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

In a smoothly bounded domain \(\Omega \subset \mathbb {R}^3\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ , the chemotaxis-Stokes system \(\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot \nabla c =\Delta c - nc, \\ u_t = \Delta u + \nabla P + n\nabla \phi , \qquad \nabla \cdot u =0 \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{n}_t+u·\mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }·(n\mathrm{\nabla }c),\hfill \\ c_t+u·\mathrm{\nabla }c=\mathrm{\Delta }c-nc,\hfill \\ u_t=\mathrm{\Delta }u+\mathrm{\nabla }P+n\mathrm{\nabla }\varphi ,\mathrm{\nabla }·u=0\hfill \end{array}\right)\end{array}$$ is considered along with the boundary conditions \(\begin{aligned} \big (\nabla n - n\nabla c\big )\cdot \nu = 0, \quad c=c_\star , \quad u=0, \quad x\in \partial \Omega , \,\, t>0, \end{aligned}\) $$\begin{array}{c}\hfill (\mathrm{\nabla }n-n\mathrm{\nabla }c)·\nu =0,c=c_{\star },u=0,x\in \partial \mathrm{\Omega },t>0,\end{array}$$ where \(c_\star \ge 0\) $c_{\star }\ge 0$ is a given constant. It is shown that under a smallness condition on \(c(\cdot ,0)\) $c(·,0)$ and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.