Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation

We consider the coupled chemotaxis–Navier–Stokes system with logistic source term [Formula: see text] in a bounded, smooth domain [Formula: see text], where [Formula: see text] and where [Formula: see text], [Formula: see text] and [Formula: see text] are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem for this system under no-flux boundary conditions for [Formula: see text] and [Formula: see text] and homogeneous Dirichlet boundary condition for [Formula: see text] possesses at least one globally defined weak solution. And this weak solution becomes smooth after some waiting time provided [Formula: see text].