Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

T. Black, C. Wu, Zeitschrift Für Angewandte Mathematik Und Physik 72 (2021).

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Journal Article | Published | English
Author
Black, TobiasLibreCat ; Wu, Chunyan
Abstract
<jats:title>Abstract</jats:title><jats:p>We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form "Equation missing"<!-- image only, no MathML or LaTex -->where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\kappa \ge 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mu &gt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Omega \subset \mathbb {R}^N$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$N\in \{2,3\}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is a prescribed time-independent nonnegative function <jats:inline-formula><jats:alternatives><jats:tex-math>$$c_*\in C^{2}\!\left( {{\,\mathrm{\overline{\Omega }}\,}}\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace /> <mml:mfenced> <mml:mrow> <mml:mspace /> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mspace /> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.</jats:p>
Publishing Year
Journal Title
Zeitschrift für angewandte Mathematik und Physik
Volume
72
Issue
4
Article Number
135
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Cite this

Black T, Wu C. Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation. Zeitschrift für angewandte Mathematik und Physik. 2021;72(4). doi:10.1007/s00033-021-01565-z
Black, T., & Wu, C. (2021). Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation. Zeitschrift Für Angewandte Mathematik Und Physik, 72(4), Article 135. https://doi.org/10.1007/s00033-021-01565-z
@article{Black_Wu_2021, title={Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation}, volume={72}, DOI={10.1007/s00033-021-01565-z}, number={4135}, journal={Zeitschrift für angewandte Mathematik und Physik}, publisher={Springer Science and Business Media LLC}, author={Black, Tobias and Wu, Chunyan}, year={2021} }
Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Weak Solvability in a Chemotaxis-Stokes System with Proliferation.” Zeitschrift Für Angewandte Mathematik Und Physik 72, no. 4 (2021). https://doi.org/10.1007/s00033-021-01565-z.
T. Black and C. Wu, “Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation,” Zeitschrift für angewandte Mathematik und Physik, vol. 72, no. 4, Art. no. 135, 2021, doi: 10.1007/s00033-021-01565-z.
Black, Tobias, and Chunyan Wu. “Prescribed Signal Concentration on the Boundary: Weak Solvability in a Chemotaxis-Stokes System with Proliferation.” Zeitschrift Für Angewandte Mathematik Und Physik, vol. 72, no. 4, 135, Springer Science and Business Media LLC, 2021, doi:10.1007/s00033-021-01565-z.

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