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

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Delta n-\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}&=\Delta u+\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 of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $\Omega\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribed time-independent nonnegative function $c_{\star}\in C^{2}\!\big(\overline{\Omega}\big)$. Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.