In this article, we study the Cauchy problem for the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics<Equation ID="Equa"> <EquationSource Format="TEX">$\begin{cases}u_t + (u \cdot \nabla) u = \Delta u + \nabla \mathbb{P} + (m+n) \nabla \phi, \quad \nabla \cdot u = 0, \\ n_t + (u \cdot \nabla) n = \Delta n - \nabla \cdot (n \nabla c) + n(1 - n - m), \\ m_t + (u \cdot \nabla) m = \Delta m - \nabla \cdot (m \nabla c) + m(1 - n - m), \\ c_t + u \cdot \nabla c = \Delta c - (n + m) c.\end{cases}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mi>t</mi> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>n</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mi>t</mi> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>m</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>m</mi> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation>The system is a model that describes the dynamics of two species and comes from a problem on account of the influence of chemotaxis, the Lotka–Volterra kinetics and fluid. By taking advantage of a scale decomposition technique together with a microlocal analysis, we prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space ℝ2.

Global Well-posedness in a Two-dimensional Two-species Chemotaxis-Navier–Stokes System with Competitive Kinetics

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发表日期：

2026年4月15日

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Minghua Yang,
Qiang Zhao,
Yatao Li,
Zunwei Fu

摘要

In this article, we study the Cauchy problem for the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics

$\begin{cases}u_t + (u \cdot \nabla) u = \Delta u + \nabla \mathbb{P} + (m+n) \nabla \phi, \quad \nabla \cdot u = 0, \\ n_t + (u \cdot \nabla) n = \Delta n - \nabla \cdot (n \nabla c) + n(1 - n - m), \\ m_t + (u \cdot \nabla) m = \Delta m - \nabla \cdot (m \nabla c) + m(1 - n - m), \\ c_t + u \cdot \nabla c = \Delta c - (n + m) c.\end{cases}$ ${\begin{matrix} u_{t} + (u \cdot \nabla) u = Δ u + \nabla P + (m + n) \nabla ϕ, \nabla \cdot u = 0, \\ n_{t} + (u \cdot \nabla) n = Δ n - \nabla \cdot (n \nabla c) + n (1 - n - m), \\ m_{t} + (u \cdot \nabla) m = Δ m - \nabla \cdot (m \nabla c) + m (1 - n - m), \\ c_{t} + u \cdot \nabla c = Δ c - (n + m) c . \end{matrix}$

The system is a model that describes the dynamics of two species and comes from a problem on account of the influence of chemotaxis, the Lotka–Volterra kinetics and fluid. By taking advantage of a scale decomposition technique together with a microlocal analysis, we prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space ℝ².