Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space $\mathbb R^d$. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension $d=3$ suggest that the effective velocity is well-defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch-Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well-defined in dimensions $d>2$, and that the variance is finite in dimensions $d>4$, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems.

中文翻译：

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Caflisch-Luke 悖论的标量版本

考虑在整个空间 $\mathbb R^d$ 中沉积在斯托克斯流中的无限硬球云。尽管在流体力学和应用数学方面有许多贡献，但迄今为止还没有相关的有效沉降速度的严格定义。Caflisch 和 Luke 在维度 $d=3$ 的计算表明，对于根据弱相关和稀点过程分布的硬球，有效速度是明确定义的，并且沉降速度的方差是无限的。这构成了卡弗利施-卢克悖论。在这个贡献中，我们考虑了这个问题的标量版本，它在微分算子和随机性之间的交互方面表现出同样的困难，但在 PDE 分析方面更简单。