[{"status":"public","volume":42,"user_id":"31496","_id":"63293","publisher":"American Institute of Mathematical Sciences (AIMS)","citation":{"ieee":"K. Kang, J. Lee, and M. Winkler, “Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $,” Discrete and Continuous Dynamical Systems, vol. 42, no. 11, Art. no. 5201, 2022, doi: 10.3934/dcds.2022091.","mla":"Kang, Kyungkeun, et al. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $  \\mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems, vol. 42, no. 11, 5201, American Institute of Mathematical Sciences (AIMS), 2022, doi:10.3934/dcds.2022091.","apa":"Kang, K., Lee, J., & Winkler, M. (2022). Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $. Discrete and Continuous Dynamical Systems, 42(11), Article 5201. https://doi.org/10.3934/dcds.2022091","bibtex":"@article{Kang_Lee_Winkler_2022, title={Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $}, volume={42}, DOI={10.3934/dcds.2022091}, number={115201}, journal={Discrete and Continuous Dynamical Systems}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Kang, Kyungkeun and Lee, Jihoon and Winkler, Michael}, year={2022} }","chicago":"Kang, Kyungkeun, Jihoon Lee, and Michael Winkler. “Global Weak Solutions to a Chemotaxis-Navier-Stokes System in $  \\mathbb{R}^3 $.” Discrete and Continuous Dynamical Systems 42, no. 11 (2022). https://doi.org/10.3934/dcds.2022091.","ama":"Kang K, Lee J, Winkler M. Global weak solutions to a chemotaxis-Navier-Stokes system in $  \\mathbb{R}^3 $. Discrete and Continuous Dynamical Systems. 2022;42(11). doi:10.3934/dcds.2022091","short":"K. Kang, J. Lee, M. Winkler, Discrete and Continuous Dynamical Systems 42 (2022)."},"intvolume":" 42","date_updated":"2025-12-18T20:08:21Z","publication_status":"published","publication_identifier":{"issn":["1078-0947","1553-5231"]},"author":[{"full_name":"Kang, Kyungkeun","first_name":"Kyungkeun","last_name":"Kang"},{"full_name":"Lee, Jihoon","first_name":"Jihoon","last_name":"Lee"},{"id":"31496","first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"title":"Global weak solutions to a chemotaxis-Navier-Stokes system in $ \\mathbb{R}^3 $","year":"2022","doi":"10.3934/dcds.2022091","language":[{"iso":"eng"}],"article_number":"5201","abstract":[{"text":"<p style='text-indent:20px;'>The Cauchy problem in <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\mathbb{R}^3 $\\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> \\begin{document}$ \\begin{eqnarray*} \\left\\{ \\begin{array}{l} n_t + u\\cdot\\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\\tc_t + u\\cdot\\nabla c = \\Delta c - nc, \\\\ \tu_t + (u\\cdot\\nabla) u = \\Delta u + \\nabla P + n\\nabla\\phi, \\qquad \\nabla \\cdot u = 0, \\ \t\\end{array} \\right. \\end{eqnarray*} $\\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered. Under suitable conditions on the initial data <inline-formula><tex-math id=\"M3\">\\begin{document}$ (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula>, with regard to the crucial first component requiring that <inline-formula><tex-math id=\"M4\">\\begin{document}$ n_0\\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula> be nonnegative and such that <inline-formula><tex-math id=\"M5\">\\begin{document}$ (n_0+1)\\ln (n_0+1) \\in L^1( \\mathbb{R}^3) $\\end{document}</tex-math></inline-formula>, a globally defined weak solution with <inline-formula><tex-math id=\"M6\">\\begin{document}$ (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\\end{document}</tex-math></inline-formula> is constructed. Apart from that, assuming that moreover <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\int_{ \\mathbb{R}^3} n_0(x) \\ln (1+|x|^2) dx $\\end{document}</tex-math></inline-formula> is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.</p>","lang":"eng"}],"issue":"11","publication":"Discrete and Continuous Dynamical Systems","type":"journal_article","date_created":"2025-12-18T19:22:04Z"},{"doi":"10.3934/dcds.2020216","user_id":"31496","volume":41,"page":"439-454","_id":"63320","language":[{"iso":"eng"}],"publisher":"American Institute of Mathematical Sciences (AIMS)","date_updated":"2025-12-18T20:04:09Z","publication_status":"published","intvolume":" 41","status":"public","title":"Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction","year":"2020","author":[{"first_name":"Youshan","last_name":"Tao","full_name":"Tao, Youshan"},{"full_name":"Winkler, Michael","first_name":"Michael","last_name":"Winkler","id":"31496"}],"publication_identifier":{"issn":["1553-5231"]},"type":"journal_article","date_created":"2025-12-18T19:33:59Z","issue":"1","publication":"Discrete & Continuous Dynamical Systems - A","citation":{"short":"Y. Tao, M. Winkler, Discrete &amp; Continuous Dynamical Systems - A 41 (2020) 439–454.","chicago":"Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up in a Haptotaxis System with Nonlinear Zero-Order Interaction.” Discrete &amp; Continuous Dynamical Systems - A 41, no. 1 (2020): 439–54. https://doi.org/10.3934/dcds.2020216.","apa":"Tao, Y., & Winkler, M. (2020). Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete &amp; Continuous Dynamical Systems - A, 41(1), 439–454. https://doi.org/10.3934/dcds.2020216","ieee":"Y. Tao and M. Winkler, “Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction,” Discrete &amp; Continuous Dynamical Systems - A, vol. 41, no. 1, pp. 439–454, 2020, doi: 10.3934/dcds.2020216.","ama":"Tao Y, Winkler M. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete &amp; Continuous Dynamical Systems - A. 2020;41(1):439-454. doi:10.3934/dcds.2020216","bibtex":"@article{Tao_Winkler_2020, title={Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction}, volume={41}, DOI={10.3934/dcds.2020216}, number={1}, journal={Discrete &amp; Continuous Dynamical Systems - A}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Tao, Youshan and Winkler, Michael}, year={2020}, pages={439–454} }","mla":"Tao, Youshan, and Michael Winkler. “Critical Mass for Infinite-Time Blow-up in a Haptotaxis System with Nonlinear Zero-Order Interaction.” Discrete &amp; Continuous Dynamical Systems - A, vol. 41, no. 1, American Institute of Mathematical Sciences (AIMS), 2020, pp. 439–54, doi:10.3934/dcds.2020216."}}]