We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott–Smith–Cowley and Cassel–Smith–Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connection between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.

我们考虑二维无粘性的非稳态Prandtl系统，这是因为它应该建模为原始粘性系统的前导分离和奇异形成。对于正则解的最大存在时间，我们给出了一个尖锐的表达式，表明奇异仅发生在边界或零涡度集上，并且它们对应于边界层分离。然后，我们展示了用于后向自相似轮廓的新的拉格朗日公式，并使用由Elliott-Smith-Cowley和Cassel-Smith-Walker发起的另一种方法来研究它们。一个特定的轮廓是所谓的Van-Dommelen和Shen奇点的核心，我们证明了其对于任何规定的欧拉外流的通用外观（即，对于一组开放且密集的爆炸溶液）。我们评论这些结果与完整的粘性Prandtl系统之间的联系。本文以新颖的方式结合了传输方程（例如拉格朗日坐标和不可压缩性）以及奇异点形成（例如自相似性和重新归一化）的思想，并设计了一种研究拟线性方程的奇异性的新方法。