Global boundedness in a chemotaxis-Stokes system with nonlinear diffusion mechanism involving gradient dependent flux limitation and indirect signal production

This paper is concerned with the Keller-Segel-Stokes system$\{\begin{array}{cc}& n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot \left(nf\right(\left|\nabla v{|}^2\right)\nabla v),\\ & v_t+u\cdot \nabla v=\Delta v-v+w,\\ & w_t+u\cdot \nabla w=\Delta w-w+n,\\ & u_t=\Delta u+\nabla P+n\nabla \varphi ,\nabla \cdot u=0,\end{array}$ under no-flux/no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain $\Omega \subset R^3$, with given suitably regular functions f and ϕ, as well as f satisfies $f\left(\xi \right)⩽K_f{(1+\xi )}^{-\frac{\alpha }{2}}$, $\xi ⩾0$ and $K_f>0$. It is shown that for all suitably regular initial data the associated initial-boundary value problem possesses at least one globally bounded weak solution provided $9m+4\alpha >10$. Our result strictly proved that the volume saturation effect is indeed conductive to the global existence and boundedness of the three-dimensional Keller-Segel-Stokes system.