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

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 \(\lambda _1, \lambda _2\).