Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation

T. Black, C. Wu, Calculus of Variations and Partial Differential Equations 61 (2022).

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Journal Article | Published | English
Author
Black, TobiasLibreCat ; Wu, Chunyan
Abstract
<jats:title>Abstract</jats:title><jats:p>We consider chemotaxis-Navier–Stokes systems with logistic proliferation and signal consumption of the form "Equation missing"<!-- image only, no MathML or LaTex -->for parameter choices <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> and <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>. Herein, we moreover impose a nonnegative and time-constant prescribed concentration <jats:inline-formula><jats:alternatives><jats:tex-math>$$c_\star \in C^2({\overline{\Omega }})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for the signal chemical on the boundary of the domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Omega \subset {\mathbb {R}}^{\mathcal {N}}$$</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>$${\mathcal {N}}\in \{2,3\}$$</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>. After first extending the previously known result on time-global existence of weak solutions for the Stokes variant to the full Navier–Stokes setting, we proceed with an investigation of eventual regularity properties in the slightly more restrictive setting of <jats:inline-formula><jats:alternatives><jats:tex-math>$$c_\star $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> being also constant in space. We show that sufficiently strong logistic influence, in the sense that for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\omega &gt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mu _0&gt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> there is some <jats:inline-formula><jats:alternatives><jats:tex-math>$$\eta =\eta (\omega ,\mu _0,c_\star )&gt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mi>η</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with the property that whenever <jats:disp-formula><jats:alternatives><jats:tex-math>$$\begin{aligned} \mu _0\le \mu \quad \text {and}\quad \frac{\kappa }{\min \{\mu ,\mu ^{\frac{{\mathcal {N}}+6}{6}+\omega }\}}&lt;\eta \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≤</mml:mo> <mml:mi>μ</mml:mi> <mml:mspace /> <mml:mtext>and</mml:mtext> <mml:mspace /> <mml:mfrac> <mml:mi>κ</mml:mi> <mml:mrow> <mml:mo>min</mml:mo> <mml:mo>{</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:mn>6</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:mo>}</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>&lt;</mml:mo> <mml:mi>η</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>are satisfied the global weak solution eventually becomes a smooth and classical solution with waiting time depending on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\omega ,\mu _0,\eta ,c_\star $$</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:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and the initial data. </jats:p>
Publishing Year
Journal Title
Calculus of Variations and Partial Differential Equations
Volume
61
Issue
3
Article Number
96
LibreCat-ID

Cite this

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
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
@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} }
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.
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.
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.

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