Global Existence and Computer-Assisted Proofs of Singularities in Incompressible Fluids

不可压缩流体运动的自由表面动力学在工程和科学中的许多问题中都有出现。在短时间内，解的行为在许多情况下是可以理解的。然而，关于奇点存在（或不存在）的理论，以及解的长期行为，还远远没有得到很好的发展。奇点的一个例子是波浪的破碎，龙卷风的形成或水滴的飞溅。这个项目有两个方向：一方面，它解决了在有限时间内是否存在光滑解的分解这一基本问题，特别关注分解的类型和爆炸的数量，另一方面，它解决了永远存在的全局解的存在。本文考虑了两个方程：光滑和斑块情况下的广义曲面准地转方程（gSQG），即在曲面准地转方程和欧拉涡度方程之间插值的一组模型；和二维自由边界欧拉方程。为了实施该项目，需要跨学科工具集中的技术组合。这些包括但不限于数学分析、高性能数值计算和导致计算机辅助证明的严格计算。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The dynamics of free surfaces moving with incompressible fluids occurs in many problems in engineering and science. For short times, the behaviour of solutions is understood in many cases. However, the theory of existence (or not) of singularities, and long term behavior of solutions, is far from well-developed. An example of singularities are the breaking of waves, the formation of a tornado or the splash of a drop.This project takes two directions: on the one hand it addresses the fundamental question of whether there exists breakdown of smooth solutions in finite time, paying particular attention to the type of breakdown and to the quantity that blows up and on the other the existence of global solutions that exist for all time. Two equations are considered: the generalized surface quasi-geostrophic (gSQG) equation- both in the smooth and the patch case -, a family of models which interpolates between the surface quasi-geostrophic (SQG) equation and the Euler vorticity equation; and the two dimensional free boundary Euler equations. In order to carry out the project, a combination of techniques among an interdisciplinary tool set is required. These include, but are not restricted to, mathematical analysis, high-performance numerical computing, and rigorous computations leading to computer-assisted proofs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.