<p>In this paper, we study a two-species chemotaxis Navier–Stokes system with Lotka–Volterra competitive kinetics: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n_t+u\cdot \nabla n=\Delta n-\chi _1\nabla \cdot (n\nabla w)+n(\lambda _1-\mu _1n^{\theta -1}-a_1v)$</EquationSource> <EquationSource Format="MATHML"><math> <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> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>μ</mi> <mn>1</mn> </msub> <msup> <mi>n</mi> <mrow> <mi>θ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>; <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$v_t+u\cdot \nabla v=\Delta v-\chi _2\nabla \cdot (v\nabla w)+v(\lambda _2-\mu _2v-a_2n)$</EquationSource> <EquationSource Format="MATHML"><math> <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> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mi>v</mi> <mo>-</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>; <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$w_t+u\cdot \nabla w=\Delta w-w+n+v$</EquationSource> <EquationSource Format="MATHML"><math> <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> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>; <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+(n+v)\nabla \phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>P</mi> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> </mrow> </math></EquationSource> </InlineEquation>; <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nabla \cdot u=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x\in \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in a bounded and smooth domain <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with no-flux/Dirichlet boundary conditions, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\chi _1, \chi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are positive constants. We present the global existence of generalized solution to a two-species chemotaxis Navier–Stokes system.</p>

Global generalized solvability in a two-species chemotaxis Navier–Stokes system with Lotka–Volterra-type competitive kinetics

期刊论文
发表日期：

2026年4月11日

查看全文

Yanxi Li,
Guoqiang Ren,
Jinyu Wei,
Qian Zhao

摘要

In this paper, we study a two-species chemotaxis Navier–Stokes system with Lotka–Volterra competitive kinetics: $n_t+u\cdot \nabla n=\Delta n-\chi _1\nabla \cdot (n\nabla w)+n(\lambda _1-\mu _1n^{\theta -1}-a_1v)$ $n_{t} + u \cdot \nabla n = Δ n - χ_{1} \nabla \cdot (n \nabla w) + n (λ_{1} - μ_{1} n^{θ - 1} - a_{1} v)$ ; $v_t+u\cdot \nabla v=\Delta v-\chi _2\nabla \cdot (v\nabla w)+v(\lambda _2-\mu _2v-a_2n)$ $v_{t} + u \cdot \nabla v = Δ v - χ_{2} \nabla \cdot (v \nabla w) + v (λ_{2} - μ_{2} v - a_{2} n)$ ; $w_t+u\cdot \nabla w=\Delta w-w+n+v$ $w_{t} + u \cdot \nabla w = Δ w - w + n + v$ ; $u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+(n+v)\nabla \phi $ $u_{t} + κ (u \cdot \nabla) u = Δ u + \nabla P + (n + v) \nabla ϕ$ ; $\nabla \cdot u=0$ $\nabla \cdot u = 0$ , $x\in \Omega $ $x \in Ω$ , $t>0$ $t > 0$ in a bounded and smooth domain $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ with no-flux/Dirichlet boundary conditions, where $\chi _1, \chi _2$ $χ_{1}, χ_{2}$ are positive constants. We present the global existence of generalized solution to a two-species chemotaxis Navier–Stokes system.