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

在工程和科学中，许多问题都涉及到与不可压缩流体一起运动的自由表面的动力学问题。在很短的时间内，溶液的行为在许多情况下都是可以理解的。然而，奇点存在（或不存在）的理论，以及长期行为的解决方案，是远远不够发达。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破裂的基本问题，特别注意破裂的类型和破裂的数量，另一方面，存在的全局解存在于所有时间。考虑两个方程：广义表面准地转（gSQG）方程--在光滑和斑块情况下--是在表面准地转（SQG）方程和欧拉涡度方程之间插值的一类模型;以及二维自由边界欧拉方程。为了执行该项目，需要在一个跨学科的工具集的技术相结合。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NSF的法定使命，并通过使用基金会的智力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.