[{"publication_status":"published","date_updated":"2025-12-18T20:12:36Z","author":[{"full_name":"Winkler, Michael","first_name":"Michael","last_name":"Winkler","id":"31496"}],"publication_identifier":{"issn":["1435-9855","1435-9863"]},"year":"2025","status":"public","title":"Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system","user_id":"31496","doi":"10.4171/jems/1607","_id":"63244","language":[{"iso":"eng"}],"publisher":"European Mathematical Society - EMS - Publishing House GmbH","abstract":[{"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>","lang":"eng"}],"citation":{"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>.","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} }","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>","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>.","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>","short":"M. Winkler, Journal of the European Mathematical Society (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>."},"publication":"Journal of the European Mathematical Society","type":"journal_article","date_created":"2025-12-18T18:59:39Z"},{"publication":"Journal of the European Mathematical Society","issue":"4","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>"}],"date_created":"2025-12-18T19:16:13Z","type":"journal_article","title":"Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?","year":"2022","author":[{"id":"31496","last_name":"Winkler","first_name":"Michael","full_name":"Winkler, Michael"}],"publication_identifier":{"issn":["1435-9855","1435-9863"]},"publication_status":"published","date_updated":"2025-12-18T20:11:51Z","intvolume":"        25","language":[{"iso":"eng"}],"doi":"10.4171/jems/1226","citation":{"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} }","short":"M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456.","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>","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>.","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>","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>."},"status":"public","page":"1423-1456","_id":"63279","publisher":"European Mathematical Society - EMS - Publishing House GmbH","user_id":"31496","volume":25},{"intvolume":"        23","publication_status":"published","date_updated":"2026-07-09T06:55:38Z","publication_identifier":{"issn":["1435-9855","1435-9863"]},"author":[{"full_name":"Moeller, Martin","last_name":"Moeller","first_name":"Martin"},{"full_name":"Ulirsch, Martin","last_name":"Ulirsch","first_name":"Martin","id":"114697"},{"full_name":"Werner, Annette","last_name":"Werner","first_name":"Annette"}],"year":"2020","title":"Realizability of tropical canonical divisors","doi":"10.4171/JEMS/1009","language":[{"iso":"eng"}],"abstract":[{"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>","lang":"eng"}],"issue":"1","publication":"Journal of the European Mathematical Society","type":"journal_article","date_created":"2026-07-08T07:06:32Z","status":"public","volume":23,"user_id":"82981","_id":"66333","publisher":"European Mathematical Society - EMS - Publishing House GmbH","page":"185-217","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>","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} }","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>.","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>.","short":"M. Moeller, M. Ulirsch, A. Werner, Journal of the European Mathematical Society 23 (2020) 185–217.","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>","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>."}}]