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

We consider the coupled chemotaxis Navier-Stokes model with logistic source terms $n_t + u\cdot \nabla n = Δn - χ\nabla \cdot (n \nabla c) + κn - μn^2$ $c_t + u\cdot \nabla c = Δc - nc$ $u_t + (u\cdot \nabla)u = Δu +\nabla P + n\nabla Φ+ f, \quad\qquad \nabla \cdot u=0$ in a bounded, smooth domain $Ω\subset \mathbb{R}^3$ under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$ and with given functions $f\in L^\infty(Ω\times(0,\infty))$ satisfying certain decay conditions and $Φ\in C^{1+β}(\barΩ)$ for some $β\in(0,1)$. We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state $(\fracκμ,0,0)$. Keywords: chemotaxis, Navier-Stokes, logistic source, boundedness, large-time behaviour

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