定常 Prandtl 方程中的反转
Reversal in the Stationary Prandtl Equations
AI总结：
本文证明了定常 Prandtl 方程存在展现反转和回流的开数据集，通过将其视为跨越 { u = 0 } 的混合型非局部拟线性自由边界问题，在自相似坐标系下提取耦合系统，结合调和分析与谱理论等技术描述了解越过 Goldstein 奇点后的演化。
摘要：
我们证明了存在一个开的数据集，使得定常 Prandtl 方程展现出“反转”和“回流”现象（由于问题中同时存在正向和反向因果性，数据取自一个适当定义的乘积空间）。反转描述了解在 Goldstein 奇点之后的演化，其特征是存在满足 u > 0 和 u < 0 的（时空）区域。将系统视为在切向方向上演化的经典观点在越过 Goldstein 奇点后完全失效。为了描述这一演化，我们将问题视为跨越曲线 { u = 0 } 的混合型、非局部、拟线性自由边界问题。在一个精心选取的非线性自相似坐标系中，我们提取了一个关于体解和几个描述自由边界的调制变量的耦合系统。我们的工作结合并引入了来自混合型问题、自由边界问题、调制理论、调和分析以及谱理论的技术。作为副产品，我们在 Prandtl 方程中获得了几个新的抵消项，并发展了几个针对具有 Airy 型核的奇异积分算子的新估计。
英文摘要：
We demonstrate the existence of an open set of data which exhibits \textit{reversal} and \textit{recirculation} for the stationary Prandtl equations (data is taken in an appropriately defined product space due to the simultaneous forward and backward causality in the problem). Reversal describes the development of the solution beyond the Goldstein singularity, and is characterized by the presence of (spatio-temporal) regions in which $u > 0$ and $u < 0$. The classical point of view of regarding the system as an evolution in the tangential direction completely breaks down past the Goldstein singularity. Instead, to describe the development, we view the problem as a \textit{mixed-type, non-local, quasilinear, free-boundary} problem across the curve $\{ u = 0 \}$. In a well-chosen nonlinear and self-similar coordinate system, we extract a coupled system for the bulk solution and several modulation variables describing the free boundary. Our work combines and introduces techniques from mixed-type problems, free-boundary problems, modulation theory, harmonic analysis, and spectral theory. As a byproduct, we obtain several new cancellations in the Prandtl equations, and develop several new estimates tailored to singular integral operators with Airy type kernels.
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