Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?
M. Winkler, Acta Applicandae Mathematicae 169 (2020) 577–591.
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<jats:title>Abstract</jats:title><jats:p>In a bounded planar domain <jats:inline-formula><jats:alternatives><jats:tex-math>$\varOmega $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Ω</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system
<jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} n_{t} + \nabla \cdot (nu) = \Delta n - \nabla \cdot (n\nabla c), & x\in \varOmega , \ t>0, \\ 0 = \Delta c -c+n, & x\in \varOmega , \ t>0, \end{array}\displaystyle \right . \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>{</mml:mo>
<mml:mtable>
<mml:mtr>
<mml:mtd>
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<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>+</mml:mo>
<mml:mi>∇</mml:mi>
<mml:mo>⋅</mml:mo>
<mml:mo>(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>Δ</mml:mi>
<mml:mi>n</mml:mi>
<mml:mo>−</mml:mo>
<mml:mi>∇</mml:mi>
<mml:mo>⋅</mml:mo>
<mml:mo>(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mi>∇</mml:mi>
<mml:mi>c</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>,</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mi>x</mml:mi>
<mml:mo>∈</mml:mo>
<mml:mi>Ω</mml:mi>
<mml:mo>,</mml:mo>
<mml:mspace/>
<mml:mi>t</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
</mml:mtd>
</mml:mtr>
<mml:mtr>
<mml:mtd>
<mml:mn>0</mml:mn>
<mml:mo>=</mml:mo>
<mml:mi>Δ</mml:mi>
<mml:mi>c</mml:mi>
<mml:mo>−</mml:mo>
<mml:mi>c</mml:mi>
<mml:mo>+</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo>,</mml:mo>
</mml:mtd>
<mml:mtd>
<mml:mi>x</mml:mi>
<mml:mo>∈</mml:mo>
<mml:mi>Ω</mml:mi>
<mml:mo>,</mml:mo>
<mml:mspace/>
<mml:mi>t</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
</mml:mtd>
</mml:mtr>
</mml:mtable>
</mml:mrow>
</mml:math></jats:alternatives></jats:disp-formula> is considered, where <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>u</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> is a given sufficiently smooth velocity field on <jats:inline-formula><jats:alternatives><jats:tex-math>$\overline {\varOmega }\times [0,\infty )$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>Ω</mml:mi>
<mml:mo>‾</mml:mo>
</mml:mover>
<mml:mo>×</mml:mo>
<mml:mo>[</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>∞</mml:mi>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> that is tangential on <jats:inline-formula><jats:alternatives><jats:tex-math>$\partial \varOmega $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>∂</mml:mi>
<mml:mi>Ω</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> but not necessarily solenoidal.</jats:p><jats:p>It is firstly shown that for any choice of <jats:inline-formula><jats:alternatives><jats:tex-math>$n_{0}\in C^{0}(\overline {\varOmega })$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>∈</mml:mo>
<mml:msup>
<mml:mi>C</mml:mi>
<mml:mn>0</mml:mn>
</mml:msup>
<mml:mo>(</mml:mo>
<mml:mover>
<mml:mi>Ω</mml:mi>
<mml:mo>‾</mml:mo>
</mml:mover>
<mml:mo>)</mml:mo>
</mml:math></jats:alternatives></jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$\int _{\varOmega}n_{0}<4\pi $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mo>∫</mml:mo>
<mml:mi>Ω</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo><</mml:mo>
<mml:mn>4</mml:mn>
<mml:mi>π</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>, this problem admits a global classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$n(\cdot ,0)=n_{0}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>n</mml:mi>
<mml:mo>(</mml:mo>
<mml:mo>⋅</mml:mo>
<mml:mo>,</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>)</mml:mo>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula>, and that this solution is even bounded whenever <jats:inline-formula><jats:alternatives><jats:tex-math>$u$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>u</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> is bounded and <jats:inline-formula><jats:alternatives><jats:tex-math>$\int _{\varOmega}n_{0}<2\pi $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mo>∫</mml:mo>
<mml:mi>Ω</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo><</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>π</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>. Secondly, it is seen that for each <jats:inline-formula><jats:alternatives><jats:tex-math>$m>4\pi $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>m</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>4</mml:mn>
<mml:mi>π</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> one can find a classical solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$\int _{\varOmega}n(\cdot ,0)=m$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mo>∫</mml:mo>
<mml:mi>Ω</mml:mi>
</mml:msub>
<mml:mi>n</mml:mi>
<mml:mo>(</mml:mo>
<mml:mo>⋅</mml:mo>
<mml:mo>,</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>m</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> which blows up in finite time, provided that <jats:inline-formula><jats:alternatives><jats:tex-math>$\varOmega $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Ω</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> satisfies a technical assumption requiring <jats:inline-formula><jats:alternatives><jats:tex-math>$\partial \varOmega $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>∂</mml:mi>
<mml:mi>Ω</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> to contain a line segment.</jats:p><jats:p>In particular, this indicates that the value <jats:inline-formula><jats:alternatives><jats:tex-math>$4\pi $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mn>4</mml:mn>
<mml:mi>π</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.</jats:p>
Publishing Year
Journal Title
Acta Applicandae Mathematicae
Volume
169
Issue
1
Page
577-591
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Cite this
Winkler M. Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains? Acta Applicandae Mathematicae. 2020;169(1):577-591. doi:10.1007/s10440-020-00312-2
Winkler, M. (2020). Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains? Acta Applicandae Mathematicae, 169(1), 577–591. https://doi.org/10.1007/s10440-020-00312-2
@article{Winkler_2020, title={Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?}, volume={169}, DOI={10.1007/s10440-020-00312-2}, number={1}, journal={Acta Applicandae Mathematicae}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={577–591} }
Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?” Acta Applicandae Mathematicae 169, no. 1 (2020): 577–91. https://doi.org/10.1007/s10440-020-00312-2.
M. Winkler, “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?,” Acta Applicandae Mathematicae, vol. 169, no. 1, pp. 577–591, 2020, doi: 10.1007/s10440-020-00312-2.
Winkler, Michael. “Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?” Acta Applicandae Mathematicae, vol. 169, no. 1, Springer Science and Business Media LLC, 2020, pp. 577–91, doi:10.1007/s10440-020-00312-2.
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