Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system

This paper deals with an initial-boundary value problem in a two-dimensional smoothly bounded domain for the Keller-Segel-Navier-Stokes system with logistic source, as given by {n(t) + u . del n = Delta n - del . (n del c) + rn - mu n(2), c(t) + u . del c = Delta c - c + n, u(t) + u . del u = Delta u - del P + n del phi + g, del . u = 0, which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid. It is shown that whenever mu > 0, r >= 0, g is an element of C-1((Omega) over bar x [0, infinity)) boolean AND L-infinity(Omega x (0, infinity)) and the initial data (n(0), c(0), u(0)) are sufficiently smooth fulfilling n(0) (sic) 0, the considered problem possesses a global classical solution which is bounded. Moreover, if r = 0, then this solution satisfies n(., t) -> 0 and c(., t) -> 0 in L-infinity(Omega) as t -> 8, and if additionally integral(0) integral(Omega)vertical bar g(x, t)vertical bar(2)dxdt < infinity, then all solution components decay in the sense that n(., t) -> 0, c(., t) -> 0 and u(., t) -> 0 in L infinity(Omega) as t -> infinity.