The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System

Abstract

We consider initial-boundary value problems for the $\kappa$-dependent family of chemotaxis-(Navier–)Stokes systems

$$\begin{aligned} \left\{ \begin{array}{lllllll} n_{t}&{}+&{}u\cdot \!\nabla n&{}=\Delta n-\nabla \!\cdot (n\nabla c), &{}x\in \Omega ,&{} t>0,\\ c_{t}&{}+&{}u\cdot \!\nabla c&{}=\Delta c-cn, &{}x\in \Omega ,&{} t>0,\\ u_{t}&{}+&{} \kappa (u\cdot \nabla )u&{}=\Delta u+\nabla P+n\nabla \phi , &{}x\in \Omega ,&{} t>0,\\ &{}&{} \nabla \cdot u&{}=0, &{}x\in \Omega ,&{} t>0, \end{array}\right. \end{aligned}$$

in a bounded domain $\Omega \subset {\mathbb {R}}^3$ with smooth boundary and given potential function $\phi \in C^{1+\beta }\!\left( {{\,\mathrm{{\overline{\Omega }}}\,}}\right)$ for some $\beta >0$. It is known that for fixed $\kappa \in {\mathbb {R}}$ an associated initial-boundary value problem possesses at least one global weak solution $(n^{(\kappa )},c^{(\kappa )},u^{(\kappa )})$, which after some waiting time becomes a classical solution of the system. In this work we will show that upon letting $\kappa \rightarrow 0$ the solutions $(n^{(\kappa )},c^{(\kappa )},u^{(\kappa )})$ converge towards a weak solution of the Stokes variant $(\kappa =0)$ of the systems above with respect to the strong topology in certain Lebesgue and Sobolev spaces. We thereby extend the recently obtained result on the Stokes limit process for classical solutions in the two-dimensional setting to the more intricate three-dimensional case.