A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization

The Keller-Segel-Navier-Stokes system {n(t) + u . del n = Delta n - chi del . (n del c) + rho n - mu n(2), c(t) + u . del c = Delta c - c + n, (*) u(t) + (u . del)u = Delta u + del P + n del phi+ f (x,t), del . u = 0, is considered in a bounded convex domain Omega subset of R-3 with smooth boundary, where phi is an element of W-1,W-infinity(Omega) and f is an element of C-1 ((Omega) over bar x [0, infinity)), and where chi > 0,rho is an element of R and mu > 0 are given parameters. It is proved that under the assumption that sup(t>o) integral(t+1)(t) parallel to f (. , s)parallel to(L6/6(Omega)) ds be finite, for any sufficiently regular initial data (n(0), c(0), u(0)) satisfying n(0) >= 0 and c(0) >= 0, the initial-value problem for (*) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L-1(Omega) x L-6(Omega) x L-2(Omega;R-3). Moreover, under the explicit hypothesis that mu > chi root rho+/4, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying (n(. , t), c(. , t)) -> (rho+/mu , rho+/mu) in L-1(Omega) x L-P(Omega) all p is an element of [1, 6) as t ->infinity. Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that u(. , t) -> 0 in L-2(Omega;R-3) as t -> infinity. (C) 2018 Elsevier Inc. All rights reserved.