发表机构
NYU Abu-Dhabi; University of California, Davis; Graduate School of Science, Kyoto University（纽约大学阿布扎比分校; 加州大学戴维斯分校; 京都大学理学研究科）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了定常Prandtl系统在逆压梯度下具有多种分离规律的解，确定了分离点附近的渐近结构，并证明了平方根分离情形的动态稳定性，解决了Oleinik和Samokhin的开放问题。
AI 中文摘要
我们在恒定逆压梯度下构造了二维定常Prandtl系统的解，这些解展现出不同的分离规律族。更精确地说，对于每个整数 $\ell\ge1$，我们构造光滑初始数据，使得分离发生在有限位置 $x=x^\ast$，且满足 \\[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \\] 其中 $\mathbf C_\ell>0$ 依赖于初始数据。我们通过基于 von Mises 变量中基态线性化的匹配内、外展开，确定了分离点附近解的局部渐近结构。这一描述解决了 Oleinik 和 Samokhin 提出的开放问题5（见文献 \cite{OleinikSamok-book-99}）。我们还证明了平方根分离情形（对应 $\ell=1$）在适当拓扑下对初始数据的小容许扰动是动态稳定的。
英文摘要
We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer $\ell\ge1$, we construct smooth initial data for which separation occurs at a finite location $x=x^\ast$, with \[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \] where $\mathbf C_\ell>0$ depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to $\ell=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.
