In this paper, we study a two-species chemotaxis–Navier–Stokes system with Lotka–Volterra type competitive kinetics: nt + u · ∇n = Δn − χ1∇ · (n∇w) + n(λ1 − μ1nθ−1 − a1v); vt + u · ∇v = Δv − χ2∇ · (v∇w) + v(λ2 − μ2v − a2n); wt + u · ∇w = Δw − w + n + v; ut + κ(u · ∇)u = Δu + ΔP + (n + v)∇ϕ; <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nabla \cdot u=0\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(x\in \Omega \)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq7.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(t>0\)</EquationSource> </InlineEquation> in a bounded and smooth domain <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \subset {\mathbb {R}}^2\)</EquationSource> </InlineEquation> with no-flux/Dirichlet boundary conditions, where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq9.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi _1, \chi _2\)</EquationSource> </InlineEquation> are positive constants. We present the global existence of generalized solution to a two-species chemotaxis–Navier–Stokes system and the eventual smoothness already occurs in systems with much weaker degradation <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((\theta >1)\)</EquationSource> </InlineEquation>, again under a smallness condition on <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="245_2025_10295_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1, \lambda _2\)</EquationSource> </InlineEquation>.

Global Solvability and Eventual Smoothness in a Two-Species Chemotaxis–Navier–Stokes System with Lotka–Volterra Type Competitive Kinetics

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发表日期：

2025年8月13日

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Guoqiang Ren,
Bin Liu,
Jianshe Yu

摘要

In this paper, we study a two-species chemotaxis–Navier–Stokes system with Lotka–Volterra type competitive kinetics: n_t + u · ∇n = Δn − χ₁∇ · (n∇w) + n(λ₁ − μ₁n^θ−1 − a₁v); v_t + u · ∇v = Δv − χ₂∇ · (v∇w) + v(λ₂ − μ₂v − a₂n); w_t + u · ∇w = Δw − w + n + v; u_t + κ(u · ∇)u = Δu + ΔP + (n + v)∇ϕ; \(\nabla \cdot u=0\) , \(x\in \Omega \) , \(t>0\) in a bounded and smooth domain \(\Omega \subset {\mathbb {R}}^2\) with no-flux/Dirichlet boundary conditions, where \(\chi _1, \chi _2\) are positive constants. We present the global existence of generalized solution to a two-species chemotaxis–Navier–Stokes system and the eventual smoothness already occurs in systems with much weaker degradation \((\theta >1)\) , again under a smallness condition on \(\lambda _1, \lambda _2\) .