<p>This paper deals with the chemotaxis-Navier–Stokes system with signal-dependent motility and indirect signal production <Equation ID="Equ110"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta (\varphi (v)n),&\qquad \quad x\in \Omega ,\,t>0,\\&v_t+u\cdot \nabla v=\Delta v-v+ w,&\qquad \quad x\in \Omega ,\,t>0,\\&w_t+u\cdot \nabla w=\Delta w-w+n,&\qquad \quad x\in \Omega ,\,t>0,\\&u_t+(u\cdot \nabla )u=\Delta u-\nabla P+n\nabla \Phi ,\nabla \cdot u=0,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd /> <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> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>-</mo> <mi>w</mi> <mo>+</mo> <mi>n</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <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> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>subject to no-flux/no-flux/no-flux/Dirichlet boundary conditions in a bounded and smooth domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \Omega \subset \mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Phi \in W^{2,\infty }(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The motility function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi (v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \varphi (v)\in C^3([0,+\infty )), \varphi _1\le \varphi (v)\le \varphi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi>C</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>φ</mi> <mn>1</mn> </msub> <mo>≤</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|\varphi '(v)|\le \varphi _3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> </mrow> <msub> <mi>φ</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varphi _1,\varphi _2,\varphi _3>0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mn>3</mn> </msub> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The purpose of this paper is to prove that the problem possesses a globally bounded classical solution.</p>

Boundedness in a two-dimensional chemotaxis-Navier–Stokes system with signal-dependent motility and indirect signal production

期刊论文
发表日期：

2025年10月11日

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Dan Li

摘要

This paper deals with the chemotaxis-Navier–Stokes system with signal-dependent motility and indirect signal production $\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta (\varphi (v)n),&\qquad \quad x\in \Omega ,\,t>0,\\&v_t+u\cdot \nabla v=\Delta v-v+ w,&\qquad \quad x\in \Omega ,\,t>0,\\&w_t+u\cdot \nabla w=\Delta w-w+n,&\qquad \quad x\in \Omega ,\,t>0,\\&u_t+(u\cdot \nabla )u=\Delta u-\nabla P+n\nabla \Phi ,\nabla \cdot u=0,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} n_{t} + u \cdot \nabla n = Δ (φ (v) n), & x \in Ω, t > 0, \\ v_{t} + u \cdot \nabla v = Δ v - v + w, & x \in Ω, t > 0, \\ w_{t} + u \cdot \nabla w = Δ w - w + n, & x \in Ω, t > 0, \\ u_{t} + (u \cdot \nabla) u = Δ u - \nabla P + n \nabla Φ, \nabla \cdot u = 0, & x \in Ω, t > 0 \end{matrix}) \end{matrix}$ subject to no-flux/no-flux/no-flux/Dirichlet boundary conditions in a bounded and smooth domain $ \Omega \subset \mathbb {R}^2$ $Ω \subset R^{2}$ , where $\Phi \in W^{2,\infty }(\Omega )$ $Φ \in W^{2, \infty} (Ω)$ . The motility function $\varphi (v)$ $φ (v)$ satisfies $ \varphi (v)\in C^3([0,+\infty )), \varphi _1\le \varphi (v)\le \varphi _2$ $φ (v) \in C^{3} ([0, + \infty)), φ_{1} \leq φ (v) \leq φ_{2}$ and $|\varphi '(v)|\le \varphi _3$ $| φ^{'} (v) | \leq φ_{3}$ with $\varphi _1,\varphi _2,\varphi _3>0.$ $φ_{1}, φ_{2}, φ_{3} > 0 .$ The purpose of this paper is to prove that the problem possesses a globally bounded classical solution.