<p>This paper is concerned with an initial-boundary problem associated with the following system <Equation ID="Equ115"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1047_Article_Equ115.gif" Format="GIF" Height="64" Rendition="HTML" Resolution="72" Type="Linedraw" Width="358" /> </MediaObject> <EquationSource Format="TEX">$\left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (n\chi (c)\nabla c),\\ c_t+u\cdot \nabla c=\Delta c-nc,\\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,\quad \nabla \cdot u=0, \end{array}\right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>n</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>n</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>c</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>c</mi> <mo>-</mo> <mi>n</mi> <mi>c</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>P</mi> <mo>+</mo> <mi>n</mi> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>under no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded planar domain, where <Equation ID="Equ116"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1047_Article_Equ116.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </MediaObject> <EquationSource Format="TEX">$\chi (c)=\frac{\chi _0}{c^{\theta }}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <msub> <mi>χ</mi> <mn>0</mn> </msub> <msup> <mi>c</mi> <mi>θ</mi> </msup> </mfrac> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1047_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi _0>0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is shown that for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1047_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta \in [0,\frac{1}{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and suitably small initial data corresponding initial-boundary problem possesses a unique classical solution which is globally bounded and stabilizes to some constant equilibria exponentially with a certain convergence rate.</p>

Boundedness and asymptotic stabilization in a two-dimensional chemotaxis-Navier–Stokes system with sub-logarithmic sensitivity

期刊论文
发表日期：

2025年3月24日

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Ji Liu

摘要

This paper is concerned with an initial-boundary problem associated with the following system $\left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (n\chi (c)\nabla c),\\ c_t+u\cdot \nabla c=\Delta c-nc,\\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,\quad \nabla \cdot u=0, \end{array}\right. $ $\{\begin{matrix} n_{t} + u \cdot \nabla n = Δ n - \nabla \cdot (n χ (c) \nabla c), \\ c_{t} + u \cdot \nabla c = Δ c - n c, \\ u_{t} + (u \cdot \nabla) u = Δ u + \nabla P + n \nabla Φ, \nabla \cdot u = 0, \end{matrix})$ under no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded planar domain, where $\chi (c)=\frac{\chi _0}{c^{\theta }}$ $χ (c) = \frac{χ_{0}}{c^{θ}}$ with $\chi _0>0.$ $χ_{0} > 0 .$ It is shown that for $\theta \in [0,\frac{1}{2})$ $θ \in [0, \frac{1}{2})$ and suitably small initial data corresponding initial-boundary problem possesses a unique classical solution which is globally bounded and stabilizes to some constant equilibria exponentially with a certain convergence rate.