Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity

This paper deals with the Keller-Segel-Navier-Stokes system {n(t) +u .del(n) =Delta(n) - del.(nS(x,n,c). del c), c(t) +u .del(c) =Delta(c) - c+n, u(t)+(u . del)(u) = Delta(u) - del P + n del phi, del . u =0, in a bounded domain Omega subset of R-3 with smooth boundary, where phi is an element of W-2,W-infinity(Omega) and S is an element of C-2((Omega) over bar x [ 0,infinity)(2); R-3x3) are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component n. Under the assumption that there exist S-0 > 0 and alpha > 1/3 such that |S(x, n, c)| <= S-0 . (1 + n)(-alpha) for all x is an element of(Omega) over bar, n >= 0 and c >= 0, it is finally shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined weak solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on a here is optimal. This result extends previous studies on global solvability for this system in the two-dimensional domain and for the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.