无穷远处的完全刚性与莱文森空腔的存在性
Complete Rigidity at infinity and Existence of the Levinson Cavity
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中文总结 AI 辅助
本文通过位势理论方法，在三维轴对称稳态不可压缩空腔流中，解决了莱文森渐近行为的存在性及无穷远处完全刚性的80年公开问题，证明了满足自然假设的解收敛到莱文森轮廓。
中文摘要 AI 辅助
我们提出一种位势理论方法，该方法将自由边界的渐近形状分析简化为对归约过程所得的精确常微分方程的分析。尽管该方法主要依赖于偏微分方程（PDE）算子的主部以获得表示公式，因此并不局限于椭圆型问题，但我们以三维轴对称稳态不可压缩空腔流这一清晰示例来呈现该方法——这类流属于诺伊曼型伯努利自由边界问题，且频率公式未知，即便存在，也不足以得到我们在此证明的极为精确的渐近行为。1946年，诺曼·莱文森（Norman Levinson）通过带有慢变修正的幂律假设，推导出这类空腔渐近形状的精确公式。然而，他的结果要求极强的假设，因此，加莱巴迪安-刘易斯-希弗（Garabedian-Lewy-Schiffer）[12]的结果表明存在的空腔解，是否真的具有该渐近行为，或者是否至少存在一个具有莱文森渐近行为的解，这一问题已悬而未决80年。在此，我们对这两个问题均给出肯定回答，并在轴对称解类中得到莱文森解在无穷远处的完全刚性，即任何在固定边界和无穷远处满足温和且自然假设的解，都会渐近收敛到莱文森轮廓$(\log r)^{-1/4}\sqrt{r}$。
英文摘要
We present a potential theoretic approach reducing the analysis of the asymptotic shape of free surfaces to the analysis of a precise ordinary differential equation resulting from the reduction process. Although the approach relies mainly on the principal part of the PDE operator to allow for a representation formula and is thus not restricted to problems of elliptic type, we present it at the clean-cut example of three-dimensional axially symmetric steady incompressible cavity flows, which are Neumann-type Bernoulli free boundary problems and for which frequency formulas are unknown and, if they do exist, insufficient to yield the very precise asymptotic behavior we prove here. In 1946 Norman Levinson derived by a power-law ansatz with a slowly varying correction a precise formula for the asymptotic shape of such cavities. However his result requires very strong assumptions such that it has remained an open problem for 80 years whether the cavity solutions we know to exist by a result by Garabedian-Lewy-Schiffer [12] actually share this asymptotic behavior, or whether at least one solution possessing the Levinson asymptotics exists. Here we answer both questions affirmatively, and we obtain complete rigidity at infinity of the Levinson solution in the class of axially symmetric solutions, that is, any solution satisfying mild and natural assumptions at the fixed boundary and infinity converges asymptotically to the Levinson profile $(\log r)^{-1/4}\sqrt{r}$.
