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

Abstract

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