<p>This paper is concerned with a three-dimensional Keller–Segel–Navier–Stokes system incorporating singular flux limitation and superlinear production. The primary goal is to establish global existence of weak solutions under conditions ensuring that flux limitations suppress the blow-up tendencies induced by superlinear growth. More precisely, this paper focuses on the system in a bounded domain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with smooth boundary, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$0< \alpha < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Under the assumption <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha > 1 - \frac{1}{3\beta -1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>β</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we prove global existence of weak solutions to the Neumann problem for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$(*)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow /> <mo>∗</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This study extends the previous work by Winkler [<CitationRef CitationID="CR27">27</CitationRef>], in which the corresponding system with the regular sensitivity <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$(|\nabla c|^2+1)^{-\frac{\alpha }{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and the linear production <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_958_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$(\beta =1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> was considered, and highlights how strong flux limitation can control the effects of superlinear growth.</p>

Global Weak Solutions in a Three-dimensional Keller–Segel–Navier–Stokes System with Flux Limitation and Superlinear Production

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发表日期：

2025年7月11日

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Jiyuan Guo,
Shohei Kohatsu,
Tomomi Yokota

摘要

This paper is concerned with a three-dimensional Keller–Segel–Navier–Stokes system incorporating singular flux limitation and superlinear production. The primary goal is to establish global existence of weak solutions under conditions ensuring that flux limitations suppress the blow-up tendencies induced by superlinear growth. More precisely, this paper focuses on the system in a bounded domain $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ with smooth boundary, where $0< \alpha < 1$ $0 < α < 1$ and $\beta \ge 1$ $β \geq 1$ . Under the assumption $\alpha > 1 - \frac{1}{3\beta -1}$ $α > 1 - \frac{1}{3 β - 1}$ , we prove global existence of weak solutions to the Neumann problem for $(*)$ $(*)$ . This study extends the previous work by Winkler [27], in which the corresponding system with the regular sensitivity $(|\nabla c|^2+1)^{-\frac{\alpha }{2}}$ ${(| \nabla c |}^{2} {+ 1)}^{- \frac{α}{2}}$ and the linear production $(\beta =1)$ $(β = 1)$ was considered, and highlights how strong flux limitation can control the effects of superlinear growth.