Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

A class of chemotaxis-Stokes systems generalizing the prototype $$\{\begin{array}{rclllll}{n}_t+u\cdot \mathrm{\nabla }n& =& \mathrm{\nabla }\cdot \left(n^{m-1}\mathrm{\nabla }n\right)-\mathrm{\nabla }\cdot \left(n\mathrm{\nabla }c\right),c_t+u\cdot \mathrm{\nabla }c& =& \mathrm{\Delta }c-nc,u_t+\mathrm{\nabla }P& =& \mathrm{\Delta }u+n\mathrm{\nabla }\varphi ,\mathrm{\nabla }\cdot u=0,\end{array}$$ is considered in bounded convex three-dimensional domains, where $\varphi \in W^{2,\mathrm{\infty }}(\mathrm{\Omega })$ is given. The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that $$m>\frac{9}{8}\mathrm{.}(\star )$$ Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}{n}_0,0,0)$ in the large time limit. This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than ($\star$).