Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal

The chemotaxis-Stokes system {n(t) + u . del n = del . (n(m-1)del n) - del . (n del c), c(t )+ u . del c = Delta c - nc, u(t) = Delta u + del p + n del phi, del . u = 0 is considered subject to the boundary condition (n(m-1)del n - n del c) . v = 0, c = c(* )(x, t), u = 0, x is an element of partial derivative Omega, t > 0, with m >= 1 and a given nonnegative function c(*) is an element of C-2 ((Omega) over bar x [0, infinity)). In contrast to the well-studied case when the second requirement herein is replaced by a homogeneous Neumann boundary condition for c, the Dirichlet condition imposed here seems to destroy a natural energy-like property that has formed a core ingredient in the literature by providing comprehensive regularity features of the latter problem. This paper attempts to suitably cope with accordingly poor regularity information in order to nevertheless derive a statement on global existence within a generalized framework of solvability which involves appropriately mild requirements on regularity, but which maintains mass conservation in the first component as a key solution property.