Schapeler T, Anant V, Fedder H, Silberhorn C, Bartley T. Single-channel
    electronic readout of a multipixel superconducting nanowire single photon detector.
    <i>Optics Express</i>. 2020;28(4). doi:<a href="https://doi.org/10.1364/oe.383111">10.1364/oe.383111</a>
  apa: Tiedau, J., Schapeler, T., Anant, V., Fedder, H., Silberhorn, C., &#38; Bartley,
    T. (2020). Single-channel electronic readout of a multipixel superconducting nanowire
    single photon detector. <i>Optics Express</i>, <i>28</i>(4), Article 5528. <a
    href="https://doi.org/10.1364/oe.383111">https://doi.org/10.1364/oe.383111</a>
  bibtex: '@article{Tiedau_Schapeler_Anant_Fedder_Silberhorn_Bartley_2020, title={Single-channel
    electronic readout of a multipixel superconducting nanowire single photon detector},
    volume={28}, DOI={<a href="https://doi.org/10.1364/oe.383111">10.1364/oe.383111</a>},
    number={45528}, journal={Optics Express}, publisher={Optica Publishing Group},
    author={Tiedau, Johannes and Schapeler, Timon and Anant, Vikas and Fedder, Helmut
    and Silberhorn, Christine and Bartley, Tim}, year={2020} }'
  chicago: Tiedau, Johannes, Timon Schapeler, Vikas Anant, Helmut Fedder, Christine
    Silberhorn, and Tim Bartley. “Single-Channel Electronic Readout of a Multipixel
    Superconducting Nanowire Single Photon Detector.” <i>Optics Express</i> 28, no.
    4 (2020). <a href="https://doi.org/10.1364/oe.383111">https://doi.org/10.1364/oe.383111</a>.
  ieee: 'J. Tiedau, T. Schapeler, V. Anant, H. Fedder, C. Silberhorn, and T. Bartley,
    “Single-channel electronic readout of a multipixel superconducting nanowire single
    photon detector,” <i>Optics Express</i>, vol. 28, no. 4, Art. no. 5528, 2020,
    doi: <a href="https://doi.org/10.1364/oe.383111">10.1364/oe.383111</a>.'
  mla: Tiedau, Johannes, et al. “Single-Channel Electronic Readout of a Multipixel
    Superconducting Nanowire Single Photon Detector.” <i>Optics Express</i>, vol.
    28, no. 4, 5528, Optica Publishing Group, 2020, doi:<a href="https://doi.org/10.1364/oe.383111">10.1364/oe.383111</a>.
  short: J. Tiedau, T. Schapeler, V. Anant, H. Fedder, C. Silberhorn, T. Bartley,
    Optics Express 28 (2020).
date_created: 2023-01-22T17:13:35Z
date_updated: 2025-12-18T17:10:24Z
department:
- _id: '288'
- _id: '15'
- _id: '623'
- _id: '230'
doi: 10.1364/oe.383111
intvolume: '        28'
issue: '4'
keyword:
- Atomic and Molecular Physics
- and Optics
language:
- iso: eng
project:
- _id: '237'
  name: 'PhoG: Sub-Poissonian Photon Gun by Coherent Diffusive Photonics - EU Flagship
    Project'
- _id: '209'
  name: 'ISOQC: Quantenkommunikation mit integrierter Optik im Zusammenhang mit supraleitender
    Elektronik'
publication: Optics Express
publication_identifier:
  issn:
  - 1094-4087
publication_status: published
publisher: Optica Publishing Group
status: public
title: Single-channel electronic readout of a multipixel superconducting nanowire
  single photon detector
type: journal_article
user_id: '55629'
volume: 28
year: '2020'
...
---
_id: '20156'
article_number: '33035'
author:
- first_name: Timon
  full_name: Schapeler, Timon
  id: '55629'
  last_name: Schapeler
  orcid: 0000-0001-7652-1716
- first_name: Jan Philipp
  full_name: Höpker, Jan Philipp
  id: '33913'
  last_name: Höpker
- first_name: Tim
  full_name: Bartley, Tim
  id: '49683'
  last_name: Bartley
citation:
  ama: Schapeler T, Höpker JP, Bartley T. Quantum detector tomography of a 2×2 multi-pixel
    array of superconducting nanowire single photon detectors. <i>Optics Express</i>.
    Published online 2020. doi:<a href="https://doi.org/10.1364/oe.404285">10.1364/oe.404285</a>
  apa: Schapeler, T., Höpker, J. P., &#38; Bartley, T. (2020). Quantum detector tomography
    of a 2×2 multi-pixel array of superconducting nanowire single photon detectors.
    <i>Optics Express</i>, Article 33035. <a href="https://doi.org/10.1364/oe.404285">https://doi.org/10.1364/oe.404285</a>
  bibtex: '@article{Schapeler_Höpker_Bartley_2020, title={Quantum detector tomography
    of a 2×2 multi-pixel array of superconducting nanowire single photon detectors},
    DOI={<a href="https://doi.org/10.1364/oe.404285">10.1364/oe.404285</a>}, number={33035},
    journal={Optics Express}, author={Schapeler, Timon and Höpker, Jan Philipp and
    Bartley, Tim}, year={2020} }'
  chicago: Schapeler, Timon, Jan Philipp Höpker, and Tim Bartley. “Quantum Detector
    Tomography of a 2×2 Multi-Pixel Array of Superconducting Nanowire Single Photon
    Detectors.” <i>Optics Express</i>, 2020. <a href="https://doi.org/10.1364/oe.404285">https://doi.org/10.1364/oe.404285</a>.
  ieee: 'T. Schapeler, J. P. Höpker, and T. Bartley, “Quantum detector tomography
    of a 2×2 multi-pixel array of superconducting nanowire single photon detectors,”
    <i>Optics Express</i>, Art. no. 33035, 2020, doi: <a href="https://doi.org/10.1364/oe.404285">10.1364/oe.404285</a>.'
  mla: Schapeler, Timon, et al. “Quantum Detector Tomography of a 2×2 Multi-Pixel
    Array of Superconducting Nanowire Single Photon Detectors.” <i>Optics Express</i>,
    33035, 2020, doi:<a href="https://doi.org/10.1364/oe.404285">10.1364/oe.404285</a>.
  short: T. Schapeler, J.P. Höpker, T. Bartley, Optics Express (2020).
date_created: 2020-10-21T11:02:41Z
date_updated: 2025-12-18T17:08:01Z
department:
- _id: '15'
- _id: '230'
doi: 10.1364/oe.404285
language:
- iso: eng
project:
- _id: '209'
  name: 'ISOQC: Quantenkommunikation mit integrierter Optik im Zusammenhang mit supraleitender
    Elektronik'
publication: Optics Express
publication_identifier:
  issn:
  - 1094-4087
publication_status: published
status: public
title: Quantum detector tomography of a 2×2 multi-pixel array of superconducting nanowire
  single photon detectors
type: journal_article
user_id: '55629'
year: '2020'
...
---
_id: '63239'
abstract:
- lang: eng
  text: <jats:p>A quartz crystal microbalance (QCM) is described, which simultaneously
    determines resonance frequency and bandwidth on four different overtones. The
    time resolution is 10 milliseconds. This fast, multi-overtone QCM is based on
    multi-frequency lockin amplification. Synchronous interrogation of overtones is
    needed, when the sample changes quickly and when information on the sample is
    to be extracted from the comparison between overtones. The application example
    is thermal inkjet-printing. At impact, the resonance frequencies change over a
    time shorter than 10 milliseconds. There is a further increase in the contact
    area, evidenced by an increasing common prefactor to the shifts in frequency,
    Δf, and half-bandwidth, ΔΓ. The ratio ΔΓ/(−Δf), which quantifies the energy dissipated
    per time and unit area, decreases with time. Often, there is a fast initial decrease,
    lasting for about 100 milliseconds, followed by a slower decrease, persisting
    over the entire drying time (a few seconds). Fitting the overtone dependence of
    Δf(n) and ΔΓ(n) with power laws, one finds power-law exponents of about 1/2, characteristic
    of semi-infinite Newtonian liquids. The power-law exponents corresponding to Δf(n)
    slightly increase with time. The decrease of ΔΓ/(−Δf) and the increase of the
    exponents are explained by evaporation and formation of a solid film at the resonator
    surface.</jats:p>
article_number: '5915'
author:
- first_name: Christian
  full_name: Leppin, Christian
  id: '117722'
  last_name: Leppin
- first_name: Sven
  full_name: Hampel, Sven
  last_name: Hampel
- first_name: Frederick Sebastian
  full_name: Meyer, Frederick Sebastian
  last_name: Meyer
- first_name: Arne
  full_name: Langhoff, Arne
  last_name: Langhoff
- first_name: Ursula Elisabeth Adriane
  full_name: Fittschen, Ursula Elisabeth Adriane
  last_name: Fittschen
- first_name: Diethelm
  full_name: Johannsmann, Diethelm
  last_name: Johannsmann
citation:
  ama: 'Leppin C, Hampel S, Meyer FS, Langhoff A, Fittschen UEA, Johannsmann D. A
    Quartz Crystal Microbalance, Which Tracks Four Overtones in Parallel with a Time
    Resolution of 10 Milliseconds: Application to Inkjet Printing. <i>Sensors</i>.
    2020;20(20). doi:<a href="https://doi.org/10.3390/s20205915">10.3390/s20205915</a>'
  apa: 'Leppin, C., Hampel, S., Meyer, F. S., Langhoff, A., Fittschen, U. E. A., &#38;
    Johannsmann, D. (2020). A Quartz Crystal Microbalance, Which Tracks Four Overtones
    in Parallel with a Time Resolution of 10 Milliseconds: Application to Inkjet Printing.
    <i>Sensors</i>, <i>20</i>(20), Article 5915. <a href="https://doi.org/10.3390/s20205915">https://doi.org/10.3390/s20205915</a>'
  bibtex: '@article{Leppin_Hampel_Meyer_Langhoff_Fittschen_Johannsmann_2020, title={A
    Quartz Crystal Microbalance, Which Tracks Four Overtones in Parallel with a Time
    Resolution of 10 Milliseconds: Application to Inkjet Printing}, volume={20}, DOI={<a
    href="https://doi.org/10.3390/s20205915">10.3390/s20205915</a>}, number={205915},
    journal={Sensors}, publisher={MDPI AG}, author={Leppin, Christian and Hampel,
    Sven and Meyer, Frederick Sebastian and Langhoff, Arne and Fittschen, Ursula Elisabeth
    Adriane and Johannsmann, Diethelm}, year={2020} }'
  chicago: 'Leppin, Christian, Sven Hampel, Frederick Sebastian Meyer, Arne Langhoff,
    Ursula Elisabeth Adriane Fittschen, and Diethelm Johannsmann. “A Quartz Crystal
    Microbalance, Which Tracks Four Overtones in Parallel with a Time Resolution of
    10 Milliseconds: Application to Inkjet Printing.” <i>Sensors</i> 20, no. 20 (2020).
    <a href="https://doi.org/10.3390/s20205915">https://doi.org/10.3390/s20205915</a>.'
  ieee: 'C. Leppin, S. Hampel, F. S. Meyer, A. Langhoff, U. E. A. Fittschen, and D.
    Johannsmann, “A Quartz Crystal Microbalance, Which Tracks Four Overtones in Parallel
    with a Time Resolution of 10 Milliseconds: Application to Inkjet Printing,” <i>Sensors</i>,
    vol. 20, no. 20, Art. no. 5915, 2020, doi: <a href="https://doi.org/10.3390/s20205915">10.3390/s20205915</a>.'
  mla: 'Leppin, Christian, et al. “A Quartz Crystal Microbalance, Which Tracks Four
    Overtones in Parallel with a Time Resolution of 10 Milliseconds: Application to
    Inkjet Printing.” <i>Sensors</i>, vol. 20, no. 20, 5915, MDPI AG, 2020, doi:<a
    href="https://doi.org/10.3390/s20205915">10.3390/s20205915</a>.'
  short: C. Leppin, S. Hampel, F.S. Meyer, A. Langhoff, U.E.A. Fittschen, D. Johannsmann,
    Sensors 20 (2020).
date_created: 2025-12-18T17:29:29Z
date_updated: 2025-12-18T17:33:50Z
doi: 10.3390/s20205915
extern: '1'
intvolume: '        20'
issue: '20'
language:
- iso: eng
publication: Sensors
publication_identifier:
  issn:
  - 1424-8220
publication_status: published
publisher: MDPI AG
quality_controlled: '1'
status: public
title: 'A Quartz Crystal Microbalance, Which Tracks Four Overtones in Parallel with
  a Time Resolution of 10 Milliseconds: Application to Inkjet Printing'
type: journal_article
user_id: '117722'
volume: 20
year: '2020'
...
---
_id: '63240'
abstract:
- lang: eng
  text: <jats:p>An electrochemical quartz crystal microbalance is described, which
    achieves a time resolution down to 100 μs. Accumulation and averaging over a few
    hours bring the noise down to about 30 mHz. The application examples are pH-driven
    viscosity changes in albumin solutions. The pH was switched with the electrode
    potential. The characteristic response time is in the millisecond range. The focus
    is on experimental aspects as well as advantages and limitations of the technique.</jats:p>
article_number: '021004'
author:
- first_name: Alexander
  full_name: Gödde, Alexander
  last_name: Gödde
- first_name: Christian
  full_name: Leppin, Christian
  id: '117722'
  last_name: Leppin
- first_name: Frederick S.
  full_name: Meyer, Frederick S.
  last_name: Meyer
- first_name: Arne
  full_name: Langhoff, Arne
  last_name: Langhoff
- first_name: Josef
  full_name: Hartl, Josef
  last_name: Hartl
- first_name: Patrick
  full_name: Garidel, Patrick
  last_name: Garidel
- first_name: Diethelm
  full_name: Johannsmann, Diethelm
  last_name: Johannsmann
citation:
  ama: Gödde A, Leppin C, Meyer FS, et al. Fast <i>p</i>H-mediated changes of the
    viscosity of protein solutions studied with a voltage-modulated quartz crystal
    microbalance. <i>Biointerphases</i>. 2020;15(2). doi:<a href="https://doi.org/10.1116/1.5140619">10.1116/1.5140619</a>
  apa: Gödde, A., Leppin, C., Meyer, F. S., Langhoff, A., Hartl, J., Garidel, P.,
    &#38; Johannsmann, D. (2020). Fast <i>p</i>H-mediated changes of the viscosity
    of protein solutions studied with a voltage-modulated quartz crystal microbalance.
    <i>Biointerphases</i>, <i>15</i>(2), Article 021004. <a href="https://doi.org/10.1116/1.5140619">https://doi.org/10.1116/1.5140619</a>
  bibtex: '@article{Gödde_Leppin_Meyer_Langhoff_Hartl_Garidel_Johannsmann_2020, title={Fast
    <i>p</i>H-mediated changes of the viscosity of protein solutions studied with
    a voltage-modulated quartz crystal microbalance}, volume={15}, DOI={<a href="https://doi.org/10.1116/1.5140619">10.1116/1.5140619</a>},
    number={2021004}, journal={Biointerphases}, publisher={American Vacuum Society},
    author={Gödde, Alexander and Leppin, Christian and Meyer, Frederick S. and Langhoff,
    Arne and Hartl, Josef and Garidel, Patrick and Johannsmann, Diethelm}, year={2020}
    }'
  chicago: Gödde, Alexander, Christian Leppin, Frederick S. Meyer, Arne Langhoff,
    Josef Hartl, Patrick Garidel, and Diethelm Johannsmann. “Fast <i>p</i>H-Mediated
    Changes of the Viscosity of Protein Solutions Studied with a Voltage-Modulated
    Quartz Crystal Microbalance.” <i>Biointerphases</i> 15, no. 2 (2020). <a href="https://doi.org/10.1116/1.5140619">https://doi.org/10.1116/1.5140619</a>.
  ieee: 'A. Gödde <i>et al.</i>, “Fast <i>p</i>H-mediated changes of the viscosity
    of protein solutions studied with a voltage-modulated quartz crystal microbalance,”
    <i>Biointerphases</i>, vol. 15, no. 2, Art. no. 021004, 2020, doi: <a href="https://doi.org/10.1116/1.5140619">10.1116/1.5140619</a>.'
  mla: Gödde, Alexander, et al. “Fast <i>p</i>H-Mediated Changes of the Viscosity
    of Protein Solutions Studied with a Voltage-Modulated Quartz Crystal Microbalance.”
    <i>Biointerphases</i>, vol. 15, no. 2, 021004, American Vacuum Society, 2020,
    doi:<a href="https://doi.org/10.1116/1.5140619">10.1116/1.5140619</a>.
  short: A. Gödde, C. Leppin, F.S. Meyer, A. Langhoff, J. Hartl, P. Garidel, D. Johannsmann,
    Biointerphases 15 (2020).
date_created: 2025-12-18T17:31:15Z
date_updated: 2025-12-18T17:35:30Z
doi: 10.1116/1.5140619
extern: '1'
intvolume: '        15'
issue: '2'
language:
- iso: eng
publication: Biointerphases
publication_identifier:
  issn:
  - 1934-8630
  - 1559-4106
publication_status: published
publisher: American Vacuum Society
quality_controlled: '1'
status: public
title: Fast <i>p</i>H-mediated changes of the viscosity of protein solutions studied
  with a voltage-modulated quartz crystal microbalance
type: journal_article
user_id: '117722'
volume: 15
year: '2020'
...
---
_id: '63340'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>The chemotaxis-growth
    system</jats:p>\r\n               <jats:p>\r\n                  <jats:disp-formula
    id=\"j_ans-2020-2107_eq_0001\">\r\n                     <jats:label>($\\star$)</jats:label>\r\n
    \                    <jats:alternatives>\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mo>{</m:mo>\r\n
    \                             <m:mtable columnspacing=\"0pt\" displaystyle=\"true\"
    rowspacing=\"0pt\">\r\n                                 <m:mtr>\r\n                                    <m:mtd
    columnalign=\"right\">\r\n                                       <m:msub>\r\n
    \                                         <m:mi>u</m:mi>\r\n                                          <m:mi>t</m:mi>\r\n
    \                                      </m:msub>\r\n                                    </m:mtd>\r\n
    \                                   <m:mtd columnalign=\"left\">\r\n                                       <m:mrow>\r\n
    \                                         <m:mrow>\r\n                                             <m:mi
    />\r\n                                             <m:mo>=</m:mo>\r\n                                             <m:mrow>\r\n
    \                                               <m:mrow>\r\n                                                   <m:mrow>\r\n
    \                                                     <m:mrow>\r\n                                                         <m:mi>D</m:mi>\r\n
    \                                                        <m:mo>⁢</m:mo>\r\n                                                         <m:mi
    mathvariant=\"normal\">Δ</m:mi>\r\n                                                         <m:mo>⁢</m:mo>\r\n
    \                                                        <m:mi>u</m:mi>\r\n                                                      </m:mrow>\r\n
    \                                                     <m:mo>-</m:mo>\r\n                                                      <m:mrow>\r\n
    \                                                        <m:mrow>\r\n                                                            <m:mi>χ</m:mi>\r\n
    \                                                           <m:mo>⁢</m:mo>\r\n
    \                                                           <m:mo>∇</m:mo>\r\n
    \                                                        </m:mrow>\r\n                                                         <m:mo>⋅</m:mo>\r\n
    \                                                        <m:mrow>\r\n                                                            <m:mo
    stretchy=\"false\">(</m:mo>\r\n                                                            <m:mrow>\r\n
    \                                                              <m:mi>u</m:mi>\r\n
    \                                                              <m:mo>⁢</m:mo>\r\n
    \                                                              <m:mrow>\r\n                                                                  <m:mo>∇</m:mo>\r\n
    \                                                                 <m:mo>⁡</m:mo>\r\n
    \                                                                 <m:mi>v</m:mi>\r\n
    \                                                              </m:mrow>\r\n                                                            </m:mrow>\r\n
    \                                                           <m:mo stretchy=\"false\">)</m:mo>\r\n
    \                                                        </m:mrow>\r\n                                                      </m:mrow>\r\n
    \                                                  </m:mrow>\r\n                                                   <m:mo>+</m:mo>\r\n
    \                                                  <m:mrow>\r\n                                                      <m:mi>ρ</m:mi>\r\n
    \                                                     <m:mo>⁢</m:mo>\r\n                                                      <m:mi>u</m:mi>\r\n
    \                                                  </m:mrow>\r\n                                                </m:mrow>\r\n
    \                                               <m:mo>-</m:mo>\r\n                                                <m:mrow>\r\n
    \                                                  <m:mi>μ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:msup>\r\n                                                      <m:mi>u</m:mi>\r\n
    \                                                     <m:mi>α</m:mi>\r\n                                                   </m:msup>\r\n
    \                                               </m:mrow>\r\n                                             </m:mrow>\r\n
    \                                         </m:mrow>\r\n                                          <m:mo>,</m:mo>\r\n
    \                                      </m:mrow>\r\n                                    </m:mtd>\r\n
    \                                </m:mtr>\r\n                                 <m:mtr>\r\n
    \                                   <m:mtd columnalign=\"right\">\r\n                                       <m:msub>\r\n
    \                                         <m:mi>v</m:mi>\r\n                                          <m:mi>t</m:mi>\r\n
    \                                      </m:msub>\r\n                                    </m:mtd>\r\n
    \                                   <m:mtd columnalign=\"left\">\r\n                                       <m:mrow>\r\n
    \                                         <m:mi />\r\n                                          <m:mo>=</m:mo>\r\n
    \                                         <m:mrow>\r\n                                             <m:mrow>\r\n
    \                                               <m:mrow>\r\n                                                   <m:mi>d</m:mi>\r\n
    \                                                  <m:mo>⁢</m:mo>\r\n                                                   <m:mi
    mathvariant=\"normal\">Δ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:mi>v</m:mi>\r\n                                                </m:mrow>\r\n
    \                                               <m:mo>-</m:mo>\r\n                                                <m:mrow>\r\n
    \                                                  <m:mi>κ</m:mi>\r\n                                                   <m:mo>⁢</m:mo>\r\n
    \                                                  <m:mi>v</m:mi>\r\n                                                </m:mrow>\r\n
    \                                            </m:mrow>\r\n                                             <m:mo>+</m:mo>\r\n
    \                                            <m:mrow>\r\n                                                <m:mi>λ</m:mi>\r\n
    \                                               <m:mo>⁢</m:mo>\r\n                                                <m:mi>u</m:mi>\r\n
    \                                            </m:mrow>\r\n                                          </m:mrow>\r\n
    \                                      </m:mrow>\r\n                                    </m:mtd>\r\n
    \                                </m:mtr>\r\n                              </m:mtable>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_fig_001.png\" />\r\n                        <jats:tex-math>{}\\left\\{\\begin{aligned}
    \\displaystyle{}u_{t}&amp;\\displaystyle=D\\Delta u-\\chi% \\nabla\\cdot(u\\nabla
    v)+\\rho u-\\mu u^{\\alpha},\\\\ \\displaystyle v_{t}&amp;\\displaystyle=d\\Delta
    v-\\kappa v+\\lambda u\\end{aligned}\\right.</jats:tex-math>\r\n                     </jats:alternatives>\r\n
    \                 </jats:disp-formula>\r\n               </jats:p>\r\n               <jats:p>is
    considered under homogeneous Neumann boundary conditions in smoothly bounded domains
    <jats:inline-formula id=\"j_ans-2020-2107_ineq_9999\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi mathvariant=\"normal\">Ω</m:mi>\r\n
    \                             <m:mo>⊂</m:mo>\r\n                              <m:msup>\r\n
    \                                <m:mi>ℝ</m:mi>\r\n                                 <m:mi>n</m:mi>\r\n
    \                             </m:msup>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2020-2107_inl_001.png\"
    />\r\n                        <jats:tex-math>{\\Omega\\subset\\mathbb{R}^{n}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    <jats:inline-formula id=\"j_ans-2020-2107_ineq_9998\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>n</m:mi>\r\n
    \                             <m:mo>≥</m:mo>\r\n                              <m:mn>1</m:mn>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_002.png\" />\r\n                        <jats:tex-math>{n\\geq
    1}</jats:tex-math>\r\n                     </jats:alternatives>\r\n                  </jats:inline-formula>.
    For any choice of <jats:inline-formula id=\"j_ans-2020-2107_ineq_9997\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>α</m:mi>\r\n
    \                             <m:mo>&gt;</m:mo>\r\n                              <m:mn>1</m:mn>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_003.png\" />\r\n                        <jats:tex-math>{\\alpha&gt;1}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    the literature provides a comprehensive result on global existence for widely
    arbitrary initial data within a suitably generalized solution concept, but the
    regularity properties of such solutions may be rather poor, as indicated by precedent
    results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic
    simplifications. Based on the analysis of a certain eventual Lyapunov-type feature
    of ($\\star$), the present work shows that, whenever <jats:inline-formula id=\"j_ans-2020-2107_ineq_9996\">\r\n
    \                    <jats:alternatives>\r\n                        <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mi>α</m:mi>\r\n
    \                             <m:mo>≥</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mn>2</m:mn>\r\n                                 <m:mo>-</m:mo>\r\n
    \                                <m:mfrac>\r\n                                    <m:mn>2</m:mn>\r\n
    \                                   <m:mi>n</m:mi>\r\n                                 </m:mfrac>\r\n
    \                             </m:mrow>\r\n                           </m:mrow>\r\n
    \                       </m:math>\r\n                        <jats:inline-graphic
    xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2020-2107_inl_004.png\"
    />\r\n                        <jats:tex-math>{\\alpha\\geq 2-\\frac{2}{n}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>,
    under an appropriate smallness assumption on χ, any such solution at least asymptotically
    exhibits relaxation by approaching the nontrivial spatially homogeneous steady
    state <jats:inline-formula id=\"j_ans-2020-2107_ineq_9995\">\r\n                     <jats:alternatives>\r\n
    \                       <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                          <m:mrow>\r\n                              <m:mo maxsize=\"120%\"
    minsize=\"120%\">(</m:mo>\r\n                              <m:msup>\r\n                                 <m:mrow>\r\n
    \                                   <m:mo maxsize=\"120%\" minsize=\"120%\">(</m:mo>\r\n
    \                                   <m:mfrac>\r\n                                       <m:mi>ρ</m:mi>\r\n
    \                                      <m:mi>μ</m:mi>\r\n                                    </m:mfrac>\r\n
    \                                   <m:mo maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n
    \                                </m:mrow>\r\n                                 <m:mfrac>\r\n
    \                                   <m:mn>1</m:mn>\r\n                                    <m:mrow>\r\n
    \                                      <m:mi>α</m:mi>\r\n                                       <m:mo>-</m:mo>\r\n
    \                                      <m:mn>1</m:mn>\r\n                                    </m:mrow>\r\n
    \                                </m:mfrac>\r\n                              </m:msup>\r\n
    \                             <m:mo>,</m:mo>\r\n                              <m:mrow>\r\n
    \                                <m:mfrac>\r\n                                    <m:mi>λ</m:mi>\r\n
    \                                   <m:mi>κ</m:mi>\r\n                                 </m:mfrac>\r\n
    \                                <m:mo>⁢</m:mo>\r\n                                 <m:msup>\r\n
    \                                   <m:mrow>\r\n                                       <m:mo
    maxsize=\"120%\" minsize=\"120%\">(</m:mo>\r\n                                       <m:mfrac>\r\n
    \                                         <m:mi>ρ</m:mi>\r\n                                          <m:mi>μ</m:mi>\r\n
    \                                      </m:mfrac>\r\n                                       <m:mo
    maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n                                    </m:mrow>\r\n
    \                                   <m:mfrac>\r\n                                       <m:mn>1</m:mn>\r\n
    \                                      <m:mrow>\r\n                                          <m:mi>α</m:mi>\r\n
    \                                         <m:mo>-</m:mo>\r\n                                          <m:mn>1</m:mn>\r\n
    \                                      </m:mrow>\r\n                                    </m:mfrac>\r\n
    \                                </m:msup>\r\n                              </m:mrow>\r\n
    \                             <m:mo maxsize=\"120%\" minsize=\"120%\">)</m:mo>\r\n
    \                          </m:mrow>\r\n                        </m:math>\r\n
    \                       <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\"
    xlink:href=\"graphic/j_ans-2020-2107_inl_005.png\" />\r\n                        <jats:tex-math>{\\bigl{(}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}},\\frac{\\lambda}{%
    \\kappa}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}}\\bigr{)}}</jats:tex-math>\r\n
    \                    </jats:alternatives>\r\n                  </jats:inline-formula>
    in the large time limit.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions. <i>Advanced Nonlinear Studies</i>.
    2020;20(4):795-817. doi:<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>
  apa: Winkler, M. (2020). Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions. <i>Advanced Nonlinear Studies</i>,
    <i>20</i>(4), 795–817. <a href="https://doi.org/10.1515/ans-2020-2107">https://doi.org/10.1515/ans-2020-2107</a>
  bibtex: '@article{Winkler_2020, title={Attractiveness of Constant States in Logistic-Type
    Keller–Segel Systems Involving Subquadratic Growth Restrictions}, volume={20},
    DOI={<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>},
    number={4}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2020}, pages={795–817} }'
  chicago: 'Winkler, Michael. “Attractiveness of Constant States in Logistic-Type
    Keller–Segel Systems Involving Subquadratic Growth Restrictions.” <i>Advanced
    Nonlinear Studies</i> 20, no. 4 (2020): 795–817. <a href="https://doi.org/10.1515/ans-2020-2107">https://doi.org/10.1515/ans-2020-2107</a>.'
  ieee: 'M. Winkler, “Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions,” <i>Advanced Nonlinear Studies</i>,
    vol. 20, no. 4, pp. 795–817, 2020, doi: <a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>.'
  mla: Winkler, Michael. “Attractiveness of Constant States in Logistic-Type Keller–Segel
    Systems Involving Subquadratic Growth Restrictions.” <i>Advanced Nonlinear Studies</i>,
    vol. 20, no. 4, Walter de Gruyter GmbH, 2020, pp. 795–817, doi:<a href="https://doi.org/10.1515/ans-2020-2107">10.1515/ans-2020-2107</a>.
  short: M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.
date_created: 2025-12-18T19:46:54Z
date_updated: 2025-12-18T19:58:22Z
doi: 10.1515/ans-2020-2107
intvolume: '        20'
issue: '4'
language:
- iso: eng
page: 795-817
publication: Advanced Nonlinear Studies
publication_identifier:
  issn:
  - 1536-1365
  - 2169-0375
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving
  Subquadratic Growth Restrictions
type: journal_article
user_id: '31496'
volume: 20
year: '2020'
...
---
_id: '63342'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a bounded planar domain <jats:inline-formula><jats:alternatives><jats:tex-math>$\\varOmega
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>Ω</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with smooth boundary, the initial-boundary value problem of homogeneous Neumann
    type for the Keller-Segel-fluid system \r\n\t\t\t<jats:disp-formula><jats:alternatives><jats:tex-math>
    $$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t} +
    \\nabla \\cdot (nu) = \\Delta n - \\nabla \\cdot (n\\nabla c), &amp; x\\in \\varOmega
    , \\ t&gt;0, \\\\ 0 = \\Delta c -c+n, &amp; x\\in \\varOmega , \\ t&gt;0, \\end{array}\\displaystyle
    \\right . \\end{aligned}$$ </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:mtable>\r\n
    \                     <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:msub>\r\n
    \                           <mml:mi>n</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:mo>(</mml:mo>\r\n                          <mml:mi>n</mml:mi>\r\n
    \                         <mml:mi>u</mml:mi>\r\n                          <mml:mo>)</mml:mo>\r\n
    \                         <mml:mo>=</mml:mo>\r\n                          <mml:mi>Δ</mml:mi>\r\n
    \                         <mml:mi>n</mml:mi>\r\n                          <mml:mo>−</mml:mo>\r\n
    \                         <mml:mi>∇</mml:mi>\r\n                          <mml:mo>⋅</mml:mo>\r\n
    \                         <mml:mo>(</mml:mo>\r\n                          <mml:mi>n</mml:mi>\r\n
    \                         <mml:mi>∇</mml:mi>\r\n                          <mml:mi>c</mml:mi>\r\n
    \                         <mml:mo>)</mml:mo>\r\n                          <mml:mo>,</mml:mo>\r\n
    \                       </mml:mtd>\r\n                        <mml:mtd>\r\n                          <mml:mi>x</mml:mi>\r\n
    \                         <mml:mo>∈</mml:mo>\r\n                          <mml:mi>Ω</mml:mi>\r\n
    \                         <mml:mo>,</mml:mo>\r\n                          <mml:mspace/>\r\n
    \                         <mml:mi>t</mml:mi>\r\n                          <mml:mo>&gt;</mml:mo>\r\n
    \                         <mml:mn>0</mml:mn>\r\n                          <mml:mo>,</mml:mo>\r\n
    \                       </mml:mtd>\r\n                      </mml:mtr>\r\n                      <mml:mtr>\r\n
    \                       <mml:mtd>\r\n                          <mml:mn>0</mml:mn>\r\n
    \                         <mml:mo>=</mml:mo>\r\n                          <mml:mi>Δ</mml:mi>\r\n
    \                         <mml:mi>c</mml:mi>\r\n                          <mml:mo>−</mml:mo>\r\n
    \                         <mml:mi>c</mml:mi>\r\n                          <mml:mo>+</mml:mo>\r\n
    \                         <mml:mi>n</mml:mi>\r\n                          <mml:mo>,</mml:mo>\r\n
    \                       </mml:mtd>\r\n                        <mml:mtd>\r\n                          <mml:mi>x</mml:mi>\r\n
    \                         <mml:mo>∈</mml:mo>\r\n                          <mml:mi>Ω</mml:mi>\r\n
    \                         <mml:mo>,</mml:mo>\r\n                          <mml:mspace/>\r\n
    \                         <mml:mi>t</mml:mi>\r\n                          <mml:mo>&gt;</mml:mo>\r\n
    \                         <mml:mn>0</mml:mn>\r\n                          <mml:mo>,</mml:mo>\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, where <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>u</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> is a given
    sufficiently smooth velocity field on <jats:inline-formula><jats:alternatives><jats:tex-math>$\\overline
    {\\varOmega }\\times [0,\\infty )$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\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: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:math></jats:alternatives></jats:inline-formula> that is
    tangential on <jats:inline-formula><jats:alternatives><jats:tex-math>$\\partial
    \\varOmega $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>∂</mml:mi>\r\n                  <mml:mi>Ω</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> but not
    necessarily solenoidal.</jats:p><jats:p>It is firstly shown that for any choice
    of <jats:inline-formula><jats:alternatives><jats:tex-math>$n_{0}\\in C^{0}(\\overline
    {\\varOmega })$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msub>\r\n                  <mml:mo>∈</mml:mo>\r\n                  <mml:msup>\r\n
    \                   <mml:mi>C</mml:mi>\r\n                    <mml:mn>0</mml:mn>\r\n
    \                 </mml:msup>\r\n                  <mml:mo>(</mml:mo>\r\n                  <mml:mover>\r\n
    \                   <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>‾</mml:mo>\r\n
    \                 </mml:mover>\r\n                  <mml:mo>)</mml:mo>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    with <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int _{\\varOmega}n_{0}&lt;4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                  <mml:mo>&lt;</mml:mo>\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>, this problem
    admits a global classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$n(\\cdot
    ,0)=n_{0}$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>n</mml:mi>\r\n                  <mml:mo>(</mml:mo>\r\n
    \                 <mml:mo>⋅</mml:mo>\r\n                  <mml:mo>,</mml:mo>\r\n
    \                 <mml:mn>0</mml:mn>\r\n                  <mml:mo>)</mml:mo>\r\n
    \                 <mml:mo>=</mml:mo>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>,
    and that this solution is even bounded whenever <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>u</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> is bounded
    and <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int _{\\varOmega}n_{0}&lt;2\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:msub>\r\n                    <mml:mi>n</mml:mi>\r\n
    \                   <mml:mn>0</mml:mn>\r\n                  </mml:msub>\r\n                  <mml:mo>&lt;</mml:mo>\r\n
    \                 <mml:mn>2</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula>. Secondly,
    it is seen that for each <jats:inline-formula><jats:alternatives><jats:tex-math>$m&gt;4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>m</mml:mi>\r\n                  <mml:mo>&gt;</mml:mo>\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> one can
    find a classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$\\int
    _{\\varOmega}n(\\cdot ,0)=m$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:msub>\r\n                    <mml:mo>∫</mml:mo>\r\n                    <mml:mi>Ω</mml:mi>\r\n
    \                 </mml:msub>\r\n                  <mml:mi>n</mml:mi>\r\n                  <mml:mo>(</mml:mo>\r\n
    \                 <mml:mo>⋅</mml:mo>\r\n                  <mml:mo>,</mml:mo>\r\n
    \                 <mml:mn>0</mml:mn>\r\n                  <mml:mo>)</mml:mo>\r\n
    \                 <mml:mo>=</mml:mo>\r\n                  <mml:mi>m</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> which blows
    up in finite time, provided that <jats:inline-formula><jats:alternatives><jats:tex-math>$\\varOmega
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>Ω</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>
    satisfies a technical assumption requiring <jats:inline-formula><jats:alternatives><jats:tex-math>$\\partial
    \\varOmega $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>∂</mml:mi>\r\n                  <mml:mi>Ω</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> to contain
    a line segment.</jats:p><jats:p>In particular, this indicates that the value <jats:inline-formula><jats:alternatives><jats:tex-math>$4\\pi
    $</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mn>4</mml:mn>\r\n                  <mml:mi>π</mml:mi>\r\n
    \               </mml:math></jats:alternatives></jats:inline-formula> of the critical
    mass for the corresponding fluid-free Keller-Segel system is left unchanged by
    any fluid interaction of the considered type, thus marking a considerable contrast
    to a recent result revealing some fluid-induced increase of critical blow-up masses
    in a related Cauchy problem in the entire plane.</jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains? <i>Acta Applicandae Mathematicae</i>. 2020;169(1):577-591.
    doi:<a href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>
  apa: Winkler, M. (2020). Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains? <i>Acta Applicandae Mathematicae</i>, <i>169</i>(1),
    577–591. <a href="https://doi.org/10.1007/s10440-020-00312-2">https://doi.org/10.1007/s10440-020-00312-2</a>
  bibtex: '@article{Winkler_2020, title={Can Fluid Interaction Influence the Critical
    Mass for Taxis-Driven Blow-up in Bounded Planar Domains?}, volume={169}, DOI={<a
    href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>},
    number={1}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science
    and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={577–591}
    }'
  chicago: 'Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for
    Taxis-Driven Blow-up in Bounded Planar Domains?” <i>Acta Applicandae Mathematicae</i>
    169, no. 1 (2020): 577–91. <a href="https://doi.org/10.1007/s10440-020-00312-2">https://doi.org/10.1007/s10440-020-00312-2</a>.'
  ieee: 'M. Winkler, “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains?,” <i>Acta Applicandae Mathematicae</i>, vol.
    169, no. 1, pp. 577–591, 2020, doi: <a href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>.'
  mla: Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven
    Blow-up in Bounded Planar Domains?” <i>Acta Applicandae Mathematicae</i>, vol.
    169, no. 1, Springer Science and Business Media LLC, 2020, pp. 577–91, doi:<a
    href="https://doi.org/10.1007/s10440-020-00312-2">10.1007/s10440-020-00312-2</a>.
  short: M. Winkler, Acta Applicandae Mathematicae 169 (2020) 577–591.
date_created: 2025-12-18T19:47:51Z
date_updated: 2025-12-18T19:57:40Z
doi: 10.1007/s10440-020-00312-2
intvolume: '       169'
issue: '1'
language:
- iso: eng
page: 577-591
publication: Acta Applicandae Mathematicae
publication_identifier:
  issn:
  - 0167-8019
  - 1572-9036
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up
  in Bounded Planar Domains?
type: journal_article
user_id: '31496'
volume: 169
year: '2020'
...
---
_id: '63336'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system. <i>Journal d’Analyse Mathématique</i>. 2020;141(2):585-624.
    doi:<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>
  apa: Winkler, M. (2020). Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system. <i>Journal d’Analyse Mathématique</i>, <i>141</i>(2), 585–624.
    <a href="https://doi.org/10.1007/s11854-020-0109-4">https://doi.org/10.1007/s11854-020-0109-4</a>
  bibtex: '@article{Winkler_2020, title={Blow-up profiles and life beyond blow-up
    in the fully parabolic Keller-Segel system}, volume={141}, DOI={<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>},
    number={2}, journal={Journal d’Analyse Mathématique}, publisher={Springer Science
    and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={585–624}
    }'
  chicago: 'Winkler, Michael. “Blow-up Profiles and Life beyond Blow-up in the Fully
    Parabolic Keller-Segel System.” <i>Journal d’Analyse Mathématique</i> 141, no.
    2 (2020): 585–624. <a href="https://doi.org/10.1007/s11854-020-0109-4">https://doi.org/10.1007/s11854-020-0109-4</a>.'
  ieee: 'M. Winkler, “Blow-up profiles and life beyond blow-up in the fully parabolic
    Keller-Segel system,” <i>Journal d’Analyse Mathématique</i>, vol. 141, no. 2,
    pp. 585–624, 2020, doi: <a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>.'
  mla: Winkler, Michael. “Blow-up Profiles and Life beyond Blow-up in the Fully Parabolic
    Keller-Segel System.” <i>Journal d’Analyse Mathématique</i>, vol. 141, no. 2,
    Springer Science and Business Media LLC, 2020, pp. 585–624, doi:<a href="https://doi.org/10.1007/s11854-020-0109-4">10.1007/s11854-020-0109-4</a>.
  short: M. Winkler, Journal d’Analyse Mathématique 141 (2020) 585–624.
date_created: 2025-12-18T19:44:38Z
date_updated: 2025-12-18T19:56:40Z
doi: 10.1007/s11854-020-0109-4
intvolume: '       141'
issue: '2'
language:
- iso: eng
page: 585-624
publication: Journal d'Analyse Mathématique
publication_identifier:
  issn:
  - 0021-7670
  - 1565-8538
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel
  system
type: journal_article
user_id: '31496'
volume: 141
year: '2020'
...
---
_id: '63329'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>Recent experimental work has revealed
    that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms.
    One is a chemotactic-driven mechanism where chemokine (chemical component) bounded
    to the extracellular matrix (ECM) is released and skewed in the flow direction.
    This leads to higher chemical concentrations downstream which the tumor cells
    can sense and migrate toward. The other is a mechanism where the flowing fluid
    imposes a stress on the tumor cells which triggers them to go in the upstream
    direction. Researchers have suggested that these two migration modes possibly
    can play a role in metastatic behavior, i.e., the process where tumor cells are
    able to break loose from the primary tumor and move to nearby lymphatic vessels.
    In Waldeland and Evje (J Biomech 81:22–35, 2018), a mathematical cell–fluid model
    was put forward based on a mixture theory formulation. It was demonstrated that
    the model was able to capture the main characteristics of the two competing migration
    mechanisms. The objective of the current work is to seek deeper insight into certain
    qualitative aspects of these competing mechanisms by means of mathematical methods.
    For that purpose, we propose a simpler version of the cell–fluid model mentioned
    above but such that the two competing migration mechanisms are retained. An initial
    cell distribution in a one-dimensional slab is exposed to a constant fluid flow
    from one end to the other, consistent with the experimental setup. Then, we explore
    by means of analytical estimates the long-time behavior of the two competing migration
    mechanisms for two different scenarios: (i) when the initial cell volume fraction
    is low and (ii) when the initial cell volume fraction is high. In particular,
    it is demonstrated in a strict mathematical sense that for a sufficiently low
    initial cell volume fraction, the downstream migration dominates in the sense
    that the solution converges to a downstream-dominated steady state as time elapses.
    On the other hand, with a sufficiently high initial cell volume fraction, the
    upstream migration mechanism is the stronger in the sense that the solution converges
    to an upstream-dominated steady state.\r\n</jats:p>"
author:
- first_name: Steinar
  full_name: Evje, Steinar
  last_name: Evje
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Evje S, Winkler M. Mathematical Analysis of Two Competing Cancer Cell Migration
    Mechanisms Driven by Interstitial Fluid Flow. <i>Journal of Nonlinear Science</i>.
    2020;30(4):1809-1847. doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>
  apa: Evje, S., &#38; Winkler, M. (2020). Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow. <i>Journal
    of Nonlinear Science</i>, <i>30</i>(4), 1809–1847. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>
  bibtex: '@article{Evje_Winkler_2020, title={Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow}, volume={30},
    DOI={<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>},
    number={4}, journal={Journal of Nonlinear Science}, publisher={Springer Science
    and Business Media LLC}, author={Evje, Steinar and Winkler, Michael}, year={2020},
    pages={1809–1847} }'
  chicago: 'Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i> 30, no. 4 (2020): 1809–47. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>.'
  ieee: 'S. Evje and M. Winkler, “Mathematical Analysis of Two Competing Cancer Cell
    Migration Mechanisms Driven by Interstitial Fluid Flow,” <i>Journal of Nonlinear
    Science</i>, vol. 30, no. 4, pp. 1809–1847, 2020, doi: <a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.'
  mla: Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i>, vol. 30, no. 4, Springer Science and Business Media
    LLC, 2020, pp. 1809–47, doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.
  short: S. Evje, M. Winkler, Journal of Nonlinear Science 30 (2020) 1809–1847.
date_created: 2025-12-18T19:38:02Z
date_updated: 2025-12-18T20:00:26Z
doi: 10.1007/s00332-020-09625-w
intvolume: '        30'
issue: '4'
language:
- iso: eng
page: 1809-1847
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Mathematical Analysis of Two Competing Cancer Cell Migration Mechanisms Driven
  by Interstitial Fluid Flow
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...
---
_id: '63331'
abstract:
- lang: eng
  text: <jats:p> We consider a class of macroscopic models for the spatio-temporal
    evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M.
    R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and
    L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl.
    Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain
    porous medium enhancement in the random diffusion of criminal agents may exert
    visible relaxation effects. It is shown that sufficient regularity of the non-negative
    source terms in the system and a sufficiently strong nonlinear enhancement ensure
    that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly
    bounded planar convex domain, admits locally bounded solutions for a wide class
    of arbitrary initial data. Furthermore, this solution is globally bounded under
    mild additional conditions on the source terms. These results are supplemented
    by numerical evidence which illustrates smoothing effects in solutions with sharply
    structured initial data in the presence of such porous medium-type diffusion and
    support the existence of singular structures in the linear diffusion case, which
    is the type of diffusion proposed in Ref. 29. </jats:p>
author:
- first_name: Nancy
  full_name: Rodríguez, Nancy
  last_name: Rodríguez
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Rodríguez N, Winkler M. Relaxation by nonlinear diffusion enhancement in a
    two-dimensional cross-diffusion model for urban crime propagation. <i>Mathematical
    Models and Methods in Applied Sciences</i>. 2020;30(11):2105-2137. doi:<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>
  apa: Rodríguez, N., &#38; Winkler, M. (2020). Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation.
    <i>Mathematical Models and Methods in Applied Sciences</i>, <i>30</i>(11), 2105–2137.
    <a href="https://doi.org/10.1142/s0218202520500396">https://doi.org/10.1142/s0218202520500396</a>
  bibtex: '@article{Rodríguez_Winkler_2020, title={Relaxation by nonlinear diffusion
    enhancement in a two-dimensional cross-diffusion model for urban crime propagation},
    volume={30}, DOI={<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>},
    number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Rodríguez, Nancy and Winkler, Michael}, year={2020},
    pages={2105–2137} }'
  chicago: 'Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion
    Enhancement in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.”
    <i>Mathematical Models and Methods in Applied Sciences</i> 30, no. 11 (2020):
    2105–37. <a href="https://doi.org/10.1142/s0218202520500396">https://doi.org/10.1142/s0218202520500396</a>.'
  ieee: 'N. Rodríguez and M. Winkler, “Relaxation by nonlinear diffusion enhancement
    in a two-dimensional cross-diffusion model for urban crime propagation,” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 11, pp. 2105–2137, 2020,
    doi: <a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>.'
  mla: Rodríguez, Nancy, and Michael Winkler. “Relaxation by Nonlinear Diffusion Enhancement
    in a Two-Dimensional Cross-Diffusion Model for Urban Crime Propagation.” <i>Mathematical
    Models and Methods in Applied Sciences</i>, vol. 30, no. 11, World Scientific
    Pub Co Pte Ltd, 2020, pp. 2105–37, doi:<a href="https://doi.org/10.1142/s0218202520500396">10.1142/s0218202520500396</a>.
  short: N. Rodríguez, M. Winkler, Mathematical Models and Methods in Applied Sciences
    30 (2020) 2105–2137.
date_created: 2025-12-18T19:38:42Z
date_updated: 2025-12-18T20:00:53Z
doi: 10.1142/s0218202520500396
intvolume: '        30'
issue: '11'
language:
- iso: eng
page: 2105-2137
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: Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion
  model for urban crime propagation
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...
---
_id: '63330'
author:
- first_name: Genglin
  full_name: Li, Genglin
  last_name: Li
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Li G, Tao Y, Winkler M. Large time behavior in a predator-prey system with
    indirect pursuit-evasion interaction. <i>Discrete and Continuous Dynamical Systems
    - B</i>. 2020;25(11):4383-4396. doi:<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>
  apa: Li, G., Tao, Y., &#38; Winkler, M. (2020). Large time behavior in a predator-prey
    system with indirect pursuit-evasion interaction. <i>Discrete and Continuous Dynamical
    Systems - B</i>, <i>25</i>(11), 4383–4396. <a href="https://doi.org/10.3934/dcdsb.2020102">https://doi.org/10.3934/dcdsb.2020102</a>
  bibtex: '@article{Li_Tao_Winkler_2020, title={Large time behavior in a predator-prey
    system with indirect pursuit-evasion interaction}, volume={25}, DOI={<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>},
    number={11}, journal={Discrete and Continuous Dynamical Systems - B}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Li, Genglin and Tao, Youshan
    and Winkler, Michael}, year={2020}, pages={4383–4396} }'
  chicago: 'Li, Genglin, Youshan Tao, and Michael Winkler. “Large Time Behavior in
    a Predator-Prey System with Indirect Pursuit-Evasion Interaction.” <i>Discrete
    and Continuous Dynamical Systems - B</i> 25, no. 11 (2020): 4383–96. <a href="https://doi.org/10.3934/dcdsb.2020102">https://doi.org/10.3934/dcdsb.2020102</a>.'
  ieee: 'G. Li, Y. Tao, and M. Winkler, “Large time behavior in a predator-prey system
    with indirect pursuit-evasion interaction,” <i>Discrete and Continuous Dynamical
    Systems - B</i>, vol. 25, no. 11, pp. 4383–4396, 2020, doi: <a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>.'
  mla: Li, Genglin, et al. “Large Time Behavior in a Predator-Prey System with Indirect
    Pursuit-Evasion Interaction.” <i>Discrete and Continuous Dynamical Systems - B</i>,
    vol. 25, no. 11, American Institute of Mathematical Sciences (AIMS), 2020, pp.
    4383–96, doi:<a href="https://doi.org/10.3934/dcdsb.2020102">10.3934/dcdsb.2020102</a>.
  short: G. Li, Y. Tao, M. Winkler, Discrete and Continuous Dynamical Systems - B
    25 (2020) 4383–4396.
date_created: 2025-12-18T19:38:22Z
date_updated: 2025-12-18T20:00:40Z
doi: 10.3934/dcdsb.2020102
intvolume: '        25'
issue: '11'
language:
- iso: eng
page: 4383-4396
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: Large time behavior in a predator-prey system with indirect pursuit-evasion
  interaction
type: journal_article
user_id: '31496'
volume: 25
year: '2020'
...
---
_id: '63327'
article_number: '103257'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: 'Winkler M. Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation. <i>Nonlinear Analysis: Real World
    Applications</i>. 2020;59. doi:<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>'
  apa: 'Winkler, M. (2020). Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation. <i>Nonlinear Analysis: Real World
    Applications</i>, <i>59</i>, Article 103257. <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">https://doi.org/10.1016/j.nonrwa.2020.103257</a>'
  bibtex: '@article{Winkler_2020, title={Global weak solutions in a three-dimensional
    Keller–Segel–Navier–Stokes system with gradient-dependent flux limitation}, volume={59},
    DOI={<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>},
    number={103257}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier
    BV}, author={Winkler, Michael}, year={2020} }'
  chicago: 'Winkler, Michael. “Global Weak Solutions in a Three-Dimensional Keller–Segel–Navier–Stokes
    System with Gradient-Dependent Flux Limitation.” <i>Nonlinear Analysis: Real World
    Applications</i> 59 (2020). <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">https://doi.org/10.1016/j.nonrwa.2020.103257</a>.'
  ieee: 'M. Winkler, “Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes
    system with gradient-dependent flux limitation,” <i>Nonlinear Analysis: Real World
    Applications</i>, vol. 59, Art. no. 103257, 2020, doi: <a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>.'
  mla: 'Winkler, Michael. “Global Weak Solutions in a Three-Dimensional Keller–Segel–Navier–Stokes
    System with Gradient-Dependent Flux Limitation.” <i>Nonlinear Analysis: Real World
    Applications</i>, vol. 59, 103257, Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.nonrwa.2020.103257">10.1016/j.nonrwa.2020.103257</a>.'
  short: 'M. Winkler, Nonlinear Analysis: Real World Applications 59 (2020).'
date_created: 2025-12-18T19:36:51Z
date_updated: 2025-12-18T19:59:57Z
doi: 10.1016/j.nonrwa.2020.103257
intvolume: '        59'
language:
- iso: eng
publication: 'Nonlinear Analysis: Real World Applications'
publication_identifier:
  issn:
  - 1468-1218
publication_status: published
publisher: Elsevier BV
status: public
title: Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system
  with gradient-dependent flux limitation
type: journal_article
user_id: '31496'
volume: 59
year: '2020'
...
---
_id: '63333'
article_number: '111870'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. A critical virus production rate for blow-up suppression
    in a haptotaxis model for oncolytic virotherapy. <i>Nonlinear Analysis</i>. 2020;198.
    doi:<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>
  apa: Tao, Y., &#38; Winkler, M. (2020). A critical virus production rate for blow-up
    suppression in a haptotaxis model for oncolytic virotherapy. <i>Nonlinear Analysis</i>,
    <i>198</i>, Article 111870. <a href="https://doi.org/10.1016/j.na.2020.111870">https://doi.org/10.1016/j.na.2020.111870</a>
  bibtex: '@article{Tao_Winkler_2020, title={A critical virus production rate for
    blow-up suppression in a haptotaxis model for oncolytic virotherapy}, volume={198},
    DOI={<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>},
    number={111870}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Tao,
    Youshan and Winkler, Michael}, year={2020} }'
  chicago: Tao, Youshan, and Michael Winkler. “A Critical Virus Production Rate for
    Blow-up Suppression in a Haptotaxis Model for Oncolytic Virotherapy.” <i>Nonlinear
    Analysis</i> 198 (2020). <a href="https://doi.org/10.1016/j.na.2020.111870">https://doi.org/10.1016/j.na.2020.111870</a>.
  ieee: 'Y. Tao and M. Winkler, “A critical virus production rate for blow-up suppression
    in a haptotaxis model for oncolytic virotherapy,” <i>Nonlinear Analysis</i>, vol.
    198, Art. no. 111870, 2020, doi: <a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>.'
  mla: Tao, Youshan, and Michael Winkler. “A Critical Virus Production Rate for Blow-up
    Suppression in a Haptotaxis Model for Oncolytic Virotherapy.” <i>Nonlinear Analysis</i>,
    vol. 198, 111870, Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.na.2020.111870">10.1016/j.na.2020.111870</a>.
  short: Y. Tao, M. Winkler, Nonlinear Analysis 198 (2020).
date_created: 2025-12-18T19:39:40Z
date_updated: 2025-12-18T20:01:18Z
doi: 10.1016/j.na.2020.111870
intvolume: '       198'
language:
- iso: eng
publication: Nonlinear Analysis
publication_identifier:
  issn:
  - 0362-546X
publication_status: published
publisher: Elsevier BV
status: public
title: A critical virus production rate for blow-up suppression in a haptotaxis model
  for oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 198
year: '2020'
...
---
_id: '63328'
article_number: '106785'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Boundedness in a three-dimensional Keller–Segel–Stokes system with
    subcritical sensitivity. <i>Applied Mathematics Letters</i>. 2020;112. doi:<a
    href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>
  apa: Winkler, M. (2020). Boundedness in a three-dimensional Keller–Segel–Stokes
    system with subcritical sensitivity. <i>Applied Mathematics Letters</i>, <i>112</i>,
    Article 106785. <a href="https://doi.org/10.1016/j.aml.2020.106785">https://doi.org/10.1016/j.aml.2020.106785</a>
  bibtex: '@article{Winkler_2020, title={Boundedness in a three-dimensional Keller–Segel–Stokes
    system with subcritical sensitivity}, volume={112}, DOI={<a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>},
    number={106785}, journal={Applied Mathematics Letters}, publisher={Elsevier BV},
    author={Winkler, Michael}, year={2020} }'
  chicago: Winkler, Michael. “Boundedness in a Three-Dimensional Keller–Segel–Stokes
    System with Subcritical Sensitivity.” <i>Applied Mathematics Letters</i> 112 (2020).
    <a href="https://doi.org/10.1016/j.aml.2020.106785">https://doi.org/10.1016/j.aml.2020.106785</a>.
  ieee: 'M. Winkler, “Boundedness in a three-dimensional Keller–Segel–Stokes system
    with subcritical sensitivity,” <i>Applied Mathematics Letters</i>, vol. 112, Art.
    no. 106785, 2020, doi: <a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>.'
  mla: Winkler, Michael. “Boundedness in a Three-Dimensional Keller–Segel–Stokes System
    with Subcritical Sensitivity.” <i>Applied Mathematics Letters</i>, vol. 112, 106785,
    Elsevier BV, 2020, doi:<a href="https://doi.org/10.1016/j.aml.2020.106785">10.1016/j.aml.2020.106785</a>.
  short: M. Winkler, Applied Mathematics Letters 112 (2020).
date_created: 2025-12-18T19:37:32Z
date_updated: 2025-12-18T20:00:10Z
doi: 10.1016/j.aml.2020.106785
intvolume: '       112'
language:
- iso: eng
publication: Applied Mathematics Letters
publication_identifier:
  issn:
  - 0893-9659
publication_status: published
publisher: Elsevier BV
status: public
title: Boundedness in a three-dimensional Keller–Segel–Stokes system with subcritical
  sensitivity
type: journal_article
user_id: '31496'
volume: 112
year: '2020'
...
---
_id: '63320'
author:
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Tao Y, Winkler M. Critical mass for infinite-time blow-up in a haptotaxis system
    with nonlinear zero-order interaction. <i>Discrete &#38;amp; Continuous Dynamical
    Systems - A</i>. 2020;41(1):439-454. doi:<a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>
  apa: Tao, Y., &#38; Winkler, M. (2020). Critical mass for infinite-time blow-up
    in a haptotaxis system with nonlinear zero-order interaction. <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i>, <i>41</i>(1), 439–454. <a href="https://doi.org/10.3934/dcds.2020216">https://doi.org/10.3934/dcds.2020216</a>
  bibtex: '@article{Tao_Winkler_2020, title={Critical mass for infinite-time blow-up
    in a haptotaxis system with nonlinear zero-order interaction}, volume={41}, DOI={<a
    href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>}, number={1},
    journal={Discrete &#38;amp; Continuous Dynamical Systems - A}, publisher={American
    Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler,
    Michael}, year={2020}, pages={439–454} }'
  chicago: 'Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
    in a Haptotaxis System with Nonlinear Zero-Order Interaction.” <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i> 41, no. 1 (2020): 439–54. <a href="https://doi.org/10.3934/dcds.2020216">https://doi.org/10.3934/dcds.2020216</a>.'
  ieee: 'Y. Tao and M. Winkler, “Critical mass for infinite-time blow-up in a haptotaxis
    system with nonlinear zero-order interaction,” <i>Discrete &#38;amp; Continuous
    Dynamical Systems - A</i>, vol. 41, no. 1, pp. 439–454, 2020, doi: <a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>.'
  mla: Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up
    in a Haptotaxis System with Nonlinear Zero-Order Interaction.” <i>Discrete &#38;amp;
    Continuous Dynamical Systems - A</i>, vol. 41, no. 1, American Institute of Mathematical
    Sciences (AIMS), 2020, pp. 439–54, doi:<a href="https://doi.org/10.3934/dcds.2020216">10.3934/dcds.2020216</a>.
  short: Y. Tao, M. Winkler, Discrete &#38;amp; Continuous Dynamical Systems - A 41
    (2020) 439–454.
date_created: 2025-12-18T19:33:59Z
date_updated: 2025-12-18T20:04:09Z
doi: 10.3934/dcds.2020216
intvolume: '        41'
issue: '1'
language:
- iso: eng
page: 439-454
publication: Discrete &amp; Continuous Dynamical Systems - A
publication_identifier:
  issn:
  - 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear
  zero-order interaction
type: journal_article
user_id: '31496'
volume: 41
year: '2020'
...
---
_id: '63335'
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Small-Mass Solutions in the Two-Dimensional Keller--Segel System
    Coupled to the Navier--Stokes Equations. <i>SIAM Journal on Mathematical Analysis</i>.
    2020;52(2):2041-2080. doi:<a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>
  apa: Winkler, M. (2020). Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations. <i>SIAM Journal on Mathematical
    Analysis</i>, <i>52</i>(2), 2041–2080. <a href="https://doi.org/10.1137/19m1264199">https://doi.org/10.1137/19m1264199</a>
  bibtex: '@article{Winkler_2020, title={Small-Mass Solutions in the Two-Dimensional
    Keller--Segel System Coupled to the Navier--Stokes Equations}, volume={52}, DOI={<a
    href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>}, number={2},
    journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Winkler, Michael}, year={2020}, pages={2041–2080}
    }'
  chicago: 'Winkler, Michael. “Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations.” <i>SIAM Journal on Mathematical
    Analysis</i> 52, no. 2 (2020): 2041–80. <a href="https://doi.org/10.1137/19m1264199">https://doi.org/10.1137/19m1264199</a>.'
  ieee: 'M. Winkler, “Small-Mass Solutions in the Two-Dimensional Keller--Segel System
    Coupled to the Navier--Stokes Equations,” <i>SIAM Journal on Mathematical Analysis</i>,
    vol. 52, no. 2, pp. 2041–2080, 2020, doi: <a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>.'
  mla: Winkler, Michael. “Small-Mass Solutions in the Two-Dimensional Keller--Segel
    System Coupled to the Navier--Stokes Equations.” <i>SIAM Journal on Mathematical
    Analysis</i>, vol. 52, no. 2, Society for Industrial &#38; Applied Mathematics
    (SIAM), 2020, pp. 2041–80, doi:<a href="https://doi.org/10.1137/19m1264199">10.1137/19m1264199</a>.
  short: M. Winkler, SIAM Journal on Mathematical Analysis 52 (2020) 2041–2080.
date_created: 2025-12-18T19:40:35Z
date_updated: 2025-12-18T20:01:42Z
doi: 10.1137/19m1264199
intvolume: '        52'
issue: '2'
language:
- iso: eng
page: 2041-2080
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: Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to
  the Navier--Stokes Equations
type: journal_article
user_id: '31496'
volume: 52
year: '2020'
...
---
_id: '63318'
abstract:
- lang: eng
  text: <jats:p>In a planar smoothly bounded domain<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline1.png"
    /><jats:tex-math>$\Omega$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    we consider the model for oncolytic virotherapy given by<jats:disp-formula id="S0956792520000133_udisp1"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu1.png" /><jats:tex-math>$$\left\{ \begin{array}{l}
    u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm]
    w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array}
    \right.$$</jats:tex-math></jats:alternatives></jats:disp-formula>with positive
    parameters<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    mime-subtype="png" xlink:href="S0956792520000133_inline2.png" /><jats:tex-math>$
    D_w $</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline3.png"
    /><jats:tex-math>$ D_z $</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline4.png"
    /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>.
    It is firstly shown that whenever<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline5.png"
    /><jats:tex-math>$\beta \lt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline6.png"
    /><jats:tex-math>$M \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    one can find initial data such that the solution of an associated no-flux initial-boundary
    value problem, well known to exist globally actually for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline7.png"
    /><jats:tex-math>$\beta \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    satisfies<jats:disp-formula id="S0956792520000133_udisp2"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu2.png" /><jats:tex-math>$$u\ge M \qquad \mbox{in
    } \Omega\times (0,\infty).$$</jats:tex-math></jats:alternatives></jats:disp-formula>If<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline8.png"
    /><jats:tex-math>$\beta \gt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>,
    however, then for arbitrary initial data the corresponding is seen to have the
    property that<jats:disp-formula id="S0956792520000133_udisp3"><jats:alternatives><jats:graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image"
    xlink:href="S0956792520000133_eqnu3.png" /><jats:tex-math>$$\liminf_{t\to\infty}
    \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$</jats:tex-math></jats:alternatives></jats:disp-formula>This
    may be interpreted as indicating that<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline9.png"
    /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>plays
    the role of a critical virus replication rate with regard to efficiency of the
    considered virotherapy, with corresponding threshold value given by<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline10.png"
    /><jats:tex-math>$\beta = 1$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>
author:
- first_name: YOUSHAN
  full_name: TAO, YOUSHAN
  last_name: TAO
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: TAO Y, Winkler M. A critical virus production rate for efficiency of oncolytic
    virotherapy. <i>European Journal of Applied Mathematics</i>. 2020;32(2):301-316.
    doi:<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>
  apa: TAO, Y., &#38; Winkler, M. (2020). A critical virus production rate for efficiency
    of oncolytic virotherapy. <i>European Journal of Applied Mathematics</i>, <i>32</i>(2),
    301–316. <a href="https://doi.org/10.1017/s0956792520000133">https://doi.org/10.1017/s0956792520000133</a>
  bibtex: '@article{TAO_Winkler_2020, title={A critical virus production rate for
    efficiency of oncolytic virotherapy}, volume={32}, DOI={<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>},
    number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge
    University Press (CUP)}, author={TAO, YOUSHAN and Winkler, Michael}, year={2020},
    pages={301–316} }'
  chicago: 'TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for
    Efficiency of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i>
    32, no. 2 (2020): 301–16. <a href="https://doi.org/10.1017/s0956792520000133">https://doi.org/10.1017/s0956792520000133</a>.'
  ieee: 'Y. TAO and M. Winkler, “A critical virus production rate for efficiency of
    oncolytic virotherapy,” <i>European Journal of Applied Mathematics</i>, vol. 32,
    no. 2, pp. 301–316, 2020, doi: <a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>.'
  mla: TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency
    of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i>, vol.
    32, no. 2, Cambridge University Press (CUP), 2020, pp. 301–16, doi:<a href="https://doi.org/10.1017/s0956792520000133">10.1017/s0956792520000133</a>.
  short: Y. TAO, M. Winkler, European Journal of Applied Mathematics 32 (2020) 301–316.
date_created: 2025-12-18T19:33:01Z
date_updated: 2025-12-18T20:06:35Z
doi: 10.1017/s0956792520000133
intvolume: '        32'
issue: '2'
language:
- iso: eng
page: 301-316
publication: European Journal of Applied Mathematics
publication_identifier:
  issn:
  - 0956-7925
  - 1469-4425
publication_status: published
publisher: Cambridge University Press (CUP)
status: public
title: A critical virus production rate for efficiency of oncolytic virotherapy
type: journal_article
user_id: '31496'
volume: 32
year: '2020'
...