Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form {n(t) + u . del n = Delta n - del . (n del c) + kappa n - mu n(2), x is an element of Omega, t > 0, c(t) + u . del c = Delta c - nc, x is an element of Omega, t > 0, u(t) = Delta u + del P + n del phi, x is an element of Omega O, t > 0, del . u = 0, x is an element of Omega, t > 0, (del n - n del c) . nu = 0, c = c(*)(x), u = 0, x is an element of partial derivative Omega, t > 0, where kappa >= 0, mu > 0 and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain Omega subset of R-N with N is an element of {2, 3} is a prescribed time-independent nonnegative function c(*) is an element of C-2((Omega) over bar) Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.