Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity

Abstract In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m \Delta {n}^{m} for m ≥ 65 63 m\ge \frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m > 7 6 m\gt \frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m > 9 8 m\gt \frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.