--- _id: '63290' abstract: - lang: eng text: This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale. author: - first_name: N. full_name: Bellomo, N. last_name: Bellomo - first_name: N. full_name: Outada, N. last_name: Outada - first_name: J. full_name: Soler, J. last_name: Soler - first_name: Y. full_name: Tao, Y. last_name: Tao - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: 'Bellomo N, Outada N, Soler J, Tao Y, Winkler M. Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences. 2022;32(04):713-792. doi:10.1142/s0218202522500166' apa: 'Bellomo, N., Outada, N., Soler, J., Tao, Y., & Winkler, M. (2022). Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Mathematical Models and Methods in Applied Sciences, 32(04), 713–792. https://doi.org/10.1142/s0218202522500166' bibtex: '@article{Bellomo_Outada_Soler_Tao_Winkler_2022, title={Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision}, volume={32}, DOI={10.1142/s0218202522500166}, number={04}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Bellomo, N. and Outada, N. and Soler, J. and Tao, Y. and Winkler, Michael}, year={2022}, pages={713–792} }' chicago: 'Bellomo, N., N. Outada, J. Soler, Y. Tao, and Michael Winkler. “Chemotaxis and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences 32, no. 04 (2022): 713–92. https://doi.org/10.1142/s0218202522500166.' ieee: 'N. Bellomo, N. Outada, J. Soler, Y. Tao, and M. Winkler, “Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision,” Mathematical Models and Methods in Applied Sciences, vol. 32, no. 04, pp. 713–792, 2022, doi: 10.1142/s0218202522500166.' mla: 'Bellomo, N., et al. “Chemotaxis and Cross-Diffusion Models in Complex Environments: Models and Analytic Problems toward a Multiscale Vision.” Mathematical Models and Methods in Applied Sciences, vol. 32, no. 04, World Scientific Pub Co Pte Ltd, 2022, pp. 713–92, doi:10.1142/s0218202522500166.' short: N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler, Mathematical Models and Methods in Applied Sciences 32 (2022) 713–792. date_created: 2025-12-18T19:20:25Z date_updated: 2025-12-18T20:07:51Z doi: 10.1142/s0218202522500166 intvolume: ' 32' issue: '04' language: - iso: eng page: 713-792 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: 'Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision' type: journal_article user_id: '31496' volume: 32 year: '2022' ... --- _id: '63295' abstract: - lang: eng text: AbstractWe introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate. article_number: '14' author: - first_name: Johannes full_name: Lankeit, Johannes last_name: Lankeit - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Lankeit J, Winkler M. Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations. 2022;22(1). doi:10.1007/s00028-022-00768-9 apa: Lankeit, J., & Winkler, M. (2022). Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects. Journal of Evolution Equations, 22(1), Article 14. https://doi.org/10.1007/s00028-022-00768-9 bibtex: '@article{Lankeit_Winkler_2022, title={Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects}, volume={22}, DOI={10.1007/s00028-022-00768-9}, number={114}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Lankeit, Johannes and Winkler, Michael}, year={2022} }' chicago: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations 22, no. 1 (2022). https://doi.org/10.1007/s00028-022-00768-9. ieee: 'J. Lankeit and M. Winkler, “Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects,” Journal of Evolution Equations, vol. 22, no. 1, Art. no. 14, 2022, doi: 10.1007/s00028-022-00768-9.' mla: Lankeit, Johannes, and Michael Winkler. “Global Existence in Reaction–Diffusion Systems with Mass Control under Relaxed Assumptions Merely Referring to Cross-Absorptive Effects.” Journal of Evolution Equations, vol. 22, no. 1, 14, Springer Science and Business Media LLC, 2022, doi:10.1007/s00028-022-00768-9. short: J. Lankeit, M. Winkler, Journal of Evolution Equations 22 (2022). date_created: 2025-12-18T19:22:46Z date_updated: 2025-12-18T20:08:35Z doi: 10.1007/s00028-022-00768-9 intvolume: ' 22' issue: '1' language: - iso: eng publication: Journal of Evolution Equations publication_identifier: issn: - 1424-3199 - 1424-3202 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects type: journal_article user_id: '31496' volume: 22 year: '2022' ... --- _id: '63299' 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. Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical Analysis. 2022;54(4):4806-4864. doi:10.1137/21m1449841 apa: Tao, Y., & Winkler, M. (2022). Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System. SIAM Journal on Mathematical Analysis, 54(4), 4806–4864. https://doi.org/10.1137/21m1449841 bibtex: '@article{Tao_Winkler_2022, title={Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System}, volume={54}, DOI={10.1137/21m1449841}, number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tao, Youshan and Winkler, Michael}, year={2022}, pages={4806–4864} }' chicago: 'Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical Analysis 54, no. 4 (2022): 4806–64. https://doi.org/10.1137/21m1449841.' ieee: 'Y. Tao and M. Winkler, “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System,” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, pp. 4806–4864, 2022, doi: 10.1137/21m1449841.' mla: Tao, Youshan, and Michael Winkler. “Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System.” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial & Applied Mathematics (SIAM), 2022, pp. 4806–64, doi:10.1137/21m1449841. short: Y. Tao, M. Winkler, SIAM Journal on Mathematical Analysis 54 (2022) 4806–4864. date_created: 2025-12-18T19:24:16Z date_updated: 2025-12-18T20:09:05Z doi: 10.1137/21m1449841 intvolume: ' 54' issue: '4' language: - iso: eng page: 4806-4864 publication: SIAM Journal on Mathematical Analysis publication_identifier: issn: - 0036-1410 - 1095-7154 publication_status: published publisher: Society for Industrial & Applied Mathematics (SIAM) status: public title: Existence Theory and Qualitative Analysis for a Fully Cross-Diffusive Predator-Prey System type: journal_article user_id: '31496' volume: 54 year: '2022' ... --- _id: '63298' author: - first_name: Angela full_name: Stevens, Angela last_name: Stevens - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Stevens A, Winkler M. Taxis-driven persistent localization in a degenerate Keller-Segel system. Communications in Partial Differential Equations. 2022;47(12):2341-2362. doi:10.1080/03605302.2022.2122836 apa: Stevens, A., & Winkler, M. (2022). Taxis-driven persistent localization in a degenerate Keller-Segel system. Communications in Partial Differential Equations, 47(12), 2341–2362. https://doi.org/10.1080/03605302.2022.2122836 bibtex: '@article{Stevens_Winkler_2022, title={Taxis-driven persistent localization in a degenerate Keller-Segel system}, volume={47}, DOI={10.1080/03605302.2022.2122836}, number={12}, journal={Communications in Partial Differential Equations}, publisher={Informa UK Limited}, author={Stevens, Angela and Winkler, Michael}, year={2022}, pages={2341–2362} }' chicago: 'Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization in a Degenerate Keller-Segel System.” Communications in Partial Differential Equations 47, no. 12 (2022): 2341–62. https://doi.org/10.1080/03605302.2022.2122836.' ieee: 'A. Stevens and M. Winkler, “Taxis-driven persistent localization in a degenerate Keller-Segel system,” Communications in Partial Differential Equations, vol. 47, no. 12, pp. 2341–2362, 2022, doi: 10.1080/03605302.2022.2122836.' mla: Stevens, Angela, and Michael Winkler. “Taxis-Driven Persistent Localization in a Degenerate Keller-Segel System.” Communications in Partial Differential Equations, vol. 47, no. 12, Informa UK Limited, 2022, pp. 2341–62, doi:10.1080/03605302.2022.2122836. short: A. Stevens, M. Winkler, Communications in Partial Differential Equations 47 (2022) 2341–2362. date_created: 2025-12-18T19:23:52Z date_updated: 2025-12-18T20:08:58Z doi: 10.1080/03605302.2022.2122836 intvolume: ' 47' issue: '12' language: - iso: eng page: 2341-2362 publication: Communications in Partial Differential Equations publication_identifier: issn: - 0360-5302 - 1532-4133 publication_status: published publisher: Informa UK Limited status: public title: Taxis-driven persistent localization in a degenerate Keller-Segel system type: journal_article user_id: '31496' volume: 47 year: '2022' ... --- _id: '63266' abstract: - lang: eng text: "AbstractIn a ball

$$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$

\r\n \ \r\n Ω\r\n =\r\n \ \r\n B\r\n \ R\r\n \r\n \ \r\n (\r\n \ 0\r\n )\r\n \ \r\n ⊂\r\n \ \r\n \r\n R\r\n \ \r\n n\r\n \ \r\n \r\n ,

$$n\\ge 2$$

\r\n \r\n \ n\r\n ≥\r\n \ 2\r\n \r\n , the chemotaxis system

$$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big ) - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$

\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 \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 (\r\n \ \r\n u\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 \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 \ 1\r\n \r\n \ |\r\n Ω\r\n \ |\r\n \r\n \ \r\n \r\n \ ∫\r\n Ω\r\n \ \r\n u\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 \ \r\n \r\n \r\n \ is considered under no-flux boundary conditions, with a focus on nonlinearities

$$S\\in C^2([0,\\infty ))$$

\r\n \ \r\n S\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 which exhibit super-algebraically fast decay in the sense that with some

$$K_S>0, \\beta \\in [0,1)$$

\r\n \ \r\n \r\n K\r\n \ S\r\n \r\n \ >\r\n 0\r\n \ ,\r\n β\r\n \ ∈\r\n \r\n [\r\n \ 0\r\n ,\r\n \ 1\r\n )\r\n \ \r\n \r\n and

$$\\xi _0>0$$

\r\n \r\n \ \r\n ξ\r\n \ 0\r\n \r\n \ >\r\n 0\r\n \ \r\n ,

$$\\begin{aligned} S(\\xi )>0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$

\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 \ 0\r\n \r\n \ \r\n and\r\n \ \r\n \r\n \ \r\n S\r\n \ ′\r\n \r\n \ \r\n (\r\n \ ξ\r\n )\r\n \ \r\n ≤\r\n \ -\r\n \r\n \ K\r\n S\r\n \ \r\n \r\n \ ξ\r\n \r\n \ -\r\n β\r\n \ \r\n \r\n \ S\r\n \r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ \r\n \r\n \ for\r\n \r\n \ \r\n all\r\n \ \r\n ξ\r\n \ ≥\r\n \r\n \ ξ\r\n 0\r\n \ \r\n .\r\n \ \r\n \r\n \ \r\n \r\n \r\n \ It is, inter alia, shown that if furthermore

$$D\\in C^2((0,\\infty ))$$

\r\n \ \r\n D\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 is positive and suitably small in relation to S by satisfying

$$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$

\r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ ξ\r\n S\r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ \r\n D\r\n \ (\r\n ξ\r\n \ )\r\n \r\n \ \r\n ≥\r\n \ \r\n K\r\n \ \r\n SD\r\n \ \r\n \r\n \ \r\n ξ\r\n \ λ\r\n \r\n \ \r\n \r\n \ for\r\n \r\n \ \r\n all\r\n \ \r\n ξ\r\n \ ≥\r\n \r\n \ ξ\r\n 0\r\n \ \r\n \r\n \ \r\n \r\n \r\n \ \r\n with some

$$K_{SD}>0$$

\r\n \r\n \ \r\n K\r\n \ \r\n SD\r\n \ \r\n \r\n >\r\n \ 0\r\n \r\n and

$$\\lambda >\\frac{2}{n}$$

\r\n \r\n \ λ\r\n >\r\n \ \r\n 2\r\n \ n\r\n \r\n \ \r\n , then throughout a considerably large set of initial data, (

$$\\star $$

\r\n \ ⋆\r\n ) admits global classical solutions (u, v) fulfilling

$$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \\text{ for } \\text{ all } t>0, \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n z\r\n \ (\r\n t\r\n \ )\r\n \r\n \ C\r\n \r\n \ ≤\r\n \r\n \ \r\n ‖\r\n \ u\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 \ L\r\n ∞\r\n \ \r\n \r\n \ (\r\n Ω\r\n \ )\r\n \r\n \ \r\n \r\n \ ≤\r\n C\r\n \ z\r\n \r\n \ (\r\n t\r\n \ )\r\n \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 with some

$$C=C^{(u,v)}\\ge 1$$

\r\n \ \r\n C\r\n =\r\n \ \r\n C\r\n \ \r\n (\r\n \ u\r\n ,\r\n \ v\r\n )\r\n \ \r\n \r\n ≥\r\n \ 1\r\n \r\n , where z denotes the solution of

$$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$

\r\n \r\n \ \r\n \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n \r\n \ \r\n z\r\n \ ′\r\n \r\n \ \r\n (\r\n \ t\r\n )\r\n \ \r\n =\r\n \ \r\n z\r\n \ 2\r\n \r\n \ \r\n (\r\n \ t\r\n )\r\n \ \r\n ·\r\n \ S\r\n \r\n \ (\r\n \r\n \ z\r\n \r\n \ (\r\n t\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 \ \r\n \r\n \ z\r\n \r\n \ (\r\n 0\r\n \ )\r\n \r\n \ =\r\n \r\n \ ξ\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 which is seen to exist globally, and to satisfy

$$z(t)\\rightarrow +\\infty $$

\r\n \ \r\n z\r\n (\r\n \ t\r\n )\r\n \ →\r\n +\r\n \ ∞\r\n \r\n as

$$t\\rightarrow \\infty $$

\r\n \ \r\n t\r\n →\r\n \ ∞\r\n \r\n . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in (

$$\\star $$

\r\n \ ⋆\r\n ) at logarithmic and doubly logarithmic rates, respectively." author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations. 2022;36(2):1677-1702. doi:10.1007/s10884-022-10167-w apa: Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations, 36(2), 1677–1702. https://doi.org/10.1007/s10884-022-10167-w bibtex: '@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel System}, volume={36}, DOI={10.1007/s10884-022-10167-w}, number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702} }' chicago: 'Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations 36, no. 2 (2022): 1677–1702. https://doi.org/10.1007/s10884-022-10167-w.' ieee: 'M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” Journal of Dynamics and Differential Equations, vol. 36, no. 2, pp. 1677–1702, 2022, doi: 10.1007/s10884-022-10167-w.' mla: Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations, vol. 36, no. 2, Springer Science and Business Media LLC, 2022, pp. 1677–702, doi:10.1007/s10884-022-10167-w. short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702. date_created: 2025-12-18T19:10:32Z date_updated: 2025-12-18T20:10:14Z doi: 10.1007/s10884-022-10167-w intvolume: ' 36' issue: '2' language: - iso: eng page: 1677-1702 publication: Journal of Dynamics and Differential Equations publication_identifier: issn: - 1040-7294 - 1572-9222 publication_status: published publisher: Springer Science and Business Media LLC status: public title: Slow Grow-up in a Quasilinear Keller–Segel System type: journal_article user_id: '31496' volume: 36 year: '2022' ... --- _id: '63272' 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 solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension. Journal of Differential Equations. 2022;343:390-418. doi:10.1016/j.jde.2022.10.022 apa: Tao, Y., & Winkler, M. (2022). Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension. Journal of Differential Equations, 343, 390–418. https://doi.org/10.1016/j.jde.2022.10.022 bibtex: '@article{Tao_Winkler_2022, title={Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension}, volume={343}, DOI={10.1016/j.jde.2022.10.022}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Tao, Youshan and Winkler, Michael}, year={2022}, pages={390–418} }' chicago: 'Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary Dimension.” Journal of Differential Equations 343 (2022): 390–418. https://doi.org/10.1016/j.jde.2022.10.022.' ieee: 'Y. Tao and M. Winkler, “Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension,” Journal of Differential Equations, vol. 343, pp. 390–418, 2022, doi: 10.1016/j.jde.2022.10.022.' mla: Tao, Youshan, and Michael Winkler. “Global Solutions to a Keller-Segel-Consumption System Involving Singularly Signal-Dependent Motilities in Domains of Arbitrary Dimension.” Journal of Differential Equations, vol. 343, Elsevier BV, 2022, pp. 390–418, doi:10.1016/j.jde.2022.10.022. short: Y. Tao, M. Winkler, Journal of Differential Equations 343 (2022) 390–418. date_created: 2025-12-18T19:13:04Z date_updated: 2025-12-18T20:11:02Z doi: 10.1016/j.jde.2022.10.022 intvolume: ' 343' language: - iso: eng page: 390-418 publication: Journal of Differential Equations publication_identifier: issn: - 0022-0396 publication_status: published publisher: Elsevier BV status: public title: Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension type: journal_article user_id: '31496' volume: 343 year: '2022' ... --- _id: '63268' article_number: '113153' author: - first_name: Laurent full_name: Desvillettes, Laurent last_name: Desvillettes - first_name: Philippe full_name: Laurençot, Philippe last_name: Laurençot - first_name: Ariane full_name: Trescases, Ariane last_name: Trescases - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Desvillettes L, Laurençot P, Trescases A, Winkler M. Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing. Nonlinear Analysis. 2022;226. doi:10.1016/j.na.2022.113153 apa: Desvillettes, L., Laurençot, P., Trescases, A., & Winkler, M. (2022). Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing. Nonlinear Analysis, 226, Article 113153. https://doi.org/10.1016/j.na.2022.113153 bibtex: '@article{Desvillettes_Laurençot_Trescases_Winkler_2022, title={Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing}, volume={226}, DOI={10.1016/j.na.2022.113153}, number={113153}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Desvillettes, Laurent and Laurençot, Philippe and Trescases, Ariane and Winkler, Michael}, year={2022} }' chicago: Desvillettes, Laurent, Philippe Laurençot, Ariane Trescases, and Michael Winkler. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis with Local Sensing.” Nonlinear Analysis 226 (2022). https://doi.org/10.1016/j.na.2022.113153. ieee: 'L. Desvillettes, P. Laurençot, A. Trescases, and M. Winkler, “Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing,” Nonlinear Analysis, vol. 226, Art. no. 113153, 2022, doi: 10.1016/j.na.2022.113153.' mla: Desvillettes, Laurent, et al. “Weak Solutions to Triangular Cross Diffusion Systems Modeling Chemotaxis with Local Sensing.” Nonlinear Analysis, vol. 226, 113153, Elsevier BV, 2022, doi:10.1016/j.na.2022.113153. short: L. Desvillettes, P. Laurençot, A. Trescases, M. Winkler, Nonlinear Analysis 226 (2022). date_created: 2025-12-18T19:11:16Z date_updated: 2025-12-18T20:10:32Z doi: 10.1016/j.na.2022.113153 intvolume: ' 226' language: - iso: eng publication: Nonlinear Analysis publication_identifier: issn: - 0362-546X publication_status: published publisher: Elsevier BV status: public title: Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing type: journal_article user_id: '31496' volume: 226 year: '2022' ... --- _id: '63278' abstract: - lang: eng text: "Abstract\r\n The Neumann problem for (0.1)$$ \\begin{align}& V_t = \\Delta V-aV+f(x,t) \\end{align}$$is considered in bounded domains $\\Omega \\subset {\\mathbb {R}}^n$ with smooth boundary, where $n\\ge 1$ and $a\\in {\\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \\begin{eqnarray*} \\sup_{t\\in (0,T)} \\int_\\Omega \\big|\\nabla V(\\cdot,t)\\big|^p L^{\\frac{n+p}{n+2}} \\Big( \\big|\\nabla V(\\cdot,t)\\big| \\Big) \\end{eqnarray*}$$in dependence on the expressions (0.2)$$ \\begin{align}& \\sup_{t\\in (0,T-\\tau)} \\int_t^{t+\\tau} \\int_\\Omega |f|^{\\frac{(n+2)p}{n+p}} L\\big( |f|\\big) \\end{align}$$is derived for $p\\ge 2$, $\\tau>0$, and $T\\ge 2\\tau $, provided that $L\\in C^0([0,\\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\\limsup _{s\\to \\infty } \\frac {L(s^{\\lambda _0})}{L(s)} <\\infty $ for some $\\lambda _0>1$. In the particular case when $p=n\\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\\xi ):=\\ln ^\\alpha (\\xi +b)$ whenever $b>0$ and $\\alpha>\\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big( D(v)\\nabla u\\big) - \\nabla \\cdot \\big( uS(v)\\nabla v\\big) + ru - \\mu u^2, \\\\[1mm] v_t = \\Delta v-v+u, \\end{array} \\right. \\end{eqnarray*}$$shows that when $n=2$, $r\\in {\\mathbb {R}}$, $0<D\\in C^2([0,\\infty ))$, and $S\\in C^2([0,\\infty )) \\cap W^{1,\\infty }((0,\\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\\mu>0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution." author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. International Mathematics Research Notices. 2022;2023(19):16336-16393. doi:10.1093/imrn/rnac286 apa: Winkler, M. (2022). A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System. International Mathematics Research Notices, 2023(19), 16336–16393. https://doi.org/10.1093/imrn/rnac286 bibtex: '@article{Winkler_2022, title={A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System}, volume={2023}, DOI={10.1093/imrn/rnac286}, number={19}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Winkler, Michael}, year={2022}, pages={16336–16393} }' chicago: 'Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” International Mathematics Research Notices 2023, no. 19 (2022): 16336–93. https://doi.org/10.1093/imrn/rnac286.' ieee: 'M. Winkler, “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System,” International Mathematics Research Notices, vol. 2023, no. 19, pp. 16336–16393, 2022, doi: 10.1093/imrn/rnac286.' mla: Winkler, Michael. “A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System.” International Mathematics Research Notices, vol. 2023, no. 19, Oxford University Press (OUP), 2022, pp. 16336–93, doi:10.1093/imrn/rnac286. short: M. Winkler, International Mathematics Research Notices 2023 (2022) 16336–16393. date_created: 2025-12-18T19:15:52Z date_updated: 2025-12-18T20:11:43Z doi: 10.1093/imrn/rnac286 intvolume: ' 2023' issue: '19' language: - iso: eng page: 16336-16393 publication: International Mathematics Research Notices publication_identifier: issn: - 1073-7928 - 1687-0247 publication_status: published publisher: Oxford University Press (OUP) status: public title: A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System type: journal_article user_id: '31496' volume: 2023 year: '2022' ... --- _id: '63279' abstract: - lang: eng text: "\r\n In a smoothly bounded convex domain\r\n \r\n \ \\Omega \\subset \\mathbb{R}^3\r\n \ \r\n , we consider the chemotaxis-Navier–Stokes model\r\n \r\n \r\n \ \r\n \\begin{cases} n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in \\Omega, \\, t>0, \\\\ c_t + u\\cdot\\nabla c = \\Delta c -nc, & x\\in \\Omega, \\, t>0, \\\\ u_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\Phi, \\quad \\nabla\\cdot u=0, & x\\in \\Omega, \\, t>0, \\end{cases} \\quad (\\star)\r\n \r\n \r\n \ \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 \r\n \ \\Phi\r\n \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 \r\n \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 \r\n (0,\\infty)\r\n \ \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 \ \r\n \\overline{\\Omega}\\times E\r\n \r\n , where\r\n \r\n E\r\n \ \r\n denotes a countable union of open intervals which is such that\r\n \r\n \ |(0,\\infty)\\setminus E|=0\r\n \ \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 \r\n \\star\r\n \ \r\n ).\r\n " author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? Journal of the European Mathematical Society. 2022;25(4):1423-1456. doi:10.4171/jems/1226 apa: Winkler, M. (2022). Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? Journal of the European Mathematical Society, 25(4), 1423–1456. https://doi.org/10.4171/jems/1226 bibtex: '@article{Winkler_2022, title={Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?}, volume={25}, DOI={10.4171/jems/1226}, number={4}, journal={Journal of the European Mathematical Society}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Winkler, Michael}, year={2022}, pages={1423–1456} }' chicago: 'Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society 25, no. 4 (2022): 1423–56. https://doi.org/10.4171/jems/1226.' ieee: 'M. Winkler, “Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?,” Journal of the European Mathematical Society, vol. 25, no. 4, pp. 1423–1456, 2022, doi: 10.4171/jems/1226.' mla: Winkler, Michael. “Does Leray’s Structure Theorem Withstand Buoyancy-Driven Chemotaxis-Fluid Interaction?” Journal of the European Mathematical Society, vol. 25, no. 4, European Mathematical Society - EMS - Publishing House GmbH, 2022, pp. 1423–56, doi:10.4171/jems/1226. short: M. Winkler, Journal of the European Mathematical Society 25 (2022) 1423–1456. date_created: 2025-12-18T19:16:13Z date_updated: 2025-12-18T20:11:51Z doi: 10.4171/jems/1226 intvolume: ' 25' issue: '4' language: - iso: eng page: 1423-1456 publication: Journal of the European Mathematical Society publication_identifier: issn: - 1435-9855 - 1435-9863 publication_status: published publisher: European Mathematical Society - EMS - Publishing House GmbH status: public title: Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction? type: journal_article user_id: '31496' volume: 25 year: '2022' ... --- _id: '63274' abstract: - lang: eng text: "In a ball

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

with

$n\\ge 2$

, the chemotaxis system\r\n

\\[ \\left\\{ \\begin{array}{@{}l} u_t = \\nabla \\cdot \\big( D(u)\\nabla u\\big) + \\nabla\\cdot \\big(\\dfrac{u}{v} \\nabla v\\big), \\\\ 0=\\Delta v - uv \\end{array} \\right. \\]

is considered along with no-flux boundary conditions for

$u$

and with prescribed constant positive Dirichlet boundary data for

$v$

. It is shown that if

$D\\in C^{3}([0,\\infty ))$

is such that

$0< D(\\xi ) \\le {K_D} (\\xi +1)^{-\\alpha }$

for all

$\\xi >0$

with some

${K_D}>0$

and

$\\alpha >0$

, then for all initial data from a considerably large set of radial functions on

$\\Omega$

, the corresponding initial-boundary value problem admits a solution blowing up in finite time." 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. Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2022;153(4):1150-1166. doi:10.1017/prm.2022.39' apa: 'Wang, Y., & Winkler, M. (2022). Finite-time blow-up in a repulsive chemotaxis-consumption system. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153(4), 1150–1166. https://doi.org/10.1017/prm.2022.39' bibtex: '@article{Wang_Winkler_2022, title={Finite-time blow-up in a repulsive chemotaxis-consumption system}, volume={153}, DOI={10.1017/prm.2022.39}, number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Wang, Yulan and Winkler, Michael}, year={2022}, pages={1150–1166} }' chicago: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153, no. 4 (2022): 1150–66. https://doi.org/10.1017/prm.2022.39.' ieee: 'Y. Wang and M. Winkler, “Finite-time blow-up in a repulsive chemotaxis-consumption system,” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, pp. 1150–1166, 2022, doi: 10.1017/prm.2022.39.' mla: 'Wang, Yulan, and Michael Winkler. “Finite-Time Blow-up in a Repulsive Chemotaxis-Consumption System.” Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol. 153, no. 4, Cambridge University Press (CUP), 2022, pp. 1150–66, doi:10.1017/prm.2022.39.' short: 'Y. Wang, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 153 (2022) 1150–1166.' date_created: 2025-12-18T19:14:20Z date_updated: 2025-12-18T20:11:15Z doi: 10.1017/prm.2022.39 intvolume: ' 153' issue: '4' language: - iso: eng page: 1150-1166 publication: 'Proceedings of the Royal Society of Edinburgh: Section A Mathematics' publication_identifier: issn: - 0308-2105 - 1473-7124 publication_status: published publisher: Cambridge University Press (CUP) status: public title: Finite-time blow-up in a repulsive chemotaxis-consumption system type: journal_article user_id: '31496' volume: 153 year: '2022' ... --- _id: '63282' abstract: - lang: eng text: ' The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text] ' article_number: '2250062' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems. Communications in Contemporary Mathematics. 2022;25(10). doi:10.1142/s0219199722500626 apa: Winkler, M. (2022). Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems. Communications in Contemporary Mathematics, 25(10), Article 2250062. https://doi.org/10.1142/s0219199722500626 bibtex: '@article{Winkler_2022, title={Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems}, volume={25}, DOI={10.1142/s0219199722500626}, number={102250062}, journal={Communications in Contemporary Mathematics}, publisher={World Scientific Pub Co Pte Ltd}, author={Winkler, Michael}, year={2022} }' chicago: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel Systems.” Communications in Contemporary Mathematics 25, no. 10 (2022). https://doi.org/10.1142/s0219199722500626. ieee: 'M. Winkler, “Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems,” Communications in Contemporary Mathematics, vol. 25, no. 10, Art. no. 2250062, 2022, doi: 10.1142/s0219199722500626.' mla: Winkler, Michael. “Arbitrarily Fast Grow-up Rates in Quasilinear Keller–Segel Systems.” Communications in Contemporary Mathematics, vol. 25, no. 10, 2250062, World Scientific Pub Co Pte Ltd, 2022, doi:10.1142/s0219199722500626. short: M. Winkler, Communications in Contemporary Mathematics 25 (2022). date_created: 2025-12-18T19:17:23Z date_updated: 2025-12-18T20:12:13Z doi: 10.1142/s0219199722500626 intvolume: ' 25' issue: '10' language: - iso: eng publication: Communications in Contemporary Mathematics publication_identifier: issn: - 0219-1997 - 1793-6683 publication_status: published publisher: World Scientific Pub Co Pte Ltd status: public title: Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems type: journal_article user_id: '31496' volume: 25 year: '2022' ... --- _id: '35615' article_number: '72' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system. Zeitschrift für Angewandte Mathematik und Physik. 2021;72. apa: Winkler, M. (2021). Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system. Zeitschrift Für Angewandte Mathematik Und Physik, 72, Article 72. bibtex: '@article{Winkler_2021, title={Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system.}, volume={72}, number={72}, journal={Zeitschrift für Angewandte Mathematik und Physik}, author={Winkler, Michael}, year={2021} }' chicago: Winkler, Michael. “Suppressing Blow-up by Gradient-Dependent Flux Limitation in a Planar Keller-Segel-Navier-Stokes System.” Zeitschrift Für Angewandte Mathematik Und Physik 72 (2021). ieee: M. Winkler, “Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system.,” Zeitschrift für Angewandte Mathematik und Physik, vol. 72, Art. no. 72, 2021. mla: Winkler, Michael. “Suppressing Blow-up by Gradient-Dependent Flux Limitation in a Planar Keller-Segel-Navier-Stokes System.” Zeitschrift Für Angewandte Mathematik Und Physik, vol. 72, 72, 2021. short: M. Winkler, Zeitschrift Für Angewandte Mathematik Und Physik 72 (2021). date_created: 2023-01-09T18:16:56Z date_updated: 2023-01-20T13:14:53Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 72' language: - iso: eng publication: Zeitschrift für Angewandte Mathematik und Physik status: public title: Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system. type: journal_article user_id: '15645' volume: 72 year: '2021' ... --- _id: '35614' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems? International Mathematics Research Notices. 2021;2021:8106-8152. apa: Winkler, M. (2021). Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems? International Mathematics Research Notices, 2021, 8106–8152. bibtex: '@article{Winkler_2021, title={Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems?}, volume={2021}, journal={International Mathematics Research Notices}, author={Winkler, Michael}, year={2021}, pages={8106–8152} }' chicago: 'Winkler, Michael. “Can Rotational Fluxes Impede the Tendency toward Spatial Homogeneity in Nutrient Taxis (-Stokes) Systems?” International Mathematics Research Notices 2021 (2021): 8106–52.' ieee: M. Winkler, “Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems?,” International Mathematics Research Notices, vol. 2021, pp. 8106–8152, 2021. mla: Winkler, Michael. “Can Rotational Fluxes Impede the Tendency toward Spatial Homogeneity in Nutrient Taxis (-Stokes) Systems?” International Mathematics Research Notices, vol. 2021, 2021, pp. 8106–52. short: M. Winkler, International Mathematics Research Notices 2021 (2021) 8106–8152. date_created: 2023-01-09T18:14:20Z date_updated: 2023-01-20T13:14:58Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 2021' language: - iso: eng page: 8106-8152 publication: International Mathematics Research Notices status: public title: Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems? type: journal_article user_id: '15645' volume: 2021 year: '2021' ... --- _id: '35617' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. Applied Mathematics Letters. 2021;112:106785. apa: Winkler, M. (2021). Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. Applied Mathematics Letters, 112, 106785. bibtex: '@article{Winkler_2021, title={Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity.}, volume={112}, journal={Applied Mathematics Letters}, author={Winkler, Michael}, year={2021}, pages={106785} }' chicago: 'Winkler, Michael. “Boundedness in a Three-Dimensional Keller-Segel-Stokes System with Subcritical Sensitivity.” Applied Mathematics Letters 112 (2021): 106785.' ieee: M. Winkler, “Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity.,” Applied Mathematics Letters, vol. 112, p. 106785, 2021. mla: Winkler, Michael. “Boundedness in a Three-Dimensional Keller-Segel-Stokes System with Subcritical Sensitivity.” Applied Mathematics Letters, vol. 112, 2021, p. 106785. short: M. Winkler, Applied Mathematics Letters 112 (2021) 106785. date_created: 2023-01-09T18:21:39Z date_updated: 2023-01-20T13:14:48Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 112' language: - iso: eng page: '106785' publication: Applied Mathematics Letters status: public title: Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. type: journal_article user_id: '15645' volume: 112 year: '2021' ... --- _id: '35613' author: - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler citation: ama: Winkler M. Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. Transactions of the American Mathematical Society. 2021;374:219-268. apa: Winkler, M. (2021). Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. Transactions of the American Mathematical Society, 374, 219–268. bibtex: '@article{Winkler_2021, title={Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow.}, volume={374}, journal={Transactions of the American Mathematical Society}, author={Winkler, Michael}, year={2021}, pages={219–268} }' chicago: 'Winkler, Michael. “Does Spatial Homogeneity Ultimately Prevail in Nutrient Taxis Systems? A Paradigm for Structure Support by Rapid Diffusion Decay in an Autonomous Parabolic Flow.” Transactions of the American Mathematical Society 374 (2021): 219–68.' ieee: M. Winkler, “Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow.,” Transactions of the American Mathematical Society, vol. 374, pp. 219–268, 2021. mla: Winkler, Michael. “Does Spatial Homogeneity Ultimately Prevail in Nutrient Taxis Systems? A Paradigm for Structure Support by Rapid Diffusion Decay in an Autonomous Parabolic Flow.” Transactions of the American Mathematical Society, vol. 374, 2021, pp. 219–68. short: M. Winkler, Transactions of the American Mathematical Society 374 (2021) 219–268. date_created: 2023-01-09T18:11:15Z date_updated: 2023-01-20T13:15:03Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 374' language: - iso: eng page: 219-268 publication: Transactions of the American Mathematical Society status: public title: Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. type: journal_article user_id: '15645' volume: 374 year: '2021' ... --- _id: '35603' article_number: '109069' 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 fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. Journal of Functional Analysis. 2021;281.' apa: 'Tao, Y., & Winkler, M. (2021). A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. Journal of Functional Analysis, 281, Article 109069.' bibtex: '@article{Tao_Winkler_2021, title={A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation.}, volume={281}, number={109069}, journal={Journal of Functional Analysis}, author={Tao, Youshan and Winkler, Michael}, year={2021} }' chicago: 'Tao, Youshan, and Michael Winkler. “A Fully Cross-Diffusive Two-Component Evolution System: Existence and Qualitative Analysis via Entropy-Consistent Thin-Film-Type Approximation.” Journal of Functional Analysis 281 (2021).' ieee: 'Y. Tao and M. Winkler, “A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation.,” Journal of Functional Analysis, vol. 281, Art. no. 109069, 2021.' mla: 'Tao, Youshan, and Michael Winkler. “A Fully Cross-Diffusive Two-Component Evolution System: Existence and Qualitative Analysis via Entropy-Consistent Thin-Film-Type Approximation.” Journal of Functional Analysis, vol. 281, 109069, 2021.' short: Y. Tao, M. Winkler, Journal of Functional Analysis 281 (2021). date_created: 2023-01-09T17:28:08Z date_updated: 2023-02-01T10:18:18Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 281' language: - iso: eng publication: Journal of Functional Analysis status: public title: 'A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation.' type: journal_article user_id: '15645' volume: 281 year: '2021' ... --- _id: '35612' author: - first_name: Yulan full_name: Wang, Yulan last_name: Wang - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler - first_name: Zhaoyin full_name: Xiang, Zhaoyin last_name: Xiang citation: ama: Wang Y, Winkler M, Xiang Z. Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. Science China Mathematics. 2021;64:725-746. apa: Wang, Y., Winkler, M., & Xiang, Z. (2021). Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. Science China Mathematics, 64, 725–746. bibtex: '@article{Wang_Winkler_Xiang_2021, title={Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption.}, volume={64}, journal={Science China Mathematics}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2021}, pages={725–746} }' chicago: 'Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “Immediate Regularization of Measure-Type Population Densities in a Two-Dimensional Chemotaxis System with Signal Consumption.” Science China Mathematics 64 (2021): 725–46.' ieee: Y. Wang, M. Winkler, and Z. Xiang, “Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption.,” Science China Mathematics, vol. 64, pp. 725–746, 2021. mla: Wang, Yulan, et al. “Immediate Regularization of Measure-Type Population Densities in a Two-Dimensional Chemotaxis System with Signal Consumption.” Science China Mathematics, vol. 64, 2021, pp. 725–46. short: Y. Wang, M. Winkler, Z. Xiang, Science China Mathematics 64 (2021) 725–746. date_created: 2023-01-09T18:06:08Z date_updated: 2023-02-01T10:22:22Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 64' language: - iso: eng page: 725-746 publication: Science China Mathematics status: public title: Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. type: journal_article user_id: '15645' volume: 64 year: '2021' ... --- _id: '35605' 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 two-dimensional cross-diffusion system modeling propagation of urban crime. Communications in Mathematical Sciences. 2021;19:829-849. apa: Tao, Y., & Winkler, M. (2021). Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. Communications in Mathematical Sciences, 19, 829–849. bibtex: '@article{Tao_Winkler_2021, title={Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime.}, volume={19}, journal={Communications in Mathematical Sciences}, author={Tao, Youshan and Winkler, Michael}, year={2021}, pages={829–849} }' chicago: 'Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Two-Dimensional Cross-Diffusion System Modeling Propagation of Urban Crime.” Communications in Mathematical Sciences 19 (2021): 829–49.' ieee: Y. Tao and M. Winkler, “Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime.,” Communications in Mathematical Sciences, vol. 19, pp. 829–849, 2021. mla: Tao, Youshan, and Michael Winkler. “Global Smooth Solutions in a Two-Dimensional Cross-Diffusion System Modeling Propagation of Urban Crime.” Communications in Mathematical Sciences, vol. 19, 2021, pp. 829–49. short: Y. Tao, M. Winkler, Communications in Mathematical Sciences 19 (2021) 829–849. date_created: 2023-01-09T17:32:36Z date_updated: 2023-02-01T10:23:03Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 19' language: - iso: eng page: 829-849 publication: Communications in Mathematical Sciences status: public title: Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. type: journal_article user_id: '15645' volume: 19 year: '2021' ... --- _id: '35610' author: - first_name: Yulan full_name: Wang, Yulan last_name: Wang - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler - first_name: Zhaoyin full_name: Xiang, Zhaoyin last_name: Xiang citation: ama: Wang Y, Winkler M, Xiang Z. Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary. Communications in Partial Differential Equations. 2021;46:1058-1091. apa: Wang, Y., Winkler, M., & Xiang, Z. (2021). Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary. Communications in Partial Differential Equations, 46, 1058–1091. bibtex: '@article{Wang_Winkler_Xiang_2021, title={Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary.}, volume={46}, journal={Communications in Partial Differential Equations}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2021}, pages={1058–1091} }' chicago: 'Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “Local Energy Estimates and Global Solvability in a Threee-Dimensional Chemotaxis-Fluid System with Prescribed Signal on the Boundary.” Communications in Partial Differential Equations 46 (2021): 1058–91.' ieee: Y. Wang, M. Winkler, and Z. Xiang, “Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary.,” Communications in Partial Differential Equations, vol. 46, pp. 1058–1091, 2021. mla: Wang, Yulan, et al. “Local Energy Estimates and Global Solvability in a Threee-Dimensional Chemotaxis-Fluid System with Prescribed Signal on the Boundary.” Communications in Partial Differential Equations, vol. 46, 2021, pp. 1058–91. short: Y. Wang, M. Winkler, Z. Xiang, Communications in Partial Differential Equations 46 (2021) 1058–1091. date_created: 2023-01-09T18:00:41Z date_updated: 2023-02-01T10:25:34Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 46' language: - iso: eng page: 1058-1091 publication: Communications in Partial Differential Equations status: public title: Local energy estimates and global solvability in a threee-dimensional chemotaxis-fluid system with prescribed signal on the boundary. type: journal_article user_id: '15645' volume: 46 year: '2021' ... --- _id: '35611' author: - first_name: Yulan full_name: Wang, Yulan last_name: Wang - first_name: Michael full_name: Winkler, Michael id: '31496' last_name: Winkler - first_name: Zhaoyin full_name: Xiang, Zhaoyin last_name: Xiang citation: ama: Wang Y, Winkler M, Xiang Z. Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation. Advances in Nonlinear Analysis. 2021;10:707-731. apa: Wang, Y., Winkler, M., & Xiang, Z. (2021). Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation. Advances in Nonlinear Analysis, 10, 707–731. bibtex: '@article{Wang_Winkler_Xiang_2021, title={Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation}, volume={10}, journal={Advances in Nonlinear Analysis}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2021}, pages={707–731} }' chicago: 'Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “Global Solvability in a Threee-Dimensional Keller-Segel.Stokes System Involving Arbitrary Superlinear Logistic Degradation.” Advances in Nonlinear Analysis 10 (2021): 707–31.' ieee: Y. Wang, M. Winkler, and Z. Xiang, “Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation,” Advances in Nonlinear Analysis, vol. 10, pp. 707–731, 2021. mla: Wang, Yulan, et al. “Global Solvability in a Threee-Dimensional Keller-Segel.Stokes System Involving Arbitrary Superlinear Logistic Degradation.” Advances in Nonlinear Analysis, vol. 10, 2021, pp. 707–31. short: Y. Wang, M. Winkler, Z. Xiang, Advances in Nonlinear Analysis 10 (2021) 707–731. date_created: 2023-01-09T18:03:45Z date_updated: 2023-02-01T10:27:17Z department: - _id: '34' - _id: '10' - _id: '90' intvolume: ' 10' language: - iso: eng page: 707-731 publication: Advances in Nonlinear Analysis status: public title: Global solvability in a threee-dimensional Keller-Segel.Stokes system involving arbitrary superlinear logistic degradation type: journal_article user_id: '15645' volume: 10 year: '2021' ...