Boundedness in the 3D Keller–Segel–Stokes system with nonlinear diffusion and indirect signal production

This paper considers the Keller–Segel–Stokes system with indirect signal production \(\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\nabla \cdot (n^{m-1}\nabla n)-\chi \nabla \cdot (n\nabla v)+rn-\mu n^\alpha ,\\&v_{t}+u\cdot \nabla v=\Delta v-v+w,\\&w_{t}+u\cdot \nabla w=\Delta w-w+n,\\&u_{t}=\Delta u-\nabla P+n\nabla \phi ,\nabla \cdot u=0 \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{n}_t+u·\mathrm{\nabla }n=\mathrm{\nabla }·(n^{m-1}\mathrm{\nabla }n)-\chi \mathrm{\nabla }·(n\mathrm{\nabla }v)+rn-\mu n^{\alpha },\hfill \\ \hfill & v_t+u·\mathrm{\nabla }v=\mathrm{\Delta }v-v+w,\hfill \\ \hfill & w_t+u·\mathrm{\nabla }w=\mathrm{\Delta }w-w+n,\hfill \\ \hfill & u_t=\mathrm{\Delta }u-\mathrm{\nabla }P+n\mathrm{\nabla }\varphi ,\mathrm{\nabla }·u=0\hfill \end{array}\right)\end{array}$$ in a bounded domain \(\Omega \subset \mathbb {R}^{3}\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with smooth boundary, where \(\chi >0\) $\chi >0$ , \(r\in \mathbb {R}\) $r\in \mathbb{R}$ , \(\mu >0\) $\mu >0$ and \(\phi \in W^{2,\infty }(\Omega )\) $\varphi \in W^{2,\infty }(\mathrm{\Omega })$ . We prove that under the conditions \(m>1 \) $m>1$ and \(\alpha \in \left( \frac{5}{4},2\right) \) $\alpha \in \left(\frac{5}{4},,,2\right)$ , the initial-boundary value problem of the 3D Keller–Segel–Stokes system with nonlinear diffusion possesses a globally bounded weak solution for all reasonable regular initial data.