Smooth solutions in a three-dimensional chemotaxis-Stokes system involving Dirichlet boundary conditions for the signal

In a smoothly bounded domain \(\Omega \subset \mathbb {R}^3\), the chemotaxis-Stokes system

$$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot \nabla c =\Delta c - nc, \\ u_t = \Delta u + \nabla P + n\nabla \phi , \qquad \nabla \cdot u =0 \end{array} \right. \end{aligned}$$

is considered along with the boundary conditions

$$\begin{aligned} \big (\nabla n - n\nabla c\big )\cdot \nu = 0, \quad c=c_\star , \quad u=0, \quad x\in \partial \Omega , \,\, t>0, \end{aligned}$$

where \(c_\star \ge 0\) is a given constant. It is shown that under a smallness condition on \(c(\cdot ,0)\) and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.