Title: In search of local singularities in ideal potential flows with free surface Show Chinese title

Title: 寻找理想势流中的局部奇点的自由表面

Abstract: For ideal fluid flow with zero surface tension and gravity, it remains unknown whether local singularities on the free surface can develop in well-posed initial value problems with smooth initial data. This is so despite great advances over the last 25 years in the mathematical analysis of the Euler equations for water waves. Here we expand our earlier work (Chin. Ann. Math. Ser. B 40 (2019) 925) and review the mathematical literature and some of the history concerning Dirichlet's ellipsoids and related hyperboloids associated with jet formation and "flip-through," "splash singularities," and recent constructions of singular free surfaces that however violate the Taylor sign condition for linear well-posedness. We illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation (whose derivation is detailed and discussed in an appendix). Additional numerical evidence strongly suggests that corner singularities may form in an unstable self-similar way from specially prepared initial data. Show Chinese abstract

Abstract: 对于无表面张力和重力的理想流体流动，即使在过去的25年中，水波欧拉方程的数学分析取得了巨大进展，但仍然不清楚在具有光滑初始数据的适定初值问题中，自由表面上是否会出现局部奇点。 在此，我们扩展了我们之前的工作（Chin. Ann. Math. Ser. B 40 (2019) 925），并回顾了与喷流形成和“翻转通过”、“溅射奇点”相关的狄利克雷椭球体和相关双曲面的数学文献和一些历史背景，以及最近构建的违反线性适定性泰勒符号条件的奇异自由表面。 我们通过基于保角映射公式的二维流动数值计算来说明其中一些现象（其推导在附录中详细讨论）。 额外的数值证据强烈表明，角奇点可能从特别准备的初始数据以不稳定的自相似方式形成。