We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in two dimensions, the system has a unique global classical solution, while the existence of a global weak solution is shown in three dimensions. In the latter case, we show that the energy is bounded uniformly in time. A key idea is to utilise a boundary energy to derive suitable {\it a priori} estimates. Moreover, we are able to remove the convexity assumption on the domain.

中文翻译：

---

具有罗宾边界条件的趋化-纳维-斯托克斯系统的全局解

我们考虑了一个趋化性-Navier-Stokes 系统，该系统模拟了液滴中的细胞游泳，其中考虑了液滴与其环境之间的氧气交换。这种现象导致不均匀的罗宾型边界条件。此外，在没有细菌种群的逻辑增长的情况下研究该系统。我们证明了在二维上，系统具有唯一的全局经典解，而在三个维度上证明了全局弱解的存在。在后一种情况下，我们表明能量在时间上是均匀有界的。一个关键的想法是利用边界能量来推导出合适的 {\it aprii} 估计。此外，我们能够去除域上的凸性假设。