Global Large-Data Solutions in a Chemotaxis-(Navier-)Stokes System Modeling Cellular Swimming in Fluid Drops

In the modeling of collective effects arising in bacterial suspensions in fluid drops, coupled chemotaxis-(Navier-) Stokes systems generalizing the prototype {n(t) + u . del n = Delta n - del . (n del c), 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 + kappa(u . del)u + del P + n del phi, x is an element of Omega, t > 0, del . u = 0, x is an element of Omega, t > 0, have been proposed to describe the spontaneous emergence of patterns in populations of oxygen-driven swimming bacteria. Here, kappa is an element of R and the gravitational potential phi are given and Omega subset of R-N is a bounded convex domain with smooth boundary. Under the boundary conditions partial derivative n/partial derivative v = partial derivative c/partial derivative v = 0 and u = 0 on partial derivative Omega, it is shown in this paper that suitable regularity assumptions on the initial data entail the following: If N = 2, then the full chemotaxis-Navier-Stokes system (with any kappa is an element of R) admits a unique global classical solution. If N = 3, then the simplified chemotaxis-Stokes system (with kappa = 0) possesses at least one global weak solution. In particular, no smallness condition on either phi or on the initial data needs to be fulfilled here, as required in a related recent work by Duan et al. [5].