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A class of chemotaxis-Stokes systems generalizing the prototype \[\left\{\begin{array}{rcl} n_t + u\cdot\nabla n &=& \nabla \cdot \big(n^{m-1}\nablan\big) - \nabla \cdot \big(n\nabla c\big), c_t + u\cdot\nabla c &=& \Deltac-nc, u_t +\nabla P &=& \Delta u + n \nabla \phi, \qquad \nabla\cdot u =0,\end{array} \right. \] is considered in bounded convex three-dimensionaldomains, where $\phi\in W^{2,\infty}(\Omega)$ is given. The paper develops ananalytical approach which consists in a combination of energy-based argumentsand maximal Sobolev regularity theory, and which allows for the construction ofglobal bounded weak solutions to an associated initial-boundary value problemunder the assumption that \[ m>\frac{9}{8}. \qquad (\star) \] Moreover, theobtained solutions are shown to approach the spatially homogeneous steady state$(\frac{1}{|\Omega|} \int_\Omega n_0,0,0)$ in the large time limit. Thisextends previous results which either relied on different and apparently lesssignificant energy-type structures, or on completely alternative approaches,and thereby exclusively achieved comparable results under hypotheses strongerthan ($\star$).

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