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We consider initial-boundary value problems for the $\kappa$-dependent familyof chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{\begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c}n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla c),\ &x\in\Omega,&t>0,\\ c_{t}&+&u\cdot\!\nabla c&=\Delta c-cn,\ &x\in\Omega,& t>0,\\u_{t}&+&\kappa(u\cdot\nabla)u&=\Delta u+\nabla P+n\nabla\phi,\ &x\in\Omega,&t>0,\\ &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*}in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary and givenpotential function $\phi\in C^{1+\beta}(\overline{\Omega})$ for some $\beta>0$.It is known that for fixed $\kappa\in\mathbb{R}$ an associated initial-boundaryvalue problem possesses at least one global weak solution$(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$, which after some waiting timebecomes a classical solution of the system. In this work we will show that uponletting $\kappa\to0$ the solutions $(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$converge towards a weak solution of the Stokes variant $(\kappa=0)$ of thesystems above with respect to the strong topology in certain Lebesgue andSobolev spaces.  We thereby extend the recently obtained result on the Stokes limit processfor classical solutions in the two-dimensional setting to the more intricatethree-dimensional case.

The Stokes limit in a three-dimensional chemotaxis-Navier-Stokes system