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Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation. <i>Calculus of Variations and Partial Differential Equations</i>, <i>61</i>(3), Article 96. <a href=\"https://doi.org/10.1007/s00526-022-02201-y\">https://doi.org/10.1007/s00526-022-02201-y</a>","mla":"Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Eventual Smoothness in a Chemotaxis-Navier–Stokes System with Logistic Proliferation.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 3, 96, Springer Science and Business Media LLC, 2022, doi:<a href=\"https://doi.org/10.1007/s00526-022-02201-y\">10.1007/s00526-022-02201-y</a>.","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={<a href=\"https://doi.org/10.1007/s00526-022-02201-y\">10.1007/s00526-022-02201-y</a>}, 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} }","short":"T. 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Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation. <i>Zeitschrift für angewandte Mathematik und Physik</i>. 2021;72(4). doi:<a href=\"https://doi.org/10.1007/s00033-021-01565-z\">10.1007/s00033-021-01565-z</a>","ieee":"T. Black and C. Wu, “Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation,” <i>Zeitschrift für angewandte Mathematik und Physik</i>, vol. 72, no. 4, Art. no. 135, 2021, doi: <a href=\"https://doi.org/10.1007/s00033-021-01565-z\">10.1007/s00033-021-01565-z</a>.","chicago":"Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Weak Solvability in a Chemotaxis-Stokes System with Proliferation.” <i>Zeitschrift Für Angewandte Mathematik Und Physik</i> 72, no. 4 (2021). <a href=\"https://doi.org/10.1007/s00033-021-01565-z\">https://doi.org/10.1007/s00033-021-01565-z</a>.","apa":"Black, T., &#38; Wu, C. (2021). Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation. <i>Zeitschrift Für Angewandte Mathematik Und Physik</i>, <i>72</i>(4), Article 135. <a href=\"https://doi.org/10.1007/s00033-021-01565-z\">https://doi.org/10.1007/s00033-021-01565-z</a>","bibtex":"@article{Black_Wu_2021, title={Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation}, volume={72}, DOI={<a href=\"https://doi.org/10.1007/s00033-021-01565-z\">10.1007/s00033-021-01565-z</a>}, number={4135}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Wu, Chunyan}, year={2021} }","mla":"Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Weak Solvability in a Chemotaxis-Stokes System with Proliferation.” <i>Zeitschrift Für Angewandte Mathematik Und Physik</i>, vol. 72, no. 4, 135, Springer Science and Business Media LLC, 2021, doi:<a href=\"https://doi.org/10.1007/s00033-021-01565-z\">10.1007/s00033-021-01565-z</a>.","short":"T. Black, C. Wu, Zeitschrift Für Angewandte Mathematik Und Physik 72 (2021)."},"publication_identifier":{"issn":["0044-2275","1420-9039"]},"publication_status":"published","doi":"10.1007/s00033-021-01565-z","volume":72,"author":[{"first_name":"Tobias","last_name":"Black","orcid":"0000-0001-9963-0800","full_name":"Black, Tobias","id":"23686"},{"full_name":"Wu, Chunyan","last_name":"Wu","first_name":"Chunyan"}],"date_updated":"2023-07-10T11:39:30Z","status":"public","type":"journal_article","article_number":"135","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","_id":"34675"},{"issue":"01","publication_status":"published","publication_identifier":{"issn":["0218-2025","1793-6314"]},"citation":{"chicago":"Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing Sets in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration on the Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i> 32, no. 01 (2021): 137–73. <a href=\"https://doi.org/10.1142/s021820252250004x\">https://doi.org/10.1142/s021820252250004x</a>.","ieee":"T. Black and M. Winkler, “Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 01, pp. 137–173, 2021, doi: <a href=\"https://doi.org/10.1142/s021820252250004x\">10.1142/s021820252250004x</a>.","ama":"Black T, Winkler M. Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary. <i>Mathematical Models and Methods in Applied Sciences</i>. 2021;32(01):137-173. doi:<a href=\"https://doi.org/10.1142/s021820252250004x\">10.1142/s021820252250004x</a>","short":"T. Black, M. Winkler, Mathematical Models and Methods in Applied Sciences 32 (2021) 137–173.","mla":"Black, Tobias, and Michael Winkler. “Global Weak Solutions and Absorbing Sets in a Chemotaxis-Navier–Stokes System with Prescribed Signal Concentration on the Boundary.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 01, World Scientific Pub Co Pte Ltd, 2021, pp. 137–73, doi:<a href=\"https://doi.org/10.1142/s021820252250004x\">10.1142/s021820252250004x</a>.","bibtex":"@article{Black_Winkler_2021, title={Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary}, volume={32}, DOI={<a href=\"https://doi.org/10.1142/s021820252250004x\">10.1142/s021820252250004x</a>}, number={01}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Black, Tobias and Winkler, Michael}, year={2021}, pages={137–173} }","apa":"Black, T., &#38; Winkler, M. (2021). Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>32</i>(01), 137–173. <a href=\"https://doi.org/10.1142/s021820252250004x\">https://doi.org/10.1142/s021820252250004x</a>"},"intvolume":"        32","page":"137-173","year":"2021","author":[{"first_name":"Tobias","orcid":"0000-0001-9963-0800","last_name":"Black","full_name":"Black, Tobias","id":"23686"},{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"date_created":"2025-12-18T19:20:48Z","volume":32,"date_updated":"2025-12-18T20:08:01Z","publisher":"World Scientific Pub Co Pte Ltd","doi":"10.1142/s021820252250004x","title":"Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary","type":"journal_article","publication":"Mathematical Models and Methods in Applied Sciences","status":"public","abstract":[{"lang":"eng","text":"<jats:p> An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes model with porous medium type diffusion is considered. Previous related literature has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip boundary conditions for the density of cells, the chemical concentration and the fluid velocity field, respectively; in particular, available qualitative results strongly indicate that only trivial solution behavior can be expected on large time scales. In line with refined modeling approaches to oxygen evolution near fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant concentration on the boundary. Despite an apparent loss of mathematically favorable energy structures thereby induced, by means of an alternative variational approach a basic theory of global existence is developed in a natural framework of weak solvability. Beyond this, some additional qualitative information on the large time behavior of these solutions is derived by identifying a certain global relaxation property. Specifically, a second result asserts, within a suitable topological setting, the existence of a bounded set which eventually absorbs each individual of the obtained trajectories, and the diameter of which is bounded only by the physically relevant quantities of total population size and prescribed boundary concentration of the chemical signal. </jats:p>"}],"user_id":"31496","_id":"63291","language":[{"iso":"eng"}]},{"type":"journal_article","publication":"Mathematical Models and Methods in Applied Sciences","status":"public","_id":"34670","user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"keyword":["Applied Mathematics","Modeling and Simulation"],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0218-2025","1793-6314"]},"issue":"06","year":"2020","citation":{"apa":"Black, T. (2020). Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>30</i>(06), 1075–1117. <a href=\"https://doi.org/10.1142/s0218202520400072\">https://doi.org/10.1142/s0218202520400072</a>","bibtex":"@article{Black_2020, title={Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties}, volume={30}, DOI={<a href=\"https://doi.org/10.1142/s0218202520400072\">10.1142/s0218202520400072</a>}, number={06}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Lt}, author={Black, Tobias}, year={2020}, pages={1075–1117} }","short":"T. Black, Mathematical Models and Methods in Applied Sciences 30 (2020) 1075–1117.","mla":"Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model with Superlinear Degradation and Their Eventual Regularity Properties.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 30, no. 06, World Scientific Pub Co Pte Lt, 2020, pp. 1075–117, doi:<a href=\"https://doi.org/10.1142/s0218202520400072\">10.1142/s0218202520400072</a>.","ieee":"T. Black, “Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 30, no. 06, pp. 1075–1117, 2020, doi: <a href=\"https://doi.org/10.1142/s0218202520400072\">10.1142/s0218202520400072</a>.","chicago":"Black, Tobias. “Global Generalized Solutions to a Forager–Exploiter Model with Superlinear Degradation and Their Eventual Regularity Properties.” <i>Mathematical Models and Methods in Applied Sciences</i> 30, no. 06 (2020): 1075–1117. <a href=\"https://doi.org/10.1142/s0218202520400072\">https://doi.org/10.1142/s0218202520400072</a>.","ama":"Black T. Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties. <i>Mathematical Models and Methods in Applied Sciences</i>. 2020;30(06):1075-1117. doi:<a href=\"https://doi.org/10.1142/s0218202520400072\">10.1142/s0218202520400072</a>"},"intvolume":"        30","page":"1075-1117","date_updated":"2023-07-10T11:39:59Z","publisher":"World Scientific Pub Co Pte Lt","author":[{"first_name":"Tobias","id":"23686","full_name":"Black, Tobias","last_name":"Black","orcid":"0000-0001-9963-0800"}],"date_created":"2022-12-21T09:48:11Z","volume":30,"title":"Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties","doi":"10.1142/s0218202520400072"},{"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Discrete Mathematics and Combinatorics","Analysis"],"publication":"Discrete &amp; Continuous Dynamical Systems - S","title":"Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity","date_created":"2022-12-21T09:48:28Z","publisher":"American Institute of Mathematical Sciences (AIMS)","year":"2019","issue":"2","user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"34672","status":"public","type":"journal_article","doi":"10.3934/dcdss.2020007","author":[{"full_name":"Black, Tobias","id":"23686","orcid":"0000-0001-9963-0800","last_name":"Black","first_name":"Tobias"}],"volume":13,"date_updated":"2022-12-21T10:04:44Z","citation":{"chicago":"Black, Tobias. “Global Generalized Solutions to a Parabolic-Elliptic Keller-Segel System with Singular Sensitivity.” <i>Discrete &#38;amp; Continuous Dynamical Systems - S</i> 13, no. 2 (2019): 119–37. <a href=\"https://doi.org/10.3934/dcdss.2020007\">https://doi.org/10.3934/dcdss.2020007</a>.","ieee":"T. Black, “Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity,” <i>Discrete &#38;amp; Continuous Dynamical Systems - S</i>, vol. 13, no. 2, pp. 119–137, 2019, doi: <a href=\"https://doi.org/10.3934/dcdss.2020007\">10.3934/dcdss.2020007</a>.","ama":"Black T. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. <i>Discrete &#38;amp; Continuous Dynamical Systems - S</i>. 2019;13(2):119-137. doi:<a href=\"https://doi.org/10.3934/dcdss.2020007\">10.3934/dcdss.2020007</a>","apa":"Black, T. (2019). Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. <i>Discrete &#38;amp; Continuous Dynamical Systems - S</i>, <i>13</i>(2), 119–137. <a href=\"https://doi.org/10.3934/dcdss.2020007\">https://doi.org/10.3934/dcdss.2020007</a>","short":"T. Black, Discrete &#38;amp; Continuous Dynamical Systems - S 13 (2019) 119–137.","mla":"Black, Tobias. “Global Generalized Solutions to a Parabolic-Elliptic Keller-Segel System with Singular Sensitivity.” <i>Discrete &#38;amp; Continuous Dynamical Systems - S</i>, vol. 13, no. 2, American Institute of Mathematical Sciences (AIMS), 2019, pp. 119–37, doi:<a href=\"https://doi.org/10.3934/dcdss.2020007\">10.3934/dcdss.2020007</a>.","bibtex":"@article{Black_2019, title={Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity}, volume={13}, DOI={<a href=\"https://doi.org/10.3934/dcdss.2020007\">10.3934/dcdss.2020007</a>}, number={2}, journal={Discrete &#38;amp; Continuous Dynamical Systems - S}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Black, Tobias}, year={2019}, pages={119–137} }"},"intvolume":"        13","page":"119-137","publication_status":"published","publication_identifier":{"issn":["1937-1179"]}},{"publication":"Journal of Mathematical Fluid Mechanics","language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Computational Mathematics","Condensed Matter Physics","Mathematical Physics"],"issue":"1","year":"2019","date_created":"2022-12-21T09:47:56Z","publisher":"Springer Science and Business Media LLC","title":"The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System","type":"journal_article","status":"public","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","_id":"34669","article_number":"1","publication_identifier":{"issn":["1422-6928","1422-6952"]},"publication_status":"published","intvolume":"        22","citation":{"chicago":"Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System.” <i>Journal of Mathematical Fluid Mechanics</i> 22, no. 1 (2019). <a href=\"https://doi.org/10.1007/s00021-019-0464-z\">https://doi.org/10.1007/s00021-019-0464-z</a>.","ieee":"T. Black, “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System,” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 22, no. 1, Art. no. 1, 2019, doi: <a href=\"https://doi.org/10.1007/s00021-019-0464-z\">10.1007/s00021-019-0464-z</a>.","ama":"Black T. The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System. <i>Journal of Mathematical Fluid Mechanics</i>. 2019;22(1). doi:<a href=\"https://doi.org/10.1007/s00021-019-0464-z\">10.1007/s00021-019-0464-z</a>","short":"T. Black, Journal of Mathematical Fluid Mechanics 22 (2019).","bibtex":"@article{Black_2019, title={The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System}, volume={22}, DOI={<a href=\"https://doi.org/10.1007/s00021-019-0464-z\">10.1007/s00021-019-0464-z</a>}, number={11}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias}, year={2019} }","mla":"Black, Tobias. “The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System.” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 22, no. 1, 1, Springer Science and Business Media LLC, 2019, doi:<a href=\"https://doi.org/10.1007/s00021-019-0464-z\">10.1007/s00021-019-0464-z</a>.","apa":"Black, T. (2019). The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System. <i>Journal of Mathematical Fluid Mechanics</i>, <i>22</i>(1), Article 1. <a href=\"https://doi.org/10.1007/s00021-019-0464-z\">https://doi.org/10.1007/s00021-019-0464-z</a>"},"volume":22,"author":[{"first_name":"Tobias","orcid":"0000-0001-9963-0800","last_name":"Black","full_name":"Black, Tobias","id":"23686"}],"date_updated":"2022-12-21T10:04:29Z","doi":"10.1007/s00021-019-0464-z"},{"language":[{"iso":"eng"}],"keyword":["General Engineering","General Mathematics"],"publication":"Mathematical Methods in the Applied Sciences","title":"A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions","date_created":"2022-12-21T09:47:47Z","publisher":"Wiley","year":"2019","issue":"9","user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"34668","status":"public","type":"journal_article","doi":"10.1002/mma.5561","author":[{"first_name":"Tobias","full_name":"Black, Tobias","id":"23686","last_name":"Black","orcid":"0000-0001-9963-0800"},{"first_name":"Johannes","last_name":"Lankeit","full_name":"Lankeit, Johannes"},{"first_name":"Masaaki","last_name":"Mizukami","full_name":"Mizukami, Masaaki"}],"volume":42,"date_updated":"2022-12-21T10:04:36Z","citation":{"mla":"Black, Tobias, et al. “A Keller‐Segel‐fluid System with Singular Sensitivity: Generalized Solutions.” <i>Mathematical Methods in the Applied Sciences</i>, vol. 42, no. 9, Wiley, 2019, pp. 3002–20, doi:<a href=\"https://doi.org/10.1002/mma.5561\">10.1002/mma.5561</a>.","bibtex":"@article{Black_Lankeit_Mizukami_2019, title={A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions}, volume={42}, DOI={<a href=\"https://doi.org/10.1002/mma.5561\">10.1002/mma.5561</a>}, number={9}, journal={Mathematical Methods in the Applied Sciences}, publisher={Wiley}, author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2019}, pages={3002–3020} }","short":"T. Black, J. Lankeit, M. Mizukami, Mathematical Methods in the Applied Sciences 42 (2019) 3002–3020.","apa":"Black, T., Lankeit, J., &#38; Mizukami, M. (2019). A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions. <i>Mathematical Methods in the Applied Sciences</i>, <i>42</i>(9), 3002–3020. <a href=\"https://doi.org/10.1002/mma.5561\">https://doi.org/10.1002/mma.5561</a>","ama":"Black T, Lankeit J, Mizukami M. A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions. <i>Mathematical Methods in the Applied Sciences</i>. 2019;42(9):3002-3020. doi:<a href=\"https://doi.org/10.1002/mma.5561\">10.1002/mma.5561</a>","chicago":"Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “A Keller‐Segel‐fluid System with Singular Sensitivity: Generalized Solutions.” <i>Mathematical Methods in the Applied Sciences</i> 42, no. 9 (2019): 3002–20. <a href=\"https://doi.org/10.1002/mma.5561\">https://doi.org/10.1002/mma.5561</a>.","ieee":"T. Black, J. Lankeit, and M. Mizukami, “A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions,” <i>Mathematical Methods in the Applied Sciences</i>, vol. 42, no. 9, pp. 3002–3020, 2019, doi: <a href=\"https://doi.org/10.1002/mma.5561\">10.1002/mma.5561</a>."},"intvolume":"        42","page":"3002-3020","publication_status":"published","publication_identifier":{"issn":["0170-4214","1099-1476"]}},{"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Analysis"],"user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"34671","status":"public","type":"journal_article","publication":"Applicable Analysis","doi":"10.1080/00036811.2019.1585534","title":"Stabilization in the Keller–Segel system with signal-dependent sensitivity","date_created":"2022-12-21T09:48:19Z","author":[{"full_name":"Black, Tobias","id":"23686","last_name":"Black","orcid":"0000-0001-9963-0800","first_name":"Tobias"},{"first_name":"Johannes","full_name":"Lankeit, Johannes","last_name":"Lankeit"},{"first_name":"Masaaki","last_name":"Mizukami","full_name":"Mizukami, Masaaki"}],"volume":99,"publisher":"Informa UK Limited","date_updated":"2022-12-21T10:04:50Z","citation":{"bibtex":"@article{Black_Lankeit_Mizukami_2019, title={Stabilization in the Keller–Segel system with signal-dependent sensitivity}, volume={99}, DOI={<a href=\"https://doi.org/10.1080/00036811.2019.1585534\">10.1080/00036811.2019.1585534</a>}, number={16}, journal={Applicable Analysis}, publisher={Informa UK Limited}, author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2019}, pages={2877–2891} }","short":"T. Black, J. Lankeit, M. Mizukami, Applicable Analysis 99 (2019) 2877–2891.","mla":"Black, Tobias, et al. “Stabilization in the Keller–Segel System with Signal-Dependent Sensitivity.” <i>Applicable Analysis</i>, vol. 99, no. 16, Informa UK Limited, 2019, pp. 2877–91, doi:<a href=\"https://doi.org/10.1080/00036811.2019.1585534\">10.1080/00036811.2019.1585534</a>.","apa":"Black, T., Lankeit, J., &#38; Mizukami, M. (2019). Stabilization in the Keller–Segel system with signal-dependent sensitivity. <i>Applicable Analysis</i>, <i>99</i>(16), 2877–2891. <a href=\"https://doi.org/10.1080/00036811.2019.1585534\">https://doi.org/10.1080/00036811.2019.1585534</a>","chicago":"Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “Stabilization in the Keller–Segel System with Signal-Dependent Sensitivity.” <i>Applicable Analysis</i> 99, no. 16 (2019): 2877–91. <a href=\"https://doi.org/10.1080/00036811.2019.1585534\">https://doi.org/10.1080/00036811.2019.1585534</a>.","ieee":"T. Black, J. Lankeit, and M. Mizukami, “Stabilization in the Keller–Segel system with signal-dependent sensitivity,” <i>Applicable Analysis</i>, vol. 99, no. 16, pp. 2877–2891, 2019, doi: <a href=\"https://doi.org/10.1080/00036811.2019.1585534\">10.1080/00036811.2019.1585534</a>.","ama":"Black T, Lankeit J, Mizukami M. Stabilization in the Keller–Segel system with signal-dependent sensitivity. <i>Applicable Analysis</i>. 2019;99(16):2877-2891. doi:<a href=\"https://doi.org/10.1080/00036811.2019.1585534\">10.1080/00036811.2019.1585534</a>"},"intvolume":"        99","page":"2877-2891","year":"2019","issue":"16","publication_status":"published","publication_identifier":{"issn":["0003-6811","1563-504X"]}},{"issue":"4","publication_identifier":{"issn":["0036-1410","1095-7154"]},"publication_status":"published","intvolume":"        50","page":"4087-4116","citation":{"ama":"Black T. Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion. <i>SIAM Journal on Mathematical Analysis</i>. 2018;50(4):4087-4116. doi:<a href=\"https://doi.org/10.1137/17m1159488\">10.1137/17m1159488</a>","chicago":"Black, Tobias. “Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion.” <i>SIAM Journal on Mathematical Analysis</i> 50, no. 4 (2018): 4087–4116. <a href=\"https://doi.org/10.1137/17m1159488\">https://doi.org/10.1137/17m1159488</a>.","ieee":"T. Black, “Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 4, pp. 4087–4116, 2018, doi: <a href=\"https://doi.org/10.1137/17m1159488\">10.1137/17m1159488</a>.","short":"T. Black, SIAM Journal on Mathematical Analysis 50 (2018) 4087–4116.","mla":"Black, Tobias. “Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 4, Society for Industrial &#38; Applied Mathematics (SIAM), 2018, pp. 4087–116, doi:<a href=\"https://doi.org/10.1137/17m1159488\">10.1137/17m1159488</a>.","bibtex":"@article{Black_2018, title={Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion}, volume={50}, DOI={<a href=\"https://doi.org/10.1137/17m1159488\">10.1137/17m1159488</a>}, number={4}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={Black, Tobias}, year={2018}, pages={4087–4116} }","apa":"Black, T. (2018). Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion. <i>SIAM Journal on Mathematical Analysis</i>, <i>50</i>(4), 4087–4116. <a href=\"https://doi.org/10.1137/17m1159488\">https://doi.org/10.1137/17m1159488</a>"},"year":"2018","volume":50,"author":[{"first_name":"Tobias","orcid":"0000-0001-9963-0800","last_name":"Black","full_name":"Black, Tobias","id":"23686"}],"date_created":"2022-12-21T09:47:01Z","publisher":"Society for Industrial & Applied Mathematics (SIAM)","date_updated":"2022-12-21T10:04:57Z","doi":"10.1137/17m1159488","title":"Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion","publication":"SIAM Journal on Mathematical Analysis","type":"journal_article","status":"public","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","_id":"34664","language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Computational Mathematics","Analysis"]},{"doi":"10.1016/j.jde.2018.04.035","date_updated":"2022-12-21T10:05:11Z","volume":265,"author":[{"id":"23686","full_name":"Black, Tobias","orcid":"0000-0001-9963-0800","last_name":"Black","first_name":"Tobias"}],"intvolume":"       265","page":"2296-2339","citation":{"apa":"Black, T. (2018). Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D. <i>Journal of Differential Equations</i>, <i>265</i>(5), 2296–2339. <a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">https://doi.org/10.1016/j.jde.2018.04.035</a>","short":"T. Black, Journal of Differential Equations 265 (2018) 2296–2339.","bibtex":"@article{Black_2018, title={Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D}, volume={265}, DOI={<a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">10.1016/j.jde.2018.04.035</a>}, number={5}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Black, Tobias}, year={2018}, pages={2296–2339} }","mla":"Black, Tobias. “Eventual Smoothness of Generalized Solutions to a Singular Chemotaxis-Stokes System in 2D.” <i>Journal of Differential Equations</i>, vol. 265, no. 5, Elsevier BV, 2018, pp. 2296–339, doi:<a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">10.1016/j.jde.2018.04.035</a>.","ama":"Black T. Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D. <i>Journal of Differential Equations</i>. 2018;265(5):2296-2339. doi:<a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">10.1016/j.jde.2018.04.035</a>","ieee":"T. Black, “Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D,” <i>Journal of Differential Equations</i>, vol. 265, no. 5, pp. 2296–2339, 2018, doi: <a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">10.1016/j.jde.2018.04.035</a>.","chicago":"Black, Tobias. “Eventual Smoothness of Generalized Solutions to a Singular Chemotaxis-Stokes System in 2D.” <i>Journal of Differential Equations</i> 265, no. 5 (2018): 2296–2339. <a href=\"https://doi.org/10.1016/j.jde.2018.04.035\">https://doi.org/10.1016/j.jde.2018.04.035</a>."},"publication_identifier":{"issn":["0022-0396"]},"publication_status":"published","_id":"34666","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","status":"public","type":"journal_article","title":"Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D","publisher":"Elsevier BV","date_created":"2022-12-21T09:47:19Z","year":"2018","issue":"5","keyword":["Analysis","Applied Mathematics"],"language":[{"iso":"eng"}],"publication":"Journal of Differential Equations"},{"doi":"10.1016/j.na.2018.10.003","title":"Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions","volume":180,"date_created":"2022-12-21T09:47:30Z","author":[{"full_name":"Black, Tobias","id":"23686","orcid":"0000-0001-9963-0800","last_name":"Black","first_name":"Tobias"}],"publisher":"Elsevier BV","date_updated":"2022-12-21T10:05:04Z","intvolume":"       180","page":"129-153","citation":{"mla":"Black, Tobias. “Global Solvability of Chemotaxis–Fluid Systems with Nonlinear Diffusion and Matrix-Valued Sensitivities in Three Dimensions.” <i>Nonlinear Analysis</i>, vol. 180, Elsevier BV, 2018, pp. 129–53, doi:<a href=\"https://doi.org/10.1016/j.na.2018.10.003\">10.1016/j.na.2018.10.003</a>.","short":"T. Black, Nonlinear Analysis 180 (2018) 129–153.","bibtex":"@article{Black_2018, title={Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions}, volume={180}, DOI={<a href=\"https://doi.org/10.1016/j.na.2018.10.003\">10.1016/j.na.2018.10.003</a>}, journal={Nonlinear Analysis}, publisher={Elsevier BV}, author={Black, Tobias}, year={2018}, pages={129–153} }","apa":"Black, T. (2018). Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions. <i>Nonlinear Analysis</i>, <i>180</i>, 129–153. <a href=\"https://doi.org/10.1016/j.na.2018.10.003\">https://doi.org/10.1016/j.na.2018.10.003</a>","ieee":"T. Black, “Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions,” <i>Nonlinear Analysis</i>, vol. 180, pp. 129–153, 2018, doi: <a href=\"https://doi.org/10.1016/j.na.2018.10.003\">10.1016/j.na.2018.10.003</a>.","chicago":"Black, Tobias. “Global Solvability of Chemotaxis–Fluid Systems with Nonlinear Diffusion and Matrix-Valued Sensitivities in Three Dimensions.” <i>Nonlinear Analysis</i> 180 (2018): 129–53. <a href=\"https://doi.org/10.1016/j.na.2018.10.003\">https://doi.org/10.1016/j.na.2018.10.003</a>.","ama":"Black T. Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions. <i>Nonlinear Analysis</i>. 2018;180:129-153. doi:<a href=\"https://doi.org/10.1016/j.na.2018.10.003\">10.1016/j.na.2018.10.003</a>"},"year":"2018","publication_identifier":{"issn":["0362-546X"]},"publication_status":"published","language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Analysis"],"department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","_id":"34667","status":"public","publication":"Nonlinear Analysis","type":"journal_article"},{"date_updated":"2022-12-21T10:05:19Z","author":[{"first_name":"Tobias","full_name":"Black, Tobias","id":"23686","last_name":"Black","orcid":"0000-0001-9963-0800"}],"volume":22,"doi":"10.3934/dcdsb.2017061","publication_status":"published","publication_identifier":{"issn":["1553-524X"]},"citation":{"short":"T. Black, Discrete &#38;amp; Continuous Dynamical Systems - B 22 (2017) 1253–1272.","bibtex":"@article{Black_2017, title={Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals}, volume={22}, DOI={<a href=\"https://doi.org/10.3934/dcdsb.2017061\">10.3934/dcdsb.2017061</a>}, number={4}, journal={Discrete &#38;amp; Continuous Dynamical Systems - B}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Black, Tobias}, year={2017}, pages={1253–1272} }","mla":"Black, Tobias. “Global Existence and Asymptotic Stability in a Competitive Two-Species Chemotaxis System with Two Signals.” <i>Discrete &#38;amp; Continuous Dynamical Systems - B</i>, vol. 22, no. 4, American Institute of Mathematical Sciences (AIMS), 2017, pp. 1253–72, doi:<a href=\"https://doi.org/10.3934/dcdsb.2017061\">10.3934/dcdsb.2017061</a>.","apa":"Black, T. (2017). Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. <i>Discrete &#38;amp; Continuous Dynamical Systems - B</i>, <i>22</i>(4), 1253–1272. <a href=\"https://doi.org/10.3934/dcdsb.2017061\">https://doi.org/10.3934/dcdsb.2017061</a>","ama":"Black T. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. <i>Discrete &#38;amp; Continuous Dynamical Systems - B</i>. 2017;22(4):1253-1272. doi:<a href=\"https://doi.org/10.3934/dcdsb.2017061\">10.3934/dcdsb.2017061</a>","ieee":"T. Black, “Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals,” <i>Discrete &#38;amp; Continuous Dynamical Systems - B</i>, vol. 22, no. 4, pp. 1253–1272, 2017, doi: <a href=\"https://doi.org/10.3934/dcdsb.2017061\">10.3934/dcdsb.2017061</a>.","chicago":"Black, Tobias. “Global Existence and Asymptotic Stability in a Competitive Two-Species Chemotaxis System with Two Signals.” <i>Discrete &#38;amp; Continuous Dynamical Systems - B</i> 22, no. 4 (2017): 1253–72. <a href=\"https://doi.org/10.3934/dcdsb.2017061\">https://doi.org/10.3934/dcdsb.2017061</a>."},"page":"1253-1272","intvolume":"        22","_id":"34663","user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"type":"journal_article","status":"public","publisher":"American Institute of Mathematical Sciences (AIMS)","date_created":"2022-12-21T09:46:50Z","title":"Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals","issue":"4","year":"2017","keyword":["Applied Mathematics","Discrete Mathematics and Combinatorics"],"language":[{"iso":"eng"}],"publication":"Discrete &amp; Continuous Dynamical Systems - B"},{"user_id":"23686","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"_id":"34665","type":"journal_article","status":"public","author":[{"last_name":"Black","orcid":"0000-0001-9963-0800","full_name":"Black, Tobias","id":"23686","first_name":"Tobias"},{"last_name":"Lankeit","full_name":"Lankeit, Johannes","first_name":"Johannes"},{"first_name":"Masaaki","full_name":"Mizukami, Masaaki","last_name":"Mizukami"}],"volume":18,"date_updated":"2022-12-21T10:05:25Z","doi":"10.1007/s00028-017-0411-5","publication_status":"published","publication_identifier":{"issn":["1424-3199","1424-3202"]},"citation":{"ama":"Black T, Lankeit J, Mizukami M. 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Singular sensitivity in a Keller–Segel-fluid system. <i>Journal of Evolution Equations</i>, <i>18</i>(2), 561–581. <a href=\"https://doi.org/10.1007/s00028-017-0411-5\">https://doi.org/10.1007/s00028-017-0411-5</a>","bibtex":"@article{Black_Lankeit_Mizukami_2017, title={Singular sensitivity in a Keller–Segel-fluid system}, volume={18}, DOI={<a href=\"https://doi.org/10.1007/s00028-017-0411-5\">10.1007/s00028-017-0411-5</a>}, number={2}, journal={Journal of Evolution Equations}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Lankeit, Johannes and Mizukami, Masaaki}, year={2017}, pages={561–581} }","mla":"Black, Tobias, et al. “Singular Sensitivity in a Keller–Segel-Fluid System.” <i>Journal of Evolution Equations</i>, vol. 18, no. 2, Springer Science and Business Media LLC, 2017, pp. 561–81, doi:<a href=\"https://doi.org/10.1007/s00028-017-0411-5\">10.1007/s00028-017-0411-5</a>.","short":"T. Black, J. Lankeit, M. 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Black, J. Lankeit, M. Mizukami, IMA Journal of Applied Mathematics 81 (2016) 860–876.","apa":"Black, T., Lankeit, J., &#38; Mizukami, M. (2016). On the weakly competitive case in a two-species chemotaxis model. <i>IMA Journal of Applied Mathematics</i>, <i>81</i>(5), 860–876. <a href=\"https://doi.org/10.1093/imamat/hxw036\">https://doi.org/10.1093/imamat/hxw036</a>","ama":"Black T, Lankeit J, Mizukami M. On the weakly competitive case in a two-species chemotaxis model. <i>IMA Journal of Applied Mathematics</i>. 2016;81(5):860-876. doi:<a href=\"https://doi.org/10.1093/imamat/hxw036\">10.1093/imamat/hxw036</a>","chicago":"Black, Tobias, Johannes Lankeit, and Masaaki Mizukami. “On the Weakly Competitive Case in a Two-Species Chemotaxis Model.” <i>IMA Journal of Applied Mathematics</i> 81, no. 5 (2016): 860–76. <a href=\"https://doi.org/10.1093/imamat/hxw036\">https://doi.org/10.1093/imamat/hxw036</a>.","ieee":"T. Black, J. Lankeit, and M. 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