arXiv 2012.095122020-12-17math.AP

$L^r$框架下叶栅定常Stokes问题的最大正则性

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework

Tomáš Neustupa

备注：: arXiv admin note: text overlap with arXiv:2009.08234, arXiv:2006.15651

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摘要：

本文研究粘性不可压缩流体通过空间周期叶栅的Stokes问题。我们使用[32]（$L^2$ -框架中的最大正则性）和[33]（$W^{1,r}$中的弱可解性）的结果，并将最大正则性的发现扩展到一般的$L^r$ -框架（对于$1<r<\infty$）。使用减少到一个空间周期$Ω$，该问题是制定了三种类型的边界条件的手段：周期性的条件曲线$Γ_0$和$Γ_1$，狄利克雷边界条件$Γ_{\rm in}$和$Γ_P$和人工“不做”型边界条件$Γ_{\rm out}$（见图1）。我们发现，虽然域$Ω$是不光滑的，不同类型的边界条件“满足”在$\partialΩ$的顶点，所考虑的问题有一个强的解决方案，“光滑”的数据具有最大的正则性。我们解释的意义上，“什么都不做”的边界条件是满足弱和强的解决方案。

英文摘要：

The paper deals with the Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. We use results from [32] (the maximum regularity property in the $L^2$-framework) and [33] (the weak solvability in $W^{1,r}$), and extend the findings on the maximum regularity property to the general $L^r$-framework (for $1<r<\infty$). Using the reduction to one spatial period $Ω$, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $Γ_0$ and $Γ_1$, the Dirichlet boundary conditions on $Γ_{\rm in}$ and $Γ_P$ and an artificial "do nothing"-type boundary condition on $Γ_{\rm out}$ (see Fig. 1). We show that, although domain $Ω$ is not smooth and different types of boundary conditions "meet" in the vertices of $\partialΩ$, the considered problem has a strong solution with the maximum regularity property for "smooth" data. We explain the sense in which the "do nothing" boundary condition is satisfied for both weak and strong solutions.

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