In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at \(x=0\). We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at \(x=0\), there exists \(x^{*}>0\) such that \(\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}\) as \(x\to x^{*}\) for some positive constant \(C\), where \(u\) is the solution of the stationary Prandtl equation in the domain \(\{0< x< x^{*},\ y> 0\}\). Our proof relies on three main ingredients: the computation of a “stable” approximate solution, using modulation theory arguments; a new formulation of the Prandtl equation, for which we derive energy estimates, relying heavily on the structure of the equation; and maximum principle and comparison principle techniques to handle some of the nonlinear terms.

在本文中，我们证明了在逆压梯度的情况下，对于\(x=0\)处的一大类边界数据，平稳普朗特方程会发生分离。我们证明了 Goldstein 奇点：更准确地说，我们证明在对\(x=0\)处的边界数据进行适当假设的情况下，存在\(x^{*}>0\)使得\(\partial _{y } u_{|y=0}(x) \sim C \sqrt{x^{*} -x}\)作为\(x\to x^{*}\)对于某些正常数\(C\)，其中\(u\)是域\(\{0< x< x^{*},\ y> 0\}\)中平稳 Prandtl 方程的解。我们的证明依赖于三个主要成分：使用调制理论参数计算“稳定”近似解；普朗特方程的新表述，我们在很大程度上依赖于方程的结构来推导能量估计；以及处理一些非线性项的最大原理和比较原理技术。