Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion

We consider an initial boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model {n(t) + u center dot del n = Delta n(m) - del center dot(n chi(c)del c), x is an element of Omega, t > 0, c(t) + u center dot del c = Delta c - nf(c), x is an element of Omega, t > 0, u(t) + k(u center dot del)u= Delta u +del P + N del phi, x is an element of Omega, t > 0, del center dot u = 0. x is an element of Omega, t > 0 in a bounded convex domain Omega subset of R-3. Here K is an element of R, phi is an element of W-1,W-infinity(Omega) , 0 < chi is an element of C-2 ([0, infinity)) and 0 <= f is an element of C-1 ([0, infinity)) with f(0) = 0. It is proved that under appropriate structural assumptions on f and chi, for any choice of m >= 2/3 and all sufficiently smooth initial data (n(0), C-0, u(0)) the model possesses at least one global weak solution. (C) 2015 Elsevier Inc. All rights reserved.