[{"user_id":"31496","volume":65,"publisher":"Springer Science and Business Media LLC","_id":"63246","status":"public","project":[{"_id":"245","name":"FOR 5208: Modellbasierte Bestimmung nichtlinearer Eigenschaften von Piezokeramiken für Leistungsschallanwendungen (NEPTUN)"}],"citation":{"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} }","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","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.","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.","short":"M. Winkler, Calculus of Variations and Partial Differential Equations 65 (2025).","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.","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"},"doi":"10.1007/s00526-025-03170-8","article_number":"1","language":[{"iso":"eng"}],"date_updated":"2026-04-23T12:18:59Z","publication_status":"published","intvolume":" 65","year":"2025","title":"Rough solutions in one-dimensional nonlinear thermoelasticity","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"publication_identifier":{"issn":["0944-2669","1432-0835"]},"type":"journal_article","date_created":"2025-12-18T19:01:02Z","abstract":[{"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 ","lang":"eng"}],"publication":"Calculus of Variations and Partial Differential Equations","issue":"1"},{"citation":{"bibtex":"@article{Ahn_Winkler_2023, title={A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system}, volume={62}, DOI={10.1007/s00526-023-02523-5}, number={6180}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Ahn, Jaewook and Winkler, Michael}, year={2023} }","ama":"Ahn J, Winkler M. A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system. Calculus of Variations and Partial Differential Equations. 2023;62(6). doi:10.1007/s00526-023-02523-5","mla":"Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential Equations, vol. 62, no. 6, 180, Springer Science and Business Media LLC, 2023, doi:10.1007/s00526-023-02523-5.","short":"J. Ahn, M. Winkler, Calculus of Variations and Partial Differential Equations 62 (2023).","chicago":"Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential Equations 62, no. 6 (2023). https://doi.org/10.1007/s00526-023-02523-5.","ieee":"J. Ahn and M. Winkler, “A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system,” Calculus of Variations and Partial Differential Equations, vol. 62, no. 6, Art. no. 180, 2023, doi: 10.1007/s00526-023-02523-5.","apa":"Ahn, J., & Winkler, M. (2023). A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system. Calculus of Variations and Partial Differential Equations, 62(6), Article 180. https://doi.org/10.1007/s00526-023-02523-5"},"volume":62,"user_id":"31496","_id":"53324","publisher":"Springer Science and Business Media LLC","status":"public","keyword":["Applied Mathematics","Analysis"],"type":"journal_article","date_created":"2024-04-07T12:40:02Z","publication":"Calculus of Variations and Partial Differential Equations","issue":"6","doi":"10.1007/s00526-023-02523-5","language":[{"iso":"eng"}],"article_number":"180","intvolume":" 62","date_updated":"2024-04-07T12:40:06Z","publication_status":"published","publication_identifier":{"issn":["0944-2669","1432-0835"]},"author":[{"full_name":"Ahn, Jaewook","last_name":"Ahn","first_name":"Jaewook"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"title":"A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system","year":"2023"},{"issue":"6","publication":"Calculus of Variations and Partial Differential Equations","citation":{"apa":"Ahn, J., & Winkler, M. (2023). A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system. Calculus of Variations and Partial Differential Equations, 62(6), Article 180. https://doi.org/10.1007/s00526-023-02523-5","mla":"Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential Equations, vol. 62, no. 6, 180, Springer Science and Business Media LLC, 2023, doi:10.1007/s00526-023-02523-5.","ieee":"J. Ahn and M. Winkler, “A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system,” Calculus of Variations and Partial Differential Equations, vol. 62, no. 6, Art. no. 180, 2023, doi: 10.1007/s00526-023-02523-5.","ama":"Ahn J, Winkler M. A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system. Calculus of Variations and Partial Differential Equations. 2023;62(6). doi:10.1007/s00526-023-02523-5","short":"J. Ahn, M. Winkler, Calculus of Variations and Partial Differential Equations 62 (2023).","chicago":"Ahn, Jaewook, and Michael Winkler. “A Critical Exponent for Blow-up in a Two-Dimensional Chemotaxis-Consumption System.” Calculus of Variations and Partial Differential Equations 62, no. 6 (2023). https://doi.org/10.1007/s00526-023-02523-5.","bibtex":"@article{Ahn_Winkler_2023, title={A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system}, volume={62}, DOI={10.1007/s00526-023-02523-5}, number={6180}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Ahn, Jaewook and Winkler, Michael}, year={2023} }"},"date_created":"2025-12-18T19:10:55Z","type":"journal_article","year":"2023","title":"A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system","status":"public","publication_identifier":{"issn":["0944-2669","1432-0835"]},"author":[{"full_name":"Ahn, Jaewook","last_name":"Ahn","first_name":"Jaewook"},{"full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael","id":"31496"}],"publication_status":"published","date_updated":"2025-12-18T20:10:21Z","intvolume":" 62","article_number":"180","language":[{"iso":"eng"}],"_id":"63267","publisher":"Springer Science and Business Media LLC","user_id":"31496","doi":"10.1007/s00526-023-02523-5","volume":62},{"date_created":"2022-12-21T09:50:59Z","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"keyword":["Applied Mathematics","Analysis"],"type":"journal_article","publication":"Calculus of Variations and Partial Differential Equations","issue":"3","language":[{"iso":"eng"}],"article_number":"96","doi":"10.1007/s00526-022-02201-y","author":[{"id":"23686","orcid":"0000-0001-9963-0800","first_name":"Tobias","last_name":"Black","full_name":"Black, Tobias"},{"full_name":"Wu, Chunyan","last_name":"Wu","first_name":"Chunyan"}],"publication_identifier":{"issn":["0944-2669","1432-0835"]},"title":"Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation","year":"2022","intvolume":" 61","date_updated":"2023-07-10T11:37:27Z","publication_status":"published","citation":{"ieee":"T. Black and C. Wu, “Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation,” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, Art. no. 96, 2022, doi: 10.1007/s00526-022-02201-y.","apa":"Black, T., & Wu, C. (2022). Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation. Calculus of Variations and Partial Differential Equations, 61(3), Article 96. https://doi.org/10.1007/s00526-022-02201-y","short":"T. Black, C. Wu, Calculus of Variations and Partial Differential Equations 61 (2022).","chicago":"Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic Proliferation.” Calculus of Variations and Partial Differential Equations 61, no. 3 (2022). https://doi.org/10.1007/s00526-022-02201-y.","mla":"Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic Proliferation.” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, 96, Springer Science and Business Media LLC, 2022, doi:10.1007/s00526-022-02201-y.","bibtex":"@article{Black_Wu_2022, title={Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation}, volume={61}, DOI={10.1007/s00526-022-02201-y}, number={396}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Wu, Chunyan}, year={2022} }","ama":"Black T, Wu C. Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation. Calculus of Variations and Partial Differential Equations. 2022;61(3). doi:10.1007/s00526-022-02201-y"},"publisher":"Springer Science and Business Media LLC","_id":"34677","volume":61,"user_id":"23686","status":"public"},{"doi":"10.1007/s00526-021-02168-2","article_number":"108","language":[{"iso":"eng"}],"date_updated":"2025-12-18T20:04:43Z","publication_status":"published","intvolume":" 61","year":"2022","title":"Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar","publication_identifier":{"issn":["0944-2669","1432-0835"]},"author":[{"full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael","id":"31496"}],"type":"journal_article","date_created":"2025-12-18T19:26:32Z","abstract":[{"lang":"eng","text":"AbstractA no-flux initial-boundary value problem for the doubly degenrate parabolic system $$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot \\big ( uv\\nabla u\\big ) + \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n v\r\n ∇\r\n u\r\n \r\n )\r\n \r\n +\r\n ℓ\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 v\r\n t\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 \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 convex domain $$\\Omega \\subset \\mathbb {R}^n$$\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 , with $$n\\ge 1$$\r\n \r\n n\r\n ≥\r\n 1\r\n \r\n and $$\\ell \\ge 0$$\r\n \r\n ℓ\r\n ≥\r\n 0\r\n \r\n . The first of the main results asserts that for nonnegative initial data $$(u_0,v_0)\\in (L^\\infty (\\Omega ))^2$$\r\n \r\n \r\n (\r\n \r\n u\r\n 0\r\n \r\n ,\r\n \r\n v\r\n 0\r\n \r\n )\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 2\r\n \r\n \r\n with $$u_0\\not \\equiv 0$$\r\n \r\n \r\n u\r\n 0\r\n \r\n ≢\r\n 0\r\n \r\n , $$v_0\\not \\equiv 0$$\r\n \r\n \r\n v\r\n 0\r\n \r\n ≢\r\n 0\r\n \r\n and $$\\sqrt{v_0}\\in W^{1,2}(\\Omega )$$\r\n \r\n \r\n \r\n v\r\n 0\r\n \r\n \r\n ∈\r\n \r\n W\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 , there exists a global weak solution (u, v) which, inter alia, belongs to $$C^0(\\overline{\\Omega }\\times (0,\\infty )) \\times C^{2,1}(\\overline{\\Omega }\\times (0,\\infty ))$$\r\n \r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n Ω\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 C\r\n \r\n 2\r\n ,\r\n 1\r\n \r\n \r\n \r\n (\r\n \r\n Ω\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 and satisfies $$\\sup _{t>0} \\Vert u(\\cdot ,t)\\Vert _{L^p(\\Omega )}<\\infty $$\r\n \r\n \r\n sup\r\n \r\n t\r\n >\r\n 0\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 p\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 for all $$p\\in [1,p_0)$$\r\n \r\n p\r\n ∈\r\n [\r\n 1\r\n ,\r\n \r\n p\r\n 0\r\n \r\n )\r\n \r\n with $$p_0:=\\frac{n}{(n-2)_+}$$\r\n \r\n \r\n p\r\n 0\r\n \r\n :\r\n =\r\n \r\n n\r\n \r\n \r\n (\r\n n\r\n -\r\n 2\r\n )\r\n \r\n +\r\n \r\n \r\n \r\n . It is next seen that for each of these solutions one can find $$u_\\infty \\in \\bigcap _{p\\in [1,p_0)} L^p(\\Omega )$$\r\n \r\n \r\n u\r\n ∞\r\n \r\n ∈\r\n \r\n ⋂\r\n \r\n p\r\n ∈\r\n [\r\n 1\r\n ,\r\n \r\n p\r\n 0\r\n \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 such that, within an appropriate topological setting, $$(u(\\cdot ,t),v(\\cdot ,t))$$\r\n \r\n (\r\n u\r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n ,\r\n v\r\n (\r\n ·\r\n ,\r\n t\r\n )\r\n )\r\n \r\n approaches the equilibrium $$(u_\\infty ,0)$$\r\n \r\n (\r\n \r\n u\r\n ∞\r\n \r\n ,\r\n 0\r\n )\r\n \r\n in the large time limit. Finally, in the case $$n\\le 5$$\r\n \r\n n\r\n ≤\r\n 5\r\n \r\n a result ensuring a certain stability property of any member in the uncountably large family of steady states $$(u_0,0)$$\r\n \r\n (\r\n \r\n u\r\n 0\r\n \r\n ,\r\n 0\r\n )\r\n \r\n , with arbitrary and suitably regular $$u_0:\\Omega \\rightarrow [0,\\infty )$$\r\n \r\n \r\n u\r\n 0\r\n \r\n :\r\n Ω\r\n →\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n \r\n , is derived. This provides some rigorous evidence for the appropriateness of ($$\\star $$\r\n ⋆\r\n ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in ($$\\star $$\r\n ⋆\r\n )."}],"issue":"3","publication":"Calculus of Variations and Partial Differential Equations","user_id":"31496","volume":61,"_id":"63305","publisher":"Springer Science and Business Media LLC","status":"public","citation":{"apa":"Winkler, M. (2022). Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calculus of Variations and Partial Differential Equations, 61(3), Article 108. https://doi.org/10.1007/s00526-021-02168-2","ieee":"M. Winkler, “Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar,” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, Art. no. 108, 2022, doi: 10.1007/s00526-021-02168-2.","short":"M. Winkler, Calculus of Variations and Partial Differential Equations 61 (2022).","chicago":"Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.” Calculus of Variations and Partial Differential Equations 61, no. 3 (2022). https://doi.org/10.1007/s00526-021-02168-2.","mla":"Winkler, Michael. “Stabilization of Arbitrary Structures in a Doubly Degenerate Reaction-Diffusion System Modeling Bacterial Motion on a Nutrient-Poor Agar.” Calculus of Variations and Partial Differential Equations, vol. 61, no. 3, 108, Springer Science and Business Media LLC, 2022, doi:10.1007/s00526-021-02168-2.","ama":"Winkler M. Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. Calculus of Variations and Partial Differential Equations. 2022;61(3). doi:10.1007/s00526-021-02168-2","bibtex":"@article{Winkler_2022, title={Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar}, volume={61}, DOI={10.1007/s00526-021-02168-2}, number={3108}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }"}},{"year":"2019","status":"public","title":"The fast signal diffusion limit in Keller–Segel(-fluid) systems","publication_identifier":{"issn":["0944-2669","1432-0835"]},"author":[{"full_name":"Wang, Yulan","last_name":"Wang","first_name":"Yulan"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"},{"first_name":"Zhaoyin","last_name":"Xiang","full_name":"Xiang, Zhaoyin"}],"date_updated":"2025-12-19T10:57:05Z","publication_status":"published","intvolume":" 58","article_number":"196","language":[{"iso":"eng"}],"_id":"63359","publisher":"Springer Science and Business Media LLC","doi":"10.1007/s00526-019-1656-3","user_id":"31496","volume":58,"publication":"Calculus of Variations and Partial Differential Equations","issue":"6","citation":{"ieee":"Y. Wang, M. Winkler, and Z. Xiang, “The fast signal diffusion limit in Keller–Segel(-fluid) systems,” Calculus of Variations and Partial Differential Equations, vol. 58, no. 6, Art. no. 196, 2019, doi: 10.1007/s00526-019-1656-3.","apa":"Wang, Y., Winkler, M., & Xiang, Z. (2019). The fast signal diffusion limit in Keller–Segel(-fluid) systems. Calculus of Variations and Partial Differential Equations, 58(6), Article 196. https://doi.org/10.1007/s00526-019-1656-3","chicago":"Wang, Yulan, Michael Winkler, and Zhaoyin Xiang. “The Fast Signal Diffusion Limit in Keller–Segel(-Fluid) Systems.” Calculus of Variations and Partial Differential Equations 58, no. 6 (2019). https://doi.org/10.1007/s00526-019-1656-3.","short":"Y. Wang, M. Winkler, Z. Xiang, Calculus of Variations and Partial Differential Equations 58 (2019).","mla":"Wang, Yulan, et al. “The Fast Signal Diffusion Limit in Keller–Segel(-Fluid) Systems.” Calculus of Variations and Partial Differential Equations, vol. 58, no. 6, 196, Springer Science and Business Media LLC, 2019, doi:10.1007/s00526-019-1656-3.","bibtex":"@article{Wang_Winkler_Xiang_2019, title={The fast signal diffusion limit in Keller–Segel(-fluid) systems}, volume={58}, DOI={10.1007/s00526-019-1656-3}, number={6196}, journal={Calculus of Variations and Partial Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}, year={2019} }","ama":"Wang Y, Winkler M, Xiang Z. The fast signal diffusion limit in Keller–Segel(-fluid) systems. Calculus of Variations and Partial Differential Equations. 2019;58(6). doi:10.1007/s00526-019-1656-3"},"date_created":"2025-12-19T10:56:58Z","type":"journal_article"}]