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We consider the coupled chemotaxis Navier–Stokes model with logistic source terms: nt + u ⋅∇n = Δn − χ∇⋅ (n∇c) + κn − μn2, ct + u ⋅∇c = Δc − nc, ut + (u ⋅∇)u = Δu + ∇P + n∇Φ + f,∇⋅ u = 0 in a bounded, smooth domain Ω ⊂ ℝ3 under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u and with given functions f ∈ L∞(Ω × (0,∞)) satisfying certain decay conditions and Φ ∈ C1+β(Ω) for some β ∈ (0, 1). We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state (κ μ, 0, 0).

Long-term behaviour in a chemotaxis-fluid system with logistic source