---
_id: '63244'
abstract:
- lang: eng
  text: "<jats:p>\r\n            The Cauchy problem in \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\mathbb{R}^{n}</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            for the cross-diffusion system \r\n          </jats:p>\r\n          <jats:p>\r\n
    \           <jats:disp-formula>\r\n              <jats:tex-math>\\begin{cases}u_{t}
    = \\nabla \\cdot (D(u)\\nabla u) - \\nabla\\cdot (u\\nabla v), \\\\ 0 = \\Delta
    v +u,\\end{cases}</jats:tex-math>\r\n            </jats:disp-formula>\r\n          </jats:p>\r\n
    \         <jats:p>\r\n             is considered for \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>n\\ge 2</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            and under assumptions ensuring that \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>D</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            suitably generalizes the prototype given by \r\n          </jats:p>\r\n
    \         <jats:p>\r\n            <jats:disp-formula>\r\n              <jats:tex-math>D(\\xi)=(\\xi+1)^{-\\alpha},
    \\quad \\xi\\ge 0.</jats:tex-math>\r\n            </jats:disp-formula>\r\n          </jats:p>\r\n
    \         <jats:p>\r\n             Under the assumption that \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\alpha&gt;1</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           , it is shown that for any \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>r_{\\star}&gt;0</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \            and \r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\delta\\in
    (0,1)</jats:tex-math>\r\n            </jats:inline-formula>\r\n             one
    can find radially symmetric initial data from \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>C_{0}^{\\infty}(\\mathbb{R}^{n})</jats:tex-math>\r\n
    \           </jats:inline-formula>\r\n             such that the corresponding
    solution blows up within some finite time, and that this explosion occurs throughout
    certain spheres in an appropriate sense, with any such sphere being located in
    the annulus \r\n            <jats:inline-formula>\r\n              <jats:tex-math>\\overline{B}_{r_\\star+\\delta}(0)\\setminus
    B_{(1-\\delta)r_\\star}(0)</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           .This is complemented by a result revealing that when \r\n            <jats:inline-formula>\r\n
    \             <jats:tex-math>\\alpha&lt;1</jats:tex-math>\r\n            </jats:inline-formula>\r\n
    \           , any finite-mass unbounded radial solution must blow up exclusively
    at the spatial origin.\r\n          </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Can diffusion degeneracies enhance complexity in chemotactic aggregation?
    Finite-time blow-up on spheres in a quasilinear Keller–Segel system. <i>Journal
    of the European Mathematical Society</i>. Published online 2025. doi:<a href="https://doi.org/10.4171/jems/1607">10.4171/jems/1607</a>
  apa: Winkler, M. (2025). Can diffusion degeneracies enhance complexity in chemotactic
    aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system.
    <i>Journal of the European Mathematical Society</i>. <a href="https://doi.org/10.4171/jems/1607">https://doi.org/10.4171/jems/1607</a>
  bibtex: '@article{Winkler_2025, title={Can diffusion degeneracies enhance complexity
    in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel
    system}, DOI={<a href="https://doi.org/10.4171/jems/1607">10.4171/jems/1607</a>},
    journal={Journal of the European Mathematical Society}, publisher={European Mathematical
    Society - EMS - Publishing House GmbH}, author={Winkler, Michael}, year={2025}
    }'
  chicago: Winkler, Michael. “Can Diffusion Degeneracies Enhance Complexity in Chemotactic
    Aggregation? Finite-Time Blow-up on Spheres in a Quasilinear Keller–Segel System.”
    <i>Journal of the European Mathematical Society</i>, 2025. <a href="https://doi.org/10.4171/jems/1607">https://doi.org/10.4171/jems/1607</a>.
  ieee: 'M. Winkler, “Can diffusion degeneracies enhance complexity in chemotactic
    aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system,”
    <i>Journal of the European Mathematical Society</i>, 2025, doi: <a href="https://doi.org/10.4171/jems/1607">10.4171/jems/1607</a>.'
  mla: Winkler, Michael. “Can Diffusion Degeneracies Enhance Complexity in Chemotactic
    Aggregation? Finite-Time Blow-up on Spheres in a Quasilinear Keller–Segel System.”
    <i>Journal of the European Mathematical Society</i>, European Mathematical Society
    - EMS - Publishing House GmbH, 2025, doi:<a href="https://doi.org/10.4171/jems/1607">10.4171/jems/1607</a>.
  short: M. Winkler, Journal of the European Mathematical Society (2025).
date_created: 2025-12-18T18:59:39Z
date_updated: 2025-12-18T20:12:36Z
doi: 10.4171/jems/1607
language:
- iso: eng
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
  - 1435-9863
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time
  blow-up on spheres in a quasilinear Keller–Segel system
type: journal_article
user_id: '31496'
year: '2025'
...
---
_id: '63279'
abstract:
- lang: eng
  text: "<jats:p>\r\n                    In a smoothly bounded convex domain\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Omega \\subset \\mathbb{R}^3</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    , we consider
    the chemotaxis-Navier–Stokes model\r\n                  </jats:p>\r\n                  <jats:p>\r\n
    \                   <jats:disp-formula>\r\n                      <jats:tex-math>\\begin{cases}
    n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), &amp; x\\in
    \\Omega, \\, t&gt;0, \\\\ c_t + u\\cdot\\nabla c = \\Delta c -nc, &amp; x\\in
    \\Omega, \\, t&gt;0, \\\\ u_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\Phi,
    \\quad \\nabla\\cdot u=0, &amp; x\\in \\Omega, \\, t&gt;0, \\end{cases} \\quad
    (\\star)</jats:tex-math>\r\n                    </jats:disp-formula>\r\n                  </jats:p>\r\n
    \                 <jats:p>\r\n                    proposed by Goldstein et al.
    to describe pattern formation in populations of aerobic bacteria interacting with
    their liquid environment via transport and buoyancy. Known results have asserted
    that under appropriate regularity assumptions on\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Phi</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   and the initial data, a corresponding no-flux/no-flux/Dirichlet
    initial-boundary value problem is globally solvable in a framework of so-called
    weak energy solutions, and that any such solution eventually becomes smooth and
    classical.\r\n                  </jats:p>\r\n                  <jats:p>\r\n                    Going
    beyond this, the present work focuses on the possible extent of unboundedness
    phenomena also on short timescales, and hence investigates in more detail the
    set of times in\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>(0,\\infty)</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    at which solutions
    may develop singularities. The main results in this direction reveal the existence
    of a global weak energy solution which coincides with a smooth function throughout\r\n
    \                   <jats:inline-formula>\r\n                      <jats:tex-math>\\overline{\\Omega}\\times
    E</jats:tex-math>\r\n                    </jats:inline-formula>\r\n                    ,
    where\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>E</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    denotes a countable
    union of open intervals which is such that\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>|(0,\\infty)\\setminus E|=0</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    . In particular,
    this indicates that a similar feature of the unperturbed Navie–Stokes equations,
    known as Leray’s structure theorem, persists even in the presence of the coupling
    to the attractive and hence potentially destabilizing cross-diffusive mechanism
    in the full system (\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>\\star</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    ).\r\n                  </jats:p>"
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
    interaction? <i>Journal of the European Mathematical Society</i>. 2022;25(4):1423-1456.
    doi:<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>
  apa: Winkler, M. (2022). Does Leray’s structure theorem withstand buoyancy-driven
    chemotaxis-fluid interaction? <i>Journal of the European Mathematical Society</i>,
    <i>25</i>(4), 1423–1456. <a href="https://doi.org/10.4171/jems/1226">https://doi.org/10.4171/jems/1226</a>
  bibtex: '@article{Winkler_2022, title={Does Leray’s structure theorem withstand
    buoyancy-driven chemotaxis-fluid interaction?}, volume={25}, DOI={<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>},
    number={4}, journal={Journal of the European Mathematical Society}, publisher={European
    Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael},
    year={2022}, pages={1423–1456} }'
  chicago: 'Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
    Chemotaxis-Fluid Interaction?” <i>Journal of the European Mathematical Society</i>
    25, no. 4 (2022): 1423–56. <a href="https://doi.org/10.4171/jems/1226">https://doi.org/10.4171/jems/1226</a>.'
  ieee: 'M. Winkler, “Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid
    interaction?,” <i>Journal of the European Mathematical Society</i>, vol. 25, no.
    4, pp. 1423–1456, 2022, doi: <a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>.'
  mla: Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven
    Chemotaxis-Fluid Interaction?” <i>Journal of the European Mathematical Society</i>,
    vol. 25, no. 4, European Mathematical Society - EMS - Publishing House GmbH, 2022,
    pp. 1423–56, doi:<a href="https://doi.org/10.4171/jems/1226">10.4171/jems/1226</a>.
  short: M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456.
date_created: 2025-12-18T19:16:13Z
date_updated: 2025-12-18T20:11:51Z
doi: 10.4171/jems/1226
intvolume: '        25'
issue: '4'
language:
- iso: eng
page: 1423-1456
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
  - 1435-9863
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?
type: journal_article
user_id: '31496'
volume: 25
year: '2022'
...
---
_id: '66333'
abstract:
- lang: eng
  text: "<jats:p>\r\n                    We use recent results by Bainbridge–Chen–Gendron–Grushevsky–Möller
    on compactifications of strata of abelian differentials to give a comprehensive
    solution to the realizability problem for effective tropical canonical divisors
    in equicharacteristic zero. Given a pair\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>(\\Gamma, D)</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   consisting of a stable tropical curve\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Gamma</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   and a divisor\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>D</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   in the canonical linear system on\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Gamma</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   , we give a purely combinatorial condition to decide whether
    there is a smooth curve\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>X</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    over a non-Archimedean
    field whose stable reduction has\r\n                    <jats:inline-formula>\r\n
    \                     <jats:tex-math>\\Gamma</jats:tex-math>\r\n                    </jats:inline-formula>\r\n
    \                   as its dual tropical curve together with an effective canonical
    divisor\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>K_X</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    that specializes
    to\r\n                    <jats:inline-formula>\r\n                      <jats:tex-math>D</jats:tex-math>\r\n
    \                   </jats:inline-formula>\r\n                    .\r\n                  </jats:p>"
author:
- first_name: Martin
  full_name: Moeller, Martin
  last_name: Moeller
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
- first_name: Annette
  full_name: Werner, Annette
  last_name: Werner
citation:
  ama: Moeller M, Ulirsch M, Werner A. Realizability of tropical canonical divisors.
    <i>Journal of the European Mathematical Society</i>. 2020;23(1):185-217. doi:<a
    href="https://doi.org/10.4171/JEMS/1009">10.4171/JEMS/1009</a>
  apa: Moeller, M., Ulirsch, M., &#38; Werner, A. (2020). Realizability of tropical
    canonical divisors. <i>Journal of the European Mathematical Society</i>, <i>23</i>(1),
    185–217. <a href="https://doi.org/10.4171/JEMS/1009">https://doi.org/10.4171/JEMS/1009</a>
  bibtex: '@article{Moeller_Ulirsch_Werner_2020, title={Realizability of tropical
    canonical divisors}, volume={23}, DOI={<a href="https://doi.org/10.4171/JEMS/1009">10.4171/JEMS/1009</a>},
    number={1}, journal={Journal of the European Mathematical Society}, publisher={European
    Mathematical Society - EMS - Publishing House GmbH}, author={Moeller, Martin and
    Ulirsch, Martin and Werner, Annette}, year={2020}, pages={185–217} }'
  chicago: 'Moeller, Martin, Martin Ulirsch, and Annette Werner. “Realizability of
    Tropical Canonical Divisors.” <i>Journal of the European Mathematical Society</i>
    23, no. 1 (2020): 185–217. <a href="https://doi.org/10.4171/JEMS/1009">https://doi.org/10.4171/JEMS/1009</a>.'
  ieee: 'M. Moeller, M. Ulirsch, and A. Werner, “Realizability of tropical canonical
    divisors,” <i>Journal of the European Mathematical Society</i>, vol. 23, no. 1,
    pp. 185–217, 2020, doi: <a href="https://doi.org/10.4171/JEMS/1009">10.4171/JEMS/1009</a>.'
  mla: Moeller, Martin, et al. “Realizability of Tropical Canonical Divisors.” <i>Journal
    of the European Mathematical Society</i>, vol. 23, no. 1, European Mathematical
    Society - EMS - Publishing House GmbH, 2020, pp. 185–217, doi:<a href="https://doi.org/10.4171/JEMS/1009">10.4171/JEMS/1009</a>.
  short: M. Moeller, M. Ulirsch, A. Werner, Journal of the European Mathematical Society
    23 (2020) 185–217.
date_created: 2026-07-08T07:06:32Z
date_updated: 2026-07-09T06:55:38Z
doi: 10.4171/JEMS/1009
intvolume: '        23'
issue: '1'
language:
- iso: eng
page: 185-217
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
  - 1435-9863
publication_status: published
publisher: European Mathematical Society - EMS - Publishing House GmbH
status: public
title: Realizability of tropical canonical divisors
type: journal_article
user_id: '82981'
volume: 23
year: '2020'
...