--- _id: '63435' author: - first_name: Leander full_name: Claes, Leander id: '11829' last_name: Claes orcid: 0000-0002-4393-268X - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: 'Claes L, Winkler M. Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis. Nonlinear Analysis: Real World Applications. 2026;91:104580. doi:10.1016/j.nonrwa.2025.104580' apa: 'Claes, L., & Winkler, M. (2026). Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis. Nonlinear Analysis: Real World Applications, 91, 104580. https://doi.org/10.1016/j.nonrwa.2025.104580' bibtex: '@article{Claes_Winkler_2026, title={Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis}, volume={91}, DOI={10.1016/j.nonrwa.2025.104580}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier BV}, author={Claes, Leander and Winkler, Michael}, year={2026}, pages={104580} }' chicago: 'Claes, Leander, and Michael Winkler. “Describing Smooth Small-Data Solutions to a Quasilinear Hyperbolic-Parabolic System by W 1,P Energy Analysis.” Nonlinear Analysis: Real World Applications 91 (2026): 104580. https://doi.org/10.1016/j.nonrwa.2025.104580.' ieee: 'L. Claes and M. Winkler, “Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis,” Nonlinear Analysis: Real World Applications, vol. 91, p. 104580, 2026, doi: 10.1016/j.nonrwa.2025.104580.' mla: 'Claes, Leander, and Michael Winkler. “Describing Smooth Small-Data Solutions to a Quasilinear Hyperbolic-Parabolic System by W 1,P Energy Analysis.” Nonlinear Analysis: Real World Applications, vol. 91, Elsevier BV, 2026, p. 104580, doi:10.1016/j.nonrwa.2025.104580.' short: 'L. Claes, M. Winkler, Nonlinear Analysis: Real World Applications 91 (2026) 104580.' date_created: 2026-01-05T07:32:00Z date_updated: 2026-01-05T07:40:49Z department: - _id: '49' - _id: '90' doi: 10.1016/j.nonrwa.2025.104580 intvolume: ' 91' language: - iso: eng page: '104580' project: - _id: '245' name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken für Leistungsschallanwendungen (NEPTUN)' publication: 'Nonlinear Analysis: Real World Applications' publication_identifier: issn: - 1468-1218 publisher: Elsevier BV status: public title: Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis type: journal_article user_id: '11829' volume: 91 year: '2026' ... --- _id: '59258' article_number: '44' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters. Applied Mathematics & Optimization. 2025;91(2). doi:10.1007/s00245-025-10243-9 apa: Winkler, M. (2025). Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters. Applied Mathematics & Optimization, 91(2), Article 44. https://doi.org/10.1007/s00245-025-10243-9 bibtex: '@article{Winkler_2025, title={Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters}, volume={91}, DOI={10.1007/s00245-025-10243-9}, number={244}, journal={Applied Mathematics & Optimization}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters.” Applied Mathematics & Optimization 91, no. 2 (2025). https://doi.org/10.1007/s00245-025-10243-9. ieee: 'M. Winkler, “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters,” Applied Mathematics & Optimization, vol. 91, no. 2, Art. no. 44, 2025, doi: 10.1007/s00245-025-10243-9.' mla: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters.” Applied Mathematics & Optimization, vol. 91, no. 2, 44, Springer Science and Business Media LLC, 2025, doi:10.1007/s00245-025-10243-9. short: M. Winkler, Applied Mathematics & Optimization 91 (2025). date_created: 2025-04-02T11:23:25Z date_updated: 2026-02-26T15:59:30Z department: - _id: '90' doi: 10.1007/s00245-025-10243-9 intvolume: ' 91' issue: '2' language: - iso: eng project: - _id: '245' name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken für Leistungsschallanwendungen (NEPTUN)' publication: Applied Mathematics & Optimization publication_identifier: issn: - 0095-4616 - 1432-0606 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters type: journal_article user_id: '11829' volume: 91 year: '2025' ... --- _id: '63250' abstract: - lang: eng text: "Abstract\r\n \r\n An initial-boundary value problem for\r\n \r\n \ \r\n $$\\begin{aligned} \\left\\{ \\begin{array}{ll}u_{tt} = \\big (\\gamma (\\Theta ) u_{xt}\\big )_x + au_{xx} - \\big (f(\\Theta )\\big )_x, \\qquad & x\\in \\Omega , \\ t>0, \\\\[1mm] \\Theta _t = \\Theta _{xx} + \\gamma (\\Theta ) u_{xt}^2 - f(\\Theta ) u_{xt}, \\qquad & x\\in \\Omega , \\ t>0, \\end{array} \\right. \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ \r\n tt\r\n \ \r\n \r\n \ =\r\n \r\n \ (\r\n \r\n \ γ\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n u\r\n \ \r\n xt\r\n \ \r\n \r\n \ \r\n \r\n \ )\r\n \r\n \ x\r\n \r\n \ +\r\n a\r\n \ \r\n u\r\n \ \r\n xx\r\n \ \r\n \r\n \ -\r\n \r\n \ (\r\n \r\n \ f\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n \r\n \ )\r\n \r\n \ x\r\n \r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ x\r\n ∈\r\n \ Ω\r\n ,\r\n \ \r\n t\r\n \ >\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n [\r\n \ 1\r\n m\r\n \ m\r\n ]\r\n \ \r\n \r\n \ Θ\r\n t\r\n \ \r\n =\r\n \ \r\n Θ\r\n \ \r\n xx\r\n \ \r\n \r\n \ +\r\n γ\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ u\r\n \r\n \ xt\r\n \r\n \ 2\r\n \r\n \ -\r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ u\r\n \r\n \ xt\r\n \r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n x\r\n \ ∈\r\n Ω\r\n \ ,\r\n \r\n \ t\r\n >\r\n \ 0\r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n is considered in an open bounded real interval\r\n \r\n \ \r\n $$\\Omega $$\r\n \r\n \ Ω\r\n \r\n \ \r\n \r\n \ . Under the assumption that\r\n \r\n \ \r\n $$\\gamma \\in C^0([0,\\infty ))$$\r\n \r\n \r\n \ γ\r\n ∈\r\n \ \r\n C\r\n \ 0\r\n \r\n \ \r\n (\r\n \ \r\n [\r\n \ 0\r\n ,\r\n \ ∞\r\n )\r\n \ \r\n )\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$f\\in C^0([0,\\infty ))$$\r\n \r\n \ \r\n f\r\n \ ∈\r\n \r\n \ C\r\n 0\r\n \ \r\n \r\n \ (\r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ are such that\r\n \r\n \ \r\n $$f(0)=0$$\r\n \ \r\n \ \r\n f\r\n \ (\r\n 0\r\n \ )\r\n =\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n , and\r\n \r\n \ \r\n $$k_\\gamma \\le \\gamma \\le K_\\gamma $$\r\n \r\n \r\n \ \r\n k\r\n \ γ\r\n \r\n \ ≤\r\n γ\r\n \ ≤\r\n \r\n \ K\r\n γ\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n as well as\r\n \ \r\n \r\n \ $$\\begin{aligned} |f(\\xi )| \\le K_f \\cdot (\\xi +1)^\\alpha \\qquad \\hbox {for all } \\xi \\ge 0 \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ |\r\n f\r\n \ \r\n (\r\n \ ξ\r\n )\r\n \ \r\n |\r\n \ \r\n ≤\r\n \ \r\n K\r\n \ f\r\n \r\n \ ·\r\n \r\n \ \r\n (\r\n \ ξ\r\n +\r\n \ 1\r\n )\r\n \ \r\n α\r\n \ \r\n \r\n \ for all\r\n \r\n \ ξ\r\n ≥\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n with some\r\n \ \r\n \r\n \ $$k_\\gamma>0, K_\\gamma>0, K_f>0$$\r\n \ \r\n \ \r\n \r\n \ k\r\n γ\r\n \ \r\n >\r\n \ 0\r\n ,\r\n \ \r\n K\r\n \ γ\r\n \r\n \ >\r\n 0\r\n \ ,\r\n \r\n \ K\r\n f\r\n \ \r\n >\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$\\alpha <\\frac{3}{2}$$\r\n \r\n \ \r\n α\r\n \ <\r\n \r\n \ 3\r\n 2\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n , for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived. By particularly covering the thermodynamically consistent choice\r\n \r\n \r\n \ $$f\\equiv id$$\r\n \r\n \r\n \ f\r\n ≡\r\n \ i\r\n d\r\n \ \r\n \r\n \ \r\n \r\n \ of predominant physical relevance, this appears to go beyond previous related literature which seems to either rely on independence of\r\n \ \r\n \r\n \ $$\\gamma $$\r\n \r\n γ\r\n \ \r\n \r\n \ \r\n on\r\n \r\n \ \r\n $$\\Theta $$\r\n \r\n \ Θ\r\n \r\n \ \r\n \r\n \ , or to operate on finite time intervals.\r\n " article_number: '192' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities. Zeitschrift für angewandte Mathematik und Physik. 2025;76(5). doi:10.1007/s00033-025-02582-y apa: Winkler, M. (2025). Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities. Zeitschrift Für Angewandte Mathematik Und Physik, 76(5), Article 192. https://doi.org/10.1007/s00033-025-02582-y bibtex: '@article{Winkler_2025, title={Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities}, volume={76}, DOI={10.1007/s00033-025-02582-y}, number={5192}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Large-Data Solutions in One-Dimensional Thermoviscoelasticity Involving Temperature-Dependent Viscosities.” Zeitschrift Für Angewandte Mathematik Und Physik 76, no. 5 (2025). https://doi.org/10.1007/s00033-025-02582-y. ieee: 'M. Winkler, “Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities,” Zeitschrift für angewandte Mathematik und Physik, vol. 76, no. 5, Art. no. 192, 2025, doi: 10.1007/s00033-025-02582-y.' mla: Winkler, Michael. “Large-Data Solutions in One-Dimensional Thermoviscoelasticity Involving Temperature-Dependent Viscosities.” Zeitschrift Für Angewandte Mathematik Und Physik, vol. 76, no. 5, 192, Springer Science and Business Media LLC, 2025, doi:10.1007/s00033-025-02582-y. short: M. Winkler, Zeitschrift Für Angewandte Mathematik Und Physik 76 (2025). date_created: 2025-12-18T19:03:19Z date_updated: 2025-12-18T20:13:25Z doi: 10.1007/s00033-025-02582-y intvolume: ' 76' issue: '5' language: - iso: eng publication: Zeitschrift für angewandte Mathematik und Physik publication_identifier: issn: - 0044-2275 - 1420-9039 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities type: journal_article user_id: '31496' volume: 76 year: '2025' ... --- _id: '63249' abstract: - lang: eng text: "Abstract\r\n \r\n The model\r\n \r\n \r\n \ $$\\begin{aligned} \\left\\{ \\begin{array}{l}u_{tt} = \\big (\\gamma (\\Theta ) u_{xt}\\big )_x + au_{xx} - \\big (f(\\Theta )\\big )_x, \\\\[1mm] \\Theta _t = \\Theta _{xx} + \\gamma (\\Theta ) u_{xt}^2 - f(\\Theta ) u_{xt}, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ u\r\n \r\n \ tt\r\n \r\n \ \r\n =\r\n \ \r\n (\r\n \ \r\n γ\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ u\r\n \r\n \ xt\r\n \r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n x\r\n \ \r\n +\r\n \ a\r\n \r\n \ u\r\n \r\n \ xx\r\n \r\n \ \r\n -\r\n \ \r\n (\r\n \ \r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n x\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ [\r\n 1\r\n \ m\r\n m\r\n \ ]\r\n \r\n \ \r\n Θ\r\n \ t\r\n \r\n \ =\r\n \r\n \ Θ\r\n \r\n \ xx\r\n \r\n \ \r\n +\r\n \ γ\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n u\r\n \ \r\n xt\r\n \ \r\n 2\r\n \ \r\n -\r\n \ f\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n u\r\n \ \r\n xt\r\n \ \r\n \r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ for thermoviscoelastic evolution in one-dimensional Kelvin–Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if\r\n \r\n \ \r\n $$\\gamma _0>0$$\r\n \r\n \ \r\n \r\n \ γ\r\n 0\r\n \ \r\n >\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n is fixed, then there exists\r\n \r\n \r\n \ $$\\delta =\\delta (\\gamma _0)>0$$\r\n \ \r\n \ \r\n δ\r\n \ =\r\n δ\r\n \ (\r\n \r\n \ γ\r\n 0\r\n \ \r\n )\r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ with the property that for suitably regular initial data of arbitrary size an associated initial boundary value problem posed in an open bounded interval admits a global classical solution whenever\r\n \r\n \ \r\n $$\\gamma \\in C^2([0,\\infty ))$$\r\n \r\n \r\n \ γ\r\n ∈\r\n \ \r\n C\r\n \ 2\r\n \r\n \ \r\n (\r\n \ \r\n [\r\n \ 0\r\n ,\r\n \ ∞\r\n )\r\n \ \r\n )\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$f\\in C^2([0,\\infty ))$$\r\n \r\n \ \r\n f\r\n \ ∈\r\n \r\n \ C\r\n 2\r\n \ \r\n \r\n \ (\r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ are such that\r\n \r\n \ \r\n $$f(0)=0$$\r\n \ \r\n \ \r\n f\r\n \ (\r\n 0\r\n \ )\r\n =\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n and\r\n \r\n \ \r\n $$|f(\\xi )| \\le K_f \\cdot (\\xi +1)^\\alpha $$\r\n \r\n \r\n \ \r\n |\r\n \ f\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ |\r\n \r\n \ ≤\r\n \r\n \ K\r\n f\r\n \ \r\n ·\r\n \ \r\n \r\n \ (\r\n ξ\r\n \ +\r\n 1\r\n \ )\r\n \r\n \ α\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ for all\r\n \r\n \r\n \ $$\\xi \\ge 0$$\r\n \r\n \r\n \ ξ\r\n ≥\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n and some\r\n \ \r\n \r\n \ $$K_f>0$$\r\n \r\n \r\n \ \r\n K\r\n \ f\r\n \r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ and\r\n \r\n \r\n \ $$\\alpha <\\frac{3}{2}$$\r\n \ \r\n \ \r\n α\r\n \ <\r\n \r\n \ 3\r\n 2\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n , and that\r\n \ \r\n \r\n \ $$\\begin{aligned} \\gamma _0 \\le \\gamma (\\xi ) \\le \\gamma _0 + \\delta \\qquad \\hbox {for all } \\xi \\ge 0. \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ γ\r\n 0\r\n \ \r\n ≤\r\n \ γ\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ ≤\r\n \r\n \ γ\r\n 0\r\n \ \r\n +\r\n \ δ\r\n \r\n \ for all\r\n \r\n \ ξ\r\n ≥\r\n \ 0\r\n .\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.\r\n " article_number: '108' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities. Journal of Evolution Equations. 2025;25(4). doi:10.1007/s00028-025-01144-z apa: Winkler, M. (2025). Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities. Journal of Evolution Equations, 25(4), Article 108. https://doi.org/10.1007/s00028-025-01144-z bibtex: '@article{Winkler_2025, title={Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities}, volume={25}, DOI={10.1007/s00028-025-01144-z}, number={4108}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Large-Data Regular Solutions in a One-Dimensional Thermoviscoelastic Evolution Problem Involving Temperature-Dependent Viscosities.” Journal of Evolution Equations 25, no. 4 (2025). https://doi.org/10.1007/s00028-025-01144-z. ieee: 'M. Winkler, “Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities,” Journal of Evolution Equations, vol. 25, no. 4, Art. no. 108, 2025, doi: 10.1007/s00028-025-01144-z.' mla: Winkler, Michael. “Large-Data Regular Solutions in a One-Dimensional Thermoviscoelastic Evolution Problem Involving Temperature-Dependent Viscosities.” Journal of Evolution Equations, vol. 25, no. 4, 108, Springer Science and Business Media LLC, 2025, doi:10.1007/s00028-025-01144-z. short: M. Winkler, Journal of Evolution Equations 25 (2025). date_created: 2025-12-18T19:02:51Z date_updated: 2025-12-18T20:13:11Z doi: 10.1007/s00028-025-01144-z intvolume: ' 25' issue: '4' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities type: journal_article user_id: '31496' volume: 25 year: '2025' ... --- _id: '63246' abstract: - lang: eng text: "Abstract\r\n \r\n The hyperbolic-parabolic model\r\n \r\n \r\n \ $$\\begin{aligned} \\left\\{ \\begin{array}{ll} u_{tt} = u_{xx} - \\big (f(\\Theta )\\big )_x, \\qquad & x\\in \\Omega , \\ t>0, \\\\ \\Theta _t = \\Theta _{xx} - f(\\Theta ) u_{xt}, \\qquad & \ x\\in \\Omega , \\ t>0, \\end{array} \\right. \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ \r\n tt\r\n \ \r\n \r\n \ =\r\n \r\n \ u\r\n \r\n \ xx\r\n \r\n \ \r\n -\r\n \ \r\n (\r\n \ \r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n x\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n x\r\n \ ∈\r\n Ω\r\n \ ,\r\n \r\n \ t\r\n >\r\n \ 0\r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ Θ\r\n t\r\n \ \r\n =\r\n \ \r\n Θ\r\n \ \r\n xx\r\n \ \r\n \r\n \ -\r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n \r\n \ u\r\n \r\n \ xt\r\n \r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n x\r\n \ ∈\r\n Ω\r\n \ ,\r\n \r\n \ t\r\n >\r\n \ 0\r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n for the evolution of the displacement variable\r\n u\r\n \ and the temperature\r\n \r\n \ \r\n $$\\Theta \\ge 0$$\r\n \r\n \ \r\n Θ\r\n \ ≥\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ during thermoelastic interaction in a one-dimensional bounded interval\r\n \r\n \r\n \ $$\\Omega $$\r\n \r\n Ω\r\n \ \r\n \r\n \ \r\n is considered. Whereas the literature has provided comprehensive results on global solutions for sufficiently regular initial data\r\n \r\n \ \r\n $$(u_0,u_{0t},\\Theta _0)=(u,u_t,\\Theta )|_{t=0}$$\r\n \r\n \r\n \ \r\n (\r\n \ \r\n u\r\n \ 0\r\n \r\n \ ,\r\n \r\n \ u\r\n \r\n \ 0\r\n t\r\n \ \r\n \r\n \ ,\r\n \r\n \ Θ\r\n 0\r\n \ \r\n )\r\n \ \r\n =\r\n \ \r\n (\r\n \ u\r\n ,\r\n \ \r\n u\r\n \ t\r\n \r\n \ ,\r\n Θ\r\n \ )\r\n \r\n \ \r\n \r\n \ |\r\n \r\n \ \r\n t\r\n \ =\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ when\r\n \r\n \r\n \ $$f\\equiv id$$\r\n \r\n \r\n \ f\r\n ≡\r\n \ i\r\n d\r\n \ \r\n \r\n \ \r\n \r\n \ , it seems to have remained open so far how far a solution theory can be built solely on the two fundamental physical principles of energy conservation and entropy nondecrease. The present manuscript addresses this by asserting global existence of weak solutions under assumptions which are energy- and entropy-minimal in the sense of allowing for any initial data\r\n \r\n \ \r\n $$u_0\\in W_0^{1,2}(\\Omega )$$\r\n \r\n \ \r\n \r\n \ u\r\n 0\r\n \ \r\n ∈\r\n \ \r\n W\r\n \ 0\r\n \r\n \ 1\r\n ,\r\n \ 2\r\n \r\n \ \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ ,\r\n \r\n \r\n \ $$u_{0t} \\in L^2(\\Omega )$$\r\n \ \r\n \ \r\n \r\n \ u\r\n \r\n \ 0\r\n t\r\n \ \r\n \r\n \ ∈\r\n \r\n \ L\r\n 2\r\n \ \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ and\r\n \r\n \r\n \ $$0\\le \\Theta _0\\in L^1(\\Omega )$$\r\n \ \r\n \ \r\n 0\r\n \ ≤\r\n \r\n \ Θ\r\n 0\r\n \ \r\n ∈\r\n \ \r\n L\r\n \ 1\r\n \r\n \ \r\n (\r\n \ Ω\r\n )\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n , and which apply to arbitrary\r\n \r\n \r\n \ $$f\\in C^1([0,\\infty ))$$\r\n \ \r\n \ \r\n f\r\n \ ∈\r\n \r\n \ C\r\n 1\r\n \ \r\n \r\n \ (\r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ with\r\n \r\n \r\n \ $$f(0)=0$$\r\n \r\n \r\n \ f\r\n (\r\n \ 0\r\n )\r\n \ =\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ and\r\n \r\n \r\n \ $$f'>0$$\r\n \r\n \r\n \ \r\n f\r\n \ ′\r\n \r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ on\r\n \r\n \r\n \ $$[0,\\infty )$$\r\n \r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n .\r\n " article_number: '1' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Rough solutions in one-dimensional nonlinear thermoelasticity. Calculus of Variations and Partial Differential Equations. 2025;65(1). doi:10.1007/s00526-025-03170-8 apa: Winkler, M. (2025). Rough solutions in one-dimensional nonlinear thermoelasticity. Calculus of Variations and Partial Differential Equations, 65(1), Article 1. https://doi.org/10.1007/s00526-025-03170-8 bibtex: '@article{Winkler_2025, title={Rough solutions in one-dimensional nonlinear thermoelasticity}, volume={65}, DOI={10.1007/s00526-025-03170-8}, number={11}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.” Calculus of Variations and Partial Differential Equations 65, no. 1 (2025). https://doi.org/10.1007/s00526-025-03170-8. ieee: 'M. Winkler, “Rough solutions in one-dimensional nonlinear thermoelasticity,” Calculus of Variations and Partial Differential Equations, vol. 65, no. 1, Art. no. 1, 2025, doi: 10.1007/s00526-025-03170-8.' mla: Winkler, Michael. “Rough Solutions in One-Dimensional Nonlinear Thermoelasticity.” Calculus of Variations and Partial Differential Equations, vol. 65, no. 1, 1, Springer Science and Business Media LLC, 2025, doi:10.1007/s00526-025-03170-8. short: M. Winkler, Calculus of Variations and Partial Differential Equations 65 (2025). date_created: 2025-12-18T19:01:02Z date_updated: 2025-12-18T20:12:50Z doi: 10.1007/s00526-025-03170-8 intvolume: ' 65' issue: '1' language: - iso: eng publication: Calculus of Variations and Partial Differential Equations publication_identifier: issn: - 0944-2669 - 1432-0835 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Rough solutions in one-dimensional nonlinear thermoelasticity type: journal_article user_id: '31496' volume: 65 year: '2025' ... --- _id: '63244' abstract: - lang: eng text: "\r\n The Cauchy problem in \r\n \r\n \ \\mathbb{R}^{n}\r\n \r\n \ for the cross-diffusion system \r\n \r\n \r\n \ \r\n \\begin{cases}u_{t} = \\nabla \\cdot (D(u)\\nabla u) - \\nabla\\cdot (u\\nabla v), \\\\ 0 = \\Delta v +u,\\end{cases}\r\n \r\n \r\n \ \r\n is considered for \r\n \r\n \ n\\ge 2\r\n \r\n \ and under assumptions ensuring that \r\n \r\n \ D\r\n \r\n \ suitably generalizes the prototype given by \r\n \r\n \ \r\n \r\n D(\\xi)=(\\xi+1)^{-\\alpha}, \\quad \\xi\\ge 0.\r\n \r\n \r\n \ \r\n Under the assumption that \r\n \r\n \ \\alpha>1\r\n \r\n \ , it is shown that for any \r\n \r\n \ r_{\\star}>0\r\n \r\n \ and \r\n \r\n \\delta\\in (0,1)\r\n \r\n one can find radially symmetric initial data from \r\n \r\n \ C_{0}^{\\infty}(\\mathbb{R}^{n})\r\n \ \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 \r\n \\overline{B}_{r_\\star+\\delta}(0)\\setminus B_{(1-\\delta)r_\\star}(0)\r\n \r\n \ .This is complemented by a result revealing that when \r\n \r\n \ \\alpha<1\r\n \r\n \ , any finite-mass unbounded radial solution must blow up exclusively at the spatial origin.\r\n " 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. Journal of the European Mathematical Society. Published online 2025. doi:10.4171/jems/1607 apa: Winkler, M. (2025). Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system. Journal of the European Mathematical Society. https://doi.org/10.4171/jems/1607 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={10.4171/jems/1607}, 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.” Journal of the European Mathematical Society, 2025. https://doi.org/10.4171/jems/1607. ieee: 'M. Winkler, “Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system,” Journal of the European Mathematical Society, 2025, doi: 10.4171/jems/1607.' mla: Winkler, Michael. “Can Diffusion Degeneracies Enhance Complexity in Chemotactic Aggregation? Finite-Time Blow-up on Spheres in a Quasilinear Keller–Segel System.” Journal of the European Mathematical Society, European Mathematical Society - EMS - Publishing House GmbH, 2025, doi:10.4171/jems/1607. 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: '63247' 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 switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production. Journal of Differential Equations. 2025;423:197-239. doi:10.1016/j.jde.2024.12.040 apa: Tao, Y., & Winkler, M. (2025). A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production. Journal of Differential Equations, 423, 197–239. https://doi.org/10.1016/j.jde.2024.12.040 bibtex: '@article{Tao_Winkler_2025, title={A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production}, volume={423}, DOI={10.1016/j.jde.2024.12.040}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao, Youshan and Winkler, Michael}, year={2025}, pages={197–239} }' chicago: 'Tao, Youshan, and Michael Winkler. “A Switch in Dimension Dependence of Critical Blow-up Exponents in a Keller-Segel System Involving Indirect Signal Production.” Journal of Differential Equations 423 (2025): 197–239. https://doi.org/10.1016/j.jde.2024.12.040.' ieee: 'Y. Tao and M. Winkler, “A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production,” Journal of Differential Equations, vol. 423, pp. 197–239, 2025, doi: 10.1016/j.jde.2024.12.040.' mla: Tao, Youshan, and Michael Winkler. “A Switch in Dimension Dependence of Critical Blow-up Exponents in a Keller-Segel System Involving Indirect Signal Production.” Journal of Differential Equations, vol. 423, Elsevier BV, 2025, pp. 197–239, doi:10.1016/j.jde.2024.12.040. short: Y. Tao, M. Winkler, Journal of Differential Equations 423 (2025) 197–239. date_created: 2025-12-18T19:01:40Z date_updated: 2025-12-18T20:12:58Z doi: 10.1016/j.jde.2024.12.040 intvolume: ' 423' language: - iso: eng page: 197-239 publication: Journal of Differential Equations publication_identifier: issn: - 0022-0396 publication_status: published publisher: Elsevier BV status: public title: A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production type: journal_article user_id: '31496' volume: 423 year: '2025' ... --- _id: '63252' 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 unified approach to existence theories for singular chemotaxis systems with nonlinear signal production. Science China Mathematics. 2025;68(12):2867-2900. doi:10.1007/s11425-023-2397-y apa: Tao, Y., & Winkler, M. (2025). A unified approach to existence theories for singular chemotaxis systems with nonlinear signal production. Science China Mathematics, 68(12), 2867–2900. https://doi.org/10.1007/s11425-023-2397-y bibtex: '@article{Tao_Winkler_2025, title={A unified approach to existence theories for singular chemotaxis systems with nonlinear signal production}, volume={68}, DOI={10.1007/s11425-023-2397-y}, number={12}, journal={Science China Mathematics}, publisher={Springer Science and Business Media LLC}, author={Tao, Youshan and Winkler, Michael}, year={2025}, pages={2867–2900} }' chicago: 'Tao, Youshan, and Michael Winkler. “A Unified Approach to Existence Theories for Singular Chemotaxis Systems with Nonlinear Signal Production.” Science China Mathematics 68, no. 12 (2025): 2867–2900. https://doi.org/10.1007/s11425-023-2397-y.' ieee: 'Y. Tao and M. Winkler, “A unified approach to existence theories for singular chemotaxis systems with nonlinear signal production,” Science China Mathematics, vol. 68, no. 12, pp. 2867–2900, 2025, doi: 10.1007/s11425-023-2397-y.' mla: Tao, Youshan, and Michael Winkler. “A Unified Approach to Existence Theories for Singular Chemotaxis Systems with Nonlinear Signal Production.” Science China Mathematics, vol. 68, no. 12, Springer Science and Business Media LLC, 2025, pp. 2867–900, doi:10.1007/s11425-023-2397-y. short: Y. Tao, M. Winkler, Science China Mathematics 68 (2025) 2867–2900. date_created: 2025-12-18T19:04:17Z date_updated: 2025-12-18T20:13:40Z doi: 10.1007/s11425-023-2397-y intvolume: ' 68' issue: '12' language: - iso: eng page: 2867-2900 publication: Science China Mathematics publication_identifier: issn: - 1674-7283 - 1869-1862 publication_status: published publisher: Springer Science and Business Media LLC status: public title: A unified approach to existence theories for singular chemotaxis systems with nonlinear signal production type: journal_article user_id: '31496' volume: 68 year: '2025' ... --- _id: '63344' abstract: - lang: eng text: "Abstract\r\n A Neumann-type initial-boundary value problem for \r\n \r\n \ $$\\begin{aligned} \\left\\{ \\begin{array}{l} u_{tt} = \\nabla \\cdot (\\gamma (\\Theta ) \\nabla u_t) + a \\nabla \\cdot (\\gamma (\\Theta ) \\nabla u) + \\nabla \\cdot f(\\Theta ), \\\\ \\Theta _t = D\\Delta \\Theta + \\Gamma (\\Theta ) |\\nabla u_t|^2 + F(\\Theta )\\cdot \\nabla u_t, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ \r\n tt\r\n \ \r\n \r\n \ =\r\n ∇\r\n \ ·\r\n \r\n \ (\r\n γ\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n ∇\r\n \ \r\n u\r\n \ t\r\n \r\n \ )\r\n \r\n \ +\r\n a\r\n \ ∇\r\n ·\r\n \ \r\n (\r\n \ γ\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ ∇\r\n u\r\n \ )\r\n \r\n \ +\r\n ∇\r\n \ ·\r\n f\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ Θ\r\n t\r\n \ \r\n =\r\n \ D\r\n Δ\r\n \ Θ\r\n +\r\n \ Γ\r\n \r\n \ (\r\n Θ\r\n \ )\r\n \r\n \ \r\n \r\n \ |\r\n ∇\r\n \ \r\n u\r\n \ t\r\n \r\n \ |\r\n \r\n \ 2\r\n \r\n \ +\r\n F\r\n \ \r\n (\r\n \ Θ\r\n )\r\n \ \r\n ·\r\n \ ∇\r\n \r\n \ u\r\n t\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \r\n \ is considered in a smoothly bounded domain \r\n \ \r\n $$\\Omega \\subset \\mathbb {R}^n$$\r\n \r\n \ \r\n Ω\r\n ⊂\r\n \ \r\n \r\n R\r\n \ \r\n n\r\n \ \r\n \r\n \r\n \ \r\n , \r\n \ \r\n $$n\\ge 1$$\r\n \ \r\n \ \r\n n\r\n ≥\r\n \ 1\r\n \r\n \r\n \ \r\n . In the case when \r\n \r\n $$n=1$$\r\n \ \r\n \ \r\n n\r\n =\r\n \ 1\r\n \r\n \r\n \ \r\n , \r\n \ \r\n $$\\gamma \\equiv \\Gamma $$\r\n \r\n \ \r\n γ\r\n ≡\r\n \ Γ\r\n \r\n \r\n \ \r\n and \r\n \ \r\n $$f\\equiv F$$\r\n \r\n \ \r\n f\r\n ≡\r\n \ F\r\n \r\n \r\n \ \r\n , this system coincides with the standard model for heat generation in a viscoelastic material of Kelvin-Voigt type, well-understood in situations in which \r\n \ \r\n $$\\gamma =const$$\r\n \r\n \ \r\n γ\r\n =\r\n \ c\r\n o\r\n \ n\r\n s\r\n \ t\r\n \r\n \r\n \ \r\n . Covering scenarios in which all key ingredients \r\n \r\n \ $$\\gamma ,\\Gamma ,f$$\r\n \r\n \r\n \ γ\r\n ,\r\n \ Γ\r\n ,\r\n \ f\r\n \r\n \r\n \ \r\n and F may depend on the temperature \r\n \r\n \ $$\\Theta $$\r\n \r\n Θ\r\n \ \r\n \r\n here, for initial data which merely satisfy \r\n \r\n \ $$u_0\\in W^{1,p+2}(\\Omega )$$\r\n \ \r\n \ \r\n \r\n u\r\n \ 0\r\n \r\n \ ∈\r\n \r\n W\r\n \ \r\n 1\r\n \ ,\r\n p\r\n \ +\r\n 2\r\n \ \r\n \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \r\n \ , \r\n \r\n \ $$u_{0t}\\in W^{1,p}(\\Omega )$$\r\n \ \r\n \ \r\n \r\n u\r\n \ \r\n 0\r\n \ t\r\n \r\n \ \r\n ∈\r\n \ \r\n W\r\n \ \r\n 1\r\n \ ,\r\n p\r\n \ \r\n \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \r\n \ and \r\n \r\n \ $$\\Theta _0\\in W^{1,p}(\\Omega )$$\r\n \ \r\n \ \r\n \r\n Θ\r\n \ 0\r\n \r\n \ ∈\r\n \r\n W\r\n \ \r\n 1\r\n \ ,\r\n p\r\n \ \r\n \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \r\n \ with some \r\n \r\n \ $$p\\ge 2$$\r\n \r\n \r\n \ p\r\n ≥\r\n \ 2\r\n \r\n \r\n \ \r\n such that \r\n \r\n $$p>n$$\r\n \ \r\n \ \r\n p\r\n >\r\n \ n\r\n \r\n \r\n \ \r\n , a result on local-in-time existence and uniqueness is derived in a natural framework of weak solvability." article_number: '44' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters. Applied Mathematics & Optimization. 2025;91(2). doi:10.1007/s00245-025-10243-9 apa: Winkler, M. (2025). Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters. Applied Mathematics & Optimization, 91(2), Article 44. https://doi.org/10.1007/s00245-025-10243-9 bibtex: '@article{Winkler_2025, title={Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters}, volume={91}, DOI={10.1007/s00245-025-10243-9}, number={244}, journal={Applied Mathematics & Optimization}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2025} }' chicago: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters.” Applied Mathematics & Optimization 91, no. 2 (2025). https://doi.org/10.1007/s00245-025-10243-9. ieee: 'M. Winkler, “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters,” Applied Mathematics & Optimization, vol. 91, no. 2, Art. no. 44, 2025, doi: 10.1007/s00245-025-10243-9.' mla: Winkler, Michael. “Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters.” Applied Mathematics & Optimization, vol. 91, no. 2, 44, Springer Science and Business Media LLC, 2025, doi:10.1007/s00245-025-10243-9. short: M. Winkler, Applied Mathematics & Optimization 91 (2025). date_created: 2025-12-18T20:20:06Z date_updated: 2025-12-18T20:20:16Z doi: 10.1007/s00245-025-10243-9 intvolume: ' 91' issue: '2' language: - iso: eng publication: Applied Mathematics & Optimization publication_identifier: issn: - 0095-4616 - 1432-0606 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters type: journal_article user_id: '31496' volume: 91 year: '2025' ... --- _id: '63242' abstract: - lang: eng text: "Abstract\r\n \r\n For\r\n \ \r\n \r\n \ $$p>2$$\r\n \r\n \r\n \ p\r\n >\r\n \ 2\r\n \r\n \ \r\n \r\n \ \r\n , the equation\r\n \ \r\n \r\n \ $$\\begin{aligned} u_t = u^p u_{xx}, \\qquad x\\in \\mathbb {R}, \\ t\\in \\mathbb {R}, \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ u\r\n t\r\n \ \r\n =\r\n \ \r\n u\r\n \ p\r\n \r\n \ \r\n u\r\n \ \r\n xx\r\n \ \r\n \r\n \ ,\r\n \r\n \ x\r\n ∈\r\n \ R\r\n ,\r\n \ \r\n t\r\n \ ∈\r\n R\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n is shown to admit positive and spatially increasing smooth solutions on all of\r\n \r\n \ \r\n $$\\mathbb {R}\\times \\mathbb {R}$$\r\n \r\n \r\n \ R\r\n ×\r\n \ R\r\n \r\n \ \r\n \r\n \ \r\n which are precisely of the form of an accelerating wave for\r\n \r\n \ \r\n $$t<0$$\r\n \ \r\n \ \r\n t\r\n \ <\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ , and of a wave slowing down for\r\n \r\n \ \r\n $$t>0$$\r\n \ \r\n \ \r\n t\r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ . These solutions satisfy\r\n \r\n \ \r\n $$u(\\cdot ,t)\\rightarrow 0$$\r\n \r\n \ \r\n u\r\n \ (\r\n ·\r\n \ ,\r\n t\r\n \ )\r\n →\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n in\r\n \r\n \ \r\n $$L^\\infty _{loc}(\\mathbb {R})$$\r\n \r\n \ \r\n \r\n \ L\r\n \r\n \ loc\r\n \r\n \ ∞\r\n \r\n \ \r\n (\r\n \ R\r\n )\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n as\r\n \r\n \ \r\n $$t\\rightarrow + \\infty $$\r\n \r\n \ \r\n t\r\n \ →\r\n +\r\n \ ∞\r\n \r\n \ \r\n \r\n \ \r\n and as\r\n \r\n \ \r\n $$t\\rightarrow -\\infty $$\r\n \r\n \ \r\n t\r\n \ →\r\n -\r\n \ ∞\r\n \r\n \ \r\n \r\n \ \r\n , and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that\r\n \r\n \r\n \ $$\\begin{aligned} \\lim _{x\\rightarrow +\\infty } x^{-1} u(x,t) \\quad \\text{ exists } \\text{ for } \\text{ all } t<0, \\end{aligned}$$\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ lim\r\n \r\n \ x\r\n →\r\n \ +\r\n ∞\r\n \ \r\n \r\n \ \r\n x\r\n \ \r\n -\r\n \ 1\r\n \r\n \ \r\n u\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ t\r\n )\r\n \ \r\n \r\n \ \r\n exists\r\n \ \r\n \r\n \ for\r\n \r\n \ \r\n all\r\n \ \r\n t\r\n \ <\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n that\r\n \r\n \ \r\n $$\\begin{aligned} \\lim _{x\\rightarrow +\\infty } x^{-\\frac{2}{p}} u(x,t) \\quad \\text{ exists } \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n lim\r\n \ \r\n x\r\n \ →\r\n +\r\n \ ∞\r\n \r\n \ \r\n \r\n \ x\r\n \r\n \ -\r\n \r\n \ 2\r\n p\r\n \ \r\n \r\n \ \r\n u\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ t\r\n )\r\n \ \r\n \r\n \ \r\n exists\r\n \ \r\n \r\n \ for\r\n \r\n \ \r\n all\r\n \ \r\n t\r\n \ >\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n but that\r\n \r\n \ \r\n $$\\begin{aligned} u(x,0)=K e^{\\alpha x} \\qquad \\text{ for } \\text{ all } x\\in \\mathbb {R}\\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ 0\r\n )\r\n \ \r\n =\r\n \ K\r\n \r\n \ e\r\n \r\n \ α\r\n x\r\n \ \r\n \r\n \ \r\n \r\n \ for\r\n \r\n \ \r\n all\r\n \ \r\n x\r\n \ ∈\r\n R\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ with some\r\n \r\n \ \r\n $$K>0$$\r\n \ \r\n \ \r\n K\r\n \ >\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \ and\r\n \r\n \r\n \ $$\\alpha >0$$\r\n \r\n \r\n \ α\r\n >\r\n \ 0\r\n \r\n \ \r\n \r\n \ \r\n .\r\n " author: - first_name: Celina full_name: Hanfland, Celina last_name: Hanfland - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Hanfland C, Winkler M. Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation. Journal of Elliptic and Parabolic Equations. 2025;11(3):2041-2063. doi:10.1007/s41808-025-00316-9 apa: Hanfland, C., & Winkler, M. (2025). Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation. Journal of Elliptic and Parabolic Equations, 11(3), 2041–2063. https://doi.org/10.1007/s41808-025-00316-9 bibtex: '@article{Hanfland_Winkler_2025, title={Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation}, volume={11}, DOI={10.1007/s41808-025-00316-9}, number={3}, journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer Science and Business Media LLC}, author={Hanfland, Celina and Winkler, Michael}, year={2025}, pages={2041–2063} }' chicago: 'Hanfland, Celina, and Michael Winkler. “Exactly Wave-Type Homoclinic Orbits and Emergence of Transient Exponential Growth in a Super-Fast Diffusion Equation.” Journal of Elliptic and Parabolic Equations 11, no. 3 (2025): 2041–63. https://doi.org/10.1007/s41808-025-00316-9.' ieee: 'C. Hanfland and M. Winkler, “Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation,” Journal of Elliptic and Parabolic Equations, vol. 11, no. 3, pp. 2041–2063, 2025, doi: 10.1007/s41808-025-00316-9.' mla: Hanfland, Celina, and Michael Winkler. “Exactly Wave-Type Homoclinic Orbits and Emergence of Transient Exponential Growth in a Super-Fast Diffusion Equation.” Journal of Elliptic and Parabolic Equations, vol. 11, no. 3, Springer Science and Business Media LLC, 2025, pp. 2041–63, doi:10.1007/s41808-025-00316-9. short: C. Hanfland, M. Winkler, Journal of Elliptic and Parabolic Equations 11 (2025) 2041–2063. date_created: 2025-12-18T18:57:21Z date_updated: 2025-12-18T20:16:49Z doi: 10.1007/s41808-025-00316-9 intvolume: ' 11' issue: '3' language: - iso: eng page: 2041-2063 publication: Journal of Elliptic and Parabolic Equations publication_identifier: issn: - 2296-9020 - 2296-9039 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation type: journal_article user_id: '31496' volume: 11 year: '2025' ... --- _id: '63164' abstract: - lang: eng text: Refined investigation of chemotaxis processes has revealed a significant role of degeneracies in corresponding motilities in a number of application contexts. A rapidly growing literature concerned with the analysis of resulting mathematical models has been capable of solving fundamental issues, but various problems have remained open, or even newly arisen. The goal of the paper consists in a summary of some developments in this area, and particularly in the discussion of the question how far the introduction of degeneracies may influence the behavior of solutions to chemotaxis systems. author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Effects of degeneracies in taxis-driven evolution. Mathematical Models and Methods in Applied Sciences. 2025;35(02):283-343. doi:10.1142/s0218202525400020 apa: Winkler, M. (2025). Effects of degeneracies in taxis-driven evolution. Mathematical Models and Methods in Applied Sciences, 35(02), 283–343. https://doi.org/10.1142/s0218202525400020 bibtex: '@article{Winkler_2025, title={Effects of degeneracies in taxis-driven evolution}, volume={35}, DOI={10.1142/s0218202525400020}, number={02}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2025}, pages={283–343} }' chicago: 'Winkler, Michael. “Effects of Degeneracies in Taxis-Driven Evolution.” Mathematical Models and Methods in Applied Sciences 35, no. 02 (2025): 283–343. https://doi.org/10.1142/s0218202525400020.' ieee: 'M. Winkler, “Effects of degeneracies in taxis-driven evolution,” Mathematical Models and Methods in Applied Sciences, vol. 35, no. 02, pp. 283–343, 2025, doi: 10.1142/s0218202525400020.' mla: Winkler, Michael. “Effects of Degeneracies in Taxis-Driven Evolution.” Mathematical Models and Methods in Applied Sciences, vol. 35, no. 02, World Scientific Pub Co Pte Ltd, 2025, pp. 283–343, doi:10.1142/s0218202525400020. short: M. Winkler, Mathematical Models and Methods in Applied Sciences 35 (2025) 283–343. date_created: 2025-12-16T19:23:40Z date_updated: 2025-12-18T20:16:23Z doi: 10.1142/s0218202525400020 intvolume: ' 35' issue: '02' language: - iso: eng page: 283-343 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: Effects of degeneracies in taxis-driven evolution type: journal_article user_id: '31496' volume: 35 year: '2025' ... --- _id: '54837' author: - first_name: Leander full_name: Claes, Leander id: '11829' last_name: Claes orcid: 0000-0002-4393-268X - first_name: Johannes full_name: Lankeit, Johannes last_name: Lankeit - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: 'Claes L, Lankeit J, Winkler M. A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions. Mathematical Models and Methods in Applied Sciences. 2025;35(11):2465-2512. doi:10.1142/s0218202525500447' apa: 'Claes, L., Lankeit, J., & Winkler, M. (2025). A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions. Mathematical Models and Methods in Applied Sciences, 35(11), 2465–2512. https://doi.org/10.1142/s0218202525500447' bibtex: '@article{Claes_Lankeit_Winkler_2025, title={A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions}, volume={35}, DOI={10.1142/s0218202525500447}, number={11}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Claes, Leander and Lankeit, Johannes and Winkler, Michael}, year={2025}, pages={2465–2512} }' chicago: 'Claes, Leander, Johannes Lankeit, and Michael Winkler. “A Model for Heat Generation by Acoustic Waves in Piezoelectric Materials: Global Large-Data Solutions.” Mathematical Models and Methods in Applied Sciences 35, no. 11 (2025): 2465–2512. https://doi.org/10.1142/s0218202525500447.' ieee: 'L. Claes, J. Lankeit, and M. Winkler, “A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions,” Mathematical Models and Methods in Applied Sciences, vol. 35, no. 11, pp. 2465–2512, 2025, doi: 10.1142/s0218202525500447.' mla: 'Claes, Leander, et al. “A Model for Heat Generation by Acoustic Waves in Piezoelectric Materials: Global Large-Data Solutions.” Mathematical Models and Methods in Applied Sciences, vol. 35, no. 11, World Scientific Pub Co Pte Ltd, 2025, pp. 2465–512, doi:10.1142/s0218202525500447.' short: L. Claes, J. Lankeit, M. Winkler, Mathematical Models and Methods in Applied Sciences 35 (2025) 2465–2512. date_created: 2024-06-20T13:43:42Z date_updated: 2026-01-05T07:59:41Z department: - _id: '90' - _id: '49' doi: 10.1142/s0218202525500447 external_id: arxiv: - '2411.14900' intvolume: ' 35' issue: '11' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/pdf/2411.14900 oa: '1' page: 2465-2512 project: - _id: '245' name: 'FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken für Leistungsschallanwendungen (NEPTUN)' publication: Mathematical Models and Methods in Applied Sciences publication_identifier: issn: - 1793-6314 publisher: World Scientific Pub Co Pte Ltd status: public title: 'A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions' type: journal_article user_id: '11829' volume: 35 year: '2025' ... --- _id: '63264' abstract: - lang: eng text: "Abstract\r\n In a smoothly bounded convex domain \r\n \ \r\n \r\n Ω\r\n \ ⊂\r\n \r\n \ \r\n R\r\n \r\n \ \r\n n\r\n \ \r\n \r\n \ \r\n \r\n${\\Omega}\\subset {\\mathbb{R}}^{n}$\r\n\r\n \r\n \ \r\n with n ≥ 1, a no-flux initial-boundary value problem for\r\n \r\n \ \r\n \r\n \r\n \r\n \r\n \r\n \r\n \ \r\n u\r\n \ \r\n \r\n \ t\r\n \r\n \ \r\n =\r\n \ Δ\r\n \ \r\n \ \r\n u\r\n \ ϕ\r\n \r\n \ (\r\n \ \r\n v\r\n \ \r\n )\r\n \r\n \ \r\n \r\n \ ,\r\n \r\n \r\n \r\n \ \r\n \r\n \r\n \ \r\n v\r\n \ \r\n \r\n \ t\r\n \r\n \ \r\n =\r\n \ Δ\r\n \ v\r\n −\r\n \ u\r\n v\r\n \ ,\r\n \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n$$\\begin{cases}_{t}={\\Delta}\\left(u\\phi \\left(v\\right)\\right),\\quad \\hfill \\\\ {v}_{t}={\\Delta}v-uv,\\quad \\hfill \\end{cases}$$\r\n\r\n \r\n \r\n \ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by\r\n \r\n \ \r\n ϕ\r\n \ \r\n (\r\n \ \r\n ξ\r\n \ \r\n )\r\n \r\n =\r\n \ \r\n \r\n \ ξ\r\n \r\n \ \r\n α\r\n \ \r\n \r\n \ ,\r\n \r\n ξ\r\n ∈\r\n \ \r\n [\r\n \ \r\n 0\r\n \ ,\r\n \r\n \ \r\n ξ\r\n \ \r\n \r\n \ 0\r\n \r\n \ \r\n \r\n \ ]\r\n \r\n \ .\r\n \r\n \ \r\n$$\\phi \\left(\\xi \\right)={\\xi }^{\\alpha },\\qquad \\xi \\in \\left[0,{\\xi }_{0}\\right].$$\r\n\r\n \ \r\n \r\n \ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy\r\n \r\n \ \r\n C\r\n \ \r\n (\r\n \ \r\n T\r\n \ \r\n )\r\n \r\n ≔\r\n \ \r\n \r\n \ ess sup\r\n \r\n \ \r\n t\r\n \ ∈\r\n \r\n \ (\r\n \r\n \ 0\r\n ,\r\n \ T\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ ∫\r\n \r\n \ \r\n Ω\r\n \r\n \ \r\n u\r\n \ \r\n (\r\n \ \r\n ⋅\r\n \ ,\r\n t\r\n \ \r\n )\r\n \r\n ln\r\n \ ⁡\r\n u\r\n \ \r\n (\r\n \ \r\n ⋅\r\n \ ,\r\n t\r\n \ \r\n )\r\n \r\n <\r\n \ ∞\r\n \r\n for all \r\n \ T\r\n >\r\n \ 0\r\n ,\r\n \ \r\n \r\n$$C\\left(T\\right){:=}\\underset{t\\in \\left(0,T\\right)}{\\text{ess\\,sup}}{\\int }_{{\\Omega}}u\\left(\\cdot ,t\\right)\\mathrm{ln}u\\left(\\cdot ,t\\right){< }\\infty \\qquad \\text{for\\,all\\,}T{ >}0,$$\r\n\r\n \ \r\n \r\n \ with sup\r\n T>0\r\n \ C(T) < ∞ if α ∈ [1, 2]." author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. A degenerate migration-consumption model in domains of arbitrary dimension. Advanced Nonlinear Studies. 2024;24(3):592-615. doi:10.1515/ans-2023-0131 apa: Winkler, M. (2024). A degenerate migration-consumption model in domains of arbitrary dimension. Advanced Nonlinear Studies, 24(3), 592–615. https://doi.org/10.1515/ans-2023-0131 bibtex: '@article{Winkler_2024, title={A degenerate migration-consumption model in domains of arbitrary dimension}, volume={24}, DOI={10.1515/ans-2023-0131}, number={3}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2024}, pages={592–615} }' chicago: 'Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains of Arbitrary Dimension.” Advanced Nonlinear Studies 24, no. 3 (2024): 592–615. https://doi.org/10.1515/ans-2023-0131.' ieee: 'M. Winkler, “A degenerate migration-consumption model in domains of arbitrary dimension,” Advanced Nonlinear Studies, vol. 24, no. 3, pp. 592–615, 2024, doi: 10.1515/ans-2023-0131.' mla: Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains of Arbitrary Dimension.” Advanced Nonlinear Studies, vol. 24, no. 3, Walter de Gruyter GmbH, 2024, pp. 592–615, doi:10.1515/ans-2023-0131. short: M. Winkler, Advanced Nonlinear Studies 24 (2024) 592–615. date_created: 2025-12-18T19:09:41Z date_updated: 2025-12-18T20:10:00Z doi: 10.1515/ans-2023-0131 intvolume: ' 24' issue: '3' language: - iso: eng page: 592-615 publication: Advanced Nonlinear Studies publication_identifier: issn: - 2169-0375 publication_status: published publisher: Walter de Gruyter GmbH status: public title: A degenerate migration-consumption model in domains of arbitrary dimension type: journal_article user_id: '31496' volume: 24 year: '2024' ... --- _id: '63248' abstract: - lang: eng text: "Abstract\r\n The Navier–Stokes system \r\n \r\n $$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t + (u\\cdot \\nabla ) u =\\Delta u+\\nabla P + f(x,t), \\\\ \\nabla \\cdot u=0, \\end{array} \\right. \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ u\r\n t\r\n \ \r\n +\r\n \ \r\n (\r\n \ u\r\n ·\r\n \ ∇\r\n )\r\n \ \r\n u\r\n \ =\r\n Δ\r\n \ u\r\n +\r\n \ ∇\r\n P\r\n \ +\r\n f\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ t\r\n )\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n ∇\r\n \ ·\r\n u\r\n \ =\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain \r\n \r\n \ $$\\Omega $$\r\n \r\n Ω\r\n \ \r\n \r\n . It is firstly, inter alia, observed that if \r\n \r\n \ $$T>0$$\r\n \r\n \r\n \ T\r\n >\r\n \ 0\r\n \r\n \r\n \ \r\n and \r\n \ \r\n $$\\begin{aligned} \\int _0^T \\bigg \\{ \\int _\\Omega |f(x,t)| \\cdot \\ln ^\\frac{1}{2} \\big (|f(x,t)|+1\\big ) dx \\bigg \\}^2 dt <\\infty , \\end{aligned}$$\r\n \ \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n ∫\r\n \ 0\r\n T\r\n \ \r\n \r\n \ {\r\n \r\n \ \r\n ∫\r\n \ Ω\r\n \r\n \ \r\n |\r\n \ f\r\n \r\n \ (\r\n x\r\n \ ,\r\n t\r\n \ )\r\n \r\n \ |\r\n \r\n \ ·\r\n \r\n \ ln\r\n \r\n \ 1\r\n 2\r\n \ \r\n \r\n \ \r\n (\r\n \ \r\n \r\n \ |\r\n f\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ t\r\n )\r\n \ \r\n |\r\n \ \r\n +\r\n \ 1\r\n \r\n \ )\r\n \r\n \ d\r\n x\r\n \ \r\n \r\n \ }\r\n \r\n \ 2\r\n \r\n \ d\r\n t\r\n \ <\r\n ∞\r\n \ ,\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \r\n \ then for all divergence-free \r\n \ \r\n $$u_0\\in L^2(\\Omega ;{\\mathbb {R}}^2)$$\r\n \r\n \r\n \ \r\n u\r\n \ 0\r\n \r\n \ ∈\r\n \r\n L\r\n \ 2\r\n \r\n \ \r\n (\r\n \ Ω\r\n ;\r\n \ \r\n \r\n R\r\n \ \r\n 2\r\n \ \r\n )\r\n \ \r\n \r\n \r\n \ \r\n , a corresponding initial-boundary value problem admits a weak solution u with \r\n \r\n $$u|_{t=0}=u_0$$\r\n \ \r\n \ \r\n \r\n \r\n \ u\r\n |\r\n \ \r\n \r\n t\r\n \ =\r\n 0\r\n \ \r\n \r\n =\r\n \ \r\n u\r\n \ 0\r\n \r\n \ \r\n \r\n \r\n \ . For any positive and nondecreasing \r\n \ \r\n $$L\\in C^0([0,\\infty ))$$\r\n \r\n \ \r\n L\r\n ∈\r\n \ \r\n C\r\n \ 0\r\n \r\n \ \r\n (\r\n \ \r\n [\r\n \ 0\r\n ,\r\n \ ∞\r\n )\r\n \ \r\n )\r\n \ \r\n \r\n \r\n \ \r\n such that \r\n \r\n $$\\begin{aligned} \\frac{L(\\xi )}{\\ln ^\\frac{1}{2} \\xi } \\rightarrow 0 \\qquad \\text{ as } \\xi \\rightarrow \\infty , \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n L\r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ \r\n \r\n \ ln\r\n \r\n \ 1\r\n 2\r\n \ \r\n \r\n \ ξ\r\n \r\n \ \r\n →\r\n \ 0\r\n \r\n \ \r\n as\r\n \ \r\n ξ\r\n \ →\r\n ∞\r\n \ ,\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \r\n \ this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed \r\n \r\n $$f:\\Omega \\times (0,T)\\rightarrow {\\mathbb {R}}^2$$\r\n \r\n \r\n \ f\r\n :\r\n \ Ω\r\n ×\r\n \ \r\n (\r\n \ 0\r\n ,\r\n \ T\r\n )\r\n \ \r\n →\r\n \ \r\n \r\n R\r\n \ \r\n 2\r\n \ \r\n \r\n \r\n \ \r\n fulfilling \r\n \r\n $$\\begin{aligned} \\int _0^T \\bigg \\{ \\int _\\Omega |f(x,t)| \\cdot L\\big (|f(x,t)|\\big ) dx \\bigg \\}^2 dt < \\infty . \\end{aligned}$$\r\n \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ ∫\r\n 0\r\n \ T\r\n \r\n \ \r\n {\r\n \ \r\n \r\n \ ∫\r\n Ω\r\n \ \r\n \r\n \ |\r\n f\r\n \ \r\n (\r\n \ x\r\n ,\r\n \ t\r\n )\r\n \ \r\n |\r\n \ \r\n ·\r\n \ \r\n L\r\n \ \r\n (\r\n \ \r\n |\r\n \ f\r\n \r\n \ (\r\n x\r\n \ ,\r\n t\r\n \ )\r\n \r\n \ |\r\n \r\n \ )\r\n \r\n \ d\r\n x\r\n \ \r\n \r\n \ \r\n }\r\n \ \r\n 2\r\n \ \r\n d\r\n \ t\r\n <\r\n \ ∞\r\n .\r\n \ \r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n This resolves a fine structure in the borderline case \r\n \r\n \ $$p=1$$\r\n \r\n \r\n \ p\r\n =\r\n \ 1\r\n \r\n \r\n \ \r\n and \r\n \ \r\n $$q=2$$\r\n \ \r\n \ \r\n q\r\n =\r\n \ 2\r\n \r\n \r\n \ \r\n appearing in results on existence of weak solutions for sources in \r\n \ \r\n $$L^q((0,T);L^p(\\Omega ;{\\mathbb {R}}^2))$$\r\n \r\n \ \r\n \r\n L\r\n \ q\r\n \r\n \ \r\n (\r\n \ \r\n (\r\n \ 0\r\n ,\r\n \ T\r\n )\r\n \ \r\n ;\r\n \ \r\n L\r\n \ p\r\n \r\n \ \r\n (\r\n \ Ω\r\n ;\r\n \ \r\n \r\n \ R\r\n \r\n \ 2\r\n \r\n \ )\r\n \r\n \ )\r\n \r\n \ \r\n \r\n \r\n \ when \r\n \r\n \ $$p\\in (1,\\infty ]$$\r\n \r\n \r\n \ p\r\n ∈\r\n \ (\r\n 1\r\n \ ,\r\n ∞\r\n \ ]\r\n \r\n \r\n \ \r\n and \r\n \ \r\n $$q\\in [1,\\infty ]$$\r\n \r\n \ \r\n q\r\n ∈\r\n \ [\r\n 1\r\n \ ,\r\n ∞\r\n \ ]\r\n \r\n \r\n \ \r\n satisfy \r\n \r\n $$\\frac{1}{p}+\\frac{1}{q}\\le \\frac{3}{2}$$\r\n \r\n \ \r\n \r\n 1\r\n \ p\r\n \r\n \ +\r\n \r\n \ 1\r\n q\r\n \ \r\n ≤\r\n \ \r\n 3\r\n \ 2\r\n \r\n \ \r\n \r\n \r\n \ , and on nonexistence if here \r\n \ \r\n $$p\\in [1,\\infty )$$\r\n \r\n \ \r\n p\r\n ∈\r\n \ [\r\n 1\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \r\n \ \r\n and \r\n \ \r\n $$q\\in [1,\\infty )$$\r\n \r\n \ \r\n q\r\n ∈\r\n \ [\r\n 1\r\n \ ,\r\n ∞\r\n \ )\r\n \r\n \r\n \ \r\n are such that \r\n \r\n $$\\frac{1}{p}+\\frac{1}{q}>\\frac{3}{2}$$\r\n \ \r\n \ \r\n \r\n 1\r\n \ p\r\n \r\n \ +\r\n \r\n \ 1\r\n q\r\n \ \r\n >\r\n \ \r\n 3\r\n \ 2\r\n \r\n \ \r\n \r\n \r\n \ ." author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system. Mathematische Annalen. 2024;391(2):3023-3054. doi:10.1007/s00208-024-02987-6 apa: Winkler, M. (2024). Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system. Mathematische Annalen, 391(2), 3023–3054. https://doi.org/10.1007/s00208-024-02987-6 bibtex: '@article{Winkler_2024, title={Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system}, volume={391}, DOI={10.1007/s00208-024-02987-6}, number={2}, journal={Mathematische Annalen}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2024}, pages={3023–3054} }' chicago: 'Winkler, Michael. “Externally Forced Blow-up and Optimal Spaces for Source Regularity in the Two-Dimensional Navier–Stokes System.” Mathematische Annalen 391, no. 2 (2024): 3023–54. https://doi.org/10.1007/s00208-024-02987-6.' ieee: 'M. Winkler, “Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system,” Mathematische Annalen, vol. 391, no. 2, pp. 3023–3054, 2024, doi: 10.1007/s00208-024-02987-6.' mla: Winkler, Michael. “Externally Forced Blow-up and Optimal Spaces for Source Regularity in the Two-Dimensional Navier–Stokes System.” Mathematische Annalen, vol. 391, no. 2, Springer Science and Business Media LLC, 2024, pp. 3023–54, doi:10.1007/s00208-024-02987-6. short: M. Winkler, Mathematische Annalen 391 (2024) 3023–3054. date_created: 2025-12-18T19:02:09Z date_updated: 2025-12-18T20:13:05Z doi: 10.1007/s00208-024-02987-6 intvolume: ' 391' issue: '2' language: - iso: eng page: 3023-3054 publication: Mathematische Annalen publication_identifier: issn: - 0025-5831 - 1432-1807 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system type: journal_article user_id: '31496' volume: 391 year: '2024' ... --- _id: '63245' abstract: - lang: eng text: "\r\n A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \r\n \r\n \\alpha>0\r\n \ \r\n the Keller–Segel-type migration–consumption system \r\n \r\n u_{t} = \\Delta (uv^{-\\alpha})\r\n \r\n \ , \r\n \r\n v_{t} = \\Delta v-uv\r\n \r\n , admits a global classical solution.\r\n " author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system. Annales de l’Institut Henri Poincaré C, Analyse non linéaire. 2024;42(6):1601-1630. doi:10.4171/aihpc/141 apa: Winkler, M. (2024). Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 42(6), 1601–1630. https://doi.org/10.4171/aihpc/141 bibtex: '@article{Winkler_2024, title={Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system}, volume={42}, DOI={10.4171/aihpc/141}, number={6}, journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael}, year={2024}, pages={1601–1630} }' chicago: 'Winkler, Michael. “Logarithmically Refined Gagliardo–Nirenberg Interpolation and Application to Blow-up Exclusion in a Singular Chemotaxis–Consumption System.” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 42, no. 6 (2024): 1601–30. https://doi.org/10.4171/aihpc/141.' ieee: 'M. Winkler, “Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system,” Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 42, no. 6, pp. 1601–1630, 2024, doi: 10.4171/aihpc/141.' mla: Winkler, Michael. “Logarithmically Refined Gagliardo–Nirenberg Interpolation and Application to Blow-up Exclusion in a Singular Chemotaxis–Consumption System.” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, vol. 42, no. 6, European Mathematical Society - EMS - Publishing House GmbH, 2024, pp. 1601–30, doi:10.4171/aihpc/141. short: M. Winkler, Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire 42 (2024) 1601–1630. date_created: 2025-12-18T19:00:24Z date_updated: 2025-12-18T20:12:43Z doi: 10.4171/aihpc/141 intvolume: ' 42' issue: '6' language: - iso: eng page: 1601-1630 publication: Annales de l'Institut Henri Poincaré C, Analyse non linéaire publication_identifier: issn: - 0294-1449 - 1873-1430 publication_status: published publisher: European Mathematical Society - EMS - Publishing House GmbH status: public title: Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system type: journal_article user_id: '31496' volume: 42 year: '2024' ... --- _id: '63257' abstract: - lang: eng text: AbstractThe quasilinear Keller–Segel system

$$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$

ut=∇·(D(u)∇u)-∇·(S(u)∇v),vt=Δv-v+u,endowed with homogeneous Neumann boundary conditions is considered in a bounded domain

$$\Omega \subset {\mathbb {R}}^n$$

Ω⊂Rn,

$$n \ge 3$$

n≥3, with smooth boundary for sufficiently regular functionsDandSsatisfying

$$D>0$$

D>0on

$$[0,\infty )$$

[0,∞),

$$S>0$$

S>0on

$$(0,\infty )$$

(0,∞)and

$$S(0)=0$$

S(0)=0. On the one hand, it is shown that if

$$\frac{S}{D}$$

SDsatisfies the subcritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$

S(s)D(s)≤Csαforalls≥1withsomeα<2nand

$$C>0$$

C>0, then for any sufficiently regular initial data there exists a global weak energy solution such that

$${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$

esssupt>0‖u(t)‖Lp(Ω)<∞for some

$$p > \frac{2n}{n+2}$$

p>2nn+2. On the other hand, if

$$\frac{S}{D}$$

SDsatisfies the supercritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$

S(s)D(s)≥csαforalls≥1withsomeα>2nand

$$c>0$$

c>0, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value

$$\alpha = \frac{2}{n}$$

α=2nfor

$$n \ge 3$$

n≥3, without any additional assumption on the behavior ofD(s) as

$$s \rightarrow \infty $$

s→∞, in particular without requiring any algebraic lower bound forD. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type

$$Q(s) = \exp (-s^\beta )$$

Q(s)=exp(-sβ),

$$s \ge 0$$

s≥0, for global solvability the exponent

$$\beta = \frac{n-2}{n}$$

β=n-2nis seen to be critical. article_number: '26' author: - first_name: Christian full_name: Stinner, Christian last_name: Stinner - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Stinner C, Winkler M. A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations. 2024;24(2). doi:10.1007/s00028-024-00954-x apa: Stinner, C., & Winkler, M. (2024). A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects. Journal of Evolution Equations, 24(2), Article 26. https://doi.org/10.1007/s00028-024-00954-x bibtex: '@article{Stinner_Winkler_2024, title={A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects}, volume={24}, DOI={10.1007/s00028-024-00954-x}, number={226}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Stinner, Christian and Winkler, Michael}, year={2024} }' chicago: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations 24, no. 2 (2024). https://doi.org/10.1007/s00028-024-00954-x. ieee: 'C. Stinner and M. Winkler, “A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects,” Journal of Evolution Equations, vol. 24, no. 2, Art. no. 26, 2024, doi: 10.1007/s00028-024-00954-x.' mla: Stinner, Christian, and Michael Winkler. “A Critical Exponent in a Quasilinear Keller–Segel System with Arbitrarily Fast Decaying Diffusivities Accounting for Volume-Filling Effects.” Journal of Evolution Equations, vol. 24, no. 2, 26, Springer Science and Business Media LLC, 2024, doi:10.1007/s00028-024-00954-x. short: C. Stinner, M. Winkler, Journal of Evolution Equations 24 (2024). date_created: 2025-12-18T19:06:36Z date_updated: 2025-12-18T20:14:21Z doi: 10.1007/s00028-024-00954-x intvolume: ' 24' issue: '2' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects type: journal_article user_id: '31496' volume: 24 year: '2024' ... --- _id: '63253' abstract: - lang: eng text: "Abstract\r\n The Neumann problem for the Keller-Segel system \r\n \r\n \ \r\n \r\n \r\n \r\n \ \r\n \r\n \ {\r\n \r\n \r\n \ \r\n \r\n \ u\r\n t\r\n \ \r\n =\r\n \ ∇\r\n \ ⋅\r\n \r\n \ (\r\n D\r\n \ \r\n (\r\n \ u\r\n )\r\n \ \r\n ∇\r\n u\r\n \ )\r\n \r\n \ −\r\n ∇\r\n ⋅\r\n \ \r\n (\r\n \ S\r\n \r\n \ (\r\n u\r\n \ )\r\n \r\n \ ∇\r\n \ v\r\n )\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ 0\r\n =\r\n \ Δ\r\n \ v\r\n −\r\n \ μ\r\n +\r\n \ u\r\n ,\r\n \ \r\n \ μ\r\n =\r\n \ \r\n −\r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n \ \r\n ∫\r\n \ Ω\r\n \ \r\n u\r\n \ d\r\n \ x\r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n is considered in n-dimensional balls Ω with \r\n \ \r\n \r\n \r\n n\r\n \ ⩾\r\n 2\r\n \ \r\n \r\n , with suitably regular and radially symmetric, radially nonincreasing initial data u\r\n 0. The functions D and S are only assumed to belong to \r\n \r\n \ \r\n \r\n \r\n \ C\r\n 2\r\n \ \r\n (\r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n )\r\n \ )\r\n \r\n \ \r\n and to satisfy D > 0 and \r\n \r\n \ \r\n \r\n S\r\n \ ⩾\r\n 0\r\n \ \r\n \r\n on \r\n \r\n \r\n \r\n \ [\r\n 0\r\n \ ,\r\n ∞\r\n )\r\n \ \r\n \r\n as well as \r\n \r\n \ \r\n \r\n S\r\n \ (\r\n 0\r\n \ )\r\n =\r\n \ 0\r\n \r\n \ \r\n ; in particular, diffusivities with arbitrarily fast decay are included.\r\n \ In this general context, it is shown that it is merely the asymptotic behavior as \r\n \r\n \ \r\n \r\n ξ\r\n \ →\r\n \ ∞\r\n \r\n \ \r\n of the expression \r\n \r\n \ \r\n \r\n \r\n \r\n \ \r\n I\r\n \ \r\n (\r\n \ ξ\r\n )\r\n \ \r\n :=\r\n \ \r\n \r\n \ S\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ \r\n \r\n \ \r\n ξ\r\n \ \r\n 2\r\n \ n\r\n \r\n \ \r\n D\r\n \ \r\n (\r\n \ ξ\r\n )\r\n \ \r\n \r\n \ \r\n ,\r\n \ \r\n ξ\r\n \ >\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n which decides about the occurrence of blow-up: Namely, it is seen that\r\n\r\n \r\n •\r\n \ if \r\n \r\n \ \r\n \r\n \r\n \ lim\r\n \ \r\n ξ\r\n \ →\r\n \ ∞\r\n \ \r\n \r\n \ I\r\n (\r\n ξ\r\n \ )\r\n \ =\r\n 0\r\n \ \r\n \r\n \ , then any such solution is global and bounded, that\r\n \r\n \r\n •\r\n \ if \r\n \r\n \ \r\n \r\n \r\n \ lim sup\r\n \ \r\n ξ\r\n \ →\r\n \ ∞\r\n \ \r\n \r\n \ I\r\n (\r\n ξ\r\n \ )\r\n \ <\r\n ∞\r\n \r\n \ \r\n and \r\n \r\n \ \r\n \r\n \r\n \ ∫\r\n Ω\r\n \r\n \ \r\n u\r\n \ 0\r\n \r\n \ \r\n \r\n \ is suitably small, then the corresponding solution is global and bounded, and that\r\n \r\n \ \r\n •\r\n \ if \r\n \r\n \ \r\n \r\n \r\n \ lim inf\r\n \ \r\n ξ\r\n \ →\r\n \ ∞\r\n \ \r\n \r\n \ I\r\n (\r\n ξ\r\n \ )\r\n \ >\r\n 0\r\n \ \r\n \r\n \ , then at each appropriately large mass level m, there exist radial initial data u\r\n 0 such that \r\n \r\n \ \r\n \r\n \r\n \ ∫\r\n Ω\r\n \r\n \ \r\n u\r\n \ 0\r\n \r\n \ =\r\n m\r\n \ \r\n \r\n \ , and that the associated solution blows up either in finite or in infinite time.\r\n \r\n \ \r\n \r\n This especially reveals the presence of critical mass phenomena whenever \r\n \ \r\n \r\n \r\n \r\n \ lim\r\n \ \r\n ξ\r\n \ →\r\n \ ∞\r\n \ \r\n \r\n \ I\r\n (\r\n ξ\r\n \ )\r\n ∈\r\n \ (\r\n 0\r\n \ ,\r\n ∞\r\n )\r\n \ \r\n \r\n exists." article_number: '125006' author: - first_name: Mengyao full_name: Ding, Mengyao last_name: Ding - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: 'Ding M, Winkler M. Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture. Nonlinearity. 2024;37(12). doi:10.1088/1361-6544/ad871a' apa: 'Ding, M., & Winkler, M. (2024). Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture. Nonlinearity, 37(12), Article 125006. https://doi.org/10.1088/1361-6544/ad871a' bibtex: '@article{Ding_Winkler_2024, title={Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture}, volume={37}, DOI={10.1088/1361-6544/ad871a}, number={12125006}, journal={Nonlinearity}, publisher={IOP Publishing}, author={Ding, Mengyao and Winkler, Michael}, year={2024} }' chicago: 'Ding, Mengyao, and Michael Winkler. “Radial Blow-up in Quasilinear Keller-Segel Systems: Approaching the Full Picture.” Nonlinearity 37, no. 12 (2024). https://doi.org/10.1088/1361-6544/ad871a.' ieee: 'M. Ding and M. Winkler, “Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture,” Nonlinearity, vol. 37, no. 12, Art. no. 125006, 2024, doi: 10.1088/1361-6544/ad871a.' mla: 'Ding, Mengyao, and Michael Winkler. “Radial Blow-up in Quasilinear Keller-Segel Systems: Approaching the Full Picture.” Nonlinearity, vol. 37, no. 12, 125006, IOP Publishing, 2024, doi:10.1088/1361-6544/ad871a.' short: M. Ding, M. Winkler, Nonlinearity 37 (2024). date_created: 2025-12-18T19:04:45Z date_updated: 2025-12-18T20:13:49Z doi: 10.1088/1361-6544/ad871a intvolume: ' 37' issue: '12' language: - iso: eng publication: Nonlinearity publication_identifier: issn: - 0951-7715 - 1361-6544 publication_status: published publisher: IOP Publishing status: public title: 'Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture' type: journal_article user_id: '31496' volume: 37 year: '2024' ... --- _id: '63254' abstract: - lang: eng text: "AbstractThe chemotaxis-Navier–Stokes system

$$\\begin{aligned} \\left\\{ \\begin{array}{rcl} n_t+u\\cdot \\nabla n & =& \\Delta \\big (n c^{-\\alpha } \\big ), \\\\ c_t+ u\\cdot \\nabla c & =& \\Delta c -nc,\\\\ u_t + (u\\cdot \\nabla ) u & =& \\Delta u+\\nabla P + n\\nabla \\Phi , \\qquad \\nabla \\cdot u=0, \\end{array} \\right. \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n n\r\n \ t\r\n \r\n \ +\r\n u\r\n \ ·\r\n ∇\r\n \ n\r\n \r\n \ \r\n \r\n \ =\r\n \r\n \ \r\n \r\n \ Δ\r\n \r\n \ (\r\n \r\n \ n\r\n \r\n \ c\r\n \r\n \ -\r\n α\r\n \ \r\n \r\n \ \r\n )\r\n \ \r\n ,\r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ c\r\n t\r\n \ \r\n +\r\n \ u\r\n ·\r\n \ ∇\r\n c\r\n \ \r\n \r\n \ \r\n =\r\n \ \r\n \r\n \ \r\n Δ\r\n \ c\r\n -\r\n \ n\r\n c\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n u\r\n \ t\r\n \r\n \ +\r\n \r\n \ (\r\n u\r\n \ ·\r\n ∇\r\n \ )\r\n \r\n \ u\r\n \r\n \ \r\n \r\n \ =\r\n \r\n \ \r\n \r\n \ Δ\r\n u\r\n \ +\r\n ∇\r\n \ P\r\n +\r\n \ n\r\n ∇\r\n \ Φ\r\n ,\r\n \ \r\n ∇\r\n \ ·\r\n u\r\n \ =\r\n 0\r\n \ ,\r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \r\n \ modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain

$$\\Omega \\subset \\mathbb R^2$$

\r\n \ \r\n Ω\r\n ⊂\r\n \ \r\n R\r\n \ 2\r\n \r\n \ \r\n . For all

$$\\alpha > 0$$

\r\n \ \r\n α\r\n >\r\n \ 0\r\n \r\n and all sufficiently regular

$$\\Phi $$

\r\n \ Φ\r\n , we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for u at a point in the proof when knowledge about n is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time

$$L^p$$

\r\n \r\n \ L\r\n p\r\n \ \r\n estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time

$$L^2$$

\r\n \r\n \ L\r\n 2\r\n \ \r\n norm of the force term raised to an arbitrary small power." article_number: '60' author: - first_name: Mario full_name: Fuest, Mario last_name: Fuest - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Fuest M, Winkler M. Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing. Journal of Mathematical Fluid Mechanics. 2024;26(4). doi:10.1007/s00021-024-00899-8 apa: Fuest, M., & Winkler, M. (2024). Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing. Journal of Mathematical Fluid Mechanics, 26(4), Article 60. https://doi.org/10.1007/s00021-024-00899-8 bibtex: '@article{Fuest_Winkler_2024, title={Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing}, volume={26}, DOI={10.1007/s00021-024-00899-8}, number={460}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer Science and Business Media LLC}, author={Fuest, Mario and Winkler, Michael}, year={2024} }' chicago: Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing.” Journal of Mathematical Fluid Mechanics 26, no. 4 (2024). https://doi.org/10.1007/s00021-024-00899-8. ieee: 'M. Fuest and M. Winkler, “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing,” Journal of Mathematical Fluid Mechanics, vol. 26, no. 4, Art. no. 60, 2024, doi: 10.1007/s00021-024-00899-8.' mla: Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing.” Journal of Mathematical Fluid Mechanics, vol. 26, no. 4, 60, Springer Science and Business Media LLC, 2024, doi:10.1007/s00021-024-00899-8. short: M. Fuest, M. Winkler, Journal of Mathematical Fluid Mechanics 26 (2024). date_created: 2025-12-18T19:05:09Z date_updated: 2025-12-18T20:13:58Z doi: 10.1007/s00021-024-00899-8 intvolume: ' 26' issue: '4' language: - iso: eng publication: Journal of Mathematical Fluid Mechanics publication_identifier: issn: - 1422-6928 - 1422-6952 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing type: journal_article user_id: '31496' volume: 26 year: '2024' ... --- _id: '63259' abstract: - lang: eng text: "AbstractIn a smoothly bounded two‐dimensional domain and for a given nondecreasing positive unbounded , for each and the inequality\r\nis shown to hold for any positive fulfilling\r\nThis is thereafter applied to nonglobal solutions of the Keller–Segel system coupled to the incompressible Navier–Stokes equations through transport and buoyancy, and it is seen that in any such blow‐up event the corresponding population density cannot remain uniformly integrable over near its explosion time." article_number: e12885 author: - first_name: Yulan full_name: Wang, Yulan last_name: Wang - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Wang Y, Winkler M. An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system. Journal of the London Mathematical Society. 2024;109(3). doi:10.1112/jlms.12885 apa: Wang, Y., & Winkler, M. (2024). An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system. Journal of the London Mathematical Society, 109(3), Article e12885. https://doi.org/10.1112/jlms.12885 bibtex: '@article{Wang_Winkler_2024, title={An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system}, volume={109}, DOI={10.1112/jlms.12885}, number={3e12885}, journal={Journal of the London Mathematical Society}, publisher={Wiley}, author={Wang, Yulan and Winkler, Michael}, year={2024} }' chicago: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving LlogL$L\log L$ Spaces and Application to the Characterization of Blow‐up Behavior in a Two‐dimensional Keller–Segel–Navier–Stokes System.” Journal of the London Mathematical Society 109, no. 3 (2024). https://doi.org/10.1112/jlms.12885. ieee: 'Y. Wang and M. Winkler, “An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system,” Journal of the London Mathematical Society, vol. 109, no. 3, Art. no. e12885, 2024, doi: 10.1112/jlms.12885.' mla: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving LlogL$L\log L$ Spaces and Application to the Characterization of Blow‐up Behavior in a Two‐dimensional Keller–Segel–Navier–Stokes System.” Journal of the London Mathematical Society, vol. 109, no. 3, e12885, Wiley, 2024, doi:10.1112/jlms.12885. short: Y. Wang, M. Winkler, Journal of the London Mathematical Society 109 (2024). date_created: 2025-12-18T19:07:25Z date_updated: 2025-12-18T20:14:39Z doi: 10.1112/jlms.12885 intvolume: ' 109' issue: '3' language: - iso: eng publication: Journal of the London Mathematical Society publication_identifier: issn: - 0024-6107 - 1469-7750 publication_status: published publisher: Wiley status: public title: An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system type: journal_article user_id: '31496' volume: 109 year: '2024' ... --- _id: '63258' abstract: - lang: eng text:

This manuscript studies a no-flux initial-boundary value problem for a four-component chemotaxis system that has been proposed as a model for the response of cytotoxic T-lymphocytes to a solid tumor. In contrast to classical Keller-Segel type situations focusing on two-component interplay of chemotaxing populations with a signal directly secreted by themselves, the presently considered system accounts for a certain indirect mechanism of attractant evolution. Despite the presence of a zero-order exciting nonlinearity of quadratic type that forms a core mathematical feature of the model, the manuscript asserts the global existence of classical solutions for initial data of arbitrary size in three-dimensional domains.

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. Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor. Proceedings of the American Mathematical Society. 2024;152(10):4325-4341. doi:10.1090/proc/16867 apa: Tao, Y., & Winkler, M. (2024). Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor. Proceedings of the American Mathematical Society, 152(10), 4325–4341. https://doi.org/10.1090/proc/16867 bibtex: '@article{Tao_Winkler_2024, title={Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor}, volume={152}, DOI={10.1090/proc/16867}, number={10}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Tao, Youshan and Winkler, Michael}, year={2024}, pages={4325–4341} }' chicago: 'Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Chemotaxis System Modeling Immune Response to a Solid Tumor.” Proceedings of the American Mathematical Society 152, no. 10 (2024): 4325–41. https://doi.org/10.1090/proc/16867.' ieee: 'Y. Tao and M. Winkler, “Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor,” Proceedings of the American Mathematical Society, vol. 152, no. 10, pp. 4325–4341, 2024, doi: 10.1090/proc/16867.' mla: Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Chemotaxis System Modeling Immune Response to a Solid Tumor.” Proceedings of the American Mathematical Society, vol. 152, no. 10, American Mathematical Society (AMS), 2024, pp. 4325–41, doi:10.1090/proc/16867. short: Y. Tao, M. Winkler, Proceedings of the American Mathematical Society 152 (2024) 4325–4341. date_created: 2025-12-18T19:07:03Z date_updated: 2025-12-18T20:14:30Z doi: 10.1090/proc/16867 intvolume: ' 152' issue: '10' language: - iso: eng page: 4325-4341 publication: Proceedings of the American Mathematical Society publication_identifier: issn: - 0002-9939 - 1088-6826 publication_status: published publisher: American Mathematical Society (AMS) status: public title: Global smooth solutions in a chemotaxis system modeling immune response to a solid tumor type: journal_article user_id: '31496' volume: 152 year: '2024' ...