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.del n = Delta n - chi del.(n del c) + kappa n - mu n(2), c(t) + u.del c = Delta c - nc, u(t) + (u.del)u = Delta u + del P + n del Phi + f, del.u = 0 in a bounded, smooth domain Omega subset of R-3 under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u and with given functions f is an element of L-infinity(Omega x (0,infinity)) satisfying certain decay conditions and Phi is an element of C1+beta((Omega) over bar) for some beta is an element of (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 (kappa/mu, 0, 0).