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We study a chemotaxis-Stokes system with signal consumption and logisticsource terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\}l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Deltan-\nabla\!\cdot(n\nabla c)+\kappa n-\mu n^{2},\ &x\in\Omega,& t>0,\\c_{t}+u\cdot\!\nabla c&=\Delta c-nc,\ &x\in\Omega,& t>0,\\ u_{t}&=\Deltau+\nabla P+n\nabla\phi,\ &x\in\Omega,& t>0,\\ \nabla\cdot u&=0,\ &x\in\Omega,&t>0,\\ \big(\nabla n-n\nabla c\big)\cdot\nu&=0,\quad c=c_{\star}(x),\quad u=0,&x\in\partial\Omega,& t>0, \end{array}\right. \end{align*} where $\kappa\geq0$,$\mu>0$ and, in contrast to the commonly investigated variants ofchemotaxis-fluid systems, the signal concentration on the boundary of thedomain $\Omega\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribedtime-independent nonnegative function $c_{\star}\inC^{2}\!\big(\overline{\Omega}\big)$. Making use of the boundedness informationentailed by the quadratic decay term of the first equation, we will show thatthe system above has at least one global weak solution for any suitably regulartriplet of initial data.

Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation