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.

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三维趋化Navier-Stokes系统中的斯托克斯极限

我们考虑依赖于（\ kappa \）趋化性-（Navier–）Stokes系统族的初始边界值问题$$ \ begin {aligned} \ left \ {\ begin {array} {lllllll} n_ {t} ＆{} +＆{} u \ cdot \！\ nabla n＆{} = \ Delta n- \ nabla \！\ cdot（n \ nabla c），＆{} x \ in \ Omega，＆{} t> 0 ，\\ c_ {t}＆{} +＆{} u \ cdot \！\ nabla c＆{} = \ Delta c-cn，＆{} x \ in \ Omega，＆{} t> 0，\\ u_ {t}＆{} +＆{} \ kappa（u \ cdot \ nabla）u＆{} = \ Delta u + \ nabla P + n \ nabla \ phi，＆{} x \ in \ Omega，＆{} t> 0，\\＆{}＆{} \ nabla \ cdot u＆{} = 0，＆{} x \ in \ Omega，＆{} t> 0，\ end {array} \ right。\ end {aligned} $$在具有平滑边界和给定势函数\（\ phi \ in C ^ {1+ \ beta} \的有界域\（\ Omega \ subset {\ mathbb {R}} ^ 3 \）中！\剩下（ {{\，\（\ beta> 0 \）。众所周知，对于固定的\（\ kappa \ in {\ mathbb {R}} \），一个相关的初始边界值问题具有至少一个全局弱解\（（n ^ {（（kappa}}，c ^ { （\ kappa）}，u ^ {（（kappa}}）\），它在等待一段时间后成为系统的经典解决方案。在这项工作中，我们将证明让\（\ kappa \ rightarrow 0 \）解决方案\（（n ^ {（\ kappa}}，c ^ {（\ kappa}}，u ^ {（\ kappa}}） \）向斯托克斯变体\（（\ kappa = 0）\）的弱解收敛关于某些Lebesgue和Sobolev空间中的强拓扑的上述系统。因此，我们将二维设置中经典解的斯托克斯极限过程的最新结果扩展到更复杂的三维情况。