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

不可压缩流体奇点的整体存在性和计算机辅助证明

基本信息

批准号：
1763356
负责人：
Alexandru Ionescu
金额：
$ 12万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1763356&HistoricalAwards=false
关键词：
Global Existence Computer Assisted Proofs

项目摘要

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.

与不可压缩流体一起运动的自由表面的动力学发生在工程和科学中的许多问题中。在短时间内，许多情况下解决方案的行为是可以理解的。然而，奇点存在（或不存在）的理论以及解的长期行为还远未得到充分发展。奇点的一个例子是波浪的破裂、龙卷风的形成或水滴的飞溅。该项目有两个方向：一方面，它解决了有限时间内是否存在平滑解崩溃的基本问题，特别要注意故障的类型和爆炸的数量，另一方面要注意一直存在的全球解决方案。考虑两个方程：广义地表准地转（gSQG）方程（在光滑和斑块情况下），在地表准地转（SQG）方程和欧拉涡度方程之间进行插值的一系列模型；和二维自由边界欧拉方程。为了开展该项目，需要跨学科工具集的技术组合。这些包括但不限于数学分析、高性能数值计算以及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优势和更广泛的评估进行评估，被认为值得支持。影响审查标准。