Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux {nt + u.del n = Delta n -del .(nS(x, n, c)del c) - nm, x is an element of Omega, t > 0, c(t) + u.del c = Delta c -c + m, x is an element of Omega, t > 0, m(t) + u.del m = Delta m - nm, x is an element of Omega, t > 0, (*) u(t) + kappa(u. del)u + del P = Delta U + (n + m)del phi, x is an element of Omega, t > 0, del.u = 0, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-3 with smooth boundary, where kappa is an element of R is given constant, S is a matrix-valued sensitivity satisfying vertical bar S(x,n, c)vertical bar <= C-S(l+n)(-alpha) with some C-S > 0 and alpha >= 0. As the case kappa = 0 (with alpha >= 1/3 or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with kappa not equal 0 admits at least one global weak solution if alpha > 0. To the best of our knowledge, this is the first analytical work for the full three-dimensional four-component chemotaxis-Navier-Stokes system.