[{"publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:13:58Z","volume":26,"date_created":"2025-12-18T19:05:09Z","author":[{"first_name":"Mario","last_name":"Fuest","full_name":"Fuest, Mario"},{"first_name":"Michael","full_name":"Winkler, Michael","id":"31496","last_name":"Winkler"}],"title":"Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing","doi":"10.1007/s00021-024-00899-8","publication_identifier":{"issn":["1422-6928","1422-6952"]},"publication_status":"published","issue":"4","year":"2024","intvolume":"        26","citation":{"ama":"Fuest M, Winkler M. Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing. <i>Journal of Mathematical Fluid Mechanics</i>. 2024;26(4). doi:<a href=\"https://doi.org/10.1007/s00021-024-00899-8\">10.1007/s00021-024-00899-8</a>","chicago":"Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing.” <i>Journal of Mathematical Fluid Mechanics</i> 26, no. 4 (2024). <a href=\"https://doi.org/10.1007/s00021-024-00899-8\">https://doi.org/10.1007/s00021-024-00899-8</a>.","ieee":"M. Fuest and M. Winkler, “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing,” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 26, no. 4, Art. no. 60, 2024, doi: <a href=\"https://doi.org/10.1007/s00021-024-00899-8\">10.1007/s00021-024-00899-8</a>.","mla":"Fuest, Mario, and Michael Winkler. “Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing.” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 26, no. 4, 60, Springer Science and Business Media LLC, 2024, doi:<a href=\"https://doi.org/10.1007/s00021-024-00899-8\">10.1007/s00021-024-00899-8</a>.","bibtex":"@article{Fuest_Winkler_2024, title={Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing}, volume={26}, DOI={<a href=\"https://doi.org/10.1007/s00021-024-00899-8\">10.1007/s00021-024-00899-8</a>}, number={460}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer Science and Business Media LLC}, author={Fuest, Mario and Winkler, Michael}, year={2024} }","short":"M. Fuest, M. Winkler, Journal of Mathematical Fluid Mechanics 26 (2024).","apa":"Fuest, M., &#38; Winkler, M. (2024). Uniform $$L^p$$ Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing. <i>Journal of Mathematical Fluid Mechanics</i>, <i>26</i>(4), Article 60. <a href=\"https://doi.org/10.1007/s00021-024-00899-8\">https://doi.org/10.1007/s00021-024-00899-8</a>"},"_id":"63254","user_id":"31496","article_number":"60","language":[{"iso":"eng"}],"publication":"Journal of Mathematical Fluid Mechanics","type":"journal_article","abstract":[{"text":"<jats:title>Abstract</jats:title><jats:p>The chemotaxis-Navier–Stokes system <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\left\\{ \\begin{array}{rcl} n_t+u\\cdot \\nabla n &amp; =&amp;  \\Delta \\big (n c^{-\\alpha } \\big ), \\\\ c_t+ u\\cdot \\nabla c &amp; =&amp;  \\Delta c -nc,\\\\ u_t + (u\\cdot \\nabla ) u &amp; =&amp; \\Delta u+\\nabla P + n\\nabla \\Phi , \\qquad \\nabla \\cdot u=0, \\end{array} \\right. \\end{aligned}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mtable>\r\n                      <mml:mtr>\r\n                        <mml:mtd>\r\n                          <mml:mfenced>\r\n                            <mml:mrow>\r\n                              <mml:mtable>\r\n                                <mml:mtr>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:msub>\r\n                                        <mml:mi>n</mml:mi>\r\n                                        <mml:mi>t</mml:mi>\r\n                                      </mml:msub>\r\n                                      <mml:mo>+</mml:mo>\r\n                                      <mml:mi>u</mml:mi>\r\n                                      <mml:mo>·</mml:mo>\r\n                                      <mml:mi>∇</mml:mi>\r\n                                      <mml:mi>n</mml:mi>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mo>=</mml:mo>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>Δ</mml:mi>\r\n                                      <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n                                      </mml:mrow>\r\n                                      <mml:mi>n</mml:mi>\r\n                                      <mml:msup>\r\n                                        <mml:mi>c</mml:mi>\r\n                                        <mml:mrow>\r\n                                          <mml:mo>-</mml:mo>\r\n                                          <mml:mi>α</mml:mi>\r\n                                        </mml:mrow>\r\n                                      </mml:msup>\r\n                                      <mml:mrow>\r\n                                        <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n                                      <mml:mo>,</mml:mo>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                </mml:mtr>\r\n                                <mml:mtr>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:mrow/>\r\n                                      <mml:msub>\r\n                                        <mml:mi>c</mml:mi>\r\n                                        <mml:mi>t</mml:mi>\r\n                                      </mml:msub>\r\n                                      <mml:mo>+</mml:mo>\r\n                                      <mml:mi>u</mml:mi>\r\n                                      <mml:mo>·</mml:mo>\r\n                                      <mml:mi>∇</mml:mi>\r\n                                      <mml:mi>c</mml:mi>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mo>=</mml:mo>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>Δ</mml:mi>\r\n                                      <mml:mi>c</mml:mi>\r\n                                      <mml:mo>-</mml:mo>\r\n                                      <mml:mi>n</mml:mi>\r\n                                      <mml:mi>c</mml:mi>\r\n                                      <mml:mo>,</mml:mo>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                </mml:mtr>\r\n                                <mml:mtr>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:mrow/>\r\n                                      <mml:msub>\r\n                                        <mml:mi>u</mml:mi>\r\n                                        <mml:mi>t</mml:mi>\r\n                                      </mml:msub>\r\n                                      <mml:mo>+</mml:mo>\r\n                                      <mml:mrow>\r\n                                        <mml:mo>(</mml:mo>\r\n                                        <mml:mi>u</mml:mi>\r\n                                        <mml:mo>·</mml:mo>\r\n                                        <mml:mi>∇</mml:mi>\r\n                                        <mml:mo>)</mml:mo>\r\n                                      </mml:mrow>\r\n                                      <mml:mi>u</mml:mi>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mo>=</mml:mo>\r\n                                  </mml:mtd>\r\n                                  <mml:mtd>\r\n                                    <mml:mrow>\r\n                                      <mml:mi>Δ</mml:mi>\r\n                                      <mml:mi>u</mml:mi>\r\n                                      <mml:mo>+</mml:mo>\r\n                                      <mml:mi>∇</mml:mi>\r\n                                      <mml:mi>P</mml:mi>\r\n                                      <mml:mo>+</mml:mo>\r\n                                      <mml:mi>n</mml:mi>\r\n                                      <mml:mi>∇</mml:mi>\r\n                                      <mml:mi>Φ</mml:mi>\r\n                                      <mml:mo>,</mml:mo>\r\n                                      <mml:mspace/>\r\n                                      <mml:mi>∇</mml:mi>\r\n                                      <mml:mo>·</mml:mo>\r\n                                      <mml:mi>u</mml:mi>\r\n                                      <mml:mo>=</mml:mo>\r\n                                      <mml:mn>0</mml:mn>\r\n                                      <mml:mo>,</mml:mo>\r\n                                    </mml:mrow>\r\n                                  </mml:mtd>\r\n                                </mml:mtr>\r\n                              </mml:mtable>\r\n                            </mml:mrow>\r\n                          </mml:mfenced>\r\n                        </mml:mtd>\r\n                      </mml:mtr>\r\n                    </mml:mtable>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:disp-formula>modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega \\subset \\mathbb R^2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>Ω</mml:mi>\r\n                    <mml:mo>⊂</mml:mo>\r\n                    <mml:msup>\r\n                      <mml:mi>R</mml:mi>\r\n                      <mml:mn>2</mml:mn>\r\n                    </mml:msup>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>. For all <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha &gt; 0$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>α</mml:mi>\r\n                    <mml:mo>&gt;</mml:mo>\r\n                    <mml:mn>0</mml:mn>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> and all sufficiently regular <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Phi $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>Φ</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for <jats:italic>u</jats:italic> at a point in the proof when knowledge about <jats:italic>n</jats:italic> is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msup>\r\n                    <mml:mi>L</mml:mi>\r\n                    <mml:mi>p</mml:mi>\r\n                  </mml:msup>\r\n                </mml:math></jats:alternatives></jats:inline-formula> estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msup>\r\n                    <mml:mi>L</mml:mi>\r\n                    <mml:mn>2</mml:mn>\r\n                  </mml:msup>\r\n                </mml:math></jats:alternatives></jats:inline-formula> norm of the force term raised to an arbitrary small power.</jats:p>","lang":"eng"}],"status":"public"},{"keyword":["Applied Mathematics","Computational Mathematics","Condensed Matter Physics","Mathematical Physics"],"language":[{"iso":"eng"}],"publication":"Journal of Mathematical Fluid Mechanics","title":"The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System","publisher":"Springer Science and Business Media LLC","date_created":"2022-12-21T09:47:56Z","year":"2019","issue":"1","article_number":"1","_id":"34669","department":[{"_id":"34"},{"_id":"10"},{"_id":"90"}],"user_id":"23686","status":"public","type":"journal_article","doi":"10.1007/s00021-019-0464-z","date_updated":"2022-12-21T10:04:29Z","volume":22,"author":[{"last_name":"Black","orcid":"0000-0001-9963-0800","id":"23686","full_name":"Black, Tobias","first_name":"Tobias"}],"intvolume":"        22","citation":{"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>","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>.","short":"T. Black, Journal of Mathematical Fluid Mechanics 22 (2019).","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>.","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>.","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>"},"publication_identifier":{"issn":["1422-6928","1422-6952"]},"publication_status":"published"},{"_id":"63375","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"Journal of Mathematical Fluid Mechanics","status":"public","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-19T11:06:09Z","date_created":"2025-12-19T11:06:02Z","author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"volume":20,"title":"Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?","doi":"10.1007/s00021-018-0395-0","publication_status":"published","publication_identifier":{"issn":["1422-6928","1422-6952"]},"issue":"4","year":"2018","citation":{"ama":"Winkler M. Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity? <i>Journal of Mathematical Fluid Mechanics</i>. 2018;20(4):1889-1909. doi:<a href=\"https://doi.org/10.1007/s00021-018-0395-0\">10.1007/s00021-018-0395-0</a>","chicago":"Winkler, Michael. “Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?” <i>Journal of Mathematical Fluid Mechanics</i> 20, no. 4 (2018): 1889–1909. <a href=\"https://doi.org/10.1007/s00021-018-0395-0\">https://doi.org/10.1007/s00021-018-0395-0</a>.","ieee":"M. Winkler, “Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?,” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 20, no. 4, pp. 1889–1909, 2018, doi: <a href=\"https://doi.org/10.1007/s00021-018-0395-0\">10.1007/s00021-018-0395-0</a>.","apa":"Winkler, M. (2018). Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity? <i>Journal of Mathematical Fluid Mechanics</i>, <i>20</i>(4), 1889–1909. <a href=\"https://doi.org/10.1007/s00021-018-0395-0\">https://doi.org/10.1007/s00021-018-0395-0</a>","mla":"Winkler, Michael. “Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?” <i>Journal of Mathematical Fluid Mechanics</i>, vol. 20, no. 4, Springer Science and Business Media LLC, 2018, pp. 1889–909, doi:<a href=\"https://doi.org/10.1007/s00021-018-0395-0\">10.1007/s00021-018-0395-0</a>.","short":"M. Winkler, Journal of Mathematical Fluid Mechanics 20 (2018) 1889–1909.","bibtex":"@article{Winkler_2018, title={Does Fluid Interaction Affect Regularity in the Three-Dimensional Keller–Segel System with Saturated Sensitivity?}, volume={20}, DOI={<a href=\"https://doi.org/10.1007/s00021-018-0395-0\">10.1007/s00021-018-0395-0</a>}, number={4}, journal={Journal of Mathematical Fluid Mechanics}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2018}, pages={1889–1909} }"},"page":"1889-1909","intvolume":"        20"}]