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 ($\star$).
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机译：一类趋化性-Stokes系统，对原型\ [\ left \ {\ begin {array} {rcl}进行归纳化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。 \]在有界凸三维域中考虑，其中$ \ phi \ in W ^ {2，\ infty}（\ Omega）$。本文开发了一种分析方法，该方法包括基于能量的论点和最大Sobolev正则性理论的组合，并允许在\ [m> \ frac {9}的假设下，构造一个相关联的初边值问题的全局有界弱解。 {8}。 \ qquad（\ star）\]此外，获得的解在较大的时间范围内显示出接近空间均匀稳态$（\ frac {1} {| \ Omega |} \ int_ \ Omega n_0,0,0）$ 。这扩展了先前的结果，后者要么依赖于不同的且显然不那么重要的能量类型结构，要么依赖于完全替代的方法，从而在假设强于（$ \ star）的情况下独家获得了可比的结果。
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